%------------------------------------------------------------------------------
% File     : ITP216^1 : TPTP v9.0.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem Time_Reasoning 00074_002979
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0029_Time_Reasoning_00074_002979 [Des22]

% Status   : Theorem
% Rating   : 0.50 v9.0.0, 0.40 v8.2.0, 0.46 v8.1.0
% Syntax   : Number of formulae    : 10879 (4924 unt;2876 typ;   0 def)
%            Number of atoms       : 20486 (10777 equ;   0 cnn)
%            Maximal formula atoms :   48 (   2 avg)
%            Number of connectives : 81663 (2054   ~; 295   |;1214   &;71318   @)
%                                         (   0 <=>;6782  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   25 (   5 avg)
%            Number of types       :  306 ( 305 usr)
%            Number of type conns  : 14872 (14872   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 2574 (2571 usr; 164 con; 0-8 aty)
%            Number of variables   : 21351 (3365   ^;17684   !; 302   ?;21351   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 16:14:05.014
%------------------------------------------------------------------------------
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    option_assn: $tType ).

thf(ty_n_t__Heap__Oref_It__Product____Type__Ounit_J,type,
    ref_Product_unit: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
    set_Product_unit: $tType ).

thf(ty_n_t__Set__Oset_It__Option__Ooption_I_Eo_J_J,type,
    set_option_o: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
    set_nat_o: $tType ).

thf(ty_n_t__Option__Ooption_It__Set__Oset_I_Eo_J_J,type,
    option_set_o: $tType ).

thf(ty_n_t__Heap__Oref_It__Heap__Oarray_Itf__a_J_J,type,
    ref_array_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    product_prod_o_o: $tType ).

thf(ty_n_t__Set__Oset_It__Assertions__Oassn_J,type,
    set_assn: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
    set_set_o: $tType ).

thf(ty_n_t__Option__Ooption_It__Rat__Orat_J,type,
    option_rat: $tType ).

thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
    option_num: $tType ).

thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
    option_nat: $tType ).

thf(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
    option_int: $tType ).

thf(ty_n_t__List__Olist_It__String__Ochar_J,type,
    list_char: $tType ).

thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
    list_nat: $tType ).

thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
    list_int: $tType ).

thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
    set_rat: $tType ).

thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
    set_num: $tType ).

thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
    set_int: $tType ).

thf(ty_n_t__Code____Numeral__Onatural,type,
    code_natural: $tType ).

thf(ty_n_t__Code____Numeral__Ointeger,type,
    code_integer: $tType ).

thf(ty_n_t__Product____Type__Ounit,type,
    product_unit: $tType ).

thf(ty_n_t__Option__Ooption_I_Eo_J,type,
    option_o: $tType ).

thf(ty_n_t__Heap__Oarray_Itf__a_J,type,
    array_a: $tType ).

thf(ty_n_t__List__Olist_Itf__a_J,type,
    list_a: $tType ).

thf(ty_n_t__Typerep__Otyperep,type,
    typerep: $tType ).

thf(ty_n_t__Assertions__Oassn,type,
    assn: $tType ).

thf(ty_n_t__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

thf(ty_n_t__String__Ochar,type,
    char: $tType ).

thf(ty_n_t__Rat__Orat,type,
    rat: $tType ).

thf(ty_n_t__Num__Onum,type,
    num: $tType ).

thf(ty_n_t__Nat__Onat,type,
    nat: $tType ).

thf(ty_n_t__Int__Oint,type,
    int: $tType ).

thf(ty_n_tf__a,type,
    a: $tType ).

% Explicit typings (2571)
thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: rat > int ).

thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
    archim3151403230148437115or_rat: rat > int ).

thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
    archimedean_frac_rat: rat > rat ).

thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
    archim7778729529865785530nd_rat: rat > int ).

thf(sy_c_Array__Time_Omake_001tf__a,type,
    array_make_a: nat > ( nat > a ) > heap_T6362127782908069795rray_a ).

thf(sy_c_Array__Time_Onew_001tf__a,type,
    array_new_a: nat > a > heap_T6362127782908069795rray_a ).

thf(sy_c_Array__Time_Oof__list_001tf__a,type,
    array_of_list_a: list_a > heap_T6362127782908069795rray_a ).

thf(sy_c_Assertions_Oassn_ORep__assn,type,
    rep_assn: assn > produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Oin__range,type,
    in_range: produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Oin__range__rel,type,
    in_range_rel: produc3658429121746597890et_nat > produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Oone__assn__raw,type,
    one_assn_raw: produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Oone__assn__raw__rel,type,
    one_assn_raw_rel: produc3658429121746597890et_nat > produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Oproper,type,
    proper: ( produc3658429121746597890et_nat > $o ) > $o ).

thf(sy_c_Assertions_Opure__assn,type,
    pure_assn: $o > assn ).

thf(sy_c_Assertions_OrelH,type,
    relH: set_nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o ).

thf(sy_c_Assertions_Otimes__assn__raw,type,
    times_assn_raw: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Otimes__assn__raw__rel,type,
    times_assn_raw_rel: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o ).

thf(sy_c_Assertions_Owand__assn,type,
    wand_assn: assn > assn > assn ).

thf(sy_c_Assertions_Owand__raw,type,
    wand_raw: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Owand__raw__rel,type,
    wand_raw_rel: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o ).

thf(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
    bNF_Cardinal_ctwo: set_Product_prod_o_o ).

thf(sy_c_BNF__Cardinal__Arithmetic_Oczero_001_Eo,type,
    bNF_Cardinal_czero_o: set_Product_prod_o_o ).

thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of_001t__Nat__Onat,type,
    bNF_Ca3793111618940312692of_nat: set_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on_001_Eo,type,
    bNF_Ca8331644756375544342r_on_o: set_o > set_Product_prod_o_o > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Def_Orel__fun_001_062_I_062_It__Int__Oint_M_Eo_J_M_062_It__Int__Oint_M_Eo_J_J_001_062_I_062_It__Int__Oint_M_Eo_J_M_062_It__Int__Oint_M_Eo_J_J_001_062_I_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_M_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_J_001_062_I_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_M_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_J,type,
    bNF_re2555365631850037877_int_o: ( ( ( int > $o ) > int > $o ) > ( ( int > $o ) > int > $o ) > $o ) > ( ( ( int > int > $o ) > int > int > $o ) > ( ( int > int > $o ) > int > int > $o ) > $o ) > ( ( ( int > $o ) > int > $o ) > ( int > int > $o ) > int > int > $o ) > ( ( ( int > $o ) > int > $o ) > ( int > int > $o ) > int > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_I_062_It__Int__Oint_M_Eo_J_M_062_It__Int__Oint_Mt__Int__Oint_J_J_001_062_I_062_It__Int__Oint_M_Eo_J_M_062_It__Int__Oint_Mt__Int__Oint_J_J_001_062_I_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_M_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J_J_001_062_I_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_M_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    bNF_re5051145820034318685nt_int: ( ( ( int > $o ) > int > int ) > ( ( int > $o ) > int > int ) > $o ) > ( ( ( int > int > $o ) > int > int > int ) > ( ( int > int > $o ) > int > int > int ) > $o ) > ( ( ( int > $o ) > int > int ) > ( int > int > $o ) > int > int > int ) > ( ( ( int > $o ) > int > int ) > ( int > int > $o ) > int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_I_062_It__Int__Oint_M_Eo_J_M_062_It__Nat__Onat_M_Eo_J_J_001_062_I_062_It__Int__Oint_M_Eo_J_M_062_It__Nat__Onat_M_Eo_J_J_001_062_I_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_M_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_J_001_062_I_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_M_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_J,type,
    bNF_re615132412071444453_nat_o: ( ( ( int > $o ) > nat > $o ) > ( ( int > $o ) > nat > $o ) > $o ) > ( ( ( int > int > $o ) > int > nat > $o ) > ( ( int > int > $o ) > int > nat > $o ) > $o ) > ( ( ( int > $o ) > nat > $o ) > ( int > int > $o ) > int > nat > $o ) > ( ( ( int > $o ) > nat > $o ) > ( int > int > $o ) > int > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_M_Eo_J_J_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_M_Eo_J_J_001_062_I_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J_M_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_J_001_062_I_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J_M_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_J,type,
    bNF_re3161894838932316637_int_o: ( ( ( int > int ) > int > $o ) > ( ( int > int ) > int > $o ) > $o ) > ( ( ( int > int > int ) > int > int > $o ) > ( ( int > int > int ) > int > int > $o ) > $o ) > ( ( ( int > int ) > int > $o ) > ( int > int > int ) > int > int > $o ) > ( ( ( int > int ) > int > $o ) > ( int > int > int ) > int > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Nat__Onat_M_Eo_J_J_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Nat__Onat_M_Eo_J_J_001_062_I_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J_M_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_J_001_062_I_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J_M_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_J,type,
    bNF_re4659646225794970589_nat_o: ( ( ( int > int ) > nat > $o ) > ( ( int > int ) > nat > $o ) > $o ) > ( ( ( int > int > int ) > int > nat > $o ) > ( ( int > int > int ) > int > nat > $o ) > $o ) > ( ( ( int > int ) > nat > $o ) > ( int > int > int ) > int > nat > $o ) > ( ( ( int > int ) > nat > $o ) > ( int > int > int ) > int > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_I_062_It__Nat__Onat_M_Eo_J_M_062_It__Int__Oint_M_Eo_J_J_001_062_I_062_It__Nat__Onat_M_Eo_J_M_062_It__Int__Oint_M_Eo_J_J_001_062_I_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_M_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_J_001_062_I_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_M_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_J,type,
    bNF_re3491090157995142405_int_o: ( ( ( nat > $o ) > int > $o ) > ( ( nat > $o ) > int > $o ) > $o ) > ( ( ( int > nat > $o ) > int > int > $o ) > ( ( int > nat > $o ) > int > int > $o ) > $o ) > ( ( ( nat > $o ) > int > $o ) > ( int > nat > $o ) > int > int > $o ) > ( ( ( nat > $o ) > int > $o ) > ( int > nat > $o ) > int > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_I_062_It__Nat__Onat_M_Eo_J_M_062_It__Int__Oint_Mt__Int__Oint_J_J_001_062_I_062_It__Nat__Onat_M_Eo_J_M_062_It__Int__Oint_Mt__Int__Oint_J_J_001_062_I_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_M_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J_J_001_062_I_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_M_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    bNF_re5539255234643219293nt_int: ( ( ( nat > $o ) > int > int ) > ( ( nat > $o ) > int > int ) > $o ) > ( ( ( int > nat > $o ) > int > int > int ) > ( ( int > nat > $o ) > int > int > int ) > $o ) > ( ( ( nat > $o ) > int > int ) > ( int > nat > $o ) > int > int > int ) > ( ( ( nat > $o ) > int > int ) > ( int > nat > $o ) > int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_I_062_It__Nat__Onat_M_Eo_J_M_062_It__Nat__Onat_M_Eo_J_J_001_062_I_062_It__Nat__Onat_M_Eo_J_M_062_It__Nat__Onat_M_Eo_J_J_001_062_I_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_M_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_J_001_062_I_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_M_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_J,type,
    bNF_re1550856938216548981_nat_o: ( ( ( nat > $o ) > nat > $o ) > ( ( nat > $o ) > nat > $o ) > $o ) > ( ( ( int > nat > $o ) > int > nat > $o ) > ( ( int > nat > $o ) > int > nat > $o ) > $o ) > ( ( ( nat > $o ) > nat > $o ) > ( int > nat > $o ) > int > nat > $o ) > ( ( ( nat > $o ) > nat > $o ) > ( int > nat > $o ) > int > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_I_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_062_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_J_001_062_I_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_M_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_J,type,
    bNF_re4919612610145136063nteger: ( ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > $o ) > ( ( ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( ( num > code_integer ) > num > code_integer ) > $o ) > ( ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( ( code_integer > code_integer ) > ( num > code_integer ) > num > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_001_062_It__Nat__Onat_Mt__Num__Onum_J_001_062_I_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_062_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_J_001_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J,type,
    bNF_re1890273300020150979nt_num: ( ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( nat > num ) > $o ) > ( ( ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( ( int > nat ) > int > num ) > $o ) > ( ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( ( nat > num ) > ( int > nat ) > int > num ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_001_062_I_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_M_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_J_001_062_I_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_062_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_J,type,
    bNF_re8134695817916415679nteger: ( ( code_integer > code_integer ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ) > ( ( ( num > code_integer ) > num > code_integer ) > ( ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ) > ( ( code_integer > code_integer ) > ( num > code_integer ) > num > code_integer ) > ( ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_I_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_M_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_J_001_062_I_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_M_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_J,type,
    bNF_re5979881370344185769nteger: ( ( code_integer > code_integer ) > ( code_integer > code_integer ) > $o ) > ( ( ( num > code_integer ) > num > code_integer ) > ( ( num > code_integer ) > num > code_integer ) > $o ) > ( ( code_integer > code_integer ) > ( num > code_integer ) > num > code_integer ) > ( ( code_integer > code_integer ) > ( num > code_integer ) > num > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_001_062_It__Int__Oint_Mt__Num__Onum_J,type,
    bNF_re5190287125156074331nt_num: ( ( code_integer > code_integer ) > ( int > nat ) > $o ) > ( ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( int > num ) > $o ) > ( ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( ( int > nat ) > int > num ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Nat__Onat_Mt__Num__Onum_J_001_062_I_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_M_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_J_001_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J,type,
    bNF_re5469402333911078061nt_num: ( ( code_integer > code_integer ) > ( nat > num ) > $o ) > ( ( ( num > code_integer ) > num > code_integer ) > ( ( int > nat ) > int > num ) > $o ) > ( ( code_integer > code_integer ) > ( num > code_integer ) > num > code_integer ) > ( ( nat > num ) > ( int > nat ) > int > num ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_001_062_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_001_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J,type,
    bNF_re6724329528997265415nteger: ( ( code_integer > code_integer ) > ( num > code_integer ) > $o ) > ( ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( num > code_integer ) > $o ) > ( ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( ( num > code_integer ) > num > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_J_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_J,type,
    bNF_re1900612664124570461_int_o: ( ( int > int > $o ) > ( int > int > $o ) > $o ) > ( ( ( int > int ) > int > int > $o ) > ( ( int > int ) > int > int > $o ) > $o ) > ( ( int > int > $o ) > ( int > int ) > int > int > $o ) > ( ( int > int > $o ) > ( int > int ) > int > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
    bNF_re8651060166502312541_int_o: ( ( int > int > $o ) > ( int > int > $o ) > $o ) > ( ( int > int > $o ) > ( int > int > $o ) > $o ) > ( ( int > int > $o ) > int > int > $o ) > ( ( int > int > $o ) > int > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bNF_re7462748251485907171nt_int: ( ( int > int > $o ) > ( int > int > $o ) > $o ) > ( ( int > int > int ) > ( int > int > int ) > $o ) > ( ( int > int > $o ) > int > int > int ) > ( ( int > int > $o ) > int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J,type,
    bNF_re2446162618891206493_nat_o: ( ( int > int > $o ) > ( int > int > $o ) > $o ) > ( ( int > nat > $o ) > ( int > nat > $o ) > $o ) > ( ( int > int > $o ) > int > nat > $o ) > ( ( int > int > $o ) > int > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J_J_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
    bNF_re4984751527058004117nt_int: ( ( int > int > int ) > ( int > int > int ) > $o ) > ( ( ( int > int ) > int > int > int ) > ( ( int > int ) > int > int > int ) > $o ) > ( ( int > int > int ) > ( int > int ) > int > int > int ) > ( ( int > int > int ) > ( int > int ) > int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
    bNF_re1313934062736347107_int_o: ( ( int > int > int ) > ( int > int > int ) > $o ) > ( ( int > int > $o ) > ( int > int > $o ) > $o ) > ( ( int > int > int ) > int > int > $o ) > ( ( int > int > int ) > int > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J,type,
    bNF_re4332408551980016867_nat_o: ( ( int > int > int ) > ( int > int > int ) > $o ) > ( ( int > nat > $o ) > ( int > nat > $o ) > $o ) > ( ( int > int > int ) > int > nat > $o ) > ( ( int > int > int ) > int > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Rat__Orat_J_J_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_J_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Rat__Orat_J_J_J,type,
    bNF_re62019127700022579nt_rat: ( ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ) > ( ( ( int > int ) > int > int > product_prod_int_int ) > ( ( int > int ) > int > int > rat ) > $o ) > ( ( int > int > product_prod_int_int ) > ( int > int ) > int > int > product_prod_int_int ) > ( ( int > int > rat ) > ( int > int ) > int > int > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_J_001_062_I_062_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_J,type,
    bNF_re7908124382625962333_nat_o: ( ( int > nat > $o ) > ( int > nat > $o ) > $o ) > ( ( ( int > int ) > int > nat > $o ) > ( ( int > int ) > int > nat > $o ) > $o ) > ( ( int > nat > $o ) > ( int > int ) > int > nat > $o ) > ( ( int > nat > $o ) > ( int > int ) > int > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
    bNF_re2655111580231008093_int_o: ( ( int > nat > $o ) > ( int > nat > $o ) > $o ) > ( ( int > int > $o ) > ( int > int > $o ) > $o ) > ( ( int > nat > $o ) > int > int > $o ) > ( ( int > nat > $o ) > int > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bNF_re7395152722673581539nt_int: ( ( int > nat > $o ) > ( int > nat > $o ) > $o ) > ( ( int > int > int ) > ( int > int > int ) > $o ) > ( ( int > nat > $o ) > int > int > int ) > ( ( int > nat > $o ) > int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J,type,
    bNF_re5673586069474677853_nat_o: ( ( int > nat > $o ) > ( int > nat > $o ) > $o ) > ( ( int > nat > $o ) > ( int > nat > $o ) > $o ) > ( ( int > nat > $o ) > int > nat > $o ) > ( ( int > nat > $o ) > int > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_M_Eo_J_001_062_It__Int__Oint_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    bNF_re2535741173671728349_nat_o: ( ( int > $o ) > ( int > $o ) > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( ( int > $o ) > nat > $o ) > ( ( int > $o ) > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
    bNF_re3014639844587090765_int_o: ( ( int > int ) > ( int > int ) > $o ) > ( ( int > int > $o ) > ( int > int > $o ) > $o ) > ( ( int > int ) > int > int > $o ) > ( ( int > int ) > int > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bNF_re724639330072410579nt_int: ( ( int > int ) > ( int > int ) > $o ) > ( ( int > int > int ) > ( int > int > int ) > $o ) > ( ( int > int ) > int > int > int ) > ( ( int > int ) > int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J_001_062_It__Int__Oint_M_062_It__Int__Oint_Mt__Rat__Orat_J_J,type,
    bNF_re949775918795716016nt_rat: ( ( int > int ) > ( int > int ) > $o ) > ( ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ) > ( ( int > int ) > int > int > product_prod_int_int ) > ( ( int > int ) > int > int > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J_001_062_It__Int__Oint_M_062_It__Nat__Onat_M_Eo_J_J,type,
    bNF_re6033114333830760525_nat_o: ( ( int > int ) > ( int > int ) > $o ) > ( ( int > nat > $o ) > ( int > nat > $o ) > $o ) > ( ( int > int ) > int > nat > $o ) > ( ( int > int ) > int > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Num__Onum_J_001_062_It__Int__Oint_Mt__Num__Onum_J,type,
    bNF_re9099949428469764705nt_num: ( ( int > nat ) > ( int > nat ) > $o ) > ( ( int > num ) > ( int > num ) > $o ) > ( ( int > nat ) > int > num ) > ( ( int > nat ) > int > num ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_001_062_It__Int__Oint_Mt__Num__Onum_J_001_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J,type,
    bNF_re8894579221946749965nteger: ( ( int > nat ) > ( num > code_integer ) > $o ) > ( ( int > num ) > ( num > code_integer ) > $o ) > ( ( int > nat ) > int > num ) > ( ( num > code_integer ) > num > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Int__Oint_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
    bNF_re2724346968851715293_int_o: ( ( nat > $o ) > ( nat > $o ) > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( ( nat > $o ) > int > $o ) > ( ( nat > $o ) > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re7183634743573646861nt_int: ( ( nat > $o ) > ( nat > $o ) > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( ( nat > $o ) > int > int ) > ( ( nat > $o ) > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    bNF_re2697311982885662685_nat_o: ( ( nat > $o ) > ( nat > $o ) > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( ( nat > $o ) > nat > $o ) > ( ( nat > $o ) > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Num__Onum_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J_001_062_I_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_M_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_J,type,
    bNF_re2163612987331707565nteger: ( ( nat > num ) > ( code_integer > code_integer ) > $o ) > ( ( ( int > nat ) > int > num ) > ( ( num > code_integer ) > num > code_integer ) > $o ) > ( ( nat > num ) > ( int > nat ) > int > num ) > ( ( code_integer > code_integer ) > ( num > code_integer ) > num > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Num__Onum_J_001_062_It__Nat__Onat_Mt__Num__Onum_J_001_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J_001_062_I_062_It__Int__Oint_Mt__Nat__Onat_J_M_062_It__Int__Oint_Mt__Num__Onum_J_J,type,
    bNF_re3846787885862361009nt_num: ( ( nat > num ) > ( nat > num ) > $o ) > ( ( ( int > nat ) > int > num ) > ( ( int > nat ) > int > num ) > $o ) > ( ( nat > num ) > ( int > nat ) > int > num ) > ( ( nat > num ) > ( int > nat ) > int > num ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_001_062_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J,type,
    bNF_re2439920061672227847nteger: ( ( num > code_integer ) > ( code_integer > code_integer ) > $o ) > ( ( num > code_integer ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ) > ( ( num > code_integer ) > num > code_integer ) > ( ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_001_062_It__Int__Oint_Mt__Nat__Onat_J_001_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_001_062_It__Int__Oint_Mt__Num__Onum_J,type,
    bNF_re1151383970801924621nt_num: ( ( num > code_integer ) > ( int > nat ) > $o ) > ( ( num > code_integer ) > ( int > num ) > $o ) > ( ( num > code_integer ) > num > code_integer ) > ( ( int > nat ) > int > num ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_001_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_001_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J_001_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J,type,
    bNF_re6097285183634142137nteger: ( ( num > code_integer ) > ( num > code_integer ) > $o ) > ( ( num > code_integer ) > ( num > code_integer ) > $o ) > ( ( num > code_integer ) > num > code_integer ) > ( ( num > code_integer ) > num > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__Int__Oint_001t__Nat__Onat,type,
    bNF_re1551779926989065179nt_nat: ( $o > $o > $o ) > ( int > nat > $o ) > ( $o > int ) > ( $o > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__Int__Oint_001t__Rat__Orat,type,
    bNF_re916649866902569443nt_rat: ( $o > $o > $o ) > ( int > rat > $o ) > ( $o > int ) > ( $o > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
    bNF_re4863541036035986956nteger: ( $o > $o > $o ) > ( nat > code_integer > $o ) > ( $o > nat ) > ( $o > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__Nat__Onat_001t__Int__Oint,type,
    bNF_re549935744578960347at_int: ( $o > $o > $o ) > ( nat > int > $o ) > ( $o > nat ) > ( $o > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re552426215088010623at_nat: ( $o > $o > $o ) > ( nat > nat > $o ) > ( $o > nat ) > ( $o > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__Nat__Onat_001t__Rat__Orat,type,
    bNF_re9140668191856290695at_rat: ( $o > $o > $o ) > ( nat > rat > $o ) > ( $o > nat ) > ( $o > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__Rat__Orat_001t__Code____Numeral__Ointeger,type,
    bNF_re5635613845597506580nteger: ( $o > $o > $o ) > ( rat > code_integer > $o ) > ( $o > rat ) > ( $o > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__Rat__Orat_001t__Int__Oint,type,
    bNF_re233381100644094435at_int: ( $o > $o > $o ) > ( rat > int > $o ) > ( $o > rat ) > ( $o > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__Rat__Orat_001t__Nat__Onat,type,
    bNF_re235871571153144711at_nat: ( $o > $o > $o ) > ( rat > nat > $o ) > ( $o > rat ) > ( $o > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_Eo_001_Eo_001t__Rat__Orat_001t__Rat__Orat,type,
    bNF_re8824113547921424783at_rat: ( $o > $o > $o ) > ( rat > rat > $o ) > ( $o > rat ) > ( $o > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Assertions__Oassn_001t__Assertions__Oassn_001_062_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001_062_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    bNF_re9137223911888007311n_assn: ( assn > assn > $o ) > ( ( assn > assn ) > ( assn > assn ) > $o ) > ( assn > assn > assn ) > ( assn > assn > assn ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Assertions__Oassn_001t__Assertions__Oassn_001_062_It__Nat__Onat_Mt__Assertions__Oassn_J_001_062_It__Nat__Onat_Mt__Assertions__Oassn_J,type,
    bNF_re3650450707543776735t_assn: ( assn > assn > $o ) > ( ( nat > assn ) > ( nat > assn ) > $o ) > ( assn > nat > assn ) > ( assn > nat > assn ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Assertions__Oassn_001t__Assertions__Oassn_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
    bNF_re1143782165119297793n_assn: ( assn > assn > $o ) > ( assn > assn > $o ) > ( assn > assn ) > ( assn > assn ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Assertions__Oassn_001t__Code____Numeral__Ointeger_001_062_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    bNF_re5822757241383739566nteger: ( assn > code_integer > $o ) > ( ( assn > assn ) > ( code_integer > code_integer ) > $o ) > ( assn > assn > assn ) > ( code_integer > code_integer > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Assertions__Oassn_001t__Code____Numeral__Ointeger_001_062_It__Nat__Onat_Mt__Assertions__Oassn_J_001_062_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J,type,
    bNF_re2102180915162216137nteger: ( assn > code_integer > $o ) > ( ( nat > assn ) > ( nat > code_integer ) > $o ) > ( assn > nat > assn ) > ( code_integer > nat > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Assertions__Oassn_001t__Code____Numeral__Ointeger_001t__Assertions__Oassn_001t__Code____Numeral__Ointeger,type,
    bNF_re4134162077946812779nteger: ( assn > code_integer > $o ) > ( assn > code_integer > $o ) > ( assn > assn ) > ( code_integer > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Assertions__Oassn_001t__Int__Oint_001_062_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re7882741803053012379nt_int: ( assn > int > $o ) > ( ( assn > assn ) > ( int > int ) > $o ) > ( assn > assn > assn ) > ( int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Assertions__Oassn_001t__Int__Oint_001_062_It__Nat__Onat_Mt__Assertions__Oassn_J_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
    bNF_re185025794348816359at_int: ( assn > int > $o ) > ( ( nat > assn ) > ( nat > int ) > $o ) > ( assn > nat > assn ) > ( int > nat > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Assertions__Oassn_001t__Int__Oint_001t__Assertions__Oassn_001t__Int__Oint,type,
    bNF_re2927800259774540553sn_int: ( assn > int > $o ) > ( assn > int > $o ) > ( assn > assn ) > ( int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Assertions__Oassn_001t__Nat__Onat_001_062_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re4386305596031132807at_nat: ( assn > nat > $o ) > ( ( assn > assn ) > ( nat > nat ) > $o ) > ( assn > assn > assn ) > ( nat > nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Assertions__Oassn_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Assertions__Oassn_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re8560028565313717551at_nat: ( assn > nat > $o ) > ( ( nat > assn ) > ( nat > nat ) > $o ) > ( assn > nat > assn ) > ( nat > nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Assertions__Oassn_001t__Nat__Onat_001t__Assertions__Oassn_001t__Nat__Onat,type,
    bNF_re6742580585434904145sn_nat: ( assn > nat > $o ) > ( assn > nat > $o ) > ( assn > assn ) > ( nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Assertions__Oassn_001t__Rat__Orat_001_062_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
    bNF_re967558844708792991at_rat: ( assn > rat > $o ) > ( ( assn > assn ) > ( rat > rat ) > $o ) > ( assn > assn > assn ) > ( rat > rat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Assertions__Oassn_001t__Rat__Orat_001_062_It__Nat__Onat_Mt__Assertions__Oassn_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
    bNF_re6138836914430330175at_rat: ( assn > rat > $o ) > ( ( nat > assn ) > ( nat > rat ) > $o ) > ( assn > nat > assn ) > ( rat > nat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Assertions__Oassn_001t__Rat__Orat_001t__Assertions__Oassn_001t__Rat__Orat,type,
    bNF_re9003240792272343649sn_rat: ( assn > rat > $o ) > ( assn > rat > $o ) > ( assn > assn ) > ( rat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Assertions__Oassn_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    bNF_re7789648203153440174n_assn: ( code_integer > assn > $o ) > ( ( code_integer > code_integer ) > ( assn > assn ) > $o ) > ( code_integer > code_integer > code_integer ) > ( assn > assn > assn ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Assertions__Oassn_001_062_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001_062_It__Nat__Onat_Mt__Assertions__Oassn_J,type,
    bNF_re2546461910754330953t_assn: ( code_integer > assn > $o ) > ( ( nat > code_integer ) > ( nat > assn ) > $o ) > ( code_integer > nat > code_integer ) > ( assn > nat > assn ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Assertions__Oassn_001t__Code____Numeral__Ointeger_001t__Assertions__Oassn,type,
    bNF_re1475883931861434731r_assn: ( code_integer > assn > $o ) > ( code_integer > assn > $o ) > ( code_integer > code_integer ) > ( assn > assn ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    bNF_re122555440370955213nteger: ( code_integer > code_integer > $o ) > ( ( code_integer > code_integer ) > ( code_integer > code_integer ) > $o ) > ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer > code_integer ) > $o ).

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    bNF_re5462749793222359603nteger: ( code_integer > code_integer > $o ) > ( ( nat > code_integer ) > ( nat > code_integer ) > $o ) > ( code_integer > nat > code_integer ) > ( code_integer > nat > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
    bNF_re4341081746073172381nteger: ( code_integer > code_integer > $o ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer > $o ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    bNF_re3370092614548195229nteger: ( code_integer > code_integer > $o ) > ( code_integer > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ) > ( code_integer > code_integer ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    bNF_re1062419521925302741nteger: ( code_integer > code_integer > $o ) > ( code_integer > code_integer > $o ) > ( code_integer > code_integer ) > ( code_integer > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Int__Oint_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re8304916493157054650nt_int: ( code_integer > int > $o ) > ( ( code_integer > code_integer ) > ( int > int ) > $o ) > ( code_integer > code_integer > code_integer ) > ( int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Int__Oint_001_062_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
    bNF_re6766650671887726929at_int: ( code_integer > int > $o ) > ( ( nat > code_integer ) > ( nat > int ) > $o ) > ( code_integer > nat > code_integer ) > ( int > nat > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Num__Onum,type,
    bNF_re8127238762813979689er_num: ( code_integer > int > $o ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > num > $o ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( int > num ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
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thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    bNF_re2891348898900095767er_nat: ( code_integer > int > $o ) > ( code_integer > nat > $o ) > ( code_integer > code_integer ) > ( int > nat ) > $o ).

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    bNF_re3981674901896804262at_nat: ( code_integer > nat > $o ) > ( ( code_integer > code_integer ) > ( nat > nat ) > $o ) > ( code_integer > code_integer > code_integer ) > ( nat > nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re4269568176766901913at_nat: ( code_integer > nat > $o ) > ( ( nat > code_integer ) > ( nat > nat ) > $o ) > ( code_integer > nat > code_integer ) > ( nat > nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Num__Onum,type,
    bNF_re7119966052990372301er_num: ( code_integer > nat > $o ) > ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > num > $o ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( nat > num ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Nat__Onat_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
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thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Nat__Onat_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
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    bNF_re2976496150901874110at_rat: ( code_integer > rat > $o ) > ( ( code_integer > code_integer ) > ( rat > rat ) > $o ) > ( code_integer > code_integer > code_integer ) > ( rat > rat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Rat__Orat_001_062_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
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thf(sy_c_BNF__Def_Orel__fun_001t__Code____Numeral__Ointeger_001t__Rat__Orat_001t__Code____Numeral__Ointeger_001t__Rat__Orat,type,
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    bNF_re398004352372739002nteger: ( int > code_integer > $o ) > ( ( int > int ) > ( code_integer > code_integer ) > $o ) > ( int > int > int ) > ( code_integer > code_integer > code_integer ) > $o ).

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    bNF_re3379532845092657523nteger: ( int > code_integer > $o ) > ( int > code_integer > $o ) > ( int > int ) > ( code_integer > code_integer ) > $o ).

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    bNF_re1831474436612530402atural: ( int > code_integer > $o ) > ( nat > code_natural > $o ) > ( int > nat ) > ( code_integer > code_natural ) > $o ).

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    bNF_re2342410721167621787atural: ( int > code_natural > $o ) > ( ( int > int ) > ( code_natural > code_natural ) > $o ) > ( int > int > int ) > ( code_natural > code_natural > code_natural ) > $o ).

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    bNF_re5484556299188842761atural: ( int > code_natural > $o ) > ( int > code_natural > $o ) > ( int > int ) > ( code_natural > code_natural ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
    bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bNF_re157797125943740599nt_int: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > product_prod_int_int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Rat__Orat_J,type,
    bNF_re3461391660133120880nt_rat: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > rat ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    bNF_re3376528473927230327_nat_o: ( int > int > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( int > nat > $o ) > ( int > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
    bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    bNF_re1784066744496875891nteger: ( int > int > $o ) > ( code_integer > code_integer > $o ) > ( int > code_integer ) > ( int > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    bNF_re343602593470180162er_int: ( int > int > $o ) > ( code_integer > int > $o ) > ( int > code_integer ) > ( int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Rat__Orat,type,
    bNF_re8934335040747510510er_rat: ( int > int > $o ) > ( code_integer > rat > $o ) > ( int > code_integer ) > ( int > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    bNF_re6290619412732081552atural: ( int > int > $o ) > ( code_natural > produc7822875418678951345atural > $o ) > ( int > code_natural ) > ( int > produc7822875418678951345atural ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    bNF_re982302072995117890nteger: ( int > int > $o ) > ( int > code_integer > $o ) > ( int > int ) > ( int > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Rat__Orat,type,
    bNF_re4079880299697760445nt_rat: ( int > int > $o ) > ( int > rat > $o ) > ( int > int ) > ( int > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re3715656647883201625at_nat: ( int > int > $o ) > ( nat > nat > $o ) > ( int > nat ) > ( int > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Num__Onum_001t__Num__Onum,type,
    bNF_re7626690874201225453um_num: ( int > int > $o ) > ( num > num > $o ) > ( int > num ) > ( int > num ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re6250860962936578807nt_int: ( int > int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Rat__Orat_001t__Code____Numeral__Ointeger,type,
    bNF_re1382721322319792878nteger: ( int > int > $o ) > ( rat > code_integer > $o ) > ( int > rat ) > ( int > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Rat__Orat_001t__Int__Oint,type,
    bNF_re3396611533439285437at_int: ( int > int > $o ) > ( rat > int > $o ) > ( int > rat ) > ( int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Rat__Orat_001t__Rat__Orat,type,
    bNF_re2763971943861839977at_rat: ( int > int > $o ) > ( rat > rat > $o ) > ( int > rat ) > ( int > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Nat__Onat_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re5416042074812042131at_nat: ( int > nat > $o ) > ( ( int > int ) > ( nat > nat ) > $o ) > ( int > int > int ) > ( nat > nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat,type,
    bNF_re4343356799547411545nt_nat: ( int > nat > $o ) > ( int > nat > $o ) > ( int > int ) > ( nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Num__Onum_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
    bNF_re151304249936775380nteger: ( int > num > $o ) > ( nat > code_integer > $o ) > ( int > nat ) > ( num > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Num__Onum_001t__Num__Onum_001t__Code____Numeral__Ointeger,type,
    bNF_re7315484114762210078nteger: ( int > num > $o ) > ( num > code_integer > $o ) > ( int > num ) > ( num > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Rat__Orat_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
    bNF_re7661208983170134443at_rat: ( int > rat > $o ) > ( ( int > int ) > ( rat > rat ) > $o ) > ( int > int > int ) > ( rat > rat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Rat__Orat_001t__Int__Oint_001t__Rat__Orat,type,
    bNF_re4480299549022435433nt_rat: ( int > rat > $o ) > ( int > rat > $o ) > ( int > int ) > ( rat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Ointeger_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    bNF_re7367609096550055590nteger: ( nat > code_integer > $o ) > ( ( nat > nat ) > ( code_integer > code_integer ) > $o ) > ( nat > nat > nat ) > ( code_integer > code_integer > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Ointeger_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
    bNF_re1599192376174140603nteger: ( nat > code_integer > $o ) > ( nat > code_integer > $o ) > ( nat > nat ) > ( code_integer > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Ointeger_001t__Num__Onum_001t__Code____Numeral__Ointeger,type,
    bNF_re8763372240999575301nteger: ( nat > code_integer > $o ) > ( num > code_integer > $o ) > ( nat > num ) > ( code_integer > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Code____Numeral__Onatural_M_Eo_J,type,
    bNF_re1639080489988575423ural_o: ( nat > code_natural > $o ) > ( ( nat > $o ) > ( code_natural > $o ) > $o ) > ( nat > nat > $o ) > ( code_natural > code_natural > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    bNF_re2895713572246619415_nat_o: ( nat > code_natural > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( code_natural > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    bNF_re88643428490162567atural: ( nat > code_natural > $o ) > ( ( nat > nat ) > ( code_natural > code_natural ) > $o ) > ( nat > nat > nat ) > ( code_natural > code_natural > code_natural ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001_Eo_001_Eo,type,
    bNF_re2785088596696291543al_o_o: ( nat > code_natural > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( code_natural > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    bNF_re5252274238750452962nteger: ( nat > code_natural > $o ) > ( int > code_integer > $o ) > ( nat > int ) > ( code_natural > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
    bNF_re3704215830270325841atural: ( nat > code_natural > $o ) > ( nat > code_natural > $o ) > ( nat > nat ) > ( code_natural > code_natural ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re7333046803109187065at_nat: ( nat > code_natural > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( code_natural > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Int__Oint_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re5864104203909894035nt_int: ( nat > int > $o ) > ( ( nat > nat ) > ( int > int ) > $o ) > ( nat > nat > nat ) > ( int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
    bNF_re6022984109467102297at_int: ( nat > int > $o ) > ( nat > int > $o ) > ( nat > nat ) > ( int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    bNF_re4935368626670024657nteger: ( nat > nat > $o ) > ( ( int > int ) > ( code_integer > code_integer ) > $o ) > ( nat > int > int ) > ( nat > code_integer > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re4785983289428654063nt_int: ( nat > nat > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( nat > int > int ) > ( nat > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    bNF_re3692170009638431535atural: ( nat > nat > $o ) > ( ( nat > nat ) > ( code_natural > code_natural ) > $o ) > ( nat > nat > nat ) > ( nat > code_natural > code_natural ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
    bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
    bNF_re4051817429369991249n_assn: ( nat > nat > $o ) > ( assn > assn > $o ) > ( nat > assn ) > ( nat > assn ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Assertions__Oassn_001t__Code____Numeral__Ointeger,type,
    bNF_re1831131418152741254nteger: ( nat > nat > $o ) > ( assn > code_integer > $o ) > ( nat > assn ) > ( nat > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Assertions__Oassn_001t__Int__Oint,type,
    bNF_re4792267480158271317sn_int: ( nat > nat > $o ) > ( assn > int > $o ) > ( nat > assn ) > ( nat > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Assertions__Oassn_001t__Nat__Onat,type,
    bNF_re4794757950667321593sn_nat: ( nat > nat > $o ) > ( assn > nat > $o ) > ( nat > assn ) > ( nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Assertions__Oassn_001t__Rat__Orat,type,
    bNF_re4159627890580825857sn_rat: ( nat > nat > $o ) > ( assn > rat > $o ) > ( nat > assn ) > ( nat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Code____Numeral__Ointeger_001t__Assertions__Oassn,type,
    bNF_re2692373401909853574r_assn: ( nat > nat > $o ) > ( code_integer > assn > $o ) > ( nat > code_integer ) > ( nat > assn ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    bNF_re1236659899165691067nteger: ( nat > nat > $o ) > ( code_integer > code_integer > $o ) > ( nat > code_integer ) > ( nat > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    bNF_re3514700588913618570er_int: ( nat > nat > $o ) > ( code_integer > int > $o ) > ( nat > code_integer ) > ( nat > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    bNF_re3517191059422668846er_nat: ( nat > nat > $o ) > ( code_integer > nat > $o ) > ( nat > code_integer ) > ( nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Code____Numeral__Ointeger_001t__Rat__Orat,type,
    bNF_re2882060999336173110er_rat: ( nat > nat > $o ) > ( code_integer > rat > $o ) > ( nat > code_integer ) > ( nat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    bNF_re6101645755646507080atural: ( nat > nat > $o ) > ( code_natural > produc7822875418678951345atural > $o ) > ( nat > code_natural ) > ( nat > produc7822875418678951345atural ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    bNF_re4153400068438556298nteger: ( nat > nat > $o ) > ( int > code_integer > $o ) > ( nat > int ) > ( nat > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint_001t__Code____Numeral__Onatural,type,
    bNF_re2976995220195273813atural: ( nat > nat > $o ) > ( int > code_natural > $o ) > ( nat > int ) > ( nat > code_natural ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint_001t__Int__Oint,type,
    bNF_re6650684261131312217nt_int: ( nat > nat > $o ) > ( int > int > $o ) > ( nat > int ) > ( nat > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat,type,
    bNF_re6653174731640362493nt_nat: ( nat > nat > $o ) > ( int > nat > $o ) > ( nat > int ) > ( nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint_001t__Rat__Orat,type,
    bNF_re6018044671553866757nt_rat: ( nat > nat > $o ) > ( int > rat > $o ) > ( nat > int ) > ( nat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
    bNF_re2605341659958429177atural: ( nat > nat > $o ) > ( nat > code_natural > $o ) > ( nat > nat ) > ( nat > code_natural ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Num__Onum_001t__Num__Onum,type,
    bNF_re341483209202555957um_num: ( nat > nat > $o ) > ( num > num > $o ) > ( nat > num ) > ( nat > num ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat_001t__Code____Numeral__Ointeger,type,
    bNF_re4553819317763231286nteger: ( nat > nat > $o ) > ( rat > code_integer > $o ) > ( nat > rat ) > ( nat > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat_001t__Code____Numeral__Onatural,type,
    bNF_re3377414469519948801atural: ( nat > nat > $o ) > ( rat > code_natural > $o ) > ( nat > rat ) > ( nat > code_natural ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat_001t__Int__Oint,type,
    bNF_re5334775905295391749at_int: ( nat > nat > $o ) > ( rat > int > $o ) > ( nat > rat ) > ( nat > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat_001t__Nat__Onat,type,
    bNF_re5337266375804442025at_nat: ( nat > nat > $o ) > ( rat > nat > $o ) > ( nat > rat ) > ( nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat_001t__Rat__Orat,type,
    bNF_re4702136315717946289at_rat: ( nat > nat > $o ) > ( rat > rat > $o ) > ( nat > rat ) > ( nat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Rat__Orat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
    bNF_re3590448190763046039at_rat: ( nat > rat > $o ) > ( ( nat > nat ) > ( rat > rat ) > $o ) > ( nat > nat > nat ) > ( rat > rat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Rat__Orat_001t__Nat__Onat_001t__Rat__Orat,type,
    bNF_re5790763769214331825at_rat: ( nat > rat > $o ) > ( nat > rat > $o ) > ( nat > nat ) > ( rat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    bNF_re5031869190687302234nteger: ( num > code_integer > $o ) > ( code_integer > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ) > ( num > code_integer ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    bNF_re5202476307641864082nteger: ( num > code_integer > $o ) > ( code_integer > code_integer > $o ) > ( num > code_integer ) > ( code_integer > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    bNF_re6324211044263125204er_nat: ( num > int > $o ) > ( code_integer > nat > $o ) > ( num > code_integer ) > ( int > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
    bNF_re2881542830418903838er_num: ( num > int > $o ) > ( code_integer > num > $o ) > ( num > code_integer ) > ( int > num ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Code____Numeral__Ointeger_J,type,
    bNF_re7876454716742015248nteger: ( num > num > $o ) > ( ( num > int ) > ( num > code_integer ) > $o ) > ( num > num > int ) > ( num > num > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Int__Oint_J,type,
    bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    bNF_re6741283560065411407nteger: ( num > num > $o ) > ( code_integer > code_integer > $o ) > ( num > code_integer ) > ( num > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    bNF_re6501075790457514782nteger: ( num > num > $o ) > ( int > code_integer > $o ) > ( num > int ) > ( num > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Code____Numeral__Onatural,type,
    bNF_re5324670942214232297atural: ( num > num > $o ) > ( int > code_natural > $o ) > ( num > int ) > ( num > code_natural ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Int__Oint,type,
    bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Nat__Onat,type,
    bNF_re1824820364696572561nt_nat: ( num > num > $o ) > ( int > nat > $o ) > ( num > int ) > ( num > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Rat__Orat,type,
    bNF_re1189690304610076825nt_rat: ( num > num > $o ) > ( int > rat > $o ) > ( num > int ) > ( num > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
    bNF_re6129422230220670146nteger: ( num > num > $o ) > ( nat > code_integer > $o ) > ( num > nat ) > ( num > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
    bNF_re4953017381977387661atural: ( num > num > $o ) > ( nat > code_natural > $o ) > ( num > nat ) > ( num > code_natural ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Nat__Onat_001t__Int__Oint,type,
    bNF_re822976182286467729at_int: ( num > num > $o ) > ( nat > int > $o ) > ( num > nat ) > ( num > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_re825466652795518005at_nat: ( num > num > $o ) > ( nat > nat > $o ) > ( num > nat ) > ( num > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Nat__Onat_001t__Rat__Orat,type,
    bNF_re190336592709022269at_rat: ( num > num > $o ) > ( nat > rat > $o ) > ( num > nat ) > ( num > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
    bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
    bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
    bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_Eo_001_Eo,type,
    bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
    bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Ounit_001t__Product____Type__Ounit_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    bNF_re8417623677625516633nteger: ( product_unit > product_unit > $o ) > ( code_integer > code_integer > $o ) > ( product_unit > code_integer ) > ( product_unit > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Ounit_001t__Product____Type__Ounit_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    bNF_re5641277236161133866atural: ( product_unit > product_unit > $o ) > ( code_natural > produc7822875418678951345atural > $o ) > ( product_unit > code_natural ) > ( product_unit > produc7822875418678951345atural ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Ounit_001t__Product____Type__Ounit_001t__Int__Oint_001t__Int__Oint,type,
    bNF_re8422471777688076791nt_int: ( product_unit > product_unit > $o ) > ( int > int > $o ) > ( product_unit > int ) > ( product_unit > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Rat__Orat_001t__Code____Numeral__Ointeger_001_062_It__Rat__Orat_Mt__Rat__Orat_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    bNF_re4307939118855507134nteger: ( rat > code_integer > $o ) > ( ( rat > rat ) > ( code_integer > code_integer ) > $o ) > ( rat > rat > rat ) > ( code_integer > code_integer > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Rat__Orat_001t__Code____Numeral__Ointeger_001t__Rat__Orat_001t__Code____Numeral__Ointeger,type,
    bNF_re2328245411608324299nteger: ( rat > code_integer > $o ) > ( rat > code_integer > $o ) > ( rat > rat ) > ( code_integer > code_integer ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Rat__Orat_001t__Code____Numeral__Onatural_001_062_It__Rat__Orat_Mt__Rat__Orat_J_001_062_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    bNF_re6252345487650389919atural: ( rat > code_natural > $o ) > ( ( rat > rat ) > ( code_natural > code_natural ) > $o ) > ( rat > rat > rat ) > ( code_natural > code_natural > code_natural ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Rat__Orat_001t__Code____Numeral__Onatural_001t__Rat__Orat_001t__Code____Numeral__Onatural,type,
    bNF_re4433268865704509537atural: ( rat > code_natural > $o ) > ( rat > code_natural > $o ) > ( rat > rat ) > ( code_natural > code_natural ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Rat__Orat_001t__Int__Oint_001_062_It__Rat__Orat_Mt__Rat__Orat_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re8876735282041863595nt_int: ( rat > int > $o ) > ( ( rat > rat ) > ( int > int ) > $o ) > ( rat > rat > rat ) > ( int > int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Rat__Orat_001t__Int__Oint_001t__Rat__Orat_001t__Int__Oint,type,
    bNF_re5838490730811218025at_int: ( rat > int > $o ) > ( rat > int > $o ) > ( rat > rat ) > ( int > int ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Rat__Orat_001t__Nat__Onat_001_062_It__Rat__Orat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_re4357912360536923287at_nat: ( rat > nat > $o ) > ( ( rat > rat ) > ( nat > nat ) > $o ) > ( rat > rat > rat ) > ( nat > nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Rat__Orat_001t__Nat__Onat_001t__Rat__Orat_001t__Nat__Onat,type,
    bNF_re5469327641083423665at_nat: ( rat > nat > $o ) > ( rat > nat > $o ) > ( rat > rat ) > ( nat > nat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Rat__Orat_001t__Rat__Orat_001_062_It__Rat__Orat_Mt__Rat__Orat_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
    bNF_re6603079268895015599at_rat: ( rat > rat > $o ) > ( ( rat > rat ) > ( rat > rat ) > $o ) > ( rat > rat > rat ) > ( rat > rat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001t__Rat__Orat_001t__Rat__Orat_001t__Rat__Orat_001t__Rat__Orat,type,
    bNF_re5606270390558447553at_rat: ( rat > rat > $o ) > ( rat > rat > $o ) > ( rat > rat ) > ( rat > rat ) > $o ).

thf(sy_c_BNF__Wellorder__Constructions_OordIso_001_Eo_001_Eo,type,
    bNF_We2654380646378065620so_o_o: set_Pr1932065953672099015od_o_o ).

thf(sy_c_BNF__Wellorder__Constructions_OordIso_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_We5258908940166488438at_nat: set_Pr4329608150637261639at_nat ).

thf(sy_c_BNF__Wellorder__Constructions_OordLess_001_Eo_001t__Nat__Onat,type,
    bNF_We8182288985678559134_o_nat: set_Pr457366540195662369at_nat ).

thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    basic_7565498685090299632nteger: ( code_integer > nat ) > ( code_integer > nat ) > produc8923325533196201883nteger > nat ).

thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    basic_8185638495112405971atural: ( code_natural > nat ) > ( produc7822875418678951345atural > nat ) > produc5835291356934675326atural > nat ).

thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod_001t__Int__Oint_001t__Int__Oint,type,
    basic_1872990034501187214nt_int: ( int > nat ) > ( int > nat ) > product_prod_int_int > nat ).

thf(sy_c_Binomial_Obinomial,type,
    binomial: nat > nat > nat ).

thf(sy_c_Binomial_Ogbinomial_001t__Code____Numeral__Ointeger,type,
    gbinom8545251970709558553nteger: code_integer > nat > code_integer ).

thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
    gbinomial_int: int > nat > int ).

thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
    gbinomial_nat: nat > nat > nat ).

thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
    gbinomial_rat: rat > nat > rat ).

thf(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).

thf(sy_c_Bit__Operations_Oand__not__num,type,
    bit_and_not_num: num > num > option_num ).

thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
    bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: nat > int > int > int ).

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    comp_n3090169619561918849nt_int: ( ( nat > $o ) > int > int ) > ( int > nat > $o ) > int > int > int ).

thf(sy_c_Fun_Ocomp_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    comp_n5656820349209612572_nat_o: ( ( nat > $o ) > nat > $o ) > ( ( nat > $o ) > nat > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Fun_Ocomp_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J_001t__Int__Oint,type,
    comp_nat_o_nat_o_int: ( ( nat > $o ) > nat > $o ) > ( int > nat > $o ) > int > nat > $o ).

thf(sy_c_Fun_Ocomp_001_062_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001_062_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_001_062_It__Num__Onum_Mt__Num__Onum_J,type,
    comp_n3323228797031117228um_num: ( ( nat > code_integer ) > int > code_integer ) > ( ( num > num ) > nat > code_integer ) > ( num > num ) > int > code_integer ).

thf(sy_c_Fun_Ocomp_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Num__Onum_J_001_062_It__Num__Onum_Mt__Int__Oint_J,type,
    comp_n4765151935410462062um_int: ( ( nat > nat ) > int > num ) > ( ( num > int ) > nat > nat ) > ( num > int ) > int > num ).

thf(sy_c_Fun_Ocomp_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
    comp_P3928908779438215820nteger: ( ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_Fun_Ocomp_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    comp_P286844350285437056_nat_o: ( ( product_prod_nat_nat > $o ) > int > $o ) > ( ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > int > $o ).

thf(sy_c_Fun_Ocomp_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    comp_P9039549498699531069at_nat: ( ( product_prod_nat_nat > $o ) > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > int > $o ).

thf(sy_c_Fun_Ocomp_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    comp_P1872448293794610368at_nat: ( ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > int > int ).

thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_I_Eo_J_Mt__Assertions__Oassn_J_001_062_It__Set__Oset_It__Set__Oset_I_Eo_J_J_Mt__Assertions__Oassn_J_001_062_I_Eo_Mt__Assertions__Oassn_J,type,
    comp_s518380907012905326o_assn: ( ( set_o > assn ) > set_set_o > assn ) > ( ( $o > assn ) > set_o > assn ) > ( $o > assn ) > set_set_o > assn ).

thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_I_Eo_J_Mt__Code____Numeral__Ointeger_J_001_062_It__Set__Oset_It__Set__Oset_I_Eo_J_J_Mt__Code____Numeral__Ointeger_J_001_062_I_Eo_Mt__Code____Numeral__Ointeger_J,type,
    comp_s262121120021782499nteger: ( ( set_o > code_integer ) > set_set_o > code_integer ) > ( ( $o > code_integer ) > set_o > code_integer ) > ( $o > code_integer ) > set_set_o > code_integer ).

thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_I_Eo_J_Mt__Nat__Onat_J_001_062_It__Set__Oset_It__Set__Oset_I_Eo_J_J_Mt__Nat__Onat_J_001_062_I_Eo_Mt__Nat__Onat_J,type,
    comp_s6084864194472624150_o_nat: ( ( set_o > nat ) > set_set_o > nat ) > ( ( $o > nat ) > set_o > nat ) > ( $o > nat ) > set_set_o > nat ).

thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_I_Eo_J_Mt__Rat__Orat_J_001_062_It__Set__Oset_It__Set__Oset_I_Eo_J_J_Mt__Rat__Orat_J_001_062_I_Eo_Mt__Rat__Orat_J,type,
    comp_s1265507639253661726_o_rat: ( ( set_o > rat ) > set_set_o > rat ) > ( ( $o > rat ) > set_o > rat ) > ( $o > rat ) > set_set_o > rat ).

thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Code____Numeral__Ointeger_J_Mt__Assertions__Oassn_J_001_062_It__Set__Oset_It__Set__Oset_It__Code____Numeral__Ointeger_J_J_Mt__Assertions__Oassn_J_001_062_It__Code____Numeral__Ointeger_Mt__Assertions__Oassn_J,type,
    comp_s3531641609786562035r_assn: ( ( set_Code_integer > assn ) > set_set_Code_integer > assn ) > ( ( code_integer > assn ) > set_Code_integer > assn ) > ( code_integer > assn ) > set_set_Code_integer > assn ).

thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Code____Numeral__Ointeger_J_Mt__Code____Numeral__Ointeger_J_001_062_It__Set__Oset_It__Set__Oset_It__Code____Numeral__Ointeger_J_J_Mt__Code____Numeral__Ointeger_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    comp_s8484400498494363538nteger: ( ( set_Code_integer > code_integer ) > set_set_Code_integer > code_integer ) > ( ( code_integer > code_integer ) > set_Code_integer > code_integer ) > ( code_integer > code_integer ) > set_set_Code_integer > code_integer ).

thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Code____Numeral__Ointeger_J_Mt__Rat__Orat_J_001_062_It__Set__Oset_It__Set__Oset_It__Code____Numeral__Ointeger_J_J_Mt__Rat__Orat_J_001_062_It__Code____Numeral__Ointeger_Mt__Rat__Orat_J,type,
    comp_s9045439038509510147er_rat: ( ( set_Code_integer > rat ) > set_set_Code_integer > rat ) > ( ( code_integer > rat ) > set_Code_integer > rat ) > ( code_integer > rat ) > set_set_Code_integer > rat ).

thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Assertions__Oassn_J_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Assertions__Oassn_J_001_062_It__Int__Oint_Mt__Assertions__Oassn_J,type,
    comp_s4595925057662634336t_assn: ( ( set_int > assn ) > set_set_int > assn ) > ( ( int > assn ) > set_int > assn ) > ( int > assn ) > set_set_int > assn ).

thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Code____Numeral__Ointeger_J_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Code____Numeral__Ointeger_J_001_062_It__Int__Oint_Mt__Code____Numeral__Ointeger_J,type,
    comp_s7646638485285982975nteger: ( ( set_int > code_integer ) > set_set_int > code_integer ) > ( ( int > code_integer ) > set_int > code_integer ) > ( int > code_integer ) > set_set_int > code_integer ).

thf(sy_c_Fun_Ocomp_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Rat__Orat_J_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Rat__Orat_J_001_062_It__Int__Oint_Mt__Rat__Orat_J,type,
    comp_s4088261402702255472nt_rat: ( ( set_int > rat ) > set_set_int > rat ) > ( ( int > rat ) > set_int > rat ) > ( int > rat ) > set_set_int > rat ).

thf(sy_c_Fun_Ocomp_001_Eo_001_Eo_001_Eo,type,
    comp_o_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).

thf(sy_c_Fun_Ocomp_001_Eo_001_Eo_001t__Nat__Onat,type,
    comp_o_o_nat: ( $o > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Fun_Ocomp_001_Eo_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    comp_o8022916681612831582at_nat: ( $o > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger,type,
    comp_C2931745410781319416nteger: ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer ) > code_integer > code_integer > code_integer ).

thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Num__Onum,type,
    comp_C3662112809048933045er_num: ( code_integer > code_integer > code_integer ) > ( num > code_integer ) > num > code_integer > code_integer ).

thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    comp_C95226695241045696nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( ( code_integer > code_integer ) > code_integer ) > ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
    comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Num__Onum,type,
    comp_C6617437204949736321er_num: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( num > code_integer ) > num > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Assertions__Oassn_001t__Nat__Onat,type,
    comp_C8462447471145125178sn_nat: ( code_integer > assn ) > ( nat > code_integer ) > nat > assn ).

thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    comp_C7449957260575251196nteger: ( code_integer > code_integer ) > ( code_integer > code_integer ) > code_integer > code_integer ).

thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    comp_C6974050283906349679er_nat: ( code_integer > code_integer ) > ( nat > code_integer ) > nat > code_integer ).

thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
    comp_C3531382070062128313er_num: ( code_integer > code_integer ) > ( num > code_integer ) > num > code_integer ).

thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    comp_C1043215000170864188nteger: ( code_integer > code_integer ) > ( produc8923325533196201883nteger > code_integer ) > produc8923325533196201883nteger > code_integer ).

thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Int__Oint,type,
    comp_C5046419837055054490nt_int: ( code_integer > int ) > ( int > code_integer ) > int > int ).

thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Int__Oint_001t__Nat__Onat,type,
    comp_C5048910307564104766nt_nat: ( code_integer > int ) > ( nat > code_integer ) > nat > int ).

thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_C4049556595663050210at_nat: ( code_integer > nat ) > ( nat > code_integer ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    comp_C2277306503614464128nteger: ( code_integer > produc8923325533196201883nteger ) > ( produc8923325533196201883nteger > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Product____Type__Ounit_001t__Nat__Onat,type,
    comp_C8086623346886033437it_nat: ( code_integer > product_unit ) > ( nat > code_integer ) > nat > product_unit ).

thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Rat__Orat_001t__Nat__Onat,type,
    comp_C3733001951728184298at_nat: ( code_integer > rat ) > ( nat > code_integer ) > nat > rat ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001_062_It__Int__Oint_M_Eo_J_001t__Int__Oint,type,
    comp_int_int_o_int: ( int > int > $o ) > ( int > int ) > int > int > $o ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001_062_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint,type,
    comp_int_int_int_int: ( int > int > int ) > ( int > int ) > int > int > int ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001_062_It__Int__Oint_Mt__Int__Oint_J_001t__Nat__Onat,type,
    comp_int_int_int_nat: ( int > int > int ) > ( nat > int ) > nat > int > int ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__Int__Oint,type,
    comp_i87437609017934369nt_int: ( int > int > product_prod_int_int ) > ( int > int ) > int > int > product_prod_int_int ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001_062_It__Int__Oint_Mt__Rat__Orat_J_001t__Int__Oint,type,
    comp_int_int_rat_int: ( int > int > rat ) > ( int > int ) > int > int > rat ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001_062_It__Nat__Onat_M_Eo_J_001t__Int__Oint,type,
    comp_int_nat_o_int: ( int > nat > $o ) > ( int > int ) > int > nat > $o ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Assertions__Oassn_001t__Int__Oint,type,
    comp_int_assn_int: ( int > assn ) > ( int > int ) > int > assn ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Assertions__Oassn_001t__Nat__Onat,type,
    comp_int_assn_nat: ( int > assn ) > ( nat > int ) > nat > assn ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    comp_i1585864551200866970er_int: ( int > code_integer ) > ( int > int ) > int > code_integer ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    comp_i1588355021709917246er_nat: ( int > code_integer ) > ( nat > int ) > nat > code_integer ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    comp_int_int_int: ( int > int ) > ( int > int ) > int > int ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Nat__Onat,type,
    comp_int_int_nat: ( int > int ) > ( nat > int ) > nat > int ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    comp_i4330622015127193916nt_int: ( int > int ) > ( product_prod_int_int > int ) > product_prod_int_int > int ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
    comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_int_nat_nat: ( int > nat ) > ( nat > int ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Num__Onum_001t__Int__Oint,type,
    comp_int_num_int: ( int > num ) > ( int > int ) > int > num ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    comp_i5941103828406351439nt_int: ( int > product_prod_int_int ) > ( product_prod_int_int > int ) > product_prod_int_int > product_prod_int_int ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    comp_i1108739374719436894at_int: ( int > product_prod_nat_nat ) > ( int > int ) > int > product_prod_nat_nat ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Rat__Orat_001t__Int__Oint,type,
    comp_int_rat_int: ( int > rat ) > ( int > int ) > int > rat ).

thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Rat__Orat_001t__Nat__Onat,type,
    comp_int_rat_nat: ( int > rat ) > ( nat > int ) > nat > rat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    comp_nat_nat_nat_nat: ( nat > nat > nat ) > ( nat > nat ) > nat > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001_Eo_001t__Nat__Onat,type,
    comp_nat_o_nat: ( nat > $o ) > ( nat > nat ) > nat > $o ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Assertions__Oassn_001t__Nat__Onat,type,
    comp_nat_assn_nat: ( nat > assn ) > ( nat > nat ) > nat > assn ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    comp_n3898172953802868194er_nat: ( nat > code_integer ) > ( nat > nat ) > nat > code_integer ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Int__Oint_001t__Int__Oint,type,
    comp_nat_int_int: ( nat > int ) > ( int > nat ) > int > int ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat,type,
    comp_nat_int_nat: ( nat > int ) > ( nat > nat ) > nat > int ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    comp_n8698576032424989604at_nat: ( nat > multis2468970476368604999at_nat ) > ( nat > nat ) > nat > multis2468970476368604999at_nat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
    comp_nat_nat_int: ( nat > nat ) > ( int > nat ) > int > nat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Num__Onum_001t__Int__Oint,type,
    comp_nat_num_int: ( nat > num ) > ( int > nat ) > int > num ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Num__Onum_001t__Nat__Onat,type,
    comp_nat_num_nat: ( nat > num ) > ( nat > nat ) > nat > num ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Product____Type__Ounit_001t__Nat__Onat,type,
    comp_n9025697279074352746it_nat: ( nat > product_unit ) > ( nat > nat ) > nat > product_unit ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Rat__Orat_001t__Nat__Onat,type,
    comp_nat_rat_nat: ( nat > rat ) > ( nat > nat ) > nat > rat ).

thf(sy_c_Fun_Ocomp_001t__Num__Onum_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
    comp_n6909714581898973814er_num: ( num > code_integer ) > ( num > num ) > num > code_integer ).

thf(sy_c_Fun_Ocomp_001t__Num__Onum_001t__Int__Oint_001t__Num__Onum,type,
    comp_num_int_num: ( num > int ) > ( num > num ) > num > int ).

thf(sy_c_Fun_Ocomp_001t__Num__Onum_001t__Nat__Onat_001t__Num__Onum,type,
    comp_num_nat_num: ( num > nat ) > ( num > num ) > num > nat ).

thf(sy_c_Fun_Ocomp_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint,type,
    comp_num_num_int: ( num > num ) > ( int > num ) > int > num ).

thf(sy_c_Fun_Ocomp_001t__Num__Onum_001t__Num__Onum_001t__Nat__Onat,type,
    comp_num_num_nat: ( num > num ) > ( nat > num ) > nat > num ).

thf(sy_c_Fun_Ocomp_001t__Num__Onum_001t__Num__Onum_001t__Num__Onum,type,
    comp_num_num_num: ( num > num ) > ( num > num ) > num > num ).

thf(sy_c_Fun_Ocomp_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
    comp_n6731957995704128387um_num: ( num > option_num ) > ( num > num ) > num > option_num ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_I_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Option__Ooption_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_001t__Option__Ooption_It__Code____Numeral__Ointeger_J,type,
    comp_o7188567781464924208nteger: ( option8057788054806935849nteger > option7428179582072448501nteger ) > ( option_Code_integer > option8057788054806935849nteger ) > option_Code_integer > option7428179582072448501nteger ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_I_062_It__Nat__Onat_M_Eo_J_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    comp_o8860755393669798242_nat_o: ( option_nat_o > nat > $o ) > ( ( nat > $o ) > option_nat_o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_I_Eo_J_001_Eo_001_Eo,type,
    comp_option_o_o_o: ( option_o > $o ) > ( $o > option_o ) > $o > $o ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Code____Numeral__Ointeger_J_001t__Option__Ooption_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_J_001t__Option__Ooption_It__Code____Numeral__Ointeger_J,type,
    comp_o9011007105448239796nteger: ( option_Code_integer > option7428179582072448501nteger ) > ( option_Code_integer > option_Code_integer ) > option_Code_integer > option7428179582072448501nteger ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Code____Numeral__Ointeger_J_001t__Option__Ooption_It__Code____Numeral__Ointeger_J_001t__Option__Ooption_It__Num__Onum_J,type,
    comp_o5321900959263616553on_num: ( option_Code_integer > option_Code_integer ) > ( option_num > option_Code_integer ) > option_num > option_Code_integer ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Int__Oint_J_001t__Int__Oint_001t__Int__Oint,type,
    comp_o2446264010603541305nt_int: ( option_int > int ) > ( int > option_int ) > int > int ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_o4663963155070737957at_nat: ( option_nat > nat ) > ( nat > option_nat ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Int__Oint_J,type,
    comp_o3128663659858294251on_int: ( option_nat > option_num ) > ( option_int > option_nat ) > option_int > option_num ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Num__Onum_J_001t__Nat__Onat_001t__Option__Ooption_It__Num__Onum_J,type,
    comp_o6878144249584144265on_num: ( option_num > nat ) > ( option_num > option_num ) > option_num > nat ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum_001t__Num__Onum,type,
    comp_o6342398210560591875um_num: ( option_num > num ) > ( num > option_num ) > num > num ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
    comp_o8931257242830428707on_num: ( option_num > option_num ) > ( option_num > option_num ) > option_num > option_num ).

thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    comp_o279924169626513613it_nat: ( option8956607266484857688it_nat > produc8664842809031399944it_nat ) > ( produc8664842809031399944it_nat > option8956607266484857688it_nat ) > produc8664842809031399944it_nat > produc8664842809031399944it_nat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    comp_P1875353012086385820e_term: ( produc7773217078559923341nt_int > int > option6357759511663192854e_term ) > ( produc3832618445565238089e_term > produc7773217078559923341nt_int ) > produc3832618445565238089e_term > int > option6357759511663192854e_term ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    comp_P5363713469694551954e_term: ( produc7773217078559923341nt_int > product_prod_int_int ) > ( produc3832618445565238089e_term > produc7773217078559923341nt_int ) > produc3832618445565238089e_term > product_prod_int_int ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_Eo_J_M_062_It__Nat__Onat_M_Eo_J_J_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
    comp_P8312345242777556868_nat_o: ( produc7360885484439884007_nat_o > nat > $o ) > ( ( nat > $o ) > produc7360885484439884007_nat_o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_Eo_J_M_062_It__Nat__Onat_M_Eo_J_J_001_062_It__Nat__Onat_M_Eo_J_001t__Int__Oint,type,
    comp_P1075760313067033319_o_int: ( produc7360885484439884007_nat_o > nat > $o ) > ( int > produc7360885484439884007_nat_o ) > int > nat > $o ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    comp_P8650103696410638905e_term: ( produc1908205239877642774nteger > produc6241069584506657477e_term > option6357759511663192854e_term ) > ( produc998928222018024388e_term > produc1908205239877642774nteger ) > produc998928222018024388e_term > produc6241069584506657477e_term > option6357759511663192854e_term ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    comp_P1663382703091044646e_term: ( produc1908205239877642774nteger > produc8923325533196201883nteger ) > ( produc998928222018024388e_term > produc1908205239877642774nteger ) > produc998928222018024388e_term > produc8923325533196201883nteger ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_M_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
    comp_P4355506512671352436_nat_o: ( produc2732055786443039994et_nat > produc3658429121746597890et_nat > $o ) > ( produc4962441903510821572_nat_o > produc2732055786443039994et_nat ) > produc4962441903510821572_nat_o > produc3658429121746597890et_nat > $o ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_M_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
    comp_P9183373265949749025_nat_o: ( produc2732055786443039994et_nat > produc3925858234332021118et_nat ) > ( produc4962441903510821572_nat_o > produc2732055786443039994et_nat ) > produc4962441903510821572_nat_o > produc3925858234332021118et_nat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
    comp_P1110778729668777324_nat_o: ( produc3925858234332021118et_nat > produc3658429121746597890et_nat > $o ) > ( produc6907119039986491712_nat_o > produc3925858234332021118et_nat ) > produc6907119039986491712_nat_o > produc3658429121746597890et_nat > $o ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
    comp_P5089495125153697957_nat_o: ( produc3925858234332021118et_nat > produc3658429121746597890et_nat ) > ( produc6907119039986491712_nat_o > produc3925858234332021118et_nat ) > produc6907119039986491712_nat_o > produc3658429121746597890et_nat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    comp_P8577347034251779110e_term: ( produc2285326912895808259nt_int > produc8551481072490612790e_term > option6357759511663192854e_term ) > ( produc1821397396195235155e_term > produc2285326912895808259nt_int ) > produc1821397396195235155e_term > produc8551481072490612790e_term > option6357759511663192854e_term ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_J,type,
    comp_P4057098934159210662e_term: ( produc2285326912895808259nt_int > product_prod_int_int ) > ( produc1821397396195235155e_term > produc2285326912895808259nt_int ) > produc1821397396195235155e_term > product_prod_int_int ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_I_Eo_M_Eo_J_001_Eo_001_Eo,type,
    comp_P2137996956111167876_o_o_o: ( product_prod_o_o > $o ) > ( $o > product_prod_o_o ) > $o > $o ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    comp_P7111796090210262324nteger: ( produc8923325533196201883nteger > code_integer ) > ( code_integer > produc8923325533196201883nteger ) > code_integer > code_integer ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    comp_P3107242062624629507er_int: ( produc8923325533196201883nteger > code_integer ) > ( int > produc8923325533196201883nteger ) > int > code_integer ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    comp_P3109732533133679783er_nat: ( produc8923325533196201883nteger > code_integer ) > ( nat > produc8923325533196201883nteger ) > nat > code_integer ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    comp_P3428181540658402820nteger: ( produc8923325533196201883nteger > code_integer ) > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > produc8923325533196201883nteger > code_integer ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Product____Type__Ounit,type,
    comp_P659088277599099894t_unit: ( produc8923325533196201883nteger > code_integer ) > ( product_unit > produc8923325533196201883nteger ) > product_unit > code_integer ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Int__Oint_J_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    comp_P2475326853202991862er_nat: ( produc2256096406371053034er_int > code_integer ) > ( nat > produc2256096406371053034er_int ) > nat > code_integer ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Int__Oint_J_001t__Int__Oint_001t__Nat__Onat,type,
    comp_P8345930540558018757nt_nat: ( produc2256096406371053034er_int > int ) > ( nat > produc2256096406371053034er_int ) > nat > int ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    comp_P5700178199414968986er_nat: ( produc6433947425880249742er_nat > code_integer ) > ( nat > produc6433947425880249742er_nat ) > nat > code_integer ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_P1337767177661389325at_nat: ( produc6433947425880249742er_nat > nat ) > ( nat > produc6433947425880249742er_nat ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    comp_P3067963646784605345atural: ( produc7822875418678951345atural > produc7822875418678951345atural ) > ( produc5835291356934675326atural > produc7822875418678951345atural ) > produc5835291356934675326atural > produc7822875418678951345atural ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    comp_P4943825737035310295atural: ( produc7822875418678951345atural > produc6756470014200349799atural ) > ( produc5835291356934675326atural > produc7822875418678951345atural ) > produc5835291356934675326atural > produc6756470014200349799atural ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_001t__Code____Numeral__Onatural_001t__Int__Oint,type,
    comp_P5589825499870570807al_int: ( produc5835291356934675326atural > code_natural ) > ( int > produc5835291356934675326atural ) > int > code_natural ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_001t__Code____Numeral__Onatural_001t__Nat__Onat,type,
    comp_P5592315970379621083al_nat: ( produc5835291356934675326atural > code_natural ) > ( nat > produc5835291356934675326atural ) > nat > code_natural ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_001t__Code____Numeral__Onatural_001t__Product____Type__Ounit,type,
    comp_P210409258367696578t_unit: ( produc5835291356934675326atural > code_natural ) > ( product_unit > produc5835291356934675326atural ) > product_unit > code_natural ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Int__Oint,type,
    comp_P8475394669922486716al_int: ( produc5835291356934675326atural > produc7822875418678951345atural ) > ( int > produc5835291356934675326atural ) > int > produc7822875418678951345atural ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Nat__Onat,type,
    comp_P8477885140431536992al_nat: ( produc5835291356934675326atural > produc7822875418678951345atural ) > ( nat > produc5835291356934675326atural ) > nat > produc7822875418678951345atural ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    comp_P7610810914149544718atural: ( produc5835291356934675326atural > produc7822875418678951345atural ) > ( produc5835291356934675326atural > produc5835291356934675326atural ) > produc5835291356934675326atural > produc7822875418678951345atural ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__Code____Numeral__Onatural_J,type,
    comp_P5756991293874479000atural: ( produc5835291356934675326atural > produc7822875418678951345atural ) > ( produc3981471736659609608atural > produc5835291356934675326atural ) > produc3981471736659609608atural > produc7822875418678951345atural ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Ounit,type,
    comp_P1945487127105810045t_unit: ( produc5835291356934675326atural > produc7822875418678951345atural ) > ( product_unit > produc5835291356934675326atural ) > product_unit > produc7822875418678951345atural ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Heap__Oarray_Itf__a_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Heap__Oarray_Itf__a_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
    comp_P2097125496263478628t_unit: ( produc6714399278180371524it_nat > option3216581401566546698it_nat ) > ( heap_e7401611519738050253t_unit > produc6714399278180371524it_nat ) > heap_e7401611519738050253t_unit > option3216581401566546698it_nat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    comp_P6870028373966156406er_nat: ( produc2963269191615333738nteger > code_integer ) > ( nat > produc2963269191615333738nteger ) > nat > code_integer ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Code____Numeral__Ointeger_J_001t__Int__Oint_001t__Nat__Onat,type,
    comp_P59025373045620805nt_nat: ( produc2963269191615333738nteger > int ) > ( nat > produc2963269191615333738nteger ) > nat > int ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint_001t__Int__Oint,type,
    comp_P6651465171681656048nt_int: ( product_prod_int_int > int ) > ( int > product_prod_int_int ) > int > int ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint_001t__Nat__Onat,type,
    comp_P6653955642190706324nt_nat: ( product_prod_int_int > int ) > ( nat > product_prod_int_int ) > nat > int ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    comp_P5356232565778667509nt_int: ( product_prod_int_int > int ) > ( product_prod_int_int > product_prod_int_int ) > product_prod_int_int > int ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint_001t__Product____Type__Ounit,type,
    comp_P6796073005956588617t_unit: ( product_prod_int_int > int ) > ( product_unit > product_prod_int_int ) > product_unit > int ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Int__Oint,type,
    comp_P5335556815845735580at_int: ( product_prod_int_int > rat ) > ( int > product_prod_int_int ) > int > rat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    comp_P5648219952011055433nt_int: ( product_prod_int_int > rat ) > ( product_prod_int_int > product_prod_int_int ) > product_prod_int_int > rat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J_001t__Int__Oint_001t__Nat__Onat,type,
    comp_P645145991195131448nt_nat: ( product_prod_int_nat > int ) > ( nat > product_prod_int_nat ) > nat > int ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_P8869164316148852700at_nat: ( product_prod_int_nat > nat ) > ( nat > product_prod_int_nat ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    comp_P5461341465284484122er_nat: ( produc2591615631378489102nteger > code_integer ) > ( nat > produc2591615631378489102nteger ) > nat > code_integer ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Code____Numeral__Ointeger_J_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_P9144323761349829517at_nat: ( produc2591615631378489102nteger > nat ) > ( nat > produc2591615631378489102nteger ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_P7964419862382602716at_nat: ( product_prod_nat_int > nat ) > ( nat > product_prod_nat_int ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Int__Oint_M_Eo_J_001t__Int__Oint,type,
    comp_P7507668834397697393_o_int: ( product_prod_nat_nat > int > $o ) > ( int > product_prod_nat_nat ) > int > int > $o ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Int__Oint_Mt__Int__Oint_J_001t__Int__Oint,type,
    comp_P1029605119844863363nt_int: ( product_prod_nat_nat > int > int ) > ( int > product_prod_nat_nat ) > int > int > int ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001t__Int__Oint,type,
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thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_P1955610211387027840at_nat: ( product_prod_nat_nat > nat ) > ( nat > product_prod_nat_nat ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Product____Type__Ounit,type,
    comp_P2980510830910040157t_unit: ( product_prod_nat_nat > nat ) > ( product_unit > product_prod_nat_nat ) > product_unit > nat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
    comp_P7428279379633703887at_int: ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > product_prod_nat_nat ) > int > product_prod_nat_nat ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_001t__Num__Onum_001t__Num__Onum,type,
    comp_P864883071962432936um_num: ( product_prod_num_num > num ) > ( num > product_prod_num_num ) > num > num ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    comp_P3887058762840424708atural: ( produc3981471736659609608atural > produc7822875418678951345atural ) > ( produc5835291356934675326atural > produc3981471736659609608atural ) > produc5835291356934675326atural > produc7822875418678951345atural ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    comp_P230220204662055659atural: ( produc6756470014200349799atural > produc7822875418678951345atural ) > ( produc5835291356934675326atural > produc6756470014200349799atural ) > produc5835291356934675326atural > produc7822875418678951345atural ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
    comp_P3118722334806803912t_unit: ( produc8664842809031399944it_nat > option8956607266484857688it_nat ) > ( heap_e7401611519738050253t_unit > produc8664842809031399944it_nat ) > heap_e7401611519738050253t_unit > option8956607266484857688it_nat ).

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thf(sy_c_Fun_Ocomp_001t__Product____Type__Ounit_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
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thf(sy_c_Fun_Ocomp_001t__Rat__Orat_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Ocomp_001t__Rat__Orat_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Fun_Ocomp_001t__Rat__Orat_001t__Rat__Orat_001t__Rat__Orat,type,
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thf(sy_c_Fun_Ofun__upd_001_Eo_001_Eo,type,
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thf(sy_c_Fun_Ofun__upd_001_Eo_001t__Int__Oint,type,
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thf(sy_c_Fun_Ofun__upd_001_Eo_001t__Nat__Onat,type,
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thf(sy_c_Fun_Ofun__upd_001_Eo_001t__Option__Ooption_It__Num__Onum_J,type,
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    fun_up8139772397987072257nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).

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thf(sy_c_Fun_Ofun__upd_001t__Code____Numeral__Onatural_001t__Option__Ooption_It__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Int__Oint_001_Eo,type,
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thf(sy_c_Fun_Ofun__upd_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Fun_Ofun__upd_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_Fun_Ofun__upd_001t__Int__Oint_001t__Num__Onum,type,
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thf(sy_c_Fun_Ofun__upd_001t__Int__Oint_001t__Option__Ooption_It__Int__Oint_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Int__Oint_001t__Option__Ooption_It__Num__Onum_J,type,
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    fun_up6992348036167380867it_nat: ( int > option8956607266484857688it_nat ) > int > option8956607266484857688it_nat > int > option8956607266484857688it_nat ).

thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001_Eo,type,
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thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Int__Oint,type,
    fun_upd_nat_int: ( nat > int ) > nat > int > nat > int ).

thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Nat__Onat,type,
    fun_upd_nat_nat: ( nat > nat ) > nat > nat > nat > nat ).

thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Option__Ooption_It__Num__Onum_J,type,
    fun_up2201401324907169337on_num: ( nat > option_num ) > nat > option_num > nat > option_num ).

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thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Num__Onum_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Product____Type__Ounit_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Fun_Oid_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Fun_Oid_001t__Int__Oint,type,
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thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oid_001t__Num__Onum,type,
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thf(sy_c_Fun_Oid_001t__Option__Ooption_I_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Fun_Oid_001t__Option__Ooption_I_Eo_J,type,
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thf(sy_c_Fun_Oid_001t__Option__Ooption_It__Int__Oint_J,type,
    id_option_int: option_int > option_int ).

thf(sy_c_Fun_Oid_001t__Option__Ooption_It__Nat__Onat_J,type,
    id_option_nat: option_nat > option_nat ).

thf(sy_c_Fun_Oid_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Fun_Oid_001t__Product____Type__Ounit,type,
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thf(sy_c_Fun_Oid_001t__Rat__Orat,type,
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    id_set_nat_o: set_nat_o > set_nat_o ).

thf(sy_c_Fun_Oid_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Fun_Oid_001t__Set__Oset_It__Int__Oint_J,type,
    id_set_int: set_int > set_int ).

thf(sy_c_Fun_Oid_001t__Set__Oset_It__Nat__Onat_J,type,
    id_set_nat: set_nat > set_nat ).

thf(sy_c_Fun_Oid_001t__Set__Oset_It__Num__Onum_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Assertions__Oassn,type,
    sup_sup_assn: assn > assn > assn ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_It__Assertions__Oassn_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Product____Type__Ounit,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Num__Onum_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_M_Eo_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Ounit_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Rat__Orat_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
    lattic8265883725875713057ax_nat: set_nat > nat ).

thf(sy_c_Lattices__Big_Osemilattice__neutr__set_OF_001t__Nat__Onat,type,
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thf(sy_c_Lifting_OQuotient_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Lifting_OQuotient_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
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thf(sy_c_List_Oappend_001t__Int__Oint,type,
    append_int: list_int > list_int > list_int ).

thf(sy_c_List_Oappend_001t__Nat__Onat,type,
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thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
    butlast_nat: list_nat > list_nat ).

thf(sy_c_List_Odrop_001t__Nat__Onat,type,
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thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
    filter_nat: ( nat > $o ) > list_nat > list_nat ).

thf(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
    fold_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).

thf(sy_c_List_Olast_001t__Nat__Onat,type,
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thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
    linord2614967742042102400et_nat: set_nat > list_nat ).

thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
    cons_int: int > list_int > list_int ).

thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
    cons_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
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thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
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thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
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thf(sy_c_List_Onth_001t__Int__Oint,type,
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thf(sy_c_List_Onth_001t__Nat__Onat,type,
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thf(sy_c_List_Otake_001t__Nat__Onat,type,
    take_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Oupt,type,
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thf(sy_c_List_Oupto,type,
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thf(sy_c_List_Oupto__rel,type,
    upto_rel: product_prod_int_int > product_prod_int_int > $o ).

thf(sy_c_Misc_OEps__Opt_001t__Nat__Onat,type,
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thf(sy_c_Misc_OEps__Opt_001t__Num__Onum,type,
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    map_mm5942422175790858651it_nat: ( int > option8956607266484857688it_nat ) > set_int > produc8664842809031399944it_nat > int > option8956607266484857688it_nat ).

thf(sy_c_Misc_Omap__mmupd_001t__Nat__Onat_001t__Num__Onum,type,
    map_mmupd_nat_num: ( nat > option_num ) > set_nat > num > nat > option_num ).

thf(sy_c_Misc_Omap__mmupd_001t__Nat__Onat_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    map_mm4316137065318800191it_nat: ( nat > option8956607266484857688it_nat ) > set_nat > produc8664842809031399944it_nat > nat > option8956607266484857688it_nat ).

thf(sy_c_Misc_Omap__mmupd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Num__Onum,type,
    map_mm6272435474268545910at_num: ( product_prod_nat_nat > option_num ) > set_Pr1261947904930325089at_nat > num > product_prod_nat_nat > option_num ).

thf(sy_c_Misc_Omap__mmupd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    map_mm8925508573636269924it_nat: ( product_prod_nat_nat > option8956607266484857688it_nat ) > set_Pr1261947904930325089at_nat > produc8664842809031399944it_nat > product_prod_nat_nat > option8956607266484857688it_nat ).

thf(sy_c_Misc_Omap__mmupd_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Num__Onum,type,
    map_mm2124855372588256220at_num: ( produc3843707927480180839at_nat > option_num ) > set_Pr4329608150637261639at_nat > num > produc3843707927480180839at_nat > option_num ).

thf(sy_c_Misc_Omap__mmupd_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    map_mm6227321356191524810it_nat: ( produc3843707927480180839at_nat > option8956607266484857688it_nat ) > set_Pr4329608150637261639at_nat > produc8664842809031399944it_nat > produc3843707927480180839at_nat > option8956607266484857688it_nat ).

thf(sy_c_Misc_Opairself_001t__Int__Oint_001t__Int__Oint,type,
    pairself_int_int: ( int > int ) > product_prod_int_int > product_prod_int_int ).

thf(sy_c_Misc_Opairself_001t__Int__Oint_001t__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J,type,
    pairse3534876335208825186e_term: ( int > option6357759511663192854e_term ) > product_prod_int_int > produc6576344331059438605e_term ).

thf(sy_c_Misc_Opairself_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Misc_Opairself_001t__Nat__Onat_001t__Int__Oint,type,
    pairself_nat_int: ( nat > int ) > product_prod_nat_nat > product_prod_int_int ).

thf(sy_c_Misc_Opairself_001t__Nat__Onat_001t__Nat__Onat,type,
    pairself_nat_nat: ( nat > nat ) > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Misc_Opairself_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    pairself_nat_set_nat: ( nat > set_nat ) > product_prod_nat_nat > produc7819656566062154093et_nat ).

thf(sy_c_Misc_Opairself__rel_001t__Int__Oint_001t__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J,type,
    pairse6848479906795794847e_term: produc7773217078559923341nt_int > produc7773217078559923341nt_int > $o ).

thf(sy_c_Misc_Orel__of_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    rel_of5543720577181062686nt_int: ( ( int > option6357759511663192854e_term ) > option4624381673175914239nt_int ) > ( produc7773217078559923341nt_int > $o ) > set_Pr1872883991513573699nt_int ).

thf(sy_c_Misc_Orel__of_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    rel_of7835217753297429671nteger: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > option2651255830984564193nteger ) > ( produc1908205239877642774nteger > $o ) > set_Pr1281608226676607948nteger ).

thf(sy_c_Misc_Orel__of_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    rel_of7774016450764239315et_nat: ( ( produc3658429121746597890et_nat > $o ) > option5190343406534369742et_nat ) > ( produc2732055786443039994et_nat > $o ) > set_Pr8536935166611901872et_nat ).

thf(sy_c_Misc_Orel__of_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    rel_of4838799251197538391et_nat: ( ( produc3658429121746597890et_nat > $o ) > option936205604648967762et_nat ) > ( produc3925858234332021118et_nat > $o ) > set_Pr3286484037609594932et_nat ).

thf(sy_c_Misc_Orel__of_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    rel_of8306664904814525588nt_int: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > option4624381673175914239nt_int ) > ( produc2285326912895808259nt_int > $o ) > set_Pr9222295170931077689nt_int ).

thf(sy_c_Misc_Orel__of_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Misc_Orel__of_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Misc_Oset__to__map_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    set_to1653371902132773855nt_int: set_Pr1872883991513573699nt_int > ( int > option6357759511663192854e_term ) > option4624381673175914239nt_int ).

thf(sy_c_Misc_Oset__to__map_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    set_to680829756418416360nteger: set_Pr1281608226676607948nteger > ( produc6241069584506657477e_term > option6357759511663192854e_term ) > option2651255830984564193nteger ).

thf(sy_c_Misc_Oset__to__map_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    set_to2047380710992656148et_nat: set_Pr8536935166611901872et_nat > ( produc3658429121746597890et_nat > $o ) > option5190343406534369742et_nat ).

thf(sy_c_Misc_Oset__to__map_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    set_to6040779677306527128et_nat: set_Pr3286484037609594932et_nat > ( produc3658429121746597890et_nat > $o ) > option936205604648967762et_nat ).

thf(sy_c_Misc_Oset__to__map_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    set_to6585427216324510421nt_int: set_Pr9222295170931077689nt_int > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > option4624381673175914239nt_int ).

thf(sy_c_Misc_Oset__to__map_001_Eo_001t__Num__Onum,type,
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thf(sy_c_Misc_Oset__to__map_001_Eo_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    set_to3023635543527487849it_nat: set_Pr6777250647868329861it_nat > $o > option8956607266484857688it_nat ).

thf(sy_c_Misc_Oset__to__map_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Misc_Oset__to__map_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
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thf(sy_c_Misc_Oset__to__map_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Misc_Oset__to__map_001t__Int__Oint_001t__Num__Onum,type,
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thf(sy_c_Misc_Oset__to__map_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Misc_Oset__to__map_001t__Nat__Onat_001t__Num__Onum,type,
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thf(sy_c_Misc_Oset__to__map_001t__Nat__Onat_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Misc_Oset__to__map_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Num__Onum,type,
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thf(sy_c_Misc_Oset__to__map_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Misc_Othe__default_001t__Num__Onum,type,
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thf(sy_c_Misc_Othe__default_001t__Product____Type__Oprod_It__Heap__Oarray_Itf__a_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Misc_Othe__default_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Misc_Othe__default_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Misc_Othe__default_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Misc_Othe__default_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Misc_Ouncurry_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Misc_Ouncurry_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Misc_Ouncurry_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Multiset_Oms__strict,type,
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thf(sy_c_Multiset_Opw__leq,type,
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thf(sy_c_Nat_OSuc,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
    size_size_num: num > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
    size_size_option_num: option_num > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Heap__Oarray_Itf__a_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    size_s6797157356856800118it_nat: option3216581401566546698it_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    size_s8766407808098229740it_nat: option8956607266484857688it_nat > nat ).

thf(sy_c_Nat__Bijection_Olist__decode,type,
    nat_list_decode: nat > list_nat ).

thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
    nat_list_decode_rel: nat > nat > $o ).

thf(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: list_nat > nat ).

thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: list_nat > list_nat > $o ).

thf(sy_c_Nat__Bijection_Oprod__decode,type,
    nat_prod_decode: nat > product_prod_nat_nat ).

thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).

thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Nat__Bijection_Oprod__encode,type,
    nat_prod_encode: product_prod_nat_nat > nat ).

thf(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set_nat ).

thf(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: set_nat > nat ).

thf(sy_c_Nat__Bijection_Osum__decode,type,
    nat_sum_decode: nat > sum_sum_nat_nat ).

thf(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: nat > nat ).

thf(sy_c_Num_OBitM,type,
    bitM: num > num ).

thf(sy_c_Num_Oinc,type,
    inc: num > num ).

thf(sy_c_Num_Onat__of__num,type,
    nat_of_num: num > nat ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
    neg_nu8804712462038260780nteger: code_integer > code_integer ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
    neg_numeral_dbl_int: int > int ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
    neg_numeral_dbl_rat: rat > rat ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
    neg_nu7757733837767384882nteger: code_integer > code_integer ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
    neg_nu3811975205180677377ec_int: int > int ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
    neg_nu3179335615603231917ec_rat: rat > rat ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
    neg_nu5831290666863070958nteger: code_integer > code_integer ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
    neg_nu5851722552734809277nc_int: int > int ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
    neg_nu5219082963157363817nc_rat: rat > rat ).

thf(sy_c_Num_Oneg__numeral__class_Ois__num_001t__Code____Numeral__Ointeger,type,
    neg_nu2435721316971906650nteger: code_integer > $o ).

thf(sy_c_Num_Oneg__numeral__class_Ois__num_001t__Int__Oint,type,
    neg_nu7441568979088483369um_int: int > $o ).

thf(sy_c_Num_Oneg__numeral__class_Ois__num_001t__Rat__Orat,type,
    neg_nu6808929389511037909um_rat: rat > $o ).

thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Code____Numeral__Ointeger,type,
    neg_nu5755505904847501662nteger: num > num > code_integer ).

thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
    neg_numeral_sub_int: num > num > int ).

thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Rat__Orat,type,
    neg_numeral_sub_rat: num > num > rat ).

thf(sy_c_Num_Onum_OBit0,type,
    bit0: num > num ).

thf(sy_c_Num_Onum_OBit1,type,
    bit1: num > num ).

thf(sy_c_Num_Onum_OOne,type,
    one: num ).

thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
    case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).

thf(sy_c_Num_Onum_Osize__num,type,
    size_num: num > nat ).

thf(sy_c_Num_Onum__of__nat,type,
    num_of_nat: nat > num ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
    numera6620942414471956472nteger: num > code_integer ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Onatural,type,
    numera5444537566228673987atural: num > code_natural ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
    numeral_numeral_int: num > int ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: num > nat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
    numeral_numeral_rat: num > rat ).

thf(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > nat ).

thf(sy_c_Num_Oring__1__class_Oiszero_001t__Code____Numeral__Ointeger,type,
    ring_16219924574208605041nteger: code_integer > $o ).

thf(sy_c_Num_Oring__1__class_Oiszero_001t__Int__Oint,type,
    ring_1_iszero_int: int > $o ).

thf(sy_c_Num_Oring__1__class_Oiszero_001t__Rat__Orat,type,
    ring_1_iszero_rat: rat > $o ).

thf(sy_c_Num_Osqr,type,
    sqr: num > num ).

thf(sy_c_Option_Obind_001t__Num__Onum_001t__Num__Onum,type,
    bind_num_num: option_num > ( num > option_num ) > option_num ).

thf(sy_c_Option_Obind_001t__Num__Onum_001t__Product____Type__Oprod_It__Heap__Oarray_Itf__a_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    bind_n6186876254402743274it_nat: option_num > ( num > option3216581401566546698it_nat ) > option3216581401566546698it_nat ).

thf(sy_c_Option_Obind_001t__Num__Onum_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    bind_n550516262659884002it_nat: option_num > ( num > option8956607266484857688it_nat ) > option8956607266484857688it_nat ).

thf(sy_c_Option_Obind_001t__Product____Type__Oprod_It__Heap__Oarray_Itf__a_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_001t__Num__Onum,type,
    bind_P7816469540982926368at_num: option3216581401566546698it_nat > ( produc6714399278180371524it_nat > option_num ) > option_num ).

thf(sy_c_Option_Obind_001t__Product____Type__Oprod_It__Heap__Oarray_Itf__a_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Heap__Oarray_Itf__a_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    bind_P1787608729393679550it_nat: option3216581401566546698it_nat > ( produc6714399278180371524it_nat > option3216581401566546698it_nat ) > option3216581401566546698it_nat ).

thf(sy_c_Option_Obind_001t__Product____Type__Oprod_It__Heap__Oarray_Itf__a_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    bind_P3087804342168909838it_nat: option3216581401566546698it_nat > ( produc6714399278180371524it_nat > option8956607266484857688it_nat ) > option8956607266484857688it_nat ).

thf(sy_c_Option_Obind_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_001t__Num__Onum,type,
    bind_P4387644102537648098at_num: option8956607266484857688it_nat > ( produc8664842809031399944it_nat > option_num ) > option_num ).

thf(sy_c_Option_Obind_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Heap__Oarray_Itf__a_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    bind_P1106314856888375420it_nat: option8956607266484857688it_nat > ( produc8664842809031399944it_nat > option3216581401566546698it_nat ) > option3216581401566546698it_nat ).

thf(sy_c_Option_Obind_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    bind_P7351952464966851536it_nat: option8956607266484857688it_nat > ( produc8664842809031399944it_nat > option8956607266484857688it_nat ) > option8956607266484857688it_nat ).

thf(sy_c_Option_Ocombine__options_001t__Num__Onum,type,
    combine_options_num: ( num > num > num ) > option_num > option_num > option_num ).

thf(sy_c_Option_Ocombine__options_001t__Product____Type__Oprod_It__Heap__Oarray_Itf__a_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    combin3867838909898076099it_nat: ( produc6714399278180371524it_nat > produc6714399278180371524it_nat > produc6714399278180371524it_nat ) > option3216581401566546698it_nat > option3216581401566546698it_nat > option3216581401566546698it_nat ).

thf(sy_c_Option_Ocombine__options_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    combin4318129983670048329it_nat: ( produc8664842809031399944it_nat > produc8664842809031399944it_nat > produc8664842809031399944it_nat ) > option8956607266484857688it_nat > option8956607266484857688it_nat > option8956607266484857688it_nat ).

thf(sy_c_Option_Ooption_ONone_001_Eo,type,
    none_o: option_o ).

thf(sy_c_Option_Ooption_ONone_001t__Int__Oint,type,
    none_int: option_int ).

thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    none_P3773570700014501484nt_int: option4256020574406277085nt_int ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    none_P4442379456014020469nteger: option8051342751916580710nteger ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    none_P199884684680593241et_nat: option2860828798490689354et_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    none_P4972525538344268765et_nat: option5190343406534369742et_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    none_P1286213070022356066nt_int: option7541221861074943443nt_int ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Heap__Oarray_Itf__a_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P2587165478634041381it_nat: option3216581401566546698it_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P9117596204409417319it_nat: option8956607266484857688it_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    none_P5811896708078067656at_nat: option8699557107911914541at_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Ounit,type,
    none_Product_unit: option_Product_unit ).

thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_I_Eo_J,type,
    none_set_o: option_set_o ).

thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Int__Oint_J,type,
    none_set_int: option_set_int ).

thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Nat__Onat_J,type,
    none_set_nat: option_set_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    none_s625347054029921090at_nat: option8963830502488799655at_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    none_s6518873173954064040at_nat: option1583680563626158861at_nat ).

thf(sy_c_Option_Ooption_OSome_001_062_It__Nat__Onat_M_Eo_J,type,
    some_nat_o: ( nat > $o ) > option_nat_o ).

thf(sy_c_Option_Ooption_OSome_001_Eo,type,
    some_o: $o > option_o ).

thf(sy_c_Option_Ooption_OSome_001t__Assertions__Oassn,type,
    some_assn: assn > option_assn ).

thf(sy_c_Option_Ooption_OSome_001t__Code____Numeral__Ointeger,type,
    some_Code_integer: code_integer > option_Code_integer ).

thf(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
    some_int: int > option_int ).

thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
    some_nat: nat > option_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
    some_num: num > option_num ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    some_P7455497367792166888nt_int: produc7773217078559923341nt_int > option4256020574406277085nt_int ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    some_P1462369734362851057nteger: produc1908205239877642774nteger > option8051342751916580710nteger ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    some_P1630309045189364437et_nat: produc2732055786443039994et_nat > option2860828798490689354et_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    some_P750831030444334937et_nat: produc3925858234332021118et_nat > option5190343406534369742et_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    some_P2355398578364412894nt_int: produc2285326912895808259nt_int > option7541221861074943443nt_int ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    some_P6772290148444788224nteger: produc8923325533196201883nteger > option2651255830984564193nteger ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    some_P5671840033927537686atural: produc7822875418678951345atural > option5403585828896915703atural ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Heap__Oarray_Itf__a_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    some_P6105628898608307625it_nat: produc6714399278180371524it_nat > option3216581401566546698it_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Ounit,type,
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thf(sy_c_Option_Ooption_OSome_001t__Rat__Orat,type,
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thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Heap__Oarray_Itf__a_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Num__Onum,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Num__Onum,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Num__Onum_001t__Num__Onum,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
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thf(sy_c_Product__Type_Ounit_OAbs__unit,type,
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thf(sy_c_Product__Type_Ounit_ORep__unit,type,
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thf(sy_c_Random_Oinc__shift,type,
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thf(sy_c_Random_Olog,type,
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thf(sy_c_Random_Olog__rel,type,
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thf(sy_c_Random_Ominus__shift,type,
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thf(sy_c_Random_Onext,type,
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thf(sy_c_Random_Orange,type,
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thf(sy_c_Random_Osplit__seed,type,
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thf(sy_c_Rat_OAbs__Rat,type,
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thf(sy_c_Rat_OFract,type,
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thf(sy_c_Rat_OFrct,type,
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thf(sy_c_Rat_ORep__Rat,type,
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thf(sy_c_Rat_Onormalize,type,
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thf(sy_c_Rat_Oof__int,type,
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thf(sy_c_Rat_Opcr__rat,type,
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thf(sy_c_Rat_Opositive,type,
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thf(sy_c_Rat_Oquotient__of,type,
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thf(sy_c_Rat_Oratrel,type,
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thf(sy_c_Refine__Imp__Hol_Oassert_H,type,
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    member6582457606847315088nt_int: produc2432570611225516007nt_int > set_Pr4708930517165415495nt_int > $o ).

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    member335077259223437328nteger: produc3334582477557816039nteger > set_Pr5052581263052047815nteger > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    member6099555550032318734et_nat: produc8111630337999740517et_nat > set_Pr719794911490849221et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member8566619992076573584nt_int: produc1219242969750017639nt_int > set_Pr2560585780119916871nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    member2200362271989506215_nat_o: produc652263959640188542_nat_o > set_Pr6558897299634722526_nat_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member4107301959146914710at_nat: produc2714363965264676845at_nat > set_Pr2072254264782222413at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
    member3480088626692287306at_nat: produc2087150632810049441at_nat > set_Pr5070339090747523585at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member6009668390874684176at_nat: produc1906917888121415911at_nat > set_Pr5564308138774400199at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    member3421417852355157663it_nat: produc8664842809031399944it_nat > set_Pr1354866905816374718it_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_I_Eo_J_Mt__Set__Oset_I_Eo_J_J,type,
    member9116954335612470352_set_o: produc7369051934464679207_set_o > set_Pr4577759397028426247_set_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Code____Numeral__Ointeger_J_Mt__Set__Oset_It__Code____Numeral__Ointeger_J_J,type,
    member4307123515891402160nteger: produc6491284506569428743nteger > set_Pr7577011563204128103nteger > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
    member2572552093476627150et_int: produc2115011035271226405et_int > set_Pr2522554150109002629et_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
    member444158400953824016od_o_o: produc2934264451710624999od_o_o > set_Pr1932065953672099015od_o_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member4095101504841534314at_nat: produc732395585841259969at_nat > set_Pr457366540195662369at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
    member1466754251312161552at_nat: produc1319942482725812455at_nat > set_Pr7459493094073627847at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member4300370950623927696at_nat: produc7299876168958306407at_nat > set_Pr1604112560894399303at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Ounit,type,
    member_Product_unit: product_unit > set_Product_unit > $o ).

thf(sy_c_member_001t__Rat__Orat,type,
    member_rat: rat > set_rat > $o ).

thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
    member_set_o: set_o > set_set_o > $o ).

thf(sy_c_member_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    member1834298987299537450nteger: set_Code_integer > set_set_Code_integer > $o ).

thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
    member_set_int: set_int > set_set_int > $o ).

thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    member_set_list_nat: set_list_nat > set_set_list_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > set_set_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    member5877623283571906838t_unit: set_Product_unit > set_set_Product_unit > $o ).

thf(sy_v_h,type,
    h: heap_e7401611519738050253t_unit ).

thf(sy_v_xs,type,
    xs: list_a ).

% Relevant facts (7951)
thf(fact_0_fails__def,axiom,
    ( time_fails_array_a
    = ( ^ [M: heap_T6362127782908069795rray_a,H: heap_e7401611519738050253t_unit] :
          ( ( heap_T5591663578999710211rray_a @ M @ H )
          = none_P2587165478634041381it_nat ) ) ) ).

% fails_def
thf(fact_1_fails__def,axiom,
    ( time_f8834461667527620124t_unit
    = ( ^ [M: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit] :
          ( ( heap_T875086893843062177t_unit @ M @ H )
          = none_P9117596204409417319it_nat ) ) ) ).

% fails_def
thf(fact_2_Heap__eqI,axiom,
    ! [F: heap_T6362127782908069795rray_a,G: heap_T6362127782908069795rray_a] :
      ( ! [H2: heap_e7401611519738050253t_unit] :
          ( ( heap_T5591663578999710211rray_a @ F @ H2 )
          = ( heap_T5591663578999710211rray_a @ G @ H2 ) )
     => ( F = G ) ) ).

% Heap_eqI
thf(fact_3_Heap__eqI,axiom,
    ! [F: heap_T5738788834812785303t_unit,G: heap_T5738788834812785303t_unit] :
      ( ! [H2: heap_e7401611519738050253t_unit] :
          ( ( heap_T875086893843062177t_unit @ F @ H2 )
          = ( heap_T875086893843062177t_unit @ G @ H2 ) )
     => ( F = G ) ) ).

% Heap_eqI
thf(fact_4_execute__raise,axiom,
    ! [S: list_char] :
      ( ( heap_T5591663578999710211rray_a @ ( heap_T1552190017797416154rray_a @ S ) )
      = ( ^ [Uu: heap_e7401611519738050253t_unit] : none_P2587165478634041381it_nat ) ) ).

% execute_raise
thf(fact_5_execute__raise,axiom,
    ! [S: list_char] :
      ( ( heap_T875086893843062177t_unit @ ( heap_T2927564422264180874t_unit @ S ) )
      = ( ^ [Uu: heap_e7401611519738050253t_unit] : none_P9117596204409417319it_nat ) ) ).

% execute_raise
thf(fact_6_execute__guard_I1_J,axiom,
    ! [P: heap_e7401611519738050253t_unit > $o,H3: heap_e7401611519738050253t_unit,F: heap_e7401611519738050253t_unit > produc6714399278180371524it_nat] :
      ( ~ ( P @ H3 )
     => ( ( heap_T5591663578999710211rray_a @ ( heap_T7685628758766326643rray_a @ P @ F ) @ H3 )
        = none_P2587165478634041381it_nat ) ) ).

% execute_guard(1)
thf(fact_7_execute__guard_I1_J,axiom,
    ! [P: heap_e7401611519738050253t_unit > $o,H3: heap_e7401611519738050253t_unit,F: heap_e7401611519738050253t_unit > produc8664842809031399944it_nat] :
      ( ~ ( P @ H3 )
     => ( ( heap_T875086893843062177t_unit @ ( heap_T8440541562793052209t_unit @ P @ F ) @ H3 )
        = none_P9117596204409417319it_nat ) ) ).

% execute_guard(1)
thf(fact_8_execute__assert_I2_J,axiom,
    ! [P: array_a > $o,X: array_a,H3: heap_e7401611519738050253t_unit] :
      ( ~ ( P @ X )
     => ( ( heap_T5591663578999710211rray_a @ ( heap_T7971689983404357896rray_a @ P @ X ) @ H3 )
        = none_P2587165478634041381it_nat ) ) ).

% execute_assert(2)
thf(fact_9_execute__assert_I2_J,axiom,
    ! [P: product_unit > $o,X: product_unit,H3: heap_e7401611519738050253t_unit] :
      ( ~ ( P @ X )
     => ( ( heap_T875086893843062177t_unit @ ( heap_T4208721593536448476t_unit @ P @ X ) @ H3 )
        = none_P9117596204409417319it_nat ) ) ).

% execute_assert(2)
thf(fact_10_successI,axiom,
    ! [F: heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit] :
      ( ( ( heap_T5591663578999710211rray_a @ F @ H3 )
       != none_P2587165478634041381it_nat )
     => ( heap_T6932021131025641745rray_a @ F @ H3 ) ) ).

% successI
thf(fact_11_successI,axiom,
    ! [F: heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( ( heap_T875086893843062177t_unit @ F @ H3 )
       != none_P9117596204409417319it_nat )
     => ( heap_T584514906347983379t_unit @ F @ H3 ) ) ).

% successI
thf(fact_12_success__def,axiom,
    ( heap_T6932021131025641745rray_a
    = ( ^ [F2: heap_T6362127782908069795rray_a,H: heap_e7401611519738050253t_unit] :
          ( ( heap_T5591663578999710211rray_a @ F2 @ H )
         != none_P2587165478634041381it_nat ) ) ) ).

% success_def
thf(fact_13_success__def,axiom,
    ( heap_T584514906347983379t_unit
    = ( ^ [F2: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit] :
          ( ( heap_T875086893843062177t_unit @ F2 @ H )
         != none_P9117596204409417319it_nat ) ) ) ).

% success_def
thf(fact_14_Heap__ordE,axiom,
    ! [X: heap_T6362127782908069795rray_a,Y: heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit] :
      ( ( heap_T3039585018531920163rray_a @ X @ Y )
     => ( ( ( heap_T5591663578999710211rray_a @ X @ H3 )
         != none_P2587165478634041381it_nat )
       => ( ( heap_T5591663578999710211rray_a @ X @ H3 )
          = ( heap_T5591663578999710211rray_a @ Y @ H3 ) ) ) ) ).

% Heap_ordE
thf(fact_15_Heap__ordE,axiom,
    ! [X: heap_T5738788834812785303t_unit,Y: heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( heap_T499439718319394433t_unit @ X @ Y )
     => ( ( ( heap_T875086893843062177t_unit @ X @ H3 )
         != none_P9117596204409417319it_nat )
       => ( ( heap_T875086893843062177t_unit @ X @ H3 )
          = ( heap_T875086893843062177t_unit @ Y @ H3 ) ) ) ) ).

% Heap_ordE
thf(fact_16_Heap__ordI,axiom,
    ! [X: heap_T6362127782908069795rray_a,Y: heap_T6362127782908069795rray_a] :
      ( ! [H2: heap_e7401611519738050253t_unit] :
          ( ( ( heap_T5591663578999710211rray_a @ X @ H2 )
            = none_P2587165478634041381it_nat )
          | ( ( heap_T5591663578999710211rray_a @ X @ H2 )
            = ( heap_T5591663578999710211rray_a @ Y @ H2 ) ) )
     => ( heap_T3039585018531920163rray_a @ X @ Y ) ) ).

% Heap_ordI
thf(fact_17_Heap__ordI,axiom,
    ! [X: heap_T5738788834812785303t_unit,Y: heap_T5738788834812785303t_unit] :
      ( ! [H2: heap_e7401611519738050253t_unit] :
          ( ( ( heap_T875086893843062177t_unit @ X @ H2 )
            = none_P9117596204409417319it_nat )
          | ( ( heap_T875086893843062177t_unit @ X @ H2 )
            = ( heap_T875086893843062177t_unit @ Y @ H2 ) ) )
     => ( heap_T499439718319394433t_unit @ X @ Y ) ) ).

% Heap_ordI
thf(fact_18_execute__bind_I2_J,axiom,
    ! [F: heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit,G: array_a > heap_T6362127782908069795rray_a] :
      ( ( ( heap_T5591663578999710211rray_a @ F @ H3 )
        = none_P2587165478634041381it_nat )
     => ( ( heap_T5591663578999710211rray_a @ ( heap_T2407400398799471159rray_a @ F @ G ) @ H3 )
        = none_P2587165478634041381it_nat ) ) ).

% execute_bind(2)
thf(fact_19_execute__bind_I2_J,axiom,
    ! [F: heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit,G: array_a > heap_T5738788834812785303t_unit] :
      ( ( ( heap_T5591663578999710211rray_a @ F @ H3 )
        = none_P2587165478634041381it_nat )
     => ( ( heap_T875086893843062177t_unit @ ( heap_T7324619625323989613t_unit @ F @ G ) @ H3 )
        = none_P9117596204409417319it_nat ) ) ).

% execute_bind(2)
thf(fact_20_execute__bind_I2_J,axiom,
    ! [F: heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit,G: product_unit > heap_T6362127782908069795rray_a] :
      ( ( ( heap_T875086893843062177t_unit @ F @ H3 )
        = none_P9117596204409417319it_nat )
     => ( ( heap_T5591663578999710211rray_a @ ( heap_T7158140910958840029rray_a @ F @ G ) @ H3 )
        = none_P2587165478634041381it_nat ) ) ).

% execute_bind(2)
thf(fact_21_execute__bind_I2_J,axiom,
    ! [F: heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit,G: product_unit > heap_T5738788834812785303t_unit] :
      ( ( ( heap_T875086893843062177t_unit @ F @ H3 )
        = none_P9117596204409417319it_nat )
     => ( ( heap_T875086893843062177t_unit @ ( heap_T2633723481742716231t_unit @ F @ G ) @ H3 )
        = none_P9117596204409417319it_nat ) ) ).

% execute_bind(2)
thf(fact_22_the__default_Osimps_I2_J,axiom,
    ! [X: produc6714399278180371524it_nat] :
      ( ( the_de8559148246674365659it_nat @ X @ none_P2587165478634041381it_nat )
      = X ) ).

% the_default.simps(2)
thf(fact_23_the__default_Osimps_I2_J,axiom,
    ! [X: num] :
      ( ( the_default_num @ X @ none_num )
      = X ) ).

% the_default.simps(2)
thf(fact_24_the__default_Osimps_I2_J,axiom,
    ! [X: produc8664842809031399944it_nat] :
      ( ( the_de2487931475039285041it_nat @ X @ none_P9117596204409417319it_nat )
      = X ) ).

% the_default.simps(2)
thf(fact_25_raise__bind,axiom,
    ! [E: list_char,F: product_unit > heap_T5738788834812785303t_unit] :
      ( ( heap_T2633723481742716231t_unit @ ( heap_T2927564422264180874t_unit @ E ) @ F )
      = ( heap_T2927564422264180874t_unit @ E ) ) ).

% raise_bind
thf(fact_26_raise__bind,axiom,
    ! [E: list_char,F: product_unit > heap_T6362127782908069795rray_a] :
      ( ( heap_T7158140910958840029rray_a @ ( heap_T2927564422264180874t_unit @ E ) @ F )
      = ( heap_T1552190017797416154rray_a @ E ) ) ).

% raise_bind
thf(fact_27_raise__bind,axiom,
    ! [E: list_char,F: array_a > heap_T5738788834812785303t_unit] :
      ( ( heap_T7324619625323989613t_unit @ ( heap_T1552190017797416154rray_a @ E ) @ F )
      = ( heap_T2927564422264180874t_unit @ E ) ) ).

% raise_bind
thf(fact_28_raise__bind,axiom,
    ! [E: list_char,F: array_a > heap_T6362127782908069795rray_a] :
      ( ( heap_T2407400398799471159rray_a @ ( heap_T1552190017797416154rray_a @ E ) @ F )
      = ( heap_T1552190017797416154rray_a @ E ) ) ).

% raise_bind
thf(fact_29_heap_Oleq__refl,axiom,
    ! [X: heap_T5738788834812785303t_unit] : ( heap_T499439718319394433t_unit @ X @ X ) ).

% heap.leq_refl
thf(fact_30_heap_Oleq__refl,axiom,
    ! [X: heap_T6362127782908069795rray_a] : ( heap_T3039585018531920163rray_a @ X @ X ) ).

% heap.leq_refl
thf(fact_31_heap_Oleq__trans,axiom,
    ! [X: heap_T5738788834812785303t_unit,Y: heap_T5738788834812785303t_unit,Z: heap_T5738788834812785303t_unit] :
      ( ( heap_T499439718319394433t_unit @ X @ Y )
     => ( ( heap_T499439718319394433t_unit @ Y @ Z )
       => ( heap_T499439718319394433t_unit @ X @ Z ) ) ) ).

% heap.leq_trans
thf(fact_32_heap_Oleq__trans,axiom,
    ! [X: heap_T6362127782908069795rray_a,Y: heap_T6362127782908069795rray_a,Z: heap_T6362127782908069795rray_a] :
      ( ( heap_T3039585018531920163rray_a @ X @ Y )
     => ( ( heap_T3039585018531920163rray_a @ Y @ Z )
       => ( heap_T3039585018531920163rray_a @ X @ Z ) ) ) ).

% heap.leq_trans
thf(fact_33_heap_Oleq__antisym,axiom,
    ! [X: heap_T5738788834812785303t_unit,Y: heap_T5738788834812785303t_unit] :
      ( ( heap_T499439718319394433t_unit @ X @ Y )
     => ( ( heap_T499439718319394433t_unit @ Y @ X )
       => ( X = Y ) ) ) ).

% heap.leq_antisym
thf(fact_34_heap_Oleq__antisym,axiom,
    ! [X: heap_T6362127782908069795rray_a,Y: heap_T6362127782908069795rray_a] :
      ( ( heap_T3039585018531920163rray_a @ X @ Y )
     => ( ( heap_T3039585018531920163rray_a @ Y @ X )
       => ( X = Y ) ) ) ).

% heap.leq_antisym
thf(fact_35_success__ifI,axiom,
    ! [C: $o,T: heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit,E: heap_T5738788834812785303t_unit] :
      ( ( C
       => ( heap_T584514906347983379t_unit @ T @ H3 ) )
     => ( ( ~ C
         => ( heap_T584514906347983379t_unit @ E @ H3 ) )
       => ( heap_T584514906347983379t_unit @ ( if_Hea8138950348631371857t_unit @ C @ T @ E ) @ H3 ) ) ) ).

% success_ifI
thf(fact_36_success__ifI,axiom,
    ! [C: $o,T: heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit,E: heap_T6362127782908069795rray_a] :
      ( ( C
       => ( heap_T6932021131025641745rray_a @ T @ H3 ) )
     => ( ( ~ C
         => ( heap_T6932021131025641745rray_a @ E @ H3 ) )
       => ( heap_T6932021131025641745rray_a @ ( if_Hea5896630067203373929rray_a @ C @ T @ E ) @ H3 ) ) ) ).

% success_ifI
thf(fact_37_success__guardI,axiom,
    ! [P: heap_e7401611519738050253t_unit > $o,H3: heap_e7401611519738050253t_unit,F: heap_e7401611519738050253t_unit > produc8664842809031399944it_nat] :
      ( ( P @ H3 )
     => ( heap_T584514906347983379t_unit @ ( heap_T8440541562793052209t_unit @ P @ F ) @ H3 ) ) ).

% success_guardI
thf(fact_38_success__guardI,axiom,
    ! [P: heap_e7401611519738050253t_unit > $o,H3: heap_e7401611519738050253t_unit,F: heap_e7401611519738050253t_unit > produc6714399278180371524it_nat] :
      ( ( P @ H3 )
     => ( heap_T6932021131025641745rray_a @ ( heap_T7685628758766326643rray_a @ P @ F ) @ H3 ) ) ).

% success_guardI
thf(fact_39_success__assertI,axiom,
    ! [P: product_unit > $o,X: product_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( P @ X )
     => ( heap_T584514906347983379t_unit @ ( heap_T4208721593536448476t_unit @ P @ X ) @ H3 ) ) ).

% success_assertI
thf(fact_40_success__assertI,axiom,
    ! [P: array_a > $o,X: array_a,H3: heap_e7401611519738050253t_unit] :
      ( ( P @ X )
     => ( heap_T6932021131025641745rray_a @ ( heap_T7971689983404357896rray_a @ P @ X ) @ H3 ) ) ).

% success_assertI
thf(fact_41_fails__bind,axiom,
    ! [M2: heap_T5738788834812785303t_unit,F: product_unit > heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( time_f8834461667527620124t_unit @ ( heap_T2633723481742716231t_unit @ M2 @ F ) @ H3 )
      = ( ~ ( time_f8834461667527620124t_unit @ M2 @ H3 )
       => ( time_f8834461667527620124t_unit @ ( F @ ( time_t2745601220510131320t_unit @ M2 @ H3 ) ) @ ( time_t3350536682685678164t_unit @ M2 @ H3 ) ) ) ) ).

% fails_bind
thf(fact_42_fails__bind,axiom,
    ! [M2: heap_T6362127782908069795rray_a,F: array_a > heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( time_f8834461667527620124t_unit @ ( heap_T7324619625323989613t_unit @ M2 @ F ) @ H3 )
      = ( ~ ( time_fails_array_a @ M2 @ H3 )
       => ( time_f8834461667527620124t_unit @ ( F @ ( time_the_res_array_a @ M2 @ H3 ) ) @ ( time_t490086488869527696rray_a @ M2 @ H3 ) ) ) ) ).

% fails_bind
thf(fact_43_fails__bind,axiom,
    ! [M2: heap_T5738788834812785303t_unit,F: product_unit > heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit] :
      ( ( time_fails_array_a @ ( heap_T7158140910958840029rray_a @ M2 @ F ) @ H3 )
      = ( ~ ( time_f8834461667527620124t_unit @ M2 @ H3 )
       => ( time_fails_array_a @ ( F @ ( time_t2745601220510131320t_unit @ M2 @ H3 ) ) @ ( time_t3350536682685678164t_unit @ M2 @ H3 ) ) ) ) ).

% fails_bind
thf(fact_44_fails__bind,axiom,
    ! [M2: heap_T6362127782908069795rray_a,F: array_a > heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit] :
      ( ( time_fails_array_a @ ( heap_T2407400398799471159rray_a @ M2 @ F ) @ H3 )
      = ( ~ ( time_fails_array_a @ M2 @ H3 )
       => ( time_fails_array_a @ ( F @ ( time_the_res_array_a @ M2 @ H3 ) ) @ ( time_t490086488869527696rray_a @ M2 @ H3 ) ) ) ) ).

% fails_bind
thf(fact_45_success__of__listI,axiom,
    ! [Xs: list_a,H3: heap_e7401611519738050253t_unit] : ( heap_T6932021131025641745rray_a @ ( array_of_list_a @ Xs ) @ H3 ) ).

% success_of_listI
thf(fact_46_success__tapI,axiom,
    ! [F: heap_e7401611519738050253t_unit > product_unit,H3: heap_e7401611519738050253t_unit] : ( heap_T584514906347983379t_unit @ ( heap_T560649228465745139t_unit @ F ) @ H3 ) ).

% success_tapI
thf(fact_47_success__tapI,axiom,
    ! [F: heap_e7401611519738050253t_unit > array_a,H3: heap_e7401611519738050253t_unit] : ( heap_T6932021131025641745rray_a @ ( heap_T2055650868554030641rray_a @ F ) @ H3 ) ).

% success_tapI
thf(fact_48_success__heapI,axiom,
    ! [F: heap_e7401611519738050253t_unit > produc8664842809031399944it_nat,H3: heap_e7401611519738050253t_unit] : ( heap_T584514906347983379t_unit @ ( heap_T6927113302350381334t_unit @ F ) @ H3 ) ).

% success_heapI
thf(fact_49_success__heapI,axiom,
    ! [F: heap_e7401611519738050253t_unit > produc6714399278180371524it_nat,H3: heap_e7401611519738050253t_unit] : ( heap_T6932021131025641745rray_a @ ( heap_T7473999186002612558rray_a @ F ) @ H3 ) ).

% success_heapI
thf(fact_50_success__ureturnI,axiom,
    ! [X: product_unit,H3: heap_e7401611519738050253t_unit] : ( heap_T584514906347983379t_unit @ ( heap_T4284346855313245393t_unit @ X ) @ H3 ) ).

% success_ureturnI
thf(fact_51_success__ureturnI,axiom,
    ! [X: array_a,H3: heap_e7401611519738050253t_unit] : ( heap_T6932021131025641745rray_a @ ( heap_T4328122439115681491rray_a @ X ) @ H3 ) ).

% success_ureturnI
thf(fact_52_execute__guard_I2_J,axiom,
    ! [P: heap_e7401611519738050253t_unit > $o,H3: heap_e7401611519738050253t_unit,F: heap_e7401611519738050253t_unit > produc6714399278180371524it_nat] :
      ( ( P @ H3 )
     => ( ( heap_T5591663578999710211rray_a @ ( heap_T7685628758766326643rray_a @ P @ F ) @ H3 )
        = ( some_P6105628898608307625it_nat @ ( F @ H3 ) ) ) ) ).

% execute_guard(2)
thf(fact_53_execute__guard_I2_J,axiom,
    ! [P: heap_e7401611519738050253t_unit > $o,H3: heap_e7401611519738050253t_unit,F: heap_e7401611519738050253t_unit > produc8664842809031399944it_nat] :
      ( ( P @ H3 )
     => ( ( heap_T875086893843062177t_unit @ ( heap_T8440541562793052209t_unit @ P @ F ) @ H3 )
        = ( some_P1914260805536162275it_nat @ ( F @ H3 ) ) ) ) ).

% execute_guard(2)
thf(fact_54_combine__options__simps_I1_J,axiom,
    ! [F: produc6714399278180371524it_nat > produc6714399278180371524it_nat > produc6714399278180371524it_nat,Y: option3216581401566546698it_nat] :
      ( ( combin3867838909898076099it_nat @ F @ none_P2587165478634041381it_nat @ Y )
      = Y ) ).

% combine_options_simps(1)
thf(fact_55_combine__options__simps_I1_J,axiom,
    ! [F: num > num > num,Y: option_num] :
      ( ( combine_options_num @ F @ none_num @ Y )
      = Y ) ).

% combine_options_simps(1)
thf(fact_56_combine__options__simps_I1_J,axiom,
    ! [F: produc8664842809031399944it_nat > produc8664842809031399944it_nat > produc8664842809031399944it_nat,Y: option8956607266484857688it_nat] :
      ( ( combin4318129983670048329it_nat @ F @ none_P9117596204409417319it_nat @ Y )
      = Y ) ).

% combine_options_simps(1)
thf(fact_57_combine__options__simps_I2_J,axiom,
    ! [F: produc6714399278180371524it_nat > produc6714399278180371524it_nat > produc6714399278180371524it_nat,X: option3216581401566546698it_nat] :
      ( ( combin3867838909898076099it_nat @ F @ X @ none_P2587165478634041381it_nat )
      = X ) ).

% combine_options_simps(2)
thf(fact_58_combine__options__simps_I2_J,axiom,
    ! [F: num > num > num,X: option_num] :
      ( ( combine_options_num @ F @ X @ none_num )
      = X ) ).

% combine_options_simps(2)
thf(fact_59_combine__options__simps_I2_J,axiom,
    ! [F: produc8664842809031399944it_nat > produc8664842809031399944it_nat > produc8664842809031399944it_nat,X: option8956607266484857688it_nat] :
      ( ( combin4318129983670048329it_nat @ F @ X @ none_P9117596204409417319it_nat )
      = X ) ).

% combine_options_simps(2)
thf(fact_60_option_Oinject,axiom,
    ! [X2: num,Y2: num] :
      ( ( ( some_num @ X2 )
        = ( some_num @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_61_option_Oinject,axiom,
    ! [X2: produc8664842809031399944it_nat,Y2: produc8664842809031399944it_nat] :
      ( ( ( some_P1914260805536162275it_nat @ X2 )
        = ( some_P1914260805536162275it_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_62_not__None__eq,axiom,
    ! [X: option3216581401566546698it_nat] :
      ( ( X != none_P2587165478634041381it_nat )
      = ( ? [Y3: produc6714399278180371524it_nat] :
            ( X
            = ( some_P6105628898608307625it_nat @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_63_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
      = ( ? [Y3: num] :
            ( X
            = ( some_num @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_64_not__None__eq,axiom,
    ! [X: option8956607266484857688it_nat] :
      ( ( X != none_P9117596204409417319it_nat )
      = ( ? [Y3: produc8664842809031399944it_nat] :
            ( X
            = ( some_P1914260805536162275it_nat @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_65_not__Some__eq,axiom,
    ! [X: option3216581401566546698it_nat] :
      ( ( ! [Y3: produc6714399278180371524it_nat] :
            ( X
           != ( some_P6105628898608307625it_nat @ Y3 ) ) )
      = ( X = none_P2587165478634041381it_nat ) ) ).

% not_Some_eq
thf(fact_66_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ( ! [Y3: num] :
            ( X
           != ( some_num @ Y3 ) ) )
      = ( X = none_num ) ) ).

% not_Some_eq
thf(fact_67_not__Some__eq,axiom,
    ! [X: option8956607266484857688it_nat] :
      ( ( ! [Y3: produc8664842809031399944it_nat] :
            ( X
           != ( some_P1914260805536162275it_nat @ Y3 ) ) )
      = ( X = none_P9117596204409417319it_nat ) ) ).

% not_Some_eq
thf(fact_68_combine__options__simps_I3_J,axiom,
    ! [F: num > num > num,A: num,B: num] :
      ( ( combine_options_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
      = ( some_num @ ( F @ A @ B ) ) ) ).

% combine_options_simps(3)
thf(fact_69_combine__options__simps_I3_J,axiom,
    ! [F: produc8664842809031399944it_nat > produc8664842809031399944it_nat > produc8664842809031399944it_nat,A: produc8664842809031399944it_nat,B: produc8664842809031399944it_nat] :
      ( ( combin4318129983670048329it_nat @ F @ ( some_P1914260805536162275it_nat @ A ) @ ( some_P1914260805536162275it_nat @ B ) )
      = ( some_P1914260805536162275it_nat @ ( F @ A @ B ) ) ) ).

% combine_options_simps(3)
thf(fact_70_mem__Collect__eq,axiom,
    ! [A: produc3843707927480180839at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( member8757157785044589968at_nat @ A @ ( collec6321179662152712658at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_71_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_72_mem__Collect__eq,axiom,
    ! [A: $o,P: $o > $o] :
      ( ( member_o @ A @ ( collect_o @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_73_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_74_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_75_mem__Collect__eq,axiom,
    ! [A: list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_76_Collect__mem__eq,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_77_Collect__mem__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_78_Collect__mem__eq,axiom,
    ! [A2: set_o] :
      ( ( collect_o
        @ ^ [X3: $o] : ( member_o @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_79_Collect__mem__eq,axiom,
    ! [A2: set_int] :
      ( ( collect_int
        @ ^ [X3: int] : ( member_int @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_80_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_81_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_82_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_83_Collect__cong,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X4: list_nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_list_nat @ P )
        = ( collect_list_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_84_option_Odistinct_I1_J,axiom,
    ! [X2: produc6714399278180371524it_nat] :
      ( none_P2587165478634041381it_nat
     != ( some_P6105628898608307625it_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_85_option_Odistinct_I1_J,axiom,
    ! [X2: num] :
      ( none_num
     != ( some_num @ X2 ) ) ).

% option.distinct(1)
thf(fact_86_option_Odistinct_I1_J,axiom,
    ! [X2: produc8664842809031399944it_nat] :
      ( none_P9117596204409417319it_nat
     != ( some_P1914260805536162275it_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_87_option_OdiscI,axiom,
    ! [Option: option3216581401566546698it_nat,X2: produc6714399278180371524it_nat] :
      ( ( Option
        = ( some_P6105628898608307625it_nat @ X2 ) )
     => ( Option != none_P2587165478634041381it_nat ) ) ).

% option.discI
thf(fact_88_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option
        = ( some_num @ X2 ) )
     => ( Option != none_num ) ) ).

% option.discI
thf(fact_89_option_OdiscI,axiom,
    ! [Option: option8956607266484857688it_nat,X2: produc8664842809031399944it_nat] :
      ( ( Option
        = ( some_P1914260805536162275it_nat @ X2 ) )
     => ( Option != none_P9117596204409417319it_nat ) ) ).

% option.discI
thf(fact_90_option_Oexhaust,axiom,
    ! [Y: option3216581401566546698it_nat] :
      ( ( Y != none_P2587165478634041381it_nat )
     => ~ ! [X22: produc6714399278180371524it_nat] :
            ( Y
           != ( some_P6105628898608307625it_nat @ X22 ) ) ) ).

% option.exhaust
thf(fact_91_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X22: num] :
            ( Y
           != ( some_num @ X22 ) ) ) ).

% option.exhaust
thf(fact_92_option_Oexhaust,axiom,
    ! [Y: option8956607266484857688it_nat] :
      ( ( Y != none_P9117596204409417319it_nat )
     => ~ ! [X22: produc8664842809031399944it_nat] :
            ( Y
           != ( some_P1914260805536162275it_nat @ X22 ) ) ) ).

% option.exhaust
thf(fact_93_split__option__ex,axiom,
    ( ( ^ [P2: option3216581401566546698it_nat > $o] :
        ? [X5: option3216581401566546698it_nat] : ( P2 @ X5 ) )
    = ( ^ [P3: option3216581401566546698it_nat > $o] :
          ( ( P3 @ none_P2587165478634041381it_nat )
          | ? [X3: produc6714399278180371524it_nat] : ( P3 @ ( some_P6105628898608307625it_nat @ X3 ) ) ) ) ) ).

% split_option_ex
thf(fact_94_split__option__ex,axiom,
    ( ( ^ [P2: option_num > $o] :
        ? [X5: option_num] : ( P2 @ X5 ) )
    = ( ^ [P3: option_num > $o] :
          ( ( P3 @ none_num )
          | ? [X3: num] : ( P3 @ ( some_num @ X3 ) ) ) ) ) ).

% split_option_ex
thf(fact_95_split__option__ex,axiom,
    ( ( ^ [P2: option8956607266484857688it_nat > $o] :
        ? [X5: option8956607266484857688it_nat] : ( P2 @ X5 ) )
    = ( ^ [P3: option8956607266484857688it_nat > $o] :
          ( ( P3 @ none_P9117596204409417319it_nat )
          | ? [X3: produc8664842809031399944it_nat] : ( P3 @ ( some_P1914260805536162275it_nat @ X3 ) ) ) ) ) ).

% split_option_ex
thf(fact_96_split__option__all,axiom,
    ( ( ^ [P2: option3216581401566546698it_nat > $o] :
        ! [X5: option3216581401566546698it_nat] : ( P2 @ X5 ) )
    = ( ^ [P3: option3216581401566546698it_nat > $o] :
          ( ( P3 @ none_P2587165478634041381it_nat )
          & ! [X3: produc6714399278180371524it_nat] : ( P3 @ ( some_P6105628898608307625it_nat @ X3 ) ) ) ) ) ).

% split_option_all
thf(fact_97_split__option__all,axiom,
    ( ( ^ [P2: option_num > $o] :
        ! [X5: option_num] : ( P2 @ X5 ) )
    = ( ^ [P3: option_num > $o] :
          ( ( P3 @ none_num )
          & ! [X3: num] : ( P3 @ ( some_num @ X3 ) ) ) ) ) ).

% split_option_all
thf(fact_98_split__option__all,axiom,
    ( ( ^ [P2: option8956607266484857688it_nat > $o] :
        ! [X5: option8956607266484857688it_nat] : ( P2 @ X5 ) )
    = ( ^ [P3: option8956607266484857688it_nat > $o] :
          ( ( P3 @ none_P9117596204409417319it_nat )
          & ! [X3: produc8664842809031399944it_nat] : ( P3 @ ( some_P1914260805536162275it_nat @ X3 ) ) ) ) ) ).

% split_option_all
thf(fact_99_combine__options__cases,axiom,
    ! [X: option3216581401566546698it_nat,P: option3216581401566546698it_nat > option3216581401566546698it_nat > $o,Y: option3216581401566546698it_nat] :
      ( ( ( X = none_P2587165478634041381it_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P2587165478634041381it_nat )
         => ( P @ X @ Y ) )
       => ( ! [A3: produc6714399278180371524it_nat,B2: produc6714399278180371524it_nat] :
              ( ( X
                = ( some_P6105628898608307625it_nat @ A3 ) )
             => ( ( Y
                  = ( some_P6105628898608307625it_nat @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_100_combine__options__cases,axiom,
    ! [X: option3216581401566546698it_nat,P: option3216581401566546698it_nat > option_num > $o,Y: option_num] :
      ( ( ( X = none_P2587165478634041381it_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X @ Y ) )
       => ( ! [A3: produc6714399278180371524it_nat,B2: num] :
              ( ( X
                = ( some_P6105628898608307625it_nat @ A3 ) )
             => ( ( Y
                  = ( some_num @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_101_combine__options__cases,axiom,
    ! [X: option3216581401566546698it_nat,P: option3216581401566546698it_nat > option8956607266484857688it_nat > $o,Y: option8956607266484857688it_nat] :
      ( ( ( X = none_P2587165478634041381it_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P9117596204409417319it_nat )
         => ( P @ X @ Y ) )
       => ( ! [A3: produc6714399278180371524it_nat,B2: produc8664842809031399944it_nat] :
              ( ( X
                = ( some_P6105628898608307625it_nat @ A3 ) )
             => ( ( Y
                  = ( some_P1914260805536162275it_nat @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_102_combine__options__cases,axiom,
    ! [X: option_num,P: option_num > option3216581401566546698it_nat > $o,Y: option3216581401566546698it_nat] :
      ( ( ( X = none_num )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P2587165478634041381it_nat )
         => ( P @ X @ Y ) )
       => ( ! [A3: num,B2: produc6714399278180371524it_nat] :
              ( ( X
                = ( some_num @ A3 ) )
             => ( ( Y
                  = ( some_P6105628898608307625it_nat @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_103_combine__options__cases,axiom,
    ! [X: option_num,P: option_num > option_num > $o,Y: option_num] :
      ( ( ( X = none_num )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X @ Y ) )
       => ( ! [A3: num,B2: num] :
              ( ( X
                = ( some_num @ A3 ) )
             => ( ( Y
                  = ( some_num @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_104_combine__options__cases,axiom,
    ! [X: option_num,P: option_num > option8956607266484857688it_nat > $o,Y: option8956607266484857688it_nat] :
      ( ( ( X = none_num )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P9117596204409417319it_nat )
         => ( P @ X @ Y ) )
       => ( ! [A3: num,B2: produc8664842809031399944it_nat] :
              ( ( X
                = ( some_num @ A3 ) )
             => ( ( Y
                  = ( some_P1914260805536162275it_nat @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_105_combine__options__cases,axiom,
    ! [X: option8956607266484857688it_nat,P: option8956607266484857688it_nat > option3216581401566546698it_nat > $o,Y: option3216581401566546698it_nat] :
      ( ( ( X = none_P9117596204409417319it_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P2587165478634041381it_nat )
         => ( P @ X @ Y ) )
       => ( ! [A3: produc8664842809031399944it_nat,B2: produc6714399278180371524it_nat] :
              ( ( X
                = ( some_P1914260805536162275it_nat @ A3 ) )
             => ( ( Y
                  = ( some_P6105628898608307625it_nat @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_106_combine__options__cases,axiom,
    ! [X: option8956607266484857688it_nat,P: option8956607266484857688it_nat > option_num > $o,Y: option_num] :
      ( ( ( X = none_P9117596204409417319it_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X @ Y ) )
       => ( ! [A3: produc8664842809031399944it_nat,B2: num] :
              ( ( X
                = ( some_P1914260805536162275it_nat @ A3 ) )
             => ( ( Y
                  = ( some_num @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_107_combine__options__cases,axiom,
    ! [X: option8956607266484857688it_nat,P: option8956607266484857688it_nat > option8956607266484857688it_nat > $o,Y: option8956607266484857688it_nat] :
      ( ( ( X = none_P9117596204409417319it_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P9117596204409417319it_nat )
         => ( P @ X @ Y ) )
       => ( ! [A3: produc8664842809031399944it_nat,B2: produc8664842809031399944it_nat] :
              ( ( X
                = ( some_P1914260805536162275it_nat @ A3 ) )
             => ( ( Y
                  = ( some_P1914260805536162275it_nat @ B2 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_108_success__effectE,axiom,
    ! [C: heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( heap_T584514906347983379t_unit @ C @ H3 )
     => ~ ! [R: product_unit,H4: heap_e7401611519738050253t_unit,N: nat] :
            ~ ( heap_T6553295506729943825t_unit @ C @ H3 @ H4 @ R @ N ) ) ).

% success_effectE
thf(fact_109_success__effectE,axiom,
    ! [C: heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit] :
      ( ( heap_T6932021131025641745rray_a @ C @ H3 )
     => ~ ! [R: array_a,H4: heap_e7401611519738050253t_unit,N: nat] :
            ~ ( heap_T5830633491639066771rray_a @ C @ H3 @ H4 @ R @ N ) ) ).

% success_effectE
thf(fact_110_effect__success,axiom,
    ! [C: heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,R2: product_unit,N2: nat] :
      ( ( heap_T6553295506729943825t_unit @ C @ H3 @ H5 @ R2 @ N2 )
     => ( heap_T584514906347983379t_unit @ C @ H3 ) ) ).

% effect_success
thf(fact_111_effect__success,axiom,
    ! [C: heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,R2: array_a,N2: nat] :
      ( ( heap_T5830633491639066771rray_a @ C @ H3 @ H5 @ R2 @ N2 )
     => ( heap_T6932021131025641745rray_a @ C @ H3 ) ) ).

% effect_success
thf(fact_112_effect__raiseE,axiom,
    ! [X: list_char,H3: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,R2: product_unit,N2: nat] :
      ~ ( heap_T6553295506729943825t_unit @ ( heap_T2927564422264180874t_unit @ X ) @ H3 @ H5 @ R2 @ N2 ) ).

% effect_raiseE
thf(fact_113_effect__raiseE,axiom,
    ! [X: list_char,H3: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,R2: array_a,N2: nat] :
      ~ ( heap_T5830633491639066771rray_a @ ( heap_T1552190017797416154rray_a @ X ) @ H3 @ H5 @ R2 @ N2 ) ).

% effect_raiseE
thf(fact_114_the__default_Osimps_I1_J,axiom,
    ! [Uu2: num,X: num] :
      ( ( the_default_num @ Uu2 @ ( some_num @ X ) )
      = X ) ).

% the_default.simps(1)
thf(fact_115_the__default_Osimps_I1_J,axiom,
    ! [Uu2: produc8664842809031399944it_nat,X: produc8664842809031399944it_nat] :
      ( ( the_de2487931475039285041it_nat @ Uu2 @ ( some_P1914260805536162275it_nat @ X ) )
      = X ) ).

% the_default.simps(1)
thf(fact_116_Heap__cases,axiom,
    ! [F: heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit] :
      ( ! [X4: array_a,H4: produc6653097349344004940it_nat] :
          ( ( heap_T5591663578999710211rray_a @ F @ H3 )
         != ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ X4 @ H4 ) ) )
     => ( ( heap_T5591663578999710211rray_a @ F @ H3 )
        = none_P2587165478634041381it_nat ) ) ).

% Heap_cases
thf(fact_117_Heap__cases,axiom,
    ! [F: heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ! [X4: product_unit,H4: produc6653097349344004940it_nat] :
          ( ( heap_T875086893843062177t_unit @ F @ H3 )
         != ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ X4 @ H4 ) ) )
     => ( ( heap_T875086893843062177t_unit @ F @ H3 )
        = none_P9117596204409417319it_nat ) ) ).

% Heap_cases
thf(fact_118_successE,axiom,
    ! [F: heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( heap_T584514906347983379t_unit @ F @ H3 )
     => ~ ! [R: product_unit,H4: produc6653097349344004940it_nat] :
            ( ( heap_T875086893843062177t_unit @ F @ H3 )
           != ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R @ H4 ) ) ) ) ).

% successE
thf(fact_119_successE,axiom,
    ! [F: heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit] :
      ( ( heap_T6932021131025641745rray_a @ F @ H3 )
     => ~ ! [R: array_a,H4: produc6653097349344004940it_nat] :
            ( ( heap_T5591663578999710211rray_a @ F @ H3 )
           != ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ R @ H4 ) ) ) ) ).

% successE
thf(fact_120_execute__heap,axiom,
    ! [F: heap_e7401611519738050253t_unit > produc6714399278180371524it_nat] :
      ( ( heap_T5591663578999710211rray_a @ ( heap_T7473999186002612558rray_a @ F ) )
      = ( comp_P2097125496263478628t_unit @ some_P6105628898608307625it_nat @ F ) ) ).

% execute_heap
thf(fact_121_execute__heap,axiom,
    ! [F: heap_e7401611519738050253t_unit > produc8664842809031399944it_nat] :
      ( ( heap_T875086893843062177t_unit @ ( heap_T6927113302350381334t_unit @ F ) )
      = ( comp_P3118722334806803912t_unit @ some_P1914260805536162275it_nat @ F ) ) ).

% execute_heap
thf(fact_122_some__opt__sym__eq__trivial,axiom,
    ! [X: num] :
      ( ( eps_Opt_num
        @ ( ^ [Y4: num,Z2: num] : ( Y4 = Z2 )
          @ X ) )
      = ( some_num @ X ) ) ).

% some_opt_sym_eq_trivial
thf(fact_123_some__opt__sym__eq__trivial,axiom,
    ! [X: produc8664842809031399944it_nat] :
      ( ( eps_Op3393321821070424684it_nat
        @ ( ^ [Y4: produc8664842809031399944it_nat,Z2: produc8664842809031399944it_nat] : ( Y4 = Z2 )
          @ X ) )
      = ( some_P1914260805536162275it_nat @ X ) ) ).

% some_opt_sym_eq_trivial
thf(fact_124_success__newI,axiom,
    ! [N2: nat,X: a,H3: heap_e7401611519738050253t_unit] : ( heap_T6932021131025641745rray_a @ ( array_new_a @ N2 @ X ) @ H3 ) ).

% success_newI
thf(fact_125_success__makeI,axiom,
    ! [N2: nat,F: nat > a,H3: heap_e7401611519738050253t_unit] : ( heap_T6932021131025641745rray_a @ ( array_make_a @ N2 @ F ) @ H3 ) ).

% success_makeI
thf(fact_126_mmupd__in__upd,axiom,
    ! [K: produc3843707927480180839at_nat,K2: set_Pr4329608150637261639at_nat,M2: produc3843707927480180839at_nat > option_num,V: num] :
      ( ( member8757157785044589968at_nat @ K @ K2 )
     => ( ( map_mm2124855372588256220at_num @ M2 @ K2 @ V @ K )
        = ( some_num @ V ) ) ) ).

% mmupd_in_upd
thf(fact_127_mmupd__in__upd,axiom,
    ! [K: product_prod_nat_nat,K2: set_Pr1261947904930325089at_nat,M2: product_prod_nat_nat > option_num,V: num] :
      ( ( member8440522571783428010at_nat @ K @ K2 )
     => ( ( map_mm6272435474268545910at_num @ M2 @ K2 @ V @ K )
        = ( some_num @ V ) ) ) ).

% mmupd_in_upd
thf(fact_128_mmupd__in__upd,axiom,
    ! [K: $o,K2: set_o,M2: $o > option_num,V: num] :
      ( ( member_o @ K @ K2 )
     => ( ( map_mmupd_o_num @ M2 @ K2 @ V @ K )
        = ( some_num @ V ) ) ) ).

% mmupd_in_upd
thf(fact_129_mmupd__in__upd,axiom,
    ! [K: nat,K2: set_nat,M2: nat > option_num,V: num] :
      ( ( member_nat @ K @ K2 )
     => ( ( map_mmupd_nat_num @ M2 @ K2 @ V @ K )
        = ( some_num @ V ) ) ) ).

% mmupd_in_upd
thf(fact_130_mmupd__in__upd,axiom,
    ! [K: int,K2: set_int,M2: int > option_num,V: num] :
      ( ( member_int @ K @ K2 )
     => ( ( map_mmupd_int_num @ M2 @ K2 @ V @ K )
        = ( some_num @ V ) ) ) ).

% mmupd_in_upd
thf(fact_131_mmupd__in__upd,axiom,
    ! [K: produc3843707927480180839at_nat,K2: set_Pr4329608150637261639at_nat,M2: produc3843707927480180839at_nat > option8956607266484857688it_nat,V: produc8664842809031399944it_nat] :
      ( ( member8757157785044589968at_nat @ K @ K2 )
     => ( ( map_mm6227321356191524810it_nat @ M2 @ K2 @ V @ K )
        = ( some_P1914260805536162275it_nat @ V ) ) ) ).

% mmupd_in_upd
thf(fact_132_mmupd__in__upd,axiom,
    ! [K: product_prod_nat_nat,K2: set_Pr1261947904930325089at_nat,M2: product_prod_nat_nat > option8956607266484857688it_nat,V: produc8664842809031399944it_nat] :
      ( ( member8440522571783428010at_nat @ K @ K2 )
     => ( ( map_mm8925508573636269924it_nat @ M2 @ K2 @ V @ K )
        = ( some_P1914260805536162275it_nat @ V ) ) ) ).

% mmupd_in_upd
thf(fact_133_mmupd__in__upd,axiom,
    ! [K: $o,K2: set_o,M2: $o > option8956607266484857688it_nat,V: produc8664842809031399944it_nat] :
      ( ( member_o @ K @ K2 )
     => ( ( map_mm4340940675810032635it_nat @ M2 @ K2 @ V @ K )
        = ( some_P1914260805536162275it_nat @ V ) ) ) ).

% mmupd_in_upd
thf(fact_134_mmupd__in__upd,axiom,
    ! [K: nat,K2: set_nat,M2: nat > option8956607266484857688it_nat,V: produc8664842809031399944it_nat] :
      ( ( member_nat @ K @ K2 )
     => ( ( map_mm4316137065318800191it_nat @ M2 @ K2 @ V @ K )
        = ( some_P1914260805536162275it_nat @ V ) ) ) ).

% mmupd_in_upd
thf(fact_135_mmupd__in__upd,axiom,
    ! [K: int,K2: set_int,M2: int > option8956607266484857688it_nat,V: produc8664842809031399944it_nat] :
      ( ( member_int @ K @ K2 )
     => ( ( map_mm5942422175790858651it_nat @ M2 @ K2 @ V @ K )
        = ( some_P1914260805536162275it_nat @ V ) ) ) ).

% mmupd_in_upd
thf(fact_136_option_Ocollapse,axiom,
    ! [Option: option3216581401566546698it_nat] :
      ( ( Option != none_P2587165478634041381it_nat )
     => ( ( some_P6105628898608307625it_nat @ ( the_Pr5856008525237892824it_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_137_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( ( some_num @ ( the_num @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_138_option_Ocollapse,axiom,
    ! [Option: option8956607266484857688it_nat] :
      ( ( Option != none_P9117596204409417319it_nat )
     => ( ( some_P1914260805536162275it_nat @ ( the_Pr7776125533777466484it_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_139_not__Some__eq2,axiom,
    ! [V: option8051342751916580710nteger] :
      ( ( ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
            ( V
           != ( some_P1462369734362851057nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) ) ) )
      = ( V = none_P4442379456014020469nteger ) ) ).

% not_Some_eq2
thf(fact_140_not__Some__eq2,axiom,
    ! [V: option5190343406534369742et_nat] :
      ( ( ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] :
            ( V
           != ( some_P750831030444334937et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) ) ) )
      = ( V = none_P4972525538344268765et_nat ) ) ).

% not_Some_eq2
thf(fact_141_not__Some__eq2,axiom,
    ! [V: option2860828798490689354et_nat] :
      ( ( ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] :
            ( V
           != ( some_P1630309045189364437et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) ) ) )
      = ( V = none_P199884684680593241et_nat ) ) ).

% not_Some_eq2
thf(fact_142_not__Some__eq2,axiom,
    ! [V: option7541221861074943443nt_int] :
      ( ( ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( V
           != ( some_P2355398578364412894nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) ) ) )
      = ( V = none_P1286213070022356066nt_int ) ) ).

% not_Some_eq2
thf(fact_143_not__Some__eq2,axiom,
    ! [V: option4256020574406277085nt_int] :
      ( ( ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
            ( V
           != ( some_P7455497367792166888nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) ) ) )
      = ( V = none_P3773570700014501484nt_int ) ) ).

% not_Some_eq2
thf(fact_144_not__Some__eq2,axiom,
    ! [V: option3216581401566546698it_nat] :
      ( ( ! [X3: array_a,Y3: produc6653097349344004940it_nat] :
            ( V
           != ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ X3 @ Y3 ) ) ) )
      = ( V = none_P2587165478634041381it_nat ) ) ).

% not_Some_eq2
thf(fact_145_not__Some__eq2,axiom,
    ! [V: option8956607266484857688it_nat] :
      ( ( ! [X3: product_unit,Y3: produc6653097349344004940it_nat] :
            ( V
           != ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ X3 @ Y3 ) ) ) )
      = ( V = none_P9117596204409417319it_nat ) ) ).

% not_Some_eq2
thf(fact_146_effect__assertI,axiom,
    ! [P: product_unit > $o,X: product_unit,H5: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: product_unit,N2: nat] :
      ( ( P @ X )
     => ( ( H5 = H3 )
       => ( ( R2 = X )
         => ( ( N2 = one_one_nat )
           => ( heap_T6553295506729943825t_unit @ ( heap_T4208721593536448476t_unit @ P @ X ) @ H3 @ H5 @ R2 @ N2 ) ) ) ) ) ).

% effect_assertI
thf(fact_147_effect__assertI,axiom,
    ! [P: array_a > $o,X: array_a,H5: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R2: array_a,N2: nat] :
      ( ( P @ X )
     => ( ( H5 = H3 )
       => ( ( R2 = X )
         => ( ( N2 = one_one_nat )
           => ( heap_T5830633491639066771rray_a @ ( heap_T7971689983404357896rray_a @ P @ X ) @ H3 @ H5 @ R2 @ N2 ) ) ) ) ) ).

% effect_assertI
thf(fact_148_fun__comp__eq__conv,axiom,
    ! [F: code_integer > code_integer,G: num > code_integer,Fg: num > code_integer] :
      ( ( ( comp_C3531382070062128313er_num @ F @ G )
        = Fg )
      = ( ! [X3: num] :
            ( ( F @ ( G @ X3 ) )
            = ( Fg @ X3 ) ) ) ) ).

% fun_comp_eq_conv
thf(fact_149_fun__comp__eq__conv,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,Fg: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( comp_C1593894019821074884nteger @ F @ G )
        = Fg )
      = ( ! [X3: code_integer] :
            ( ( F @ ( G @ X3 ) )
            = ( Fg @ X3 ) ) ) ) ).

% fun_comp_eq_conv
thf(fact_150_fun__comp__eq__conv,axiom,
    ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,Fg: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( comp_C8797469213163452608nteger @ F @ G )
        = Fg )
      = ( ! [X3: code_integer] :
            ( ( F @ ( G @ X3 ) )
            = ( Fg @ X3 ) ) ) ) ).

% fun_comp_eq_conv
thf(fact_151_fun__comp__eq__conv,axiom,
    ! [F: nat > num,G: int > nat,Fg: int > num] :
      ( ( ( comp_nat_num_int @ F @ G )
        = Fg )
      = ( ! [X3: int] :
            ( ( F @ ( G @ X3 ) )
            = ( Fg @ X3 ) ) ) ) ).

% fun_comp_eq_conv
thf(fact_152_comp__cong__right,axiom,
    ! [X: num > code_integer,Y: num > code_integer,F: code_integer > code_integer] :
      ( ( X = Y )
     => ( ( comp_C3531382070062128313er_num @ F @ X )
        = ( comp_C3531382070062128313er_num @ F @ Y ) ) ) ).

% comp_cong_right
thf(fact_153_comp__cong__right,axiom,
    ! [X: code_integer > code_integer,Y: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( X = Y )
     => ( ( comp_C1593894019821074884nteger @ F @ X )
        = ( comp_C1593894019821074884nteger @ F @ Y ) ) ) ).

% comp_cong_right
thf(fact_154_comp__cong__right,axiom,
    ! [X: code_integer > code_integer > code_integer,Y: code_integer > code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( X = Y )
     => ( ( comp_C8797469213163452608nteger @ F @ X )
        = ( comp_C8797469213163452608nteger @ F @ Y ) ) ) ).

% comp_cong_right
thf(fact_155_comp__cong__right,axiom,
    ! [X: int > nat,Y: int > nat,F: nat > num] :
      ( ( X = Y )
     => ( ( comp_nat_num_int @ F @ X )
        = ( comp_nat_num_int @ F @ Y ) ) ) ).

% comp_cong_right
thf(fact_156_comp__cong__left,axiom,
    ! [X: code_integer > code_integer,Y: code_integer > code_integer,F: num > code_integer] :
      ( ( X = Y )
     => ( ( comp_C3531382070062128313er_num @ X @ F )
        = ( comp_C3531382070062128313er_num @ Y @ F ) ) ) ).

% comp_cong_left
thf(fact_157_comp__cong__left,axiom,
    ! [X: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,Y: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer] :
      ( ( X = Y )
     => ( ( comp_C1593894019821074884nteger @ X @ F )
        = ( comp_C1593894019821074884nteger @ Y @ F ) ) ) ).

% comp_cong_left
thf(fact_158_comp__cong__left,axiom,
    ! [X: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,Y: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer > code_integer] :
      ( ( X = Y )
     => ( ( comp_C8797469213163452608nteger @ X @ F )
        = ( comp_C8797469213163452608nteger @ Y @ F ) ) ) ).

% comp_cong_left
thf(fact_159_comp__cong__left,axiom,
    ! [X: nat > num,Y: nat > num,F: int > nat] :
      ( ( X = Y )
     => ( ( comp_nat_num_int @ X @ F )
        = ( comp_nat_num_int @ Y @ F ) ) ) ).

% comp_cong_left
thf(fact_160_bex2I,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,S2: set_Pr4329608150637261639at_nat,P: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ S2 )
     => ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ S2 )
         => ( P @ A @ B ) )
       => ? [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
            ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ B2 ) @ S2 )
            & ( P @ A3 @ B2 ) ) ) ) ).

% bex2I
thf(fact_161_bex2I,axiom,
    ! [A: nat,B: nat,S2: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ S2 )
     => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ S2 )
         => ( P @ A @ B ) )
       => ? [A3: nat,B2: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B2 ) @ S2 )
            & ( P @ A3 @ B2 ) ) ) ) ).

% bex2I
thf(fact_162_bex2I,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,S2: set_Pr1281608226676607948nteger,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A @ B ) @ S2 )
     => ( ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A @ B ) @ S2 )
         => ( P @ A @ B ) )
       => ? [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
            ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A3 @ B2 ) @ S2 )
            & ( P @ A3 @ B2 ) ) ) ) ).

% bex2I
thf(fact_163_bex2I,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,S2: set_Pr3286484037609594932et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ S2 )
     => ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ S2 )
         => ( P @ A @ B ) )
       => ? [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
            ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A3 @ B2 ) @ S2 )
            & ( P @ A3 @ B2 ) ) ) ) ).

% bex2I
thf(fact_164_bex2I,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,S2: set_Pr8536935166611901872et_nat,P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ S2 )
     => ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ S2 )
         => ( P @ A @ B ) )
       => ? [A3: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
            ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A3 @ B2 ) @ S2 )
            & ( P @ A3 @ B2 ) ) ) ) ).

% bex2I
thf(fact_165_bex2I,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,S2: set_Pr9222295170931077689nt_int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ S2 )
     => ( ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ S2 )
         => ( P @ A @ B ) )
       => ? [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
            ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A3 @ B2 ) @ S2 )
            & ( P @ A3 @ B2 ) ) ) ) ).

% bex2I
thf(fact_166_bex2I,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,S2: set_Pr1872883991513573699nt_int,P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ S2 )
     => ( ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ S2 )
         => ( P @ A @ B ) )
       => ? [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
            ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A3 @ B2 ) @ S2 )
            & ( P @ A3 @ B2 ) ) ) ) ).

% bex2I
thf(fact_167_pairself_Ocases,axiom,
    ! [X: produc7773217078559923341nt_int] :
      ~ ! [F3: int > option6357759511663192854e_term,A3: int,B2: int] :
          ( X
         != ( produc4305682042979456191nt_int @ F3 @ ( product_Pair_int_int @ A3 @ B2 ) ) ) ).

% pairself.cases
thf(fact_168_fails__array__new,axiom,
    ! [N2: nat,X: a,H3: heap_e7401611519738050253t_unit] :
      ~ ( time_fails_array_a @ ( array_new_a @ N2 @ X ) @ H3 ) ).

% fails_array_new
thf(fact_169_timeFrame_Ocases,axiom,
    ! [X: produc2867611627572283311it_nat] :
      ( ! [N: nat,R: array_a,H2: heap_e7401611519738050253t_unit,N3: nat] :
          ( X
         != ( produc3378006678175723489it_nat @ N @ ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ R @ ( produc584006145561248582it_nat @ H2 @ N3 ) ) ) ) )
     => ~ ! [N: nat] :
            ( X
           != ( produc3378006678175723489it_nat @ N @ none_P2587165478634041381it_nat ) ) ) ).

% timeFrame.cases
thf(fact_170_timeFrame_Ocases,axiom,
    ! [X: produc3911288613690379145it_nat] :
      ( ! [N: nat,R: product_unit,H2: heap_e7401611519738050253t_unit,N3: nat] :
          ( X
         != ( produc638857205735767105it_nat @ N @ ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R @ ( produc584006145561248582it_nat @ H2 @ N3 ) ) ) ) )
     => ~ ! [N: nat] :
            ( X
           != ( produc638857205735767105it_nat @ N @ none_P9117596204409417319it_nat ) ) ) ).

% timeFrame.cases
thf(fact_171_execute__assert_I1_J,axiom,
    ! [P: array_a > $o,X: array_a,H3: heap_e7401611519738050253t_unit] :
      ( ( P @ X )
     => ( ( heap_T5591663578999710211rray_a @ ( heap_T7971689983404357896rray_a @ P @ X ) @ H3 )
        = ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ X @ ( produc584006145561248582it_nat @ H3 @ one_one_nat ) ) ) ) ) ).

% execute_assert(1)
thf(fact_172_execute__assert_I1_J,axiom,
    ! [P: product_unit > $o,X: product_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( P @ X )
     => ( ( heap_T875086893843062177t_unit @ ( heap_T4208721593536448476t_unit @ P @ X ) @ H3 )
        = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ X @ ( produc584006145561248582it_nat @ H3 @ one_one_nat ) ) ) ) ) ).

% execute_assert(1)
thf(fact_173_execute__tap,axiom,
    ! [F: heap_e7401611519738050253t_unit > array_a,H3: heap_e7401611519738050253t_unit] :
      ( ( heap_T5591663578999710211rray_a @ ( heap_T2055650868554030641rray_a @ F ) @ H3 )
      = ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ ( F @ H3 ) @ ( produc584006145561248582it_nat @ H3 @ one_one_nat ) ) ) ) ).

% execute_tap
thf(fact_174_execute__tap,axiom,
    ! [F: heap_e7401611519738050253t_unit > product_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( heap_T875086893843062177t_unit @ ( heap_T560649228465745139t_unit @ F ) @ H3 )
      = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ ( F @ H3 ) @ ( produc584006145561248582it_nat @ H3 @ one_one_nat ) ) ) ) ).

% execute_tap
thf(fact_175_option_Osel,axiom,
    ! [X2: num] :
      ( ( the_num @ ( some_num @ X2 ) )
      = X2 ) ).

% option.sel
thf(fact_176_option_Osel,axiom,
    ! [X2: produc8664842809031399944it_nat] :
      ( ( the_Pr7776125533777466484it_nat @ ( some_P1914260805536162275it_nat @ X2 ) )
      = X2 ) ).

% option.sel
thf(fact_177_option_Oexpand,axiom,
    ! [Option: option3216581401566546698it_nat,Option2: option3216581401566546698it_nat] :
      ( ( ( Option = none_P2587165478634041381it_nat )
        = ( Option2 = none_P2587165478634041381it_nat ) )
     => ( ( ( Option != none_P2587165478634041381it_nat )
         => ( ( Option2 != none_P2587165478634041381it_nat )
           => ( ( the_Pr5856008525237892824it_nat @ Option )
              = ( the_Pr5856008525237892824it_nat @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_178_option_Oexpand,axiom,
    ! [Option: option_num,Option2: option_num] :
      ( ( ( Option = none_num )
        = ( Option2 = none_num ) )
     => ( ( ( Option != none_num )
         => ( ( Option2 != none_num )
           => ( ( the_num @ Option )
              = ( the_num @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_179_option_Oexpand,axiom,
    ! [Option: option8956607266484857688it_nat,Option2: option8956607266484857688it_nat] :
      ( ( ( Option = none_P9117596204409417319it_nat )
        = ( Option2 = none_P9117596204409417319it_nat ) )
     => ( ( ( Option != none_P9117596204409417319it_nat )
         => ( ( Option2 != none_P9117596204409417319it_nat )
           => ( ( the_Pr7776125533777466484it_nat @ Option )
              = ( the_Pr7776125533777466484it_nat @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_180_effect__def,axiom,
    ( heap_T5830633491639066771rray_a
    = ( ^ [C2: heap_T6362127782908069795rray_a,H: heap_e7401611519738050253t_unit,H6: heap_e7401611519738050253t_unit,R3: array_a,N4: nat] :
          ( ( heap_T5591663578999710211rray_a @ C2 @ H )
          = ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ R3 @ ( produc584006145561248582it_nat @ H6 @ N4 ) ) ) ) ) ) ).

% effect_def
thf(fact_181_effect__def,axiom,
    ( heap_T6553295506729943825t_unit
    = ( ^ [C2: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit,H6: heap_e7401611519738050253t_unit,R3: product_unit,N4: nat] :
          ( ( heap_T875086893843062177t_unit @ C2 @ H )
          = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R3 @ ( produc584006145561248582it_nat @ H6 @ N4 ) ) ) ) ) ) ).

% effect_def
thf(fact_182_effectI,axiom,
    ! [C: heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit,R2: array_a,H5: heap_e7401611519738050253t_unit,N2: nat] :
      ( ( ( heap_T5591663578999710211rray_a @ C @ H3 )
        = ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ R2 @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( heap_T5830633491639066771rray_a @ C @ H3 @ H5 @ R2 @ N2 ) ) ).

% effectI
thf(fact_183_effectI,axiom,
    ! [C: heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit,R2: product_unit,H5: heap_e7401611519738050253t_unit,N2: nat] :
      ( ( ( heap_T875086893843062177t_unit @ C @ H3 )
        = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R2 @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( heap_T6553295506729943825t_unit @ C @ H3 @ H5 @ R2 @ N2 ) ) ).

% effectI
thf(fact_184_map__mmupd__def,axiom,
    ( map_mm2124855372588256220at_num
    = ( ^ [M: produc3843707927480180839at_nat > option_num,K3: set_Pr4329608150637261639at_nat,V2: num,K4: produc3843707927480180839at_nat] : ( if_option_num @ ( member8757157785044589968at_nat @ K4 @ K3 ) @ ( some_num @ V2 ) @ ( M @ K4 ) ) ) ) ).

% map_mmupd_def
thf(fact_185_map__mmupd__def,axiom,
    ( map_mm6272435474268545910at_num
    = ( ^ [M: product_prod_nat_nat > option_num,K3: set_Pr1261947904930325089at_nat,V2: num,K4: product_prod_nat_nat] : ( if_option_num @ ( member8440522571783428010at_nat @ K4 @ K3 ) @ ( some_num @ V2 ) @ ( M @ K4 ) ) ) ) ).

% map_mmupd_def
thf(fact_186_map__mmupd__def,axiom,
    ( map_mmupd_o_num
    = ( ^ [M: $o > option_num,K3: set_o,V2: num,K4: $o] : ( if_option_num @ ( member_o @ K4 @ K3 ) @ ( some_num @ V2 ) @ ( M @ K4 ) ) ) ) ).

% map_mmupd_def
thf(fact_187_map__mmupd__def,axiom,
    ( map_mmupd_nat_num
    = ( ^ [M: nat > option_num,K3: set_nat,V2: num,K4: nat] : ( if_option_num @ ( member_nat @ K4 @ K3 ) @ ( some_num @ V2 ) @ ( M @ K4 ) ) ) ) ).

% map_mmupd_def
thf(fact_188_map__mmupd__def,axiom,
    ( map_mmupd_int_num
    = ( ^ [M: int > option_num,K3: set_int,V2: num,K4: int] : ( if_option_num @ ( member_int @ K4 @ K3 ) @ ( some_num @ V2 ) @ ( M @ K4 ) ) ) ) ).

% map_mmupd_def
thf(fact_189_map__mmupd__def,axiom,
    ( map_mm6227321356191524810it_nat
    = ( ^ [M: produc3843707927480180839at_nat > option8956607266484857688it_nat,K3: set_Pr4329608150637261639at_nat,V2: produc8664842809031399944it_nat,K4: produc3843707927480180839at_nat] : ( if_opt1729522071442692626it_nat @ ( member8757157785044589968at_nat @ K4 @ K3 ) @ ( some_P1914260805536162275it_nat @ V2 ) @ ( M @ K4 ) ) ) ) ).

% map_mmupd_def
thf(fact_190_map__mmupd__def,axiom,
    ( map_mm8925508573636269924it_nat
    = ( ^ [M: product_prod_nat_nat > option8956607266484857688it_nat,K3: set_Pr1261947904930325089at_nat,V2: produc8664842809031399944it_nat,K4: product_prod_nat_nat] : ( if_opt1729522071442692626it_nat @ ( member8440522571783428010at_nat @ K4 @ K3 ) @ ( some_P1914260805536162275it_nat @ V2 ) @ ( M @ K4 ) ) ) ) ).

% map_mmupd_def
thf(fact_191_map__mmupd__def,axiom,
    ( map_mm4340940675810032635it_nat
    = ( ^ [M: $o > option8956607266484857688it_nat,K3: set_o,V2: produc8664842809031399944it_nat,K4: $o] : ( if_opt1729522071442692626it_nat @ ( member_o @ K4 @ K3 ) @ ( some_P1914260805536162275it_nat @ V2 ) @ ( M @ K4 ) ) ) ) ).

% map_mmupd_def
thf(fact_192_map__mmupd__def,axiom,
    ( map_mm4316137065318800191it_nat
    = ( ^ [M: nat > option8956607266484857688it_nat,K3: set_nat,V2: produc8664842809031399944it_nat,K4: nat] : ( if_opt1729522071442692626it_nat @ ( member_nat @ K4 @ K3 ) @ ( some_P1914260805536162275it_nat @ V2 ) @ ( M @ K4 ) ) ) ) ).

% map_mmupd_def
thf(fact_193_map__mmupd__def,axiom,
    ( map_mm5942422175790858651it_nat
    = ( ^ [M: int > option8956607266484857688it_nat,K3: set_int,V2: produc8664842809031399944it_nat,K4: int] : ( if_opt1729522071442692626it_nat @ ( member_int @ K4 @ K3 ) @ ( some_P1914260805536162275it_nat @ V2 ) @ ( M @ K4 ) ) ) ) ).

% map_mmupd_def
thf(fact_194_map__mmupdE,axiom,
    ! [M2: produc3843707927480180839at_nat > option_num,K2: set_Pr4329608150637261639at_nat,V: num,K: produc3843707927480180839at_nat,X: num] :
      ( ( ( map_mm2124855372588256220at_num @ M2 @ K2 @ V @ K )
        = ( some_num @ X ) )
     => ( ( ~ ( member8757157785044589968at_nat @ K @ K2 )
         => ( ( M2 @ K )
           != ( some_num @ X ) ) )
       => ~ ( ( member8757157785044589968at_nat @ K @ K2 )
           => ( X != V ) ) ) ) ).

% map_mmupdE
thf(fact_195_map__mmupdE,axiom,
    ! [M2: product_prod_nat_nat > option_num,K2: set_Pr1261947904930325089at_nat,V: num,K: product_prod_nat_nat,X: num] :
      ( ( ( map_mm6272435474268545910at_num @ M2 @ K2 @ V @ K )
        = ( some_num @ X ) )
     => ( ( ~ ( member8440522571783428010at_nat @ K @ K2 )
         => ( ( M2 @ K )
           != ( some_num @ X ) ) )
       => ~ ( ( member8440522571783428010at_nat @ K @ K2 )
           => ( X != V ) ) ) ) ).

% map_mmupdE
thf(fact_196_map__mmupdE,axiom,
    ! [M2: $o > option_num,K2: set_o,V: num,K: $o,X: num] :
      ( ( ( map_mmupd_o_num @ M2 @ K2 @ V @ K )
        = ( some_num @ X ) )
     => ( ( ~ ( member_o @ K @ K2 )
         => ( ( M2 @ K )
           != ( some_num @ X ) ) )
       => ~ ( ( member_o @ K @ K2 )
           => ( X != V ) ) ) ) ).

% map_mmupdE
thf(fact_197_map__mmupdE,axiom,
    ! [M2: nat > option_num,K2: set_nat,V: num,K: nat,X: num] :
      ( ( ( map_mmupd_nat_num @ M2 @ K2 @ V @ K )
        = ( some_num @ X ) )
     => ( ( ~ ( member_nat @ K @ K2 )
         => ( ( M2 @ K )
           != ( some_num @ X ) ) )
       => ~ ( ( member_nat @ K @ K2 )
           => ( X != V ) ) ) ) ).

% map_mmupdE
thf(fact_198_map__mmupdE,axiom,
    ! [M2: int > option_num,K2: set_int,V: num,K: int,X: num] :
      ( ( ( map_mmupd_int_num @ M2 @ K2 @ V @ K )
        = ( some_num @ X ) )
     => ( ( ~ ( member_int @ K @ K2 )
         => ( ( M2 @ K )
           != ( some_num @ X ) ) )
       => ~ ( ( member_int @ K @ K2 )
           => ( X != V ) ) ) ) ).

% map_mmupdE
thf(fact_199_map__mmupdE,axiom,
    ! [M2: produc3843707927480180839at_nat > option8956607266484857688it_nat,K2: set_Pr4329608150637261639at_nat,V: produc8664842809031399944it_nat,K: produc3843707927480180839at_nat,X: produc8664842809031399944it_nat] :
      ( ( ( map_mm6227321356191524810it_nat @ M2 @ K2 @ V @ K )
        = ( some_P1914260805536162275it_nat @ X ) )
     => ( ( ~ ( member8757157785044589968at_nat @ K @ K2 )
         => ( ( M2 @ K )
           != ( some_P1914260805536162275it_nat @ X ) ) )
       => ~ ( ( member8757157785044589968at_nat @ K @ K2 )
           => ( X != V ) ) ) ) ).

% map_mmupdE
thf(fact_200_map__mmupdE,axiom,
    ! [M2: product_prod_nat_nat > option8956607266484857688it_nat,K2: set_Pr1261947904930325089at_nat,V: produc8664842809031399944it_nat,K: product_prod_nat_nat,X: produc8664842809031399944it_nat] :
      ( ( ( map_mm8925508573636269924it_nat @ M2 @ K2 @ V @ K )
        = ( some_P1914260805536162275it_nat @ X ) )
     => ( ( ~ ( member8440522571783428010at_nat @ K @ K2 )
         => ( ( M2 @ K )
           != ( some_P1914260805536162275it_nat @ X ) ) )
       => ~ ( ( member8440522571783428010at_nat @ K @ K2 )
           => ( X != V ) ) ) ) ).

% map_mmupdE
thf(fact_201_map__mmupdE,axiom,
    ! [M2: $o > option8956607266484857688it_nat,K2: set_o,V: produc8664842809031399944it_nat,K: $o,X: produc8664842809031399944it_nat] :
      ( ( ( map_mm4340940675810032635it_nat @ M2 @ K2 @ V @ K )
        = ( some_P1914260805536162275it_nat @ X ) )
     => ( ( ~ ( member_o @ K @ K2 )
         => ( ( M2 @ K )
           != ( some_P1914260805536162275it_nat @ X ) ) )
       => ~ ( ( member_o @ K @ K2 )
           => ( X != V ) ) ) ) ).

% map_mmupdE
thf(fact_202_map__mmupdE,axiom,
    ! [M2: nat > option8956607266484857688it_nat,K2: set_nat,V: produc8664842809031399944it_nat,K: nat,X: produc8664842809031399944it_nat] :
      ( ( ( map_mm4316137065318800191it_nat @ M2 @ K2 @ V @ K )
        = ( some_P1914260805536162275it_nat @ X ) )
     => ( ( ~ ( member_nat @ K @ K2 )
         => ( ( M2 @ K )
           != ( some_P1914260805536162275it_nat @ X ) ) )
       => ~ ( ( member_nat @ K @ K2 )
           => ( X != V ) ) ) ) ).

% map_mmupdE
thf(fact_203_map__mmupdE,axiom,
    ! [M2: int > option8956607266484857688it_nat,K2: set_int,V: produc8664842809031399944it_nat,K: int,X: produc8664842809031399944it_nat] :
      ( ( ( map_mm5942422175790858651it_nat @ M2 @ K2 @ V @ K )
        = ( some_P1914260805536162275it_nat @ X ) )
     => ( ( ~ ( member_int @ K @ K2 )
         => ( ( M2 @ K )
           != ( some_P1914260805536162275it_nat @ X ) ) )
       => ~ ( ( member_int @ K @ K2 )
           => ( X != V ) ) ) ) ).

% map_mmupdE
thf(fact_204_success__bind__executeI,axiom,
    ! [F: heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit,X: array_a,H5: heap_e7401611519738050253t_unit,N2: nat,G: array_a > heap_T5738788834812785303t_unit] :
      ( ( ( heap_T5591663578999710211rray_a @ F @ H3 )
        = ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( heap_T584514906347983379t_unit @ ( G @ X ) @ H5 )
       => ( heap_T584514906347983379t_unit @ ( heap_T7324619625323989613t_unit @ F @ G ) @ H3 ) ) ) ).

% success_bind_executeI
thf(fact_205_success__bind__executeI,axiom,
    ! [F: heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit,X: product_unit,H5: heap_e7401611519738050253t_unit,N2: nat,G: product_unit > heap_T5738788834812785303t_unit] :
      ( ( ( heap_T875086893843062177t_unit @ F @ H3 )
        = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( heap_T584514906347983379t_unit @ ( G @ X ) @ H5 )
       => ( heap_T584514906347983379t_unit @ ( heap_T2633723481742716231t_unit @ F @ G ) @ H3 ) ) ) ).

% success_bind_executeI
thf(fact_206_success__bind__executeI,axiom,
    ! [F: heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit,X: array_a,H5: heap_e7401611519738050253t_unit,N2: nat,G: array_a > heap_T6362127782908069795rray_a] :
      ( ( ( heap_T5591663578999710211rray_a @ F @ H3 )
        = ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( heap_T6932021131025641745rray_a @ ( G @ X ) @ H5 )
       => ( heap_T6932021131025641745rray_a @ ( heap_T2407400398799471159rray_a @ F @ G ) @ H3 ) ) ) ).

% success_bind_executeI
thf(fact_207_success__bind__executeI,axiom,
    ! [F: heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit,X: product_unit,H5: heap_e7401611519738050253t_unit,N2: nat,G: product_unit > heap_T6362127782908069795rray_a] :
      ( ( ( heap_T875086893843062177t_unit @ F @ H3 )
        = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( heap_T6932021131025641745rray_a @ ( G @ X ) @ H5 )
       => ( heap_T6932021131025641745rray_a @ ( heap_T7158140910958840029rray_a @ F @ G ) @ H3 ) ) ) ).

% success_bind_executeI
thf(fact_208_Eps__Opt__eq__Some__implies,axiom,
    ! [P: num > $o,X: num] :
      ( ( ( eps_Opt_num @ P )
        = ( some_num @ X ) )
     => ( P @ X ) ) ).

% Eps_Opt_eq_Some_implies
thf(fact_209_Eps__Opt__eq__Some__implies,axiom,
    ! [P: produc8664842809031399944it_nat > $o,X: produc8664842809031399944it_nat] :
      ( ( ( eps_Op3393321821070424684it_nat @ P )
        = ( some_P1914260805536162275it_nat @ X ) )
     => ( P @ X ) ) ).

% Eps_Opt_eq_Some_implies
thf(fact_210_Eps__Opt__eq__Some,axiom,
    ! [P: num > $o,X: num] :
      ( ! [X6: num] :
          ( ( P @ X )
         => ( ( P @ X6 )
           => ( X6 = X ) ) )
     => ( ( ( eps_Opt_num @ P )
          = ( some_num @ X ) )
        = ( P @ X ) ) ) ).

% Eps_Opt_eq_Some
thf(fact_211_Eps__Opt__eq__Some,axiom,
    ! [P: produc8664842809031399944it_nat > $o,X: produc8664842809031399944it_nat] :
      ( ! [X6: produc8664842809031399944it_nat] :
          ( ( P @ X )
         => ( ( P @ X6 )
           => ( X6 = X ) ) )
     => ( ( ( eps_Op3393321821070424684it_nat @ P )
          = ( some_P1914260805536162275it_nat @ X ) )
        = ( P @ X ) ) ) ).

% Eps_Opt_eq_Some
thf(fact_212_option_Oexhaust__sel,axiom,
    ! [Option: option3216581401566546698it_nat] :
      ( ( Option != none_P2587165478634041381it_nat )
     => ( Option
        = ( some_P6105628898608307625it_nat @ ( the_Pr5856008525237892824it_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_213_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option
        = ( some_num @ ( the_num @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_214_option_Oexhaust__sel,axiom,
    ! [Option: option8956607266484857688it_nat] :
      ( ( Option != none_P9117596204409417319it_nat )
     => ( Option
        = ( some_P1914260805536162275it_nat @ ( the_Pr7776125533777466484it_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_215_effect__assertE,axiom,
    ! [P: product_unit > $o,X: product_unit,H3: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,R2: product_unit,N2: nat] :
      ( ( heap_T6553295506729943825t_unit @ ( heap_T4208721593536448476t_unit @ P @ X ) @ H3 @ H5 @ R2 @ N2 )
     => ~ ( ( P @ X )
         => ( ( R2 = X )
           => ( ( H5 = H3 )
             => ( N2 != one_one_nat ) ) ) ) ) ).

% effect_assertE
thf(fact_216_effect__assertE,axiom,
    ! [P: array_a > $o,X: array_a,H3: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,R2: array_a,N2: nat] :
      ( ( heap_T5830633491639066771rray_a @ ( heap_T7971689983404357896rray_a @ P @ X ) @ H3 @ H5 @ R2 @ N2 )
     => ~ ( ( P @ X )
         => ( ( R2 = X )
           => ( ( H5 = H3 )
             => ( N2 != one_one_nat ) ) ) ) ) ).

% effect_assertE
thf(fact_217_comp__apply,axiom,
    ( comp_C3531382070062128313er_num
    = ( ^ [F2: code_integer > code_integer,G2: num > code_integer,X3: num] : ( F2 @ ( G2 @ X3 ) ) ) ) ).

% comp_apply
thf(fact_218_comp__apply,axiom,
    ( comp_C1593894019821074884nteger
    = ( ^ [F2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer,X3: code_integer] : ( F2 @ ( G2 @ X3 ) ) ) ) ).

% comp_apply
thf(fact_219_comp__apply,axiom,
    ( comp_C8797469213163452608nteger
    = ( ^ [F2: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer > code_integer,X3: code_integer] : ( F2 @ ( G2 @ X3 ) ) ) ) ).

% comp_apply
thf(fact_220_comp__apply,axiom,
    ( comp_nat_num_int
    = ( ^ [F2: nat > num,G2: int > nat,X3: int] : ( F2 @ ( G2 @ X3 ) ) ) ) ).

% comp_apply
thf(fact_221_prod_Oinject,axiom,
    ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X2: produc8923325533196201883nteger,Y1: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ X1 @ X2 )
        = ( produc8603105652947943368nteger @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_222_prod_Oinject,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X2: produc3658429121746597890et_nat,Y1: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ X1 @ X2 )
        = ( produc5001842942810119800et_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_223_prod_Oinject,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X2: produc3925858234332021118et_nat,Y1: produc3658429121746597890et_nat > $o,Y2: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ X1 @ X2 )
        = ( produc2245416461498447860et_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_224_prod_Oinject,axiom,
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X2: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ X1 @ X2 )
        = ( produc5700946648718959541nt_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_225_prod_Oinject,axiom,
    ! [X1: int > option6357759511663192854e_term,X2: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ X1 @ X2 )
        = ( produc4305682042979456191nt_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_226_old_Oprod_Oinject,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A4: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ A @ B )
        = ( produc8603105652947943368nteger @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_227_old_Oprod_Oinject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A4: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ A @ B )
        = ( produc5001842942810119800et_nat @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_228_old_Oprod_Oinject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A4: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ A @ B )
        = ( produc2245416461498447860et_nat @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_229_old_Oprod_Oinject,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A4: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A @ B )
        = ( produc5700946648718959541nt_int @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_230_old_Oprod_Oinject,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A4: int > option6357759511663192854e_term,B3: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A @ B )
        = ( produc4305682042979456191nt_int @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B = B3 ) ) ) ).

% old.prod.inject
thf(fact_231_execute__bind_I1_J,axiom,
    ! [F: heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit,X: array_a,H5: heap_e7401611519738050253t_unit,N2: nat,G: array_a > heap_T6362127782908069795rray_a] :
      ( ( ( heap_T5591663578999710211rray_a @ F @ H3 )
        = ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( heap_T5591663578999710211rray_a @ ( heap_T2407400398799471159rray_a @ F @ G ) @ H3 )
        = ( heap_T531568236820715630rray_a @ N2 @ ( heap_T5591663578999710211rray_a @ ( G @ X ) @ H5 ) ) ) ) ).

% execute_bind(1)
thf(fact_232_execute__bind_I1_J,axiom,
    ! [F: heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit,X: array_a,H5: heap_e7401611519738050253t_unit,N2: nat,G: array_a > heap_T5738788834812785303t_unit] :
      ( ( ( heap_T5591663578999710211rray_a @ F @ H3 )
        = ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( heap_T875086893843062177t_unit @ ( heap_T7324619625323989613t_unit @ F @ G ) @ H3 )
        = ( heap_T3616969660504097270t_unit @ N2 @ ( heap_T875086893843062177t_unit @ ( G @ X ) @ H5 ) ) ) ) ).

% execute_bind(1)
thf(fact_233_execute__bind_I1_J,axiom,
    ! [F: heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit,X: product_unit,H5: heap_e7401611519738050253t_unit,N2: nat,G: product_unit > heap_T6362127782908069795rray_a] :
      ( ( ( heap_T875086893843062177t_unit @ F @ H3 )
        = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( heap_T5591663578999710211rray_a @ ( heap_T7158140910958840029rray_a @ F @ G ) @ H3 )
        = ( heap_T531568236820715630rray_a @ N2 @ ( heap_T5591663578999710211rray_a @ ( G @ X ) @ H5 ) ) ) ) ).

% execute_bind(1)
thf(fact_234_execute__bind_I1_J,axiom,
    ! [F: heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit,X: product_unit,H5: heap_e7401611519738050253t_unit,N2: nat,G: product_unit > heap_T5738788834812785303t_unit] :
      ( ( ( heap_T875086893843062177t_unit @ F @ H3 )
        = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( heap_T875086893843062177t_unit @ ( heap_T2633723481742716231t_unit @ F @ G ) @ H3 )
        = ( heap_T3616969660504097270t_unit @ N2 @ ( heap_T875086893843062177t_unit @ ( G @ X ) @ H5 ) ) ) ) ).

% execute_bind(1)
thf(fact_235_prod__cases3,axiom,
    ! [Y: produc1908205239877642774nteger] :
      ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: code_integer,C3: code_integer] :
          ( Y
         != ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B2 @ C3 ) ) ) ).

% prod_cases3
thf(fact_236_prod__cases3,axiom,
    ! [Y: produc3925858234332021118et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B2: heap_e7401611519738050253t_unit,C3: set_nat] :
          ( Y
         != ( produc5001842942810119800et_nat @ A3 @ ( produc7507926704131184380et_nat @ B2 @ C3 ) ) ) ).

% prod_cases3
thf(fact_237_prod__cases3,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C3: produc3658429121746597890et_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B2 @ C3 ) ) ) ).

% prod_cases3
thf(fact_238_prod__cases3,axiom,
    ! [Y: produc2285326912895808259nt_int] :
      ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: int,C3: int] :
          ( Y
         != ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C3 ) ) ) ).

% prod_cases3
thf(fact_239_prod__cases3,axiom,
    ! [Y: produc7773217078559923341nt_int] :
      ~ ! [A3: int > option6357759511663192854e_term,B2: int,C3: int] :
          ( Y
         != ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C3 ) ) ) ).

% prod_cases3
thf(fact_240_prod__cases4,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C3: heap_e7401611519738050253t_unit,D: set_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B2 @ ( produc7507926704131184380et_nat @ C3 @ D ) ) ) ) ).

% prod_cases4
thf(fact_241_prod__induct3,axiom,
    ! [P: produc1908205239877642774nteger > $o,X: produc1908205239877642774nteger] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: code_integer,C3: code_integer] : ( P @ ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B2 @ C3 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_242_prod__induct3,axiom,
    ! [P: produc3925858234332021118et_nat > $o,X: produc3925858234332021118et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: heap_e7401611519738050253t_unit,C3: set_nat] : ( P @ ( produc5001842942810119800et_nat @ A3 @ ( produc7507926704131184380et_nat @ B2 @ C3 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_243_prod__induct3,axiom,
    ! [P: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C3: produc3658429121746597890et_nat] : ( P @ ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B2 @ C3 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_244_prod__induct3,axiom,
    ! [P: produc2285326912895808259nt_int > $o,X: produc2285326912895808259nt_int] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: int,C3: int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C3 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_245_prod__induct3,axiom,
    ! [P: produc7773217078559923341nt_int > $o,X: produc7773217078559923341nt_int] :
      ( ! [A3: int > option6357759511663192854e_term,B2: int,C3: int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C3 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_246_prod__induct4,axiom,
    ! [P: produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat > $o,C3: heap_e7401611519738050253t_unit,D: set_nat] : ( P @ ( produc2245416461498447860et_nat @ A3 @ ( produc5001842942810119800et_nat @ B2 @ ( produc7507926704131184380et_nat @ C3 @ D ) ) ) )
     => ( P @ X ) ) ).

% prod_induct4
thf(fact_247_timeFrame_Osimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( heap_T531568236820715630rray_a @ N2 @ none_P2587165478634041381it_nat )
      = none_P2587165478634041381it_nat ) ).

% timeFrame.simps(2)
thf(fact_248_timeFrame_Osimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( heap_T3616969660504097270t_unit @ N2 @ none_P9117596204409417319it_nat )
      = none_P9117596204409417319it_nat ) ).

% timeFrame.simps(2)
thf(fact_249_Pair__inject,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A4: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ A @ B )
        = ( produc8603105652947943368nteger @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_250_Pair__inject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A4: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat] :
      ( ( ( produc5001842942810119800et_nat @ A @ B )
        = ( produc5001842942810119800et_nat @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_251_Pair__inject,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A4: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat] :
      ( ( ( produc2245416461498447860et_nat @ A @ B )
        = ( produc2245416461498447860et_nat @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_252_Pair__inject,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A4: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A @ B )
        = ( produc5700946648718959541nt_int @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_253_Pair__inject,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A4: int > option6357759511663192854e_term,B3: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A @ B )
        = ( produc4305682042979456191nt_int @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B != B3 ) ) ) ).

% Pair_inject
thf(fact_254_prod__cases,axiom,
    ! [P: produc1908205239877642774nteger > $o,P4: produc1908205239877642774nteger] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] : ( P @ ( produc8603105652947943368nteger @ A3 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_255_prod__cases,axiom,
    ! [P: produc3925858234332021118et_nat > $o,P4: produc3925858234332021118et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] : ( P @ ( produc5001842942810119800et_nat @ A3 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_256_prod__cases,axiom,
    ! [P: produc2732055786443039994et_nat > $o,P4: produc2732055786443039994et_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] : ( P @ ( produc2245416461498447860et_nat @ A3 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_257_prod__cases,axiom,
    ! [P: produc2285326912895808259nt_int > $o,P4: produc2285326912895808259nt_int] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_258_prod__cases,axiom,
    ! [P: produc7773217078559923341nt_int > $o,P4: produc7773217078559923341nt_int] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ B2 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_259_surj__pair,axiom,
    ! [P4: produc1908205239877642774nteger] :
    ? [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
      ( P4
      = ( produc8603105652947943368nteger @ X4 @ Y5 ) ) ).

% surj_pair
thf(fact_260_surj__pair,axiom,
    ! [P4: produc3925858234332021118et_nat] :
    ? [X4: produc3658429121746597890et_nat > $o,Y5: produc3658429121746597890et_nat] :
      ( P4
      = ( produc5001842942810119800et_nat @ X4 @ Y5 ) ) ).

% surj_pair
thf(fact_261_surj__pair,axiom,
    ! [P4: produc2732055786443039994et_nat] :
    ? [X4: produc3658429121746597890et_nat > $o,Y5: produc3925858234332021118et_nat] :
      ( P4
      = ( produc2245416461498447860et_nat @ X4 @ Y5 ) ) ).

% surj_pair
thf(fact_262_surj__pair,axiom,
    ! [P4: produc2285326912895808259nt_int] :
    ? [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] :
      ( P4
      = ( produc5700946648718959541nt_int @ X4 @ Y5 ) ) ).

% surj_pair
thf(fact_263_surj__pair,axiom,
    ! [P4: produc7773217078559923341nt_int] :
    ? [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
      ( P4
      = ( produc4305682042979456191nt_int @ X4 @ Y5 ) ) ).

% surj_pair
thf(fact_264_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1908205239877642774nteger] :
      ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
          ( Y
         != ( produc8603105652947943368nteger @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_265_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3925858234332021118et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( Y
         != ( produc5001842942810119800et_nat @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_266_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2732055786443039994et_nat] :
      ~ ! [A3: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
          ( Y
         != ( produc2245416461498447860et_nat @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_267_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2285326912895808259nt_int] :
      ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( Y
         != ( produc5700946648718959541nt_int @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_268_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7773217078559923341nt_int] :
      ~ ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( Y
         != ( produc4305682042979456191nt_int @ A3 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_269_one__reorient,axiom,
    ! [X: code_integer] :
      ( ( one_one_Code_integer = X )
      = ( X = one_one_Code_integer ) ) ).

% one_reorient
thf(fact_270_one__reorient,axiom,
    ! [X: assn] :
      ( ( one_one_assn = X )
      = ( X = one_one_assn ) ) ).

% one_reorient
thf(fact_271_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
      = ( X = one_one_rat ) ) ).

% one_reorient
thf(fact_272_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_273_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_274_comp__eq__dest__lhs,axiom,
    ! [A: code_integer > code_integer,B: num > code_integer,C: num > code_integer,V: num] :
      ( ( ( comp_C3531382070062128313er_num @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_275_comp__eq__dest__lhs,axiom,
    ! [A: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,B: code_integer > code_integer,C: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,V: code_integer] :
      ( ( ( comp_C1593894019821074884nteger @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_276_comp__eq__dest__lhs,axiom,
    ! [A: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,B: code_integer > code_integer > code_integer,C: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,V: code_integer] :
      ( ( ( comp_C8797469213163452608nteger @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_277_comp__eq__dest__lhs,axiom,
    ! [A: nat > num,B: int > nat,C: int > num,V: int] :
      ( ( ( comp_nat_num_int @ A @ B )
        = C )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_278_comp__eq__elim,axiom,
    ! [A: code_integer > code_integer,B: num > code_integer,C: code_integer > code_integer,D2: num > code_integer] :
      ( ( ( comp_C3531382070062128313er_num @ A @ B )
        = ( comp_C3531382070062128313er_num @ C @ D2 ) )
     => ! [V3: num] :
          ( ( A @ ( B @ V3 ) )
          = ( C @ ( D2 @ V3 ) ) ) ) ).

% comp_eq_elim
thf(fact_279_comp__eq__elim,axiom,
    ! [A: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,B: code_integer > code_integer,C: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,D2: code_integer > code_integer] :
      ( ( ( comp_C1593894019821074884nteger @ A @ B )
        = ( comp_C1593894019821074884nteger @ C @ D2 ) )
     => ! [V3: code_integer] :
          ( ( A @ ( B @ V3 ) )
          = ( C @ ( D2 @ V3 ) ) ) ) ).

% comp_eq_elim
thf(fact_280_comp__eq__elim,axiom,
    ! [A: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,B: code_integer > code_integer,C: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,D2: code_integer > code_integer > code_integer] :
      ( ( ( comp_C1593894019821074884nteger @ A @ B )
        = ( comp_C8797469213163452608nteger @ C @ D2 ) )
     => ! [V3: code_integer] :
          ( ( A @ ( B @ V3 ) )
          = ( C @ ( D2 @ V3 ) ) ) ) ).

% comp_eq_elim
thf(fact_281_comp__eq__elim,axiom,
    ! [A: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,B: code_integer > code_integer > code_integer,C: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,D2: code_integer > code_integer] :
      ( ( ( comp_C8797469213163452608nteger @ A @ B )
        = ( comp_C1593894019821074884nteger @ C @ D2 ) )
     => ! [V3: code_integer] :
          ( ( A @ ( B @ V3 ) )
          = ( C @ ( D2 @ V3 ) ) ) ) ).

% comp_eq_elim
thf(fact_282_comp__eq__elim,axiom,
    ! [A: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,B: code_integer > code_integer > code_integer,C: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,D2: code_integer > code_integer > code_integer] :
      ( ( ( comp_C8797469213163452608nteger @ A @ B )
        = ( comp_C8797469213163452608nteger @ C @ D2 ) )
     => ! [V3: code_integer] :
          ( ( A @ ( B @ V3 ) )
          = ( C @ ( D2 @ V3 ) ) ) ) ).

% comp_eq_elim
thf(fact_283_comp__eq__elim,axiom,
    ! [A: nat > num,B: int > nat,C: nat > num,D2: int > nat] :
      ( ( ( comp_nat_num_int @ A @ B )
        = ( comp_nat_num_int @ C @ D2 ) )
     => ! [V3: int] :
          ( ( A @ ( B @ V3 ) )
          = ( C @ ( D2 @ V3 ) ) ) ) ).

% comp_eq_elim
thf(fact_284_comp__eq__dest,axiom,
    ! [A: code_integer > code_integer,B: num > code_integer,C: code_integer > code_integer,D2: num > code_integer,V: num] :
      ( ( ( comp_C3531382070062128313er_num @ A @ B )
        = ( comp_C3531382070062128313er_num @ C @ D2 ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D2 @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_285_comp__eq__dest,axiom,
    ! [A: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,B: code_integer > code_integer,C: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,D2: code_integer > code_integer,V: code_integer] :
      ( ( ( comp_C1593894019821074884nteger @ A @ B )
        = ( comp_C1593894019821074884nteger @ C @ D2 ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D2 @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_286_comp__eq__dest,axiom,
    ! [A: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,B: code_integer > code_integer,C: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,D2: code_integer > code_integer > code_integer,V: code_integer] :
      ( ( ( comp_C1593894019821074884nteger @ A @ B )
        = ( comp_C8797469213163452608nteger @ C @ D2 ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D2 @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_287_comp__eq__dest,axiom,
    ! [A: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,B: code_integer > code_integer > code_integer,C: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,D2: code_integer > code_integer,V: code_integer] :
      ( ( ( comp_C8797469213163452608nteger @ A @ B )
        = ( comp_C1593894019821074884nteger @ C @ D2 ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D2 @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_288_comp__eq__dest,axiom,
    ! [A: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,B: code_integer > code_integer > code_integer,C: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,D2: code_integer > code_integer > code_integer,V: code_integer] :
      ( ( ( comp_C8797469213163452608nteger @ A @ B )
        = ( comp_C8797469213163452608nteger @ C @ D2 ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D2 @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_289_comp__eq__dest,axiom,
    ! [A: nat > num,B: int > nat,C: nat > num,D2: int > nat,V: int] :
      ( ( ( comp_nat_num_int @ A @ B )
        = ( comp_nat_num_int @ C @ D2 ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ ( D2 @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_290_comp__assoc,axiom,
    ! [F: code_integer > code_integer,G: num > code_integer,H3: num > num] :
      ( ( comp_n6909714581898973814er_num @ ( comp_C3531382070062128313er_num @ F @ G ) @ H3 )
      = ( comp_C3531382070062128313er_num @ F @ ( comp_n6909714581898973814er_num @ G @ H3 ) ) ) ).

% comp_assoc
thf(fact_291_comp__assoc,axiom,
    ! [F: nat > num,G: int > nat,H3: int > int] :
      ( ( comp_int_num_int @ ( comp_nat_num_int @ F @ G ) @ H3 )
      = ( comp_nat_num_int @ F @ ( comp_int_nat_int @ G @ H3 ) ) ) ).

% comp_assoc
thf(fact_292_comp__assoc,axiom,
    ! [F: code_integer > code_integer,G: code_integer > code_integer,H3: num > code_integer] :
      ( ( comp_C3531382070062128313er_num @ ( comp_C7449957260575251196nteger @ F @ G ) @ H3 )
      = ( comp_C3531382070062128313er_num @ F @ ( comp_C3531382070062128313er_num @ G @ H3 ) ) ) ).

% comp_assoc
thf(fact_293_comp__assoc,axiom,
    ! [F: num > num,G: nat > num,H3: int > nat] :
      ( ( comp_nat_num_int @ ( comp_num_num_nat @ F @ G ) @ H3 )
      = ( comp_num_num_int @ F @ ( comp_nat_num_int @ G @ H3 ) ) ) ).

% comp_assoc
thf(fact_294_comp__assoc,axiom,
    ! [F: nat > num,G: nat > nat,H3: int > nat] :
      ( ( comp_nat_num_int @ ( comp_nat_num_nat @ F @ G ) @ H3 )
      = ( comp_nat_num_int @ F @ ( comp_nat_nat_int @ G @ H3 ) ) ) ).

% comp_assoc
thf(fact_295_comp__assoc,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,H3: num > code_integer] :
      ( ( comp_C6617437204949736321er_num @ ( comp_C1593894019821074884nteger @ F @ G ) @ H3 )
      = ( comp_C6617437204949736321er_num @ F @ ( comp_C3531382070062128313er_num @ G @ H3 ) ) ) ).

% comp_assoc
thf(fact_296_comp__assoc,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,H3: code_integer > code_integer] :
      ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ G ) @ H3 )
      = ( comp_C1593894019821074884nteger @ F @ ( comp_C7449957260575251196nteger @ G @ H3 ) ) ) ).

% comp_assoc
thf(fact_297_comp__assoc,axiom,
    ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,H3: code_integer > code_integer] :
      ( ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ F @ G ) @ H3 )
      = ( comp_C8797469213163452608nteger @ F @ ( comp_C2931745410781319416nteger @ G @ H3 ) ) ) ).

% comp_assoc
thf(fact_298_comp__assoc,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > code_integer,H3: code_integer > code_integer > code_integer] :
      ( ( comp_C8797469213163452608nteger @ ( comp_C95226695241045696nteger @ F @ G ) @ H3 )
      = ( comp_C1593894019821074884nteger @ F @ ( comp_C5076291335850163704nteger @ G @ H3 ) ) ) ).

% comp_assoc
thf(fact_299_comp__assoc,axiom,
    ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > code_integer > code_integer,H3: code_integer > code_integer > code_integer] :
      ( ( comp_C8797469213163452608nteger @ ( comp_C3983183376822107068nteger @ F @ G ) @ H3 )
      = ( comp_C8797469213163452608nteger @ F @ ( comp_C2822359487432382964nteger @ G @ H3 ) ) ) ).

% comp_assoc
thf(fact_300_comp__def,axiom,
    ( comp_C3531382070062128313er_num
    = ( ^ [F2: code_integer > code_integer,G2: num > code_integer,X3: num] : ( F2 @ ( G2 @ X3 ) ) ) ) ).

% comp_def
thf(fact_301_comp__def,axiom,
    ( comp_C1593894019821074884nteger
    = ( ^ [F2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer,X3: code_integer] : ( F2 @ ( G2 @ X3 ) ) ) ) ).

% comp_def
thf(fact_302_comp__def,axiom,
    ( comp_C8797469213163452608nteger
    = ( ^ [F2: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G2: code_integer > code_integer > code_integer,X3: code_integer] : ( F2 @ ( G2 @ X3 ) ) ) ) ).

% comp_def
thf(fact_303_comp__def,axiom,
    ( comp_nat_num_int
    = ( ^ [F2: nat > num,G2: int > nat,X3: int] : ( F2 @ ( G2 @ X3 ) ) ) ) ).

% comp_def
thf(fact_304_timeFrame_Oelims,axiom,
    ! [X: nat,Xa: option3216581401566546698it_nat,Y: option3216581401566546698it_nat] :
      ( ( ( heap_T531568236820715630rray_a @ X @ Xa )
        = Y )
     => ( ! [R: array_a,H2: heap_e7401611519738050253t_unit,N3: nat] :
            ( ( Xa
              = ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ R @ ( produc584006145561248582it_nat @ H2 @ N3 ) ) ) )
           => ( Y
             != ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ R @ ( produc584006145561248582it_nat @ H2 @ ( plus_plus_nat @ X @ N3 ) ) ) ) ) )
       => ~ ( ( Xa = none_P2587165478634041381it_nat )
           => ( Y != none_P2587165478634041381it_nat ) ) ) ) ).

% timeFrame.elims
thf(fact_305_timeFrame_Oelims,axiom,
    ! [X: nat,Xa: option8956607266484857688it_nat,Y: option8956607266484857688it_nat] :
      ( ( ( heap_T3616969660504097270t_unit @ X @ Xa )
        = Y )
     => ( ! [R: product_unit,H2: heap_e7401611519738050253t_unit,N3: nat] :
            ( ( Xa
              = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R @ ( produc584006145561248582it_nat @ H2 @ N3 ) ) ) )
           => ( Y
             != ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R @ ( produc584006145561248582it_nat @ H2 @ ( plus_plus_nat @ X @ N3 ) ) ) ) ) )
       => ~ ( ( Xa = none_P9117596204409417319it_nat )
           => ( Y != none_P9117596204409417319it_nat ) ) ) ) ).

% timeFrame.elims
thf(fact_306_execute__bind__eq__SomeI,axiom,
    ! [F: heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit,X: array_a,H5: heap_e7401611519738050253t_unit,N2: nat,G: array_a > heap_T6362127782908069795rray_a,Y: array_a,H7: heap_e7401611519738050253t_unit,N5: nat] :
      ( ( ( heap_T5591663578999710211rray_a @ F @ H3 )
        = ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( ( heap_T5591663578999710211rray_a @ ( G @ X ) @ H5 )
          = ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ Y @ ( produc584006145561248582it_nat @ H7 @ N5 ) ) ) )
       => ( ( heap_T5591663578999710211rray_a @ ( heap_T2407400398799471159rray_a @ F @ G ) @ H3 )
          = ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ Y @ ( produc584006145561248582it_nat @ H7 @ ( plus_plus_nat @ N2 @ N5 ) ) ) ) ) ) ) ).

% execute_bind_eq_SomeI
thf(fact_307_execute__bind__eq__SomeI,axiom,
    ! [F: heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit,X: array_a,H5: heap_e7401611519738050253t_unit,N2: nat,G: array_a > heap_T5738788834812785303t_unit,Y: product_unit,H7: heap_e7401611519738050253t_unit,N5: nat] :
      ( ( ( heap_T5591663578999710211rray_a @ F @ H3 )
        = ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( ( heap_T875086893843062177t_unit @ ( G @ X ) @ H5 )
          = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ Y @ ( produc584006145561248582it_nat @ H7 @ N5 ) ) ) )
       => ( ( heap_T875086893843062177t_unit @ ( heap_T7324619625323989613t_unit @ F @ G ) @ H3 )
          = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ Y @ ( produc584006145561248582it_nat @ H7 @ ( plus_plus_nat @ N2 @ N5 ) ) ) ) ) ) ) ).

% execute_bind_eq_SomeI
thf(fact_308_execute__bind__eq__SomeI,axiom,
    ! [F: heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit,X: product_unit,H5: heap_e7401611519738050253t_unit,N2: nat,G: product_unit > heap_T6362127782908069795rray_a,Y: array_a,H7: heap_e7401611519738050253t_unit,N5: nat] :
      ( ( ( heap_T875086893843062177t_unit @ F @ H3 )
        = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( ( heap_T5591663578999710211rray_a @ ( G @ X ) @ H5 )
          = ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ Y @ ( produc584006145561248582it_nat @ H7 @ N5 ) ) ) )
       => ( ( heap_T5591663578999710211rray_a @ ( heap_T7158140910958840029rray_a @ F @ G ) @ H3 )
          = ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ Y @ ( produc584006145561248582it_nat @ H7 @ ( plus_plus_nat @ N2 @ N5 ) ) ) ) ) ) ) ).

% execute_bind_eq_SomeI
thf(fact_309_execute__bind__eq__SomeI,axiom,
    ! [F: heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit,X: product_unit,H5: heap_e7401611519738050253t_unit,N2: nat,G: product_unit > heap_T5738788834812785303t_unit,Y: product_unit,H7: heap_e7401611519738050253t_unit,N5: nat] :
      ( ( ( heap_T875086893843062177t_unit @ F @ H3 )
        = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ X @ ( produc584006145561248582it_nat @ H5 @ N2 ) ) ) )
     => ( ( ( heap_T875086893843062177t_unit @ ( G @ X ) @ H5 )
          = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ Y @ ( produc584006145561248582it_nat @ H7 @ N5 ) ) ) )
       => ( ( heap_T875086893843062177t_unit @ ( heap_T2633723481742716231t_unit @ F @ G ) @ H3 )
          = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ Y @ ( produc584006145561248582it_nat @ H7 @ ( plus_plus_nat @ N2 @ N5 ) ) ) ) ) ) ) ).

% execute_bind_eq_SomeI
thf(fact_310_timeFrame_Osimps_I1_J,axiom,
    ! [N2: nat,R2: product_unit,H3: heap_e7401611519738050253t_unit,N5: nat] :
      ( ( heap_T3616969660504097270t_unit @ N2 @ ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R2 @ ( produc584006145561248582it_nat @ H3 @ N5 ) ) ) )
      = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R2 @ ( produc584006145561248582it_nat @ H3 @ ( plus_plus_nat @ N2 @ N5 ) ) ) ) ) ).

% timeFrame.simps(1)
thf(fact_311_heap__def,axiom,
    ( heap_T6927113302350381334t_unit
    = ( ^ [F2: heap_e7401611519738050253t_unit > produc8664842809031399944it_nat] : ( heap_T6183433275982383450t_unit @ ( comp_P3118722334806803912t_unit @ some_P1914260805536162275it_nat @ F2 ) ) ) ) ).

% heap_def
thf(fact_312_option_Osplit__sel,axiom,
    ! [P: int > $o,F1: int,F22: num > int,Option: option_num] :
      ( ( P @ ( case_option_int_num @ F1 @ F22 @ Option ) )
      = ( ( ( Option = none_num )
         => ( P @ F1 ) )
        & ( ( Option
            = ( some_num @ ( the_num @ Option ) ) )
         => ( P @ ( F22 @ ( the_num @ Option ) ) ) ) ) ) ).

% option.split_sel
thf(fact_313_option_Osplit__sel,axiom,
    ! [P: option_num > $o,F1: option_num,F22: num > option_num,Option: option_num] :
      ( ( P @ ( case_o6005452278849405969um_num @ F1 @ F22 @ Option ) )
      = ( ( ( Option = none_num )
         => ( P @ F1 ) )
        & ( ( Option
            = ( some_num @ ( the_num @ Option ) ) )
         => ( P @ ( F22 @ ( the_num @ Option ) ) ) ) ) ) ).

% option.split_sel
thf(fact_314_option_Osplit__sel,axiom,
    ! [P: num > $o,F1: num,F22: num > num,Option: option_num] :
      ( ( P @ ( case_option_num_num @ F1 @ F22 @ Option ) )
      = ( ( ( Option = none_num )
         => ( P @ F1 ) )
        & ( ( Option
            = ( some_num @ ( the_num @ Option ) ) )
         => ( P @ ( F22 @ ( the_num @ Option ) ) ) ) ) ) ).

% option.split_sel
thf(fact_315_option_Osplit__sel__asm,axiom,
    ! [P: int > $o,F1: int,F22: num > int,Option: option_num] :
      ( ( P @ ( case_option_int_num @ F1 @ F22 @ Option ) )
      = ( ~ ( ( ( Option = none_num )
              & ~ ( P @ F1 ) )
            | ( ( Option
                = ( some_num @ ( the_num @ Option ) ) )
              & ~ ( P @ ( F22 @ ( the_num @ Option ) ) ) ) ) ) ) ).

% option.split_sel_asm
thf(fact_316_option_Osplit__sel__asm,axiom,
    ! [P: option_num > $o,F1: option_num,F22: num > option_num,Option: option_num] :
      ( ( P @ ( case_o6005452278849405969um_num @ F1 @ F22 @ Option ) )
      = ( ~ ( ( ( Option = none_num )
              & ~ ( P @ F1 ) )
            | ( ( Option
                = ( some_num @ ( the_num @ Option ) ) )
              & ~ ( P @ ( F22 @ ( the_num @ Option ) ) ) ) ) ) ) ).

% option.split_sel_asm
thf(fact_317_option_Osplit__sel__asm,axiom,
    ! [P: num > $o,F1: num,F22: num > num,Option: option_num] :
      ( ( P @ ( case_option_num_num @ F1 @ F22 @ Option ) )
      = ( ~ ( ( ( Option = none_num )
              & ~ ( P @ F1 ) )
            | ( ( Option
                = ( some_num @ ( the_num @ Option ) ) )
              & ~ ( P @ ( F22 @ ( the_num @ Option ) ) ) ) ) ) ) ).

% option.split_sel_asm
thf(fact_318_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
thf(fact_319_timeFrame_Opelims,axiom,
    ! [X: nat,Xa: option3216581401566546698it_nat,Y: option3216581401566546698it_nat] :
      ( ( ( heap_T531568236820715630rray_a @ X @ Xa )
        = Y )
     => ( ( accp_P8948468358075348838it_nat @ heap_T6578553156929092987rray_a @ ( produc3378006678175723489it_nat @ X @ Xa ) )
       => ( ! [R: array_a,H2: heap_e7401611519738050253t_unit,N3: nat] :
              ( ( Xa
                = ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ R @ ( produc584006145561248582it_nat @ H2 @ N3 ) ) ) )
             => ( ( Y
                  = ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ R @ ( produc584006145561248582it_nat @ H2 @ ( plus_plus_nat @ X @ N3 ) ) ) ) )
               => ~ ( accp_P8948468358075348838it_nat @ heap_T6578553156929092987rray_a @ ( produc3378006678175723489it_nat @ X @ ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ R @ ( produc584006145561248582it_nat @ H2 @ N3 ) ) ) ) ) ) )
         => ~ ( ( Xa = none_P2587165478634041381it_nat )
             => ( ( Y = none_P2587165478634041381it_nat )
               => ~ ( accp_P8948468358075348838it_nat @ heap_T6578553156929092987rray_a @ ( produc3378006678175723489it_nat @ X @ none_P2587165478634041381it_nat ) ) ) ) ) ) ) ).

% timeFrame.pelims
thf(fact_320_timeFrame_Opelims,axiom,
    ! [X: nat,Xa: option8956607266484857688it_nat,Y: option8956607266484857688it_nat] :
      ( ( ( heap_T3616969660504097270t_unit @ X @ Xa )
        = Y )
     => ( ( accp_P414730952086964626it_nat @ heap_T996182799752388649t_unit @ ( produc638857205735767105it_nat @ X @ Xa ) )
       => ( ! [R: product_unit,H2: heap_e7401611519738050253t_unit,N3: nat] :
              ( ( Xa
                = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R @ ( produc584006145561248582it_nat @ H2 @ N3 ) ) ) )
             => ( ( Y
                  = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R @ ( produc584006145561248582it_nat @ H2 @ ( plus_plus_nat @ X @ N3 ) ) ) ) )
               => ~ ( accp_P414730952086964626it_nat @ heap_T996182799752388649t_unit @ ( produc638857205735767105it_nat @ X @ ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ R @ ( produc584006145561248582it_nat @ H2 @ N3 ) ) ) ) ) ) )
         => ~ ( ( Xa = none_P9117596204409417319it_nat )
             => ( ( Y = none_P9117596204409417319it_nat )
               => ~ ( accp_P414730952086964626it_nat @ heap_T996182799752388649t_unit @ ( produc638857205735767105it_nat @ X @ none_P9117596204409417319it_nat ) ) ) ) ) ) ) ).

% timeFrame.pelims
thf(fact_321_add__right__cancel,axiom,
    ! [B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ B @ A )
        = ( plus_p7104986032573967614at_nat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_322_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_323_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_324_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_325_add__left__cancel,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ A @ B )
        = ( plus_p7104986032573967614at_nat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_326_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_327_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_328_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_329_Heap_Oinject,axiom,
    ! [X: heap_e7401611519738050253t_unit > option8956607266484857688it_nat,Ya: heap_e7401611519738050253t_unit > option8956607266484857688it_nat] :
      ( ( ( heap_T6183433275982383450t_unit @ X )
        = ( heap_T6183433275982383450t_unit @ Ya ) )
      = ( X = Ya ) ) ).

% Heap.inject
thf(fact_330_Heap__execute,axiom,
    ! [F: heap_T6362127782908069795rray_a] :
      ( ( heap_T1488828001306802186rray_a @ ( heap_T5591663578999710211rray_a @ F ) )
      = F ) ).

% Heap_execute
thf(fact_331_Heap__execute,axiom,
    ! [F: heap_T5738788834812785303t_unit] :
      ( ( heap_T6183433275982383450t_unit @ ( heap_T875086893843062177t_unit @ F ) )
      = F ) ).

% Heap_execute
thf(fact_332_add__right__imp__eq,axiom,
    ! [B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ B @ A )
        = ( plus_p7104986032573967614at_nat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_333_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_334_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_335_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_336_add__left__imp__eq,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ A @ B )
        = ( plus_p7104986032573967614at_nat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_337_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_338_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_339_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_340_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ B @ ( plus_p7104986032573967614at_nat @ A @ C ) )
      = ( plus_p7104986032573967614at_nat @ A @ ( plus_p7104986032573967614at_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_341_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_342_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_343_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_344_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_p7104986032573967614at_nat
    = ( ^ [A5: multis2468970476368604999at_nat,B4: multis2468970476368604999at_nat] : ( plus_p7104986032573967614at_nat @ B4 @ A5 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_345_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ B4 @ A5 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_346_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_347_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_348_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_349_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_350_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_351_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_352_add_Oassoc,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ ( plus_p7104986032573967614at_nat @ A @ B ) @ C )
      = ( plus_p7104986032573967614at_nat @ A @ ( plus_p7104986032573967614at_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_353_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% add.assoc
thf(fact_354_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_355_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% add.assoc
thf(fact_356_group__cancel_Oadd2,axiom,
    ! [B5: multis2468970476368604999at_nat,K: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat] :
      ( ( B5
        = ( plus_p7104986032573967614at_nat @ K @ B ) )
     => ( ( plus_p7104986032573967614at_nat @ A @ B5 )
        = ( plus_p7104986032573967614at_nat @ K @ ( plus_p7104986032573967614at_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_357_group__cancel_Oadd2,axiom,
    ! [B5: rat,K: rat,B: rat,A: rat] :
      ( ( B5
        = ( plus_plus_rat @ K @ B ) )
     => ( ( plus_plus_rat @ A @ B5 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_358_group__cancel_Oadd2,axiom,
    ! [B5: nat,K: nat,B: nat,A: nat] :
      ( ( B5
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B5 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_359_group__cancel_Oadd2,axiom,
    ! [B5: int,K: int,B: int,A: int] :
      ( ( B5
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A @ B5 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_360_group__cancel_Oadd1,axiom,
    ! [A2: multis2468970476368604999at_nat,K: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( A2
        = ( plus_p7104986032573967614at_nat @ K @ A ) )
     => ( ( plus_p7104986032573967614at_nat @ A2 @ B )
        = ( plus_p7104986032573967614at_nat @ K @ ( plus_p7104986032573967614at_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_361_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( plus_plus_rat @ A2 @ B )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_362_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A2 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_363_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( plus_plus_int @ A2 @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_364_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_rat @ I @ K )
        = ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_365_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_366_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_int @ I @ K )
        = ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_367_Heap_Oexhaust,axiom,
    ! [Y: heap_T5738788834812785303t_unit] :
      ~ ! [X4: heap_e7401611519738050253t_unit > option8956607266484857688it_nat] :
          ( Y
         != ( heap_T6183433275982383450t_unit @ X4 ) ) ).

% Heap.exhaust
thf(fact_368_execute_Osimps,axiom,
    ! [F: heap_e7401611519738050253t_unit > option3216581401566546698it_nat] :
      ( ( heap_T5591663578999710211rray_a @ ( heap_T1488828001306802186rray_a @ F ) )
      = F ) ).

% execute.simps
thf(fact_369_execute_Osimps,axiom,
    ! [F: heap_e7401611519738050253t_unit > option8956607266484857688it_nat] :
      ( ( heap_T875086893843062177t_unit @ ( heap_T6183433275982383450t_unit @ F ) )
      = F ) ).

% execute.simps
thf(fact_370_option_Osimps_I5_J,axiom,
    ! [F1: int,F22: num > int,X2: num] :
      ( ( case_option_int_num @ F1 @ F22 @ ( some_num @ X2 ) )
      = ( F22 @ X2 ) ) ).

% option.simps(5)
thf(fact_371_option_Osimps_I5_J,axiom,
    ! [F1: option_num,F22: num > option_num,X2: num] :
      ( ( case_o6005452278849405969um_num @ F1 @ F22 @ ( some_num @ X2 ) )
      = ( F22 @ X2 ) ) ).

% option.simps(5)
thf(fact_372_option_Osimps_I5_J,axiom,
    ! [F1: num,F22: num > num,X2: num] :
      ( ( case_option_num_num @ F1 @ F22 @ ( some_num @ X2 ) )
      = ( F22 @ X2 ) ) ).

% option.simps(5)
thf(fact_373_option_Osimps_I4_J,axiom,
    ! [F1: int,F22: num > int] :
      ( ( case_option_int_num @ F1 @ F22 @ none_num )
      = F1 ) ).

% option.simps(4)
thf(fact_374_option_Osimps_I4_J,axiom,
    ! [F1: option_num,F22: num > option_num] :
      ( ( case_o6005452278849405969um_num @ F1 @ F22 @ none_num )
      = F1 ) ).

% option.simps(4)
thf(fact_375_option_Osimps_I4_J,axiom,
    ! [F1: num,F22: num > num] :
      ( ( case_option_num_num @ F1 @ F22 @ none_num )
      = F1 ) ).

% option.simps(4)
thf(fact_376_option_Ocase__eq__if,axiom,
    ( case_option_int_num
    = ( ^ [F12: int,F23: num > int,Option3: option_num] : ( if_int @ ( Option3 = none_num ) @ F12 @ ( F23 @ ( the_num @ Option3 ) ) ) ) ) ).

% option.case_eq_if
thf(fact_377_option_Ocase__eq__if,axiom,
    ( case_o6005452278849405969um_num
    = ( ^ [F12: option_num,F23: num > option_num,Option3: option_num] : ( if_option_num @ ( Option3 = none_num ) @ F12 @ ( F23 @ ( the_num @ Option3 ) ) ) ) ) ).

% option.case_eq_if
thf(fact_378_option_Ocase__eq__if,axiom,
    ( case_option_num_num
    = ( ^ [F12: num,F23: num > num,Option3: option_num] : ( if_num @ ( Option3 = none_num ) @ F12 @ ( F23 @ ( the_num @ Option3 ) ) ) ) ) ).

% option.case_eq_if
thf(fact_379_accp_Ocases,axiom,
    ! [R2: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( accp_P4275260045618599050at_nat @ R2 @ A )
     => ! [Y6: product_prod_nat_nat] :
          ( ( R2 @ Y6 @ A )
         => ( accp_P4275260045618599050at_nat @ R2 @ Y6 ) ) ) ).

% accp.cases
thf(fact_380_accp_Ocases,axiom,
    ! [R2: product_prod_int_int > product_prod_int_int > $o,A: product_prod_int_int] :
      ( ( accp_P1096762738010456898nt_int @ R2 @ A )
     => ! [Y6: product_prod_int_int] :
          ( ( R2 @ Y6 @ A )
         => ( accp_P1096762738010456898nt_int @ R2 @ Y6 ) ) ) ).

% accp.cases
thf(fact_381_accp_Ocases,axiom,
    ! [R2: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o,A: produc2732055786443039994et_nat] :
      ( ( accp_P1862375125659990705et_nat @ R2 @ A )
     => ! [Y6: produc2732055786443039994et_nat] :
          ( ( R2 @ Y6 @ A )
         => ( accp_P1862375125659990705et_nat @ R2 @ Y6 ) ) ) ).

% accp.cases
thf(fact_382_accp_Ocases,axiom,
    ! [R2: list_nat > list_nat > $o,A: list_nat] :
      ( ( accp_list_nat @ R2 @ A )
     => ! [Y6: list_nat] :
          ( ( R2 @ Y6 @ A )
         => ( accp_list_nat @ R2 @ Y6 ) ) ) ).

% accp.cases
thf(fact_383_accp_Ocases,axiom,
    ! [R2: nat > nat > $o,A: nat] :
      ( ( accp_nat @ R2 @ A )
     => ! [Y6: nat] :
          ( ( R2 @ Y6 @ A )
         => ( accp_nat @ R2 @ Y6 ) ) ) ).

% accp.cases
thf(fact_384_accp_Osimps,axiom,
    ( accp_P4275260045618599050at_nat
    = ( ^ [R3: product_prod_nat_nat > product_prod_nat_nat > $o,A5: product_prod_nat_nat] :
        ? [X3: product_prod_nat_nat] :
          ( ( A5 = X3 )
          & ! [Y3: product_prod_nat_nat] :
              ( ( R3 @ Y3 @ X3 )
             => ( accp_P4275260045618599050at_nat @ R3 @ Y3 ) ) ) ) ) ).

% accp.simps
thf(fact_385_accp_Osimps,axiom,
    ( accp_P1096762738010456898nt_int
    = ( ^ [R3: product_prod_int_int > product_prod_int_int > $o,A5: product_prod_int_int] :
        ? [X3: product_prod_int_int] :
          ( ( A5 = X3 )
          & ! [Y3: product_prod_int_int] :
              ( ( R3 @ Y3 @ X3 )
             => ( accp_P1096762738010456898nt_int @ R3 @ Y3 ) ) ) ) ) ).

% accp.simps
thf(fact_386_accp_Osimps,axiom,
    ( accp_P1862375125659990705et_nat
    = ( ^ [R3: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o,A5: produc2732055786443039994et_nat] :
        ? [X3: produc2732055786443039994et_nat] :
          ( ( A5 = X3 )
          & ! [Y3: produc2732055786443039994et_nat] :
              ( ( R3 @ Y3 @ X3 )
             => ( accp_P1862375125659990705et_nat @ R3 @ Y3 ) ) ) ) ) ).

% accp.simps
thf(fact_387_accp_Osimps,axiom,
    ( accp_list_nat
    = ( ^ [R3: list_nat > list_nat > $o,A5: list_nat] :
        ? [X3: list_nat] :
          ( ( A5 = X3 )
          & ! [Y3: list_nat] :
              ( ( R3 @ Y3 @ X3 )
             => ( accp_list_nat @ R3 @ Y3 ) ) ) ) ) ).

% accp.simps
thf(fact_388_accp_Osimps,axiom,
    ( accp_nat
    = ( ^ [R3: nat > nat > $o,A5: nat] :
        ? [X3: nat] :
          ( ( A5 = X3 )
          & ! [Y3: nat] :
              ( ( R3 @ Y3 @ X3 )
             => ( accp_nat @ R3 @ Y3 ) ) ) ) ) ).

% accp.simps
thf(fact_389_accpI,axiom,
    ! [R2: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
      ( ! [Y5: product_prod_nat_nat] :
          ( ( R2 @ Y5 @ X )
         => ( accp_P4275260045618599050at_nat @ R2 @ Y5 ) )
     => ( accp_P4275260045618599050at_nat @ R2 @ X ) ) ).

% accpI
thf(fact_390_accpI,axiom,
    ! [R2: product_prod_int_int > product_prod_int_int > $o,X: product_prod_int_int] :
      ( ! [Y5: product_prod_int_int] :
          ( ( R2 @ Y5 @ X )
         => ( accp_P1096762738010456898nt_int @ R2 @ Y5 ) )
     => ( accp_P1096762738010456898nt_int @ R2 @ X ) ) ).

% accpI
thf(fact_391_accpI,axiom,
    ! [R2: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
      ( ! [Y5: produc2732055786443039994et_nat] :
          ( ( R2 @ Y5 @ X )
         => ( accp_P1862375125659990705et_nat @ R2 @ Y5 ) )
     => ( accp_P1862375125659990705et_nat @ R2 @ X ) ) ).

% accpI
thf(fact_392_accpI,axiom,
    ! [R2: list_nat > list_nat > $o,X: list_nat] :
      ( ! [Y5: list_nat] :
          ( ( R2 @ Y5 @ X )
         => ( accp_list_nat @ R2 @ Y5 ) )
     => ( accp_list_nat @ R2 @ X ) ) ).

% accpI
thf(fact_393_accpI,axiom,
    ! [R2: nat > nat > $o,X: nat] :
      ( ! [Y5: nat] :
          ( ( R2 @ Y5 @ X )
         => ( accp_nat @ R2 @ Y5 ) )
     => ( accp_nat @ R2 @ X ) ) ).

% accpI
thf(fact_394_accp__induct,axiom,
    ! [R2: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( accp_P4275260045618599050at_nat @ R2 @ A )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( accp_P4275260045618599050at_nat @ R2 @ X4 )
           => ( ! [Y6: product_prod_nat_nat] :
                  ( ( R2 @ Y6 @ X4 )
                 => ( P @ Y6 ) )
             => ( P @ X4 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct
thf(fact_395_accp__induct,axiom,
    ! [R2: product_prod_int_int > product_prod_int_int > $o,A: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( accp_P1096762738010456898nt_int @ R2 @ A )
     => ( ! [X4: product_prod_int_int] :
            ( ( accp_P1096762738010456898nt_int @ R2 @ X4 )
           => ( ! [Y6: product_prod_int_int] :
                  ( ( R2 @ Y6 @ X4 )
                 => ( P @ Y6 ) )
             => ( P @ X4 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct
thf(fact_396_accp__induct,axiom,
    ! [R2: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o,A: produc2732055786443039994et_nat,P: produc2732055786443039994et_nat > $o] :
      ( ( accp_P1862375125659990705et_nat @ R2 @ A )
     => ( ! [X4: produc2732055786443039994et_nat] :
            ( ( accp_P1862375125659990705et_nat @ R2 @ X4 )
           => ( ! [Y6: produc2732055786443039994et_nat] :
                  ( ( R2 @ Y6 @ X4 )
                 => ( P @ Y6 ) )
             => ( P @ X4 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct
thf(fact_397_accp__induct,axiom,
    ! [R2: list_nat > list_nat > $o,A: list_nat,P: list_nat > $o] :
      ( ( accp_list_nat @ R2 @ A )
     => ( ! [X4: list_nat] :
            ( ( accp_list_nat @ R2 @ X4 )
           => ( ! [Y6: list_nat] :
                  ( ( R2 @ Y6 @ X4 )
                 => ( P @ Y6 ) )
             => ( P @ X4 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct
thf(fact_398_accp__induct,axiom,
    ! [R2: nat > nat > $o,A: nat,P: nat > $o] :
      ( ( accp_nat @ R2 @ A )
     => ( ! [X4: nat] :
            ( ( accp_nat @ R2 @ X4 )
           => ( ! [Y6: nat] :
                  ( ( R2 @ Y6 @ X4 )
                 => ( P @ Y6 ) )
             => ( P @ X4 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct
thf(fact_399_accp__downward,axiom,
    ! [R2: product_prod_nat_nat > product_prod_nat_nat > $o,B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( accp_P4275260045618599050at_nat @ R2 @ B )
     => ( ( R2 @ A @ B )
       => ( accp_P4275260045618599050at_nat @ R2 @ A ) ) ) ).

% accp_downward
thf(fact_400_accp__downward,axiom,
    ! [R2: product_prod_int_int > product_prod_int_int > $o,B: product_prod_int_int,A: product_prod_int_int] :
      ( ( accp_P1096762738010456898nt_int @ R2 @ B )
     => ( ( R2 @ A @ B )
       => ( accp_P1096762738010456898nt_int @ R2 @ A ) ) ) ).

% accp_downward
thf(fact_401_accp__downward,axiom,
    ! [R2: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o,B: produc2732055786443039994et_nat,A: produc2732055786443039994et_nat] :
      ( ( accp_P1862375125659990705et_nat @ R2 @ B )
     => ( ( R2 @ A @ B )
       => ( accp_P1862375125659990705et_nat @ R2 @ A ) ) ) ).

% accp_downward
thf(fact_402_accp__downward,axiom,
    ! [R2: list_nat > list_nat > $o,B: list_nat,A: list_nat] :
      ( ( accp_list_nat @ R2 @ B )
     => ( ( R2 @ A @ B )
       => ( accp_list_nat @ R2 @ A ) ) ) ).

% accp_downward
thf(fact_403_accp__downward,axiom,
    ! [R2: nat > nat > $o,B: nat,A: nat] :
      ( ( accp_nat @ R2 @ B )
     => ( ( R2 @ A @ B )
       => ( accp_nat @ R2 @ A ) ) ) ).

% accp_downward
thf(fact_404_not__accp__down,axiom,
    ! [R4: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
      ( ~ ( accp_P4275260045618599050at_nat @ R4 @ X )
     => ~ ! [Z3: product_prod_nat_nat] :
            ( ( R4 @ Z3 @ X )
           => ( accp_P4275260045618599050at_nat @ R4 @ Z3 ) ) ) ).

% not_accp_down
thf(fact_405_not__accp__down,axiom,
    ! [R4: product_prod_int_int > product_prod_int_int > $o,X: product_prod_int_int] :
      ( ~ ( accp_P1096762738010456898nt_int @ R4 @ X )
     => ~ ! [Z3: product_prod_int_int] :
            ( ( R4 @ Z3 @ X )
           => ( accp_P1096762738010456898nt_int @ R4 @ Z3 ) ) ) ).

% not_accp_down
thf(fact_406_not__accp__down,axiom,
    ! [R4: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat] :
      ( ~ ( accp_P1862375125659990705et_nat @ R4 @ X )
     => ~ ! [Z3: produc2732055786443039994et_nat] :
            ( ( R4 @ Z3 @ X )
           => ( accp_P1862375125659990705et_nat @ R4 @ Z3 ) ) ) ).

% not_accp_down
thf(fact_407_not__accp__down,axiom,
    ! [R4: list_nat > list_nat > $o,X: list_nat] :
      ( ~ ( accp_list_nat @ R4 @ X )
     => ~ ! [Z3: list_nat] :
            ( ( R4 @ Z3 @ X )
           => ( accp_list_nat @ R4 @ Z3 ) ) ) ).

% not_accp_down
thf(fact_408_not__accp__down,axiom,
    ! [R4: nat > nat > $o,X: nat] :
      ( ~ ( accp_nat @ R4 @ X )
     => ~ ! [Z3: nat] :
            ( ( R4 @ Z3 @ X )
           => ( accp_nat @ R4 @ Z3 ) ) ) ).

% not_accp_down
thf(fact_409_accp__induct__rule,axiom,
    ! [R2: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( accp_P4275260045618599050at_nat @ R2 @ A )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( accp_P4275260045618599050at_nat @ R2 @ X4 )
           => ( ! [Y6: product_prod_nat_nat] :
                  ( ( R2 @ Y6 @ X4 )
                 => ( P @ Y6 ) )
             => ( P @ X4 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct_rule
thf(fact_410_accp__induct__rule,axiom,
    ! [R2: product_prod_int_int > product_prod_int_int > $o,A: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( accp_P1096762738010456898nt_int @ R2 @ A )
     => ( ! [X4: product_prod_int_int] :
            ( ( accp_P1096762738010456898nt_int @ R2 @ X4 )
           => ( ! [Y6: product_prod_int_int] :
                  ( ( R2 @ Y6 @ X4 )
                 => ( P @ Y6 ) )
             => ( P @ X4 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct_rule
thf(fact_411_accp__induct__rule,axiom,
    ! [R2: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o,A: produc2732055786443039994et_nat,P: produc2732055786443039994et_nat > $o] :
      ( ( accp_P1862375125659990705et_nat @ R2 @ A )
     => ( ! [X4: produc2732055786443039994et_nat] :
            ( ( accp_P1862375125659990705et_nat @ R2 @ X4 )
           => ( ! [Y6: produc2732055786443039994et_nat] :
                  ( ( R2 @ Y6 @ X4 )
                 => ( P @ Y6 ) )
             => ( P @ X4 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct_rule
thf(fact_412_accp__induct__rule,axiom,
    ! [R2: list_nat > list_nat > $o,A: list_nat,P: list_nat > $o] :
      ( ( accp_list_nat @ R2 @ A )
     => ( ! [X4: list_nat] :
            ( ( accp_list_nat @ R2 @ X4 )
           => ( ! [Y6: list_nat] :
                  ( ( R2 @ Y6 @ X4 )
                 => ( P @ Y6 ) )
             => ( P @ X4 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct_rule
thf(fact_413_accp__induct__rule,axiom,
    ! [R2: nat > nat > $o,A: nat,P: nat > $o] :
      ( ( accp_nat @ R2 @ A )
     => ( ! [X4: nat] :
            ( ( accp_nat @ R2 @ X4 )
           => ( ! [Y6: nat] :
                  ( ( R2 @ Y6 @ X4 )
                 => ( P @ Y6 ) )
             => ( P @ X4 ) ) )
       => ( P @ A ) ) ) ).

% accp_induct_rule
thf(fact_414_fun_Omap__comp,axiom,
    ! [G: num > num,F: nat > num,V: int > nat] :
      ( ( comp_num_num_int @ G @ ( comp_nat_num_int @ F @ V ) )
      = ( comp_nat_num_int @ ( comp_num_num_nat @ G @ F ) @ V ) ) ).

% fun.map_comp
thf(fact_415_fun_Omap__comp,axiom,
    ! [G: code_integer > code_integer,F: num > code_integer,V: num > num] :
      ( ( comp_C3531382070062128313er_num @ G @ ( comp_n6909714581898973814er_num @ F @ V ) )
      = ( comp_n6909714581898973814er_num @ ( comp_C3531382070062128313er_num @ G @ F ) @ V ) ) ).

% fun.map_comp
thf(fact_416_fun_Omap__comp,axiom,
    ! [G: code_integer > code_integer,F: code_integer > code_integer,V: num > code_integer] :
      ( ( comp_C3531382070062128313er_num @ G @ ( comp_C3531382070062128313er_num @ F @ V ) )
      = ( comp_C3531382070062128313er_num @ ( comp_C7449957260575251196nteger @ G @ F ) @ V ) ) ).

% fun.map_comp
thf(fact_417_fun_Omap__comp,axiom,
    ! [G: nat > num,F: int > nat,V: int > int] :
      ( ( comp_nat_num_int @ G @ ( comp_int_nat_int @ F @ V ) )
      = ( comp_int_num_int @ ( comp_nat_num_int @ G @ F ) @ V ) ) ).

% fun.map_comp
thf(fact_418_fun_Omap__comp,axiom,
    ! [G: nat > num,F: nat > nat,V: int > nat] :
      ( ( comp_nat_num_int @ G @ ( comp_nat_nat_int @ F @ V ) )
      = ( comp_nat_num_int @ ( comp_nat_num_nat @ G @ F ) @ V ) ) ).

% fun.map_comp
thf(fact_419_fun_Omap__comp,axiom,
    ! [G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer,V: num > code_integer] :
      ( ( comp_C6617437204949736321er_num @ G @ ( comp_C3531382070062128313er_num @ F @ V ) )
      = ( comp_C6617437204949736321er_num @ ( comp_C1593894019821074884nteger @ G @ F ) @ V ) ) ).

% fun.map_comp
thf(fact_420_fun_Omap__comp,axiom,
    ! [G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer,V: code_integer > code_integer] :
      ( ( comp_C1593894019821074884nteger @ G @ ( comp_C7449957260575251196nteger @ F @ V ) )
      = ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ G @ F ) @ V ) ) ).

% fun.map_comp
thf(fact_421_fun_Omap__comp,axiom,
    ! [G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,F: ( code_integer > code_integer ) > code_integer,V: code_integer > code_integer > code_integer] :
      ( ( comp_C1593894019821074884nteger @ G @ ( comp_C5076291335850163704nteger @ F @ V ) )
      = ( comp_C8797469213163452608nteger @ ( comp_C95226695241045696nteger @ G @ F ) @ V ) ) ).

% fun.map_comp
thf(fact_422_fun_Omap__comp,axiom,
    ! [G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer > code_integer,V: code_integer > code_integer] :
      ( ( comp_C8797469213163452608nteger @ G @ ( comp_C2931745410781319416nteger @ F @ V ) )
      = ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ G @ F ) @ V ) ) ).

% fun.map_comp
thf(fact_423_fun_Omap__comp,axiom,
    ! [G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F: ( code_integer > code_integer ) > code_integer > code_integer,V: code_integer > code_integer > code_integer] :
      ( ( comp_C8797469213163452608nteger @ G @ ( comp_C2822359487432382964nteger @ F @ V ) )
      = ( comp_C8797469213163452608nteger @ ( comp_C3983183376822107068nteger @ G @ F ) @ V ) ) ).

% fun.map_comp
thf(fact_424_comp__cong,axiom,
    ! [F: code_integer > code_integer,G: num > code_integer,X: num,F4: code_integer > code_integer,G3: num > code_integer,X7: num] :
      ( ( ( F @ ( G @ X ) )
        = ( F4 @ ( G3 @ X7 ) ) )
     => ( ( comp_C3531382070062128313er_num @ F @ G @ X )
        = ( comp_C3531382070062128313er_num @ F4 @ G3 @ X7 ) ) ) ).

% comp_cong
thf(fact_425_comp__cong,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,X: code_integer,F4: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G3: code_integer > code_integer,X7: code_integer] :
      ( ( ( F @ ( G @ X ) )
        = ( F4 @ ( G3 @ X7 ) ) )
     => ( ( comp_C1593894019821074884nteger @ F @ G @ X )
        = ( comp_C1593894019821074884nteger @ F4 @ G3 @ X7 ) ) ) ).

% comp_cong
thf(fact_426_comp__cong,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,X: code_integer,F4: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G3: code_integer > code_integer > code_integer,X7: code_integer] :
      ( ( ( F @ ( G @ X ) )
        = ( F4 @ ( G3 @ X7 ) ) )
     => ( ( comp_C1593894019821074884nteger @ F @ G @ X )
        = ( comp_C8797469213163452608nteger @ F4 @ G3 @ X7 ) ) ) ).

% comp_cong
thf(fact_427_comp__cong,axiom,
    ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,X: code_integer,F4: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G3: code_integer > code_integer,X7: code_integer] :
      ( ( ( F @ ( G @ X ) )
        = ( F4 @ ( G3 @ X7 ) ) )
     => ( ( comp_C8797469213163452608nteger @ F @ G @ X )
        = ( comp_C1593894019821074884nteger @ F4 @ G3 @ X7 ) ) ) ).

% comp_cong
thf(fact_428_comp__cong,axiom,
    ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,X: code_integer,F4: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G3: code_integer > code_integer > code_integer,X7: code_integer] :
      ( ( ( F @ ( G @ X ) )
        = ( F4 @ ( G3 @ X7 ) ) )
     => ( ( comp_C8797469213163452608nteger @ F @ G @ X )
        = ( comp_C8797469213163452608nteger @ F4 @ G3 @ X7 ) ) ) ).

% comp_cong
thf(fact_429_comp__cong,axiom,
    ! [F: nat > num,G: int > nat,X: int,F4: nat > num,G3: int > nat,X7: int] :
      ( ( ( F @ ( G @ X ) )
        = ( F4 @ ( G3 @ X7 ) ) )
     => ( ( comp_nat_num_int @ F @ G @ X )
        = ( comp_nat_num_int @ F4 @ G3 @ X7 ) ) ) ).

% comp_cong
thf(fact_430_curryI,axiom,
    ! [F: produc1908205239877642774nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( F @ ( produc8603105652947943368nteger @ A @ B ) )
     => ( produc1217013493180205570eger_o @ F @ A @ B ) ) ).

% curryI
thf(fact_431_curryI,axiom,
    ! [F: produc3925858234332021118et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( F @ ( produc5001842942810119800et_nat @ A @ B ) )
     => ( produc6216949301066131538_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_432_curryI,axiom,
    ! [F: produc2732055786443039994et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
      ( ( F @ ( produc2245416461498447860et_nat @ A @ B ) )
     => ( produc5101573711933517782_nat_o @ F @ A @ B ) ) ).

% curryI
thf(fact_433_curryI,axiom,
    ! [F: produc2285326912895808259nt_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ ( produc5700946648718959541nt_int @ A @ B ) )
     => ( produc730925184835016917_int_o @ F @ A @ B ) ) ).

% curryI
thf(fact_434_curryI,axiom,
    ! [F: produc7773217078559923341nt_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ ( produc4305682042979456191nt_int @ A @ B ) )
     => ( produc9098658269643458507_int_o @ F @ A @ B ) ) ).

% curryI
thf(fact_435_curryE,axiom,
    ! [F: produc1908205239877642774nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc1217013493180205570eger_o @ F @ A @ B )
     => ( F @ ( produc8603105652947943368nteger @ A @ B ) ) ) ).

% curryE
thf(fact_436_curryE,axiom,
    ! [F: produc3925858234332021118et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( produc6216949301066131538_nat_o @ F @ A @ B )
     => ( F @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ).

% curryE
thf(fact_437_curryE,axiom,
    ! [F: produc2732055786443039994et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
      ( ( produc5101573711933517782_nat_o @ F @ A @ B )
     => ( F @ ( produc2245416461498447860et_nat @ A @ B ) ) ) ).

% curryE
thf(fact_438_curryE,axiom,
    ! [F: produc2285326912895808259nt_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc730925184835016917_int_o @ F @ A @ B )
     => ( F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).

% curryE
thf(fact_439_curryE,axiom,
    ! [F: produc7773217078559923341nt_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc9098658269643458507_int_o @ F @ A @ B )
     => ( F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).

% curryE
thf(fact_440_curryD,axiom,
    ! [F: produc1908205239877642774nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc1217013493180205570eger_o @ F @ A @ B )
     => ( F @ ( produc8603105652947943368nteger @ A @ B ) ) ) ).

% curryD
thf(fact_441_curryD,axiom,
    ! [F: produc3925858234332021118et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( produc6216949301066131538_nat_o @ F @ A @ B )
     => ( F @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ).

% curryD
thf(fact_442_curryD,axiom,
    ! [F: produc2732055786443039994et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
      ( ( produc5101573711933517782_nat_o @ F @ A @ B )
     => ( F @ ( produc2245416461498447860et_nat @ A @ B ) ) ) ).

% curryD
thf(fact_443_curryD,axiom,
    ! [F: produc2285326912895808259nt_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc730925184835016917_int_o @ F @ A @ B )
     => ( F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).

% curryD
thf(fact_444_curryD,axiom,
    ! [F: produc7773217078559923341nt_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc9098658269643458507_int_o @ F @ A @ B )
     => ( F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).

% curryD
thf(fact_445_in__lex__prod,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc2922128104949294807at_nat @ A4 @ B3 ) ) @ ( lex_pr8000699672184251119at_nat @ R2 @ S ) )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_446_in__lex__prod,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: nat,A4: set_Pr1261947904930325089at_nat,B3: nat,R2: set_Pr4329608150637261639at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member3480088626692287306at_nat @ ( produc3585720979125124569at_nat @ ( produc3149970401713881818at_nat @ A @ B ) @ ( produc3149970401713881818at_nat @ A4 @ B3 ) ) @ ( lex_pr1951919194703979714at_nat @ R2 @ S ) )
      = ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_447_in__lex__prod,axiom,
    ! [A: nat,B: set_Pr1261947904930325089at_nat,A4: nat,B3: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( member4107301959146914710at_nat @ ( produc8824474337961625893at_nat @ ( produc5463602008962177208at_nat @ A @ B ) @ ( produc5463602008962177208at_nat @ A4 @ B3 ) ) @ ( lex_pr4265550801952275104at_nat @ R2 @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_448_in__lex__prod,axiom,
    ! [A: nat,B: nat,A4: nat,B3: nat,R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ A4 @ B3 ) ) @ ( lex_prod_nat_nat @ R2 @ S ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_449_in__lex__prod,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A4: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger,R2: set_Pr6418322821503574069e_term,S: set_Pr5052581263052047815nteger] :
      ( ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ A @ B ) @ ( produc8603105652947943368nteger @ A4 @ B3 ) ) @ ( lex_pr3763927252266284464nteger @ R2 @ S ) )
      = ( ( member6786453453753086846e_term @ ( produc6915061378690470669e_term @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member335077259223437328nteger @ ( produc4911887526399440471nteger @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_450_in__lex__prod,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A4: produc3658429121746597890et_nat > $o,B3: produc3658429121746597890et_nat,R2: set_Pr2161125870931222855_nat_o,S: set_Pr719794911490849221et_nat] :
      ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ ( produc5001842942810119800et_nat @ A4 @ B3 ) ) @ ( lex_pr5340572901959109728et_nat @ R2 @ S ) )
      = ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_451_in__lex__prod,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A4: produc3658429121746597890et_nat > $o,B3: produc3925858234332021118et_nat,R2: set_Pr2161125870931222855_nat_o,S: set_Pr7928877670098842301et_nat] :
      ( ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ ( produc2245416461498447860et_nat @ A4 @ B3 ) ) @ ( lex_pr4722427456421979612et_nat @ R2 @ S ) )
      = ( ( member8781333585448626064_nat_o @ ( produc7368190662567826135_nat_o @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_452_in__lex__prod,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A4: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int,R2: set_Pr4675205556198584599e_term,S: set_Pr2560585780119916871nt_int] :
      ( ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ ( produc5700946648718959541nt_int @ A4 @ B3 ) ) @ ( lex_pr4548132154224949661nt_int @ R2 @ S ) )
      = ( ( member8796044733138646368e_term @ ( produc8124735558134971759e_term @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_453_in__lex__prod,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A4: int > option6357759511663192854e_term,B3: product_prod_int_int,R2: set_Pr7140365069852192003e_term,S: set_Pr2560585780119916871nt_int] :
      ( ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ ( produc4305682042979456191nt_int @ A4 @ B3 ) ) @ ( lex_pr7234932505866811047nt_int @ R2 @ S ) )
      = ( ( member6538663773838357324e_term @ ( produc4510790414113558107e_term @ A @ A4 ) @ R2 )
        | ( ( A = A4 )
          & ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ B @ B3 ) @ S ) ) ) ) ).

% in_lex_prod
thf(fact_454_rewriteR__comp__comp2,axiom,
    ! [G: int > nat,H3: int > int,R1: nat > nat,R22: int > nat,F: nat > num,L: nat > num] :
      ( ( ( comp_int_nat_int @ G @ H3 )
        = ( comp_nat_nat_int @ R1 @ R22 ) )
     => ( ( ( comp_nat_num_nat @ F @ R1 )
          = L )
       => ( ( comp_int_num_int @ ( comp_nat_num_int @ F @ G ) @ H3 )
          = ( comp_nat_num_int @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_455_rewriteR__comp__comp2,axiom,
    ! [G: nat > nat,H3: int > nat,R1: int > nat,R22: int > int,F: nat > num,L: int > num] :
      ( ( ( comp_nat_nat_int @ G @ H3 )
        = ( comp_int_nat_int @ R1 @ R22 ) )
     => ( ( ( comp_nat_num_int @ F @ R1 )
          = L )
       => ( ( comp_nat_num_int @ ( comp_nat_num_nat @ F @ G ) @ H3 )
          = ( comp_int_num_int @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_456_rewriteR__comp__comp2,axiom,
    ! [G: num > code_integer,H3: num > num,R1: code_integer > code_integer,R22: num > code_integer,F: code_integer > code_integer,L: code_integer > code_integer] :
      ( ( ( comp_n6909714581898973814er_num @ G @ H3 )
        = ( comp_C3531382070062128313er_num @ R1 @ R22 ) )
     => ( ( ( comp_C7449957260575251196nteger @ F @ R1 )
          = L )
       => ( ( comp_n6909714581898973814er_num @ ( comp_C3531382070062128313er_num @ F @ G ) @ H3 )
          = ( comp_C3531382070062128313er_num @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_457_rewriteR__comp__comp2,axiom,
    ! [G: code_integer > code_integer,H3: num > code_integer,R1: num > code_integer,R22: num > num,F: code_integer > code_integer,L: num > code_integer] :
      ( ( ( comp_C3531382070062128313er_num @ G @ H3 )
        = ( comp_n6909714581898973814er_num @ R1 @ R22 ) )
     => ( ( ( comp_C3531382070062128313er_num @ F @ R1 )
          = L )
       => ( ( comp_C3531382070062128313er_num @ ( comp_C7449957260575251196nteger @ F @ G ) @ H3 )
          = ( comp_n6909714581898973814er_num @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_458_rewriteR__comp__comp2,axiom,
    ! [G: code_integer > code_integer,H3: num > code_integer,R1: code_integer > code_integer,R22: num > code_integer,F: code_integer > code_integer,L: code_integer > code_integer] :
      ( ( ( comp_C3531382070062128313er_num @ G @ H3 )
        = ( comp_C3531382070062128313er_num @ R1 @ R22 ) )
     => ( ( ( comp_C7449957260575251196nteger @ F @ R1 )
          = L )
       => ( ( comp_C3531382070062128313er_num @ ( comp_C7449957260575251196nteger @ F @ G ) @ H3 )
          = ( comp_C3531382070062128313er_num @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_459_rewriteR__comp__comp2,axiom,
    ! [G: nat > num,H3: int > nat,R1: nat > num,R22: int > nat,F: num > num,L: nat > num] :
      ( ( ( comp_nat_num_int @ G @ H3 )
        = ( comp_nat_num_int @ R1 @ R22 ) )
     => ( ( ( comp_num_num_nat @ F @ R1 )
          = L )
       => ( ( comp_nat_num_int @ ( comp_num_num_nat @ F @ G ) @ H3 )
          = ( comp_nat_num_int @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_460_rewriteR__comp__comp2,axiom,
    ! [G: code_integer > code_integer,H3: code_integer > code_integer,R1: code_integer > code_integer,R22: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,L: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( comp_C7449957260575251196nteger @ G @ H3 )
        = ( comp_C7449957260575251196nteger @ R1 @ R22 ) )
     => ( ( ( comp_C1593894019821074884nteger @ F @ R1 )
          = L )
       => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ G ) @ H3 )
          = ( comp_C1593894019821074884nteger @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_461_rewriteR__comp__comp2,axiom,
    ! [G: code_integer > code_integer,H3: num > code_integer,R1: code_integer > code_integer,R22: num > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,L: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( comp_C3531382070062128313er_num @ G @ H3 )
        = ( comp_C3531382070062128313er_num @ R1 @ R22 ) )
     => ( ( ( comp_C1593894019821074884nteger @ F @ R1 )
          = L )
       => ( ( comp_C6617437204949736321er_num @ ( comp_C1593894019821074884nteger @ F @ G ) @ H3 )
          = ( comp_C6617437204949736321er_num @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_462_rewriteR__comp__comp2,axiom,
    ! [G: code_integer > code_integer,H3: code_integer > code_integer,R1: ( code_integer > code_integer ) > code_integer,R22: code_integer > code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,L: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( comp_C7449957260575251196nteger @ G @ H3 )
        = ( comp_C5076291335850163704nteger @ R1 @ R22 ) )
     => ( ( ( comp_C95226695241045696nteger @ F @ R1 )
          = L )
       => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ G ) @ H3 )
          = ( comp_C8797469213163452608nteger @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_463_rewriteR__comp__comp2,axiom,
    ! [G: ( code_integer > code_integer ) > code_integer,H3: code_integer > code_integer > code_integer,R1: code_integer > code_integer,R22: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,L: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( comp_C5076291335850163704nteger @ G @ H3 )
        = ( comp_C7449957260575251196nteger @ R1 @ R22 ) )
     => ( ( ( comp_C1593894019821074884nteger @ F @ R1 )
          = L )
       => ( ( comp_C8797469213163452608nteger @ ( comp_C95226695241045696nteger @ F @ G ) @ H3 )
          = ( comp_C1593894019821074884nteger @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_464_rewriteL__comp__comp2,axiom,
    ! [F: num > num,G: nat > num,L1: nat > num,L2: nat > nat,H3: int > nat,R2: int > nat] :
      ( ( ( comp_num_num_nat @ F @ G )
        = ( comp_nat_num_nat @ L1 @ L2 ) )
     => ( ( ( comp_nat_nat_int @ L2 @ H3 )
          = R2 )
       => ( ( comp_num_num_int @ F @ ( comp_nat_num_int @ G @ H3 ) )
          = ( comp_nat_num_int @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_465_rewriteL__comp__comp2,axiom,
    ! [F: code_integer > code_integer,G: code_integer > code_integer,L1: code_integer > code_integer,L2: code_integer > code_integer,H3: num > code_integer,R2: num > code_integer] :
      ( ( ( comp_C7449957260575251196nteger @ F @ G )
        = ( comp_C7449957260575251196nteger @ L1 @ L2 ) )
     => ( ( ( comp_C3531382070062128313er_num @ L2 @ H3 )
          = R2 )
       => ( ( comp_C3531382070062128313er_num @ F @ ( comp_C3531382070062128313er_num @ G @ H3 ) )
          = ( comp_C3531382070062128313er_num @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_466_rewriteL__comp__comp2,axiom,
    ! [F: nat > num,G: nat > nat,L1: num > num,L2: nat > num,H3: int > nat,R2: int > num] :
      ( ( ( comp_nat_num_nat @ F @ G )
        = ( comp_num_num_nat @ L1 @ L2 ) )
     => ( ( ( comp_nat_num_int @ L2 @ H3 )
          = R2 )
       => ( ( comp_nat_num_int @ F @ ( comp_nat_nat_int @ G @ H3 ) )
          = ( comp_num_num_int @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_467_rewriteL__comp__comp2,axiom,
    ! [F: code_integer > code_integer,G: num > code_integer,L1: code_integer > code_integer,L2: num > code_integer,H3: num > num,R2: num > code_integer] :
      ( ( ( comp_C3531382070062128313er_num @ F @ G )
        = ( comp_C3531382070062128313er_num @ L1 @ L2 ) )
     => ( ( ( comp_n6909714581898973814er_num @ L2 @ H3 )
          = R2 )
       => ( ( comp_C3531382070062128313er_num @ F @ ( comp_n6909714581898973814er_num @ G @ H3 ) )
          = ( comp_C3531382070062128313er_num @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_468_rewriteL__comp__comp2,axiom,
    ! [F: nat > num,G: int > nat,L1: nat > num,L2: int > nat,H3: int > int,R2: int > nat] :
      ( ( ( comp_nat_num_int @ F @ G )
        = ( comp_nat_num_int @ L1 @ L2 ) )
     => ( ( ( comp_int_nat_int @ L2 @ H3 )
          = R2 )
       => ( ( comp_nat_num_int @ F @ ( comp_int_nat_int @ G @ H3 ) )
          = ( comp_nat_num_int @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_469_rewriteL__comp__comp2,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,L1: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,L2: code_integer > code_integer,H3: code_integer > code_integer,R2: code_integer > code_integer] :
      ( ( ( comp_C1593894019821074884nteger @ F @ G )
        = ( comp_C1593894019821074884nteger @ L1 @ L2 ) )
     => ( ( ( comp_C7449957260575251196nteger @ L2 @ H3 )
          = R2 )
       => ( ( comp_C1593894019821074884nteger @ F @ ( comp_C7449957260575251196nteger @ G @ H3 ) )
          = ( comp_C1593894019821074884nteger @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_470_rewriteL__comp__comp2,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,L1: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,L2: code_integer > code_integer,H3: num > code_integer,R2: num > code_integer] :
      ( ( ( comp_C1593894019821074884nteger @ F @ G )
        = ( comp_C1593894019821074884nteger @ L1 @ L2 ) )
     => ( ( ( comp_C3531382070062128313er_num @ L2 @ H3 )
          = R2 )
       => ( ( comp_C6617437204949736321er_num @ F @ ( comp_C3531382070062128313er_num @ G @ H3 ) )
          = ( comp_C6617437204949736321er_num @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_471_rewriteL__comp__comp2,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,L1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,L2: code_integer > code_integer > code_integer,H3: num > code_integer,R2: num > code_integer > code_integer] :
      ( ( ( comp_C1593894019821074884nteger @ F @ G )
        = ( comp_C8797469213163452608nteger @ L1 @ L2 ) )
     => ( ( ( comp_C3662112809048933045er_num @ L2 @ H3 )
          = R2 )
       => ( ( comp_C6617437204949736321er_num @ F @ ( comp_C3531382070062128313er_num @ G @ H3 ) )
          = ( comp_C8645541002826910333er_num @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_472_rewriteL__comp__comp2,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,L1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,L2: code_integer > code_integer > code_integer,H3: code_integer > code_integer,R2: code_integer > code_integer > code_integer] :
      ( ( ( comp_C1593894019821074884nteger @ F @ G )
        = ( comp_C8797469213163452608nteger @ L1 @ L2 ) )
     => ( ( ( comp_C2931745410781319416nteger @ L2 @ H3 )
          = R2 )
       => ( ( comp_C1593894019821074884nteger @ F @ ( comp_C7449957260575251196nteger @ G @ H3 ) )
          = ( comp_C8797469213163452608nteger @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_473_rewriteL__comp__comp2,axiom,
    ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,L1: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,L2: code_integer > code_integer,H3: code_integer > code_integer,R2: code_integer > code_integer] :
      ( ( ( comp_C8797469213163452608nteger @ F @ G )
        = ( comp_C1593894019821074884nteger @ L1 @ L2 ) )
     => ( ( ( comp_C7449957260575251196nteger @ L2 @ H3 )
          = R2 )
       => ( ( comp_C8797469213163452608nteger @ F @ ( comp_C2931745410781319416nteger @ G @ H3 ) )
          = ( comp_C1593894019821074884nteger @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_474_rewriteR__comp__comp,axiom,
    ! [G: num > code_integer,H3: num > num,R2: num > code_integer,F: code_integer > code_integer] :
      ( ( ( comp_n6909714581898973814er_num @ G @ H3 )
        = R2 )
     => ( ( comp_n6909714581898973814er_num @ ( comp_C3531382070062128313er_num @ F @ G ) @ H3 )
        = ( comp_C3531382070062128313er_num @ F @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_475_rewriteR__comp__comp,axiom,
    ! [G: int > nat,H3: int > int,R2: int > nat,F: nat > num] :
      ( ( ( comp_int_nat_int @ G @ H3 )
        = R2 )
     => ( ( comp_int_num_int @ ( comp_nat_num_int @ F @ G ) @ H3 )
        = ( comp_nat_num_int @ F @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_476_rewriteR__comp__comp,axiom,
    ! [G: nat > nat,H3: int > nat,R2: int > nat,F: nat > num] :
      ( ( ( comp_nat_nat_int @ G @ H3 )
        = R2 )
     => ( ( comp_nat_num_int @ ( comp_nat_num_nat @ F @ G ) @ H3 )
        = ( comp_nat_num_int @ F @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_477_rewriteR__comp__comp,axiom,
    ! [G: code_integer > code_integer,H3: num > code_integer,R2: num > code_integer,F: code_integer > code_integer] :
      ( ( ( comp_C3531382070062128313er_num @ G @ H3 )
        = R2 )
     => ( ( comp_C3531382070062128313er_num @ ( comp_C7449957260575251196nteger @ F @ G ) @ H3 )
        = ( comp_C3531382070062128313er_num @ F @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_478_rewriteR__comp__comp,axiom,
    ! [G: nat > num,H3: int > nat,R2: int > num,F: num > num] :
      ( ( ( comp_nat_num_int @ G @ H3 )
        = R2 )
     => ( ( comp_nat_num_int @ ( comp_num_num_nat @ F @ G ) @ H3 )
        = ( comp_num_num_int @ F @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_479_rewriteR__comp__comp,axiom,
    ! [G: code_integer > code_integer,H3: code_integer > code_integer,R2: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( comp_C7449957260575251196nteger @ G @ H3 )
        = R2 )
     => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ G ) @ H3 )
        = ( comp_C1593894019821074884nteger @ F @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_480_rewriteR__comp__comp,axiom,
    ! [G: code_integer > code_integer,H3: num > code_integer,R2: num > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( comp_C3531382070062128313er_num @ G @ H3 )
        = R2 )
     => ( ( comp_C6617437204949736321er_num @ ( comp_C1593894019821074884nteger @ F @ G ) @ H3 )
        = ( comp_C6617437204949736321er_num @ F @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_481_rewriteR__comp__comp,axiom,
    ! [G: code_integer > code_integer > code_integer,H3: code_integer > code_integer,R2: code_integer > code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( comp_C2931745410781319416nteger @ G @ H3 )
        = R2 )
     => ( ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ F @ G ) @ H3 )
        = ( comp_C8797469213163452608nteger @ F @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_482_rewriteR__comp__comp,axiom,
    ! [G: ( code_integer > code_integer ) > code_integer,H3: code_integer > code_integer > code_integer,R2: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( comp_C5076291335850163704nteger @ G @ H3 )
        = R2 )
     => ( ( comp_C8797469213163452608nteger @ ( comp_C95226695241045696nteger @ F @ G ) @ H3 )
        = ( comp_C1593894019821074884nteger @ F @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_483_rewriteR__comp__comp,axiom,
    ! [G: ( code_integer > code_integer ) > code_integer > code_integer,H3: code_integer > code_integer > code_integer,R2: code_integer > code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( comp_C2822359487432382964nteger @ G @ H3 )
        = R2 )
     => ( ( comp_C8797469213163452608nteger @ ( comp_C3983183376822107068nteger @ F @ G ) @ H3 )
        = ( comp_C8797469213163452608nteger @ F @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_484_rewriteL__comp__comp,axiom,
    ! [F: num > num,G: nat > num,L: nat > num,H3: int > nat] :
      ( ( ( comp_num_num_nat @ F @ G )
        = L )
     => ( ( comp_num_num_int @ F @ ( comp_nat_num_int @ G @ H3 ) )
        = ( comp_nat_num_int @ L @ H3 ) ) ) ).

% rewriteL_comp_comp
thf(fact_485_rewriteL__comp__comp,axiom,
    ! [F: code_integer > code_integer,G: code_integer > code_integer,L: code_integer > code_integer,H3: num > code_integer] :
      ( ( ( comp_C7449957260575251196nteger @ F @ G )
        = L )
     => ( ( comp_C3531382070062128313er_num @ F @ ( comp_C3531382070062128313er_num @ G @ H3 ) )
        = ( comp_C3531382070062128313er_num @ L @ H3 ) ) ) ).

% rewriteL_comp_comp
thf(fact_486_rewriteL__comp__comp,axiom,
    ! [F: nat > num,G: nat > nat,L: nat > num,H3: int > nat] :
      ( ( ( comp_nat_num_nat @ F @ G )
        = L )
     => ( ( comp_nat_num_int @ F @ ( comp_nat_nat_int @ G @ H3 ) )
        = ( comp_nat_num_int @ L @ H3 ) ) ) ).

% rewriteL_comp_comp
thf(fact_487_rewriteL__comp__comp,axiom,
    ! [F: code_integer > code_integer,G: num > code_integer,L: num > code_integer,H3: num > num] :
      ( ( ( comp_C3531382070062128313er_num @ F @ G )
        = L )
     => ( ( comp_C3531382070062128313er_num @ F @ ( comp_n6909714581898973814er_num @ G @ H3 ) )
        = ( comp_n6909714581898973814er_num @ L @ H3 ) ) ) ).

% rewriteL_comp_comp
thf(fact_488_rewriteL__comp__comp,axiom,
    ! [F: nat > num,G: int > nat,L: int > num,H3: int > int] :
      ( ( ( comp_nat_num_int @ F @ G )
        = L )
     => ( ( comp_nat_num_int @ F @ ( comp_int_nat_int @ G @ H3 ) )
        = ( comp_int_num_int @ L @ H3 ) ) ) ).

% rewriteL_comp_comp
thf(fact_489_rewriteL__comp__comp,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,L: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,H3: num > code_integer] :
      ( ( ( comp_C1593894019821074884nteger @ F @ G )
        = L )
     => ( ( comp_C6617437204949736321er_num @ F @ ( comp_C3531382070062128313er_num @ G @ H3 ) )
        = ( comp_C6617437204949736321er_num @ L @ H3 ) ) ) ).

% rewriteL_comp_comp
thf(fact_490_rewriteL__comp__comp,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,L: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,H3: code_integer > code_integer] :
      ( ( ( comp_C1593894019821074884nteger @ F @ G )
        = L )
     => ( ( comp_C1593894019821074884nteger @ F @ ( comp_C7449957260575251196nteger @ G @ H3 ) )
        = ( comp_C1593894019821074884nteger @ L @ H3 ) ) ) ).

% rewriteL_comp_comp
thf(fact_491_rewriteL__comp__comp,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > code_integer,L: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,H3: code_integer > code_integer > code_integer] :
      ( ( ( comp_C95226695241045696nteger @ F @ G )
        = L )
     => ( ( comp_C1593894019821074884nteger @ F @ ( comp_C5076291335850163704nteger @ G @ H3 ) )
        = ( comp_C8797469213163452608nteger @ L @ H3 ) ) ) ).

% rewriteL_comp_comp
thf(fact_492_rewriteL__comp__comp,axiom,
    ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,L: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,H3: code_integer > code_integer] :
      ( ( ( comp_C8797469213163452608nteger @ F @ G )
        = L )
     => ( ( comp_C8797469213163452608nteger @ F @ ( comp_C2931745410781319416nteger @ G @ H3 ) )
        = ( comp_C1593894019821074884nteger @ L @ H3 ) ) ) ).

% rewriteL_comp_comp
thf(fact_493_rewriteL__comp__comp,axiom,
    ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > code_integer > code_integer,L: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,H3: code_integer > code_integer > code_integer] :
      ( ( ( comp_C3983183376822107068nteger @ F @ G )
        = L )
     => ( ( comp_C8797469213163452608nteger @ F @ ( comp_C2822359487432382964nteger @ G @ H3 ) )
        = ( comp_C8797469213163452608nteger @ L @ H3 ) ) ) ).

% rewriteL_comp_comp
thf(fact_494_type__copy__map__cong0,axiom,
    ! [M3: num > code_integer,G: num > num,X: num,N6: code_integer > code_integer,H3: num > code_integer,F: code_integer > code_integer] :
      ( ( ( M3 @ ( G @ X ) )
        = ( N6 @ ( H3 @ X ) ) )
     => ( ( comp_n6909714581898973814er_num @ ( comp_C3531382070062128313er_num @ F @ M3 ) @ G @ X )
        = ( comp_C3531382070062128313er_num @ ( comp_C7449957260575251196nteger @ F @ N6 ) @ H3 @ X ) ) ) ).

% type_copy_map_cong0
thf(fact_495_type__copy__map__cong0,axiom,
    ! [M3: int > nat,G: int > int,X: int,N6: nat > nat,H3: int > nat,F: nat > num] :
      ( ( ( M3 @ ( G @ X ) )
        = ( N6 @ ( H3 @ X ) ) )
     => ( ( comp_int_num_int @ ( comp_nat_num_int @ F @ M3 ) @ G @ X )
        = ( comp_nat_num_int @ ( comp_nat_num_nat @ F @ N6 ) @ H3 @ X ) ) ) ).

% type_copy_map_cong0
thf(fact_496_type__copy__map__cong0,axiom,
    ! [M3: code_integer > code_integer,G: num > code_integer,X: num,N6: num > code_integer,H3: num > num,F: code_integer > code_integer] :
      ( ( ( M3 @ ( G @ X ) )
        = ( N6 @ ( H3 @ X ) ) )
     => ( ( comp_C3531382070062128313er_num @ ( comp_C7449957260575251196nteger @ F @ M3 ) @ G @ X )
        = ( comp_n6909714581898973814er_num @ ( comp_C3531382070062128313er_num @ F @ N6 ) @ H3 @ X ) ) ) ).

% type_copy_map_cong0
thf(fact_497_type__copy__map__cong0,axiom,
    ! [M3: code_integer > code_integer,G: code_integer > code_integer,X: code_integer,N6: code_integer > code_integer,H3: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( M3 @ ( G @ X ) )
        = ( N6 @ ( H3 @ X ) ) )
     => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M3 ) @ G @ X )
        = ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ N6 ) @ H3 @ X ) ) ) ).

% type_copy_map_cong0
thf(fact_498_type__copy__map__cong0,axiom,
    ! [M3: code_integer > code_integer,G: code_integer > code_integer,X: code_integer,N6: ( code_integer > code_integer ) > code_integer,H3: code_integer > code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( M3 @ ( G @ X ) )
        = ( N6 @ ( H3 @ X ) ) )
     => ( ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ M3 ) @ G @ X )
        = ( comp_C8797469213163452608nteger @ ( comp_C95226695241045696nteger @ F @ N6 ) @ H3 @ X ) ) ) ).

% type_copy_map_cong0
thf(fact_499_type__copy__map__cong0,axiom,
    ! [M3: code_integer > code_integer > code_integer,G: code_integer > code_integer,X: code_integer,N6: code_integer > code_integer > code_integer,H3: code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( M3 @ ( G @ X ) )
        = ( N6 @ ( H3 @ X ) ) )
     => ( ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ F @ M3 ) @ G @ X )
        = ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ F @ N6 ) @ H3 @ X ) ) ) ).

% type_copy_map_cong0
thf(fact_500_type__copy__map__cong0,axiom,
    ! [M3: code_integer > code_integer > code_integer,G: code_integer > code_integer,X: code_integer,N6: ( code_integer > code_integer ) > code_integer > code_integer,H3: code_integer > code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( M3 @ ( G @ X ) )
        = ( N6 @ ( H3 @ X ) ) )
     => ( ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ F @ M3 ) @ G @ X )
        = ( comp_C8797469213163452608nteger @ ( comp_C3983183376822107068nteger @ F @ N6 ) @ H3 @ X ) ) ) ).

% type_copy_map_cong0
thf(fact_501_type__copy__map__cong0,axiom,
    ! [M3: ( code_integer > code_integer ) > code_integer,G: code_integer > code_integer > code_integer,X: code_integer,N6: code_integer > code_integer,H3: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( M3 @ ( G @ X ) )
        = ( N6 @ ( H3 @ X ) ) )
     => ( ( comp_C8797469213163452608nteger @ ( comp_C95226695241045696nteger @ F @ M3 ) @ G @ X )
        = ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ F @ N6 ) @ H3 @ X ) ) ) ).

% type_copy_map_cong0
thf(fact_502_type__copy__map__cong0,axiom,
    ! [M3: ( code_integer > code_integer ) > code_integer > code_integer,G: code_integer > code_integer > code_integer,X: code_integer,N6: code_integer > code_integer > code_integer,H3: code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( M3 @ ( G @ X ) )
        = ( N6 @ ( H3 @ X ) ) )
     => ( ( comp_C8797469213163452608nteger @ ( comp_C3983183376822107068nteger @ F @ M3 ) @ G @ X )
        = ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ F @ N6 ) @ H3 @ X ) ) ) ).

% type_copy_map_cong0
thf(fact_503_type__copy__map__cong0,axiom,
    ! [M3: nat > nat,G: int > nat,X: int,N6: int > nat,H3: int > int,F: nat > num] :
      ( ( ( M3 @ ( G @ X ) )
        = ( N6 @ ( H3 @ X ) ) )
     => ( ( comp_nat_num_int @ ( comp_nat_num_nat @ F @ M3 ) @ G @ X )
        = ( comp_int_num_int @ ( comp_nat_num_int @ F @ N6 ) @ H3 @ X ) ) ) ).

% type_copy_map_cong0
thf(fact_504_comp__apply__eq,axiom,
    ! [F: code_integer > code_integer,G: num > code_integer,X: num,H3: code_integer > code_integer,K: num > code_integer] :
      ( ( ( F @ ( G @ X ) )
        = ( H3 @ ( K @ X ) ) )
     => ( ( comp_C3531382070062128313er_num @ F @ G @ X )
        = ( comp_C3531382070062128313er_num @ H3 @ K @ X ) ) ) ).

% comp_apply_eq
thf(fact_505_comp__apply__eq,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,X: code_integer,H3: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,K: code_integer > code_integer] :
      ( ( ( F @ ( G @ X ) )
        = ( H3 @ ( K @ X ) ) )
     => ( ( comp_C1593894019821074884nteger @ F @ G @ X )
        = ( comp_C1593894019821074884nteger @ H3 @ K @ X ) ) ) ).

% comp_apply_eq
thf(fact_506_comp__apply__eq,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,X: code_integer,H3: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,K: code_integer > code_integer > code_integer] :
      ( ( ( F @ ( G @ X ) )
        = ( H3 @ ( K @ X ) ) )
     => ( ( comp_C1593894019821074884nteger @ F @ G @ X )
        = ( comp_C8797469213163452608nteger @ H3 @ K @ X ) ) ) ).

% comp_apply_eq
thf(fact_507_comp__apply__eq,axiom,
    ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,X: code_integer,H3: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,K: code_integer > code_integer] :
      ( ( ( F @ ( G @ X ) )
        = ( H3 @ ( K @ X ) ) )
     => ( ( comp_C8797469213163452608nteger @ F @ G @ X )
        = ( comp_C1593894019821074884nteger @ H3 @ K @ X ) ) ) ).

% comp_apply_eq
thf(fact_508_comp__apply__eq,axiom,
    ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,X: code_integer,H3: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,K: code_integer > code_integer > code_integer] :
      ( ( ( F @ ( G @ X ) )
        = ( H3 @ ( K @ X ) ) )
     => ( ( comp_C8797469213163452608nteger @ F @ G @ X )
        = ( comp_C8797469213163452608nteger @ H3 @ K @ X ) ) ) ).

% comp_apply_eq
thf(fact_509_comp__apply__eq,axiom,
    ! [F: nat > num,G: int > nat,X: int,H3: nat > num,K: int > nat] :
      ( ( ( F @ ( G @ X ) )
        = ( H3 @ ( K @ X ) ) )
     => ( ( comp_nat_num_int @ F @ G @ X )
        = ( comp_nat_num_int @ H3 @ K @ X ) ) ) ).

% comp_apply_eq
thf(fact_510_add_Oright__commute,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ ( plus_p7104986032573967614at_nat @ A @ B ) @ C )
      = ( plus_p7104986032573967614at_nat @ ( plus_p7104986032573967614at_nat @ A @ C ) @ B ) ) ).

% add.right_commute
thf(fact_511_add_Oright__commute,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% add.right_commute
thf(fact_512_add_Oright__commute,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ).

% add.right_commute
thf(fact_513_add_Oright__commute,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% add.right_commute
thf(fact_514_add__less__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
      = ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_515_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_516_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_517_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_518_add__less__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
      = ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_519_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_520_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_521_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_522_add__less__imp__less__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
     => ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_523_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_524_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_525_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_526_add__less__imp__less__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
     => ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_527_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_528_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_529_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_530_add__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_531_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_532_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_533_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_534_add__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ C @ A ) @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_535_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_536_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_537_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_538_add__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_539_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_540_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_541_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_542_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le6747313008572928689nteger @ I @ J )
        & ( K = L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_543_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( K = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_544_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_545_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_546_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( I = J )
        & ( ord_le6747313008572928689nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_547_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_548_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_549_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_550_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le6747313008572928689nteger @ I @ J )
        & ( ord_le6747313008572928689nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_551_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_552_add__mono__thms__linordered__field_I5_J,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 ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_553_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_554_success__returnI,axiom,
    ! [X: product_unit,H3: heap_e7401611519738050253t_unit] : ( heap_T584514906347983379t_unit @ ( heap_T7507251653302230130t_unit @ X ) @ H3 ) ).

% success_returnI
thf(fact_555_success__returnI,axiom,
    ! [X: array_a,H3: heap_e7401611519738050253t_unit] : ( heap_T6932021131025641745rray_a @ ( heap_T4896204969850773234rray_a @ X ) @ H3 ) ).

% success_returnI
thf(fact_556_fails__return,axiom,
    ! [X: product_unit,H3: heap_e7401611519738050253t_unit] :
      ~ ( time_f8834461667527620124t_unit @ ( heap_T7507251653302230130t_unit @ X ) @ H3 ) ).

% fails_return
thf(fact_557_fails__return,axiom,
    ! [X: array_a,H3: heap_e7401611519738050253t_unit] :
      ~ ( time_fails_array_a @ ( heap_T4896204969850773234rray_a @ X ) @ H3 ) ).

% fails_return
thf(fact_558_add_Oright__assoc,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ ( plus_p7104986032573967614at_nat @ A @ B ) @ C )
      = ( plus_p7104986032573967614at_nat @ A @ ( plus_p7104986032573967614at_nat @ B @ C ) ) ) ).

% add.right_assoc
thf(fact_559_add_Oright__assoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% add.right_assoc
thf(fact_560_add_Oright__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.right_assoc
thf(fact_561_add_Oright__assoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% add.right_assoc
thf(fact_562_nat__add__left__cancel__less,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% nat_add_left_cancel_less
thf(fact_563_less__option__Some,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le7113747843092208513nteger @ ( some_Code_integer @ X ) @ ( some_Code_integer @ Y ) )
      = ( ord_le6747313008572928689nteger @ X @ Y ) ) ).

% less_option_Some
thf(fact_564_less__option__Some,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_option_rat @ ( some_rat @ X ) @ ( some_rat @ Y ) )
      = ( ord_less_rat @ X @ Y ) ) ).

% less_option_Some
thf(fact_565_less__option__Some,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_option_num @ ( some_num @ X ) @ ( some_num @ Y ) )
      = ( ord_less_num @ X @ Y ) ) ).

% less_option_Some
thf(fact_566_less__option__Some,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_option_nat @ ( some_nat @ X ) @ ( some_nat @ Y ) )
      = ( ord_less_nat @ X @ Y ) ) ).

% less_option_Some
thf(fact_567_less__option__Some,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_option_int @ ( some_int @ X ) @ ( some_int @ Y ) )
      = ( ord_less_int @ X @ Y ) ) ).

% less_option_Some
thf(fact_568_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X @ Xa @ Xb @ Xc )
        = Y )
     => ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) )
       => ~ ( ( ( ( ord_less_nat @ Xb @ Xa )
               => ( Y = Xc ) )
              & ( ~ ( ord_less_nat @ Xb @ Xa )
               => ( Y
                  = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) )
           => ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X @ ( produc487386426758144856at_nat @ Xa @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
thf(fact_569_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: nat > nat > nat,A: nat,B: nat,Acc: nat] :
      ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc ) ) ) )
     => ( ( ( ord_less_nat @ B @ A )
         => ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc )
            = Acc ) )
        & ( ~ ( ord_less_nat @ B @ A )
         => ( ( set_fo2584398358068434914at_nat @ F @ A @ B @ Acc )
            = ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F @ A @ Acc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
thf(fact_570_in__measure,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,F: set_Pr1261947904930325089at_nat > nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( measur1827424007717751593at_nat @ F ) )
      = ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_571_in__measure,axiom,
    ! [X: nat,Y: nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measure_nat @ F ) )
      = ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).

% in_measure
thf(fact_572_less__add__one,axiom,
    ! [A: code_integer] : ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) ) ).

% less_add_one
thf(fact_573_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_574_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_575_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_576_add__mono1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ B @ one_one_Code_integer ) ) ) ).

% add_mono1
thf(fact_577_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).

% add_mono1
thf(fact_578_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_579_add__mono1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).

% add_mono1
thf(fact_580_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
     => ( ord_less_nat @ I @ K ) ) ).

% add_lessD1
thf(fact_581_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 ) ) ) ) ).

% add_less_mono
thf(fact_582_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_583_less__option__None,axiom,
    ! [X: option_num] :
      ~ ( ord_less_option_num @ X @ none_num ) ).

% less_option_None
thf(fact_584_less__option__None__Some__code,axiom,
    ! [X: num] : ( ord_less_option_num @ none_num @ ( some_num @ X ) ) ).

% less_option_None_Some_code
thf(fact_585_less__option__None__Some,axiom,
    ! [X: num] : ( ord_less_option_num @ none_num @ ( some_num @ X ) ) ).

% less_option_None_Some
thf(fact_586_less__option__None__is__Some,axiom,
    ! [X: option_num] :
      ( ( ord_less_option_num @ none_num @ X )
     => ? [Z3: num] :
          ( X
          = ( some_num @ Z3 ) ) ) ).

% less_option_None_is_Some
thf(fact_587_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F2: nat > nat > nat,A5: nat,B4: nat,Acc2: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A5 ) @ Acc2 @ ( set_fo2584398358068434914at_nat @ F2 @ ( plus_plus_nat @ A5 @ one_one_nat ) @ B4 @ ( F2 @ A5 @ Acc2 ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_588_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X @ Xa @ Xb @ Xc )
        = Y )
     => ( ( ( ord_less_nat @ Xb @ Xa )
         => ( Y = Xc ) )
        & ( ~ ( ord_less_nat @ Xb @ Xa )
         => ( Y
            = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
thf(fact_589_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M2 @ L )
          = ( plus_plus_nat @ K @ N2 ) )
       => ( ord_less_nat @ M2 @ N2 ) ) ) ).

% less_add_eq_less
thf(fact_590_trans__less__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_less_add2
thf(fact_591_trans__less__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_less_add1
thf(fact_592_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 ) ) ) ).

% add_less_mono1
thf(fact_593_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_594_less__numeral__extra_I4_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ one_one_Code_integer ) ).

% less_numeral_extra(4)
thf(fact_595_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_596_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_597_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_598_mlex__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,F: set_Pr1261947904930325089at_nat > nat,R4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( mlex_p3502713205431853936at_nat @ F @ R4 ) )
      = ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
        | ( ( ( F @ X )
            = ( F @ Y ) )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R4 ) ) ) ) ).

% mlex_iff
thf(fact_599_mlex__iff,axiom,
    ! [X: nat,Y: nat,F: nat > nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( mlex_prod_nat @ F @ R4 ) )
      = ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
        | ( ( ( F @ X )
            = ( F @ Y ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R4 ) ) ) ) ).

% mlex_iff
thf(fact_600_mlex__less,axiom,
    ! [F: set_Pr1261947904930325089at_nat > nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R4: set_Pr4329608150637261639at_nat] :
      ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( mlex_p3502713205431853936at_nat @ F @ R4 ) ) ) ).

% mlex_less
thf(fact_601_mlex__less,axiom,
    ! [F: nat > nat,X: nat,Y: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( mlex_prod_nat @ F @ R4 ) ) ) ).

% mlex_less
thf(fact_602_pairself_Opelims,axiom,
    ! [X: int > option6357759511663192854e_term,Xa: product_prod_int_int,Y: produc6576344331059438605e_term] :
      ( ( ( pairse3534876335208825186e_term @ X @ Xa )
        = Y )
     => ( ( accp_P8928870874622223812nt_int @ pairse6848479906795794847e_term @ ( produc4305682042979456191nt_int @ X @ Xa ) )
       => ~ ! [A3: int,B2: int] :
              ( ( Xa
                = ( product_Pair_int_int @ A3 @ B2 ) )
             => ( ( Y
                  = ( produc5950683997804057413e_term @ ( X @ A3 ) @ ( X @ B2 ) ) )
               => ~ ( accp_P8928870874622223812nt_int @ pairse6848479906795794847e_term @ ( produc4305682042979456191nt_int @ X @ ( product_Pair_int_int @ A3 @ B2 ) ) ) ) ) ) ) ).

% pairself.pelims
thf(fact_603_execute__return,axiom,
    ! [X: array_a] :
      ( ( heap_T5591663578999710211rray_a @ ( heap_T4896204969850773234rray_a @ X ) )
      = ( comp_P2097125496263478628t_unit @ some_P6105628898608307625it_nat
        @ ^ [H: heap_e7401611519738050253t_unit] : ( produc1733996804261680252it_nat @ X @ ( produc584006145561248582it_nat @ H @ one_one_nat ) ) ) ) ).

% execute_return
thf(fact_604_execute__return,axiom,
    ! [X: product_unit] :
      ( ( heap_T875086893843062177t_unit @ ( heap_T7507251653302230130t_unit @ X ) )
      = ( comp_P3118722334806803912t_unit @ some_P1914260805536162275it_nat
        @ ^ [H: heap_e7401611519738050253t_unit] : ( produc7133225469290080770it_nat @ X @ ( produc584006145561248582it_nat @ H @ one_one_nat ) ) ) ) ).

% execute_return
thf(fact_605_same__fstI,axiom,
    ! [P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > $o,X: produc6241069584506657477e_term > option6357759511663192854e_term,Y7: produc8923325533196201883nteger,Y: produc8923325533196201883nteger,R4: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > set_Pr5052581263052047815nteger] :
      ( ( P @ X )
     => ( ( member335077259223437328nteger @ ( produc4911887526399440471nteger @ Y7 @ Y ) @ ( R4 @ X ) )
       => ( member2446009503569356342nteger @ ( produc7954516348491987909nteger @ ( produc8603105652947943368nteger @ X @ Y7 ) @ ( produc8603105652947943368nteger @ X @ Y ) ) @ ( same_f8038803414157382067nteger @ P @ R4 ) ) ) ) ).

% same_fstI
thf(fact_606_same__fstI,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o,X: produc3658429121746597890et_nat > $o,Y7: produc3658429121746597890et_nat,Y: produc3658429121746597890et_nat,R4: ( produc3658429121746597890et_nat > $o ) > set_Pr719794911490849221et_nat] :
      ( ( P @ X )
     => ( ( member6099555550032318734et_nat @ ( produc8199053930788261021et_nat @ Y7 @ Y ) @ ( R4 @ X ) )
       => ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ ( produc5001842942810119800et_nat @ X @ Y7 ) @ ( produc5001842942810119800et_nat @ X @ Y ) ) @ ( same_f7251492184213700963et_nat @ P @ R4 ) ) ) ) ).

% same_fstI
thf(fact_607_same__fstI,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o,X: produc3658429121746597890et_nat > $o,Y7: produc3925858234332021118et_nat,Y: produc3925858234332021118et_nat,R4: ( produc3658429121746597890et_nat > $o ) > set_Pr7928877670098842301et_nat] :
      ( ( P @ X )
     => ( ( member4763271486408492550et_nat @ ( produc8599840265553166229et_nat @ Y7 @ Y ) @ ( R4 @ X ) )
       => ( member6341495586645257982et_nat @ ( produc1940133919992309389et_nat @ ( produc2245416461498447860et_nat @ X @ Y7 ) @ ( produc2245416461498447860et_nat @ X @ Y ) ) @ ( same_f1912051055550046943et_nat @ P @ R4 ) ) ) ) ).

% same_fstI
thf(fact_608_same__fstI,axiom,
    ! [P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y7: product_prod_int_int,Y: product_prod_int_int,R4: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr2560585780119916871nt_int] :
      ( ( P @ X )
     => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y7 @ Y ) @ ( R4 @ X ) )
       => ( member6582457606847315088nt_int @ ( produc7601053194514725023nt_int @ ( produc5700946648718959541nt_int @ X @ Y7 ) @ ( produc5700946648718959541nt_int @ X @ Y ) ) @ ( same_f5472592420709775776nt_int @ P @ R4 ) ) ) ) ).

% same_fstI
thf(fact_609_same__fstI,axiom,
    ! [P: ( int > option6357759511663192854e_term ) > $o,X: int > option6357759511663192854e_term,Y7: product_prod_int_int,Y: product_prod_int_int,R4: ( int > option6357759511663192854e_term ) > set_Pr2560585780119916871nt_int] :
      ( ( P @ X )
     => ( ( member8566619992076573584nt_int @ ( produc3646306378393792727nt_int @ Y7 @ Y ) @ ( R4 @ X ) )
       => ( member4085533954029916580nt_int @ ( produc8406175334058502835nt_int @ ( produc4305682042979456191nt_int @ X @ Y7 ) @ ( produc4305682042979456191nt_int @ X @ Y ) ) @ ( same_f6662725367016992042nt_int @ P @ R4 ) ) ) ) ).

% same_fstI
thf(fact_610_zero__less__two,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer ) ).

% zero_less_two
thf(fact_611_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_612_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_613_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_614_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_615_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_616_add_Osafe__commute,axiom,
    ! [X: multis2468970476368604999at_nat,Y: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( syntax5084084251851430274at_nat @ ( plus_p7104986032573967614at_nat @ X @ Y ) @ A )
     => ( ( plus_p7104986032573967614at_nat @ A @ B )
        = ( plus_p7104986032573967614at_nat @ B @ A ) ) ) ).

% add.safe_commute
thf(fact_617_add_Osafe__commute,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] :
      ( ( syntax3730441303064801268at_rat @ ( plus_plus_rat @ X @ Y ) @ A )
     => ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ B @ A ) ) ) ).

% add.safe_commute
thf(fact_618_add_Osafe__commute,axiom,
    ! [X: nat,Y: nat,A: nat,B: nat] :
      ( ( syntax4682126007086162916at_nat @ ( plus_plus_nat @ X @ Y ) @ A )
     => ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ B @ A ) ) ) ).

% add.safe_commute
thf(fact_619_add_Osafe__commute,axiom,
    ! [X: int,Y: int,A: int,B: int] :
      ( ( syntax5678989248478167196nt_int @ ( plus_plus_int @ X @ Y ) @ A )
     => ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ B @ A ) ) ) ).

% add.safe_commute
thf(fact_620_bijective__def,axiom,
    ( biject3388518990857851981at_nat
    = ( ^ [R5: set_Pr4329608150637261639at_nat] :
          ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z4: set_Pr1261947904930325089at_nat] :
              ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R5 )
                & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Z4 ) @ R5 ) )
             => ( Y3 = Z4 ) )
          & ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z4: set_Pr1261947904930325089at_nat] :
              ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Z4 ) @ R5 )
                & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ Z4 ) @ R5 ) )
             => ( X3 = Y3 ) ) ) ) ) ).

% bijective_def
thf(fact_621_bijective__def,axiom,
    ( bijective_nat_nat
    = ( ^ [R5: set_Pr1261947904930325089at_nat] :
          ( ! [X3: nat,Y3: nat,Z4: nat] :
              ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R5 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z4 ) @ R5 ) )
             => ( Y3 = Z4 ) )
          & ! [X3: nat,Y3: nat,Z4: nat] :
              ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z4 ) @ R5 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z4 ) @ R5 ) )
             => ( X3 = Y3 ) ) ) ) ) ).

% bijective_def
thf(fact_622_bijective__def,axiom,
    ( biject9051520373387432658nteger
    = ( ^ [R5: set_Pr1281608226676607948nteger] :
          ( ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger,Z4: produc8923325533196201883nteger] :
              ( ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ R5 )
                & ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Z4 ) @ R5 ) )
             => ( Y3 = Z4 ) )
          & ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc6241069584506657477e_term > option6357759511663192854e_term,Z4: produc8923325533196201883nteger] :
              ( ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Z4 ) @ R5 )
                & ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ Y3 @ Z4 ) @ R5 ) )
             => ( X3 = Y3 ) ) ) ) ) ).

% bijective_def
thf(fact_623_bijective__def,axiom,
    ( biject2615096655818420098et_nat
    = ( ^ [R5: set_Pr3286484037609594932et_nat] :
          ( ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat,Z4: produc3658429121746597890et_nat] :
              ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ R5 )
                & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Z4 ) @ R5 ) )
             => ( Y3 = Z4 ) )
          & ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat > $o,Z4: produc3658429121746597890et_nat] :
              ( ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Z4 ) @ R5 )
                & ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ Y3 @ Z4 ) @ R5 ) )
             => ( X3 = Y3 ) ) ) ) ) ).

% bijective_def
thf(fact_624_bijective__def,axiom,
    ( biject1468766312547416318et_nat
    = ( ^ [R5: set_Pr8536935166611901872et_nat] :
          ( ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat,Z4: produc3925858234332021118et_nat] :
              ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ R5 )
                & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Z4 ) @ R5 ) )
             => ( Y3 = Z4 ) )
          & ! [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat > $o,Z4: produc3925858234332021118et_nat] :
              ( ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Z4 ) @ R5 )
                & ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ Y3 @ Z4 ) @ R5 ) )
             => ( X3 = Y3 ) ) ) ) ) ).

% bijective_def
thf(fact_625_bijective__def,axiom,
    ( biject383251550997737151nt_int
    = ( ^ [R5: set_Pr9222295170931077689nt_int] :
          ( ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int,Z4: product_prod_int_int] :
              ( ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R5 )
                & ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Z4 ) @ R5 ) )
             => ( Y3 = Z4 ) )
          & ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: produc8551481072490612790e_term > option6357759511663192854e_term,Z4: product_prod_int_int] :
              ( ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Z4 ) @ R5 )
                & ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ Y3 @ Z4 ) @ R5 ) )
             => ( X3 = Y3 ) ) ) ) ) ).

% bijective_def
thf(fact_626_bijective__def,axiom,
    ( biject576505603616484041nt_int
    = ( ^ [R5: set_Pr1872883991513573699nt_int] :
          ( ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int,Z4: product_prod_int_int] :
              ( ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R5 )
                & ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Z4 ) @ R5 ) )
             => ( Y3 = Z4 ) )
          & ! [X3: int > option6357759511663192854e_term,Y3: int > option6357759511663192854e_term,Z4: product_prod_int_int] :
              ( ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Z4 ) @ R5 )
                & ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ Y3 @ Z4 ) @ R5 ) )
             => ( X3 = Y3 ) ) ) ) ) ).

% bijective_def
thf(fact_627_ssubst__Pair__rhs,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,R4: set_Pr4329608150637261639at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ S ) @ R4 )
     => ( ( S3 = S )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ S3 ) @ R4 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_628_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S: nat,R4: set_Pr1261947904930325089at_nat,S3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S ) @ R4 )
     => ( ( S3 = S )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S3 ) @ R4 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_629_ssubst__Pair__rhs,axiom,
    ! [R2: produc6241069584506657477e_term > option6357759511663192854e_term,S: produc8923325533196201883nteger,R4: set_Pr1281608226676607948nteger,S3: produc8923325533196201883nteger] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R2 @ S ) @ R4 )
     => ( ( S3 = S )
       => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R2 @ S3 ) @ R4 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_630_ssubst__Pair__rhs,axiom,
    ! [R2: produc3658429121746597890et_nat > $o,S: produc3658429121746597890et_nat,R4: set_Pr3286484037609594932et_nat,S3: produc3658429121746597890et_nat] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R2 @ S ) @ R4 )
     => ( ( S3 = S )
       => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ R2 @ S3 ) @ R4 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_631_ssubst__Pair__rhs,axiom,
    ! [R2: produc3658429121746597890et_nat > $o,S: produc3925858234332021118et_nat,R4: set_Pr8536935166611901872et_nat,S3: produc3925858234332021118et_nat] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R2 @ S ) @ R4 )
     => ( ( S3 = S )
       => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ R2 @ S3 ) @ R4 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_632_ssubst__Pair__rhs,axiom,
    ! [R2: produc8551481072490612790e_term > option6357759511663192854e_term,S: product_prod_int_int,R4: set_Pr9222295170931077689nt_int,S3: product_prod_int_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R2 @ S ) @ R4 )
     => ( ( S3 = S )
       => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R2 @ S3 ) @ R4 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_633_ssubst__Pair__rhs,axiom,
    ! [R2: int > option6357759511663192854e_term,S: product_prod_int_int,R4: set_Pr1872883991513573699nt_int,S3: product_prod_int_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R2 @ S ) @ R4 )
     => ( ( S3 = S )
       => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R2 @ S3 ) @ R4 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_634_add__is__0,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M2 @ N2 )
        = zero_zero_nat )
      = ( ( M2 = zero_zero_nat )
        & ( N2 = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_635_Nat_Oadd__0__right,axiom,
    ! [M2: nat] :
      ( ( plus_plus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% Nat.add_0_right
thf(fact_636_not__gr__zero,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_637_add__0,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A )
      = A ) ).

% add_0
thf(fact_638_add__0,axiom,
    ! [A: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat @ A )
      = A ) ).

% add_0
thf(fact_639_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_640_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_641_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_642_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y ) )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_643_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_644_add__cancel__right__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_right
thf(fact_645_add__cancel__right__right,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( A
        = ( plus_p7104986032573967614at_nat @ A @ B ) )
      = ( B = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_right_right
thf(fact_646_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( plus_plus_rat @ A @ B ) )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
thf(fact_647_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_648_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ A @ B ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_649_add__cancel__right__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_left
thf(fact_650_add__cancel__right__left,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( A
        = ( plus_p7104986032573967614at_nat @ B @ A ) )
      = ( B = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_right_left
thf(fact_651_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( plus_plus_rat @ B @ A ) )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
thf(fact_652_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ A ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_653_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ B @ A ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_654_add__cancel__left__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = A )
      = ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_right
thf(fact_655_add__cancel__left__right,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ A @ B )
        = A )
      = ( B = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_left_right
thf(fact_656_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = A )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
thf(fact_657_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_658_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_659_add__cancel__left__left,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ B @ A )
        = A )
      = ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_left
thf(fact_660_add__cancel__left__left,axiom,
    ! [B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ B @ A )
        = A )
      = ( B = zero_z1048942125864253310at_nat ) ) ).

% add_cancel_left_left
thf(fact_661_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = A )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
thf(fact_662_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_663_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( ( plus_plus_int @ B @ A )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_664_double__zero__sym,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( plus_p5714425477246183910nteger @ A @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% double_zero_sym
thf(fact_665_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_666_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_667_add_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% add.right_neutral
thf(fact_668_add_Oright__neutral,axiom,
    ! [A: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ A @ zero_z1048942125864253310at_nat )
      = A ) ).

% add.right_neutral
thf(fact_669_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_670_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_671_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_672_add__gr__0,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M2 )
        | ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% add_gr_0
thf(fact_673_less__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ one_one_nat )
      = ( N2 = zero_zero_nat ) ) ).

% less_one
thf(fact_674_some__opt__eq__trivial,axiom,
    ! [X: num] :
      ( ( eps_Opt_num
        @ ^ [Y3: num] : ( Y3 = X ) )
      = ( some_num @ X ) ) ).

% some_opt_eq_trivial
thf(fact_675_some__opt__eq__trivial,axiom,
    ! [X: produc8664842809031399944it_nat] :
      ( ( eps_Op3393321821070424684it_nat
        @ ^ [Y3: produc8664842809031399944it_nat] : ( Y3 = X ) )
      = ( some_P1914260805536162275it_nat @ X ) ) ).

% some_opt_eq_trivial
thf(fact_676_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ A ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_677_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_678_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_679_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_680_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_681_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_682_less__add__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ).

% less_add_same_cancel2
thf(fact_683_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ B ) ) ).

% less_add_same_cancel2
thf(fact_684_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_685_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel2
thf(fact_686_less__add__same__cancel1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ).

% less_add_same_cancel1
thf(fact_687_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( ord_less_rat @ zero_zero_rat @ B ) ) ).

% less_add_same_cancel1
thf(fact_688_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_689_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel1
thf(fact_690_add__less__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% add_less_same_cancel2
thf(fact_691_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel2
thf(fact_692_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_693_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_694_add__less__same__cancel1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% add_less_same_cancel1
thf(fact_695_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel1
thf(fact_696_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_697_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_698_some__opt__false__trivial,axiom,
    ( ( eps_Op1141264490366522080it_nat
      @ ^ [Uu: produc6714399278180371524it_nat] : $false )
    = none_P2587165478634041381it_nat ) ).

% some_opt_false_trivial
thf(fact_699_some__opt__false__trivial,axiom,
    ( ( eps_Opt_num
      @ ^ [Uu: num] : $false )
    = none_num ) ).

% some_opt_false_trivial
thf(fact_700_some__opt__false__trivial,axiom,
    ( ( eps_Op3393321821070424684it_nat
      @ ^ [Uu: produc8664842809031399944it_nat] : $false )
    = none_P9117596204409417319it_nat ) ).

% some_opt_false_trivial
thf(fact_701_zero__reorient,axiom,
    ! [X: code_integer] :
      ( ( zero_z3403309356797280102nteger = X )
      = ( X = zero_z3403309356797280102nteger ) ) ).

% zero_reorient
thf(fact_702_zero__reorient,axiom,
    ! [X: multis2468970476368604999at_nat] :
      ( ( zero_z1048942125864253310at_nat = X )
      = ( X = zero_z1048942125864253310at_nat ) ) ).

% zero_reorient
thf(fact_703_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
      = ( X = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_704_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_705_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_706_fun_Omap__ident,axiom,
    ! [T: num > code_integer] :
      ( ( comp_C3531382070062128313er_num
        @ ^ [X3: code_integer] : X3
        @ T )
      = T ) ).

% fun.map_ident
thf(fact_707_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option3216581401566546698it_nat] :
      ( ( Option != none_P2587165478634041381it_nat )
      = ( case_o2169478532838437047it_nat @ $false
        @ ^ [Uu: produc6714399278180371524it_nat] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_708_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
      = ( case_option_o_num @ $false
        @ ^ [Uu: num] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_709_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option8956607266484857688it_nat] :
      ( ( Option != none_P9117596204409417319it_nat )
      = ( case_o2686588417244861013it_nat @ $false
        @ ^ [Uu: produc8664842809031399944it_nat] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_710_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option3216581401566546698it_nat] :
      ( ( Option = none_P2587165478634041381it_nat )
      = ( case_o2169478532838437047it_nat @ $true
        @ ^ [Uu: produc6714399278180371524it_nat] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_711_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_num] :
      ( ( Option = none_num )
      = ( case_option_o_num @ $true
        @ ^ [Uu: num] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_712_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option8956607266484857688it_nat] :
      ( ( Option = none_P9117596204409417319it_nat )
      = ( case_o2686588417244861013it_nat @ $true
        @ ^ [Uu: produc8664842809031399944it_nat] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_713_option_Ocase__distrib,axiom,
    ! [H3: int > int,F1: int,F22: num > int,Option: option_num] :
      ( ( H3 @ ( case_option_int_num @ F1 @ F22 @ Option ) )
      = ( case_option_int_num @ ( H3 @ F1 )
        @ ^ [X3: num] : ( H3 @ ( F22 @ X3 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_714_option_Ocase__distrib,axiom,
    ! [H3: int > option_num,F1: int,F22: num > int,Option: option_num] :
      ( ( H3 @ ( case_option_int_num @ F1 @ F22 @ Option ) )
      = ( case_o6005452278849405969um_num @ ( H3 @ F1 )
        @ ^ [X3: num] : ( H3 @ ( F22 @ X3 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_715_option_Ocase__distrib,axiom,
    ! [H3: int > num,F1: int,F22: num > int,Option: option_num] :
      ( ( H3 @ ( case_option_int_num @ F1 @ F22 @ Option ) )
      = ( case_option_num_num @ ( H3 @ F1 )
        @ ^ [X3: num] : ( H3 @ ( F22 @ X3 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_716_option_Ocase__distrib,axiom,
    ! [H3: option_num > int,F1: option_num,F22: num > option_num,Option: option_num] :
      ( ( H3 @ ( case_o6005452278849405969um_num @ F1 @ F22 @ Option ) )
      = ( case_option_int_num @ ( H3 @ F1 )
        @ ^ [X3: num] : ( H3 @ ( F22 @ X3 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_717_option_Ocase__distrib,axiom,
    ! [H3: option_num > option_num,F1: option_num,F22: num > option_num,Option: option_num] :
      ( ( H3 @ ( case_o6005452278849405969um_num @ F1 @ F22 @ Option ) )
      = ( case_o6005452278849405969um_num @ ( H3 @ F1 )
        @ ^ [X3: num] : ( H3 @ ( F22 @ X3 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_718_option_Ocase__distrib,axiom,
    ! [H3: option_num > num,F1: option_num,F22: num > option_num,Option: option_num] :
      ( ( H3 @ ( case_o6005452278849405969um_num @ F1 @ F22 @ Option ) )
      = ( case_option_num_num @ ( H3 @ F1 )
        @ ^ [X3: num] : ( H3 @ ( F22 @ X3 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_719_option_Ocase__distrib,axiom,
    ! [H3: num > int,F1: num,F22: num > num,Option: option_num] :
      ( ( H3 @ ( case_option_num_num @ F1 @ F22 @ Option ) )
      = ( case_option_int_num @ ( H3 @ F1 )
        @ ^ [X3: num] : ( H3 @ ( F22 @ X3 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_720_option_Ocase__distrib,axiom,
    ! [H3: num > option_num,F1: num,F22: num > num,Option: option_num] :
      ( ( H3 @ ( case_option_num_num @ F1 @ F22 @ Option ) )
      = ( case_o6005452278849405969um_num @ ( H3 @ F1 )
        @ ^ [X3: num] : ( H3 @ ( F22 @ X3 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_721_option_Ocase__distrib,axiom,
    ! [H3: num > num,F1: num,F22: num > num,Option: option_num] :
      ( ( H3 @ ( case_option_num_num @ F1 @ F22 @ Option ) )
      = ( case_option_num_num @ ( H3 @ F1 )
        @ ^ [X3: num] : ( H3 @ ( F22 @ X3 ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_722_case__optionE,axiom,
    ! [P: $o,Q: produc6714399278180371524it_nat > $o,X: option3216581401566546698it_nat] :
      ( ( case_o2169478532838437047it_nat @ P @ Q @ X )
     => ( ( ( X = none_P2587165478634041381it_nat )
         => ~ P )
       => ~ ! [Y5: produc6714399278180371524it_nat] :
              ( ( X
                = ( some_P6105628898608307625it_nat @ Y5 ) )
             => ~ ( Q @ Y5 ) ) ) ) ).

% case_optionE
thf(fact_723_case__optionE,axiom,
    ! [P: $o,Q: num > $o,X: option_num] :
      ( ( case_option_o_num @ P @ Q @ X )
     => ( ( ( X = none_num )
         => ~ P )
       => ~ ! [Y5: num] :
              ( ( X
                = ( some_num @ Y5 ) )
             => ~ ( Q @ Y5 ) ) ) ) ).

% case_optionE
thf(fact_724_case__optionE,axiom,
    ! [P: $o,Q: produc8664842809031399944it_nat > $o,X: option8956607266484857688it_nat] :
      ( ( case_o2686588417244861013it_nat @ P @ Q @ X )
     => ( ( ( X = none_P9117596204409417319it_nat )
         => ~ P )
       => ~ ! [Y5: produc8664842809031399944it_nat] :
              ( ( X
                = ( some_P1914260805536162275it_nat @ Y5 ) )
             => ~ ( Q @ Y5 ) ) ) ) ).

% case_optionE
thf(fact_725_combine__options__def,axiom,
    ( combin4318129983670048329it_nat
    = ( ^ [F2: produc8664842809031399944it_nat > produc8664842809031399944it_nat > produc8664842809031399944it_nat,X3: option8956607266484857688it_nat,Y3: option8956607266484857688it_nat] :
          ( case_o2963978774867076333it_nat @ Y3
          @ ^ [Z4: produc8664842809031399944it_nat] :
              ( case_o2963978774867076333it_nat @ ( some_P1914260805536162275it_nat @ Z4 )
              @ ^ [Aa: produc8664842809031399944it_nat] : ( some_P1914260805536162275it_nat @ ( F2 @ Z4 @ Aa ) )
              @ Y3 )
          @ X3 ) ) ) ).

% combine_options_def
thf(fact_726_combine__options__def,axiom,
    ( combine_options_num
    = ( ^ [F2: num > num > num,X3: option_num,Y3: option_num] :
          ( case_o6005452278849405969um_num @ Y3
          @ ^ [Z4: num] :
              ( case_o6005452278849405969um_num @ ( some_num @ Z4 )
              @ ^ [Aa: num] : ( some_num @ ( F2 @ Z4 @ Aa ) )
              @ Y3 )
          @ X3 ) ) ) ).

% combine_options_def
thf(fact_727_less__option__def,axiom,
    ( ord_le7113747843092208513nteger
    = ( ^ [X3: option_Code_integer] :
          ( case_o43118980978444458nteger @ $false
          @ ^ [Y3: code_integer] :
              ( case_o43118980978444458nteger @ $true
              @ ^ [Z4: code_integer] : ( ord_le6747313008572928689nteger @ Z4 @ Y3 )
              @ X3 ) ) ) ) ).

% less_option_def
thf(fact_728_less__option__def,axiom,
    ( ord_less_option_rat
    = ( ^ [X3: option_rat] :
          ( case_option_o_rat @ $false
          @ ^ [Y3: rat] :
              ( case_option_o_rat @ $true
              @ ^ [Z4: rat] : ( ord_less_rat @ Z4 @ Y3 )
              @ X3 ) ) ) ) ).

% less_option_def
thf(fact_729_less__option__def,axiom,
    ( ord_less_option_num
    = ( ^ [X3: option_num] :
          ( case_option_o_num @ $false
          @ ^ [Y3: num] :
              ( case_option_o_num @ $true
              @ ^ [Z4: num] : ( ord_less_num @ Z4 @ Y3 )
              @ X3 ) ) ) ) ).

% less_option_def
thf(fact_730_less__option__def,axiom,
    ( ord_less_option_nat
    = ( ^ [X3: option_nat] :
          ( case_option_o_nat @ $false
          @ ^ [Y3: nat] :
              ( case_option_o_nat @ $true
              @ ^ [Z4: nat] : ( ord_less_nat @ Z4 @ Y3 )
              @ X3 ) ) ) ) ).

% less_option_def
thf(fact_731_less__option__def,axiom,
    ( ord_less_option_int
    = ( ^ [X3: option_int] :
          ( case_option_o_int @ $false
          @ ^ [Y3: int] :
              ( case_option_o_int @ $true
              @ ^ [Z4: int] : ( ord_less_int @ Z4 @ Y3 )
              @ X3 ) ) ) ) ).

% less_option_def
thf(fact_732_zero__less__iff__neq__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( N2 != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_733_gr__implies__not__zero,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_734_not__less__zero,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less_zero
thf(fact_735_gr__zeroI,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr_zeroI
thf(fact_736_add_Ogroup__left__neutral,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_737_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_738_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_739_add_Ocomm__neutral,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% add.comm_neutral
thf(fact_740_add_Ocomm__neutral,axiom,
    ! [A: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ A @ zero_z1048942125864253310at_nat )
      = A ) ).

% add.comm_neutral
thf(fact_741_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_742_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_743_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_744_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_745_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_746_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_747_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_748_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_749_zero__neq__one,axiom,
    zero_z3403309356797280102nteger != one_one_Code_integer ).

% zero_neq_one
thf(fact_750_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_751_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_752_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_753_plus__nat_Oadd__0,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% plus_nat.add_0
thf(fact_754_add__eq__self__zero,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M2 @ N2 )
        = M2 )
     => ( N2 = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_755_less__numeral__extra_I1_J,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% less_numeral_extra(1)
thf(fact_756_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_757_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_758_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_759_raise__def,axiom,
    ( heap_T1552190017797416154rray_a
    = ( ^ [S4: list_char] :
          ( heap_T1488828001306802186rray_a
          @ ^ [Uu: heap_e7401611519738050253t_unit] : none_P2587165478634041381it_nat ) ) ) ).

% raise_def
thf(fact_760_raise__def,axiom,
    ( heap_T2927564422264180874t_unit
    = ( ^ [S4: list_char] :
          ( heap_T6183433275982383450t_unit
          @ ^ [Uu: heap_e7401611519738050253t_unit] : none_P9117596204409417319it_nat ) ) ) ).

% raise_def
thf(fact_761_guard__def,axiom,
    ( heap_T7685628758766326643rray_a
    = ( ^ [P3: heap_e7401611519738050253t_unit > $o,F2: heap_e7401611519738050253t_unit > produc6714399278180371524it_nat] :
          ( heap_T1488828001306802186rray_a
          @ ^ [H: heap_e7401611519738050253t_unit] : ( if_opt7017092727090204112it_nat @ ( P3 @ H ) @ ( some_P6105628898608307625it_nat @ ( F2 @ H ) ) @ none_P2587165478634041381it_nat ) ) ) ) ).

% guard_def
thf(fact_762_guard__def,axiom,
    ( heap_T8440541562793052209t_unit
    = ( ^ [P3: heap_e7401611519738050253t_unit > $o,F2: heap_e7401611519738050253t_unit > produc8664842809031399944it_nat] :
          ( heap_T6183433275982383450t_unit
          @ ^ [H: heap_e7401611519738050253t_unit] : ( if_opt1729522071442692626it_nat @ ( P3 @ H ) @ ( some_P1914260805536162275it_nat @ ( F2 @ H ) ) @ none_P9117596204409417319it_nat ) ) ) ) ).

% guard_def
thf(fact_763_pos__add__strict,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_764_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_765_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_766_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_767_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C3: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C3 ) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_768_add__pos__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_769_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_770_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_771_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_772_add__neg__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_neg_neg
thf(fact_773_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_neg_neg
thf(fact_774_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_775_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_776_add__less__zeroD,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ X @ Y ) @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ X @ zero_z3403309356797280102nteger )
        | ( ord_le6747313008572928689nteger @ Y @ zero_z3403309356797280102nteger ) ) ) ).

% add_less_zeroD
thf(fact_777_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X @ zero_zero_rat )
        | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_778_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
     => ( ( ord_less_int @ X @ zero_zero_int )
        | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_779_zero__less__one,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% zero_less_one
thf(fact_780_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_781_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_782_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_783_not__one__less__zero,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).

% not_one_less_zero
thf(fact_784_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_785_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_786_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_787_execute__ureturn,axiom,
    ! [X: array_a] :
      ( ( heap_T5591663578999710211rray_a @ ( heap_T4328122439115681491rray_a @ X ) )
      = ( comp_P2097125496263478628t_unit @ some_P6105628898608307625it_nat
        @ ^ [H: heap_e7401611519738050253t_unit] : ( produc1733996804261680252it_nat @ X @ ( produc584006145561248582it_nat @ H @ zero_zero_nat ) ) ) ) ).

% execute_ureturn
thf(fact_788_execute__ureturn,axiom,
    ! [X: product_unit] :
      ( ( heap_T875086893843062177t_unit @ ( heap_T4284346855313245393t_unit @ X ) )
      = ( comp_P3118722334806803912t_unit @ some_P1914260805536162275it_nat
        @ ^ [H: heap_e7401611519738050253t_unit] : ( produc7133225469290080770it_nat @ X @ ( produc584006145561248582it_nat @ H @ zero_zero_nat ) ) ) ) ).

% execute_ureturn
thf(fact_789_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K5: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K5 )
          & ( ( plus_plus_nat @ I @ K5 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_790_tap__def,axiom,
    ( heap_T560649228465745139t_unit
    = ( ^ [F2: heap_e7401611519738050253t_unit > product_unit] :
          ( heap_T6183433275982383450t_unit
          @ ^ [H: heap_e7401611519738050253t_unit] : ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ ( F2 @ H ) @ ( produc584006145561248582it_nat @ H @ one_one_nat ) ) ) ) ) ) ).

% tap_def
thf(fact_791_double__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% double_eq_0_iff
thf(fact_792_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( plus_plus_rat @ A @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
thf(fact_793_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( plus_plus_int @ A @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% double_eq_0_iff
thf(fact_794_add__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B
        = ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% add_0_iff
thf(fact_795_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B
        = ( plus_plus_rat @ B @ A ) )
      = ( A = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_796_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_797_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_798_verit__sum__simplify,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% verit_sum_simplify
thf(fact_799_verit__sum__simplify,axiom,
    ! [A: multis2468970476368604999at_nat] :
      ( ( plus_p7104986032573967614at_nat @ A @ zero_z1048942125864253310at_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_800_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_801_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_802_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_803_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ zero_z3403309356797280102nteger )
    = one_one_Code_integer ) ).

% dbl_inc_simps(2)
thf(fact_804_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_805_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_806_disjE__realizer2,axiom,
    ! [P: $o,Q: num > $o,X: option_num,R4: int > $o,F: int,G: num > int] :
      ( ( case_option_o_num @ P @ Q @ X )
     => ( ( P
         => ( R4 @ F ) )
       => ( ! [Q2: num] :
              ( ( Q @ Q2 )
             => ( R4 @ ( G @ Q2 ) ) )
         => ( R4 @ ( case_option_int_num @ F @ G @ X ) ) ) ) ) ).

% disjE_realizer2
thf(fact_807_disjE__realizer2,axiom,
    ! [P: $o,Q: num > $o,X: option_num,R4: option_num > $o,F: option_num,G: num > option_num] :
      ( ( case_option_o_num @ P @ Q @ X )
     => ( ( P
         => ( R4 @ F ) )
       => ( ! [Q2: num] :
              ( ( Q @ Q2 )
             => ( R4 @ ( G @ Q2 ) ) )
         => ( R4 @ ( case_o6005452278849405969um_num @ F @ G @ X ) ) ) ) ) ).

% disjE_realizer2
thf(fact_808_disjE__realizer2,axiom,
    ! [P: $o,Q: num > $o,X: option_num,R4: num > $o,F: num,G: num > num] :
      ( ( case_option_o_num @ P @ Q @ X )
     => ( ( P
         => ( R4 @ F ) )
       => ( ! [Q2: num] :
              ( ( Q @ Q2 )
             => ( R4 @ ( G @ Q2 ) ) )
         => ( R4 @ ( case_option_num_num @ F @ G @ X ) ) ) ) ) ).

% disjE_realizer2
thf(fact_809_set__to__map__def,axiom,
    ( set_to4447731662395764663at_nat
    = ( ^ [S5: set_Pr4329608150637261639at_nat,K4: set_Pr1261947904930325089at_nat] :
          ( eps_Op994234036888646909at_nat
          @ ^ [V2: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ K4 @ V2 ) @ S5 ) ) ) ) ).

% set_to_map_def
thf(fact_810_set__to__map__def,axiom,
    ( set_to_map_nat_nat
    = ( ^ [S5: set_Pr1261947904930325089at_nat,K4: nat] :
          ( eps_Opt_nat
          @ ^ [V2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ K4 @ V2 ) @ S5 ) ) ) ) ).

% set_to_map_def
thf(fact_811_set__to__map__def,axiom,
    ( set_to680829756418416360nteger
    = ( ^ [S5: set_Pr1281608226676607948nteger,K4: produc6241069584506657477e_term > option6357759511663192854e_term] :
          ( eps_Op9034888937550303799nteger
          @ ^ [V2: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ K4 @ V2 ) @ S5 ) ) ) ) ).

% set_to_map_def
thf(fact_812_set__to__map__def,axiom,
    ( set_to6040779677306527128et_nat
    = ( ^ [S5: set_Pr3286484037609594932et_nat,K4: produc3658429121746597890et_nat > $o] :
          ( eps_Op2013419657081471078et_nat
          @ ^ [V2: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ K4 @ V2 ) @ S5 ) ) ) ) ).

% set_to_map_def
thf(fact_813_set__to__map__def,axiom,
    ( set_to2047380710992656148et_nat
    = ( ^ [S5: set_Pr8536935166611901872et_nat,K4: produc3658429121746597890et_nat > $o] :
          ( eps_Op6423059015816587746et_nat
          @ ^ [V2: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ K4 @ V2 ) @ S5 ) ) ) ) ).

% set_to_map_def
thf(fact_814_set__to__map__def,axiom,
    ( set_to6585427216324510421nt_int
    = ( ^ [S5: set_Pr9222295170931077689nt_int,K4: produc8551481072490612790e_term > option6357759511663192854e_term] :
          ( eps_Op2446201859369042517nt_int
          @ ^ [V2: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ K4 @ V2 ) @ S5 ) ) ) ) ).

% set_to_map_def
thf(fact_815_set__to__map__def,axiom,
    ( set_to1653371902132773855nt_int
    = ( ^ [S5: set_Pr1872883991513573699nt_int,K4: int > option6357759511663192854e_term] :
          ( eps_Op2446201859369042517nt_int
          @ ^ [V2: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ K4 @ V2 ) @ S5 ) ) ) ) ).

% set_to_map_def
thf(fact_816_K__record__comp,axiom,
    ! [C: code_integer,F: num > code_integer] :
      ( ( comp_C3531382070062128313er_num
        @ ^ [X3: code_integer] : C
        @ F )
      = ( ^ [X3: num] : C ) ) ).

% K_record_comp
thf(fact_817_K__record__comp,axiom,
    ! [C: produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer] :
      ( ( comp_C1593894019821074884nteger
        @ ^ [X3: code_integer] : C
        @ F )
      = ( ^ [X3: code_integer] : C ) ) ).

% K_record_comp
thf(fact_818_K__record__comp,axiom,
    ! [C: produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer > code_integer] :
      ( ( comp_C8797469213163452608nteger
        @ ^ [X3: code_integer > code_integer] : C
        @ F )
      = ( ^ [X3: code_integer] : C ) ) ).

% K_record_comp
thf(fact_819_K__record__comp,axiom,
    ! [C: num,F: int > nat] :
      ( ( comp_nat_num_int
        @ ^ [X3: nat] : C
        @ F )
      = ( ^ [X3: int] : C ) ) ).

% K_record_comp
thf(fact_820_pred__equals__eq2,axiom,
    ! [R4: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( ( ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R4 ) )
        = ( ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ S2 ) ) )
      = ( R4 = S2 ) ) ).

% pred_equals_eq2
thf(fact_821_pred__equals__eq2,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R4 ) )
        = ( ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S2 ) ) )
      = ( R4 = S2 ) ) ).

% pred_equals_eq2
thf(fact_822_pred__equals__eq2,axiom,
    ! [R4: set_Pr1281608226676607948nteger,S2: set_Pr1281608226676607948nteger] :
      ( ( ( ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ R4 ) )
        = ( ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ S2 ) ) )
      = ( R4 = S2 ) ) ).

% pred_equals_eq2
thf(fact_823_pred__equals__eq2,axiom,
    ! [R4: set_Pr3286484037609594932et_nat,S2: set_Pr3286484037609594932et_nat] :
      ( ( ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ R4 ) )
        = ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ S2 ) ) )
      = ( R4 = S2 ) ) ).

% pred_equals_eq2
thf(fact_824_pred__equals__eq2,axiom,
    ! [R4: set_Pr8536935166611901872et_nat,S2: set_Pr8536935166611901872et_nat] :
      ( ( ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ R4 ) )
        = ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ S2 ) ) )
      = ( R4 = S2 ) ) ).

% pred_equals_eq2
thf(fact_825_pred__equals__eq2,axiom,
    ! [R4: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
      ( ( ( ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R4 ) )
        = ( ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ S2 ) ) )
      = ( R4 = S2 ) ) ).

% pred_equals_eq2
thf(fact_826_pred__equals__eq2,axiom,
    ! [R4: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
      ( ( ( ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R4 ) )
        = ( ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ S2 ) ) )
      = ( R4 = S2 ) ) ).

% pred_equals_eq2
thf(fact_827_dbl__inc__def,axiom,
    ( neg_nu5831290666863070958nteger
    = ( ^ [X3: code_integer] : ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ X3 @ X3 ) @ one_one_Code_integer ) ) ) ).

% dbl_inc_def
thf(fact_828_dbl__inc__def,axiom,
    ( neg_nu5219082963157363817nc_rat
    = ( ^ [X3: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).

% dbl_inc_def
thf(fact_829_dbl__inc__def,axiom,
    ( neg_nu5851722552734809277nc_int
    = ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).

% dbl_inc_def
thf(fact_830_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
thf(fact_831_fails__wait,axiom,
    ! [X: nat,H3: heap_e7401611519738050253t_unit] :
      ~ ( time_f8834461667527620124t_unit @ ( heap_Time_wait @ X ) @ H3 ) ).

% fails_wait
thf(fact_832_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( ( P @ A3 @ B2 )
          = ( P @ B2 @ A3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
       => ( ! [A3: nat,B2: nat] :
              ( ( P @ A3 @ B2 )
             => ( P @ A3 @ ( plus_plus_nat @ A3 @ B2 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_833_Ints__odd__less__0,axiom,
    ! [A: code_integer] :
      ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
     => ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ A ) @ zero_z3403309356797280102nteger )
        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ) ).

% Ints_odd_less_0
thf(fact_834_Ints__odd__less__0,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A ) @ zero_zero_rat )
        = ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% Ints_odd_less_0
thf(fact_835_Ints__odd__less__0,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A ) @ zero_zero_int )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% Ints_odd_less_0
thf(fact_836_num__of__nat__plus__distrib,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( plus_plus_nat @ M2 @ N2 ) )
          = ( plus_plus_num @ ( num_of_nat @ M2 ) @ ( num_of_nat @ N2 ) ) ) ) ) ).

% num_of_nat_plus_distrib
thf(fact_837_divides__aux__eq,axiom,
    ! [Q3: code_integer,R2: code_integer] :
      ( ( unique5706413561485394159nteger @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
      = ( R2 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
thf(fact_838_divides__aux__eq,axiom,
    ! [Q3: nat,R2: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_839_divides__aux__eq,axiom,
    ! [Q3: int,R2: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q3 @ R2 ) )
      = ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_840_sgn__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( sgn_sgn_Code_integer @ A )
        = one_one_Code_integer ) ) ).

% sgn_pos
thf(fact_841_sgn__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( sgn_sgn_rat @ A )
        = one_one_rat ) ) ).

% sgn_pos
thf(fact_842_sgn__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( sgn_sgn_int @ A )
        = one_one_int ) ) ).

% sgn_pos
thf(fact_843_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_844_abs__add__one__gt__zero,axiom,
    ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_845_abs__add__one__gt__zero,axiom,
    ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_846_map__option__case,axiom,
    ( map_option_num_num
    = ( ^ [F2: num > num] :
          ( case_o6005452278849405969um_num @ none_num
          @ ^ [X3: num] : ( some_num @ ( F2 @ X3 ) ) ) ) ) ).

% map_option_case
thf(fact_847_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
    = zero_zero_int ) ).

% add_neg_numeral_special(7)
thf(fact_848_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% add_neg_numeral_special(7)
thf(fact_849_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = zero_zero_rat ) ).

% add_neg_numeral_special(7)
thf(fact_850_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
    = zero_zero_int ) ).

% add_neg_numeral_special(8)
thf(fact_851_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
    = zero_z3403309356797280102nteger ) ).

% add_neg_numeral_special(8)
thf(fact_852_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
    = zero_zero_rat ) ).

% add_neg_numeral_special(8)
thf(fact_853_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% dbl_dec_simps(3)
thf(fact_854_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
    = one_one_rat ) ).

% dbl_dec_simps(3)
thf(fact_855_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_856_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_857_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_858_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = ( uminus_uminus_rat @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_859_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_860_add_Oinverse__inverse,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_861_add_Oinverse__inverse,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_862_abs__idempotent,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_idempotent
thf(fact_863_abs__idempotent,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_idempotent
thf(fact_864_abs__idempotent,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_idempotent
thf(fact_865_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_866_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
thf(fact_867_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = A )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_868_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_869_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
thf(fact_870_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A
        = ( uminus_uminus_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_871_neg__equal__0__iff__equal,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
thf(fact_872_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
thf(fact_873_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
thf(fact_874_neg__0__equal__iff__equal,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( uminus_uminus_int @ A ) )
      = ( zero_zero_int = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_875_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( uminus1351360451143612070nteger @ A ) )
      = ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_876_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( uminus_uminus_rat @ A ) )
      = ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_877_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_878_add_Oinverse__neutral,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% add.inverse_neutral
thf(fact_879_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_880_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_881_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_882_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_883_minus__add__distrib,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).

% minus_add_distrib
thf(fact_884_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).

% minus_add_distrib
thf(fact_885_minus__add__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_add_distrib
thf(fact_886_minus__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_887_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_888_minus__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_889_add__minus__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_890_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_891_add__minus__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_892_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_893_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_894_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_895_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_896_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_897_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_898_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_899_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_900_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_901_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
      = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_add_abs
thf(fact_902_abs__add__abs,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
      = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_add_abs
thf(fact_903_abs__add__abs,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
      = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_add_abs
thf(fact_904_abs__1,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_1
thf(fact_905_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_906_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_907_abs__minus__cancel,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus_cancel
thf(fact_908_abs__minus__cancel,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus_cancel
thf(fact_909_abs__minus__cancel,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus_cancel
thf(fact_910_sgn__1,axiom,
    ( ( sgn_sgn_int @ one_one_int )
    = one_one_int ) ).

% sgn_1
thf(fact_911_sgn__1,axiom,
    ( ( sgn_sgn_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% sgn_1
thf(fact_912_sgn__1,axiom,
    ( ( sgn_sgn_rat @ one_one_rat )
    = one_one_rat ) ).

% sgn_1
thf(fact_913_map__option__eq__Some,axiom,
    ! [F: produc8664842809031399944it_nat > num,Xo: option8956607266484857688it_nat,Y: num] :
      ( ( ( map_op2731889312448867759at_num @ F @ Xo )
        = ( some_num @ Y ) )
      = ( ? [Z4: produc8664842809031399944it_nat] :
            ( ( Xo
              = ( some_P1914260805536162275it_nat @ Z4 ) )
            & ( ( F @ Z4 )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_914_map__option__eq__Some,axiom,
    ! [F: num > produc8664842809031399944it_nat,Xo: option_num,Y: produc8664842809031399944it_nat] :
      ( ( ( map_op8118133509425879471it_nat @ F @ Xo )
        = ( some_P1914260805536162275it_nat @ Y ) )
      = ( ? [Z4: num] :
            ( ( Xo
              = ( some_num @ Z4 ) )
            & ( ( F @ Z4 )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_915_map__option__eq__Some,axiom,
    ! [F: produc8664842809031399944it_nat > produc8664842809031399944it_nat,Xo: option8956607266484857688it_nat,Y: produc8664842809031399944it_nat] :
      ( ( ( map_op8673683442680995229it_nat @ F @ Xo )
        = ( some_P1914260805536162275it_nat @ Y ) )
      = ( ? [Z4: produc8664842809031399944it_nat] :
            ( ( Xo
              = ( some_P1914260805536162275it_nat @ Z4 ) )
            & ( ( F @ Z4 )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_916_map__option__eq__Some,axiom,
    ! [F: num > num,Xo: option_num,Y: num] :
      ( ( ( map_option_num_num @ F @ Xo )
        = ( some_num @ Y ) )
      = ( ? [Z4: num] :
            ( ( Xo
              = ( some_num @ Z4 ) )
            & ( ( F @ Z4 )
              = Y ) ) ) ) ).

% map_option_eq_Some
thf(fact_917_None__eq__map__option__iff,axiom,
    ! [F: produc6714399278180371524it_nat > produc6714399278180371524it_nat,X: option3216581401566546698it_nat] :
      ( ( none_P2587165478634041381it_nat
        = ( map_op6002829699023054091it_nat @ F @ X ) )
      = ( X = none_P2587165478634041381it_nat ) ) ).

% None_eq_map_option_iff
thf(fact_918_None__eq__map__option__iff,axiom,
    ! [F: num > produc6714399278180371524it_nat,X: option_num] :
      ( ( none_P2587165478634041381it_nat
        = ( map_op6598675229099881181it_nat @ F @ X ) )
      = ( X = none_num ) ) ).

% None_eq_map_option_iff
thf(fact_919_None__eq__map__option__iff,axiom,
    ! [F: produc8664842809031399944it_nat > produc6714399278180371524it_nat,X: option8956607266484857688it_nat] :
      ( ( none_P2587165478634041381it_nat
        = ( map_op7916408704904942447it_nat @ F @ X ) )
      = ( X = none_P9117596204409417319it_nat ) ) ).

% None_eq_map_option_iff
thf(fact_920_None__eq__map__option__iff,axiom,
    ! [F: produc6714399278180371524it_nat > num,X: option3216581401566546698it_nat] :
      ( ( none_num
        = ( map_op8228268515680064275at_num @ F @ X ) )
      = ( X = none_P2587165478634041381it_nat ) ) ).

% None_eq_map_option_iff
thf(fact_921_None__eq__map__option__iff,axiom,
    ! [F: produc8664842809031399944it_nat > num,X: option8956607266484857688it_nat] :
      ( ( none_num
        = ( map_op2731889312448867759at_num @ F @ X ) )
      = ( X = none_P9117596204409417319it_nat ) ) ).

% None_eq_map_option_iff
thf(fact_922_None__eq__map__option__iff,axiom,
    ! [F: produc6714399278180371524it_nat > produc8664842809031399944it_nat,X: option3216581401566546698it_nat] :
      ( ( none_P9117596204409417319it_nat
        = ( map_op674526153330701057it_nat @ F @ X ) )
      = ( X = none_P2587165478634041381it_nat ) ) ).

% None_eq_map_option_iff
thf(fact_923_None__eq__map__option__iff,axiom,
    ! [F: num > produc8664842809031399944it_nat,X: option_num] :
      ( ( none_P9117596204409417319it_nat
        = ( map_op8118133509425879471it_nat @ F @ X ) )
      = ( X = none_num ) ) ).

% None_eq_map_option_iff
thf(fact_924_None__eq__map__option__iff,axiom,
    ! [F: produc8664842809031399944it_nat > produc8664842809031399944it_nat,X: option8956607266484857688it_nat] :
      ( ( none_P9117596204409417319it_nat
        = ( map_op8673683442680995229it_nat @ F @ X ) )
      = ( X = none_P9117596204409417319it_nat ) ) ).

% None_eq_map_option_iff
thf(fact_925_None__eq__map__option__iff,axiom,
    ! [F: num > num,X: option_num] :
      ( ( none_num
        = ( map_option_num_num @ F @ X ) )
      = ( X = none_num ) ) ).

% None_eq_map_option_iff
thf(fact_926_map__option__is__None,axiom,
    ! [F: produc6714399278180371524it_nat > produc6714399278180371524it_nat,Opt: option3216581401566546698it_nat] :
      ( ( ( map_op6002829699023054091it_nat @ F @ Opt )
        = none_P2587165478634041381it_nat )
      = ( Opt = none_P2587165478634041381it_nat ) ) ).

% map_option_is_None
thf(fact_927_map__option__is__None,axiom,
    ! [F: num > produc6714399278180371524it_nat,Opt: option_num] :
      ( ( ( map_op6598675229099881181it_nat @ F @ Opt )
        = none_P2587165478634041381it_nat )
      = ( Opt = none_num ) ) ).

% map_option_is_None
thf(fact_928_map__option__is__None,axiom,
    ! [F: produc8664842809031399944it_nat > produc6714399278180371524it_nat,Opt: option8956607266484857688it_nat] :
      ( ( ( map_op7916408704904942447it_nat @ F @ Opt )
        = none_P2587165478634041381it_nat )
      = ( Opt = none_P9117596204409417319it_nat ) ) ).

% map_option_is_None
thf(fact_929_map__option__is__None,axiom,
    ! [F: produc6714399278180371524it_nat > num,Opt: option3216581401566546698it_nat] :
      ( ( ( map_op8228268515680064275at_num @ F @ Opt )
        = none_num )
      = ( Opt = none_P2587165478634041381it_nat ) ) ).

% map_option_is_None
thf(fact_930_map__option__is__None,axiom,
    ! [F: produc8664842809031399944it_nat > num,Opt: option8956607266484857688it_nat] :
      ( ( ( map_op2731889312448867759at_num @ F @ Opt )
        = none_num )
      = ( Opt = none_P9117596204409417319it_nat ) ) ).

% map_option_is_None
thf(fact_931_map__option__is__None,axiom,
    ! [F: produc6714399278180371524it_nat > produc8664842809031399944it_nat,Opt: option3216581401566546698it_nat] :
      ( ( ( map_op674526153330701057it_nat @ F @ Opt )
        = none_P9117596204409417319it_nat )
      = ( Opt = none_P2587165478634041381it_nat ) ) ).

% map_option_is_None
thf(fact_932_map__option__is__None,axiom,
    ! [F: num > produc8664842809031399944it_nat,Opt: option_num] :
      ( ( ( map_op8118133509425879471it_nat @ F @ Opt )
        = none_P9117596204409417319it_nat )
      = ( Opt = none_num ) ) ).

% map_option_is_None
thf(fact_933_map__option__is__None,axiom,
    ! [F: produc8664842809031399944it_nat > produc8664842809031399944it_nat,Opt: option8956607266484857688it_nat] :
      ( ( ( map_op8673683442680995229it_nat @ F @ Opt )
        = none_P9117596204409417319it_nat )
      = ( Opt = none_P9117596204409417319it_nat ) ) ).

% map_option_is_None
thf(fact_934_map__option__is__None,axiom,
    ! [F: num > num,Opt: option_num] :
      ( ( ( map_option_num_num @ F @ Opt )
        = none_num )
      = ( Opt = none_num ) ) ).

% map_option_is_None
thf(fact_935_option_Omap__disc__iff,axiom,
    ! [F: produc6714399278180371524it_nat > produc6714399278180371524it_nat,A: option3216581401566546698it_nat] :
      ( ( ( map_op6002829699023054091it_nat @ F @ A )
        = none_P2587165478634041381it_nat )
      = ( A = none_P2587165478634041381it_nat ) ) ).

% option.map_disc_iff
thf(fact_936_option_Omap__disc__iff,axiom,
    ! [F: num > produc6714399278180371524it_nat,A: option_num] :
      ( ( ( map_op6598675229099881181it_nat @ F @ A )
        = none_P2587165478634041381it_nat )
      = ( A = none_num ) ) ).

% option.map_disc_iff
thf(fact_937_option_Omap__disc__iff,axiom,
    ! [F: produc8664842809031399944it_nat > produc6714399278180371524it_nat,A: option8956607266484857688it_nat] :
      ( ( ( map_op7916408704904942447it_nat @ F @ A )
        = none_P2587165478634041381it_nat )
      = ( A = none_P9117596204409417319it_nat ) ) ).

% option.map_disc_iff
thf(fact_938_option_Omap__disc__iff,axiom,
    ! [F: produc6714399278180371524it_nat > num,A: option3216581401566546698it_nat] :
      ( ( ( map_op8228268515680064275at_num @ F @ A )
        = none_num )
      = ( A = none_P2587165478634041381it_nat ) ) ).

% option.map_disc_iff
thf(fact_939_option_Omap__disc__iff,axiom,
    ! [F: produc8664842809031399944it_nat > num,A: option8956607266484857688it_nat] :
      ( ( ( map_op2731889312448867759at_num @ F @ A )
        = none_num )
      = ( A = none_P9117596204409417319it_nat ) ) ).

% option.map_disc_iff
thf(fact_940_option_Omap__disc__iff,axiom,
    ! [F: produc6714399278180371524it_nat > produc8664842809031399944it_nat,A: option3216581401566546698it_nat] :
      ( ( ( map_op674526153330701057it_nat @ F @ A )
        = none_P9117596204409417319it_nat )
      = ( A = none_P2587165478634041381it_nat ) ) ).

% option.map_disc_iff
thf(fact_941_option_Omap__disc__iff,axiom,
    ! [F: num > produc8664842809031399944it_nat,A: option_num] :
      ( ( ( map_op8118133509425879471it_nat @ F @ A )
        = none_P9117596204409417319it_nat )
      = ( A = none_num ) ) ).

% option.map_disc_iff
thf(fact_942_option_Omap__disc__iff,axiom,
    ! [F: produc8664842809031399944it_nat > produc8664842809031399944it_nat,A: option8956607266484857688it_nat] :
      ( ( ( map_op8673683442680995229it_nat @ F @ A )
        = none_P9117596204409417319it_nat )
      = ( A = none_P9117596204409417319it_nat ) ) ).

% option.map_disc_iff
thf(fact_943_option_Omap__disc__iff,axiom,
    ! [F: num > num,A: option_num] :
      ( ( ( map_option_num_num @ F @ A )
        = none_num )
      = ( A = none_num ) ) ).

% option.map_disc_iff
thf(fact_944_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_0_iff_less
thf(fact_945_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_0_iff_less
thf(fact_946_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% neg_less_0_iff_less
thf(fact_947_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% neg_0_less_iff_less
thf(fact_948_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% neg_0_less_iff_less
thf(fact_949_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% neg_0_less_iff_less
thf(fact_950_neg__less__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_pos
thf(fact_951_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_pos
thf(fact_952_neg__less__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% neg_less_pos
thf(fact_953_less__neg__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% less_neg_neg
thf(fact_954_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_955_less__neg__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% less_neg_neg
thf(fact_956_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_957_ab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_958_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_959_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_960_add_Oright__inverse,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_961_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_962_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
      = ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
thf(fact_963_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
      = ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
thf(fact_964_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
      = ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
thf(fact_965_abs__neg__one,axiom,
    ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
    = one_one_int ) ).

% abs_neg_one
thf(fact_966_abs__neg__one,axiom,
    ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = one_one_Code_integer ) ).

% abs_neg_one
thf(fact_967_abs__neg__one,axiom,
    ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = one_one_rat ) ).

% abs_neg_one
thf(fact_968_case__map__option,axiom,
    ! [G: code_integer,H3: code_integer > code_integer,F: num > code_integer,X: option_num] :
      ( ( case_o9049918639272118855nteger @ G @ H3 @ ( map_op2849635639313731588nteger @ F @ X ) )
      = ( case_o356765784539232260er_num @ G @ ( comp_C3531382070062128313er_num @ H3 @ F ) @ X ) ) ).

% case_map_option
thf(fact_969_case__map__option,axiom,
    ! [G: produc8923325533196201883nteger > produc8923325533196201883nteger,H3: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer,X: option_Code_integer] :
      ( ( case_o7134296353695833103nteger @ G @ H3 @ ( map_op3669829223712506439nteger @ F @ X ) )
      = ( case_o7134296353695833103nteger @ G @ ( comp_C1593894019821074884nteger @ H3 @ F ) @ X ) ) ).

% case_map_option
thf(fact_970_case__map__option,axiom,
    ! [G: produc8923325533196201883nteger > produc8923325533196201883nteger,H3: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer > code_integer,X: option_Code_integer] :
      ( ( case_o1441893360019914891nteger @ G @ H3 @ ( map_op370346339404370627nteger @ F @ X ) )
      = ( case_o7134296353695833103nteger @ G @ ( comp_C8797469213163452608nteger @ H3 @ F ) @ X ) ) ).

% case_map_option
thf(fact_971_case__map__option,axiom,
    ! [G: num,H3: nat > num,F: int > nat,X: option_int] :
      ( ( case_option_num_nat @ G @ H3 @ ( map_option_int_nat @ F @ X ) )
      = ( case_option_num_int @ G @ ( comp_nat_num_int @ H3 @ F ) @ X ) ) ).

% case_map_option
thf(fact_972_case__map__option,axiom,
    ! [G: int,H3: num > int,F: num > num,X: option_num] :
      ( ( case_option_int_num @ G @ H3 @ ( map_option_num_num @ F @ X ) )
      = ( case_option_int_num @ G @ ( comp_num_int_num @ H3 @ F ) @ X ) ) ).

% case_map_option
thf(fact_973_case__map__option,axiom,
    ! [G: option_num,H3: num > option_num,F: num > num,X: option_num] :
      ( ( case_o6005452278849405969um_num @ G @ H3 @ ( map_option_num_num @ F @ X ) )
      = ( case_o6005452278849405969um_num @ G @ ( comp_n6731957995704128387um_num @ H3 @ F ) @ X ) ) ).

% case_map_option
thf(fact_974_case__map__option,axiom,
    ! [G: num,H3: num > num,F: num > num,X: option_num] :
      ( ( case_option_num_num @ G @ H3 @ ( map_option_num_num @ F @ X ) )
      = ( case_option_num_num @ G @ ( comp_num_num_num @ H3 @ F ) @ X ) ) ).

% case_map_option
thf(fact_975_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_inc_simps(4)
thf(fact_976_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_inc_simps(4)
thf(fact_977_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_inc_simps(4)
thf(fact_978_abs__sgn__eq__1,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
        = one_one_Code_integer ) ) ).

% abs_sgn_eq_1
thf(fact_979_abs__sgn__eq__1,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
        = one_one_rat ) ) ).

% abs_sgn_eq_1
thf(fact_980_abs__sgn__eq__1,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
        = one_one_int ) ) ).

% abs_sgn_eq_1
thf(fact_981_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_982_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_dec_simps(2)
thf(fact_983_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_984_sgn__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( sgn_sgn_int @ A )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% sgn_neg
thf(fact_985_sgn__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( sgn_sgn_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% sgn_neg
thf(fact_986_sgn__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( sgn_sgn_rat @ A )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% sgn_neg
thf(fact_987_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_988_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( uminus1351360451143612070nteger @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_989_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( uminus_uminus_rat @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_990_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_991_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% equation_minus_iff
thf(fact_992_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% equation_minus_iff
thf(fact_993_same__sgn__abs__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ B )
        = ( sgn_sgn_Code_integer @ A ) )
     => ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
        = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_994_same__sgn__abs__add,axiom,
    ! [B: rat,A: rat] :
      ( ( ( sgn_sgn_rat @ B )
        = ( sgn_sgn_rat @ A ) )
     => ( ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) )
        = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_995_same__sgn__abs__add,axiom,
    ! [B: int,A: int] :
      ( ( ( sgn_sgn_int @ B )
        = ( sgn_sgn_int @ A ) )
     => ( ( abs_abs_int @ ( plus_plus_int @ A @ B ) )
        = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_996_sgn__minus__1,axiom,
    ( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% sgn_minus_1
thf(fact_997_sgn__minus__1,axiom,
    ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% sgn_minus_1
thf(fact_998_sgn__minus__1,axiom,
    ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% sgn_minus_1
thf(fact_999_option_Omap__ident,axiom,
    ! [T: option_num] :
      ( ( map_option_num_num
        @ ^ [X3: num] : X3
        @ T )
      = T ) ).

% option.map_ident
thf(fact_1000_abs__of__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( abs_abs_int @ A )
        = ( uminus_uminus_int @ A ) ) ) ).

% abs_of_neg
thf(fact_1001_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_neg
thf(fact_1002_abs__of__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( abs_abs_rat @ A )
        = ( uminus_uminus_rat @ A ) ) ) ).

% abs_of_neg
thf(fact_1003_abs__sgn__eq,axiom,
    ! [A: code_integer] :
      ( ( ( A = zero_z3403309356797280102nteger )
       => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
          = zero_z3403309356797280102nteger ) )
      & ( ( A != zero_z3403309356797280102nteger )
       => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
          = one_one_Code_integer ) ) ) ).

% abs_sgn_eq
thf(fact_1004_abs__sgn__eq,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
          = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
          = one_one_rat ) ) ) ).

% abs_sgn_eq
thf(fact_1005_abs__sgn__eq,axiom,
    ! [A: int] :
      ( ( ( A = zero_zero_int )
       => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
          = zero_zero_int ) )
      & ( ( A != zero_zero_int )
       => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
          = one_one_int ) ) ) ).

% abs_sgn_eq
thf(fact_1006_abs__one,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_one
thf(fact_1007_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_1008_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_1009_map__option_Ocompositionality,axiom,
    ! [F: code_integer > code_integer,G: num > code_integer,Option: option_num] :
      ( ( map_op3669829223712506439nteger @ F @ ( map_op2849635639313731588nteger @ G @ Option ) )
      = ( map_op2849635639313731588nteger @ ( comp_C3531382070062128313er_num @ F @ G ) @ Option ) ) ).

% map_option.compositionality
thf(fact_1010_map__option_Ocompositionality,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,Option: option_Code_integer] :
      ( ( map_op9146685272036663823nteger @ F @ ( map_op3669829223712506439nteger @ G @ Option ) )
      = ( map_op9146685272036663823nteger @ ( comp_C1593894019821074884nteger @ F @ G ) @ Option ) ) ).

% map_option.compositionality
thf(fact_1011_map__option_Ocompositionality,axiom,
    ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,Option: option_Code_integer] :
      ( ( map_op3563471509957180299nteger @ F @ ( map_op370346339404370627nteger @ G @ Option ) )
      = ( map_op9146685272036663823nteger @ ( comp_C8797469213163452608nteger @ F @ G ) @ Option ) ) ).

% map_option.compositionality
thf(fact_1012_map__option_Ocompositionality,axiom,
    ! [F: nat > num,G: int > nat,Option: option_int] :
      ( ( map_option_nat_num @ F @ ( map_option_int_nat @ G @ Option ) )
      = ( map_option_int_num @ ( comp_nat_num_int @ F @ G ) @ Option ) ) ).

% map_option.compositionality
thf(fact_1013_map__option_Ocompositionality,axiom,
    ! [F: num > num,G: num > num,Option: option_num] :
      ( ( map_option_num_num @ F @ ( map_option_num_num @ G @ Option ) )
      = ( map_option_num_num @ ( comp_num_num_num @ F @ G ) @ Option ) ) ).

% map_option.compositionality
thf(fact_1014_option_Omap__comp,axiom,
    ! [G: code_integer > code_integer,F: num > code_integer,V: option_num] :
      ( ( map_op3669829223712506439nteger @ G @ ( map_op2849635639313731588nteger @ F @ V ) )
      = ( map_op2849635639313731588nteger @ ( comp_C3531382070062128313er_num @ G @ F ) @ V ) ) ).

% option.map_comp
thf(fact_1015_option_Omap__comp,axiom,
    ! [G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer,V: option_Code_integer] :
      ( ( map_op9146685272036663823nteger @ G @ ( map_op3669829223712506439nteger @ F @ V ) )
      = ( map_op9146685272036663823nteger @ ( comp_C1593894019821074884nteger @ G @ F ) @ V ) ) ).

% option.map_comp
thf(fact_1016_option_Omap__comp,axiom,
    ! [G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,F: code_integer > code_integer > code_integer,V: option_Code_integer] :
      ( ( map_op3563471509957180299nteger @ G @ ( map_op370346339404370627nteger @ F @ V ) )
      = ( map_op9146685272036663823nteger @ ( comp_C8797469213163452608nteger @ G @ F ) @ V ) ) ).

% option.map_comp
thf(fact_1017_option_Omap__comp,axiom,
    ! [G: nat > num,F: int > nat,V: option_int] :
      ( ( map_option_nat_num @ G @ ( map_option_int_nat @ F @ V ) )
      = ( map_option_int_num @ ( comp_nat_num_int @ G @ F ) @ V ) ) ).

% option.map_comp
thf(fact_1018_option_Omap__comp,axiom,
    ! [G: num > num,F: num > num,V: option_num] :
      ( ( map_option_num_num @ G @ ( map_option_num_num @ F @ V ) )
      = ( map_option_num_num @ ( comp_num_num_num @ G @ F ) @ V ) ) ).

% option.map_comp
thf(fact_1019_map__option_Ocomp,axiom,
    ! [F: code_integer > code_integer,G: num > code_integer] :
      ( ( comp_o5321900959263616553on_num @ ( map_op3669829223712506439nteger @ F ) @ ( map_op2849635639313731588nteger @ G ) )
      = ( map_op2849635639313731588nteger @ ( comp_C3531382070062128313er_num @ F @ G ) ) ) ).

% map_option.comp
thf(fact_1020_map__option_Ocomp,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer] :
      ( ( comp_o9011007105448239796nteger @ ( map_op9146685272036663823nteger @ F ) @ ( map_op3669829223712506439nteger @ G ) )
      = ( map_op9146685272036663823nteger @ ( comp_C1593894019821074884nteger @ F @ G ) ) ) ).

% map_option.comp
thf(fact_1021_map__option_Ocomp,axiom,
    ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer] :
      ( ( comp_o7188567781464924208nteger @ ( map_op3563471509957180299nteger @ F ) @ ( map_op370346339404370627nteger @ G ) )
      = ( map_op9146685272036663823nteger @ ( comp_C8797469213163452608nteger @ F @ G ) ) ) ).

% map_option.comp
thf(fact_1022_map__option_Ocomp,axiom,
    ! [F: nat > num,G: int > nat] :
      ( ( comp_o3128663659858294251on_int @ ( map_option_nat_num @ F ) @ ( map_option_int_nat @ G ) )
      = ( map_option_int_num @ ( comp_nat_num_int @ F @ G ) ) ) ).

% map_option.comp
thf(fact_1023_map__option_Ocomp,axiom,
    ! [F: num > num,G: num > num] :
      ( ( comp_o8931257242830428707on_num @ ( map_option_num_num @ F ) @ ( map_option_num_num @ G ) )
      = ( map_option_num_num @ ( comp_num_num_num @ F @ G ) ) ) ).

% map_option.comp
thf(fact_1024_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_1025_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_1026_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_1027_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% less_minus_iff
thf(fact_1028_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% less_minus_iff
thf(fact_1029_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).

% less_minus_iff
thf(fact_1030_is__num__normalize_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_1031_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_1032_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_1033_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_1034_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_1035_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_1036_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( uminus_uminus_int @ A2 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_1037_group__cancel_Oneg1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer] :
      ( ( A2
        = ( plus_p5714425477246183910nteger @ K @ A ) )
     => ( ( uminus1351360451143612070nteger @ A2 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_1038_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( uminus_uminus_rat @ A2 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_1039_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_1040_one__neq__neg__one,axiom,
    ( one_one_Code_integer
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% one_neq_neg_one
thf(fact_1041_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_1042_map__option__cong,axiom,
    ! [X: option_num,Y: option_num,F: num > num,G: num > num] :
      ( ( X = Y )
     => ( ! [A3: num] :
            ( ( Y
              = ( some_num @ A3 ) )
           => ( ( F @ A3 )
              = ( G @ A3 ) ) )
       => ( ( map_option_num_num @ F @ X )
          = ( map_option_num_num @ G @ Y ) ) ) ) ).

% map_option_cong
thf(fact_1043_option_Osimps_I9_J,axiom,
    ! [F: num > produc8664842809031399944it_nat,X2: num] :
      ( ( map_op8118133509425879471it_nat @ F @ ( some_num @ X2 ) )
      = ( some_P1914260805536162275it_nat @ ( F @ X2 ) ) ) ).

% option.simps(9)
thf(fact_1044_option_Osimps_I9_J,axiom,
    ! [F: produc8664842809031399944it_nat > num,X2: produc8664842809031399944it_nat] :
      ( ( map_op2731889312448867759at_num @ F @ ( some_P1914260805536162275it_nat @ X2 ) )
      = ( some_num @ ( F @ X2 ) ) ) ).

% option.simps(9)
thf(fact_1045_option_Osimps_I9_J,axiom,
    ! [F: produc8664842809031399944it_nat > produc8664842809031399944it_nat,X2: produc8664842809031399944it_nat] :
      ( ( map_op8673683442680995229it_nat @ F @ ( some_P1914260805536162275it_nat @ X2 ) )
      = ( some_P1914260805536162275it_nat @ ( F @ X2 ) ) ) ).

% option.simps(9)
thf(fact_1046_option_Osimps_I9_J,axiom,
    ! [F: num > num,X2: num] :
      ( ( map_option_num_num @ F @ ( some_num @ X2 ) )
      = ( some_num @ ( F @ X2 ) ) ) ).

% option.simps(9)
thf(fact_1047_option_Osimps_I8_J,axiom,
    ! [F: produc6714399278180371524it_nat > produc6714399278180371524it_nat] :
      ( ( map_op6002829699023054091it_nat @ F @ none_P2587165478634041381it_nat )
      = none_P2587165478634041381it_nat ) ).

% option.simps(8)
thf(fact_1048_option_Osimps_I8_J,axiom,
    ! [F: produc6714399278180371524it_nat > num] :
      ( ( map_op8228268515680064275at_num @ F @ none_P2587165478634041381it_nat )
      = none_num ) ).

% option.simps(8)
thf(fact_1049_option_Osimps_I8_J,axiom,
    ! [F: produc6714399278180371524it_nat > produc8664842809031399944it_nat] :
      ( ( map_op674526153330701057it_nat @ F @ none_P2587165478634041381it_nat )
      = none_P9117596204409417319it_nat ) ).

% option.simps(8)
thf(fact_1050_option_Osimps_I8_J,axiom,
    ! [F: num > produc6714399278180371524it_nat] :
      ( ( map_op6598675229099881181it_nat @ F @ none_num )
      = none_P2587165478634041381it_nat ) ).

% option.simps(8)
thf(fact_1051_option_Osimps_I8_J,axiom,
    ! [F: num > produc8664842809031399944it_nat] :
      ( ( map_op8118133509425879471it_nat @ F @ none_num )
      = none_P9117596204409417319it_nat ) ).

% option.simps(8)
thf(fact_1052_option_Osimps_I8_J,axiom,
    ! [F: produc8664842809031399944it_nat > produc6714399278180371524it_nat] :
      ( ( map_op7916408704904942447it_nat @ F @ none_P9117596204409417319it_nat )
      = none_P2587165478634041381it_nat ) ).

% option.simps(8)
thf(fact_1053_option_Osimps_I8_J,axiom,
    ! [F: produc8664842809031399944it_nat > num] :
      ( ( map_op2731889312448867759at_num @ F @ none_P9117596204409417319it_nat )
      = none_num ) ).

% option.simps(8)
thf(fact_1054_option_Osimps_I8_J,axiom,
    ! [F: produc8664842809031399944it_nat > produc8664842809031399944it_nat] :
      ( ( map_op8673683442680995229it_nat @ F @ none_P9117596204409417319it_nat )
      = none_P9117596204409417319it_nat ) ).

% option.simps(8)
thf(fact_1055_option_Osimps_I8_J,axiom,
    ! [F: num > num] :
      ( ( map_option_num_num @ F @ none_num )
      = none_num ) ).

% option.simps(8)
thf(fact_1056_same__sgn__sgn__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ B )
        = ( sgn_sgn_Code_integer @ A ) )
     => ( ( sgn_sgn_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
        = ( sgn_sgn_Code_integer @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_1057_same__sgn__sgn__add,axiom,
    ! [B: rat,A: rat] :
      ( ( ( sgn_sgn_rat @ B )
        = ( sgn_sgn_rat @ A ) )
     => ( ( sgn_sgn_rat @ ( plus_plus_rat @ A @ B ) )
        = ( sgn_sgn_rat @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_1058_same__sgn__sgn__add,axiom,
    ! [B: int,A: int] :
      ( ( ( sgn_sgn_int @ B )
        = ( sgn_sgn_int @ A ) )
     => ( ( sgn_sgn_int @ ( plus_plus_int @ A @ B ) )
        = ( sgn_sgn_int @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_1059_Ints__add,axiom,
    ! [A: rat,B: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( member_rat @ B @ ring_1_Ints_rat )
       => ( member_rat @ ( plus_plus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).

% Ints_add
thf(fact_1060_Ints__add,axiom,
    ! [A: int,B: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( member_int @ B @ ring_1_Ints_int )
       => ( member_int @ ( plus_plus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).

% Ints_add
thf(fact_1061_success__lookupI,axiom,
    ! [R2: ref_Product_unit,H3: heap_e7401611519738050253t_unit] : ( heap_T584514906347983379t_unit @ ( ref_lo7930775386976318366t_unit @ R2 ) @ H3 ) ).

% success_lookupI
thf(fact_1062_success__lookupI,axiom,
    ! [R2: ref_array_a,H3: heap_e7401611519738050253t_unit] : ( heap_T6932021131025641745rray_a @ ( ref_lookup_array_a @ R2 ) @ H3 ) ).

% success_lookupI
thf(fact_1063_Ints__1,axiom,
    member_Code_integer @ one_one_Code_integer @ ring_11222124179247155820nteger ).

% Ints_1
thf(fact_1064_Ints__1,axiom,
    member_rat @ one_one_rat @ ring_1_Ints_rat ).

% Ints_1
thf(fact_1065_Ints__1,axiom,
    member_int @ one_one_int @ ring_1_Ints_int ).

% Ints_1
thf(fact_1066_Ints__nonzero__abs__less1,axiom,
    ! [X: code_integer] :
      ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer )
       => ( X = zero_z3403309356797280102nteger ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_1067_Ints__nonzero__abs__less1,axiom,
    ! [X: rat] :
      ( ( member_rat @ X @ ring_1_Ints_rat )
     => ( ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat )
       => ( X = zero_zero_rat ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_1068_Ints__nonzero__abs__less1,axiom,
    ! [X: int] :
      ( ( member_int @ X @ ring_1_Ints_int )
     => ( ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int )
       => ( X = zero_zero_int ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_1069_sgn__if,axiom,
    ( sgn_sgn_int
    = ( ^ [X3: int] : ( if_int @ ( X3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% sgn_if
thf(fact_1070_sgn__if,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [X3: code_integer] : ( if_Code_integer @ ( X3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X3 ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).

% sgn_if
thf(fact_1071_sgn__if,axiom,
    ( sgn_sgn_rat
    = ( ^ [X3: rat] : ( if_rat @ ( X3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_if
thf(fact_1072_sgn__1__neg,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% sgn_1_neg
thf(fact_1073_sgn__1__neg,axiom,
    ! [A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% sgn_1_neg
thf(fact_1074_sgn__1__neg,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% sgn_1_neg
thf(fact_1075_abs__not__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_1076_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_1077_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_1078_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_1079_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_1080_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_1081_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_1082_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_1083_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_1084_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_1085_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_1086_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_1087_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
     => ( ( uminus_uminus_int @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_1088_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_1089_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat )
     => ( ( uminus_uminus_rat @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_1090_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_group_add_class.ab_left_minus
thf(fact_1091_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_group_add_class.ab_left_minus
thf(fact_1092_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_group_add_class.ab_left_minus
thf(fact_1093_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% add_eq_0_iff
thf(fact_1094_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% add_eq_0_iff
thf(fact_1095_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% add_eq_0_iff
thf(fact_1096_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_1097_zero__neq__neg__one,axiom,
    ( zero_z3403309356797280102nteger
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% zero_neq_neg_one
thf(fact_1098_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_1099_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% less_minus_one_simps(4)
thf(fact_1100_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(4)
thf(fact_1101_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% less_minus_one_simps(4)
thf(fact_1102_less__minus__one__simps_I2_J,axiom,
    ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).

% less_minus_one_simps(2)
thf(fact_1103_less__minus__one__simps_I2_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).

% less_minus_one_simps(2)
thf(fact_1104_less__minus__one__simps_I2_J,axiom,
    ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).

% less_minus_one_simps(2)
thf(fact_1105_option_Omap__sel,axiom,
    ! [A: option_num,F: num > num] :
      ( ( A != none_num )
     => ( ( the_num @ ( map_option_num_num @ F @ A ) )
        = ( F @ ( the_num @ A ) ) ) ) ).

% option.map_sel
thf(fact_1106_Ints__double__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
     => ( ( ( plus_p5714425477246183910nteger @ A @ A )
          = zero_z3403309356797280102nteger )
        = ( A = zero_z3403309356797280102nteger ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_1107_Ints__double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( ( plus_plus_rat @ A @ A )
          = zero_zero_rat )
        = ( A = zero_zero_rat ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_1108_Ints__double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( ( plus_plus_int @ A @ A )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_1109_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% less_minus_one_simps(3)
thf(fact_1110_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_1111_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% less_minus_one_simps(3)
thf(fact_1112_less__minus__one__simps_I1_J,axiom,
    ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).

% less_minus_one_simps(1)
thf(fact_1113_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_1114_less__minus__one__simps_I1_J,axiom,
    ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).

% less_minus_one_simps(1)
thf(fact_1115_sgn__1__pos,axiom,
    ! [A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A )
        = one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% sgn_1_pos
thf(fact_1116_sgn__1__pos,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = one_one_rat )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% sgn_1_pos
thf(fact_1117_sgn__1__pos,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = one_one_int )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% sgn_1_pos
thf(fact_1118_Ints__odd__nonzero,axiom,
    ! [A: code_integer] :
      ( ( member_Code_integer @ A @ ring_11222124179247155820nteger )
     => ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ A )
       != zero_z3403309356797280102nteger ) ) ).

% Ints_odd_nonzero
thf(fact_1119_Ints__odd__nonzero,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A )
       != zero_zero_rat ) ) ).

% Ints_odd_nonzero
thf(fact_1120_Ints__odd__nonzero,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A )
       != zero_zero_int ) ) ).

% Ints_odd_nonzero
thf(fact_1121_execute__lookup,axiom,
    ! [R2: ref_array_a,H3: heap_e7401611519738050253t_unit] :
      ( ( heap_T5591663578999710211rray_a @ ( ref_lookup_array_a @ R2 ) @ H3 )
      = ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ ( ref_get_array_a @ H3 @ R2 ) @ ( produc584006145561248582it_nat @ H3 @ one_one_nat ) ) ) ) ).

% execute_lookup
thf(fact_1122_execute__lookup,axiom,
    ! [R2: ref_Product_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( heap_T875086893843062177t_unit @ ( ref_lo7930775386976318366t_unit @ R2 ) @ H3 )
      = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ ( ref_get_Product_unit @ H3 @ R2 ) @ ( produc584006145561248582it_nat @ H3 @ one_one_nat ) ) ) ) ).

% execute_lookup
thf(fact_1123_fails__assert_H,axiom,
    ! [P: $o,H3: heap_e7401611519738050253t_unit] :
      ( ( time_f8834461667527620124t_unit @ ( refine_Imp_assert @ P ) @ H3 )
      = ~ P ) ).

% fails_assert'
thf(fact_1124_add_Ogroup__axioms,axiom,
    group_int @ plus_plus_int @ zero_zero_int @ uminus_uminus_int ).

% add.group_axioms
thf(fact_1125_add_Ogroup__axioms,axiom,
    group_Code_integer @ plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ uminus1351360451143612070nteger ).

% add.group_axioms
thf(fact_1126_add_Ogroup__axioms,axiom,
    group_rat @ plus_plus_rat @ zero_zero_rat @ uminus_uminus_rat ).

% add.group_axioms
thf(fact_1127_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 ) ) ) ) ).

% num_of_nat_double
thf(fact_1128_Ints__nonzero__abs__ge1,axiom,
    ! [X: code_integer] :
      ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
     => ( ( X != zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_1129_Ints__nonzero__abs__ge1,axiom,
    ! [X: rat] :
      ( ( member_rat @ X @ ring_1_Ints_rat )
     => ( ( X != zero_zero_rat )
       => ( ord_less_eq_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_1130_Ints__nonzero__abs__ge1,axiom,
    ! [X: int] :
      ( ( member_int @ X @ ring_1_Ints_int )
     => ( ( X != zero_zero_int )
       => ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_1131_map__conv__bind__option,axiom,
    ( map_option_num_num
    = ( ^ [F2: num > num,X3: option_num] : ( bind_num_num @ X3 @ ( comp_n6731957995704128387um_num @ some_num @ F2 ) ) ) ) ).

% map_conv_bind_option
thf(fact_1132_nat__add__left__cancel__le,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% nat_add_left_cancel_le
thf(fact_1133_less__eq__option__None__code,axiom,
    ! [X: option_num] : ( ord_le6622620407824499402on_num @ none_num @ X ) ).

% less_eq_option_None_code
thf(fact_1134_le__zero__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_1135_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_1136_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_1137_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_1138_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_1139_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_1140_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_1141_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_1142_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_1143_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_1144_less__eq__option__Some,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_le2843612097646854710et_nat @ ( some_set_nat @ X ) @ ( some_set_nat @ Y ) )
      = ( ord_less_eq_set_nat @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1145_less__eq__option__Some,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_le2406147912482264968on_rat @ ( some_rat @ X ) @ ( some_rat @ Y ) )
      = ( ord_less_eq_rat @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1146_less__eq__option__Some,axiom,
    ! [X: num,Y: num] :
      ( ( ord_le6622620407824499402on_num @ ( some_num @ X ) @ ( some_num @ Y ) )
      = ( ord_less_eq_num @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1147_less__eq__option__Some,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_le5914376470875661696on_nat @ ( some_nat @ X ) @ ( some_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1148_less__eq__option__Some,axiom,
    ! [X: int,Y: int] :
      ( ( ord_le1736525451366464988on_int @ ( some_int @ X ) @ ( some_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_1149_less__eq__option__Some__None,axiom,
    ! [X: num] :
      ~ ( ord_le6622620407824499402on_num @ ( some_num @ X ) @ none_num ) ).

% less_eq_option_Some_None
thf(fact_1150_bind__runit,axiom,
    ! [X: option_num] :
      ( ( bind_num_num @ X @ some_num )
      = X ) ).

% bind_runit
thf(fact_1151_bind__runit,axiom,
    ! [X: option8956607266484857688it_nat] :
      ( ( bind_P7351952464966851536it_nat @ X @ some_P1914260805536162275it_nat )
      = X ) ).

% bind_runit
thf(fact_1152_add__le__same__cancel1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% add_le_same_cancel1
thf(fact_1153_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel1
thf(fact_1154_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_1155_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_1156_add__le__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% add_le_same_cancel2
thf(fact_1157_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel2
thf(fact_1158_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_1159_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_1160_le__add__same__cancel1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ).

% le_add_same_cancel1
thf(fact_1161_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).

% le_add_same_cancel1
thf(fact_1162_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_1163_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel1
thf(fact_1164_le__add__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ).

% le_add_same_cancel2
thf(fact_1165_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).

% le_add_same_cancel2
thf(fact_1166_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_1167_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel2
thf(fact_1168_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_1169_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_1170_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_1171_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ A ) )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_1172_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_1173_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_1174_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% neg_0_le_iff_le
thf(fact_1175_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% neg_0_le_iff_le
thf(fact_1176_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% neg_0_le_iff_le
thf(fact_1177_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_le_0_iff_le
thf(fact_1178_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% neg_le_0_iff_le
thf(fact_1179_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_le_0_iff_le
thf(fact_1180_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_1181_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% less_eq_neg_nonpos
thf(fact_1182_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% less_eq_neg_nonpos
thf(fact_1183_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_1184_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_1185_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_1186_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_1187_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_1188_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_1189_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% abs_le_self_iff
thf(fact_1190_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% abs_le_self_iff
thf(fact_1191_abs__le__self__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% abs_le_self_iff
thf(fact_1192_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
thf(fact_1193_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
thf(fact_1194_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
thf(fact_1195_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_nonpos
thf(fact_1196_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( abs_abs_rat @ A )
        = ( uminus_uminus_rat @ A ) ) ) ).

% abs_of_nonpos
thf(fact_1197_abs__of__nonpos,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( abs_abs_int @ A )
        = ( uminus_uminus_int @ A ) ) ) ).

% abs_of_nonpos
thf(fact_1198_ord__eq__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D2: set_nat] :
      ( ( A = B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ( C = D2 )
         => ( ord_less_eq_set_nat @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_1199_ord__eq__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( A = B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ( C = D2 )
         => ( ord_less_eq_rat @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_1200_ord__eq__le__eq__trans,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( A = B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ( C = D2 )
         => ( ord_less_eq_num @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_1201_ord__eq__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( C = D2 )
         => ( ord_less_eq_nat @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_1202_ord__eq__le__eq__trans,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ( C = D2 )
         => ( ord_less_eq_int @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_1203_accp__subset,axiom,
    ! [R12: product_prod_nat_nat > product_prod_nat_nat > $o,R23: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( ord_le5604493270027003598_nat_o @ R12 @ R23 )
     => ( ord_le704812498762024988_nat_o @ ( accp_P4275260045618599050at_nat @ R23 ) @ ( accp_P4275260045618599050at_nat @ R12 ) ) ) ).

% accp_subset
thf(fact_1204_accp__subset,axiom,
    ! [R12: product_prod_int_int > product_prod_int_int > $o,R23: product_prod_int_int > product_prod_int_int > $o] :
      ( ( ord_le1598226405681992910_int_o @ R12 @ R23 )
     => ( ord_le8369615600986905444_int_o @ ( accp_P1096762738010456898nt_int @ R23 ) @ ( accp_P1096762738010456898nt_int @ R12 ) ) ) ).

% accp_subset
thf(fact_1205_accp__subset,axiom,
    ! [R12: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o,R23: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o] :
      ( ( ord_le7226304311989208114_nat_o @ R12 @ R23 )
     => ( ord_le1658592502415435381_nat_o @ ( accp_P1862375125659990705et_nat @ R23 ) @ ( accp_P1862375125659990705et_nat @ R12 ) ) ) ).

% accp_subset
thf(fact_1206_accp__subset,axiom,
    ! [R12: list_nat > list_nat > $o,R23: list_nat > list_nat > $o] :
      ( ( ord_le6558929396352911974_nat_o @ R12 @ R23 )
     => ( ord_le1520216061033275535_nat_o @ ( accp_list_nat @ R23 ) @ ( accp_list_nat @ R12 ) ) ) ).

% accp_subset
thf(fact_1207_accp__subset,axiom,
    ! [R12: nat > nat > $o,R23: nat > nat > $o] :
      ( ( ord_le2646555220125990790_nat_o @ R12 @ R23 )
     => ( ord_less_eq_nat_o @ ( accp_nat @ R23 ) @ ( accp_nat @ R12 ) ) ) ).

% accp_subset
thf(fact_1208_le__some__optE,axiom,
    ! [M2: set_nat,X: option_set_nat] :
      ( ( ord_le2843612097646854710et_nat @ ( some_set_nat @ M2 ) @ X )
     => ~ ! [M4: set_nat] :
            ( ( X
              = ( some_set_nat @ M4 ) )
           => ~ ( ord_less_eq_set_nat @ M2 @ M4 ) ) ) ).

% le_some_optE
thf(fact_1209_le__some__optE,axiom,
    ! [M2: rat,X: option_rat] :
      ( ( ord_le2406147912482264968on_rat @ ( some_rat @ M2 ) @ X )
     => ~ ! [M4: rat] :
            ( ( X
              = ( some_rat @ M4 ) )
           => ~ ( ord_less_eq_rat @ M2 @ M4 ) ) ) ).

% le_some_optE
thf(fact_1210_le__some__optE,axiom,
    ! [M2: num,X: option_num] :
      ( ( ord_le6622620407824499402on_num @ ( some_num @ M2 ) @ X )
     => ~ ! [M4: num] :
            ( ( X
              = ( some_num @ M4 ) )
           => ~ ( ord_less_eq_num @ M2 @ M4 ) ) ) ).

% le_some_optE
thf(fact_1211_le__some__optE,axiom,
    ! [M2: nat,X: option_nat] :
      ( ( ord_le5914376470875661696on_nat @ ( some_nat @ M2 ) @ X )
     => ~ ! [M4: nat] :
            ( ( X
              = ( some_nat @ M4 ) )
           => ~ ( ord_less_eq_nat @ M2 @ M4 ) ) ) ).

% le_some_optE
thf(fact_1212_le__some__optE,axiom,
    ! [M2: int,X: option_int] :
      ( ( ord_le1736525451366464988on_int @ ( some_int @ M2 ) @ X )
     => ~ ! [M4: int] :
            ( ( X
              = ( some_int @ M4 ) )
           => ~ ( ord_less_eq_int @ M2 @ M4 ) ) ) ).

% le_some_optE
thf(fact_1213_pred__subset__eq2,axiom,
    ! [R4: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le3935385432712749774_nat_o
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R4 )
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ S2 ) )
      = ( ord_le1268244103169919719at_nat @ R4 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_1214_pred__subset__eq2,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R4 )
        @ ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S2 ) )
      = ( ord_le3146513528884898305at_nat @ R4 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_1215_pred__subset__eq2,axiom,
    ! [R4: set_Pr1281608226676607948nteger,S2: set_Pr1281608226676607948nteger] :
      ( ( ord_le4340812435750786203eger_o
        @ ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ R4 )
        @ ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ S2 ) )
      = ( ord_le653643898420964396nteger @ R4 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_1216_pred__subset__eq2,axiom,
    ! [R4: set_Pr3286484037609594932et_nat,S2: set_Pr3286484037609594932et_nat] :
      ( ( ord_le8000401564054156549_nat_o
        @ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ R4 )
        @ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ S2 ) )
      = ( ord_le5966269811547037844et_nat @ R4 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_1217_pred__subset__eq2,axiom,
    ! [R4: set_Pr8536935166611901872et_nat,S2: set_Pr8536935166611901872et_nat] :
      ( ( ord_le6753239538765779593_nat_o
        @ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ R4 )
        @ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ S2 ) )
      = ( ord_le4763372923235995152et_nat @ R4 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_1218_pred__subset__eq2,axiom,
    ! [R4: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
      ( ( ord_le5643404153117327598_int_o
        @ ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R4 )
        @ ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ S2 ) )
      = ( ord_le8725513860283290265nt_int @ R4 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_1219_pred__subset__eq2,axiom,
    ! [R4: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
      ( ( ord_le2124322318746777828_int_o
        @ ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R4 )
        @ ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ S2 ) )
      = ( ord_le135402666524580259nt_int @ R4 @ S2 ) ) ).

% pred_subset_eq2
thf(fact_1220_less__eq__option__def,axiom,
    ( ord_le2843612097646854710et_nat
    = ( ^ [X3: option_set_nat,Y3: option_set_nat] :
          ( case_o4401850862724306899et_nat @ $true
          @ ^ [Z4: set_nat] : ( case_o4401850862724306899et_nat @ $false @ ( ord_less_eq_set_nat @ Z4 ) @ Y3 )
          @ X3 ) ) ) ).

% less_eq_option_def
thf(fact_1221_less__eq__option__def,axiom,
    ( ord_le2406147912482264968on_rat
    = ( ^ [X3: option_rat,Y3: option_rat] :
          ( case_option_o_rat @ $true
          @ ^ [Z4: rat] : ( case_option_o_rat @ $false @ ( ord_less_eq_rat @ Z4 ) @ Y3 )
          @ X3 ) ) ) ).

% less_eq_option_def
thf(fact_1222_less__eq__option__def,axiom,
    ( ord_le6622620407824499402on_num
    = ( ^ [X3: option_num,Y3: option_num] :
          ( case_option_o_num @ $true
          @ ^ [Z4: num] : ( case_option_o_num @ $false @ ( ord_less_eq_num @ Z4 ) @ Y3 )
          @ X3 ) ) ) ).

% less_eq_option_def
thf(fact_1223_less__eq__option__def,axiom,
    ( ord_le5914376470875661696on_nat
    = ( ^ [X3: option_nat,Y3: option_nat] :
          ( case_option_o_nat @ $true
          @ ^ [Z4: nat] : ( case_option_o_nat @ $false @ ( ord_less_eq_nat @ Z4 ) @ Y3 )
          @ X3 ) ) ) ).

% less_eq_option_def
thf(fact_1224_less__eq__option__def,axiom,
    ( ord_le1736525451366464988on_int
    = ( ^ [X3: option_int,Y3: option_int] :
          ( case_option_o_int @ $true
          @ ^ [Z4: int] : ( case_option_o_int @ $false @ ( ord_less_eq_int @ Z4 ) @ Y3 )
          @ X3 ) ) ) ).

% less_eq_option_def
thf(fact_1225_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_1226_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1227_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1228_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1229_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1230_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1231_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1232_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1233_add__mono__thms__linordered__semiring_I1_J,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 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1234_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1235_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D2 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).

% add_mono
thf(fact_1236_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_mono
thf(fact_1237_add__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_mono
thf(fact_1238_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_1239_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_1240_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_left_mono
thf(fact_1241_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C3: nat] :
            ( B
           != ( plus_plus_nat @ A @ C3 ) ) ) ).

% less_eqE
thf(fact_1242_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_1243_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_1244_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_right_mono
thf(fact_1245_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
        ? [C2: nat] :
          ( B4
          = ( plus_plus_nat @ A5 @ C2 ) ) ) ) ).

% le_iff_add
thf(fact_1246_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_1247_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_1248_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_1249_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_1250_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_1251_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_1252_accp__subset__induct,axiom,
    ! [D3: product_prod_nat_nat > $o,R4: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( ord_le704812498762024988_nat_o @ D3 @ ( accp_P4275260045618599050at_nat @ R4 ) )
     => ( ! [X4: product_prod_nat_nat,Z3: product_prod_nat_nat] :
            ( ( D3 @ X4 )
           => ( ( R4 @ Z3 @ X4 )
             => ( D3 @ Z3 ) ) )
       => ( ( D3 @ X )
         => ( ! [X4: product_prod_nat_nat] :
                ( ( D3 @ X4 )
               => ( ! [Z5: product_prod_nat_nat] :
                      ( ( R4 @ Z5 @ X4 )
                     => ( P @ Z5 ) )
                 => ( P @ X4 ) ) )
           => ( P @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_1253_accp__subset__induct,axiom,
    ! [D3: product_prod_int_int > $o,R4: product_prod_int_int > product_prod_int_int > $o,X: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ D3 @ ( accp_P1096762738010456898nt_int @ R4 ) )
     => ( ! [X4: product_prod_int_int,Z3: product_prod_int_int] :
            ( ( D3 @ X4 )
           => ( ( R4 @ Z3 @ X4 )
             => ( D3 @ Z3 ) ) )
       => ( ( D3 @ X )
         => ( ! [X4: product_prod_int_int] :
                ( ( D3 @ X4 )
               => ( ! [Z5: product_prod_int_int] :
                      ( ( R4 @ Z5 @ X4 )
                     => ( P @ Z5 ) )
                 => ( P @ X4 ) ) )
           => ( P @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_1254_accp__subset__induct,axiom,
    ! [D3: produc2732055786443039994et_nat > $o,R4: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o,X: produc2732055786443039994et_nat,P: produc2732055786443039994et_nat > $o] :
      ( ( ord_le1658592502415435381_nat_o @ D3 @ ( accp_P1862375125659990705et_nat @ R4 ) )
     => ( ! [X4: produc2732055786443039994et_nat,Z3: produc2732055786443039994et_nat] :
            ( ( D3 @ X4 )
           => ( ( R4 @ Z3 @ X4 )
             => ( D3 @ Z3 ) ) )
       => ( ( D3 @ X )
         => ( ! [X4: produc2732055786443039994et_nat] :
                ( ( D3 @ X4 )
               => ( ! [Z5: produc2732055786443039994et_nat] :
                      ( ( R4 @ Z5 @ X4 )
                     => ( P @ Z5 ) )
                 => ( P @ X4 ) ) )
           => ( P @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_1255_accp__subset__induct,axiom,
    ! [D3: list_nat > $o,R4: list_nat > list_nat > $o,X: list_nat,P: list_nat > $o] :
      ( ( ord_le1520216061033275535_nat_o @ D3 @ ( accp_list_nat @ R4 ) )
     => ( ! [X4: list_nat,Z3: list_nat] :
            ( ( D3 @ X4 )
           => ( ( R4 @ Z3 @ X4 )
             => ( D3 @ Z3 ) ) )
       => ( ( D3 @ X )
         => ( ! [X4: list_nat] :
                ( ( D3 @ X4 )
               => ( ! [Z5: list_nat] :
                      ( ( R4 @ Z5 @ X4 )
                     => ( P @ Z5 ) )
                 => ( P @ X4 ) ) )
           => ( P @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_1256_accp__subset__induct,axiom,
    ! [D3: nat > $o,R4: nat > nat > $o,X: nat,P: nat > $o] :
      ( ( ord_less_eq_nat_o @ D3 @ ( accp_nat @ R4 ) )
     => ( ! [X4: nat,Z3: nat] :
            ( ( D3 @ X4 )
           => ( ( R4 @ Z3 @ X4 )
             => ( D3 @ Z3 ) ) )
       => ( ( D3 @ X )
         => ( ! [X4: nat] :
                ( ( D3 @ X4 )
               => ( ! [Z5: nat] :
                      ( ( R4 @ Z5 @ X4 )
                     => ( P @ Z5 ) )
                 => ( P @ X4 ) ) )
           => ( P @ X ) ) ) ) ) ).

% accp_subset_induct
thf(fact_1257_le__numeral__extra_I4_J,axiom,
    ord_le3102999989581377725nteger @ one_one_Code_integer @ one_one_Code_integer ).

% le_numeral_extra(4)
thf(fact_1258_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_1259_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_1260_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_1261_subrelI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ! [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ R2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ S ) )
     => ( ord_le1268244103169919719at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_1262_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y5: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ S ) )
     => ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_1263_subrelI,axiom,
    ! [R2: set_Pr1281608226676607948nteger,S: set_Pr1281608226676607948nteger] :
      ( ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
          ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y5 ) @ R2 )
         => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y5 ) @ S ) )
     => ( ord_le653643898420964396nteger @ R2 @ S ) ) ).

% subrelI
thf(fact_1264_subrelI,axiom,
    ! [R2: set_Pr3286484037609594932et_nat,S: set_Pr3286484037609594932et_nat] :
      ( ! [X4: produc3658429121746597890et_nat > $o,Y5: produc3658429121746597890et_nat] :
          ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y5 ) @ R2 )
         => ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y5 ) @ S ) )
     => ( ord_le5966269811547037844et_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_1265_subrelI,axiom,
    ! [R2: set_Pr8536935166611901872et_nat,S: set_Pr8536935166611901872et_nat] :
      ( ! [X4: produc3658429121746597890et_nat > $o,Y5: produc3925858234332021118et_nat] :
          ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y5 ) @ R2 )
         => ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y5 ) @ S ) )
     => ( ord_le4763372923235995152et_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_1266_subrelI,axiom,
    ! [R2: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
      ( ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] :
          ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y5 ) @ R2 )
         => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y5 ) @ S ) )
     => ( ord_le8725513860283290265nt_int @ R2 @ S ) ) ).

% subrelI
thf(fact_1267_subrelI,axiom,
    ! [R2: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
      ( ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
          ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y5 ) @ R2 )
         => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y5 ) @ S ) )
     => ( ord_le135402666524580259nt_int @ R2 @ S ) ) ).

% subrelI
thf(fact_1268_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_minus_iff
thf(fact_1269_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).

% le_minus_iff
thf(fact_1270_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% le_minus_iff
thf(fact_1271_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_1272_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_1273_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_1274_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_imp_neg_le
thf(fact_1275_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% le_imp_neg_le
thf(fact_1276_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% le_imp_neg_le
thf(fact_1277_exists__leI,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [N7: nat] :
            ( ( ord_less_nat @ N7 @ N2 )
           => ~ ( P @ N7 ) )
       => ( P @ N2 ) )
     => ? [N3: nat] :
          ( ( ord_less_eq_nat @ N3 @ N2 )
          & ( P @ N3 ) ) ) ).

% exists_leI
thf(fact_1278_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M: nat,N4: nat] :
        ? [K4: nat] :
          ( N4
          = ( plus_plus_nat @ M @ K4 ) ) ) ) ).

% nat_le_iff_add
thf(fact_1279_trans__le__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_le_add2
thf(fact_1280_trans__le__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_le_add1
thf(fact_1281_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 ) ) ) ).

% add_le_mono1
thf(fact_1282_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 ) ) ) ) ).

% add_le_mono
thf(fact_1283_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N: nat] :
          ( L
          = ( plus_plus_nat @ K @ N ) ) ) ).

% le_Suc_ex
thf(fact_1284_add__leD2,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
     => ( ord_less_eq_nat @ K @ N2 ) ) ).

% add_leD2
thf(fact_1285_add__leD1,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% add_leD1
thf(fact_1286_le__add2,axiom,
    ! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).

% le_add2
thf(fact_1287_le__add1,axiom,
    ! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).

% le_add1
thf(fact_1288_add__leE,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
     => ~ ( ( ord_less_eq_nat @ M2 @ N2 )
         => ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).

% add_leE
thf(fact_1289_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
     => ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% abs_le_D1
thf(fact_1290_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% abs_le_D1
thf(fact_1291_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% abs_le_D1
thf(fact_1292_abs__ge__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_self
thf(fact_1293_abs__ge__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).

% abs_ge_self
thf(fact_1294_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_1295_less__eq__option__None,axiom,
    ! [X: option_num] : ( ord_le6622620407824499402on_num @ none_num @ X ) ).

% less_eq_option_None
thf(fact_1296_less__eq__option__None__is__None,axiom,
    ! [X: option_num] :
      ( ( ord_le6622620407824499402on_num @ X @ none_num )
     => ( X = none_num ) ) ).

% less_eq_option_None_is_None
thf(fact_1297_bind__eq__Some__conv,axiom,
    ! [F: option_num,G: num > option_num,X: num] :
      ( ( ( bind_num_num @ F @ G )
        = ( some_num @ X ) )
      = ( ? [Y3: num] :
            ( ( F
              = ( some_num @ Y3 ) )
            & ( ( G @ Y3 )
              = ( some_num @ X ) ) ) ) ) ).

% bind_eq_Some_conv
thf(fact_1298_bind__eq__Some__conv,axiom,
    ! [F: option8956607266484857688it_nat,G: produc8664842809031399944it_nat > option_num,X: num] :
      ( ( ( bind_P4387644102537648098at_num @ F @ G )
        = ( some_num @ X ) )
      = ( ? [Y3: produc8664842809031399944it_nat] :
            ( ( F
              = ( some_P1914260805536162275it_nat @ Y3 ) )
            & ( ( G @ Y3 )
              = ( some_num @ X ) ) ) ) ) ).

% bind_eq_Some_conv
thf(fact_1299_bind__eq__Some__conv,axiom,
    ! [F: option_num,G: num > option8956607266484857688it_nat,X: produc8664842809031399944it_nat] :
      ( ( ( bind_n550516262659884002it_nat @ F @ G )
        = ( some_P1914260805536162275it_nat @ X ) )
      = ( ? [Y3: num] :
            ( ( F
              = ( some_num @ Y3 ) )
            & ( ( G @ Y3 )
              = ( some_P1914260805536162275it_nat @ X ) ) ) ) ) ).

% bind_eq_Some_conv
thf(fact_1300_bind__eq__Some__conv,axiom,
    ! [F: option8956607266484857688it_nat,G: produc8664842809031399944it_nat > option8956607266484857688it_nat,X: produc8664842809031399944it_nat] :
      ( ( ( bind_P7351952464966851536it_nat @ F @ G )
        = ( some_P1914260805536162275it_nat @ X ) )
      = ( ? [Y3: produc8664842809031399944it_nat] :
            ( ( F
              = ( some_P1914260805536162275it_nat @ Y3 ) )
            & ( ( G @ Y3 )
              = ( some_P1914260805536162275it_nat @ X ) ) ) ) ) ).

% bind_eq_Some_conv
thf(fact_1301_bind_Obind__lzero,axiom,
    ! [F: produc6714399278180371524it_nat > option3216581401566546698it_nat] :
      ( ( bind_P1787608729393679550it_nat @ none_P2587165478634041381it_nat @ F )
      = none_P2587165478634041381it_nat ) ).

% bind.bind_lzero
thf(fact_1302_bind_Obind__lzero,axiom,
    ! [F: produc6714399278180371524it_nat > option_num] :
      ( ( bind_P7816469540982926368at_num @ none_P2587165478634041381it_nat @ F )
      = none_num ) ).

% bind.bind_lzero
thf(fact_1303_bind_Obind__lzero,axiom,
    ! [F: produc6714399278180371524it_nat > option8956607266484857688it_nat] :
      ( ( bind_P3087804342168909838it_nat @ none_P2587165478634041381it_nat @ F )
      = none_P9117596204409417319it_nat ) ).

% bind.bind_lzero
thf(fact_1304_bind_Obind__lzero,axiom,
    ! [F: num > option3216581401566546698it_nat] :
      ( ( bind_n6186876254402743274it_nat @ none_num @ F )
      = none_P2587165478634041381it_nat ) ).

% bind.bind_lzero
thf(fact_1305_bind_Obind__lzero,axiom,
    ! [F: num > option_num] :
      ( ( bind_num_num @ none_num @ F )
      = none_num ) ).

% bind.bind_lzero
thf(fact_1306_bind_Obind__lzero,axiom,
    ! [F: num > option8956607266484857688it_nat] :
      ( ( bind_n550516262659884002it_nat @ none_num @ F )
      = none_P9117596204409417319it_nat ) ).

% bind.bind_lzero
thf(fact_1307_bind_Obind__lzero,axiom,
    ! [F: produc8664842809031399944it_nat > option3216581401566546698it_nat] :
      ( ( bind_P1106314856888375420it_nat @ none_P9117596204409417319it_nat @ F )
      = none_P2587165478634041381it_nat ) ).

% bind.bind_lzero
thf(fact_1308_bind_Obind__lzero,axiom,
    ! [F: produc8664842809031399944it_nat > option_num] :
      ( ( bind_P4387644102537648098at_num @ none_P9117596204409417319it_nat @ F )
      = none_num ) ).

% bind.bind_lzero
thf(fact_1309_bind_Obind__lzero,axiom,
    ! [F: produc8664842809031399944it_nat > option8956607266484857688it_nat] :
      ( ( bind_P7351952464966851536it_nat @ none_P9117596204409417319it_nat @ F )
      = none_P9117596204409417319it_nat ) ).

% bind.bind_lzero
thf(fact_1310_add__decreasing,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ C @ B )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_1311_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_1312_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_1313_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_1314_add__increasing,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ B @ C )
       => ( ord_le3102999989581377725nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_1315_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_1316_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_1317_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_1318_add__decreasing2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ A @ B )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1319_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1320_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1321_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1322_add__increasing2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( ord_le3102999989581377725nteger @ B @ A )
       => ( ord_le3102999989581377725nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_1323_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_1324_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_1325_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_1326_add__nonneg__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1327_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1328_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1329_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1330_add__nonpos__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_nonpos_nonpos
thf(fact_1331_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_nonpos_nonpos
thf(fact_1332_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_1333_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_1334_add__nonneg__eq__0__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ( plus_p5714425477246183910nteger @ X @ Y )
            = zero_z3403309356797280102nteger )
          = ( ( X = zero_z3403309356797280102nteger )
            & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1335_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1336_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1337_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1338_add__nonpos__eq__0__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ Y @ zero_z3403309356797280102nteger )
       => ( ( ( plus_p5714425477246183910nteger @ X @ Y )
            = zero_z3403309356797280102nteger )
          = ( ( X = zero_z3403309356797280102nteger )
            & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1339_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1340_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1341_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y @ zero_zero_int )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1342_not__one__le__zero,axiom,
    ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).

% not_one_le_zero
thf(fact_1343_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_1344_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_1345_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_1346_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1347_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1348_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1349_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_int @ zero_zero_int @ one_one_int ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1350_zero__less__one__class_Ozero__le__one,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% zero_less_one_class.zero_le_one
thf(fact_1351_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one_class.zero_le_one
thf(fact_1352_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one_class.zero_le_one
thf(fact_1353_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_int @ zero_zero_int @ one_one_int ).

% zero_less_one_class.zero_le_one
thf(fact_1354_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ I @ J )
        & ( ord_le6747313008572928689nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1355_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1356_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1357_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1358_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le6747313008572928689nteger @ I @ J )
        & ( ord_le3102999989581377725nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1359_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1360_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1361_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1362_add__le__less__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_1363_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_1364_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_1365_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_1366_add__less__le__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C @ D2 )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_1367_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_1368_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_1369_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_1370_le__minus__one__simps_I2_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).

% le_minus_one_simps(2)
thf(fact_1371_le__minus__one__simps_I2_J,axiom,
    ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).

% le_minus_one_simps(2)
thf(fact_1372_le__minus__one__simps_I2_J,axiom,
    ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).

% le_minus_one_simps(2)
thf(fact_1373_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(4)
thf(fact_1374_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% le_minus_one_simps(4)
thf(fact_1375_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% le_minus_one_simps(4)
thf(fact_1376_abs__ge__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_zero
thf(fact_1377_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_1378_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_1379_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_1380_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_1381_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_1382_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
       => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_1383_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
       => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_1384_abs__leI,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
       => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_1385_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_1386_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_1387_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_1388_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
      = ( ( ord_le3102999989581377725nteger @ A @ B )
        & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_1389_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
      = ( ( ord_less_eq_rat @ A @ B )
        & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_1390_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
      = ( ( ord_less_eq_int @ A @ B )
        & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_1391_abs__ge__minus__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_minus_self
thf(fact_1392_abs__ge__minus__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).

% abs_ge_minus_self
thf(fact_1393_abs__ge__minus__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).

% abs_ge_minus_self
thf(fact_1394_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M2: nat,K: nat] :
      ( ! [M5: nat,N: nat] :
          ( ( ord_less_nat @ M5 @ N )
         => ( ord_less_nat @ ( F @ M5 ) @ ( F @ N ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_1395_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ X )
      = ( X != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
thf(fact_1396_bind__split__asm,axiom,
    ! [P: option3216581401566546698it_nat > $o,M2: option3216581401566546698it_nat,F: produc6714399278180371524it_nat > option3216581401566546698it_nat] :
      ( ( P @ ( bind_P1787608729393679550it_nat @ M2 @ F ) )
      = ( ~ ( ( ( M2 = none_P2587165478634041381it_nat )
              & ~ ( P @ none_P2587165478634041381it_nat ) )
            | ? [X3: produc6714399278180371524it_nat] :
                ( ( M2
                  = ( some_P6105628898608307625it_nat @ X3 ) )
                & ~ ( P @ ( F @ X3 ) ) ) ) ) ) ).

% bind_split_asm
thf(fact_1397_bind__split__asm,axiom,
    ! [P: option_num > $o,M2: option3216581401566546698it_nat,F: produc6714399278180371524it_nat > option_num] :
      ( ( P @ ( bind_P7816469540982926368at_num @ M2 @ F ) )
      = ( ~ ( ( ( M2 = none_P2587165478634041381it_nat )
              & ~ ( P @ none_num ) )
            | ? [X3: produc6714399278180371524it_nat] :
                ( ( M2
                  = ( some_P6105628898608307625it_nat @ X3 ) )
                & ~ ( P @ ( F @ X3 ) ) ) ) ) ) ).

% bind_split_asm
thf(fact_1398_bind__split__asm,axiom,
    ! [P: option8956607266484857688it_nat > $o,M2: option3216581401566546698it_nat,F: produc6714399278180371524it_nat > option8956607266484857688it_nat] :
      ( ( P @ ( bind_P3087804342168909838it_nat @ M2 @ F ) )
      = ( ~ ( ( ( M2 = none_P2587165478634041381it_nat )
              & ~ ( P @ none_P9117596204409417319it_nat ) )
            | ? [X3: produc6714399278180371524it_nat] :
                ( ( M2
                  = ( some_P6105628898608307625it_nat @ X3 ) )
                & ~ ( P @ ( F @ X3 ) ) ) ) ) ) ).

% bind_split_asm
thf(fact_1399_bind__split__asm,axiom,
    ! [P: option3216581401566546698it_nat > $o,M2: option_num,F: num > option3216581401566546698it_nat] :
      ( ( P @ ( bind_n6186876254402743274it_nat @ M2 @ F ) )
      = ( ~ ( ( ( M2 = none_num )
              & ~ ( P @ none_P2587165478634041381it_nat ) )
            | ? [X3: num] :
                ( ( M2
                  = ( some_num @ X3 ) )
                & ~ ( P @ ( F @ X3 ) ) ) ) ) ) ).

% bind_split_asm
thf(fact_1400_bind__split__asm,axiom,
    ! [P: option_num > $o,M2: option_num,F: num > option_num] :
      ( ( P @ ( bind_num_num @ M2 @ F ) )
      = ( ~ ( ( ( M2 = none_num )
              & ~ ( P @ none_num ) )
            | ? [X3: num] :
                ( ( M2
                  = ( some_num @ X3 ) )
                & ~ ( P @ ( F @ X3 ) ) ) ) ) ) ).

% bind_split_asm
thf(fact_1401_bind__split__asm,axiom,
    ! [P: option8956607266484857688it_nat > $o,M2: option_num,F: num > option8956607266484857688it_nat] :
      ( ( P @ ( bind_n550516262659884002it_nat @ M2 @ F ) )
      = ( ~ ( ( ( M2 = none_num )
              & ~ ( P @ none_P9117596204409417319it_nat ) )
            | ? [X3: num] :
                ( ( M2
                  = ( some_num @ X3 ) )
                & ~ ( P @ ( F @ X3 ) ) ) ) ) ) ).

% bind_split_asm
thf(fact_1402_bind__split__asm,axiom,
    ! [P: option3216581401566546698it_nat > $o,M2: option8956607266484857688it_nat,F: produc8664842809031399944it_nat > option3216581401566546698it_nat] :
      ( ( P @ ( bind_P1106314856888375420it_nat @ M2 @ F ) )
      = ( ~ ( ( ( M2 = none_P9117596204409417319it_nat )
              & ~ ( P @ none_P2587165478634041381it_nat ) )
            | ? [X3: produc8664842809031399944it_nat] :
                ( ( M2
                  = ( some_P1914260805536162275it_nat @ X3 ) )
                & ~ ( P @ ( F @ X3 ) ) ) ) ) ) ).

% bind_split_asm
thf(fact_1403_bind__split__asm,axiom,
    ! [P: option_num > $o,M2: option8956607266484857688it_nat,F: produc8664842809031399944it_nat > option_num] :
      ( ( P @ ( bind_P4387644102537648098at_num @ M2 @ F ) )
      = ( ~ ( ( ( M2 = none_P9117596204409417319it_nat )
              & ~ ( P @ none_num ) )
            | ? [X3: produc8664842809031399944it_nat] :
                ( ( M2
                  = ( some_P1914260805536162275it_nat @ X3 ) )
                & ~ ( P @ ( F @ X3 ) ) ) ) ) ) ).

% bind_split_asm
thf(fact_1404_bind__split__asm,axiom,
    ! [P: option8956607266484857688it_nat > $o,M2: option8956607266484857688it_nat,F: produc8664842809031399944it_nat > option8956607266484857688it_nat] :
      ( ( P @ ( bind_P7351952464966851536it_nat @ M2 @ F ) )
      = ( ~ ( ( ( M2 = none_P9117596204409417319it_nat )
              & ~ ( P @ none_P9117596204409417319it_nat ) )
            | ? [X3: produc8664842809031399944it_nat] :
                ( ( M2
                  = ( some_P1914260805536162275it_nat @ X3 ) )
                & ~ ( P @ ( F @ X3 ) ) ) ) ) ) ).

% bind_split_asm
thf(fact_1405_bind__split,axiom,
    ! [P: option3216581401566546698it_nat > $o,M2: option3216581401566546698it_nat,F: produc6714399278180371524it_nat > option3216581401566546698it_nat] :
      ( ( P @ ( bind_P1787608729393679550it_nat @ M2 @ F ) )
      = ( ( ( M2 = none_P2587165478634041381it_nat )
         => ( P @ none_P2587165478634041381it_nat ) )
        & ! [V2: produc6714399278180371524it_nat] :
            ( ( M2
              = ( some_P6105628898608307625it_nat @ V2 ) )
           => ( P @ ( F @ V2 ) ) ) ) ) ).

% bind_split
thf(fact_1406_bind__split,axiom,
    ! [P: option_num > $o,M2: option3216581401566546698it_nat,F: produc6714399278180371524it_nat > option_num] :
      ( ( P @ ( bind_P7816469540982926368at_num @ M2 @ F ) )
      = ( ( ( M2 = none_P2587165478634041381it_nat )
         => ( P @ none_num ) )
        & ! [V2: produc6714399278180371524it_nat] :
            ( ( M2
              = ( some_P6105628898608307625it_nat @ V2 ) )
           => ( P @ ( F @ V2 ) ) ) ) ) ).

% bind_split
thf(fact_1407_bind__split,axiom,
    ! [P: option8956607266484857688it_nat > $o,M2: option3216581401566546698it_nat,F: produc6714399278180371524it_nat > option8956607266484857688it_nat] :
      ( ( P @ ( bind_P3087804342168909838it_nat @ M2 @ F ) )
      = ( ( ( M2 = none_P2587165478634041381it_nat )
         => ( P @ none_P9117596204409417319it_nat ) )
        & ! [V2: produc6714399278180371524it_nat] :
            ( ( M2
              = ( some_P6105628898608307625it_nat @ V2 ) )
           => ( P @ ( F @ V2 ) ) ) ) ) ).

% bind_split
thf(fact_1408_bind__split,axiom,
    ! [P: option3216581401566546698it_nat > $o,M2: option_num,F: num > option3216581401566546698it_nat] :
      ( ( P @ ( bind_n6186876254402743274it_nat @ M2 @ F ) )
      = ( ( ( M2 = none_num )
         => ( P @ none_P2587165478634041381it_nat ) )
        & ! [V2: num] :
            ( ( M2
              = ( some_num @ V2 ) )
           => ( P @ ( F @ V2 ) ) ) ) ) ).

% bind_split
thf(fact_1409_bind__split,axiom,
    ! [P: option_num > $o,M2: option_num,F: num > option_num] :
      ( ( P @ ( bind_num_num @ M2 @ F ) )
      = ( ( ( M2 = none_num )
         => ( P @ none_num ) )
        & ! [V2: num] :
            ( ( M2
              = ( some_num @ V2 ) )
           => ( P @ ( F @ V2 ) ) ) ) ) ).

% bind_split
thf(fact_1410_bind__split,axiom,
    ! [P: option8956607266484857688it_nat > $o,M2: option_num,F: num > option8956607266484857688it_nat] :
      ( ( P @ ( bind_n550516262659884002it_nat @ M2 @ F ) )
      = ( ( ( M2 = none_num )
         => ( P @ none_P9117596204409417319it_nat ) )
        & ! [V2: num] :
            ( ( M2
              = ( some_num @ V2 ) )
           => ( P @ ( F @ V2 ) ) ) ) ) ).

% bind_split
thf(fact_1411_bind__split,axiom,
    ! [P: option3216581401566546698it_nat > $o,M2: option8956607266484857688it_nat,F: produc8664842809031399944it_nat > option3216581401566546698it_nat] :
      ( ( P @ ( bind_P1106314856888375420it_nat @ M2 @ F ) )
      = ( ( ( M2 = none_P9117596204409417319it_nat )
         => ( P @ none_P2587165478634041381it_nat ) )
        & ! [V2: produc8664842809031399944it_nat] :
            ( ( M2
              = ( some_P1914260805536162275it_nat @ V2 ) )
           => ( P @ ( F @ V2 ) ) ) ) ) ).

% bind_split
thf(fact_1412_bind__split,axiom,
    ! [P: option_num > $o,M2: option8956607266484857688it_nat,F: produc8664842809031399944it_nat > option_num] :
      ( ( P @ ( bind_P4387644102537648098at_num @ M2 @ F ) )
      = ( ( ( M2 = none_P9117596204409417319it_nat )
         => ( P @ none_num ) )
        & ! [V2: produc8664842809031399944it_nat] :
            ( ( M2
              = ( some_P1914260805536162275it_nat @ V2 ) )
           => ( P @ ( F @ V2 ) ) ) ) ) ).

% bind_split
thf(fact_1413_bind__split,axiom,
    ! [P: option8956607266484857688it_nat > $o,M2: option8956607266484857688it_nat,F: produc8664842809031399944it_nat > option8956607266484857688it_nat] :
      ( ( P @ ( bind_P7351952464966851536it_nat @ M2 @ F ) )
      = ( ( ( M2 = none_P9117596204409417319it_nat )
         => ( P @ none_P9117596204409417319it_nat ) )
        & ! [V2: produc8664842809031399944it_nat] :
            ( ( M2
              = ( some_P1914260805536162275it_nat @ V2 ) )
           => ( P @ ( F @ V2 ) ) ) ) ) ).

% bind_split
thf(fact_1414_mlex__leq,axiom,
    ! [F: set_Pr1261947904930325089at_nat > nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R4: set_Pr4329608150637261639at_nat] :
      ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R4 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( mlex_p3502713205431853936at_nat @ F @ R4 ) ) ) ) ).

% mlex_leq
thf(fact_1415_mlex__leq,axiom,
    ! [F: nat > nat,X: nat,Y: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R4 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( mlex_prod_nat @ F @ R4 ) ) ) ) ).

% mlex_leq
thf(fact_1416_bind__eq__None__conv,axiom,
    ! [A: option3216581401566546698it_nat,F: produc6714399278180371524it_nat > option3216581401566546698it_nat] :
      ( ( ( bind_P1787608729393679550it_nat @ A @ F )
        = none_P2587165478634041381it_nat )
      = ( ( A = none_P2587165478634041381it_nat )
        | ( ( F @ ( the_Pr5856008525237892824it_nat @ A ) )
          = none_P2587165478634041381it_nat ) ) ) ).

% bind_eq_None_conv
thf(fact_1417_bind__eq__None__conv,axiom,
    ! [A: option_num,F: num > option3216581401566546698it_nat] :
      ( ( ( bind_n6186876254402743274it_nat @ A @ F )
        = none_P2587165478634041381it_nat )
      = ( ( A = none_num )
        | ( ( F @ ( the_num @ A ) )
          = none_P2587165478634041381it_nat ) ) ) ).

% bind_eq_None_conv
thf(fact_1418_bind__eq__None__conv,axiom,
    ! [A: option8956607266484857688it_nat,F: produc8664842809031399944it_nat > option3216581401566546698it_nat] :
      ( ( ( bind_P1106314856888375420it_nat @ A @ F )
        = none_P2587165478634041381it_nat )
      = ( ( A = none_P9117596204409417319it_nat )
        | ( ( F @ ( the_Pr7776125533777466484it_nat @ A ) )
          = none_P2587165478634041381it_nat ) ) ) ).

% bind_eq_None_conv
thf(fact_1419_bind__eq__None__conv,axiom,
    ! [A: option3216581401566546698it_nat,F: produc6714399278180371524it_nat > option_num] :
      ( ( ( bind_P7816469540982926368at_num @ A @ F )
        = none_num )
      = ( ( A = none_P2587165478634041381it_nat )
        | ( ( F @ ( the_Pr5856008525237892824it_nat @ A ) )
          = none_num ) ) ) ).

% bind_eq_None_conv
thf(fact_1420_bind__eq__None__conv,axiom,
    ! [A: option_num,F: num > option_num] :
      ( ( ( bind_num_num @ A @ F )
        = none_num )
      = ( ( A = none_num )
        | ( ( F @ ( the_num @ A ) )
          = none_num ) ) ) ).

% bind_eq_None_conv
thf(fact_1421_bind__eq__None__conv,axiom,
    ! [A: option8956607266484857688it_nat,F: produc8664842809031399944it_nat > option_num] :
      ( ( ( bind_P4387644102537648098at_num @ A @ F )
        = none_num )
      = ( ( A = none_P9117596204409417319it_nat )
        | ( ( F @ ( the_Pr7776125533777466484it_nat @ A ) )
          = none_num ) ) ) ).

% bind_eq_None_conv
thf(fact_1422_bind__eq__None__conv,axiom,
    ! [A: option3216581401566546698it_nat,F: produc6714399278180371524it_nat > option8956607266484857688it_nat] :
      ( ( ( bind_P3087804342168909838it_nat @ A @ F )
        = none_P9117596204409417319it_nat )
      = ( ( A = none_P2587165478634041381it_nat )
        | ( ( F @ ( the_Pr5856008525237892824it_nat @ A ) )
          = none_P9117596204409417319it_nat ) ) ) ).

% bind_eq_None_conv
thf(fact_1423_bind__eq__None__conv,axiom,
    ! [A: option_num,F: num > option8956607266484857688it_nat] :
      ( ( ( bind_n550516262659884002it_nat @ A @ F )
        = none_P9117596204409417319it_nat )
      = ( ( A = none_num )
        | ( ( F @ ( the_num @ A ) )
          = none_P9117596204409417319it_nat ) ) ) ).

% bind_eq_None_conv
thf(fact_1424_bind__eq__None__conv,axiom,
    ! [A: option8956607266484857688it_nat,F: produc8664842809031399944it_nat > option8956607266484857688it_nat] :
      ( ( ( bind_P7351952464966851536it_nat @ A @ F )
        = none_P9117596204409417319it_nat )
      = ( ( A = none_P9117596204409417319it_nat )
        | ( ( F @ ( the_Pr7776125533777466484it_nat @ A ) )
          = none_P9117596204409417319it_nat ) ) ) ).

% bind_eq_None_conv
thf(fact_1425_add__neg__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_neg_nonpos
thf(fact_1426_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_neg_nonpos
thf(fact_1427_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_1428_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_1429_add__nonneg__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_1430_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_1431_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_1432_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_1433_add__nonpos__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% add_nonpos_neg
thf(fact_1434_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_nonpos_neg
thf(fact_1435_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_1436_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_1437_add__pos__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_1438_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_1439_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_1440_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_1441_add__strict__increasing,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ B @ C )
       => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_1442_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_1443_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_1444_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_1445_add__strict__increasing2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ B @ C )
       => ( ord_le6747313008572928689nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_1446_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_1447_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_1448_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_1449_discrete,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A5: code_integer] : ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A5 @ one_one_Code_integer ) ) ) ) ).

% discrete
thf(fact_1450_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A5 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_1451_discrete,axiom,
    ( ord_less_int
    = ( ^ [A5: int] : ( ord_less_eq_int @ ( plus_plus_int @ A5 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_1452_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_1453_le__minus__one__simps_I1_J,axiom,
    ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).

% le_minus_one_simps(1)
thf(fact_1454_le__minus__one__simps_I1_J,axiom,
    ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).

% le_minus_one_simps(1)
thf(fact_1455_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_1456_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% le_minus_one_simps(3)
thf(fact_1457_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% le_minus_one_simps(3)
thf(fact_1458_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E2 ) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_1459_abs__minus__le__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_1460_abs__minus__le__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).

% abs_minus_le_zero
thf(fact_1461_abs__minus__le__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).

% abs_minus_le_zero
thf(fact_1462_success__changeI,axiom,
    ! [F: product_unit > product_unit,R2: ref_Product_unit,H3: heap_e7401611519738050253t_unit] : ( heap_T584514906347983379t_unit @ ( ref_ch7259622376331601608t_unit @ F @ R2 ) @ H3 ) ).

% success_changeI
thf(fact_1463_success__changeI,axiom,
    ! [F: array_a > array_a,R2: ref_array_a,H3: heap_e7401611519738050253t_unit] : ( heap_T6932021131025641745rray_a @ ( ref_change_array_a @ F @ R2 ) @ H3 ) ).

% success_changeI
thf(fact_1464_semiring__norm_I6_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_num @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) )
      = ( bit0 @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% semiring_norm(6)
thf(fact_1465_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E2 ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_1466_convex__bound__lt,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ A )
     => ( ( ord_le6747313008572928689nteger @ Y @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V )
           => ( ( ( plus_p5714425477246183910nteger @ U @ V )
                = one_one_Code_integer )
             => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_1467_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X @ A )
     => ( ( ord_less_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_1468_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_int @ X @ A )
     => ( ( ord_less_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_1469_Ints__eq__abs__less1,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( member_Code_integer @ X @ ring_11222124179247155820nteger )
     => ( ( member_Code_integer @ Y @ ring_11222124179247155820nteger )
       => ( ( X = Y )
          = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ Y ) ) @ one_one_Code_integer ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_1470_Ints__eq__abs__less1,axiom,
    ! [X: rat,Y: rat] :
      ( ( member_rat @ X @ ring_1_Ints_rat )
     => ( ( member_rat @ Y @ ring_1_Ints_rat )
       => ( ( X = Y )
          = ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ Y ) ) @ one_one_rat ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_1471_Ints__eq__abs__less1,axiom,
    ! [X: int,Y: int] :
      ( ( member_int @ X @ ring_1_Ints_int )
     => ( ( member_int @ Y @ ring_1_Ints_int )
       => ( ( X = Y )
          = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Y ) ) @ one_one_int ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_1472_nat0__intermed__int__val,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N2 )
              & ( ( F @ I2 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_1473_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 ) ) ) ).

% diff_diff_left
thf(fact_1474_nat__1__eq__mult__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M2 @ N2 ) )
      = ( ( M2 = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_1475_nat__mult__eq__1__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( times_times_nat @ M2 @ N2 )
        = one_one_nat )
      = ( ( M2 = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_1476_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ A @ A )
      = zero_z1048942125864253310at_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1477_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1478_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1479_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ A )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1480_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1481_diff__zero,axiom,
    ! [A: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ A @ zero_z1048942125864253310at_nat )
      = A ) ).

% diff_zero
thf(fact_1482_diff__zero,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% diff_zero
thf(fact_1483_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_1484_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_1485_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_1486_zero__diff,axiom,
    ! [A: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ zero_z1048942125864253310at_nat @ A )
      = zero_z1048942125864253310at_nat ) ).

% zero_diff
thf(fact_1487_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_1488_diff__0__right,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% diff_0_right
thf(fact_1489_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_1490_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_1491_diff__self,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% diff_self
thf(fact_1492_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_1493_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_1494_mult_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ one_one_Code_integer )
      = A ) ).

% mult.right_neutral
thf(fact_1495_mult_Oright__neutral,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ A @ one_one_assn )
      = A ) ).

% mult.right_neutral
thf(fact_1496_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_1497_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_1498_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_1499_mult__1,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ one_one_Code_integer @ A )
      = A ) ).

% mult_1
thf(fact_1500_mult__1,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ one_one_assn @ A )
      = A ) ).

% mult_1
thf(fact_1501_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_1502_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_1503_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_1504_add__diff__cancel__right_H,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_1505_add__diff__cancel__right_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_1506_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_1507_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_1508_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_1509_add__diff__cancel__right,axiom,
    ! [A: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ A @ C ) @ ( plus_p7104986032573967614at_nat @ B @ C ) )
      = ( minus_4286766774447292334at_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1510_add__diff__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
      = ( minus_8373710615458151222nteger @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1511_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1512_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1513_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1514_add__diff__cancel__left_H,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_1515_add__diff__cancel__left_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_1516_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_1517_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_1518_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_1519_add__diff__cancel__left,axiom,
    ! [C: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( plus_p7104986032573967614at_nat @ C @ A ) @ ( plus_p7104986032573967614at_nat @ C @ B ) )
      = ( minus_4286766774447292334at_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_1520_add__diff__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ C @ A ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
      = ( minus_8373710615458151222nteger @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_1521_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_1522_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_1523_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_1524_diff__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_1525_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_1526_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_1527_add__diff__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_1528_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_1529_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_1530_minus__diff__eq,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
      = ( minus_minus_int @ B @ A ) ) ).

% minus_diff_eq
thf(fact_1531_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% minus_diff_eq
thf(fact_1532_minus__diff__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
      = ( minus_minus_rat @ B @ A ) ) ).

% minus_diff_eq
thf(fact_1533_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 ) ) ) ).

% Nat.add_diff_assoc
thf(fact_1534_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 ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_1535_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 ) ) ) ).

% Nat.diff_diff_right
thf(fact_1536_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% zle_add1_eq_le
thf(fact_1537_diff__ge__0__iff__ge,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( ord_le3102999989581377725nteger @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_1538_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_1539_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_eq_int @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_1540_zero__comp__diff__simps_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( ord_le3102999989581377725nteger @ B @ A ) ) ).

% zero_comp_diff_simps(1)
thf(fact_1541_zero__comp__diff__simps_I1_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ B @ A ) ) ).

% zero_comp_diff_simps(1)
thf(fact_1542_zero__comp__diff__simps_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_eq_int @ B @ A ) ) ).

% zero_comp_diff_simps(1)
thf(fact_1543_diff__gt__0__iff__gt,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( ord_le6747313008572928689nteger @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_1544_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_rat @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_1545_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_int @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_1546_zero__comp__diff__simps_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( ord_le6747313008572928689nteger @ B @ A ) ) ).

% zero_comp_diff_simps(2)
thf(fact_1547_zero__comp__diff__simps_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_rat @ B @ A ) ) ).

% zero_comp_diff_simps(2)
thf(fact_1548_zero__comp__diff__simps_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_int @ B @ A ) ) ).

% zero_comp_diff_simps(2)
thf(fact_1549_le__add__diff__inverse,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( plus_p5714425477246183910nteger @ B @ ( minus_8373710615458151222nteger @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_1550_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_1551_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_1552_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_1553_le__add__diff__inverse2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_1554_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_1555_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_1556_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_1557_mult__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C
        = ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_left1
thf(fact_1558_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
thf(fact_1559_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_1560_mult__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( ( times_3573771949741848930nteger @ C @ A )
        = C )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_left2
thf(fact_1561_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ( times_times_rat @ C @ A )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
thf(fact_1562_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ( times_times_int @ C @ A )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_1563_mult__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C
        = ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_right1
thf(fact_1564_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
thf(fact_1565_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_1566_mult__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ C )
        = C )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_right2
thf(fact_1567_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ( times_times_rat @ A @ C )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
thf(fact_1568_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ( times_times_int @ A @ C )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_1569_diff__add__zero,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ A @ ( plus_p7104986032573967614at_nat @ A @ B ) )
      = zero_z1048942125864253310at_nat ) ).

% diff_add_zero
thf(fact_1570_diff__add__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_1571_diff__numeral__special_I9_J,axiom,
    ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ one_one_Code_integer )
    = zero_z3403309356797280102nteger ) ).

% diff_numeral_special(9)
thf(fact_1572_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_1573_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_1574_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_1575_diff__0,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% diff_0
thf(fact_1576_diff__0,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A )
      = ( uminus_uminus_rat @ A ) ) ).

% diff_0
thf(fact_1577_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_1578_mult__minus1,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1
thf(fact_1579_mult__minus1,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1
thf(fact_1580_mult__minus1__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1_right
thf(fact_1581_mult__minus1__right,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1_right
thf(fact_1582_mult__minus1__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1_right
thf(fact_1583_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
      = ( minus_minus_int @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_1584_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_1585_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( minus_minus_rat @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_1586_diff__minus__eq__add,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
      = ( plus_plus_int @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_1587_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( plus_p5714425477246183910nteger @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_1588_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( plus_plus_rat @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_1589_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
    = zero_zero_int ) ).

% diff_numeral_special(12)
thf(fact_1590_diff__numeral__special_I12_J,axiom,
    ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% diff_numeral_special(12)
thf(fact_1591_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
    = zero_zero_rat ) ).

% diff_numeral_special(12)
thf(fact_1592_le__imp__0__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).

% le_imp_0_less
thf(fact_1593_q__pos__lemma,axiom,
    ! [B3: int,Q4: int,R6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q4 ) @ R6 ) )
     => ( ( ord_less_int @ R6 @ B3 )
       => ( ( ord_less_int @ zero_zero_int @ B3 )
         => ( ord_less_eq_int @ zero_zero_int @ Q4 ) ) ) ) ).

% q_pos_lemma
thf(fact_1594_zdiv__mono2__lemma,axiom,
    ! [B: int,Q3: int,R2: int,B3: int,Q4: int,R6: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B3 @ Q4 ) @ R6 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q4 ) @ R6 ) )
       => ( ( ord_less_int @ R6 @ B3 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
           => ( ( ord_less_int @ zero_zero_int @ B3 )
             => ( ( ord_less_eq_int @ B3 @ B )
               => ( ord_less_eq_int @ Q3 @ Q4 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_1595_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q3: int,R2: int,B3: int,Q4: int,R6: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B3 @ Q4 ) @ R6 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q4 ) @ R6 ) @ zero_zero_int )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ R6 )
           => ( ( ord_less_int @ zero_zero_int @ B3 )
             => ( ( ord_less_eq_int @ B3 @ B )
               => ( ord_less_eq_int @ Q4 @ Q3 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_1596_unique__quotient__lemma,axiom,
    ! [B: int,Q4: int,R6: int,Q3: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R6 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R6 )
       => ( ( ord_less_int @ R6 @ B )
         => ( ( ord_less_int @ R2 @ B )
           => ( ord_less_eq_int @ Q4 @ Q3 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_1597_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q4: int,R6: int,Q3: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R6 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( ord_less_int @ B @ R6 )
           => ( ord_less_eq_int @ Q3 @ Q4 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_1598_int__ge__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I )
     => ( ( P @ K )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ K @ I2 )
             => ( ( P @ I2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_ge_induct
thf(fact_1599_int__induct,axiom,
    ! [P: int > $o,K: int,I: int] :
      ( ( P @ K )
     => ( ! [I2: int] :
            ( ( ord_less_eq_int @ K @ I2 )
           => ( ( P @ I2 )
             => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_induct
thf(fact_1600_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ Z )
     => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).

% zless_imp_add1_zle
thf(fact_1601_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
      = ( ord_less_int @ W @ Z ) ) ).

% add1_zle_eq
thf(fact_1602_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 )
          = N2 ) ) ) ).

% nat_eq_add_iff1
thf(fact_1603_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( M2
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_1604_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( 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 ) @ M2 ) @ N2 ) ) ) ).

% nat_le_add_iff1
thf(fact_1605_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_le_add_iff2
thf(fact_1606_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N2 ) ) ) ).

% nat_diff_add_eq1
thf(fact_1607_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( minus_minus_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_1608_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M3: nat] :
      ( ( P @ X )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( ord_less_eq_nat @ X4 @ M3 ) )
       => ~ ! [M5: nat] :
              ( ( P @ M5 )
             => ~ ! [X8: nat] :
                    ( ( P @ X8 )
                   => ( ord_less_eq_nat @ X8 @ M5 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1609_subset__Collect__conv,axiom,
    ! [S2: set_list_nat,P: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ S2 @ ( collect_list_nat @ P ) )
      = ( ! [X3: list_nat] :
            ( ( member_list_nat @ X3 @ S2 )
           => ( P @ X3 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1610_subset__Collect__conv,axiom,
    ! [S2: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ S2 @ ( collect_nat @ P ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ S2 )
           => ( P @ X3 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1611_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( minus_8373710615458151222nteger @ A @ C ) @ B )
      = ( minus_8373710615458151222nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1612_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1613_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1614_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
      = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1615_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: assn,A: assn,C: assn] :
      ( ( times_times_assn @ B @ ( times_times_assn @ A @ C ) )
      = ( times_times_assn @ A @ ( times_times_assn @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_1616_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_1617_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_1618_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_1619_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_assn
    = ( ^ [A5: assn,B4: assn] : ( times_times_assn @ B4 @ A5 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_1620_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A5: rat,B4: rat] : ( times_times_rat @ B4 @ A5 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_1621_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A5: nat,B4: nat] : ( times_times_nat @ B4 @ A5 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_1622_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A5: int,B4: int] : ( times_times_int @ B4 @ A5 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_1623_mult_Oassoc,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C )
      = ( times_times_assn @ A @ ( times_times_assn @ B @ C ) ) ) ).

% mult.assoc
thf(fact_1624_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_1625_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_1626_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% mult.assoc
thf(fact_1627_diff__eq__diff__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ( minus_8373710615458151222nteger @ A @ B )
        = ( minus_8373710615458151222nteger @ C @ D2 ) )
     => ( ( A = B )
        = ( C = D2 ) ) ) ).

% diff_eq_diff_eq
thf(fact_1628_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D2 ) )
     => ( ( A = B )
        = ( C = D2 ) ) ) ).

% diff_eq_diff_eq
thf(fact_1629_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D2 ) )
     => ( ( A = B )
        = ( C = D2 ) ) ) ).

% diff_eq_diff_eq
thf(fact_1630_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
thf(fact_1631_int__gr__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_int @ K @ I )
     => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I2: int] :
              ( ( ord_less_int @ K @ I2 )
             => ( ( P @ I2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_gr_induct
thf(fact_1632_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ( ord_less_int @ W @ Z )
        | ( W = Z ) ) ) ).

% zless_add1_eq
thf(fact_1633_odd__nonzero,axiom,
    ! [Z: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_1634_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% odd_less_0_iff
thf(fact_1635_eq__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D2 ) )
      = ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ E ) @ C )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_1636_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_1637_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_1638_eq__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D2 ) )
      = ( C
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_1639_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( C
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_1640_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_1641_square__diff__square__factored,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) )
      = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( minus_8373710615458151222nteger @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_1642_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
      = ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_1643_square__diff__square__factored,axiom,
    ! [X: int,Y: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
      = ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_1644_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 ) ) ).

% left_add_mult_distrib
thf(fact_1645_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N2 ) ) ) ).

% nat_less_add_iff1
thf(fact_1646_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_less_add_iff2
thf(fact_1647_le__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D2 ) )
      = ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ E ) @ C ) @ D2 ) ) ).

% le_add_iff1
thf(fact_1648_le__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D2 ) ) ).

% le_add_iff1
thf(fact_1649_le__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D2 ) ) ).

% le_add_iff1
thf(fact_1650_le__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D2 ) )
      = ( ord_le3102999989581377725nteger @ C @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ E ) @ D2 ) ) ) ).

% le_add_iff2
thf(fact_1651_le__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D2 ) ) ) ).

% le_add_iff2
thf(fact_1652_le__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).

% le_add_iff2
thf(fact_1653_less__add__iff1,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D2 ) )
      = ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ E ) @ C ) @ D2 ) ) ).

% less_add_iff1
thf(fact_1654_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D2 ) ) ).

% less_add_iff1
thf(fact_1655_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D2 ) ) ).

% less_add_iff1
thf(fact_1656_less__add__iff2,axiom,
    ! [A: code_integer,E: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E ) @ D2 ) )
      = ( ord_le6747313008572928689nteger @ C @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_1657_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D2 ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_1658_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D2 ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_1659_square__diff__one__factored,axiom,
    ! [X: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ X @ X ) @ one_one_Code_integer )
      = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ X @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ X @ one_one_Code_integer ) ) ) ).

% square_diff_one_factored
thf(fact_1660_square__diff__one__factored,axiom,
    ! [X: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
      = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).

% square_diff_one_factored
thf(fact_1661_square__diff__one__factored,axiom,
    ! [X: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
      = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).

% square_diff_one_factored
thf(fact_1662_diff__eq__diff__less__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ( minus_8373710615458151222nteger @ A @ B )
        = ( minus_8373710615458151222nteger @ C @ D2 ) )
     => ( ( ord_le3102999989581377725nteger @ A @ B )
        = ( ord_le3102999989581377725nteger @ C @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1663_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D2 ) )
     => ( ( ord_less_eq_rat @ A @ B )
        = ( ord_less_eq_rat @ C @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1664_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D2 ) )
     => ( ( ord_less_eq_int @ A @ B )
        = ( ord_less_eq_int @ C @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1665_diff__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ C ) @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_1666_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_1667_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_1668_diff__left__mono,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ C @ A ) @ ( minus_8373710615458151222nteger @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_1669_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_1670_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_1671_diff__mono,axiom,
    ! [A: code_integer,B: code_integer,D2: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ D2 @ C )
       => ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ C ) @ ( minus_8373710615458151222nteger @ B @ D2 ) ) ) ) ).

% diff_mono
thf(fact_1672_diff__mono,axiom,
    ! [A: rat,B: rat,D2: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ D2 @ C )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).

% diff_mono
thf(fact_1673_diff__mono,axiom,
    ! [A: int,B: int,D2: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ D2 @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).

% diff_mono
thf(fact_1674_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: code_integer,Z2: code_integer] : ( Y4 = Z2 ) )
    = ( ^ [A5: code_integer,B4: code_integer] :
          ( ( minus_8373710615458151222nteger @ A5 @ B4 )
          = zero_z3403309356797280102nteger ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1675_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: rat,Z2: rat] : ( Y4 = Z2 ) )
    = ( ^ [A5: rat,B4: rat] :
          ( ( minus_minus_rat @ A5 @ B4 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1676_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    = ( ^ [A5: int,B4: int] :
          ( ( minus_minus_int @ A5 @ B4 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1677_diff__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ C ) @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1678_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1679_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1680_diff__strict__left__mono,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ C @ A ) @ ( minus_8373710615458151222nteger @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1681_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1682_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1683_diff__eq__diff__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ( minus_8373710615458151222nteger @ A @ B )
        = ( minus_8373710615458151222nteger @ C @ D2 ) )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
        = ( ord_le6747313008572928689nteger @ C @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_1684_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D2 ) )
     => ( ( ord_less_rat @ A @ B )
        = ( ord_less_rat @ C @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_1685_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D2 ) )
     => ( ( ord_less_int @ A @ B )
        = ( ord_less_int @ C @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_1686_diff__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,D2: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ D2 @ C )
       => ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ C ) @ ( minus_8373710615458151222nteger @ B @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_1687_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D2: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ D2 @ C )
       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_1688_diff__strict__mono,axiom,
    ! [A: int,B: int,D2: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ D2 @ C )
       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_1689_diff__diff__eq,axiom,
    ! [A: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,C: multis2468970476368604999at_nat] :
      ( ( minus_4286766774447292334at_nat @ ( minus_4286766774447292334at_nat @ A @ B ) @ C )
      = ( minus_4286766774447292334at_nat @ A @ ( plus_p7104986032573967614at_nat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1690_diff__diff__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
      = ( minus_8373710615458151222nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1691_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1692_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1693_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1694_add__implies__diff,axiom,
    ! [C: multis2468970476368604999at_nat,B: multis2468970476368604999at_nat,A: multis2468970476368604999at_nat] :
      ( ( ( plus_p7104986032573967614at_nat @ C @ B )
        = A )
     => ( C
        = ( minus_4286766774447292334at_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1695_add__implies__diff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ C @ B )
        = A )
     => ( C
        = ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1696_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ( plus_plus_rat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_rat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1697_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1698_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ( plus_plus_int @ C @ B )
        = A )
     => ( C
        = ( minus_minus_int @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1699_diff__add__eq__diff__diff__swap,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
      = ( minus_8373710615458151222nteger @ ( minus_8373710615458151222nteger @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1700_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1701_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1702_diff__add__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_1703_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_1704_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_1705_diff__diff__eq2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_1706_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_1707_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_1708_add__diff__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_1709_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_1710_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_1711_eq__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( A
        = ( minus_8373710615458151222nteger @ C @ B ) )
      = ( ( plus_p5714425477246183910nteger @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_1712_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A
        = ( minus_minus_rat @ C @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_1713_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A
        = ( minus_minus_int @ C @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_1714_diff__eq__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ( minus_8373710615458151222nteger @ A @ B )
        = C )
      = ( A
        = ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_1715_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = C )
      = ( A
        = ( plus_plus_rat @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_1716_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( minus_minus_int @ A @ B )
        = C )
      = ( A
        = ( plus_plus_int @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_1717_group__cancel_Osub1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer,B: code_integer] :
      ( ( A2
        = ( plus_p5714425477246183910nteger @ K @ A ) )
     => ( ( minus_8373710615458151222nteger @ A2 @ B )
        = ( plus_p5714425477246183910nteger @ K @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_1718_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( minus_minus_rat @ A2 @ B )
        = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_1719_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( minus_minus_int @ A2 @ B )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_1720_minus__diff__commute,axiom,
    ! [B: int,A: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
      = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_1721_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
      = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_1722_minus__diff__commute,axiom,
    ! [B: rat,A: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
      = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_1723_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_1724_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_1725_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_1726_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_1727_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_1728_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_1729_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_1730_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% distrib_left
thf(fact_1731_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% distrib_left
thf(fact_1732_distrib__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% distrib_left
thf(fact_1733_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% distrib_right
thf(fact_1734_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% distrib_right
thf(fact_1735_distrib__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% distrib_right
thf(fact_1736_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_1737_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_1738_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_1739_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,X: rat,Z: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X @ Z ) )
        = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_1740_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X: nat,Z: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
        = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_1741_crossproduct__eq,axiom,
    ! [W: int,Y: int,X: int,Z: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
        = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_1742_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ( A != B )
        & ( C != D2 ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) )
       != ( plus_plus_rat @ ( times_times_rat @ A @ D2 ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_1743_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ( A != B )
        & ( C != D2 ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) )
       != ( plus_plus_nat @ ( times_times_nat @ A @ D2 ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_1744_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( A != B )
        & ( C != D2 ) )
      = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) )
       != ( plus_plus_int @ ( times_times_int @ A @ D2 ) @ ( times_times_int @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_1745_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ one_one_Code_integer @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1746_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ one_one_assn @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1747_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1748_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1749_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1750_mult_Ocomm__neutral,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ one_one_Code_integer )
      = A ) ).

% mult.comm_neutral
thf(fact_1751_mult_Ocomm__neutral,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ A @ one_one_assn )
      = A ) ).

% mult.comm_neutral
thf(fact_1752_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_1753_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_1754_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_1755_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_1756_abs__minus__commute,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
      = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_1757_abs__minus__commute,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
      = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_1758_Nat_Odiff__cancel,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( minus_minus_nat @ M2 @ N2 ) ) ).

% Nat.diff_cancel
thf(fact_1759_diff__cancel2,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
      = ( minus_minus_nat @ M2 @ N2 ) ) ).

% diff_cancel2
thf(fact_1760_diff__add__inverse,axiom,
    ! [N2: nat,M2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
      = M2 ) ).

% diff_add_inverse
thf(fact_1761_diff__add__inverse2,axiom,
    ! [M2: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
      = M2 ) ).

% diff_add_inverse2
thf(fact_1762_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 ) ) ) ).

% le_diff_conv
thf(fact_1763_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 ) ) ) ).

% Nat.le_diff_conv2
thf(fact_1764_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 ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_1765_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 ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_1766_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 ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_1767_add__mult__distrib,axiom,
    ! [M2: nat,N2: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M2 @ N2 ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).

% add_mult_distrib
thf(fact_1768_add__mult__distrib2,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% add_mult_distrib2
thf(fact_1769_mlex__leI,axiom,
    ! [A: nat,A4: nat,B: nat,B3: nat,N6: nat] :
      ( ( ord_less_eq_nat @ A @ A4 )
     => ( ( ord_less_eq_nat @ B @ B3 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N6 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A4 @ N6 ) @ B3 ) ) ) ) ).

% mlex_leI
thf(fact_1770_nat__mult__1,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ one_one_nat @ N2 )
      = N2 ) ).

% nat_mult_1
thf(fact_1771_nat__mult__1__right,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ N2 @ one_one_nat )
      = N2 ) ).

% nat_mult_1_right
thf(fact_1772_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M: nat,N4: nat] : ( if_nat @ ( M = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N4 ) ) ) ) ) ).

% mult_eq_if
thf(fact_1773_lambda__one,axiom,
    ( ( ^ [X3: code_integer] : X3 )
    = ( times_3573771949741848930nteger @ one_one_Code_integer ) ) ).

% lambda_one
thf(fact_1774_lambda__one,axiom,
    ( ( ^ [X3: assn] : X3 )
    = ( times_times_assn @ one_one_assn ) ) ).

% lambda_one
thf(fact_1775_lambda__one,axiom,
    ( ( ^ [X3: rat] : X3 )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_1776_lambda__one,axiom,
    ( ( ^ [X3: nat] : X3 )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_1777_lambda__one,axiom,
    ( ( ^ [X3: int] : X3 )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_1778_is__num__normalize_I6_J,axiom,
    ! [X: rat,Y: rat] :
      ( ( neg_nu6808929389511037909um_rat @ X )
     => ( ( neg_nu6808929389511037909um_rat @ Y )
       => ( neg_nu6808929389511037909um_rat @ ( plus_plus_rat @ X @ Y ) ) ) ) ).

% is_num_normalize(6)
thf(fact_1779_is__num__normalize_I6_J,axiom,
    ! [X: int,Y: int] :
      ( ( neg_nu7441568979088483369um_int @ X )
     => ( ( neg_nu7441568979088483369um_int @ Y )
       => ( neg_nu7441568979088483369um_int @ ( plus_plus_int @ X @ Y ) ) ) ) ).

% is_num_normalize(6)
thf(fact_1780_is__num__add__commute,axiom,
    ! [X: rat,Y: rat] :
      ( ( neg_nu6808929389511037909um_rat @ X )
     => ( ( neg_nu6808929389511037909um_rat @ Y )
       => ( ( plus_plus_rat @ X @ Y )
          = ( plus_plus_rat @ Y @ X ) ) ) ) ).

% is_num_add_commute
thf(fact_1781_is__num__add__commute,axiom,
    ! [X: int,Y: int] :
      ( ( neg_nu7441568979088483369um_int @ X )
     => ( ( neg_nu7441568979088483369um_int @ Y )
       => ( ( plus_plus_int @ X @ Y )
          = ( plus_plus_int @ Y @ X ) ) ) ) ).

% is_num_add_commute
thf(fact_1782_is__num__add__left__commute,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( neg_nu6808929389511037909um_rat @ X )
     => ( ( neg_nu6808929389511037909um_rat @ Y )
       => ( ( plus_plus_rat @ X @ ( plus_plus_rat @ Y @ Z ) )
          = ( plus_plus_rat @ Y @ ( plus_plus_rat @ X @ Z ) ) ) ) ) ).

% is_num_add_left_commute
thf(fact_1783_is__num__add__left__commute,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( neg_nu7441568979088483369um_int @ X )
     => ( ( neg_nu7441568979088483369um_int @ Y )
       => ( ( plus_plus_int @ X @ ( plus_plus_int @ Y @ Z ) )
          = ( plus_plus_int @ Y @ ( plus_plus_int @ X @ Z ) ) ) ) ) ).

% is_num_add_left_commute
thf(fact_1784_is__num__normalize_I4_J,axiom,
    neg_nu2435721316971906650nteger @ one_one_Code_integer ).

% is_num_normalize(4)
thf(fact_1785_is__num__normalize_I4_J,axiom,
    neg_nu6808929389511037909um_rat @ one_one_rat ).

% is_num_normalize(4)
thf(fact_1786_is__num__normalize_I4_J,axiom,
    neg_nu7441568979088483369um_int @ one_one_int ).

% is_num_normalize(4)
thf(fact_1787_le__iff__diff__le__0,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A5: code_integer,B4: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A5 @ B4 ) @ zero_z3403309356797280102nteger ) ) ) ).

% le_iff_diff_le_0
thf(fact_1788_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A5 @ B4 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_1789_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_1790_less__iff__diff__less__0,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A5: code_integer,B4: code_integer] : ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A5 @ B4 ) @ zero_z3403309356797280102nteger ) ) ) ).

% less_iff_diff_less_0
thf(fact_1791_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A5: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A5 @ B4 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_1792_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_1793_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C )
          = ( B
            = ( plus_plus_nat @ C @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1794_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1795_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1796_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1797_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1798_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1799_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1800_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1801_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% le_add_diff
thf(fact_1802_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1803_le__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
      = ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_1804_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_1805_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_1806_diff__le__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
      = ( ord_le3102999989581377725nteger @ A @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_1807_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_1808_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_1809_add__le__imp__le__diff,axiom,
    ! [I: code_integer,K: code_integer,N2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ N2 )
     => ( ord_le3102999989581377725nteger @ I @ ( minus_8373710615458151222nteger @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1810_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
     => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1811_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1812_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1813_add__le__add__imp__diff__le,axiom,
    ! [I: code_integer,K: code_integer,N2: code_integer,J: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ N2 )
     => ( ( ord_le3102999989581377725nteger @ N2 @ ( plus_p5714425477246183910nteger @ J @ K ) )
       => ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ N2 )
         => ( ( ord_le3102999989581377725nteger @ N2 @ ( plus_p5714425477246183910nteger @ J @ K ) )
           => ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1814_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N2: rat,J: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
     => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
         => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1815_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N2: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
         => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1816_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N2: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
     => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
         => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1817_less__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
      = ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_1818_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_1819_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_1820_diff__less__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
      = ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_1821_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_1822_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_1823_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( plus_p5714425477246183910nteger @ B @ ( minus_8373710615458151222nteger @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1824_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ ( ord_less_rat @ A @ B )
     => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1825_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less_nat @ A @ B )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1826_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ ( ord_less_int @ A @ B )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1827_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1828_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A5: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1829_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1830_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_1831_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A5: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_1832_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_1833_group__cancel_Osub2,axiom,
    ! [B5: int,K: int,B: int,A: int] :
      ( ( B5
        = ( plus_plus_int @ K @ B ) )
     => ( ( minus_minus_int @ A @ B5 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_1834_group__cancel_Osub2,axiom,
    ! [B5: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B5
        = ( plus_p5714425477246183910nteger @ K @ B ) )
     => ( ( minus_8373710615458151222nteger @ A @ B5 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_1835_group__cancel_Osub2,axiom,
    ! [B5: rat,K: rat,B: rat,A: rat] :
      ( ( B5
        = ( plus_plus_rat @ K @ B ) )
     => ( ( minus_minus_rat @ A @ B5 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_1836_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_1837_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_1838_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_1839_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_1840_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_1841_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_1842_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_1843_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_1844_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_1845_add__scale__eq__noteq,axiom,
    ! [R2: code_integer,A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( R2 != zero_z3403309356797280102nteger )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ R2 @ C ) )
         != ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1846_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D2: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
         != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1847_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D2: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1848_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D2: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
         != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1849_less__1__mult,axiom,
    ! [M2: code_integer,N2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ M2 )
     => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ N2 )
       => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ M2 @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_1850_less__1__mult,axiom,
    ! [M2: rat,N2: rat] :
      ( ( ord_less_rat @ one_one_rat @ M2 )
     => ( ( ord_less_rat @ one_one_rat @ N2 )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M2 @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_1851_less__1__mult,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ M2 )
     => ( ( ord_less_nat @ one_one_nat @ N2 )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_1852_less__1__mult,axiom,
    ! [M2: int,N2: int] :
      ( ( ord_less_int @ one_one_int @ M2 )
     => ( ( ord_less_int @ one_one_int @ N2 )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M2 @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_1853_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( ( times_times_int @ X @ X )
        = one_one_int )
      = ( ( X = one_one_int )
        | ( X
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% square_eq_1_iff
thf(fact_1854_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( ( times_3573771949741848930nteger @ X @ X )
        = one_one_Code_integer )
      = ( ( X = one_one_Code_integer )
        | ( X
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% square_eq_1_iff
thf(fact_1855_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( ( times_times_rat @ X @ X )
        = one_one_rat )
      = ( ( X = one_one_rat )
        | ( X
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% square_eq_1_iff
thf(fact_1856_diff__add__0,axiom,
    ! [N2: nat,M2: nat] :
      ( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_1857_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 ) ) ).

% less_diff_conv
thf(fact_1858_add__diff__inverse__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ~ ( ord_less_nat @ M2 @ N2 )
     => ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
        = M2 ) ) ).

% add_diff_inverse_nat
thf(fact_1859_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 ) ) ) ) ).

% less_diff_conv2
thf(fact_1860_mlex__snd__decrI,axiom,
    ! [A: nat,A4: nat,B: nat,B3: nat,N6: nat] :
      ( ( A = A4 )
     => ( ( ord_less_nat @ B @ B3 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N6 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A4 @ N6 ) @ B3 ) ) ) ) ).

% mlex_snd_decrI
thf(fact_1861_mlex__fst__decrI,axiom,
    ! [A: nat,A4: nat,B: nat,N6: nat,B3: nat] :
      ( ( ord_less_nat @ A @ A4 )
     => ( ( ord_less_nat @ B @ N6 )
       => ( ( ord_less_nat @ B3 @ N6 )
         => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N6 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A4 @ N6 ) @ B3 ) ) ) ) ) ).

% mlex_fst_decrI
thf(fact_1862_mlex__bound,axiom,
    ! [A: nat,A2: nat,B: nat,N6: nat] :
      ( ( ord_less_nat @ A @ A2 )
     => ( ( ord_less_nat @ B @ N6 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N6 ) @ B ) @ ( times_times_nat @ A2 @ N6 ) ) ) ) ).

% mlex_bound
thf(fact_1863_mult__eq__self__implies__10,axiom,
    ! [M2: nat,N2: nat] :
      ( ( M2
        = ( times_times_nat @ M2 @ N2 ) )
     => ( ( N2 = one_one_nat )
        | ( M2 = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_1864_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z3: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z3 )
         => ( ( ord_less_rat @ Z3 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ X ) @ Y ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_1865_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
        & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_1866_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
        & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_1867_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X )
        & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_1868_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D2 ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_1869_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D2 ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_1870_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D2 ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_1871_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_1872_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_1873_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_1874_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
        & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_1875_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_1876_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
      = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X )
        & ( ord_less_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_1877_sum__squares__ge__zero,axiom,
    ! [X: code_integer,Y: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_1878_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_1879_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_1880_mult__left__le,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ one_one_Code_integer )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_1881_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_1882_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_1883_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ C @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_1884_mult__le__one,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ) ).

% mult_le_one
thf(fact_1885_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_eq_rat @ B @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).

% mult_le_one
thf(fact_1886_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ one_one_nat )
         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_1887_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).

% mult_le_one
thf(fact_1888_mult__right__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ord_le3102999989581377725nteger @ Y @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_1889_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_1890_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_1891_mult__left__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ord_le3102999989581377725nteger @ Y @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_1892_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_1893_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_1894_not__sum__squares__lt__zero,axiom,
    ! [X: code_integer,Y: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) @ zero_z3403309356797280102nteger ) ).

% not_sum_squares_lt_zero
thf(fact_1895_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_1896_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_1897_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D4: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D4 ) )
           => ( P @ D4 ) ) ) ) ).

% nat_diff_split
thf(fact_1898_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D4: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D4 ) )
                & ~ ( P @ D4 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_1899_dbl__dec__def,axiom,
    ( neg_nu7757733837767384882nteger
    = ( ^ [X3: code_integer] : ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ X3 @ X3 ) @ one_one_Code_integer ) ) ) ).

% dbl_dec_def
thf(fact_1900_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X3: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_1901_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X3: int] : ( minus_minus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_1902_mult__le__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_1903_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_1904_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_1905_mult__le__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ C )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_1906_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_1907_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_1908_mult__le__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_1909_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_1910_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_1911_mult__le__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ C )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_1912_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_1913_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_1914_mult__less__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_1915_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_1916_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_1917_mult__less__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ C )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_1918_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_1919_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_1920_mult__less__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_1921_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_1922_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_1923_mult__less__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ C )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_1924_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_1925_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_1926_convex__bound__le,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ A )
     => ( ( ord_le3102999989581377725nteger @ Y @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V )
           => ( ( ( plus_p5714425477246183910nteger @ U @ V )
                = one_one_Code_integer )
             => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_1927_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X @ A )
     => ( ( ord_less_eq_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_1928_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X @ A )
     => ( ( ord_less_eq_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_1929_is__num_Ocases,axiom,
    ! [A: int] :
      ( ( neg_nu7441568979088483369um_int @ A )
     => ( ( A != one_one_int )
       => ( ! [X4: int] :
              ( ( A
                = ( uminus_uminus_int @ X4 ) )
             => ~ ( neg_nu7441568979088483369um_int @ X4 ) )
         => ~ ! [X4: int,Y5: int] :
                ( ( A
                  = ( plus_plus_int @ X4 @ Y5 ) )
               => ( ( neg_nu7441568979088483369um_int @ X4 )
                 => ~ ( neg_nu7441568979088483369um_int @ Y5 ) ) ) ) ) ) ).

% is_num.cases
thf(fact_1930_is__num_Ocases,axiom,
    ! [A: code_integer] :
      ( ( neg_nu2435721316971906650nteger @ A )
     => ( ( A != one_one_Code_integer )
       => ( ! [X4: code_integer] :
              ( ( A
                = ( uminus1351360451143612070nteger @ X4 ) )
             => ~ ( neg_nu2435721316971906650nteger @ X4 ) )
         => ~ ! [X4: code_integer,Y5: code_integer] :
                ( ( A
                  = ( plus_p5714425477246183910nteger @ X4 @ Y5 ) )
               => ( ( neg_nu2435721316971906650nteger @ X4 )
                 => ~ ( neg_nu2435721316971906650nteger @ Y5 ) ) ) ) ) ) ).

% is_num.cases
thf(fact_1931_is__num_Ocases,axiom,
    ! [A: rat] :
      ( ( neg_nu6808929389511037909um_rat @ A )
     => ( ( A != one_one_rat )
       => ( ! [X4: rat] :
              ( ( A
                = ( uminus_uminus_rat @ X4 ) )
             => ~ ( neg_nu6808929389511037909um_rat @ X4 ) )
         => ~ ! [X4: rat,Y5: rat] :
                ( ( A
                  = ( plus_plus_rat @ X4 @ Y5 ) )
               => ( ( neg_nu6808929389511037909um_rat @ X4 )
                 => ~ ( neg_nu6808929389511037909um_rat @ Y5 ) ) ) ) ) ) ).

% is_num.cases
thf(fact_1932_is__num_Osimps,axiom,
    ( neg_nu7441568979088483369um_int
    = ( ^ [A5: int] :
          ( ( A5 = one_one_int )
          | ? [X3: int] :
              ( ( A5
                = ( uminus_uminus_int @ X3 ) )
              & ( neg_nu7441568979088483369um_int @ X3 ) )
          | ? [X3: int,Y3: int] :
              ( ( A5
                = ( plus_plus_int @ X3 @ Y3 ) )
              & ( neg_nu7441568979088483369um_int @ X3 )
              & ( neg_nu7441568979088483369um_int @ Y3 ) ) ) ) ) ).

% is_num.simps
thf(fact_1933_is__num_Osimps,axiom,
    ( neg_nu2435721316971906650nteger
    = ( ^ [A5: code_integer] :
          ( ( A5 = one_one_Code_integer )
          | ? [X3: code_integer] :
              ( ( A5
                = ( uminus1351360451143612070nteger @ X3 ) )
              & ( neg_nu2435721316971906650nteger @ X3 ) )
          | ? [X3: code_integer,Y3: code_integer] :
              ( ( A5
                = ( plus_p5714425477246183910nteger @ X3 @ Y3 ) )
              & ( neg_nu2435721316971906650nteger @ X3 )
              & ( neg_nu2435721316971906650nteger @ Y3 ) ) ) ) ) ).

% is_num.simps
thf(fact_1934_is__num_Osimps,axiom,
    ( neg_nu6808929389511037909um_rat
    = ( ^ [A5: rat] :
          ( ( A5 = one_one_rat )
          | ? [X3: rat] :
              ( ( A5
                = ( uminus_uminus_rat @ X3 ) )
              & ( neg_nu6808929389511037909um_rat @ X3 ) )
          | ? [X3: rat,Y3: rat] :
              ( ( A5
                = ( plus_plus_rat @ X3 @ Y3 ) )
              & ( neg_nu6808929389511037909um_rat @ X3 )
              & ( neg_nu6808929389511037909um_rat @ Y3 ) ) ) ) ) ).

% is_num.simps
thf(fact_1935_sum__squares__gt__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) )
      = ( ( X != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_1936_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_1937_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_1938_sum__squares__le__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) ) @ zero_z3403309356797280102nteger )
      = ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_1939_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_1940_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_1941_incr__lemma,axiom,
    ! [D2: int,Z: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) ) ) ).

% incr_lemma
thf(fact_1942_decr__lemma,axiom,
    ! [D2: int,X: int,Z: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) @ Z ) ) ).

% decr_lemma
thf(fact_1943_sum__squares__eq__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ X @ X ) @ ( times_3573771949741848930nteger @ Y @ Y ) )
        = zero_z3403309356797280102nteger )
      = ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1944_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1945_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1946_frac__unique__iff,axiom,
    ! [X: rat,A: rat] :
      ( ( ( archimedean_frac_rat @ X )
        = A )
      = ( ( member_rat @ ( minus_minus_rat @ X @ A ) @ ring_1_Ints_rat )
        & ( ord_less_eq_rat @ zero_zero_rat @ A )
        & ( ord_less_rat @ A @ one_one_rat ) ) ) ).

% frac_unique_iff
thf(fact_1947_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S: rat] :
      ( ( ord_less_eq_rat @ U @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
       => ( ( ord_less_eq_rat @ R2 @ S )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_1948_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% divide_le_eq_1_neg
thf(fact_1949_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_1950_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_1951_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_1952_div__by__1,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ one_one_Code_integer )
      = A ) ).

% div_by_1
thf(fact_1953_div__by__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ one_one_rat )
      = A ) ).

% div_by_1
thf(fact_1954_div__by__1,axiom,
    ! [A: code_natural] :
      ( ( divide5121882707175180666atural @ A @ one_one_Code_natural )
      = A ) ).

% div_by_1
thf(fact_1955_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_1956_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_1957_div__self,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ A @ A )
        = one_one_Code_integer ) ) ).

% div_self
thf(fact_1958_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% div_self
thf(fact_1959_div__self,axiom,
    ! [A: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ A @ A )
        = one_one_Code_natural ) ) ).

% div_self
thf(fact_1960_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( divide_divide_rat @ one_one_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
thf(fact_1961_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( divide_divide_rat @ one_one_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
thf(fact_1962_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ B @ A ) )
      = ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
thf(fact_1963_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ( divide_divide_rat @ B @ A )
        = one_one_rat )
      = ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
thf(fact_1964_divide__self__if,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( ( divide_divide_rat @ A @ A )
          = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( ( divide_divide_rat @ A @ A )
          = one_one_rat ) ) ) ).

% divide_self_if
thf(fact_1965_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% divide_self
thf(fact_1966_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ A @ B ) )
      = ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_1967_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = one_one_rat )
      = ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_1968_divide__minus1,axiom,
    ! [X: rat] :
      ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X ) ) ).

% divide_minus1
thf(fact_1969_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% zero_le_divide_1_iff
thf(fact_1970_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% divide_le_0_1_iff
thf(fact_1971_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% zero_less_divide_1_iff
thf(fact_1972_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% less_divide_eq_1_pos
thf(fact_1973_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_1974_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_rat @ B @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_1975_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_rat @ A @ B ) ) ) ).

% divide_less_eq_1_neg
thf(fact_1976_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% divide_less_0_1_iff
thf(fact_1977_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
        = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_1978_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
        = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_1979_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% le_divide_eq_1_pos
thf(fact_1980_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_1981_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
      = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).

% int_distrib(2)
thf(fact_1982_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] :
      ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
      = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).

% int_distrib(1)
thf(fact_1983_plus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( plus_plus_int @ zero_zero_int @ L )
      = L ) ).

% plus_int_code(2)
thf(fact_1984_plus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( plus_plus_int @ K @ zero_zero_int )
      = K ) ).

% plus_int_code(1)
thf(fact_1985_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
thf(fact_1986_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_1987_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( divide_divide_int @ A @ B )
            = Q3 ) ) ) ) ).

% int_div_pos_eq
thf(fact_1988_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( divide_divide_int @ A @ B )
            = Q3 ) ) ) ) ).

% int_div_neg_eq
thf(fact_1989_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,J2: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
                & ( ord_less_int @ J2 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
             => ( P @ I3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I3: int,J2: int] :
              ( ( ( ord_less_int @ K @ J2 )
                & ( ord_less_eq_int @ J2 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
             => ( P @ I3 ) ) ) ) ) ).

% split_zdiv
thf(fact_1990_frac__lt__1,axiom,
    ! [X: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X ) @ one_one_rat ) ).

% frac_lt_1
thf(fact_1991_frac__1__eq,axiom,
    ! [X: rat] :
      ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
      = ( archimedean_frac_rat @ X ) ) ).

% frac_1_eq
thf(fact_1992_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( ( divide_divide_rat @ A @ B )
          = one_one_rat )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_1993_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_1994_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ A @ B ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
thf(fact_1995_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_1996_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_1997_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_1998_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_1999_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2000_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2001_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = B ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2002_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).

% less_half_sum
thf(fact_2003_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).

% gt_half_sum
thf(fact_2004_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ( B != zero_zero_rat )
        & ( A
          = ( uminus_uminus_rat @ B ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_2005_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 ) ) ) ) ).

% div_pos_geq
thf(fact_2006_frac__eq,axiom,
    ! [X: rat] :
      ( ( ( archimedean_frac_rat @ X )
        = X )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ X )
        & ( ord_less_rat @ X @ one_one_rat ) ) ) ).

% frac_eq
thf(fact_2007_frac__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat ) ) ) ) ).

% frac_add
thf(fact_2008_incr__mult__lemma,axiom,
    ! [D2: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X4: int] :
            ( ( P @ X4 )
           => ( P @ ( plus_plus_int @ X4 @ D2 ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X8: int] :
              ( ( P @ X8 )
             => ( P @ ( plus_plus_int @ X8 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_2009_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ A @ B ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
thf(fact_2010_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ A @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_2011_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_2012_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = B ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_2013_frac__neg,axiom,
    ! [X: rat] :
      ( ( ( member_rat @ X @ ring_1_Ints_rat )
       => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
          = zero_zero_rat ) )
      & ( ~ ( member_rat @ X @ ring_1_Ints_rat )
       => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X ) )
          = ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X ) ) ) ) ) ).

% frac_neg
thf(fact_2014_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_2015_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_2016_div__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_2017_div__mult__self4,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ C ) @ A ) @ B )
        = ( plus_p4538020629002901425atural @ C @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_2018_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_2019_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_2020_div__mult__self3,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_2021_div__mult__self3,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ C @ B ) @ A ) @ B )
        = ( plus_p4538020629002901425atural @ C @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_2022_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_2023_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_2024_div__mult__self2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_2025_div__mult__self2,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ B @ C ) ) @ B )
        = ( plus_p4538020629002901425atural @ C @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_2026_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_2027_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_2028_div__mult__self1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_2029_div__mult__self1,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ C @ B ) ) @ B )
        = ( plus_p4538020629002901425atural @ C @ ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_2030_div__minus1__right,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ A ) ) ).

% div_minus1_right
thf(fact_2031_div__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% div_minus1_right
thf(fact_2032_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 ) ) ) ) ).

% div_pos_neg_trivial
thf(fact_2033_bits__div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% bits_div_by_1
thf(fact_2034_bits__div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% bits_div_by_1
thf(fact_2035_bits__div__by__1,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ one_one_Code_integer )
      = A ) ).

% bits_div_by_1
thf(fact_2036_bits__div__by__1,axiom,
    ! [A: code_natural] :
      ( ( divide5121882707175180666atural @ A @ one_one_Code_natural )
      = A ) ).

% bits_div_by_1
thf(fact_2037_dividend__less__times__div,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ M2 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M2 @ N2 ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_2038_dividend__less__div__times,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ M2 @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N2 ) @ N2 ) ) ) ) ).

% dividend_less_div_times
thf(fact_2039_split__div,axiom,
    ! [P: nat > $o,M2: nat,N2: nat] :
      ( ( P @ ( divide_divide_nat @ M2 @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N2 != zero_zero_nat )
         => ! [I3: nat,J2: nat] :
              ( ( ord_less_nat @ J2 @ N2 )
             => ( ( M2
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I3 ) @ J2 ) )
               => ( P @ I3 ) ) ) ) ) ) ).

% split_div
thf(fact_2040_div__less__dividend,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_nat @ ( divide_divide_nat @ M2 @ N2 ) @ M2 ) ) ) ).

% div_less_dividend
thf(fact_2041_div__eq__dividend__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ( divide_divide_nat @ M2 @ N2 )
          = M2 )
        = ( N2 = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_2042_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% div_add_self1
thf(fact_2043_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% div_add_self1
thf(fact_2044_div__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% div_add_self1
thf(fact_2045_div__add__self1,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ B @ A ) @ B )
        = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ B ) @ one_one_Code_natural ) ) ) ).

% div_add_self1
thf(fact_2046_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% div_add_self2
thf(fact_2047_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% div_add_self2
thf(fact_2048_div__add__self2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% div_add_self2
thf(fact_2049_div__add__self2,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ B ) @ B )
        = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ B ) @ one_one_Code_natural ) ) ) ).

% div_add_self2
thf(fact_2050_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [I2: int,J3: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J3 ) )
           => ( ( ( ord_less_eq_int @ I2 @ J3 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) )
             => ( P @ I2 @ J3 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% upto.pinduct
thf(fact_2051_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A12: int,A22: int,A32: product_prod_int_int] :
          ( ? [K4: int] :
              ( ( A12 = K4 )
              & ( A22 = zero_zero_int )
              & ( A32
                = ( product_Pair_int_int @ zero_zero_int @ K4 ) ) )
          | ? [L3: int,K4: int,Q5: int] :
              ( ( A12 = K4 )
              & ( A22 = L3 )
              & ( A32
                = ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
              & ( L3 != zero_zero_int )
              & ( K4
                = ( times_times_int @ Q5 @ L3 ) ) )
          | ? [R3: int,L3: int,K4: int,Q5: int] :
              ( ( A12 = K4 )
              & ( A22 = L3 )
              & ( A32
                = ( product_Pair_int_int @ Q5 @ R3 ) )
              & ( ( sgn_sgn_int @ R3 )
                = ( sgn_sgn_int @ L3 ) )
              & ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ L3 ) )
              & ( K4
                = ( plus_plus_int @ ( times_times_int @ Q5 @ L3 ) @ R3 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_2052_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A23: int,A33: product_prod_int_int] :
      ( ( eucl_rel_int @ A1 @ A23 @ A33 )
     => ( ( ( A23 = zero_zero_int )
         => ( A33
           != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
       => ( ! [Q2: int] :
              ( ( A33
                = ( product_Pair_int_int @ Q2 @ zero_zero_int ) )
             => ( ( A23 != zero_zero_int )
               => ( A1
                 != ( times_times_int @ Q2 @ A23 ) ) ) )
         => ~ ! [R: int,Q2: int] :
                ( ( A33
                  = ( product_Pair_int_int @ Q2 @ R ) )
               => ( ( ( sgn_sgn_int @ R )
                    = ( sgn_sgn_int @ A23 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R ) @ ( abs_abs_int @ A23 ) )
                   => ( A1
                     != ( plus_plus_int @ ( times_times_int @ Q2 @ A23 ) @ R ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_2053_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L @ Q3 ) @ 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 )
             => ( Q3 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_2054_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,J2: int] :
              ( ( ( ord_less_int @ K @ J2 )
                & ( ord_less_eq_int @ J2 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
             => ( P @ I3 @ J2 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_2055_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,J2: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
                & ( ord_less_int @ J2 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
             => ( P @ I3 @ J2 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_2056_mod__add__self2,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self2
thf(fact_2057_mod__add__self2,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self2
thf(fact_2058_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self2
thf(fact_2059_mod__add__self2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_add_self2
thf(fact_2060_mod__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self1
thf(fact_2061_mod__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self1
thf(fact_2062_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self1
thf(fact_2063_mod__add__self1,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ B @ A ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_add_self1
thf(fact_2064_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_2065_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_2066_mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% mod_by_1
thf(fact_2067_mod__by__1,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural @ A @ one_one_Code_natural )
      = zero_z2226904508553997617atural ) ).

% mod_by_1
thf(fact_2068_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_2069_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_2070_bits__mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
thf(fact_2071_bits__mod__by__1,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural @ A @ one_one_Code_natural )
      = zero_z2226904508553997617atural ) ).

% bits_mod_by_1
thf(fact_2072_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self1
thf(fact_2073_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self1
thf(fact_2074_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self1
thf(fact_2075_mod__mult__self1,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ C @ B ) ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self1
thf(fact_2076_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self2
thf(fact_2077_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self2
thf(fact_2078_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self2
thf(fact_2079_mod__mult__self2,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ B @ C ) ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self2
thf(fact_2080_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self3
thf(fact_2081_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self3
thf(fact_2082_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self3
thf(fact_2083_mod__mult__self3,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ C @ B ) @ A ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self3
thf(fact_2084_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self4
thf(fact_2085_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self4
thf(fact_2086_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self4
thf(fact_2087_mod__mult__self4,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ C ) @ A ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self4
thf(fact_2088_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_2089_mod__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_2090_mod__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
     => ( ( modulo_modulo_int @ K @ L )
        = R2 ) ) ).

% mod_int_unique
thf(fact_2091_unique__quotient,axiom,
    ! [A: int,B: int,Q3: int,R2: int,Q4: int,R6: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R6 ) )
       => ( Q3 = Q4 ) ) ) ).

% unique_quotient
thf(fact_2092_unique__remainder,axiom,
    ! [A: int,B: int,Q3: int,R2: int,Q4: int,R6: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R6 ) )
       => ( R2 = R6 ) ) ) ).

% unique_remainder
thf(fact_2093_full__exhaustive__int_H_Ocases,axiom,
    ! [X: produc2285326912895808259nt_int] :
      ~ ! [F3: produc8551481072490612790e_term > option6357759511663192854e_term,D: int,I2: int] :
          ( X
         != ( produc5700946648718959541nt_int @ F3 @ ( product_Pair_int_int @ D @ I2 ) ) ) ).

% full_exhaustive_int'.cases
thf(fact_2094_exhaustive__int_H_Ocases,axiom,
    ! [X: produc7773217078559923341nt_int] :
      ~ ! [F3: int > option6357759511663192854e_term,D: int,I2: int] :
          ( X
         != ( produc4305682042979456191nt_int @ F3 @ ( product_Pair_int_int @ D @ I2 ) ) ) ).

% exhaustive_int'.cases
thf(fact_2095_small__lazy_H_Ocases,axiom,
    ! [X: product_prod_int_int] :
      ~ ! [D: int,I2: int] :
          ( X
         != ( product_Pair_int_int @ D @ I2 ) ) ).

% small_lazy'.cases
thf(fact_2096_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 ) ) ) ).

% eucl_rel_int
thf(fact_2097_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_2098_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_2099_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_2100_mod__add__right__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
      = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_2101_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_2102_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_2103_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_2104_mod__add__left__eq,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ C ) @ B ) @ C )
      = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_2105_mod__add__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B3: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A4 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B3 @ C ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( plus_plus_int @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_2106_mod__add__cong,axiom,
    ! [A: nat,C: nat,A4: nat,B: nat,B3: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A4 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B3 @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_2107_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B3: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A4 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B3 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_2108_mod__add__cong,axiom,
    ! [A: code_natural,C: code_natural,A4: code_natural,B: code_natural,B3: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A @ C )
        = ( modulo8411746178871703098atural @ A4 @ C ) )
     => ( ( ( modulo8411746178871703098atural @ B @ C )
          = ( modulo8411746178871703098atural @ B3 @ C ) )
       => ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
          = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A4 @ B3 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_2109_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_2110_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_2111_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_2112_mod__add__eq,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ C ) @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
      = ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_2113_eucl__rel__int__by0,axiom,
    ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).

% eucl_rel_int_by0
thf(fact_2114_div__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
     => ( ( divide_divide_int @ K @ L )
        = Q3 ) ) ).

% div_int_unique
thf(fact_2115_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
     => ~ ! [D: int] :
            ( B
           != ( plus_plus_int @ A @ ( times_times_int @ C @ D ) ) ) ) ).

% mod_eqE
thf(fact_2116_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B @ C ) )
     => ~ ! [D: code_integer] :
            ( B
           != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).

% mod_eqE
thf(fact_2117_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_2118_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_2119_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_2120_div__add1__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
      = ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ C ) @ ( divide5121882707175180666atural @ B @ C ) ) @ ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ C ) @ ( modulo8411746178871703098atural @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_2121_mult__div__mod__eq,axiom,
    ! [B: int,A: int] :
      ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_2122_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_2123_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_2124_mult__div__mod__eq,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) @ ( modulo8411746178871703098atural @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_2125_mod__mult__div__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_2126_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_2127_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_2128_mod__mult__div__eq,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ B ) @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_2129_mod__div__mult__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_2130_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_2131_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_2132_mod__div__mult__eq,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ B ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_2133_div__mult__mod__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_2134_div__mult__mod__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_2135_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_2136_div__mult__mod__eq,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_2137_mod__div__decomp,axiom,
    ! [A: int,B: int] :
      ( A
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_2138_mod__div__decomp,axiom,
    ! [A: nat,B: nat] :
      ( A
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_2139_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] :
      ( A
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_2140_mod__div__decomp,axiom,
    ! [A: code_natural,B: code_natural] :
      ( A
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_2141_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_2142_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_2143_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_2144_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) ) @ C )
      = ( plus_p4538020629002901425atural @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_2145_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_2146_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_2147_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_2148_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) @ ( modulo8411746178871703098atural @ A @ B ) ) @ C )
      = ( plus_p4538020629002901425atural @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_2149_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_2150_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_2151_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_2152_div__mult1__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ C )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ C ) ) @ ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ ( modulo8411746178871703098atural @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_2153_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q3: int] :
      ( ( L != zero_zero_int )
     => ( ( K
          = ( times_times_int @ Q3 @ L ) )
       => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_2154_div__mod__decomp__int,axiom,
    ! [A2: int,N2: int] :
      ( A2
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N2 ) @ N2 ) @ ( modulo_modulo_int @ A2 @ N2 ) ) ) ).

% div_mod_decomp_int
thf(fact_2155_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 ) ) ) ) ).

% mod_pos_neg_trivial
thf(fact_2156_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2157_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2158_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_2159_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( modulo_modulo_int @ A @ B )
            = R2 ) ) ) ) ).

% int_mod_pos_eq
thf(fact_2160_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( modulo_modulo_int @ A @ B )
            = R2 ) ) ) ) ).

% int_mod_neg_eq
thf(fact_2161_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,J2: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
                & ( ord_less_int @ J2 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
             => ( P @ J2 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I3: int,J2: int] :
              ( ( ( ord_less_int @ K @ J2 )
                & ( ord_less_eq_int @ J2 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J2 ) ) )
             => ( P @ J2 ) ) ) ) ) ).

% split_zmod
thf(fact_2162_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_2163_zminus1__lemma,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
     => ( ( B != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q3 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q3 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R2 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_2164_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q3: 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 @ Q3 @ L ) @ R2 ) )
         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_2165_verit__le__mono__div__int,axiom,
    ! [A2: int,B5: int,N2: int] :
      ( ( ord_less_int @ A2 @ B5 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N2 )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B5 @ N2 )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B5 @ N2 ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_2166_bezw__0,axiom,
    ! [X: nat] :
      ( ( bezw @ X @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_2167_verit__le__mono__div,axiom,
    ! [A2: nat,B5: nat,N2: nat] :
      ( ( ord_less_nat @ A2 @ B5 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N2 )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B5 @ N2 )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B5 @ N2 ) ) ) ) ).

% verit_le_mono_div
thf(fact_2168_zero__le__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X ) ) ).

% zero_le_ceiling
thf(fact_2169_ceiling__less__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% ceiling_less_zero
thf(fact_2170_floor__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ one_one_int ) ) ) ) ).

% floor_add
thf(fact_2171_cpmi,axiom,
    ! [D3: int,P: int > $o,P5: int > $o,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ? [Z5: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z5 )
           => ( ( P @ X4 )
              = ( P5 @ X4 ) ) )
       => ( ! [X4: int] :
              ( ! [Xa2: int] :
                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ B5 )
                     => ( X4
                       != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P @ X4 )
               => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
         => ( ! [X4: int,K5: int] :
                ( ( P5 @ X4 )
                = ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K5 @ D3 ) ) ) )
           => ( ( ? [X9: int] : ( P @ X9 ) )
              = ( ? [X3: int] :
                    ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ( P5 @ X3 ) )
                | ? [X3: int] :
                    ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ? [Y3: int] :
                        ( ( member_int @ Y3 @ B5 )
                        & ( P @ ( plus_plus_int @ Y3 @ X3 ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_2172_cppi,axiom,
    ! [D3: int,P: int > $o,P5: int > $o,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ? [Z5: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z5 @ X4 )
           => ( ( P @ X4 )
              = ( P5 @ X4 ) ) )
       => ( ! [X4: int] :
              ( ! [Xa2: int] :
                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ A2 )
                     => ( X4
                       != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P @ X4 )
               => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
         => ( ! [X4: int,K5: int] :
                ( ( P5 @ X4 )
                = ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K5 @ D3 ) ) ) )
           => ( ( ? [X9: int] : ( P @ X9 ) )
              = ( ? [X3: int] :
                    ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ( P5 @ X3 ) )
                | ? [X3: int] :
                    ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ? [Y3: int] :
                        ( ( member_int @ Y3 @ A2 )
                        & ( P @ ( minus_minus_int @ Y3 @ X3 ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_2173_bset_I6_J,axiom,
    ! [D3: int,B5: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ! [X8: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B5 )
                 => ( X8
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X8 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X8 @ D3 ) @ T ) ) ) ) ).

% bset(6)
thf(fact_2174_bset_I8_J,axiom,
    ! [D3: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
       => ! [X8: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X8
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X8 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X8 @ D3 ) ) ) ) ) ) ).

% bset(8)
thf(fact_2175_aset_I6_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
       => ! [X8: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X8
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X8 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X8 @ D3 ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_2176_atLeastAtMost__iff,axiom,
    ! [I: $o,L: $o,U: $o] :
      ( ( member_o @ I @ ( set_or8904488021354931149Most_o @ L @ U ) )
      = ( ( ord_less_eq_o @ L @ I )
        & ( ord_less_eq_o @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_2177_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
      = ( ( ord_less_eq_set_nat @ L @ I )
        & ( ord_less_eq_set_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_2178_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
      = ( ( ord_less_eq_rat @ L @ I )
        & ( ord_less_eq_rat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_2179_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
      = ( ( ord_less_eq_num @ L @ I )
        & ( ord_less_eq_num @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_2180_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
      = ( ( ord_less_eq_int @ L @ I )
        & ( ord_less_eq_int @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_2181_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = ( ( ord_less_eq_nat @ L @ I )
        & ( ord_less_eq_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_2182_atLeastAtMost__iff,axiom,
    ! [I: code_integer,L: code_integer,U: code_integer] :
      ( ( member_Code_integer @ I @ ( set_or189985376899183464nteger @ L @ U ) )
      = ( ( ord_le3102999989581377725nteger @ L @ I )
        & ( ord_le3102999989581377725nteger @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_2183_Icc__eq__Icc,axiom,
    ! [L: set_nat,H3: set_nat,L4: set_nat,H5: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ L @ H3 )
        = ( set_or4548717258645045905et_nat @ L4 @ H5 ) )
      = ( ( ( L = L4 )
          & ( H3 = H5 ) )
        | ( ~ ( ord_less_eq_set_nat @ L @ H3 )
          & ~ ( ord_less_eq_set_nat @ L4 @ H5 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_2184_Icc__eq__Icc,axiom,
    ! [L: rat,H3: rat,L4: rat,H5: rat] :
      ( ( ( set_or633870826150836451st_rat @ L @ H3 )
        = ( set_or633870826150836451st_rat @ L4 @ H5 ) )
      = ( ( ( L = L4 )
          & ( H3 = H5 ) )
        | ( ~ ( ord_less_eq_rat @ L @ H3 )
          & ~ ( ord_less_eq_rat @ L4 @ H5 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_2185_Icc__eq__Icc,axiom,
    ! [L: num,H3: num,L4: num,H5: num] :
      ( ( ( set_or7049704709247886629st_num @ L @ H3 )
        = ( set_or7049704709247886629st_num @ L4 @ H5 ) )
      = ( ( ( L = L4 )
          & ( H3 = H5 ) )
        | ( ~ ( ord_less_eq_num @ L @ H3 )
          & ~ ( ord_less_eq_num @ L4 @ H5 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_2186_Icc__eq__Icc,axiom,
    ! [L: int,H3: int,L4: int,H5: int] :
      ( ( ( set_or1266510415728281911st_int @ L @ H3 )
        = ( set_or1266510415728281911st_int @ L4 @ H5 ) )
      = ( ( ( L = L4 )
          & ( H3 = H5 ) )
        | ( ~ ( ord_less_eq_int @ L @ H3 )
          & ~ ( ord_less_eq_int @ L4 @ H5 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_2187_Icc__eq__Icc,axiom,
    ! [L: nat,H3: nat,L4: nat,H5: nat] :
      ( ( ( set_or1269000886237332187st_nat @ L @ H3 )
        = ( set_or1269000886237332187st_nat @ L4 @ H5 ) )
      = ( ( ( L = L4 )
          & ( H3 = H5 ) )
        | ( ~ ( ord_less_eq_nat @ L @ H3 )
          & ~ ( ord_less_eq_nat @ L4 @ H5 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_2188_Icc__eq__Icc,axiom,
    ! [L: code_integer,H3: code_integer,L4: code_integer,H5: code_integer] :
      ( ( ( set_or189985376899183464nteger @ L @ H3 )
        = ( set_or189985376899183464nteger @ L4 @ H5 ) )
      = ( ( ( L = L4 )
          & ( H3 = H5 ) )
        | ( ~ ( ord_le3102999989581377725nteger @ L @ H3 )
          & ~ ( ord_le3102999989581377725nteger @ L4 @ H5 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_2189_atLeastatMost__subset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D2: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D2 ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B )
        | ( ( ord_less_eq_set_nat @ C @ A )
          & ( ord_less_eq_set_nat @ B @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_2190_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D2 ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_2191_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D2 ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( ( ord_less_eq_num @ C @ A )
          & ( ord_less_eq_num @ B @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_2192_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D2 ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_eq_int @ B @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_2193_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D2 ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_eq_nat @ B @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_2194_atLeastatMost__subset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or189985376899183464nteger @ A @ B ) @ ( set_or189985376899183464nteger @ C @ D2 ) )
      = ( ~ ( ord_le3102999989581377725nteger @ A @ B )
        | ( ( ord_le3102999989581377725nteger @ C @ A )
          & ( ord_le3102999989581377725nteger @ B @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_2195_floor__one,axiom,
    ( ( archim3151403230148437115or_rat @ one_one_rat )
    = one_one_int ) ).

% floor_one
thf(fact_2196_ceiling__one,axiom,
    ( ( archim2889992004027027881ng_rat @ one_one_rat )
    = one_one_int ) ).

% ceiling_one
thf(fact_2197_floor__add2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( member_rat @ X @ ring_1_Ints_rat )
        | ( member_rat @ Y @ ring_1_Ints_rat ) )
     => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) )
        = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ) ).

% floor_add2
thf(fact_2198_zero__less__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).

% zero_less_floor
thf(fact_2199_floor__le__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ zero_zero_int )
      = ( ord_less_rat @ X @ one_one_rat ) ) ).

% floor_le_zero
thf(fact_2200_one__le__floor,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ one_one_rat @ X ) ) ).

% one_le_floor
thf(fact_2201_floor__less__one,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( ord_less_rat @ X @ one_one_rat ) ) ).

% floor_less_one
thf(fact_2202_ceiling__le__one,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
      = ( ord_less_eq_rat @ X @ one_one_rat ) ) ).

% ceiling_le_one
thf(fact_2203_one__less__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ one_one_rat @ X ) ) ).

% one_less_ceiling
thf(fact_2204_ceiling__add__one,axiom,
    ! [X: rat] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_2205_floor__diff__one,axiom,
    ! [X: rat] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ).

% floor_diff_one
thf(fact_2206_ceiling__diff__one,axiom,
    ! [X: rat] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).

% ceiling_diff_one
thf(fact_2207_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D2: set_nat] :
      ( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D2 ) )
      = ( ( ~ ( ord_less_eq_set_nat @ A @ B )
          | ( ( ord_less_eq_set_nat @ C @ A )
            & ( ord_less_eq_set_nat @ B @ D2 )
            & ( ( ord_less_set_nat @ C @ A )
              | ( ord_less_set_nat @ B @ D2 ) ) ) )
        & ( ord_less_eq_set_nat @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_2208_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D2 ) )
      = ( ( ~ ( ord_less_eq_rat @ A @ B )
          | ( ( ord_less_eq_rat @ C @ A )
            & ( ord_less_eq_rat @ B @ D2 )
            & ( ( ord_less_rat @ C @ A )
              | ( ord_less_rat @ B @ D2 ) ) ) )
        & ( ord_less_eq_rat @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_2209_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D2 ) )
      = ( ( ~ ( ord_less_eq_num @ A @ B )
          | ( ( ord_less_eq_num @ C @ A )
            & ( ord_less_eq_num @ B @ D2 )
            & ( ( ord_less_num @ C @ A )
              | ( ord_less_num @ B @ D2 ) ) ) )
        & ( ord_less_eq_num @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_2210_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D2 ) )
      = ( ( ~ ( ord_less_eq_int @ A @ B )
          | ( ( ord_less_eq_int @ C @ A )
            & ( ord_less_eq_int @ B @ D2 )
            & ( ( ord_less_int @ C @ A )
              | ( ord_less_int @ B @ D2 ) ) ) )
        & ( ord_less_eq_int @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_2211_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D2 ) )
      = ( ( ~ ( ord_less_eq_nat @ A @ B )
          | ( ( ord_less_eq_nat @ C @ A )
            & ( ord_less_eq_nat @ B @ D2 )
            & ( ( ord_less_nat @ C @ A )
              | ( ord_less_nat @ B @ D2 ) ) ) )
        & ( ord_less_eq_nat @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_2212_atLeastatMost__psubset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le1307284697595431911nteger @ ( set_or189985376899183464nteger @ A @ B ) @ ( set_or189985376899183464nteger @ C @ D2 ) )
      = ( ( ~ ( ord_le3102999989581377725nteger @ A @ B )
          | ( ( ord_le3102999989581377725nteger @ C @ A )
            & ( ord_le3102999989581377725nteger @ B @ D2 )
            & ( ( ord_le6747313008572928689nteger @ C @ A )
              | ( ord_le6747313008572928689nteger @ B @ D2 ) ) ) )
        & ( ord_le3102999989581377725nteger @ C @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_2213_nat__mod__eq__iff,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N2 )
        = ( modulo_modulo_nat @ Y @ N2 ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X @ ( times_times_nat @ N2 @ Q1 ) )
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_2214_le__floor__add,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ ( archim3151403230148437115or_rat @ Y ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ Y ) ) ) ).

% le_floor_add
thf(fact_2215_bset_I1_J,axiom,
    ! [D3: int,B5: set_int,P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B5 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X4 )
           => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
     => ( ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B5 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X4 )
             => ( Q @ ( minus_minus_int @ X4 @ D3 ) ) ) )
       => ! [X8: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X8
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X8 )
                & ( Q @ X8 ) )
             => ( ( P @ ( minus_minus_int @ X8 @ D3 ) )
                & ( Q @ ( minus_minus_int @ X8 @ D3 ) ) ) ) ) ) ) ).

% bset(1)
thf(fact_2216_bset_I2_J,axiom,
    ! [D3: int,B5: set_int,P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B5 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X4 )
           => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
     => ( ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B5 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X4 )
             => ( Q @ ( minus_minus_int @ X4 @ D3 ) ) ) )
       => ! [X8: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X8
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X8 )
                | ( Q @ X8 ) )
             => ( ( P @ ( minus_minus_int @ X8 @ D3 ) )
                | ( Q @ ( minus_minus_int @ X8 @ D3 ) ) ) ) ) ) ) ).

% bset(2)
thf(fact_2217_aset_I1_J,axiom,
    ! [D3: int,A2: set_int,P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X4 )
           => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
     => ( ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X4 )
             => ( Q @ ( plus_plus_int @ X4 @ D3 ) ) ) )
       => ! [X8: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X8
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X8 )
                & ( Q @ X8 ) )
             => ( ( P @ ( plus_plus_int @ X8 @ D3 ) )
                & ( Q @ ( plus_plus_int @ X8 @ D3 ) ) ) ) ) ) ) ).

% aset(1)
thf(fact_2218_aset_I2_J,axiom,
    ! [D3: int,A2: set_int,P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X4 )
           => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
     => ( ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X4 )
             => ( Q @ ( plus_plus_int @ X4 @ D3 ) ) ) )
       => ! [X8: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X8
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X8 )
                | ( Q @ X8 ) )
             => ( ( P @ ( plus_plus_int @ X8 @ D3 ) )
                | ( Q @ ( plus_plus_int @ X8 @ D3 ) ) ) ) ) ) ) ).

% aset(2)
thf(fact_2219_nat__mod__eq__lemma,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N2 )
        = ( modulo_modulo_nat @ Y @ N2 ) )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ? [Q2: nat] :
            ( X
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q2 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_2220_mod__eq__nat2E,axiom,
    ! [M2: nat,Q3: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M2 @ Q3 )
        = ( modulo_modulo_nat @ N2 @ Q3 ) )
     => ( ( ord_less_eq_nat @ M2 @ N2 )
       => ~ ! [S6: nat] :
              ( N2
             != ( plus_plus_nat @ M2 @ ( times_times_nat @ Q3 @ S6 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_2221_mod__eq__nat1E,axiom,
    ! [M2: nat,Q3: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M2 @ Q3 )
        = ( modulo_modulo_nat @ N2 @ Q3 ) )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ~ ! [S6: nat] :
              ( M2
             != ( plus_plus_nat @ N2 @ ( times_times_nat @ Q3 @ S6 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_2222_ceiling__add__le,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ Y ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) ) ) ).

% ceiling_add_le
thf(fact_2223_mod__mult2__eq,axiom,
    ! [M2: nat,N2: nat,Q3: nat] :
      ( ( modulo_modulo_nat @ M2 @ ( times_times_nat @ N2 @ Q3 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M2 @ N2 ) @ Q3 ) ) @ ( modulo_modulo_nat @ M2 @ N2 ) ) ) ).

% mod_mult2_eq
thf(fact_2224_div__mod__decomp,axiom,
    ! [A2: nat,N2: nat] :
      ( A2
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N2 ) @ N2 ) @ ( modulo_modulo_nat @ A2 @ N2 ) ) ) ).

% div_mod_decomp
thf(fact_2225_one__add__floor,axiom,
    ! [X: rat] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ one_one_rat ) ) ) ).

% one_add_floor
thf(fact_2226_split__mod,axiom,
    ! [P: nat > $o,M2: nat,N2: nat] :
      ( ( P @ ( modulo_modulo_nat @ M2 @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ M2 ) )
        & ( ( N2 != zero_zero_nat )
         => ! [I3: nat,J2: nat] :
              ( ( ord_less_nat @ J2 @ N2 )
             => ( ( M2
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I3 ) @ J2 ) )
               => ( P @ J2 ) ) ) ) ) ) ).

% split_mod
thf(fact_2227_bset_I3_J,axiom,
    ! [D3: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
       => ! [X8: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X8
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X8 = T )
             => ( ( minus_minus_int @ X8 @ D3 )
                = T ) ) ) ) ) ).

% bset(3)
thf(fact_2228_bset_I4_J,axiom,
    ! [D3: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ T @ B5 )
       => ! [X8: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X8
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X8 != T )
             => ( ( minus_minus_int @ X8 @ D3 )
               != T ) ) ) ) ) ).

% bset(4)
thf(fact_2229_bset_I5_J,axiom,
    ! [D3: int,B5: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ! [X8: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B5 )
                 => ( X8
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ X8 @ T )
           => ( ord_less_int @ ( minus_minus_int @ X8 @ D3 ) @ T ) ) ) ) ).

% bset(5)
thf(fact_2230_bset_I7_J,axiom,
    ! [D3: int,T: int,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ T @ B5 )
       => ! [X8: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B5 )
                   => ( X8
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ T @ X8 )
             => ( ord_less_int @ T @ ( minus_minus_int @ X8 @ D3 ) ) ) ) ) ) ).

% bset(7)
thf(fact_2231_aset_I3_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
       => ! [X8: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X8
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X8 = T )
             => ( ( plus_plus_int @ X8 @ D3 )
                = T ) ) ) ) ) ).

% aset(3)
thf(fact_2232_aset_I4_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ T @ A2 )
       => ! [X8: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X8
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X8 != T )
             => ( ( plus_plus_int @ X8 @ D3 )
               != T ) ) ) ) ) ).

% aset(4)
thf(fact_2233_aset_I5_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ( member_int @ T @ A2 )
       => ! [X8: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A2 )
                   => ( X8
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ X8 @ T )
             => ( ord_less_int @ ( plus_plus_int @ X8 @ D3 ) @ T ) ) ) ) ) ).

% aset(5)
thf(fact_2234_aset_I7_J,axiom,
    ! [D3: int,A2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ! [X8: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A2 )
                 => ( X8
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ T @ X8 )
           => ( ord_less_int @ T @ ( plus_plus_int @ X8 @ D3 ) ) ) ) ) ).

% aset(7)
thf(fact_2235_aset_I8_J,axiom,
    ! [D3: int,A2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ! [X8: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A2 )
                 => ( X8
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X8 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X8 @ D3 ) ) ) ) ) ).

% aset(8)
thf(fact_2236_ceiling__divide__lower,axiom,
    ! [Q3: rat,P4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P4 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) @ P4 ) ) ).

% ceiling_divide_lower
thf(fact_2237_floor__divide__upper,axiom,
    ! [Q3: rat,P4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_rat @ P4 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P4 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) ) ) ).

% floor_divide_upper
thf(fact_2238_power__decreasing__iff,axiom,
    ! [B: code_integer,M2: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
       => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ M2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_2239_power__decreasing__iff,axiom,
    ! [B: rat,M2: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M2 ) @ ( power_power_rat @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_2240_power__decreasing__iff,axiom,
    ! [B: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_2241_power__decreasing__iff,axiom,
    ! [B: int,M2: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_2242_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M: nat,N4: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M @ N4 ) @ ( modulo_modulo_nat @ M @ N4 ) ) ) ) ).

% divmod_nat_def
thf(fact_2243_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: rat,B: rat,C: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_2244_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: nat,B: nat,C: nat] :
      ( ( nO_MATCH_int_nat @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_2245_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: int,B: int,C: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_2246_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: rat,B: rat,C: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_2247_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: nat,B: nat,C: nat] :
      ( ( nO_MATCH_nat_nat @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_2248_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: int,B: int,C: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_2249_distrib__left__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,A: rat,B: rat,C: rat] :
      ( ( nO_MAT7795273704451493282er_rat @ ( divide6298287555418463151nteger @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_2250_distrib__left__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,A: nat,B: nat,C: nat] :
      ( ( nO_MAT8430403764537989018er_nat @ ( divide6298287555418463151nteger @ X @ Y ) @ A )
     => ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_2251_distrib__left__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,A: int,B: int,C: int] :
      ( ( nO_MAT8427913294028938742er_int @ ( divide6298287555418463151nteger @ X @ Y ) @ A )
     => ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_2252_distrib__left__NO__MATCH,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat,C: rat] :
      ( ( nO_MATCH_rat_rat @ ( divide_divide_rat @ X @ Y ) @ A )
     => ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_2253_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: rat,A: rat,B: rat] :
      ( ( nO_MATCH_int_rat @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_2254_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: nat,A: nat,B: nat] :
      ( ( nO_MATCH_int_nat @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_2255_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: int,A: int,B: int] :
      ( ( nO_MATCH_int_int @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_2256_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: rat,A: rat,B: rat] :
      ( ( nO_MATCH_nat_rat @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_2257_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: nat,A: nat,B: nat] :
      ( ( nO_MATCH_nat_nat @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_2258_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: int,A: int,B: int] :
      ( ( nO_MATCH_nat_int @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_2259_distrib__right__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,C: rat,A: rat,B: rat] :
      ( ( nO_MAT7795273704451493282er_rat @ ( divide6298287555418463151nteger @ X @ Y ) @ C )
     => ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_2260_distrib__right__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,C: nat,A: nat,B: nat] :
      ( ( nO_MAT8430403764537989018er_nat @ ( divide6298287555418463151nteger @ X @ Y ) @ C )
     => ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_2261_distrib__right__NO__MATCH,axiom,
    ! [X: code_integer,Y: code_integer,C: int,A: int,B: int] :
      ( ( nO_MAT8427913294028938742er_int @ ( divide6298287555418463151nteger @ X @ Y ) @ C )
     => ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_2262_distrib__right__NO__MATCH,axiom,
    ! [X: rat,Y: rat,C: rat,A: rat,B: rat] :
      ( ( nO_MATCH_rat_rat @ ( divide_divide_rat @ X @ Y ) @ C )
     => ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_2263_Suc__times__mod__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M2 @ N2 ) ) @ M2 )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_2264_power__one,axiom,
    ! [N2: nat] :
      ( ( power_8256067586552552935nteger @ one_one_Code_integer @ N2 )
      = one_one_Code_integer ) ).

% power_one
thf(fact_2265_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_assn @ one_one_assn @ N2 )
      = one_one_assn ) ).

% power_one
thf(fact_2266_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ one_one_rat @ N2 )
      = one_one_rat ) ).

% power_one
thf(fact_2267_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ one_one_nat @ N2 )
      = one_one_nat ) ).

% power_one
thf(fact_2268_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ one_one_int @ N2 )
      = one_one_int ) ).

% power_one
thf(fact_2269_add__Suc__right,axiom,
    ! [M2: nat,N2: nat] :
      ( ( plus_plus_nat @ M2 @ ( suc @ N2 ) )
      = ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).

% add_Suc_right
thf(fact_2270_power__one__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_2271_power__one__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_2272_power__inject__exp,axiom,
    ! [A: code_integer,M2: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ( power_8256067586552552935nteger @ A @ M2 )
          = ( power_8256067586552552935nteger @ A @ N2 ) )
        = ( M2 = N2 ) ) ) ).

% power_inject_exp
thf(fact_2273_power__inject__exp,axiom,
    ! [A: rat,M2: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ( power_power_rat @ A @ M2 )
          = ( power_power_rat @ A @ N2 ) )
        = ( M2 = N2 ) ) ) ).

% power_inject_exp
thf(fact_2274_power__inject__exp,axiom,
    ! [A: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M2 )
          = ( power_power_nat @ A @ N2 ) )
        = ( M2 = N2 ) ) ) ).

% power_inject_exp
thf(fact_2275_power__inject__exp,axiom,
    ! [A: int,M2: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M2 )
          = ( power_power_int @ A @ N2 ) )
        = ( M2 = N2 ) ) ) ).

% power_inject_exp
thf(fact_2276_mult__Suc__right,axiom,
    ! [M2: nat,N2: nat] :
      ( ( times_times_nat @ M2 @ ( suc @ N2 ) )
      = ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N2 ) ) ) ).

% mult_Suc_right
thf(fact_2277_diff__Suc__1,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
      = N2 ) ).

% diff_Suc_1
thf(fact_2278_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_int @ Z )
        = one_one_int )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_2279_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_18347121197199848620nteger @ Z )
        = one_one_Code_integer )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_2280_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_rat @ Z )
        = one_one_rat )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_2281_of__int__1,axiom,
    ( ( ring_1_of_int_int @ one_one_int )
    = one_one_int ) ).

% of_int_1
thf(fact_2282_of__int__1,axiom,
    ( ( ring_18347121197199848620nteger @ one_one_int )
    = one_one_Code_integer ) ).

% of_int_1
thf(fact_2283_of__int__1,axiom,
    ( ( ring_1_of_int_rat @ one_one_int )
    = one_one_rat ) ).

% of_int_1
thf(fact_2284_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_add
thf(fact_2285_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_18347121197199848620nteger @ ( plus_plus_int @ W @ Z ) )
      = ( plus_p5714425477246183910nteger @ ( ring_18347121197199848620nteger @ W ) @ ( ring_18347121197199848620nteger @ Z ) ) ) ).

% of_int_add
thf(fact_2286_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_add
thf(fact_2287_power__strict__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B )
     => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ X ) @ ( power_8256067586552552935nteger @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_2288_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_2289_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_2290_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_2291_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: int] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_2292_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_2293_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: rat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_2294_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) )
      = one_one_int ) ).

% minus_one_mult_self
thf(fact_2295_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) )
      = one_one_Code_integer ) ).

% minus_one_mult_self
thf(fact_2296_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) )
      = one_one_rat ) ).

% minus_one_mult_self
thf(fact_2297_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 ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_2298_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 ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_2299_Suc__diff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_eq_nat @ one_one_nat @ M2 )
       => ( ( suc @ ( minus_minus_nat @ N2 @ M2 ) )
          = ( minus_minus_nat @ N2 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ).

% Suc_diff
thf(fact_2300_Suc__mod__mult__self4,axiom,
    ! [N2: nat,K: nat,M2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N2 @ K ) @ M2 ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 ) ) ).

% Suc_mod_mult_self4
thf(fact_2301_Suc__mod__mult__self3,axiom,
    ! [K: nat,N2: nat,M2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N2 ) @ M2 ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 ) ) ).

% Suc_mod_mult_self3
thf(fact_2302_Suc__mod__mult__self2,axiom,
    ! [M2: nat,N2: nat,K: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M2 @ ( times_times_nat @ N2 @ K ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 ) ) ).

% Suc_mod_mult_self2
thf(fact_2303_Suc__mod__mult__self1,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M2 @ ( times_times_nat @ K @ N2 ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M2 ) @ N2 ) ) ).

% Suc_mod_mult_self1
thf(fact_2304_ceiling__add__of__int,axiom,
    ! [X: rat,Z: int] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z ) ) ).

% ceiling_add_of_int
thf(fact_2305_power__strict__decreasing__iff,axiom,
    ! [B: code_integer,M2: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
       => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ M2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2306_power__strict__decreasing__iff,axiom,
    ! [B: rat,M2: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B @ M2 ) @ ( power_power_rat @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2307_power__strict__decreasing__iff,axiom,
    ! [B: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2308_power__strict__decreasing__iff,axiom,
    ! [B: int,M2: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2309_power__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ X ) @ ( power_8256067586552552935nteger @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2310_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2311_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2312_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2313_Suc__diff__1,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
        = N2 ) ) ).

% Suc_diff_1
thf(fact_2314_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_2315_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_2316_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_2317_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ one_one_Code_integer )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_2318_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_2319_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_2320_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ one_one_Code_integer )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_2321_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_2322_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_2323_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( ring_18347121197199848620nteger @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_2324_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_2325_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_2326_power__gt1,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_2327_power__gt1,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_2328_power__gt1,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_2329_power__gt1,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_2330_add__Suc__shift,axiom,
    ! [M2: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
      = ( plus_plus_nat @ M2 @ ( suc @ N2 ) ) ) ).

% add_Suc_shift
thf(fact_2331_add__Suc,axiom,
    ! [M2: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
      = ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).

% add_Suc
thf(fact_2332_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A2 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_2333_power__Suc__le__self,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2334_power__Suc__le__self,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2335_power__Suc__le__self,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2336_power__Suc__le__self,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2337_power__Suc__less__one,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N2 ) ) @ one_one_Code_integer ) ) ) ).

% power_Suc_less_one
thf(fact_2338_power__Suc__less__one,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ one_one_rat ) ) ) ).

% power_Suc_less_one
thf(fact_2339_power__Suc__less__one,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_2340_power__Suc__less__one,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_2341_power__minus_H,axiom,
    ! [X: int,N2: nat] :
      ( ( nO_MATCH_int_int @ one_one_int @ X )
     => ( ( power_power_int @ ( uminus_uminus_int @ X ) @ N2 )
        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ X @ N2 ) ) ) ) ).

% power_minus'
thf(fact_2342_power__minus_H,axiom,
    ! [X: code_integer,N2: nat] :
      ( ( nO_MAT8252062027627875367nteger @ one_one_Code_integer @ X )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ N2 )
        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ X @ N2 ) ) ) ) ).

% power_minus'
thf(fact_2343_power__minus_H,axiom,
    ! [X: rat,N2: nat] :
      ( ( nO_MATCH_rat_rat @ one_one_rat @ X )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ N2 )
        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ X @ N2 ) ) ) ) ).

% power_minus'
thf(fact_2344_one__le__power,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_2345_one__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_2346_one__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_2347_one__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_2348_left__right__inverse__power,axiom,
    ! [X: code_integer,Y: code_integer,N2: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y )
        = one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N2 ) @ ( power_8256067586552552935nteger @ Y @ N2 ) )
        = one_one_Code_integer ) ) ).

% left_right_inverse_power
thf(fact_2349_left__right__inverse__power,axiom,
    ! [X: assn,Y: assn,N2: nat] :
      ( ( ( times_times_assn @ X @ Y )
        = one_one_assn )
     => ( ( times_times_assn @ ( power_power_assn @ X @ N2 ) @ ( power_power_assn @ Y @ N2 ) )
        = one_one_assn ) ) ).

% left_right_inverse_power
thf(fact_2350_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N2: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ Y @ N2 ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_2351_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N2: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ N2 ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_2352_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N2: nat] :
      ( ( ( times_times_int @ X @ Y )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_2353_power__0,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ zero_zero_nat )
      = one_one_Code_integer ) ).

% power_0
thf(fact_2354_power__0,axiom,
    ! [A: assn] :
      ( ( power_power_assn @ A @ zero_zero_nat )
      = one_one_assn ) ).

% power_0
thf(fact_2355_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_2356_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_2357_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_2358_power__one__over,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N2 )
      = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_one_over
thf(fact_2359_power__add,axiom,
    ! [A: assn,M2: nat,N2: nat] :
      ( ( power_power_assn @ A @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( times_times_assn @ ( power_power_assn @ A @ M2 ) @ ( power_power_assn @ A @ N2 ) ) ) ).

% power_add
thf(fact_2360_power__add,axiom,
    ! [A: rat,M2: nat,N2: nat] :
      ( ( power_power_rat @ A @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( times_times_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_add
thf(fact_2361_power__add,axiom,
    ! [A: nat,M2: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) ) ) ).

% power_add
thf(fact_2362_power__add,axiom,
    ! [A: int,M2: nat,N2: nat] :
      ( ( power_power_int @ A @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( times_times_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N2 ) ) ) ).

% power_add
thf(fact_2363_nat__compl__induct_H,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N ) ) )
       => ( P @ N2 ) ) ) ).

% nat_compl_induct'
thf(fact_2364_nat__compl__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N ) ) )
       => ( P @ N2 ) ) ) ).

% nat_compl_induct
thf(fact_2365_one__is__add,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M2 @ N2 ) )
      = ( ( ( M2
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_2366_add__is__1,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M2 @ N2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M2
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_2367_nat__in__between__eq_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_nat @ B @ ( suc @ A ) ) )
      = ( B = A ) ) ).

% nat_in_between_eq(2)
thf(fact_2368_nat__in__between__eq_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_nat @ A @ B )
        & ( ord_less_eq_nat @ B @ ( suc @ A ) ) )
      = ( B
        = ( suc @ A ) ) ) ).

% nat_in_between_eq(1)
thf(fact_2369_less__imp__Suc__add,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ? [K5: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M2 @ K5 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_2370_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M: nat,N4: nat] :
        ? [K4: nat] :
          ( N4
          = ( suc @ ( plus_plus_nat @ M @ K4 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_2371_less__add__Suc2,axiom,
    ! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).

% less_add_Suc2
thf(fact_2372_less__add__Suc1,axiom,
    ! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).

% less_add_Suc1
thf(fact_2373_less__natE,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ~ ! [Q2: nat] :
            ( N2
           != ( suc @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ) ).

% less_natE
thf(fact_2374_Suc__to__right,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ( suc @ N2 )
        = M2 )
     => ( N2
        = ( minus_minus_nat @ M2 @ ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_to_right
thf(fact_2375_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_2376_mult__Suc,axiom,
    ! [M2: nat,N2: nat] :
      ( ( times_times_nat @ ( suc @ M2 ) @ N2 )
      = ( plus_plus_nat @ N2 @ ( times_times_nat @ M2 @ N2 ) ) ) ).

% mult_Suc
thf(fact_2377_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_2378_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_2379_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_2380_diff__Suc__eq__diff__pred,axiom,
    ! [M2: nat,N2: nat] :
      ( ( minus_minus_nat @ M2 @ ( suc @ N2 ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_2381_power__le__one,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ one_one_Code_integer ) ) ) ).

% power_le_one
thf(fact_2382_power__le__one,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ one_one_rat ) ) ) ).

% power_le_one
thf(fact_2383_power__le__one,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_2384_power__le__one,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_2385_power__less__power__Suc,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_2386_power__less__power__Suc,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_2387_power__less__power__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_2388_power__less__power__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_2389_power__gt1__lemma,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_2390_power__gt1__lemma,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_2391_power__gt1__lemma,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_2392_power__gt1__lemma,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_2393_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N2 )
          = one_one_Code_integer ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N2 )
          = zero_z3403309356797280102nteger ) ) ) ).

% power_0_left
thf(fact_2394_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N2 )
          = one_one_rat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N2 )
          = zero_zero_rat ) ) ) ).

% power_0_left
thf(fact_2395_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N2 )
          = one_one_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N2 )
          = zero_zero_nat ) ) ) ).

% power_0_left
thf(fact_2396_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N2 )
          = one_one_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N2 )
          = zero_zero_int ) ) ) ).

% power_0_left
thf(fact_2397_power__strict__increasing,axiom,
    ! [N2: nat,N6: nat,A: code_integer] :
      ( ( ord_less_nat @ N2 @ N6 )
     => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ A @ N6 ) ) ) ) ).

% power_strict_increasing
thf(fact_2398_power__strict__increasing,axiom,
    ! [N2: nat,N6: nat,A: rat] :
      ( ( ord_less_nat @ N2 @ N6 )
     => ( ( ord_less_rat @ one_one_rat @ A )
       => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N6 ) ) ) ) ).

% power_strict_increasing
thf(fact_2399_power__strict__increasing,axiom,
    ! [N2: nat,N6: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ N6 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).

% power_strict_increasing
thf(fact_2400_power__strict__increasing,axiom,
    ! [N2: nat,N6: nat,A: int] :
      ( ( ord_less_nat @ N2 @ N6 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N6 ) ) ) ) ).

% power_strict_increasing
thf(fact_2401_power__less__imp__less__exp,axiom,
    ! [A: code_integer,M2: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ M2 ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
       => ( ord_less_nat @ M2 @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_2402_power__less__imp__less__exp,axiom,
    ! [A: rat,M2: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N2 ) )
       => ( ord_less_nat @ M2 @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_2403_power__less__imp__less__exp,axiom,
    ! [A: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) )
       => ( ord_less_nat @ M2 @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_2404_power__less__imp__less__exp,axiom,
    ! [A: int,M2: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N2 ) )
       => ( ord_less_nat @ M2 @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_2405_power__minus,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ A @ N2 ) ) ) ).

% power_minus
thf(fact_2406_power__minus,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% power_minus
thf(fact_2407_power__minus,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_minus
thf(fact_2408_power__increasing,axiom,
    ! [N2: nat,N6: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ N2 @ N6 )
     => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ A @ N6 ) ) ) ) ).

% power_increasing
thf(fact_2409_power__increasing,axiom,
    ! [N2: nat,N6: nat,A: rat] :
      ( ( ord_less_eq_nat @ N2 @ N6 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N6 ) ) ) ) ).

% power_increasing
thf(fact_2410_power__increasing,axiom,
    ! [N2: nat,N6: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ N6 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).

% power_increasing
thf(fact_2411_power__increasing,axiom,
    ! [N2: nat,N6: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ N6 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N6 ) ) ) ) ).

% power_increasing
thf(fact_2412_floor__add__int,axiom,
    ! [X: rat,Z: int] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ) ).

% floor_add_int
thf(fact_2413_int__add__floor,axiom,
    ! [Z: int,X: rat] :
      ( ( plus_plus_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ) ).

% int_add_floor
thf(fact_2414_nat__induct__non__zero,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( P @ one_one_nat )
       => ( ! [N: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N )
             => ( ( P @ N )
               => ( P @ ( suc @ N ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct_non_zero
thf(fact_2415_of__int__leD,axiom,
    ! [N2: int,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_leD
thf(fact_2416_of__int__leD,axiom,
    ! [N2: int,X: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).

% of_int_leD
thf(fact_2417_of__int__leD,axiom,
    ! [N2: int,X: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).

% of_int_leD
thf(fact_2418_of__int__lessD,axiom,
    ! [N2: int,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_lessD
thf(fact_2419_of__int__lessD,axiom,
    ! [N2: int,X: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_rat @ one_one_rat @ X ) ) ) ).

% of_int_lessD
thf(fact_2420_of__int__lessD,axiom,
    ! [N2: int,X: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_int @ one_one_int @ X ) ) ) ).

% of_int_lessD
thf(fact_2421_floor__exists,axiom,
    ! [X: rat] :
    ? [Z3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_2422_floor__exists1,axiom,
    ! [X: rat] :
    ? [X4: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X4 @ one_one_int ) ) )
      & ! [Y6: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y6 ) @ X )
            & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y6 @ one_one_int ) ) ) )
         => ( Y6 = X4 ) ) ) ).

% floor_exists1
thf(fact_2423_power__Suc__less,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N2 ) ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2424_power__Suc__less,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2425_power__Suc__less,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2426_power__Suc__less,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2427_power__strict__decreasing,axiom,
    ! [N2: nat,N6: nat,A: code_integer] :
      ( ( ord_less_nat @ N2 @ N6 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
         => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N6 ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2428_power__strict__decreasing,axiom,
    ! [N2: nat,N6: nat,A: rat] :
      ( ( ord_less_nat @ N2 @ N6 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ( ord_less_rat @ A @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A @ N6 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2429_power__strict__decreasing,axiom,
    ! [N2: nat,N6: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ N6 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2430_power__strict__decreasing,axiom,
    ! [N2: nat,N6: nat,A: int] :
      ( ( ord_less_nat @ N2 @ N6 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N6 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2431_power__decreasing,axiom,
    ! [N2: nat,N6: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ N2 @ N6 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N6 ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2432_power__decreasing,axiom,
    ! [N2: nat,N6: nat,A: rat] :
      ( ( ord_less_eq_nat @ N2 @ N6 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ A @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N6 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2433_power__decreasing,axiom,
    ! [N2: nat,N6: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ N6 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2434_power__decreasing,axiom,
    ! [N2: nat,N6: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ N6 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N6 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2435_power__le__imp__le__exp,axiom,
    ! [A: code_integer,M2: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ M2 ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
       => ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_2436_power__le__imp__le__exp,axiom,
    ! [A: rat,M2: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N2 ) )
       => ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_2437_power__le__imp__le__exp,axiom,
    ! [A: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) )
       => ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_2438_power__le__imp__le__exp,axiom,
    ! [A: int,M2: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N2 ) )
       => ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_2439_self__le__power,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_le3102999989581377725nteger @ A @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2440_self__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2441_self__le__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2442_self__le__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2443_one__less__power,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2444_one__less__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2445_one__less__power,axiom,
    ! [A: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2446_one__less__power,axiom,
    ! [A: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2447_nz__le__conv__less,axiom,
    ! [K: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ K @ M2 )
       => ( ord_less_nat @ ( minus_minus_nat @ K @ ( suc @ zero_zero_nat ) ) @ M2 ) ) ) ).

% nz_le_conv_less
thf(fact_2448_ceiling__altdef,axiom,
    ( archim2889992004027027881ng_rat
    = ( ^ [X3: rat] :
          ( if_int
          @ ( X3
            = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X3 ) ) )
          @ ( archim3151403230148437115or_rat @ X3 )
          @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int ) ) ) ) ).

% ceiling_altdef
thf(fact_2449_Suc__diff__eq__diff__pred,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
        = ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_2450_Suc__pred_H,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( N2
        = ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_2451_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M: nat,N4: nat] : ( if_nat @ ( M = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N4 ) ) ) ) ) ).

% add_eq_if
thf(fact_2452_Suc__n__minus__m__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_nat @ one_one_nat @ M2 )
       => ( ( suc @ ( minus_minus_nat @ N2 @ M2 ) )
          = ( minus_minus_nat @ N2 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ).

% Suc_n_minus_m_eq
thf(fact_2453_floor__unique,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X )
     => ( ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
       => ( ( archim3151403230148437115or_rat @ X )
          = Z ) ) ) ).

% floor_unique
thf(fact_2454_floor__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( ( archim3151403230148437115or_rat @ X )
        = A )
      = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).

% floor_eq_iff
thf(fact_2455_floor__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim3151403230148437115or_rat @ T ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I3 ) @ T )
              & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) ) )
           => ( P @ I3 ) ) ) ) ).

% floor_split
thf(fact_2456_ceiling__correct,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) @ one_one_rat ) @ X )
      & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ) ).

% ceiling_correct
thf(fact_2457_ceiling__unique,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X )
     => ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) )
       => ( ( archim2889992004027027881ng_rat @ X )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_2458_ceiling__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( ( archim2889992004027027881ng_rat @ X )
        = A )
      = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X )
        & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_2459_ceiling__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) @ T )
              & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I3 ) ) )
           => ( P @ I3 ) ) ) ) ).

% ceiling_split
thf(fact_2460_less__floor__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).

% less_floor_iff
thf(fact_2461_floor__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ Z )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% floor_le_iff
thf(fact_2462_ceiling__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% ceiling_less_iff
thf(fact_2463_le__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).

% le_ceiling_iff
thf(fact_2464_floor__correct,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X ) ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_2465_power__diff__power__eq,axiom,
    ! [A: int,N2: nat,M2: nat] :
      ( ( A != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N2 @ M2 )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N2 ) )
            = ( power_power_int @ A @ ( minus_minus_nat @ M2 @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M2 )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N2 ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_2466_power__diff__power__eq,axiom,
    ! [A: nat,N2: nat,M2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N2 @ M2 )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) )
            = ( power_power_nat @ A @ ( minus_minus_nat @ M2 @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M2 )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N2 ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_2467_power__diff__power__eq,axiom,
    ! [A: code_integer,N2: nat,M2: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( ( ord_less_eq_nat @ N2 @ M2 )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M2 ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
            = ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M2 @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M2 )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M2 ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
            = ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_2468_power__diff__power__eq,axiom,
    ! [A: code_natural,N2: nat,M2: nat] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( ( ord_less_eq_nat @ N2 @ M2 )
         => ( ( divide5121882707175180666atural @ ( power_7079662738309270450atural @ A @ M2 ) @ ( power_7079662738309270450atural @ A @ N2 ) )
            = ( power_7079662738309270450atural @ A @ ( minus_minus_nat @ M2 @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M2 )
         => ( ( divide5121882707175180666atural @ ( power_7079662738309270450atural @ A @ M2 ) @ ( power_7079662738309270450atural @ A @ N2 ) )
            = ( divide5121882707175180666atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_2469_power__eq__if,axiom,
    ( power_8256067586552552935nteger
    = ( ^ [P6: code_integer,M: nat] : ( if_Code_integer @ ( M = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ P6 @ ( power_8256067586552552935nteger @ P6 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2470_power__eq__if,axiom,
    ( power_power_assn
    = ( ^ [P6: assn,M: nat] : ( if_assn @ ( M = zero_zero_nat ) @ one_one_assn @ ( times_times_assn @ P6 @ ( power_power_assn @ P6 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2471_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P6: rat,M: nat] : ( if_rat @ ( M = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P6 @ ( power_power_rat @ P6 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2472_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P6: nat,M: nat] : ( if_nat @ ( M = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2473_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P6: int,M: nat] : ( if_int @ ( M = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P6 @ ( power_power_int @ P6 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2474_power__minus__mult,axiom,
    ! [N2: nat,A: assn] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_assn @ ( power_power_assn @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_assn @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2475_power__minus__mult,axiom,
    ! [N2: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_rat @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2476_power__minus__mult,axiom,
    ! [N2: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_nat @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2477_power__minus__mult,axiom,
    ! [N2: nat,A: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_int @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2478_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_2479_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_2480_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_2481_option_Osize__gen_I2_J,axiom,
    ! [X: num > nat,X2: num] :
      ( ( size_option_num @ X @ ( some_num @ X2 ) )
      = ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_2482_option_Osize__gen_I2_J,axiom,
    ! [X: produc8664842809031399944it_nat > nat,X2: produc8664842809031399944it_nat] :
      ( ( size_o29782932136985253it_nat @ X @ ( some_P1914260805536162275it_nat @ X2 ) )
      = ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_2483_Heap_Osize__gen,axiom,
    ! [Xa: product_unit > nat,X: heap_e7401611519738050253t_unit > option8956607266484857688it_nat] :
      ( ( heap_T4142866422068808505t_unit @ Xa @ ( heap_T6183433275982383450t_unit @ X ) )
      = ( suc @ zero_zero_nat ) ) ).

% Heap.size_gen
thf(fact_2484_num_Osize__gen_I2_J,axiom,
    ! [X2: num] :
      ( ( size_num @ ( bit0 @ X2 ) )
      = ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(2)
thf(fact_2485_normalize__negative,axiom,
    ! [Q3: int,P4: int] :
      ( ( ord_less_int @ Q3 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q3 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P4 ) @ ( uminus_uminus_int @ Q3 ) ) ) ) ) ).

% normalize_negative
thf(fact_2486_option_Osize__gen_I1_J,axiom,
    ! [X: produc6714399278180371524it_nat > nat] :
      ( ( size_o5312321843473198183it_nat @ X @ none_P2587165478634041381it_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_2487_option_Osize__gen_I1_J,axiom,
    ! [X: num > nat] :
      ( ( size_option_num @ X @ none_num )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_2488_option_Osize__gen_I1_J,axiom,
    ! [X: produc8664842809031399944it_nat > nat] :
      ( ( size_o29782932136985253it_nat @ X @ none_P9117596204409417319it_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_2489_Heap_Osize_I2_J,axiom,
    ! [X: heap_e7401611519738050253t_unit > option8956607266484857688it_nat] :
      ( ( size_s1564113455978345259t_unit @ ( heap_T6183433275982383450t_unit @ X ) )
      = ( suc @ zero_zero_nat ) ) ).

% Heap.size(2)
thf(fact_2490_num_Osize_I5_J,axiom,
    ! [X2: num] :
      ( ( size_size_num @ ( bit0 @ X2 ) )
      = ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(5)
thf(fact_2491_option_Osize_I3_J,axiom,
    ( ( size_s6797157356856800118it_nat @ none_P2587165478634041381it_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2492_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2493_option_Osize_I3_J,axiom,
    ( ( size_s8766407808098229740it_nat @ none_P9117596204409417319it_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2494_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 ) ) ).

% normalize_denom_zero
thf(fact_2495_option_Osize__gen__o__map,axiom,
    ! [F: num > nat,G: num > num] :
      ( ( comp_o6878144249584144265on_num @ ( size_option_num @ F ) @ ( map_option_num_num @ G ) )
      = ( size_option_num @ ( comp_num_nat_num @ F @ G ) ) ) ).

% option.size_gen_o_map
thf(fact_2496_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P4: int,Q3: int] :
      ( ( ( normalize @ R2 )
        = ( product_Pair_int_int @ P4 @ Q3 ) )
     => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).

% normalize_denom_pos
thf(fact_2497_normalize__crossproduct,axiom,
    ! [Q3: int,S: int,P4: int,R2: int] :
      ( ( Q3 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P4 @ Q3 ) )
            = ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
         => ( ( times_times_int @ P4 @ S )
            = ( times_times_int @ R2 @ Q3 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_2498_option_Osize_I4_J,axiom,
    ! [X2: num] :
      ( ( size_size_option_num @ ( some_num @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2499_option_Osize_I4_J,axiom,
    ! [X2: produc8664842809031399944it_nat] :
      ( ( size_s8766407808098229740it_nat @ ( some_P1914260805536162275it_nat @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2500_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M: nat,N4: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N4 = zero_zero_nat )
            | ( ord_less_nat @ M @ N4 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M @ N4 ) @ N4 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_2501_sum__gp__multiplied,axiom,
    ! [M2: nat,N2: nat,X: code_integer] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) )
        = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ M2 ) @ ( power_8256067586552552935nteger @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_2502_sum__gp__multiplied,axiom,
    ! [M2: nat,N2: nat,X: rat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) )
        = ( minus_minus_rat @ ( power_power_rat @ X @ M2 ) @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_2503_sum__gp__multiplied,axiom,
    ! [M2: nat,N2: nat,X: int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) )
        = ( minus_minus_int @ ( power_power_int @ X @ M2 ) @ ( power_power_int @ X @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_2504_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_2505_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z ) ) ).

% pair_less_iff1
thf(fact_2506_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 ) ) ).

% Suc_times_numeral_mod_eq
thf(fact_2507_one__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).

% one_int.transfer
thf(fact_2508_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ less_than )
      = ( ord_less_nat @ X @ Y ) ) ).

% less_than_iff
thf(fact_2509_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_2510_sum__telescope_H_H,axiom,
    ! [M2: nat,N2: nat,F: nat > code_integer] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups7501900531339628137nteger
          @ ^ [K4: nat] : ( minus_8373710615458151222nteger @ ( F @ K4 ) @ ( F @ ( minus_minus_nat @ K4 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) )
        = ( minus_8373710615458151222nteger @ ( F @ N2 ) @ ( F @ M2 ) ) ) ) ).

% sum_telescope''
thf(fact_2511_sum__telescope_H_H,axiom,
    ! [M2: nat,N2: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [K4: nat] : ( minus_minus_rat @ ( F @ K4 ) @ ( F @ ( minus_minus_nat @ K4 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) )
        = ( minus_minus_rat @ ( F @ N2 ) @ ( F @ M2 ) ) ) ) ).

% sum_telescope''
thf(fact_2512_sum__telescope_H_H,axiom,
    ! [M2: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [K4: nat] : ( minus_minus_int @ ( F @ K4 ) @ ( F @ ( minus_minus_nat @ K4 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) )
        = ( minus_minus_int @ ( F @ N2 ) @ ( F @ M2 ) ) ) ) ).

% sum_telescope''
thf(fact_2513_case__prod__conv,axiom,
    ! [F: nat > nat > nat,A: nat,B: nat] :
      ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_2514_case__prod__conv,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
      ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_2515_case__prod__conv,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_2516_case__prod__conv,axiom,
    ! [F: int > int > product_prod_int_int,A: int,B: int] :
      ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_2517_case__prod__conv,axiom,
    ! [F: int > int > int,A: int,B: int] :
      ( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_2518_case__prod__curry,axiom,
    ! [F: product_prod_nat_nat > nat] :
      ( ( produc6842872674320459806at_nat @ ( produc8871627978511607229at_nat @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_2519_case__prod__curry,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc27273713700761075at_nat @ ( produc3610074571335452306at_nat @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_2520_case__prod__curry,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( produc8739625826339149834_nat_o @ ( produc3704529784387675049_nat_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_2521_case__prod__curry,axiom,
    ! [F: product_prod_int_int > product_prod_int_int] :
      ( ( produc4245557441103728435nt_int @ ( produc8249235968001453780nt_int @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_2522_case__prod__curry,axiom,
    ! [F: product_prod_int_int > int] :
      ( ( produc8211389475949308722nt_int @ ( produc1016772743285680337nt_int @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_2523_curry__case__prod,axiom,
    ! [F: nat > nat > nat] :
      ( ( produc8871627978511607229at_nat @ ( produc6842872674320459806at_nat @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_2524_curry__case__prod,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc3610074571335452306at_nat @ ( produc27273713700761075at_nat @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_2525_curry__case__prod,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o] :
      ( ( produc3704529784387675049_nat_o @ ( produc8739625826339149834_nat_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_2526_curry__case__prod,axiom,
    ! [F: int > int > product_prod_int_int] :
      ( ( produc8249235968001453780nt_int @ ( produc4245557441103728435nt_int @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_2527_curry__case__prod,axiom,
    ! [F: int > int > int] :
      ( ( produc1016772743285680337nt_int @ ( produc8211389475949308722nt_int @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_2528_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2529_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2530_distrib__right__numeral,axiom,
    ! [A: code_integer,B: code_integer,V: num] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( numera6620942414471956472nteger @ V ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ B @ ( numera6620942414471956472nteger @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2531_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2532_distrib__right__numeral,axiom,
    ! [A: code_natural,B: code_natural,V: num] :
      ( ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ A @ B ) @ ( numera5444537566228673987atural @ V ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ ( numera5444537566228673987atural @ V ) ) @ ( times_2397367101498566445atural @ B @ ( numera5444537566228673987atural @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2533_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2534_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2535_distrib__left__numeral,axiom,
    ! [V: num,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ B ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2536_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2537_distrib__left__numeral,axiom,
    ! [V: num,B: code_natural,C: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ ( plus_p4538020629002901425atural @ B @ C ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ B ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2538_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_2539_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_2540_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_2541_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_2542_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_2543_add__numeral__left,axiom,
    ! [V: num,W: num,Z: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ V ) @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ W ) @ Z ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_2544_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_2545_add__numeral__left,axiom,
    ! [V: num,W: num,Z: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ V ) @ ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ W ) @ Z ) )
      = ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_2546_numeral__plus__numeral,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_2547_numeral__plus__numeral,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_2548_numeral__plus__numeral,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_2549_numeral__plus__numeral,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_2550_numeral__plus__numeral,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ M2 ) @ ( numera5444537566228673987atural @ N2 ) )
      = ( numera5444537566228673987atural @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_2551_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_2552_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_2553_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_2554_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N2: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_2555_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N2: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M2 @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_2556_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N2: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N2 ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_2557_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N2: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_2558_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N2: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_2559_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N2: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_2560_ceiling__add__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_2561_power__add__numeral,axiom,
    ! [A: assn,M2: num,N2: num] :
      ( ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_assn @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_assn @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_2562_power__add__numeral,axiom,
    ! [A: rat,M2: num,N2: num] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_2563_power__add__numeral,axiom,
    ! [A: nat,M2: num,N2: num] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_2564_power__add__numeral,axiom,
    ! [A: int,M2: num,N2: num] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_2565_power__add__numeral2,axiom,
    ! [A: assn,M2: num,N2: num,B: assn] :
      ( ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_2566_power__add__numeral2,axiom,
    ! [A: rat,M2: num,N2: num,B: rat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_2567_power__add__numeral2,axiom,
    ! [A: nat,M2: num,N2: num,B: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_2568_power__add__numeral2,axiom,
    ! [A: int,M2: num,N2: num,B: int] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_2569_numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).

% numeral_less_floor
thf(fact_2570_floor__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% floor_le_numeral
thf(fact_2571_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > code_integer] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups7501900531339628137nteger @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups7501900531339628137nteger @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( plus_p5714425477246183910nteger @ ( groups7501900531339628137nteger @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_2572_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = zero_z1048942125864253310at_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_2573_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_2574_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_2575_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_2576_ceiling__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% ceiling_less_numeral
thf(fact_2577_numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).

% numeral_le_ceiling
thf(fact_2578_neg__numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).

% neg_numeral_less_floor
thf(fact_2579_floor__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_rat @ X @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% floor_le_neg_numeral
thf(fact_2580_ceiling__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_2581_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).

% neg_numeral_le_ceiling
thf(fact_2582_nested__case__prod__simp,axiom,
    ( produc27273713700761075at_nat
    = ( ^ [F2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( produc2626176000494625587at_nat
          @ ^ [A5: nat,B4: nat] : ( F2 @ A5 @ B4 @ Y3 )
          @ X3 ) ) ) ).

% nested_case_prod_simp
thf(fact_2583_nested__case__prod__simp,axiom,
    ( produc8739625826339149834_nat_o
    = ( ^ [F2: nat > nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [A5: nat,B4: nat] : ( F2 @ A5 @ B4 @ Y3 )
          @ X3 ) ) ) ).

% nested_case_prod_simp
thf(fact_2584_prod_Ocase__distrib,axiom,
    ! [H3: nat > nat,F: nat > nat > nat,Prod: product_prod_nat_nat] :
      ( ( H3 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
      = ( produc6842872674320459806at_nat
        @ ^ [X12: nat,X23: nat] : ( H3 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_2585_prod_Ocase__distrib,axiom,
    ! [H3: int > int,F: int > int > int,Prod: product_prod_int_int] :
      ( ( H3 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
      = ( produc8211389475949308722nt_int
        @ ^ [X12: int,X23: int] : ( H3 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_2586_prod_Ocase__distrib,axiom,
    ! [H3: product_prod_int_int > int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( H3 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( produc8211389475949308722nt_int
        @ ^ [X12: int,X23: int] : ( H3 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_2587_prod_Ocase__distrib,axiom,
    ! [H3: int > product_prod_int_int,F: int > int > int,Prod: product_prod_int_int] :
      ( ( H3 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
      = ( produc4245557441103728435nt_int
        @ ^ [X12: int,X23: int] : ( H3 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_2588_prod_Ocase__distrib,axiom,
    ! [H3: nat > product_prod_nat_nat > $o,F: nat > nat > nat,Prod: product_prod_nat_nat] :
      ( ( H3 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X12: nat,X23: nat] : ( H3 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_2589_prod_Ocase__distrib,axiom,
    ! [H3: ( product_prod_nat_nat > $o ) > nat,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
      ( ( H3 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
      = ( produc6842872674320459806at_nat
        @ ^ [X12: nat,X23: nat] : ( H3 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_2590_prod_Ocase__distrib,axiom,
    ! [H3: product_prod_int_int > product_prod_int_int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( H3 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( produc4245557441103728435nt_int
        @ ^ [X12: int,X23: int] : ( H3 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_2591_prod_Ocase__distrib,axiom,
    ! [H3: nat > product_prod_nat_nat > product_prod_nat_nat,F: nat > nat > nat,Prod: product_prod_nat_nat] :
      ( ( H3 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
      = ( produc27273713700761075at_nat
        @ ^ [X12: nat,X23: nat] : ( H3 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_2592_prod_Ocase__distrib,axiom,
    ! [H3: ( product_prod_nat_nat > product_prod_nat_nat ) > nat,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
      ( ( H3 @ ( produc27273713700761075at_nat @ F @ Prod ) )
      = ( produc6842872674320459806at_nat
        @ ^ [X12: nat,X23: nat] : ( H3 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_2593_prod_Ocase__distrib,axiom,
    ! [H3: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
      ( ( H3 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X12: nat,X23: nat] : ( H3 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_2594_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G @ suc ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% sum.atLeast_Suc_atMost_Suc_shift
thf(fact_2595_sum_OatLeastAtMost__shift__bounds,axiom,
    ! [G: nat > nat,M2: nat,K: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G @ ( plus_plus_nat @ K ) ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% sum.atLeastAtMost_shift_bounds
thf(fact_2596_split__cong,axiom,
    ! [Q3: product_prod_nat_nat,F: nat > nat > nat,G: nat > nat > nat,P4: product_prod_nat_nat] :
      ( ! [X4: nat,Y5: nat] :
          ( ( ( product_Pair_nat_nat @ X4 @ Y5 )
            = Q3 )
         => ( ( F @ X4 @ Y5 )
            = ( G @ X4 @ Y5 ) ) )
     => ( ( P4 = Q3 )
       => ( ( produc6842872674320459806at_nat @ F @ P4 )
          = ( produc6842872674320459806at_nat @ G @ Q3 ) ) ) ) ).

% split_cong
thf(fact_2597_split__cong,axiom,
    ! [Q3: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: nat > nat > product_prod_nat_nat > product_prod_nat_nat,P4: product_prod_nat_nat] :
      ( ! [X4: nat,Y5: nat] :
          ( ( ( product_Pair_nat_nat @ X4 @ Y5 )
            = Q3 )
         => ( ( F @ X4 @ Y5 )
            = ( G @ X4 @ Y5 ) ) )
     => ( ( P4 = Q3 )
       => ( ( produc27273713700761075at_nat @ F @ P4 )
          = ( produc27273713700761075at_nat @ G @ Q3 ) ) ) ) ).

% split_cong
thf(fact_2598_split__cong,axiom,
    ! [Q3: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > $o,G: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat] :
      ( ! [X4: nat,Y5: nat] :
          ( ( ( product_Pair_nat_nat @ X4 @ Y5 )
            = Q3 )
         => ( ( F @ X4 @ Y5 )
            = ( G @ X4 @ Y5 ) ) )
     => ( ( P4 = Q3 )
       => ( ( produc8739625826339149834_nat_o @ F @ P4 )
          = ( produc8739625826339149834_nat_o @ G @ Q3 ) ) ) ) ).

% split_cong
thf(fact_2599_split__cong,axiom,
    ! [Q3: product_prod_int_int,F: int > int > product_prod_int_int,G: int > int > product_prod_int_int,P4: product_prod_int_int] :
      ( ! [X4: int,Y5: int] :
          ( ( ( product_Pair_int_int @ X4 @ Y5 )
            = Q3 )
         => ( ( F @ X4 @ Y5 )
            = ( G @ X4 @ Y5 ) ) )
     => ( ( P4 = Q3 )
       => ( ( produc4245557441103728435nt_int @ F @ P4 )
          = ( produc4245557441103728435nt_int @ G @ Q3 ) ) ) ) ).

% split_cong
thf(fact_2600_split__cong,axiom,
    ! [Q3: product_prod_int_int,F: int > int > int,G: int > int > int,P4: product_prod_int_int] :
      ( ! [X4: int,Y5: int] :
          ( ( ( product_Pair_int_int @ X4 @ Y5 )
            = Q3 )
         => ( ( F @ X4 @ Y5 )
            = ( G @ X4 @ Y5 ) ) )
     => ( ( P4 = Q3 )
       => ( ( produc8211389475949308722nt_int @ F @ P4 )
          = ( produc8211389475949308722nt_int @ G @ Q3 ) ) ) ) ).

% split_cong
thf(fact_2601_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > nat,X1: nat,X2: nat] :
      ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_2602_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X2: nat] :
      ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_2603_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X2: nat] :
      ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_2604_old_Oprod_Ocase,axiom,
    ! [F: int > int > product_prod_int_int,X1: int,X2: int] :
      ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_2605_old_Oprod_Ocase,axiom,
    ! [F: int > int > int,X1: int,X2: int] :
      ( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_2606_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc2626176000494625587at_nat
          @ ^ [X3: nat,Y3: nat] : ( product_Pair_nat_nat @ Y3 @ X3 )
          @ X ) ) ) ).

% uminus_int.abs_eq
thf(fact_2607_case__prod__Pair__iden,axiom,
    ! [P4: produc1908205239877642774nteger] :
      ( ( produc6512950862096126219nteger @ produc8603105652947943368nteger @ P4 )
      = P4 ) ).

% case_prod_Pair_iden
thf(fact_2608_case__prod__Pair__iden,axiom,
    ! [P4: produc3925858234332021118et_nat] :
      ( ( produc4058941399401191971et_nat @ produc5001842942810119800et_nat @ P4 )
      = P4 ) ).

% case_prod_Pair_iden
thf(fact_2609_case__prod__Pair__iden,axiom,
    ! [P4: produc2732055786443039994et_nat] :
      ( ( produc2377985495875741467et_nat @ produc2245416461498447860et_nat @ P4 )
      = P4 ) ).

% case_prod_Pair_iden
thf(fact_2610_case__prod__Pair__iden,axiom,
    ! [P4: produc2285326912895808259nt_int] :
      ( ( produc8492565224438309093nt_int @ produc5700946648718959541nt_int @ P4 )
      = P4 ) ).

% case_prod_Pair_iden
thf(fact_2611_case__prod__Pair__iden,axiom,
    ! [P4: produc7773217078559923341nt_int] :
      ( ( produc5122537100556696953nt_int @ produc4305682042979456191nt_int @ P4 )
      = P4 ) ).

% case_prod_Pair_iden
thf(fact_2612_case__prod__Pair__iden,axiom,
    ! [P4: product_prod_int_int] :
      ( ( produc4245557441103728435nt_int @ product_Pair_int_int @ P4 )
      = P4 ) ).

% case_prod_Pair_iden
thf(fact_2613_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_2614_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M2: nat,K: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_2615_cond__case__prod__eta,axiom,
    ! [F: nat > nat > nat,G: product_prod_nat_nat > nat] :
      ( ! [X4: nat,Y5: nat] :
          ( ( F @ X4 @ Y5 )
          = ( G @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) )
     => ( ( produc6842872674320459806at_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_2616_cond__case__prod__eta,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ! [X4: nat,Y5: nat] :
          ( ( F @ X4 @ Y5 )
          = ( G @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) )
     => ( ( produc27273713700761075at_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_2617_cond__case__prod__eta,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [X4: nat,Y5: nat] :
          ( ( F @ X4 @ Y5 )
          = ( G @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) )
     => ( ( produc8739625826339149834_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_2618_cond__case__prod__eta,axiom,
    ! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
      ( ! [X4: int,Y5: int] :
          ( ( F @ X4 @ Y5 )
          = ( G @ ( product_Pair_int_int @ X4 @ Y5 ) ) )
     => ( ( produc4245557441103728435nt_int @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_2619_cond__case__prod__eta,axiom,
    ! [F: int > int > int,G: product_prod_int_int > int] :
      ( ! [X4: int,Y5: int] :
          ( ( F @ X4 @ Y5 )
          = ( G @ ( product_Pair_int_int @ X4 @ Y5 ) ) )
     => ( ( produc8211389475949308722nt_int @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_2620_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > nat] :
      ( ( produc6842872674320459806at_nat
        @ ^ [X3: nat,Y3: nat] : ( F @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_2621_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc27273713700761075at_nat
        @ ^ [X3: nat,Y3: nat] : ( F @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_2622_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( produc8739625826339149834_nat_o
        @ ^ [X3: nat,Y3: nat] : ( F @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_2623_case__prod__eta,axiom,
    ! [F: product_prod_int_int > product_prod_int_int] :
      ( ( produc4245557441103728435nt_int
        @ ^ [X3: int,Y3: int] : ( F @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_2624_case__prod__eta,axiom,
    ! [F: product_prod_int_int > int] :
      ( ( produc8211389475949308722nt_int
        @ ^ [X3: int,Y3: int] : ( F @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_2625_case__prodE2,axiom,
    ! [Q: nat > $o,P: nat > nat > nat,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc6842872674320459806at_nat @ P @ Z ) )
     => ~ ! [X4: nat,Y5: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X4 @ Y5 ) )
           => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_2626_case__prodE2,axiom,
    ! [Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc27273713700761075at_nat @ P @ Z ) )
     => ~ ! [X4: nat,Y5: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X4 @ Y5 ) )
           => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_2627_case__prodE2,axiom,
    ! [Q: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc8739625826339149834_nat_o @ P @ Z ) )
     => ~ ! [X4: nat,Y5: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X4 @ Y5 ) )
           => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_2628_case__prodE2,axiom,
    ! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z: product_prod_int_int] :
      ( ( Q @ ( produc4245557441103728435nt_int @ P @ Z ) )
     => ~ ! [X4: int,Y5: int] :
            ( ( Z
              = ( product_Pair_int_int @ X4 @ Y5 ) )
           => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_2629_case__prodE2,axiom,
    ! [Q: int > $o,P: int > int > int,Z: product_prod_int_int] :
      ( ( Q @ ( produc8211389475949308722nt_int @ P @ Z ) )
     => ~ ! [X4: int,Y5: int] :
            ( ( Z
              = ( product_Pair_int_int @ X4 @ Y5 ) )
           => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_2630_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X4: nat,Y5: nat] :
          ( Z
         != ( abs_Integ @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) ) ).

% eq_Abs_Integ
thf(fact_2631_one__le__numeral,axiom,
    ! [N2: num] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N2 ) ) ).

% one_le_numeral
thf(fact_2632_one__le__numeral,axiom,
    ! [N2: num] : ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ N2 ) ) ).

% one_le_numeral
thf(fact_2633_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% one_le_numeral
thf(fact_2634_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% one_le_numeral
thf(fact_2635_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).

% one_le_numeral
thf(fact_2636_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_2637_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_2638_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N2 ) @ one_one_Code_integer ) ).

% not_numeral_less_one
thf(fact_2639_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_2640_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_le5570908160329646204atural @ ( numera5444537566228673987atural @ N2 ) @ one_one_Code_natural ) ).

% not_numeral_less_one
thf(fact_2641_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).

% one_plus_numeral_commute
thf(fact_2642_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).

% one_plus_numeral_commute
thf(fact_2643_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer ) ) ).

% one_plus_numeral_commute
thf(fact_2644_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).

% one_plus_numeral_commute
thf(fact_2645_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ X ) )
      = ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ X ) @ one_one_Code_natural ) ) ).

% one_plus_numeral_commute
thf(fact_2646_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_2647_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_2648_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numera6620942414471956472nteger @ ( bit0 @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_2649_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_2650_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numera5444537566228673987atural @ ( bit0 @ N2 ) )
      = ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N2 ) @ ( numera5444537566228673987atural @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_2651_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_2652_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_Code_integer
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_2653_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_2654_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ N2 )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_2655_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numera6620942414471956472nteger @ N2 )
     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% numeral_neq_neg_one
thf(fact_2656_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ N2 )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_2657_uncurry__def,axiom,
    uncurry_nat_nat_nat = produc6842872674320459806at_nat ).

% uncurry_def
thf(fact_2658_uncurry__def,axiom,
    uncurr8011562610307062878at_nat = produc27273713700761075at_nat ).

% uncurry_def
thf(fact_2659_uncurry__def,axiom,
    uncurr7511940902602773877_nat_o = produc8739625826339149834_nat_o ).

% uncurry_def
thf(fact_2660_uncurry__def,axiom,
    uncurr7650761721940715016nt_int = produc4245557441103728435nt_int ).

% uncurry_def
thf(fact_2661_uncurry__def,axiom,
    uncurry_int_int_int = produc8211389475949308722nt_int ).

% uncurry_def
thf(fact_2662_internal__case__prod__def,axiom,
    produc6557683772455589876at_nat = produc6842872674320459806at_nat ).

% internal_case_prod_def
thf(fact_2663_internal__case__prod__def,axiom,
    produc1854806715440696265at_nat = produc27273713700761075at_nat ).

% internal_case_prod_def
thf(fact_2664_internal__case__prod__def,axiom,
    produc4780622933104268256_nat_o = produc8739625826339149834_nat_o ).

% internal_case_prod_def
thf(fact_2665_internal__case__prod__def,axiom,
    produc297006045350968285nt_int = produc4245557441103728435nt_int ).

% internal_case_prod_def
thf(fact_2666_internal__case__prod__def,axiom,
    produc7926200574084438792nt_int = produc8211389475949308722nt_int ).

% internal_case_prod_def
thf(fact_2667_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N2: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M2 ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_2668_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_2669_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_2670_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numera6620942414471956472nteger @ ( bit0 @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_2671_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_2672_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numera5444537566228673987atural @ ( bit0 @ N2 ) )
      = ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N2 ) @ ( numera5444537566228673987atural @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_2673_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > code_integer,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_z3403309356797280102nteger )
     => ( ( groups7501900531339628137nteger @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups7501900531339628137nteger @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_2674_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > multis2468970476368604999at_nat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_z1048942125864253310at_nat )
     => ( ( groups6857163185585827899at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups6857163185585827899at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_2675_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > rat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_rat )
     => ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_2676_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > int,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_int )
     => ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_2677_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > nat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_nat )
     => ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_2678_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_2679_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_2680_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_2681_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_2682_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_p7104986032573967614at_nat @ ( G @ zero_zero_nat ) @ ( groups6857163185585827899at_nat @ ( comp_n8698576032424989604at_nat @ G @ suc ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc_shift
thf(fact_2683_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat ) @ ( groups2906978787729119204at_rat @ ( comp_nat_rat_nat @ G @ suc ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc_shift
thf(fact_2684_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat ) @ ( groups3539618377306564664at_int @ ( comp_nat_int_nat @ G @ suc ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc_shift
thf(fact_2685_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat ) @ ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G @ suc ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc_shift
thf(fact_2686_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N2: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( plus_p7104986032573967614at_nat @ ( G @ M2 ) @ ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_2687_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( plus_plus_rat @ ( G @ M2 ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_2688_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( plus_plus_int @ ( G @ M2 ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_2689_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( plus_plus_nat @ ( G @ M2 ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_2690_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N2: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
        = ( plus_p7104986032573967614at_nat @ ( G @ ( suc @ N2 ) ) @ ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_2691_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
        = ( plus_plus_rat @ ( G @ ( suc @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_2692_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
        = ( plus_plus_int @ ( G @ ( suc @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_2693_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
        = ( plus_plus_nat @ ( G @ ( suc @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_2694_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_2695_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat ) ).

% neg_numeral_le_one
thf(fact_2696_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) ).

% neg_numeral_le_one
thf(fact_2697_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_2698_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_2699_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_2700_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% neg_numeral_le_neg_one
thf(fact_2701_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% neg_numeral_le_neg_one
thf(fact_2702_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% neg_numeral_le_neg_one
thf(fact_2703_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_2704_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_le_neg_one
thf(fact_2705_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_le_neg_one
thf(fact_2706_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_2707_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_2708_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_2709_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_2710_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_2711_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_2712_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_2713_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_2714_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_2715_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_less_neg_one
thf(fact_2716_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_2717_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_less_neg_one
thf(fact_2718_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_2719_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_2720_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_2721_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) ).

% neg_numeral_less_one
thf(fact_2722_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_2723_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat ) ).

% neg_numeral_less_one
thf(fact_2724_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N2: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_p7104986032573967614at_nat @ ( G @ M2 )
          @ ( groups6857163185585827899at_nat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_2725_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_rat @ ( G @ M2 )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_2726_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_int @ ( G @ M2 )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_2727_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_nat @ ( G @ M2 )
          @ ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_2728_sum__Suc__diff,axiom,
    ! [M2: nat,N2: nat,F: nat > code_integer] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( minus_8373710615458151222nteger @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( minus_8373710615458151222nteger @ ( F @ ( suc @ N2 ) ) @ ( F @ M2 ) ) ) ) ).

% sum_Suc_diff
thf(fact_2729_sum__Suc__diff,axiom,
    ! [M2: nat,N2: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ M2 ) ) ) ) ).

% sum_Suc_diff
thf(fact_2730_sum__Suc__diff,axiom,
    ! [M2: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ M2 ) ) ) ) ).

% sum_Suc_diff
thf(fact_2731_sum__power__add,axiom,
    ! [X: rat,M2: nat,I4: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M2 @ I3 ) )
        @ I4 )
      = ( times_times_rat @ ( power_power_rat @ X @ M2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I4 ) ) ) ).

% sum_power_add
thf(fact_2732_sum__power__add,axiom,
    ! [X: int,M2: nat,I4: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M2 @ I3 ) )
        @ I4 )
      = ( times_times_int @ ( power_power_int @ X @ M2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I4 ) ) ) ).

% sum_power_add
thf(fact_2733_sum_OatLeast__atMost__pred__shift,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ( comp_nat_nat_nat @ G
          @ ^ [N4: nat] : ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% sum.atLeast_atMost_pred_shift
thf(fact_2734_sum__atLeastAtMost__code,axiom,
    ! [F: nat > code_integer,A: nat,B: nat] :
      ( ( groups7501900531339628137nteger @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1084959871951514735nteger
        @ ^ [A5: nat] : ( plus_p5714425477246183910nteger @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_z3403309356797280102nteger ) ) ).

% sum_atLeastAtMost_code
thf(fact_2735_sum__atLeastAtMost__code,axiom,
    ! [F: nat > multis2468970476368604999at_nat,A: nat,B: nat] :
      ( ( groups6857163185585827899at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo4497565046347964853at_nat
        @ ^ [A5: nat] : ( plus_p7104986032573967614at_nat @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_z1048942125864253310at_nat ) ) ).

% sum_atLeastAtMost_code
thf(fact_2736_sum__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B: nat] :
      ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A5: nat] : ( plus_plus_rat @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_zero_rat ) ) ).

% sum_atLeastAtMost_code
thf(fact_2737_sum__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A5: nat] : ( plus_plus_int @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_zero_int ) ) ).

% sum_atLeastAtMost_code
thf(fact_2738_sum__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A5: nat] : ( plus_plus_nat @ ( F @ A5 ) )
        @ A
        @ B
        @ zero_zero_nat ) ) ).

% sum_atLeastAtMost_code
thf(fact_2739_sum_Oub__add__nat,axiom,
    ! [M2: nat,N2: nat,G: nat > multis2468970476368604999at_nat,P4: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_2740_sum_Oub__add__nat,axiom,
    ! [M2: nat,N2: nat,G: nat > rat,P4: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_2741_sum_Oub__add__nat,axiom,
    ! [M2: nat,N2: nat,G: nat > int,P4: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_2742_sum_Oub__add__nat,axiom,
    ! [M2: nat,N2: nat,G: nat > nat,P4: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_2743_sum_OatLeastAtMost__shift__0,axiom,
    ! [M2: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G @ ( plus_plus_nat @ M2 ) ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ).

% sum.atLeastAtMost_shift_0
thf(fact_2744_zero__int__def,axiom,
    ( zero_zero_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).

% zero_int_def
thf(fact_2745_pair__lessI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_2746_zero__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).

% zero_int.transfer
thf(fact_2747_sum__natinterval__diff,axiom,
    ! [M2: nat,N2: nat,F: nat > code_integer] :
      ( ( ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( groups7501900531339628137nteger
            @ ^ [K4: nat] : ( minus_8373710615458151222nteger @ ( F @ K4 ) @ ( F @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = ( minus_8373710615458151222nteger @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( groups7501900531339628137nteger
            @ ^ [K4: nat] : ( minus_8373710615458151222nteger @ ( F @ K4 ) @ ( F @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = zero_z3403309356797280102nteger ) ) ) ).

% sum_natinterval_diff
thf(fact_2748_sum__natinterval__diff,axiom,
    ! [M2: nat,N2: nat,F: nat > rat] :
      ( ( ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K4: nat] : ( minus_minus_rat @ ( F @ K4 ) @ ( F @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = ( minus_minus_rat @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K4: nat] : ( minus_minus_rat @ ( F @ K4 ) @ ( F @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
thf(fact_2749_sum__natinterval__diff,axiom,
    ! [M2: nat,N2: nat,F: nat > int] :
      ( ( ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [K4: nat] : ( minus_minus_int @ ( F @ K4 ) @ ( F @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = ( minus_minus_int @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [K4: nat] : ( minus_minus_int @ ( F @ K4 ) @ ( F @ ( plus_plus_nat @ K4 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = zero_zero_int ) ) ) ).

% sum_natinterval_diff
thf(fact_2750_convex__sum__bound__le,axiom,
    ! [I4: set_o,X: $o > code_integer,A: $o > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ I4 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups4406642042086082107nteger @ X @ I4 )
          = one_one_Code_integer )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I4 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups4406642042086082107nteger
                  @ ^ [I3: $o] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I4 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_2751_convex__sum__bound__le,axiom,
    ! [I4: set_nat,X: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I4 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups7501900531339628137nteger @ X @ I4 )
          = one_one_Code_integer )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I4 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7501900531339628137nteger
                  @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I4 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_2752_convex__sum__bound__le,axiom,
    ! [I4: set_int,X: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I4 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups7873554091576472773nteger @ X @ I4 )
          = one_one_Code_integer )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I4 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7873554091576472773nteger
                  @ ^ [I3: int] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I4 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_2753_convex__sum__bound__le,axiom,
    ! [I4: set_o,X: $o > rat,A: $o > rat,B: rat,Delta: rat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ I4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
     => ( ( ( groups7872700643590313910_o_rat @ X @ I4 )
          = one_one_rat )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I4 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups7872700643590313910_o_rat
                  @ ^ [I3: $o] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I4 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_2754_convex__sum__bound__le,axiom,
    ! [I4: set_nat,X: nat > rat,A: nat > rat,B: rat,Delta: rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
     => ( ( ( groups2906978787729119204at_rat @ X @ I4 )
          = one_one_rat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I4 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I4 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_2755_convex__sum__bound__le,axiom,
    ! [I4: set_int,X: int > rat,A: int > rat,B: rat,Delta: rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
     => ( ( ( groups3906332499630173760nt_rat @ X @ I4 )
          = one_one_rat )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I4 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups3906332499630173760nt_rat
                  @ ^ [I3: int] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I4 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_2756_convex__sum__bound__le,axiom,
    ! [I4: set_o,X: $o > int,A: $o > int,B: int,Delta: int] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ I4 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X @ I2 ) ) )
     => ( ( ( groups8505340233167759370_o_int @ X @ I4 )
          = one_one_int )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I4 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups8505340233167759370_o_int
                  @ ^ [I3: $o] : ( times_times_int @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I4 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_2757_convex__sum__bound__le,axiom,
    ! [I4: set_nat,X: nat > int,A: nat > int,B: int,Delta: int] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I4 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X @ I2 ) ) )
     => ( ( ( groups3539618377306564664at_int @ X @ I4 )
          = one_one_int )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I4 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups3539618377306564664at_int
                  @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I4 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_2758_convex__sum__bound__le,axiom,
    ! [I4: set_int,X: int > int,A: int > int,B: int,Delta: int] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I4 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X @ I2 ) ) )
     => ( ( ( groups4538972089207619220nt_int @ X @ I4 )
          = one_one_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I4 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups4538972089207619220nt_int
                  @ ^ [I3: int] : ( times_times_int @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I4 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_2759_convex__sum__bound__le,axiom,
    ! [I4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat > code_integer,A: product_prod_nat_nat > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ I2 @ I4 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups196832835297161892nteger @ X @ I4 )
          = one_one_Code_integer )
       => ( ! [I2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ I2 @ I4 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups196832835297161892nteger
                  @ ^ [I3: product_prod_nat_nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
                  @ I4 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_2760_sum__comp__morphism,axiom,
    ! [H3: code_integer > code_integer,G: num > code_integer,A2: set_num] :
      ( ( ( H3 @ zero_z3403309356797280102nteger )
        = zero_z3403309356797280102nteger )
     => ( ! [X4: code_integer,Y5: code_integer] :
            ( ( H3 @ ( plus_p5714425477246183910nteger @ X4 @ Y5 ) )
            = ( plus_p5714425477246183910nteger @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) )
       => ( ( groups5442708359310287027nteger @ ( comp_C3531382070062128313er_num @ H3 @ G ) @ A2 )
          = ( H3 @ ( groups5442708359310287027nteger @ G @ A2 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_2761_sum__comp__morphism,axiom,
    ! [H3: nat > code_integer,G: nat > nat,A2: set_nat] :
      ( ( ( H3 @ zero_zero_nat )
        = zero_z3403309356797280102nteger )
     => ( ! [X4: nat,Y5: nat] :
            ( ( H3 @ ( plus_plus_nat @ X4 @ Y5 ) )
            = ( plus_p5714425477246183910nteger @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) )
       => ( ( groups7501900531339628137nteger @ ( comp_n3898172953802868194er_nat @ H3 @ G ) @ A2 )
          = ( H3 @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_2762_sum__comp__morphism,axiom,
    ! [H3: nat > rat,G: nat > nat,A2: set_nat] :
      ( ( ( H3 @ zero_zero_nat )
        = zero_zero_rat )
     => ( ! [X4: nat,Y5: nat] :
            ( ( H3 @ ( plus_plus_nat @ X4 @ Y5 ) )
            = ( plus_plus_rat @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) )
       => ( ( groups2906978787729119204at_rat @ ( comp_nat_rat_nat @ H3 @ G ) @ A2 )
          = ( H3 @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_2763_sum__comp__morphism,axiom,
    ! [H3: nat > int,G: nat > nat,A2: set_nat] :
      ( ( ( H3 @ zero_zero_nat )
        = zero_zero_int )
     => ( ! [X4: nat,Y5: nat] :
            ( ( H3 @ ( plus_plus_nat @ X4 @ Y5 ) )
            = ( plus_plus_int @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) )
       => ( ( groups3539618377306564664at_int @ ( comp_nat_int_nat @ H3 @ G ) @ A2 )
          = ( H3 @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_2764_sum__comp__morphism,axiom,
    ! [H3: int > code_integer,G: int > int,A2: set_int] :
      ( ( ( H3 @ zero_zero_int )
        = zero_z3403309356797280102nteger )
     => ( ! [X4: int,Y5: int] :
            ( ( H3 @ ( plus_plus_int @ X4 @ Y5 ) )
            = ( plus_p5714425477246183910nteger @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) )
       => ( ( groups7873554091576472773nteger @ ( comp_i1585864551200866970er_int @ H3 @ G ) @ A2 )
          = ( H3 @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_2765_sum__comp__morphism,axiom,
    ! [H3: int > rat,G: int > int,A2: set_int] :
      ( ( ( H3 @ zero_zero_int )
        = zero_zero_rat )
     => ( ! [X4: int,Y5: int] :
            ( ( H3 @ ( plus_plus_int @ X4 @ Y5 ) )
            = ( plus_plus_rat @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) )
       => ( ( groups3906332499630173760nt_rat @ ( comp_int_rat_int @ H3 @ G ) @ A2 )
          = ( H3 @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_2766_sum__comp__morphism,axiom,
    ! [H3: int > nat,G: int > int,A2: set_int] :
      ( ( ( H3 @ zero_zero_int )
        = zero_zero_nat )
     => ( ! [X4: int,Y5: int] :
            ( ( H3 @ ( plus_plus_int @ X4 @ Y5 ) )
            = ( plus_plus_nat @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) )
       => ( ( groups4541462559716669496nt_nat @ ( comp_int_nat_int @ H3 @ G ) @ A2 )
          = ( H3 @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_2767_sum__comp__morphism,axiom,
    ! [H3: code_integer > nat,G: nat > code_integer,A2: set_nat] :
      ( ( ( H3 @ zero_z3403309356797280102nteger )
        = zero_zero_nat )
     => ( ! [X4: code_integer,Y5: code_integer] :
            ( ( H3 @ ( plus_p5714425477246183910nteger @ X4 @ Y5 ) )
            = ( plus_plus_nat @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) )
       => ( ( groups3542108847815614940at_nat @ ( comp_C4049556595663050210at_nat @ H3 @ G ) @ A2 )
          = ( H3 @ ( groups7501900531339628137nteger @ G @ A2 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_2768_sum__comp__morphism,axiom,
    ! [H3: rat > nat,G: nat > rat,A2: set_nat] :
      ( ( ( H3 @ zero_zero_rat )
        = zero_zero_nat )
     => ( ! [X4: rat,Y5: rat] :
            ( ( H3 @ ( plus_plus_rat @ X4 @ Y5 ) )
            = ( plus_plus_nat @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) )
       => ( ( groups3542108847815614940at_nat @ ( comp_rat_nat_nat @ H3 @ G ) @ A2 )
          = ( H3 @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_2769_sum__comp__morphism,axiom,
    ! [H3: int > nat,G: nat > int,A2: set_nat] :
      ( ( ( H3 @ zero_zero_int )
        = zero_zero_nat )
     => ( ! [X4: int,Y5: int] :
            ( ( H3 @ ( plus_plus_int @ X4 @ Y5 ) )
            = ( plus_plus_nat @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) )
       => ( ( groups3542108847815614940at_nat @ ( comp_int_nat_nat @ H3 @ G ) @ A2 )
          = ( H3 @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).

% sum_comp_morphism
thf(fact_2770_sum__gp,axiom,
    ! [N2: nat,M2: nat,X: rat] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( ( X = one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
              = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M2 ) ) ) )
          & ( ( X != one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
              = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M2 ) @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_2771_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_2772_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
            = ( divide_divide_nat @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_2773_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
            = ( divide_divide_int @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_2774_sum__gp__offset,axiom,
    ! [X: rat,M2: nat,N2: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N2 ) ) )
          = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N2 ) ) )
          = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X @ M2 ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_2775_Sum__Icc__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M2 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_2776_arith__series__nat,axiom,
    ! [A: nat,D2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I3 @ D2 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N2 @ D2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_2777_mod__double__modulus,axiom,
    ! [M2: code_integer,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
       => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) )
            = ( modulo364778990260209775nteger @ X @ M2 ) )
          | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) )
            = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M2 ) @ M2 ) ) ) ) ) ).

% mod_double_modulus
thf(fact_2778_mod__double__modulus,axiom,
    ! [M2: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
            = ( modulo_modulo_nat @ X @ M2 ) )
          | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M2 ) @ M2 ) ) ) ) ) ).

% mod_double_modulus
thf(fact_2779_mod__double__modulus,axiom,
    ! [M2: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ M2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
            = ( modulo_modulo_int @ X @ M2 ) )
          | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X @ M2 ) @ M2 ) ) ) ) ) ).

% mod_double_modulus
thf(fact_2780_sum_Odistrib,axiom,
    ! [G: nat > nat,H3: nat > nat,A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : ( plus_plus_nat @ ( G @ X3 ) @ ( H3 @ X3 ) )
        @ A2 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H3 @ A2 ) ) ) ).

% sum.distrib
thf(fact_2781_sum_Odistrib,axiom,
    ! [G: int > int,H3: int > int,A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X3: int] : ( plus_plus_int @ ( G @ X3 ) @ ( H3 @ X3 ) )
        @ A2 )
      = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H3 @ A2 ) ) ) ).

% sum.distrib
thf(fact_2782_case__prodI,axiom,
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( F @ A @ B )
     => ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) ) ) ).

% case_prodI
thf(fact_2783_case__prodI,axiom,
    ! [F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( F @ A @ B )
     => ( produc1437786849005270451_nat_o @ F @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_2784_case__prodI,axiom,
    ! [F: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
      ( ( F @ A @ B )
     => ( produc838355143741117751_nat_o @ F @ ( produc2245416461498447860et_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_2785_case__prodI,axiom,
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ A @ B )
     => ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).

% case_prodI
thf(fact_2786_case__prodI,axiom,
    ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ A @ B )
     => ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).

% case_prodI
thf(fact_2787_case__prodI2,axiom,
    ! [P4: produc1908205239877642774nteger,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
          ( ( P4
            = ( produc8603105652947943368nteger @ A3 @ B2 ) )
         => ( C @ A3 @ B2 ) )
     => ( produc6253627499356882019eger_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_2788_case__prodI2,axiom,
    ! [P4: produc3925858234332021118et_nat,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( ( P4
            = ( produc5001842942810119800et_nat @ A3 @ B2 ) )
         => ( C @ A3 @ B2 ) )
     => ( produc1437786849005270451_nat_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_2789_case__prodI2,axiom,
    ! [P4: produc2732055786443039994et_nat,C: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3925858234332021118et_nat] :
          ( ( P4
            = ( produc2245416461498447860et_nat @ A3 @ B2 ) )
         => ( C @ A3 @ B2 ) )
     => ( produc838355143741117751_nat_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_2790_case__prodI2,axiom,
    ! [P4: produc2285326912895808259nt_int,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( P4
            = ( produc5700946648718959541nt_int @ A3 @ B2 ) )
         => ( C @ A3 @ B2 ) )
     => ( produc1573362020775583542_int_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_2791_case__prodI2,axiom,
    ! [P4: produc7773217078559923341nt_int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( P4
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) )
         => ( C @ A3 @ B2 ) )
     => ( produc2558449545302689196_int_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_2792_mem__case__prodI,axiom,
    ! [Z: $o,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( member_o @ Z @ ( C @ A @ B ) )
     => ( member_o @ Z @ ( produc8646739037753556108_set_o @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_2793_mem__case__prodI,axiom,
    ! [Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_2794_mem__case__prodI,axiom,
    ! [Z: int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_int,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( member_int @ Z @ ( C @ A @ B ) )
     => ( member_int @ Z @ ( produc4111701587417901774et_int @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_2795_mem__case__prodI,axiom,
    ! [Z: $o,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_o,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( member_o @ Z @ ( C @ A @ B ) )
     => ( member_o @ Z @ ( produc7715693535893868691_set_o @ C @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_2796_mem__case__prodI,axiom,
    ! [Z: nat,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_nat,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc8805491454895738411et_nat @ C @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_2797_mem__case__prodI,axiom,
    ! [Z: int,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_int,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( member_int @ Z @ ( C @ A @ B ) )
     => ( member_int @ Z @ ( produc4627640435386541703et_int @ C @ ( produc5001842942810119800et_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_2798_mem__case__prodI,axiom,
    ! [Z: product_prod_nat_nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_Pr1261947904930325089at_nat,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( member8440522571783428010at_nat @ Z @ ( C @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc5154663461364529269at_nat @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_2799_mem__case__prodI,axiom,
    ! [Z: $o,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( member_o @ Z @ ( C @ A @ B ) )
     => ( member_o @ Z @ ( produc2373554395599118147_set_o @ C @ ( produc8603105652947943368nteger @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_2800_mem__case__prodI,axiom,
    ! [Z: nat,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc654622801845291899et_nat @ C @ ( produc8603105652947943368nteger @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_2801_mem__case__prodI,axiom,
    ! [Z: int,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( member_int @ Z @ ( C @ A @ B ) )
     => ( member_int @ Z @ ( produc5700143819190870999et_int @ C @ ( produc8603105652947943368nteger @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_2802_mem__case__prodI2,axiom,
    ! [P4: produc7773217078559923341nt_int,Z: $o,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_o] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( P4
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) )
         => ( member_o @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member_o @ Z @ ( produc8646739037753556108_set_o @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_2803_mem__case__prodI2,axiom,
    ! [P4: produc7773217078559923341nt_int,Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( P4
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) )
         => ( member_nat @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_2804_mem__case__prodI2,axiom,
    ! [P4: produc7773217078559923341nt_int,Z: int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_int] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( P4
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) )
         => ( member_int @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member_int @ Z @ ( produc4111701587417901774et_int @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_2805_mem__case__prodI2,axiom,
    ! [P4: produc3925858234332021118et_nat,Z: $o,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_o] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( ( P4
            = ( produc5001842942810119800et_nat @ A3 @ B2 ) )
         => ( member_o @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member_o @ Z @ ( produc7715693535893868691_set_o @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_2806_mem__case__prodI2,axiom,
    ! [P4: produc3925858234332021118et_nat,Z: nat,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_nat] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( ( P4
            = ( produc5001842942810119800et_nat @ A3 @ B2 ) )
         => ( member_nat @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member_nat @ Z @ ( produc8805491454895738411et_nat @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_2807_mem__case__prodI2,axiom,
    ! [P4: produc3925858234332021118et_nat,Z: int,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_int] :
      ( ! [A3: produc3658429121746597890et_nat > $o,B2: produc3658429121746597890et_nat] :
          ( ( P4
            = ( produc5001842942810119800et_nat @ A3 @ B2 ) )
         => ( member_int @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member_int @ Z @ ( produc4627640435386541703et_int @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_2808_mem__case__prodI2,axiom,
    ! [P4: produc7773217078559923341nt_int,Z: product_prod_nat_nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_Pr1261947904930325089at_nat] :
      ( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
          ( ( P4
            = ( produc4305682042979456191nt_int @ A3 @ B2 ) )
         => ( member8440522571783428010at_nat @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member8440522571783428010at_nat @ Z @ ( produc5154663461364529269at_nat @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_2809_mem__case__prodI2,axiom,
    ! [P4: produc1908205239877642774nteger,Z: $o,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_o] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
          ( ( P4
            = ( produc8603105652947943368nteger @ A3 @ B2 ) )
         => ( member_o @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member_o @ Z @ ( produc2373554395599118147_set_o @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_2810_mem__case__prodI2,axiom,
    ! [P4: produc1908205239877642774nteger,Z: nat,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
          ( ( P4
            = ( produc8603105652947943368nteger @ A3 @ B2 ) )
         => ( member_nat @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member_nat @ Z @ ( produc654622801845291899et_nat @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_2811_mem__case__prodI2,axiom,
    ! [P4: produc1908205239877642774nteger,Z: int,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int] :
      ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
          ( ( P4
            = ( produc8603105652947943368nteger @ A3 @ B2 ) )
         => ( member_int @ Z @ ( C @ A3 @ B2 ) ) )
     => ( member_int @ Z @ ( produc5700143819190870999et_int @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_2812_case__prodI2_H,axiom,
    ! [P4: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
      ( ! [A3: nat,B2: nat] :
          ( ( ( product_Pair_nat_nat @ A3 @ B2 )
            = P4 )
         => ( C @ A3 @ B2 @ X ) )
     => ( produc8739625826339149834_nat_o @ C @ P4 @ X ) ) ).

% case_prodI2'
thf(fact_2813_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_2814_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_2815_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_Code_integer
        = ( numera6620942414471956472nteger @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_2816_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_2817_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_Code_natural
        = ( numera5444537566228673987atural @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_2818_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_nat @ N2 )
        = one_one_nat )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_2819_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_int @ N2 )
        = one_one_int )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_2820_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numera6620942414471956472nteger @ N2 )
        = one_one_Code_integer )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_2821_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_rat @ N2 )
        = one_one_rat )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_2822_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numera5444537566228673987atural @ N2 )
        = one_one_Code_natural )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_2823_of__nat__add,axiom,
    ! [M2: nat,N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).

% of_nat_add
thf(fact_2824_of__nat__add,axiom,
    ! [M2: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_add
thf(fact_2825_of__nat__add,axiom,
    ! [M2: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_add
thf(fact_2826_of__nat__add,axiom,
    ! [M2: nat,N2: nat] :
      ( ( semiri4939895301339042750nteger @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).

% of_nat_add
thf(fact_2827_of__nat__add,axiom,
    ! [M2: nat,N2: nat] :
      ( ( semiri3763490453095760265atural @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ M2 ) @ ( semiri3763490453095760265atural @ N2 ) ) ) ).

% of_nat_add
thf(fact_2828_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_2829_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_2830_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_2831_of__nat__1,axiom,
    ( ( semiri4939895301339042750nteger @ one_one_nat )
    = one_one_Code_integer ) ).

% of_nat_1
thf(fact_2832_of__nat__1,axiom,
    ( ( semiri3763490453095760265atural @ one_one_nat )
    = one_one_Code_natural ) ).

% of_nat_1
thf(fact_2833_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_rat
        = ( semiri681578069525770553at_rat @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_2834_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_int
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_2835_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_2836_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_Code_integer
        = ( semiri4939895301339042750nteger @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_2837_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_Code_natural
        = ( semiri3763490453095760265atural @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_2838_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri681578069525770553at_rat @ N2 )
        = one_one_rat )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_2839_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ N2 )
        = one_one_int )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_2840_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N2 )
        = one_one_nat )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_2841_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri4939895301339042750nteger @ N2 )
        = one_one_Code_integer )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_2842_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri3763490453095760265atural @ N2 )
        = one_one_Code_natural )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_2843_semiring__norm_I2_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% semiring_norm(2)
thf(fact_2844_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N2 ) @ one_one_Code_integer )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_2845_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_le1926595141338095240atural @ ( numera5444537566228673987atural @ N2 ) @ one_one_Code_natural )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_2846_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_2847_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_2848_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_2849_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_2850_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_2851_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_2852_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_2853_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_2854_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_2855_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_2856_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_2857_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_int @ one_one_int )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_2858_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_2859_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_rat @ one_one_rat )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_2860_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M2 ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_2861_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M2 ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_2862_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M2 ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_2863_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ M2 ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( semiri4939895301339042750nteger @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_2864_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri3763490453095760265atural @ ( suc @ M2 ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( semiri3763490453095760265atural @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_2865_Suc__numeral,axiom,
    ! [N2: num] :
      ( ( suc @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% Suc_numeral
thf(fact_2866_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) )
      = ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_2867_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) )
      = ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_2868_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) )
      = ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_2869_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2870_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2871_one__add__one,axiom,
    ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2872_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2873_one__add__one,axiom,
    ( ( plus_p4538020629002901425atural @ one_one_Code_natural @ one_one_Code_natural )
    = ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2874_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_2875_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_2876_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_2877_bits__one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% bits_one_mod_two_eq_one
thf(fact_2878_bits__one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% bits_one_mod_two_eq_one
thf(fact_2879_bits__one__mod__two__eq__one,axiom,
    ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% bits_one_mod_two_eq_one
thf(fact_2880_bits__one__mod__two__eq__one,axiom,
    ( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
    = one_one_Code_natural ) ).

% bits_one_mod_two_eq_one
thf(fact_2881_one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_mod_two_eq_one
thf(fact_2882_one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_mod_two_eq_one
thf(fact_2883_one__mod__two__eq__one,axiom,
    ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% one_mod_two_eq_one
thf(fact_2884_one__mod__two__eq__one,axiom,
    ( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
    = one_one_Code_natural ) ).

% one_mod_two_eq_one
thf(fact_2885_add__2__eq__Suc,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
      = ( suc @ ( suc @ N2 ) ) ) ).

% add_2_eq_Suc
thf(fact_2886_add__2__eq__Suc_H,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( suc @ N2 ) ) ) ).

% add_2_eq_Suc'
thf(fact_2887_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_2888_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2889_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
      = ( numeral_numeral_int @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2890_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N2 ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2891_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
      = ( numeral_numeral_rat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2892_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N2 ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2893_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_2894_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_2895_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N2 ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_2896_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_2897_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ N2 ) )
      = ( numera5444537566228673987atural @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_2898_add__self__div__2,axiom,
    ! [M2: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ M2 @ M2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = M2 ) ).

% add_self_div_2
thf(fact_2899_bits__1__div__2,axiom,
    ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% bits_1_div_2
thf(fact_2900_bits__1__div__2,axiom,
    ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% bits_1_div_2
thf(fact_2901_bits__1__div__2,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% bits_1_div_2
thf(fact_2902_bits__1__div__2,axiom,
    ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
    = zero_z2226904508553997617atural ) ).

% bits_1_div_2
thf(fact_2903_one__div__two__eq__zero,axiom,
    ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% one_div_two_eq_zero
thf(fact_2904_one__div__two__eq__zero,axiom,
    ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% one_div_two_eq_zero
thf(fact_2905_one__div__two__eq__zero,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% one_div_two_eq_zero
thf(fact_2906_one__div__two__eq__zero,axiom,
    ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
    = zero_z2226904508553997617atural ) ).

% one_div_two_eq_zero
thf(fact_2907_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_2908_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_2909_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_2910_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_2911_diff__numeral__special_I11_J,axiom,
    ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_2912_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_2913_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_2914_diff__numeral__special_I10_J,axiom,
    ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_2915_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_2916_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != zero_zero_int )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_2917_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != zero_zero_nat )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_2918_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_2919_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
       != zero_z2226904508553997617atural )
      = ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = one_one_Code_natural ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_2920_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != one_one_int )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_2921_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != one_one_nat )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_2922_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != one_one_Code_integer )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_2923_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
       != one_one_Code_natural )
      = ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = zero_z2226904508553997617atural ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_2924_minus__1__div__2__eq,axiom,
    ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% minus_1_div_2_eq
thf(fact_2925_minus__1__div__2__eq,axiom,
    ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% minus_1_div_2_eq
thf(fact_2926_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% bits_minus_1_mod_2_eq
thf(fact_2927_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% bits_minus_1_mod_2_eq
thf(fact_2928_minus__1__mod__2__eq,axiom,
    ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% minus_1_mod_2_eq
thf(fact_2929_minus__1__mod__2__eq,axiom,
    ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% minus_1_mod_2_eq
thf(fact_2930_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_2931_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_2932_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_2933_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_2934_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_2935_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_2936_add__self__mod__2,axiom,
    ! [M2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ M2 @ M2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% add_self_mod_2
thf(fact_2937_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_int ) ).

% power_minus1_even
thf(fact_2938_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_Code_integer ) ).

% power_minus1_even
thf(fact_2939_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_rat ) ).

% power_minus1_even
thf(fact_2940_mod2__gr__0,axiom,
    ! [M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% mod2_gr_0
thf(fact_2941_lex__prod__def,axiom,
    ( lex_pr8000699672184251119at_nat
    = ( ^ [Ra: set_Pr4329608150637261639at_nat,Rb: set_Pr4329608150637261639at_nat] :
          ( collec223960383693904210at_nat
          @ ( produc1561160019221259564_nat_o
            @ ( produc1971064617671185730_nat_o
              @ ^ [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
                  ( produc410239310623530412_nat_o
                  @ ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
                      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A5 @ A6 ) @ Ra )
                      | ( ( A5 = A6 )
                        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_2942_lex__prod__def,axiom,
    ( lex_pr1951919194703979714at_nat
    = ( ^ [Ra: set_Pr4329608150637261639at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec1461227866750739596at_nat
          @ ( produc8701595406162815018_nat_o
            @ ( produc7110675705651593384_nat_o
              @ ^ [A5: set_Pr1261947904930325089at_nat,B4: nat] :
                  ( produc3595121349294487249_nat_o
                  @ ^ [A6: set_Pr1261947904930325089at_nat,B6: nat] :
                      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A5 @ A6 ) @ Ra )
                      | ( ( A5 = A6 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_2943_lex__prod__def,axiom,
    ( lex_pr4265550801952275104at_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr4329608150637261639at_nat] :
          ( collec2088441199205367000at_nat
          @ ( produc7830147906768200158_nat_o
            @ ( produc1908958340173245860_nat_o
              @ ^ [A5: nat,B4: set_Pr1261947904930325089at_nat] :
                  ( produc7997411485876045171_nat_o
                  @ ^ [A6: nat,B6: set_Pr1261947904930325089at_nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ A6 ) @ Ra )
                      | ( ( A5 = A6 )
                        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_2944_lex__prod__def,axiom,
    ( lex_prod_nat_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [A5: nat,B4: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [A6: nat,B6: nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ A6 ) @ Ra )
                      | ( ( A5 = A6 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ B6 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_2945_same__fst__def,axiom,
    ( same_fst_nat_nat
    = ( ^ [P3: nat > $o,R5: nat > set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [X10: nat,Y8: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [X3: nat,Y3: nat] :
                      ( ( X10 = X3 )
                      & ( P3 @ X3 )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y8 @ Y3 ) @ ( R5 @ X3 ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_2946_mem__case__prodE,axiom,
    ! [Z: $o,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_o,P4: produc7773217078559923341nt_int] :
      ( ( member_o @ Z @ ( produc8646739037753556108_set_o @ C @ P4 ) )
     => ~ ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
            ( ( P4
              = ( produc4305682042979456191nt_int @ X4 @ Y5 ) )
           => ~ ( member_o @ Z @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_2947_mem__case__prodE,axiom,
    ! [Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,P4: produc7773217078559923341nt_int] :
      ( ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ P4 ) )
     => ~ ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
            ( ( P4
              = ( produc4305682042979456191nt_int @ X4 @ Y5 ) )
           => ~ ( member_nat @ Z @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_2948_mem__case__prodE,axiom,
    ! [Z: int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_int,P4: produc7773217078559923341nt_int] :
      ( ( member_int @ Z @ ( produc4111701587417901774et_int @ C @ P4 ) )
     => ~ ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
            ( ( P4
              = ( produc4305682042979456191nt_int @ X4 @ Y5 ) )
           => ~ ( member_int @ Z @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_2949_mem__case__prodE,axiom,
    ! [Z: $o,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_o,P4: produc3925858234332021118et_nat] :
      ( ( member_o @ Z @ ( produc7715693535893868691_set_o @ C @ P4 ) )
     => ~ ! [X4: produc3658429121746597890et_nat > $o,Y5: produc3658429121746597890et_nat] :
            ( ( P4
              = ( produc5001842942810119800et_nat @ X4 @ Y5 ) )
           => ~ ( member_o @ Z @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_2950_mem__case__prodE,axiom,
    ! [Z: nat,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_nat,P4: produc3925858234332021118et_nat] :
      ( ( member_nat @ Z @ ( produc8805491454895738411et_nat @ C @ P4 ) )
     => ~ ! [X4: produc3658429121746597890et_nat > $o,Y5: produc3658429121746597890et_nat] :
            ( ( P4
              = ( produc5001842942810119800et_nat @ X4 @ Y5 ) )
           => ~ ( member_nat @ Z @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_2951_mem__case__prodE,axiom,
    ! [Z: int,C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > set_int,P4: produc3925858234332021118et_nat] :
      ( ( member_int @ Z @ ( produc4627640435386541703et_int @ C @ P4 ) )
     => ~ ! [X4: produc3658429121746597890et_nat > $o,Y5: produc3658429121746597890et_nat] :
            ( ( P4
              = ( produc5001842942810119800et_nat @ X4 @ Y5 ) )
           => ~ ( member_int @ Z @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_2952_mem__case__prodE,axiom,
    ! [Z: product_prod_nat_nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_Pr1261947904930325089at_nat,P4: produc7773217078559923341nt_int] :
      ( ( member8440522571783428010at_nat @ Z @ ( produc5154663461364529269at_nat @ C @ P4 ) )
     => ~ ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
            ( ( P4
              = ( produc4305682042979456191nt_int @ X4 @ Y5 ) )
           => ~ ( member8440522571783428010at_nat @ Z @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_2953_mem__case__prodE,axiom,
    ! [Z: $o,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_o,P4: produc1908205239877642774nteger] :
      ( ( member_o @ Z @ ( produc2373554395599118147_set_o @ C @ P4 ) )
     => ~ ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
            ( ( P4
              = ( produc8603105652947943368nteger @ X4 @ Y5 ) )
           => ~ ( member_o @ Z @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_2954_mem__case__prodE,axiom,
    ! [Z: nat,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,P4: produc1908205239877642774nteger] :
      ( ( member_nat @ Z @ ( produc654622801845291899et_nat @ C @ P4 ) )
     => ~ ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
            ( ( P4
              = ( produc8603105652947943368nteger @ X4 @ Y5 ) )
           => ~ ( member_nat @ Z @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_2955_mem__case__prodE,axiom,
    ! [Z: int,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,P4: produc1908205239877642774nteger] :
      ( ( member_int @ Z @ ( produc5700143819190870999et_int @ C @ P4 ) )
     => ~ ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
            ( ( P4
              = ( produc8603105652947943368nteger @ X4 @ Y5 ) )
           => ~ ( member_int @ Z @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_2956_case__prodD,axiom,
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_2957_case__prodD,axiom,
    ! [F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat] :
      ( ( produc1437786849005270451_nat_o @ F @ ( produc5001842942810119800et_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_2958_case__prodD,axiom,
    ! [F: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat] :
      ( ( produc838355143741117751_nat_o @ F @ ( produc2245416461498447860et_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_2959_case__prodD,axiom,
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_2960_case__prodD,axiom,
    ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_2961_case__prodE,axiom,
    ! [C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc1908205239877642774nteger] :
      ( ( produc6253627499356882019eger_o @ C @ P4 )
     => ~ ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
            ( ( P4
              = ( produc8603105652947943368nteger @ X4 @ Y5 ) )
           => ~ ( C @ X4 @ Y5 ) ) ) ).

% case_prodE
thf(fact_2962_case__prodE,axiom,
    ! [C: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,P4: produc3925858234332021118et_nat] :
      ( ( produc1437786849005270451_nat_o @ C @ P4 )
     => ~ ! [X4: produc3658429121746597890et_nat > $o,Y5: produc3658429121746597890et_nat] :
            ( ( P4
              = ( produc5001842942810119800et_nat @ X4 @ Y5 ) )
           => ~ ( C @ X4 @ Y5 ) ) ) ).

% case_prodE
thf(fact_2963_case__prodE,axiom,
    ! [C: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o,P4: produc2732055786443039994et_nat] :
      ( ( produc838355143741117751_nat_o @ C @ P4 )
     => ~ ! [X4: produc3658429121746597890et_nat > $o,Y5: produc3925858234332021118et_nat] :
            ( ( P4
              = ( produc2245416461498447860et_nat @ X4 @ Y5 ) )
           => ~ ( C @ X4 @ Y5 ) ) ) ).

% case_prodE
thf(fact_2964_case__prodE,axiom,
    ! [C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc2285326912895808259nt_int] :
      ( ( produc1573362020775583542_int_o @ C @ P4 )
     => ~ ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] :
            ( ( P4
              = ( produc5700946648718959541nt_int @ X4 @ Y5 ) )
           => ~ ( C @ X4 @ Y5 ) ) ) ).

% case_prodE
thf(fact_2965_case__prodE,axiom,
    ! [C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc7773217078559923341nt_int] :
      ( ( produc2558449545302689196_int_o @ C @ P4 )
     => ~ ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
            ( ( P4
              = ( produc4305682042979456191nt_int @ X4 @ Y5 ) )
           => ~ ( C @ X4 @ Y5 ) ) ) ).

% case_prodE
thf(fact_2966_add__One__commute,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ N2 )
      = ( plus_plus_num @ N2 @ one ) ) ).

% add_One_commute
thf(fact_2967_case__prodD_H,axiom,
    ! [R4: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ R4 @ ( product_Pair_nat_nat @ A @ B ) @ C )
     => ( R4 @ A @ B @ C ) ) ).

% case_prodD'
thf(fact_2968_case__prodE_H,axiom,
    ! [C: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ C @ P4 @ Z )
     => ~ ! [X4: nat,Y5: nat] :
            ( ( P4
              = ( product_Pair_nat_nat @ X4 @ Y5 ) )
           => ~ ( C @ X4 @ Y5 @ Z ) ) ) ).

% case_prodE'
thf(fact_2969_sum_OatLeast__int__atMost__int__shift,axiom,
    ! [G: int > nat,M2: nat,N2: nat] :
      ( ( groups4541462559716669496nt_nat @ G @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( groups3542108847815614940at_nat @ ( comp_int_nat_nat @ G @ semiri1314217659103216013at_int ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% sum.atLeast_int_atMost_int_shift
thf(fact_2970_sum_OatLeast__int__atMost__int__shift,axiom,
    ! [G: int > int,M2: nat,N2: nat] :
      ( ( groups4538972089207619220nt_int @ G @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( groups3539618377306564664at_int @ ( comp_int_int_nat @ G @ semiri1314217659103216013at_int ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% sum.atLeast_int_atMost_int_shift
thf(fact_2971_less__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( ord_less_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X3: nat,Y3: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) )
        @ Xa
        @ X ) ) ).

% less_int.abs_eq
thf(fact_2972_less__eq__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( ord_less_eq_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X3: nat,Y3: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) )
        @ Xa
        @ X ) ) ).

% less_eq_int.abs_eq
thf(fact_2973_zadd__int__left,axiom,
    ! [M2: nat,N2: nat,Z: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N2 ) ) @ Z ) ) ).

% zadd_int_left
thf(fact_2974_int__plus,axiom,
    ! [N2: nat,M2: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).

% int_plus
thf(fact_2975_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(5)
thf(fact_2976_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_2977_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W2: int,Z4: int] :
        ? [N4: nat] :
          ( Z4
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_2978_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_2979_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_2980_numeral__One,axiom,
    ( ( numera6620942414471956472nteger @ one )
    = one_one_Code_integer ) ).

% numeral_One
thf(fact_2981_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_2982_numeral__One,axiom,
    ( ( numera5444537566228673987atural @ one )
    = one_one_Code_natural ) ).

% numeral_One
thf(fact_2983_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_2984_execute__bind__case,axiom,
    ! [F: heap_T6362127782908069795rray_a,G: array_a > heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit] :
      ( ( heap_T5591663578999710211rray_a @ ( heap_T2407400398799471159rray_a @ F @ G ) @ H3 )
      = ( case_o8555119740594961093it_nat @ none_P2587165478634041381it_nat
        @ ( produc8915672860170107321it_nat
          @ ^ [X3: array_a] :
              ( produc921929042997482327it_nat
              @ ^ [H6: heap_e7401611519738050253t_unit,N4: nat] : ( heap_T531568236820715630rray_a @ N4 @ ( heap_T5591663578999710211rray_a @ ( G @ X3 ) @ H6 ) ) ) )
        @ ( heap_T5591663578999710211rray_a @ F @ H3 ) ) ) ).

% execute_bind_case
thf(fact_2985_execute__bind__case,axiom,
    ! [F: heap_T5738788834812785303t_unit,G: product_unit > heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit] :
      ( ( heap_T5591663578999710211rray_a @ ( heap_T7158140910958840029rray_a @ F @ G ) @ H3 )
      = ( case_o5699750240967658439it_nat @ none_P2587165478634041381it_nat
        @ ( produc875294279294635931it_nat
          @ ^ [X3: product_unit] :
              ( produc921929042997482327it_nat
              @ ^ [H6: heap_e7401611519738050253t_unit,N4: nat] : ( heap_T531568236820715630rray_a @ N4 @ ( heap_T5591663578999710211rray_a @ ( G @ X3 ) @ H6 ) ) ) )
        @ ( heap_T875086893843062177t_unit @ F @ H3 ) ) ) ).

% execute_bind_case
thf(fact_2986_execute__bind__case,axiom,
    ! [F: heap_T6362127782908069795rray_a,G: array_a > heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( heap_T875086893843062177t_unit @ ( heap_T7324619625323989613t_unit @ F @ G ) @ H3 )
      = ( case_o3886096569520666655it_nat @ none_P9117596204409417319it_nat
        @ ( produc3495058329975564393it_nat
          @ ^ [X3: array_a] :
              ( produc7488178964372371019it_nat
              @ ^ [H6: heap_e7401611519738050253t_unit,N4: nat] : ( heap_T3616969660504097270t_unit @ N4 @ ( heap_T875086893843062177t_unit @ ( G @ X3 ) @ H6 ) ) ) )
        @ ( heap_T5591663578999710211rray_a @ F @ H3 ) ) ) ).

% execute_bind_case
thf(fact_2987_execute__bind__case,axiom,
    ! [F: heap_T5738788834812785303t_unit,G: product_unit > heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( heap_T875086893843062177t_unit @ ( heap_T2633723481742716231t_unit @ F @ G ) @ H3 )
      = ( case_o2963978774867076333it_nat @ none_P9117596204409417319it_nat
        @ ( produc1230459020282432263it_nat
          @ ^ [X3: product_unit] :
              ( produc7488178964372371019it_nat
              @ ^ [H6: heap_e7401611519738050253t_unit,N4: nat] : ( heap_T3616969660504097270t_unit @ N4 @ ( heap_T875086893843062177t_unit @ ( G @ X3 ) @ H6 ) ) ) )
        @ ( heap_T875086893843062177t_unit @ F @ H3 ) ) ) ).

% execute_bind_case
thf(fact_2988_Sum__Icc__int,axiom,
    ! [M2: int,N2: int] :
      ( ( ord_less_eq_int @ M2 @ N2 )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X3: int] : X3
          @ ( set_or1266510415728281911st_int @ M2 @ N2 ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N2 @ ( plus_plus_int @ N2 @ one_one_int ) ) @ ( times_times_int @ M2 @ ( minus_minus_int @ M2 @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_2989_Heap__Time__Monad_Obind__def,axiom,
    ( heap_T2407400398799471159rray_a
    = ( ^ [F2: heap_T6362127782908069795rray_a,G2: array_a > heap_T6362127782908069795rray_a] :
          ( heap_T1488828001306802186rray_a
          @ ^ [H: heap_e7401611519738050253t_unit] :
              ( case_o8555119740594961093it_nat @ none_P2587165478634041381it_nat
              @ ( produc8915672860170107321it_nat
                @ ^ [R3: array_a] :
                    ( produc921929042997482327it_nat
                    @ ^ [H6: heap_e7401611519738050253t_unit,N4: nat] : ( heap_T531568236820715630rray_a @ N4 @ ( heap_T5591663578999710211rray_a @ ( G2 @ R3 ) @ H6 ) ) ) )
              @ ( heap_T5591663578999710211rray_a @ F2 @ H ) ) ) ) ) ).

% Heap_Time_Monad.bind_def
thf(fact_2990_Heap__Time__Monad_Obind__def,axiom,
    ( heap_T7158140910958840029rray_a
    = ( ^ [F2: heap_T5738788834812785303t_unit,G2: product_unit > heap_T6362127782908069795rray_a] :
          ( heap_T1488828001306802186rray_a
          @ ^ [H: heap_e7401611519738050253t_unit] :
              ( case_o5699750240967658439it_nat @ none_P2587165478634041381it_nat
              @ ( produc875294279294635931it_nat
                @ ^ [R3: product_unit] :
                    ( produc921929042997482327it_nat
                    @ ^ [H6: heap_e7401611519738050253t_unit,N4: nat] : ( heap_T531568236820715630rray_a @ N4 @ ( heap_T5591663578999710211rray_a @ ( G2 @ R3 ) @ H6 ) ) ) )
              @ ( heap_T875086893843062177t_unit @ F2 @ H ) ) ) ) ) ).

% Heap_Time_Monad.bind_def
thf(fact_2991_Heap__Time__Monad_Obind__def,axiom,
    ( heap_T7324619625323989613t_unit
    = ( ^ [F2: heap_T6362127782908069795rray_a,G2: array_a > heap_T5738788834812785303t_unit] :
          ( heap_T6183433275982383450t_unit
          @ ^ [H: heap_e7401611519738050253t_unit] :
              ( case_o3886096569520666655it_nat @ none_P9117596204409417319it_nat
              @ ( produc3495058329975564393it_nat
                @ ^ [R3: array_a] :
                    ( produc7488178964372371019it_nat
                    @ ^ [H6: heap_e7401611519738050253t_unit,N4: nat] : ( heap_T3616969660504097270t_unit @ N4 @ ( heap_T875086893843062177t_unit @ ( G2 @ R3 ) @ H6 ) ) ) )
              @ ( heap_T5591663578999710211rray_a @ F2 @ H ) ) ) ) ) ).

% Heap_Time_Monad.bind_def
thf(fact_2992_Heap__Time__Monad_Obind__def,axiom,
    ( heap_T2633723481742716231t_unit
    = ( ^ [F2: heap_T5738788834812785303t_unit,G2: product_unit > heap_T5738788834812785303t_unit] :
          ( heap_T6183433275982383450t_unit
          @ ^ [H: heap_e7401611519738050253t_unit] :
              ( case_o2963978774867076333it_nat @ none_P9117596204409417319it_nat
              @ ( produc1230459020282432263it_nat
                @ ^ [R3: product_unit] :
                    ( produc7488178964372371019it_nat
                    @ ^ [H6: heap_e7401611519738050253t_unit,N4: nat] : ( heap_T3616969660504097270t_unit @ N4 @ ( heap_T875086893843062177t_unit @ ( G2 @ R3 ) @ H6 ) ) ) )
              @ ( heap_T875086893843062177t_unit @ F2 @ H ) ) ) ) ) ).

% Heap_Time_Monad.bind_def
thf(fact_2993_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W2: int,Z4: int] :
        ? [N4: nat] :
          ( Z4
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_2994_int__ops_I4_J,axiom,
    ! [A: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).

% int_ops(4)
thf(fact_2995_int__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).

% int_Suc
thf(fact_2996_int__bit__induct,axiom,
    ! [P: int > $o,K: int] :
      ( ( P @ zero_zero_int )
     => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
       => ( ! [K5: int] :
              ( ( P @ K5 )
             => ( ( K5 != zero_zero_int )
               => ( P @ ( times_times_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
         => ( ! [K5: int] :
                ( ( P @ K5 )
               => ( ( K5
                   != ( uminus_uminus_int @ one_one_int ) )
                 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
           => ( P @ K ) ) ) ) ) ).

% int_bit_induct
thf(fact_2997_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_2998_uminus__numeral__One,axiom,
    ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% uminus_numeral_One
thf(fact_2999_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_3000_int__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N4: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N4 @ zero_zero_nat ) ) ) ) ).

% int_def
thf(fact_3001_num__of__nat__One,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ one_one_nat )
     => ( ( num_of_nat @ N2 )
        = one ) ) ).

% num_of_nat_One
thf(fact_3002_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).

% neg_zdiv_mult_2
thf(fact_3003_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ B @ A ) ) ) ).

% pos_zdiv_mult_2
thf(fact_3004_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).

% pos_zmod_mult_2
thf(fact_3005_mod__mult2__eq_H,axiom,
    ! [A: int,M2: nat,N2: nat] :
      ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).

% mod_mult2_eq'
thf(fact_3006_mod__mult2__eq_H,axiom,
    ! [A: nat,M2: nat,N2: nat] :
      ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ) ).

% mod_mult2_eq'
thf(fact_3007_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M2: nat,N2: nat] :
      ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( semiri4939895301339042750nteger @ N2 ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M2 ) ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M2 ) ) ) ) ).

% mod_mult2_eq'
thf(fact_3008_mod__mult2__eq_H,axiom,
    ! [A: code_natural,M2: nat,N2: nat] :
      ( ( modulo8411746178871703098atural @ A @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M2 ) @ ( semiri3763490453095760265atural @ N2 ) ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M2 ) @ ( modulo8411746178871703098atural @ ( divide5121882707175180666atural @ A @ ( semiri3763490453095760265atural @ M2 ) ) @ ( semiri3763490453095760265atural @ N2 ) ) ) @ ( modulo8411746178871703098atural @ A @ ( semiri3763490453095760265atural @ M2 ) ) ) ) ).

% mod_mult2_eq'
thf(fact_3009_numeral__num__of__nat__unfold,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( numeral_numeral_rat @ ( num_of_nat @ N2 ) )
          = one_one_rat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( numeral_numeral_rat @ ( num_of_nat @ N2 ) )
          = ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).

% numeral_num_of_nat_unfold
thf(fact_3010_numeral__num__of__nat__unfold,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( numeral_numeral_int @ ( num_of_nat @ N2 ) )
          = one_one_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( numeral_numeral_int @ ( num_of_nat @ N2 ) )
          = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% numeral_num_of_nat_unfold
thf(fact_3011_numeral__num__of__nat__unfold,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( numeral_numeral_nat @ ( num_of_nat @ N2 ) )
          = one_one_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( numeral_numeral_nat @ ( num_of_nat @ N2 ) )
          = ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).

% numeral_num_of_nat_unfold
thf(fact_3012_numeral__num__of__nat__unfold,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( numera6620942414471956472nteger @ ( num_of_nat @ N2 ) )
          = one_one_Code_integer ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( numera6620942414471956472nteger @ ( num_of_nat @ N2 ) )
          = ( semiri4939895301339042750nteger @ N2 ) ) ) ) ).

% numeral_num_of_nat_unfold
thf(fact_3013_numeral__num__of__nat__unfold,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( numera5444537566228673987atural @ ( num_of_nat @ N2 ) )
          = one_one_Code_natural ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( numera5444537566228673987atural @ ( num_of_nat @ N2 ) )
          = ( semiri3763490453095760265atural @ N2 ) ) ) ) ).

% numeral_num_of_nat_unfold
thf(fact_3014_double__arith__series,axiom,
    ! [A: rat,D2: rat,N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I3 ) @ D2 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ D2 ) ) ) ) ).

% double_arith_series
thf(fact_3015_double__arith__series,axiom,
    ! [A: int,D2: int,N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D2 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D2 ) ) ) ) ).

% double_arith_series
thf(fact_3016_double__arith__series,axiom,
    ! [A: code_integer,D2: code_integer,N2: nat] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
        @ ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I3 ) @ D2 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ D2 ) ) ) ) ).

% double_arith_series
thf(fact_3017_double__arith__series,axiom,
    ! [A: code_natural,D2: code_natural,N2: nat] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) )
        @ ( groups6325495683096345652atural
          @ ^ [I3: nat] : ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ I3 ) @ D2 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ N2 ) @ one_one_Code_natural ) @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N2 ) @ D2 ) ) ) ) ).

% double_arith_series
thf(fact_3018_double__arith__series,axiom,
    ! [A: nat,D2: nat,N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D2 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D2 ) ) ) ) ).

% double_arith_series
thf(fact_3019_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).

% double_gauss_sum
thf(fact_3020_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).

% double_gauss_sum
thf(fact_3021_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) ) ) ).

% double_gauss_sum
thf(fact_3022_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( groups6325495683096345652atural @ semiri3763490453095760265atural @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N2 ) @ ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ N2 ) @ one_one_Code_natural ) ) ) ).

% double_gauss_sum
thf(fact_3023_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).

% double_gauss_sum
thf(fact_3024_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).

% neg_zmod_mult_2
thf(fact_3025_mult__2,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2
thf(fact_3026_mult__2,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2
thf(fact_3027_mult__2,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Z )
      = ( plus_p5714425477246183910nteger @ Z @ Z ) ) ).

% mult_2
thf(fact_3028_mult__2,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2
thf(fact_3029_mult__2,axiom,
    ! [Z: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ Z )
      = ( plus_p4538020629002901425atural @ Z @ Z ) ) ).

% mult_2
thf(fact_3030_mult__2__right,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_3031_mult__2__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2_right
thf(fact_3032_mult__2__right,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ Z @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( plus_p5714425477246183910nteger @ Z @ Z ) ) ).

% mult_2_right
thf(fact_3033_mult__2__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_3034_mult__2__right,axiom,
    ! [Z: code_natural] :
      ( ( times_2397367101498566445atural @ Z @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
      = ( plus_p4538020629002901425atural @ Z @ Z ) ) ).

% mult_2_right
thf(fact_3035_left__add__twice,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_3036_left__add__twice,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_3037_left__add__twice,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_3038_left__add__twice,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_3039_left__add__twice,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( plus_p4538020629002901425atural @ A @ ( plus_p4538020629002901425atural @ A @ B ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_3040_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q3 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_3041_one__power2,axiom,
    ( ( power_8256067586552552935nteger @ one_one_Code_integer @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% one_power2
thf(fact_3042_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_3043_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_3044_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_3045_nat__1__add__1,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% nat_1_add_1
thf(fact_3046_Suc__nat__number__of__add,axiom,
    ! [V: num,N2: nat] :
      ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N2 ) ) ).

% Suc_nat_number_of_add
thf(fact_3047_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_3048_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_3049_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_3050_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( groups6325495683096345652atural @ semiri3763490453095760265atural @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N2 ) @ ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ N2 ) @ one_one_Code_natural ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_3051_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_3052_arith__series,axiom,
    ! [A: int,D2: int,N2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D2 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D2 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_3053_arith__series,axiom,
    ! [A: code_integer,D2: code_integer,N2: nat] :
      ( ( groups7501900531339628137nteger
        @ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I3 ) @ D2 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ D2 ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_3054_arith__series,axiom,
    ! [A: code_natural,D2: code_natural,N2: nat] :
      ( ( groups6325495683096345652atural
        @ ^ [I3: nat] : ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ I3 ) @ D2 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ N2 ) @ one_one_Code_natural ) @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N2 ) @ D2 ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_3055_arith__series,axiom,
    ! [A: nat,D2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D2 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_3056_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_3057_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_3058_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups6325495683096345652atural @ semiri3763490453095760265atural @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N2 ) @ ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ N2 ) @ one_one_Code_natural ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_3059_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_3060_nat__approx__posE,axiom,
    ! [E: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E )
     => ~ ! [N: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_3061_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R2: int] :
      ( ( ord_less_eq_int @ B @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q3 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_3062_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_3063_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_3064_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups6325495683096345652atural @ semiri3763490453095760265atural @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N2 ) @ ( plus_p4538020629002901425atural @ ( semiri3763490453095760265atural @ N2 ) @ one_one_Code_natural ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_3065_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_3066_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
thf(fact_3067_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
thf(fact_3068_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
thf(fact_3069_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) )
       != zero_z2226904508553997617atural )
     => ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 )
       != zero_z2226904508553997617atural ) ) ).

% exp_add_not_zero_imp_right
thf(fact_3070_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
thf(fact_3071_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
thf(fact_3072_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
thf(fact_3073_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) )
       != zero_z2226904508553997617atural )
     => ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 )
       != zero_z2226904508553997617atural ) ) ).

% exp_add_not_zero_imp_left
thf(fact_3074_sum__power2__eq__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_3075_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_3076_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_3077_div__exp__eq,axiom,
    ! [A: int,M2: nat,N2: nat] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_3078_div__exp__eq,axiom,
    ! [A: nat,M2: nat,N2: nat] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_3079_div__exp__eq,axiom,
    ! [A: code_integer,M2: nat,N2: nat] :
      ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_3080_div__exp__eq,axiom,
    ! [A: code_natural,M2: nat,N2: nat] :
      ( ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide5121882707175180666atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_3081_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( A = one_one_int )
        | ( A
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% power2_eq_1_iff
thf(fact_3082_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( A = one_one_Code_integer )
        | ( A
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% power2_eq_1_iff
thf(fact_3083_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( A = one_one_rat )
        | ( A
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% power2_eq_1_iff
thf(fact_3084_plus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( plus_plus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X3: nat,Y3: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y3 @ V2 ) ) )
          @ Xa
          @ X ) ) ) ).

% plus_int.abs_eq
thf(fact_3085_abs__square__eq__1,axiom,
    ! [X: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( abs_abs_Code_integer @ X )
        = one_one_Code_integer ) ) ).

% abs_square_eq_1
thf(fact_3086_abs__square__eq__1,axiom,
    ! [X: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( abs_abs_rat @ X )
        = one_one_rat ) ) ).

% abs_square_eq_1
thf(fact_3087_abs__square__eq__1,axiom,
    ! [X: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( abs_abs_int @ X )
        = one_one_int ) ) ).

% abs_square_eq_1
thf(fact_3088_minus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( minus_minus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X3: nat,Y3: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ Y3 @ U2 ) ) )
          @ Xa
          @ X ) ) ) ).

% minus_int.abs_eq
thf(fact_3089_nat__induct2,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ( P @ one_one_nat )
       => ( ! [N: nat] :
              ( ( P @ N )
             => ( P @ ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct2
thf(fact_3090_times__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X3: nat,Y3: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y3 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V2 ) @ ( times_times_nat @ Y3 @ U2 ) ) ) )
          @ Xa
          @ X ) ) ) ).

% times_int.abs_eq
thf(fact_3091_sum__power2__le__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_z3403309356797280102nteger )
      = ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3092_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3093_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3094_sum__power2__ge__zero,axiom,
    ! [X: code_integer,Y: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3095_sum__power2__ge__zero,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3096_sum__power2__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3097_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = zero_zero_int ) ) ).

% bits_stable_imp_add_self
thf(fact_3098_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
thf(fact_3099_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
thf(fact_3100_bits__stable__imp__add__self,axiom,
    ! [A: code_natural] :
      ( ( ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_p4538020629002901425atural @ A @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
        = zero_z2226904508553997617atural ) ) ).

% bits_stable_imp_add_self
thf(fact_3101_not__sum__power2__lt__zero,axiom,
    ! [X: code_integer,Y: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_z3403309356797280102nteger ) ).

% not_sum_power2_lt_zero
thf(fact_3102_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).

% not_sum_power2_lt_zero
thf(fact_3103_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).

% not_sum_power2_lt_zero
thf(fact_3104_sum__power2__gt__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_3105_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_3106_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_3107_power2__sum,axiom,
    ! [X: nat,Y: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_3108_power2__sum,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_3109_power2__sum,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( power_8256067586552552935nteger @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_3110_power2__sum,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_3111_power2__sum,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( power_7079662738309270450atural @ ( plus_p4538020629002901425atural @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( power_7079662738309270450atural @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_7079662738309270450atural @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_2397367101498566445atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_3112_square__le__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
     => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% square_le_1
thf(fact_3113_square__le__1,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
     => ( ( ord_less_eq_rat @ X @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).

% square_le_1
thf(fact_3114_square__le__1,axiom,
    ! [X: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
     => ( ( ord_less_eq_int @ X @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% square_le_1
thf(fact_3115_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).

% abs_square_le_1
thf(fact_3116_abs__square__le__1,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).

% abs_square_le_1
thf(fact_3117_abs__square__le__1,axiom,
    ! [X: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).

% abs_square_le_1
thf(fact_3118_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).

% abs_square_less_1
thf(fact_3119_abs__square__less__1,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).

% abs_square_less_1
thf(fact_3120_abs__square__less__1,axiom,
    ! [X: int] :
      ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).

% abs_square_less_1
thf(fact_3121_div__exp__mod__exp__eq,axiom,
    ! [A: int,N2: nat,M2: nat] :
      ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
      = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_3122_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N2: nat,M2: nat] :
      ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
      = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M2 ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_3123_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N2: nat,M2: nat] :
      ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) )
      = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M2 ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_3124_div__exp__mod__exp__eq,axiom,
    ! [A: code_natural,N2: nat,M2: nat] :
      ( ( modulo8411746178871703098atural @ ( divide5121882707175180666atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) )
      = ( divide5121882707175180666atural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M2 ) ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_3125_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
       => ? [N: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_3126_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ one_one_nat @ K )
       => ? [N: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_3127_power2__diff,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_3128_power2__diff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( power_8256067586552552935nteger @ ( minus_8373710615458151222nteger @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_3129_power2__diff,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_3130_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% power_minus1_odd
thf(fact_3131_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% power_minus1_odd
thf(fact_3132_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus_uminus_rat @ one_one_rat ) ) ).

% power_minus1_odd
thf(fact_3133_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int )
      = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_3134_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer )
      = ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_3135_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_natural )
      = ( groups6325495683096345652atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_3136_mask__eq__sum__exp,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_3137_sum_Oin__pairs,axiom,
    ! [G: nat > multis2468970476368604999at_nat,M2: nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6857163185585827899at_nat
        @ ^ [I3: nat] : ( plus_p7104986032573967614at_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_3138_sum_Oin__pairs,axiom,
    ! [G: nat > rat,M2: nat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_3139_sum_Oin__pairs,axiom,
    ! [G: nat > int,M2: nat,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_3140_sum_Oin__pairs,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_3141_gauss__sum__nat,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_nat
thf(fact_3142_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q3: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_3143_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q3: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_3144_divmod__step__eq,axiom,
    ! [L: num,R2: code_integer,Q3: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_3145_divmod__step__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L3: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q5: nat,R3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L3 ) @ R3 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R3 @ ( numeral_numeral_nat @ L3 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R3 ) ) ) ) ) ).

% divmod_step_def
thf(fact_3146_divmod__step__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L3: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q5: int,R3: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L3 ) @ R3 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R3 @ ( numeral_numeral_int @ L3 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R3 ) ) ) ) ) ).

% divmod_step_def
thf(fact_3147_divmod__step__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L3: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q5: code_integer,R3: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L3 ) @ R3 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R3 @ ( numera6620942414471956472nteger @ L3 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R3 ) ) ) ) ) ).

% divmod_step_def
thf(fact_3148_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L3: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q5: nat,R3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L3 ) @ R3 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R3 @ ( numeral_numeral_nat @ L3 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R3 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_3149_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L3: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q5: int,R3: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L3 ) @ R3 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R3 @ ( numeral_numeral_int @ L3 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R3 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_3150_set__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se2793503036327961859nteger @ zero_zero_nat @ A )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_3151_set__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_3152_set__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_3153_set__bit__0,axiom,
    ! [A: code_natural] :
      ( ( bit_se1617098188084679374atural @ zero_zero_nat @ A )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_3154_mlex__eq,axiom,
    ( mlex_p3502713205431853936at_nat
    = ( ^ [F2: set_Pr1261947904930325089at_nat > nat,R5: set_Pr4329608150637261639at_nat] :
          ( collec6321179662152712658at_nat
          @ ( produc410239310623530412_nat_o
            @ ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
                ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
                | ( ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
                  & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R5 ) ) ) ) ) ) ) ).

% mlex_eq
thf(fact_3155_mlex__eq,axiom,
    ( mlex_prod_nat
    = ( ^ [F2: nat > nat,R5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X3: nat,Y3: nat] :
                ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
                | ( ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R5 ) ) ) ) ) ) ) ).

% mlex_eq
thf(fact_3156_unset__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se8260200283734997820nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_3157_unset__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se4203085406695923979it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_3158_unset__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se4205575877204974255it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_3159_unset__bit__Suc,axiom,
    ! [N2: nat,A: code_natural] :
      ( ( bit_se7083795435491715335atural @ ( suc @ N2 ) @ A )
      = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se7083795435491715335atural @ N2 @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_3160_flip__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_3161_flip__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se2159334234014336723it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_3162_flip__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se2161824704523386999it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_3163_flip__bit__Suc,axiom,
    ! [N2: nat,A: code_natural] :
      ( ( bit_se168947363167071951atural @ ( suc @ N2 ) @ A )
      = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se168947363167071951atural @ N2 @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_3164_set__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se2793503036327961859nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_3165_set__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se7879613467334960850it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_3166_set__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se7882103937844011126it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_3167_set__bit__Suc,axiom,
    ! [N2: nat,A: code_natural] :
      ( ( bit_se1617098188084679374atural @ ( suc @ N2 ) @ A )
      = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se1617098188084679374atural @ N2 @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_3168_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L3: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q5: code_integer,R3: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L3 ) @ R3 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R3 @ ( numera6620942414471956472nteger @ L3 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R3 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_3169_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N4: nat,A5: code_integer] : ( if_Code_integer @ ( N4 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_3170_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N4: nat,A5: int] : ( if_int @ ( N4 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_3171_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
     => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
       => ( ( archim7778729529865785530nd_rat @ X )
          = Y ) ) ) ).

% round_unique
thf(fact_3172_rel__of__def,axiom,
    ( rel_of7835217753297429671nteger
    = ( ^ [M: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > option2651255830984564193nteger,P3: produc1908205239877642774nteger > $o] :
          ( collec6950949345391930603nteger
          @ ( produc6253627499356882019eger_o
            @ ^ [K4: produc6241069584506657477e_term > option6357759511663192854e_term,V2: produc8923325533196201883nteger] :
                ( ( ( M @ K4 )
                  = ( some_P6772290148444788224nteger @ V2 ) )
                & ( P3 @ ( produc8603105652947943368nteger @ K4 @ V2 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_3173_rel__of__def,axiom,
    ( rel_of4838799251197538391et_nat
    = ( ^ [M: ( produc3658429121746597890et_nat > $o ) > option936205604648967762et_nat,P3: produc3925858234332021118et_nat > $o] :
          ( collec1402215087704437587et_nat
          @ ( produc1437786849005270451_nat_o
            @ ^ [K4: produc3658429121746597890et_nat > $o,V2: produc3658429121746597890et_nat] :
                ( ( ( M @ K4 )
                  = ( some_P624177172695371229et_nat @ V2 ) )
                & ( P3 @ ( produc5001842942810119800et_nat @ K4 @ V2 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_3174_rel__of__def,axiom,
    ( rel_of7774016450764239315et_nat
    = ( ^ [M: ( produc3658429121746597890et_nat > $o ) > option5190343406534369742et_nat,P3: produc2732055786443039994et_nat > $o] :
          ( collec5543584681430388431et_nat
          @ ( produc838355143741117751_nat_o
            @ ^ [K4: produc3658429121746597890et_nat > $o,V2: produc3925858234332021118et_nat] :
                ( ( ( M @ K4 )
                  = ( some_P750831030444334937et_nat @ V2 ) )
                & ( P3 @ ( produc2245416461498447860et_nat @ K4 @ V2 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_3175_rel__of__def,axiom,
    ( rel_of8306664904814525588nt_int
    = ( ^ [M: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > option4624381673175914239nt_int,P3: produc2285326912895808259nt_int > $o] :
          ( collec1790188477890212312nt_int
          @ ( produc1573362020775583542_int_o
            @ ^ [K4: produc8551481072490612790e_term > option6357759511663192854e_term,V2: product_prod_int_int] :
                ( ( ( M @ K4 )
                  = ( some_P4184893108420464158nt_int @ V2 ) )
                & ( P3 @ ( produc5700946648718959541nt_int @ K4 @ V2 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_3176_rel__of__def,axiom,
    ( rel_of5543720577181062686nt_int
    = ( ^ [M: ( int > option6357759511663192854e_term ) > option4624381673175914239nt_int,P3: produc7773217078559923341nt_int > $o] :
          ( collec506566255779805410nt_int
          @ ( produc2558449545302689196_int_o
            @ ^ [K4: int > option6357759511663192854e_term,V2: product_prod_int_int] :
                ( ( ( M @ K4 )
                  = ( some_P4184893108420464158nt_int @ V2 ) )
                & ( P3 @ ( produc4305682042979456191nt_int @ K4 @ V2 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_3177_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_3178_dbl__simps_I4_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_3179_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_3180_round__altdef,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X3: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X3 ) ) @ ( archim2889992004027027881ng_rat @ X3 ) @ ( archim3151403230148437115or_rat @ X3 ) ) ) ) ).

% round_altdef
thf(fact_3181_round__unique_H,axiom,
    ! [X: rat,N2: int] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N2 ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
     => ( ( archim7778729529865785530nd_rat @ X )
        = N2 ) ) ).

% round_unique'
thf(fact_3182_of__int__round__abs__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_3183_divmod__algorithm__code_I5_J,axiom,
    ! [M2: num,N2: num] :
      ( ( unique3479559517661332726nteger @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q5: code_integer,R3: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R3 ) )
        @ ( unique3479559517661332726nteger @ M2 @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_3184_divmod__algorithm__code_I5_J,axiom,
    ! [M2: num,N2: num] :
      ( ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q5: int,R3: int] : ( product_Pair_int_int @ Q5 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R3 ) )
        @ ( unique5052692396658037445od_int @ M2 @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_3185_divmod__algorithm__code_I5_J,axiom,
    ! [M2: num,N2: num] :
      ( ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q5: nat,R3: nat] : ( product_Pair_nat_nat @ Q5 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R3 ) )
        @ ( unique5055182867167087721od_nat @ M2 @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_3186_signed__take__bit__of__minus__1,axiom,
    ! [N2: nat] :
      ( ( bit_ri6519982836138164636nteger @ N2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% signed_take_bit_of_minus_1
thf(fact_3187_signed__take__bit__of__minus__1,axiom,
    ! [N2: nat] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% signed_take_bit_of_minus_1
thf(fact_3188_signed__take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N2 ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% signed_take_bit_Suc_1
thf(fact_3189_signed__take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_Suc_1
thf(fact_3190_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] :
      ( ( bit_ri6519982836138164636nteger @ ( numeral_numeral_nat @ K ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% signed_take_bit_numeral_of_1
thf(fact_3191_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_numeral_of_1
thf(fact_3192_round__1,axiom,
    ( ( archim7778729529865785530nd_rat @ one_one_rat )
    = one_one_int ) ).

% round_1
thf(fact_3193_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] :
      ( ( unique3479559517661332726nteger @ M2 @ one )
      = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M2 ) @ zero_z3403309356797280102nteger ) ) ).

% divmod_algorithm_code(2)
thf(fact_3194_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] :
      ( ( unique5052692396658037445od_int @ M2 @ one )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ M2 ) @ zero_zero_int ) ) ).

% divmod_algorithm_code(2)
thf(fact_3195_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] :
      ( ( unique5055182867167087721od_nat @ M2 @ one )
      = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M2 ) @ zero_zero_nat ) ) ).

% divmod_algorithm_code(2)
thf(fact_3196_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_3197_dbl__simps_I3_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_3198_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_3199_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N2 ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_3200_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N2 ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_3201_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N2 ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_3202_less__eq__integer__code_I1_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% less_eq_integer_code(1)
thf(fact_3203_sgn__integer__code,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [K4: code_integer] : ( if_Code_integer @ ( K4 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).

% sgn_integer_code
thf(fact_3204_exhaustive__integer_H_Ocases,axiom,
    ! [X: produc8763457246119570046nteger] :
      ~ ! [F3: code_integer > option6357759511663192854e_term,D: code_integer,I2: code_integer] :
          ( X
         != ( produc6137756002093451184nteger @ F3 @ ( produc1086072967326762835nteger @ D @ I2 ) ) ) ).

% exhaustive_integer'.cases
thf(fact_3205_full__exhaustive__integer_H_Ocases,axiom,
    ! [X: produc1908205239877642774nteger] :
      ~ ! [F3: produc6241069584506657477e_term > option6357759511663192854e_term,D: code_integer,I2: code_integer] :
          ( X
         != ( produc8603105652947943368nteger @ F3 @ ( produc1086072967326762835nteger @ D @ I2 ) ) ) ).

% full_exhaustive_integer'.cases
thf(fact_3206_times__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(1)
thf(fact_3207_times__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(2)
thf(fact_3208_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% minus_integer_code(1)
thf(fact_3209_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L )
      = L ) ).

% plus_integer_code(2)
thf(fact_3210_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% plus_integer_code(1)
thf(fact_3211_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
      = ( uminus1351360451143612070nteger @ L ) ) ).

% minus_integer_code(2)
thf(fact_3212_divmod__integer_H__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M: num,N4: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N4 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N4 ) ) ) ) ) ).

% divmod_integer'_def
thf(fact_3213_signed__take__bit__add,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ K @ L ) ) ) ).

% signed_take_bit_add
thf(fact_3214_dbl__def,axiom,
    ( neg_numeral_dbl_rat
    = ( ^ [X3: rat] : ( plus_plus_rat @ X3 @ X3 ) ) ) ).

% dbl_def
thf(fact_3215_dbl__def,axiom,
    ( neg_numeral_dbl_int
    = ( ^ [X3: int] : ( plus_plus_int @ X3 @ X3 ) ) ) ).

% dbl_def
thf(fact_3216_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M: num,N4: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N4 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N4 ) ) ) ) ) ).

% divmod_int_def
thf(fact_3217_divmod__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M: num,N4: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N4 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N4 ) ) ) ) ) ).

% divmod_def
thf(fact_3218_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M: num,N4: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N4 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N4 ) ) ) ) ) ).

% divmod_def
thf(fact_3219_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M: num,N4: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N4 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N4 ) ) ) ) ) ).

% divmod_def
thf(fact_3220_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M: num,N4: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N4 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N4 ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_3221_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 @ one ) ) @ N2 ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_3222_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M: num,N4: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M @ N4 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M ) ) @ ( unique5026877609467782581ep_nat @ N4 @ ( unique5055182867167087721od_nat @ M @ ( bit0 @ N4 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_3223_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M: num,N4: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M @ N4 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M ) ) @ ( unique5024387138958732305ep_int @ N4 @ ( unique5052692396658037445od_int @ M @ ( bit0 @ N4 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_3224_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M: num,N4: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M @ N4 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( unique4921790084139445826nteger @ N4 @ ( unique3479559517661332726nteger @ M @ ( bit0 @ N4 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_3225_round__def,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X3: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_3226_signed__take__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_3227_signed__take__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_3228_of__int__round__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_3229_of__int__round__ge,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).

% of_int_round_ge
thf(fact_3230_of__int__round__gt,axiom,
    ! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).

% of_int_round_gt
thf(fact_3231_integer__of__int__code,axiom,
    ( code_integer_of_int
    = ( ^ [K4: int] :
          ( if_Code_integer @ ( ord_less_int @ K4 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K4 ) ) )
          @ ( if_Code_integer @ ( K4 = zero_zero_int ) @ zero_z3403309356797280102nteger
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% integer_of_int_code
thf(fact_3232_divmod__algorithm__code_I6_J,axiom,
    ! [M2: num,N2: num] :
      ( ( unique3479559517661332726nteger @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q5: code_integer,R3: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R3 ) @ one_one_Code_integer ) )
        @ ( unique3479559517661332726nteger @ M2 @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_3233_divmod__algorithm__code_I6_J,axiom,
    ! [M2: num,N2: num] :
      ( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q5: int,R3: int] : ( product_Pair_int_int @ Q5 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R3 ) @ one_one_int ) )
        @ ( unique5052692396658037445od_int @ M2 @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_3234_divmod__algorithm__code_I6_J,axiom,
    ! [M2: num,N2: num] :
      ( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q5: nat,R3: nat] : ( product_Pair_nat_nat @ Q5 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R3 ) @ one_one_nat ) )
        @ ( unique5055182867167087721od_nat @ M2 @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_3235_of__nat__code,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N4: nat] :
          ( semiri7787848453975740701ux_rat
          @ ^ [I3: rat] : ( plus_plus_rat @ I3 @ one_one_rat )
          @ N4
          @ zero_zero_rat ) ) ) ).

% of_nat_code
thf(fact_3236_of__nat__code,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N4: nat] :
          ( semiri8420488043553186161ux_int
          @ ^ [I3: int] : ( plus_plus_int @ I3 @ one_one_int )
          @ N4
          @ zero_zero_int ) ) ) ).

% of_nat_code
thf(fact_3237_of__nat__code,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N4: nat] :
          ( semiri8422978514062236437ux_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ one_one_nat )
          @ N4
          @ zero_zero_nat ) ) ) ).

% of_nat_code
thf(fact_3238_of__nat__code,axiom,
    ( semiri4939895301339042750nteger
    = ( ^ [N4: nat] :
          ( semiri4055485073559036834nteger
          @ ^ [I3: code_integer] : ( plus_p5714425477246183910nteger @ I3 @ one_one_Code_integer )
          @ N4
          @ zero_z3403309356797280102nteger ) ) ) ).

% of_nat_code
thf(fact_3239_of__nat__code,axiom,
    ( semiri3763490453095760265atural
    = ( ^ [N4: nat] :
          ( semiri2879080225315754349atural
          @ ^ [I3: code_natural] : ( plus_p4538020629002901425atural @ I3 @ one_one_Code_natural )
          @ N4
          @ zero_z2226904508553997617atural ) ) ) ).

% of_nat_code
thf(fact_3240_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_minus_bit1
thf(fact_3241_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 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).

% concat_bit_Suc
thf(fact_3242_even__succ__mod__exp,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_3243_even__succ__mod__exp,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_3244_even__succ__mod__exp,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_3245_even__succ__mod__exp,axiom,
    ! [A: code_natural,N2: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_3246_even__succ__div__exp,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_3247_even__succ__div__exp,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_3248_even__succ__div__exp,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_3249_even__succ__div__exp,axiom,
    ! [A: code_natural,N2: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide5121882707175180666atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_3250_take__bit__rec,axiom,
    ( bit_se1745604003318907178nteger
    = ( ^ [N4: nat,A5: code_integer] : ( if_Code_integer @ ( N4 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_3251_take__bit__rec,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N4: nat,A5: int] : ( if_int @ ( N4 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_3252_take__bit__rec,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N4: nat,A5: nat] : ( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_3253_take__bit__rec,axiom,
    ( bit_se569199155075624693atural
    = ( ^ [N4: nat,A5: code_natural] : ( if_Code_natural @ ( N4 = zero_zero_nat ) @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide5121882707175180666atural @ A5 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( modulo8411746178871703098atural @ A5 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_3254_nat__dvd__1__iff__1,axiom,
    ! [M2: nat] :
      ( ( dvd_dvd_nat @ M2 @ one_one_nat )
      = ( M2 = one_one_nat ) ) ).

% nat_dvd_1_iff_1
thf(fact_3255_integer__of__int__eq__of__int,axiom,
    code_integer_of_int = ring_18347121197199848620nteger ).

% integer_of_int_eq_of_int
thf(fact_3256_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_3257_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_3258_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_3259_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_3260_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_3261_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_3262_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_3263_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_3264_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3265_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3266_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3267_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3268_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3269_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3270_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3271_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3272_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% unit_prod
thf(fact_3273_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).

% unit_prod
thf(fact_3274_unit__prod,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).

% unit_prod
thf(fact_3275_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_3276_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_3277_div__add,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_3278_div__add,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ A )
     => ( ( dvd_dvd_Code_natural @ C @ B )
       => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
          = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ C ) @ ( divide5121882707175180666atural @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_3279_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_3280_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_3281_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_3282_unit__div__1__div__1,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( divide5121882707175180666atural @ one_one_Code_natural @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_3283_unit__div__1__unit,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).

% unit_div_1_unit
thf(fact_3284_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).

% unit_div_1_unit
thf(fact_3285_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).

% unit_div_1_unit
thf(fact_3286_unit__div__1__unit,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( dvd_dvd_Code_natural @ ( divide5121882707175180666atural @ one_one_Code_natural @ A ) @ one_one_Code_natural ) ) ).

% unit_div_1_unit
thf(fact_3287_unit__div,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% unit_div
thf(fact_3288_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% unit_div
thf(fact_3289_unit__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% unit_div
thf(fact_3290_unit__div,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
       => ( dvd_dvd_Code_natural @ ( divide5121882707175180666atural @ A @ B ) @ one_one_Code_natural ) ) ) ).

% unit_div
thf(fact_3291_take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% take_bit_Suc_1
thf(fact_3292_take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ one_one_int )
      = one_one_int ) ).

% take_bit_Suc_1
thf(fact_3293_take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_Suc_1
thf(fact_3294_take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_se569199155075624693atural @ ( suc @ N2 ) @ one_one_Code_natural )
      = one_one_Code_natural ) ).

% take_bit_Suc_1
thf(fact_3295_take__bit__numeral__1,axiom,
    ! [L: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ L ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% take_bit_numeral_1
thf(fact_3296_take__bit__numeral__1,axiom,
    ! [L: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ one_one_int )
      = one_one_int ) ).

% take_bit_numeral_1
thf(fact_3297_take__bit__numeral__1,axiom,
    ! [L: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_numeral_1
thf(fact_3298_take__bit__numeral__1,axiom,
    ! [L: num] :
      ( ( bit_se569199155075624693atural @ ( numeral_numeral_nat @ L ) @ one_one_Code_natural )
      = one_one_Code_natural ) ).

% take_bit_numeral_1
thf(fact_3299_semiring__norm_I9_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_num @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) )
      = ( bit1 @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% semiring_norm(9)
thf(fact_3300_semiring__norm_I7_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_num @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) )
      = ( bit1 @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% semiring_norm(7)
thf(fact_3301_semiring__norm_I16_J,axiom,
    ! [M2: num,N2: num] :
      ( ( times_times_num @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) )
      = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M2 @ N2 ) @ ( bit0 @ ( times_times_num @ M2 @ N2 ) ) ) ) ) ).

% semiring_norm(16)
thf(fact_3302_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
        = ( divide_divide_int @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_3303_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
        = ( divide_divide_nat @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_3304_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
        = ( divide6298287555418463151nteger @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_3305_unit__mult__div__div,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ one_one_Code_natural @ A ) )
        = ( divide5121882707175180666atural @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_3306_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_3307_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_3308_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_3309_unit__div__mult__self,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_3310_take__bit__of__1__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se1745604003318907178nteger @ N2 @ one_one_Code_integer )
        = zero_z3403309356797280102nteger )
      = ( N2 = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_3311_take__bit__of__1__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ one_one_int )
        = zero_zero_int )
      = ( N2 = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_3312_take__bit__of__1__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ one_one_nat )
        = zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_3313_take__bit__of__1__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se569199155075624693atural @ N2 @ one_one_Code_natural )
        = zero_z2226904508553997617atural )
      = ( N2 = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_3314_semiring__norm_I10_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_num @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) )
      = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M2 @ N2 ) @ one ) ) ) ).

% semiring_norm(10)
thf(fact_3315_semiring__norm_I8_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_num @ ( bit1 @ M2 ) @ one )
      = ( bit0 @ ( plus_plus_num @ M2 @ one ) ) ) ).

% semiring_norm(8)
thf(fact_3316_semiring__norm_I5_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_num @ ( bit0 @ M2 ) @ one )
      = ( bit1 @ M2 ) ) ).

% semiring_norm(5)
thf(fact_3317_semiring__norm_I4_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N2 ) )
      = ( bit0 @ ( plus_plus_num @ N2 @ one ) ) ) ).

% semiring_norm(4)
thf(fact_3318_semiring__norm_I3_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N2 ) )
      = ( bit1 @ N2 ) ) ).

% semiring_norm(3)
thf(fact_3319_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_3320_odd__add,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_3321_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_3322_odd__add,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_p4538020629002901425atural @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_3323_even__add,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_3324_even__add,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_3325_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_3326_even__add,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_p4538020629002901425atural @ A @ B ) )
      = ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_3327_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_3328_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_3329_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_3330_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_3331_even__plus__one__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_3332_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
      = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_3333_even__plus__one__iff,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_p4538020629002901425atural @ A @ one_one_Code_natural ) )
      = ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_3334_even__diff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).

% even_diff
thf(fact_3335_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).

% even_diff
thf(fact_3336_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
        = one_one_int ) ) ).

% neg_one_even_power
thf(fact_3337_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
        = one_one_Code_integer ) ) ).

% neg_one_even_power
thf(fact_3338_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
        = one_one_rat ) ) ).

% neg_one_even_power
thf(fact_3339_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% neg_one_odd_power
thf(fact_3340_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% neg_one_odd_power
thf(fact_3341_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% neg_one_odd_power
thf(fact_3342_even__diff__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N2 ) )
      = ( ( ord_less_nat @ M2 @ N2 )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N2 ) ) ) ) ).

% even_diff_nat
thf(fact_3343_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N2 ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_3344_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N2 ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_3345_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N2 ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_3346_mod__Suc__eq__mod__add3,axiom,
    ! [M2: nat,N2: nat] :
      ( ( modulo_modulo_nat @ M2 @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
      = ( modulo_modulo_nat @ M2 @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).

% mod_Suc_eq_mod_add3
thf(fact_3347_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M2: nat,V: num] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_mod_eq_add3_mod_numeral
thf(fact_3348_div__Suc__eq__div__add3,axiom,
    ! [M2: nat,N2: nat] :
      ( ( divide_divide_nat @ M2 @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
      = ( divide_divide_nat @ M2 @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).

% div_Suc_eq_div_add3
thf(fact_3349_Suc__div__eq__add3__div__numeral,axiom,
    ! [M2: nat,V: num] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_div_eq_add3_div_numeral
thf(fact_3350_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_3351_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_3352_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_3353_even__succ__div__2,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_3354_even__succ__div__2,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_3355_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_3356_even__succ__div__2,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_3357_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% odd_succ_div_two
thf(fact_3358_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).

% odd_succ_div_two
thf(fact_3359_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% odd_succ_div_two
thf(fact_3360_odd__succ__div__two,axiom,
    ! [A: code_natural] :
      ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ one_one_Code_natural ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ one_one_Code_natural ) ) ) ).

% odd_succ_div_two
thf(fact_3361_even__succ__div__two,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_3362_even__succ__div__two,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_3363_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_3364_even__succ__div__two,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ one_one_Code_natural ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_3365_odd__two__times__div__two__nat,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).

% odd_two_times_div_two_nat
thf(fact_3366_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_3367_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_3368_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_3369_odd__two__times__div__two__succ,axiom,
    ! [A: code_natural] :
      ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ one_one_Code_natural )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_3370_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3371_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3372_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3373_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_natural ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3374_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).

% zmod_numeral_Bit1
thf(fact_3375_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_num @ M2 @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_num @ M2 @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_3376_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_num @ M2 @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_num @ M2 @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_3377_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_num @ M2 @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_num @ M2 @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_3378_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_eq_num @ M2 @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M2 @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_3379_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_eq_num @ M2 @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M2 @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_3380_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_eq_num @ M2 @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M2 @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_3381_signed__take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_bit1
thf(fact_3382_uminus__integer__code_I1_J,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% uminus_integer_code(1)
thf(fact_3383_take__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,X: int] :
      ( ( bit_se1745604003318907178nteger @ Xa @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se2923211474154528505it_int @ Xa @ X ) ) ) ).

% take_bit_integer.abs_eq
thf(fact_3384_abs__integer__code,axiom,
    ( abs_abs_Code_integer
    = ( ^ [K4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K4 ) @ K4 ) ) ) ).

% abs_integer_code
thf(fact_3385_less__integer__code_I1_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_integer_code(1)
thf(fact_3386_unset__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,X: int] :
      ( ( bit_se8260200283734997820nteger @ Xa @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se4203085406695923979it_int @ Xa @ X ) ) ) ).

% unset_bit_integer.abs_eq
thf(fact_3387_flip__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,X: int] :
      ( ( bit_se1345352211410354436nteger @ Xa @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se2159334234014336723it_int @ Xa @ X ) ) ) ).

% flip_bit_integer.abs_eq
thf(fact_3388_set__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,X: int] :
      ( ( bit_se2793503036327961859nteger @ Xa @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se7879613467334960850it_int @ Xa @ X ) ) ) ).

% set_bit_integer.abs_eq
thf(fact_3389_take__bit__add,axiom,
    ! [N2: nat,A: code_integer,B: code_integer] :
      ( ( bit_se1745604003318907178nteger @ N2 @ ( plus_p5714425477246183910nteger @ ( bit_se1745604003318907178nteger @ N2 @ A ) @ ( bit_se1745604003318907178nteger @ N2 @ B ) ) )
      = ( bit_se1745604003318907178nteger @ N2 @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).

% take_bit_add
thf(fact_3390_take__bit__add,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ A ) @ ( bit_se2923211474154528505it_int @ N2 @ B ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ A @ B ) ) ) ).

% take_bit_add
thf(fact_3391_take__bit__add,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N2 @ A ) @ ( bit_se2925701944663578781it_nat @ N2 @ B ) ) )
      = ( bit_se2925701944663578781it_nat @ N2 @ ( plus_plus_nat @ A @ B ) ) ) ).

% take_bit_add
thf(fact_3392_take__bit__add,axiom,
    ! [N2: nat,A: code_natural,B: code_natural] :
      ( ( bit_se569199155075624693atural @ N2 @ ( plus_p4538020629002901425atural @ ( bit_se569199155075624693atural @ N2 @ A ) @ ( bit_se569199155075624693atural @ N2 @ B ) ) )
      = ( bit_se569199155075624693atural @ N2 @ ( plus_p4538020629002901425atural @ A @ B ) ) ) ).

% take_bit_add
thf(fact_3393_zero__integer__def,axiom,
    ( zero_z3403309356797280102nteger
    = ( code_integer_of_int @ zero_zero_int ) ) ).

% zero_integer_def
thf(fact_3394_less__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
      = ( ord_less_int @ Xa @ X ) ) ).

% less_integer.abs_eq
thf(fact_3395_uminus__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( uminus1351360451143612070nteger @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( uminus_uminus_int @ X ) ) ) ).

% uminus_integer.abs_eq
thf(fact_3396_abs__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( abs_abs_Code_integer @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( abs_abs_int @ X ) ) ) ).

% abs_integer.abs_eq
thf(fact_3397_dvd__add__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3398_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( dvd_dvd_rat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3399_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3400_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3401_dvd__add__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ C )
     => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_3402_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_3403_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_3404_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_3405_dvd__add,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ A @ C )
       => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_3406_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ C )
       => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_3407_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_3408_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ C )
       => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_3409_divide__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( divide_divide_int @ Xa @ X ) ) ) ).

% divide_integer.abs_eq
thf(fact_3410_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).

% dvd_unit_imp_unit
thf(fact_3411_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).

% dvd_unit_imp_unit
thf(fact_3412_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).

% dvd_unit_imp_unit
thf(fact_3413_unit__imp__dvd,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_3414_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_3415_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( dvd_dvd_int @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_3416_one__dvd,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).

% one_dvd
thf(fact_3417_one__dvd,axiom,
    ! [A: assn] : ( dvd_dvd_assn @ one_one_assn @ A ) ).

% one_dvd
thf(fact_3418_one__dvd,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).

% one_dvd
thf(fact_3419_one__dvd,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).

% one_dvd
thf(fact_3420_one__dvd,axiom,
    ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).

% one_dvd
thf(fact_3421_modulo__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( ( modulo364778990260209775nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( modulo_modulo_int @ Xa @ X ) ) ) ).

% modulo_integer.abs_eq
thf(fact_3422_sgn__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( sgn_sgn_Code_integer @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( sgn_sgn_int @ X ) ) ) ).

% sgn_integer.abs_eq
thf(fact_3423_concat__bit__assoc,axiom,
    ! [N2: nat,K: int,M2: nat,L: int,R2: int] :
      ( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M2 @ L @ R2 ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M2 @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L ) @ R2 ) ) ).

% concat_bit_assoc
thf(fact_3424_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).

% not_is_unit_0
thf(fact_3425_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_3426_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_3427_pinf_I9_J,axiom,
    ! [D2: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X8: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X8 )
     => ( ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X8 @ S ) )
        = ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X8 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3428_pinf_I9_J,axiom,
    ! [D2: rat,S: rat] :
    ? [Z3: rat] :
    ! [X8: rat] :
      ( ( ord_less_rat @ Z3 @ X8 )
     => ( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X8 @ S ) )
        = ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X8 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3429_pinf_I9_J,axiom,
    ! [D2: nat,S: nat] :
    ? [Z3: nat] :
    ! [X8: nat] :
      ( ( ord_less_nat @ Z3 @ X8 )
     => ( ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X8 @ S ) )
        = ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X8 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3430_pinf_I9_J,axiom,
    ! [D2: int,S: int] :
    ? [Z3: int] :
    ! [X8: int] :
      ( ( ord_less_int @ Z3 @ X8 )
     => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X8 @ S ) )
        = ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X8 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3431_pinf_I10_J,axiom,
    ! [D2: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X8: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X8 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X8 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X8 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3432_pinf_I10_J,axiom,
    ! [D2: rat,S: rat] :
    ? [Z3: rat] :
    ! [X8: rat] :
      ( ( ord_less_rat @ Z3 @ X8 )
     => ( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X8 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X8 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3433_pinf_I10_J,axiom,
    ! [D2: nat,S: nat] :
    ? [Z3: nat] :
    ! [X8: nat] :
      ( ( ord_less_nat @ Z3 @ X8 )
     => ( ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X8 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X8 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3434_pinf_I10_J,axiom,
    ! [D2: int,S: int] :
    ? [Z3: int] :
    ! [X8: int] :
      ( ( ord_less_int @ Z3 @ X8 )
     => ( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X8 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X8 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3435_minf_I9_J,axiom,
    ! [D2: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X8: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X8 @ Z3 )
     => ( ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X8 @ S ) )
        = ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X8 @ S ) ) ) ) ).

% minf(9)
thf(fact_3436_minf_I9_J,axiom,
    ! [D2: rat,S: rat] :
    ? [Z3: rat] :
    ! [X8: rat] :
      ( ( ord_less_rat @ X8 @ Z3 )
     => ( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X8 @ S ) )
        = ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X8 @ S ) ) ) ) ).

% minf(9)
thf(fact_3437_minf_I9_J,axiom,
    ! [D2: nat,S: nat] :
    ? [Z3: nat] :
    ! [X8: nat] :
      ( ( ord_less_nat @ X8 @ Z3 )
     => ( ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X8 @ S ) )
        = ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X8 @ S ) ) ) ) ).

% minf(9)
thf(fact_3438_minf_I9_J,axiom,
    ! [D2: int,S: int] :
    ? [Z3: int] :
    ! [X8: int] :
      ( ( ord_less_int @ X8 @ Z3 )
     => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X8 @ S ) )
        = ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X8 @ S ) ) ) ) ).

% minf(9)
thf(fact_3439_minf_I10_J,axiom,
    ! [D2: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X8: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X8 @ Z3 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X8 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X8 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3440_minf_I10_J,axiom,
    ! [D2: rat,S: rat] :
    ? [Z3: rat] :
    ! [X8: rat] :
      ( ( ord_less_rat @ X8 @ Z3 )
     => ( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X8 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X8 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3441_minf_I10_J,axiom,
    ! [D2: nat,S: nat] :
    ? [Z3: nat] :
    ! [X8: nat] :
      ( ( ord_less_nat @ X8 @ Z3 )
     => ( ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X8 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X8 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3442_minf_I10_J,axiom,
    ! [D2: int,S: int] :
    ? [Z3: int] :
    ! [X8: int] :
      ( ( ord_less_int @ X8 @ Z3 )
     => ( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X8 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X8 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3443_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ B @ A )
          = ( times_3573771949741848930nteger @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_3444_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ B @ A )
          = ( times_times_nat @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_3445_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ B @ A )
          = ( times_times_int @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_3446_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ A @ B )
          = ( times_3573771949741848930nteger @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_3447_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ A @ B )
          = ( times_times_nat @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_3448_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ A @ B )
          = ( times_times_int @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_3449_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_3450_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_3451_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_3452_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_3453_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_3454_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_3455_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_3456_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_3457_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_3458_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_3459_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_3460_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_3461_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).

% is_unit_mult_iff
thf(fact_3462_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).

% is_unit_mult_iff
thf(fact_3463_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).

% is_unit_mult_iff
thf(fact_3464_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_3465_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_3466_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_3467_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ A )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
        = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ C ) @ ( divide5121882707175180666atural @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_3468_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_3469_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_3470_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_3471_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ B )
     => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
        = ( plus_p4538020629002901425atural @ ( divide5121882707175180666atural @ A @ C ) @ ( divide5121882707175180666atural @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_3472_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( divide_divide_int @ B @ A )
          = ( divide_divide_int @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_3473_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( divide_divide_nat @ B @ A )
          = ( divide_divide_nat @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_3474_unit__div__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ B @ A )
          = ( divide6298287555418463151nteger @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_3475_unit__div__cancel,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( ( divide5121882707175180666atural @ B @ A )
          = ( divide5121882707175180666atural @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_3476_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_3477_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_3478_div__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_3479_div__unit__dvd__iff,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ ( divide5121882707175180666atural @ A @ B ) @ C )
        = ( dvd_dvd_Code_natural @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_3480_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_3481_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_3482_dvd__div__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_3483_dvd__div__unit__iff,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ A @ ( divide5121882707175180666atural @ C @ B ) )
        = ( dvd_dvd_Code_natural @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_3484_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X
       != ( product_Pair_num_num @ one @ one ) )
     => ( ! [N: num] :
            ( X
           != ( product_Pair_num_num @ one @ ( bit0 @ N ) ) )
       => ( ! [N: num] :
              ( X
             != ( product_Pair_num_num @ one @ ( bit1 @ N ) ) )
         => ( ! [M5: num] :
                ( X
               != ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) )
           => ( ! [M5: num,N: num] :
                  ( X
                 != ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N ) ) )
             => ( ! [M5: num,N: num] :
                    ( X
                   != ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N ) ) )
               => ( ! [M5: num] :
                      ( X
                     != ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) )
                 => ( ! [M5: num,N: num] :
                        ( X
                       != ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N ) ) )
                   => ~ ! [M5: num,N: num] :
                          ( X
                         != ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_3485_plus__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( plus_plus_int @ Xa @ X ) ) ) ).

% plus_integer.abs_eq
thf(fact_3486_bezout__lemma__nat,axiom,
    ! [D2: nat,A: nat,B: nat,X: nat,Y: nat] :
      ( ( dvd_dvd_nat @ D2 @ A )
     => ( ( dvd_dvd_nat @ D2 @ B )
       => ( ( ( ( times_times_nat @ A @ X )
              = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D2 ) )
            | ( ( times_times_nat @ B @ X )
              = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D2 ) ) )
         => ? [X4: nat,Y5: nat] :
              ( ( dvd_dvd_nat @ D2 @ A )
              & ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ A @ B ) )
              & ( ( ( times_times_nat @ A @ X4 )
                  = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y5 ) @ D2 ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X4 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y5 ) @ D2 ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_3487_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D: nat,X4: nat,Y5: nat] :
      ( ( dvd_dvd_nat @ D @ A )
      & ( dvd_dvd_nat @ D @ B )
      & ( ( ( times_times_nat @ A @ X4 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ D ) )
        | ( ( times_times_nat @ B @ X4 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y5 ) @ D ) ) ) ) ).

% bezout_add_nat
thf(fact_3488_one__integer__def,axiom,
    ( one_one_Code_integer
    = ( code_integer_of_int @ one_one_int ) ) ).

% one_integer_def
thf(fact_3489_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
      = ( ord_less_eq_int @ Xa @ X ) ) ).

% less_eq_integer.abs_eq
thf(fact_3490_times__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( times_times_int @ Xa @ X ) ) ) ).

% times_integer.abs_eq
thf(fact_3491_zdvd__period,axiom,
    ! [A: int,D2: int,X: int,T: int,C: int] :
      ( ( dvd_dvd_int @ A @ D2 )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
        = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D2 ) ) @ T ) ) ) ) ).

% zdvd_period
thf(fact_3492_zdvd__reduce,axiom,
    ! [K: int,N2: int,M2: int] :
      ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N2 @ ( times_times_int @ K @ M2 ) ) )
      = ( dvd_dvd_int @ K @ N2 ) ) ).

% zdvd_reduce
thf(fact_3493_minus__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( minus_minus_int @ Xa @ X ) ) ) ).

% minus_integer.abs_eq
thf(fact_3494_unity__coeff__ex,axiom,
    ! [P: code_integer > $o,L: code_integer] :
      ( ( ? [X3: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X3 ) ) )
      = ( ? [X3: code_integer] :
            ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X3 @ zero_z3403309356797280102nteger ) )
            & ( P @ X3 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3495_unity__coeff__ex,axiom,
    ! [P: rat > $o,L: rat] :
      ( ( ? [X3: rat] : ( P @ ( times_times_rat @ L @ X3 ) ) )
      = ( ? [X3: rat] :
            ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X3 @ zero_zero_rat ) )
            & ( P @ X3 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3496_unity__coeff__ex,axiom,
    ! [P: nat > $o,L: nat] :
      ( ( ? [X3: nat] : ( P @ ( times_times_nat @ L @ X3 ) ) )
      = ( ? [X3: nat] :
            ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X3 @ zero_zero_nat ) )
            & ( P @ X3 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3497_unity__coeff__ex,axiom,
    ! [P: int > $o,L: int] :
      ( ( ? [X3: int] : ( P @ ( times_times_int @ L @ X3 ) ) )
      = ( ? [X3: int] :
            ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X3 @ zero_zero_int ) )
            & ( P @ X3 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3498_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C3: code_integer] :
              ( B
             != ( times_3573771949741848930nteger @ A @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_3499_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [C3: nat] :
              ( B
             != ( times_times_nat @ A @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_3500_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [C3: int] :
              ( B
             != ( times_times_int @ A @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_3501_inf__period_I4_J,axiom,
    ! [D2: code_integer,D3: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D2 @ D3 )
     => ! [X8: code_integer,K6: code_integer] :
          ( ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X8 @ T ) ) )
          = ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X8 @ ( times_3573771949741848930nteger @ K6 @ D3 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3502_inf__period_I4_J,axiom,
    ! [D2: rat,D3: rat,T: rat] :
      ( ( dvd_dvd_rat @ D2 @ D3 )
     => ! [X8: rat,K6: rat] :
          ( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X8 @ T ) ) )
          = ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X8 @ ( times_times_rat @ K6 @ D3 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3503_inf__period_I4_J,axiom,
    ! [D2: int,D3: int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D3 )
     => ! [X8: int,K6: int] :
          ( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X8 @ T ) ) )
          = ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X8 @ ( times_times_int @ K6 @ D3 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3504_inf__period_I3_J,axiom,
    ! [D2: code_integer,D3: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D2 @ D3 )
     => ! [X8: code_integer,K6: code_integer] :
          ( ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X8 @ T ) )
          = ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X8 @ ( times_3573771949741848930nteger @ K6 @ D3 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3505_inf__period_I3_J,axiom,
    ! [D2: rat,D3: rat,T: rat] :
      ( ( dvd_dvd_rat @ D2 @ D3 )
     => ! [X8: rat,K6: rat] :
          ( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X8 @ T ) )
          = ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X8 @ ( times_times_rat @ K6 @ D3 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3506_inf__period_I3_J,axiom,
    ! [D2: int,D3: int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D3 )
     => ! [X8: int,K6: int] :
          ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X8 @ T ) )
          = ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X8 @ ( times_times_int @ K6 @ D3 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3507_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
thf(fact_3508_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
thf(fact_3509_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = zero_z3403309356797280102nteger )
        = ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
thf(fact_3510_unit__div__eq__0__iff,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( ( divide5121882707175180666atural @ A @ B )
          = zero_z2226904508553997617atural )
        = ( A = zero_z2226904508553997617atural ) ) ) ).

% unit_div_eq_0_iff
thf(fact_3511_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).

% numeral_Bit1
thf(fact_3512_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).

% numeral_Bit1
thf(fact_3513_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numera6620942414471956472nteger @ ( bit1 @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ one_one_Code_integer ) ) ).

% numeral_Bit1
thf(fact_3514_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).

% numeral_Bit1
thf(fact_3515_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numera5444537566228673987atural @ ( bit1 @ N2 ) )
      = ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N2 ) @ ( numera5444537566228673987atural @ N2 ) ) @ one_one_Code_natural ) ) ).

% numeral_Bit1
thf(fact_3516_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B )
          = C )
        = ( A
          = ( times_times_int @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_3517_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B )
          = C )
        = ( A
          = ( times_times_nat @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_3518_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = C )
        = ( A
          = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_3519_unit__eq__div1,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( ( divide5121882707175180666atural @ A @ B )
          = C )
        = ( A
          = ( times_2397367101498566445atural @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_3520_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( A
          = ( divide_divide_int @ C @ B ) )
        = ( ( times_times_int @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_3521_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( A
          = ( divide_divide_nat @ C @ B ) )
        = ( ( times_times_nat @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_3522_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( A
          = ( divide6298287555418463151nteger @ C @ B ) )
        = ( ( times_3573771949741848930nteger @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_3523_unit__eq__div2,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( A
          = ( divide5121882707175180666atural @ C @ B ) )
        = ( ( times_2397367101498566445atural @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_3524_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_3525_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_3526_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_3527_div__mult__unit2,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ B @ A )
       => ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ B @ C ) )
          = ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_3528_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
        = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_3529_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
        = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_3530_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_3531_unit__div__commute,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ C )
        = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_3532_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_3533_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_3534_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_3535_unit__div__mult__swap,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ one_one_Code_natural )
     => ( ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ C ) )
        = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_3536_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ C @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_3537_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ C @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_3538_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_3539_is__unit__div__mult2__eq,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ C @ one_one_Code_natural )
       => ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ B @ C ) )
          = ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_3540_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
thf(fact_3541_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
thf(fact_3542_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
thf(fact_3543_unit__imp__mod__eq__0,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( modulo8411746178871703098atural @ A @ B )
        = zero_z2226904508553997617atural ) ) ).

% unit_imp_mod_eq_0
thf(fact_3544_is__unit__power__iff,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_3545_is__unit__power__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_3546_is__unit__power__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_3547_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D: nat,X4: nat,Y5: nat] :
          ( ( dvd_dvd_nat @ D @ A )
          & ( dvd_dvd_nat @ D @ B )
          & ( ( times_times_nat @ A @ X4 )
            = ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ D ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_3548_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ ( numera6620942414471956472nteger @ ( bit1 @ K ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N2 @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ).

% take_bit_Suc_bit1
thf(fact_3549_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_bit1
thf(fact_3550_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_Suc_bit1
thf(fact_3551_take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se569199155075624693atural @ ( suc @ N2 ) @ ( numera5444537566228673987atural @ ( bit1 @ K ) ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N2 @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ one_one_Code_natural ) ) ).

% take_bit_Suc_bit1
thf(fact_3552_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).

% numeral_code(3)
thf(fact_3553_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).

% numeral_code(3)
thf(fact_3554_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numera6620942414471956472nteger @ ( bit1 @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ one_one_Code_integer ) ) ).

% numeral_code(3)
thf(fact_3555_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).

% numeral_code(3)
thf(fact_3556_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numera5444537566228673987atural @ ( bit1 @ N2 ) )
      = ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N2 ) @ ( numera5444537566228673987atural @ N2 ) ) @ one_one_Code_natural ) ) ).

% numeral_code(3)
thf(fact_3557_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se1745604003318907178nteger @ N2 @ A )
            = zero_z3403309356797280102nteger ) )
        & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se1745604003318907178nteger @ N2 @ A )
            = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_3558_stable__imp__take__bit__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2923211474154528505it_int @ N2 @ A )
            = zero_zero_int ) )
        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2923211474154528505it_int @ N2 @ A )
            = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_3559_stable__imp__take__bit__eq,axiom,
    ! [A: nat,N2: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2925701944663578781it_nat @ N2 @ A )
            = zero_zero_nat ) )
        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2925701944663578781it_nat @ N2 @ A )
            = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_3560_stable__imp__take__bit__eq,axiom,
    ! [A: code_natural,N2: nat] :
      ( ( ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se569199155075624693atural @ N2 @ A )
            = zero_z2226904508553997617atural ) )
        & ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se569199155075624693atural @ N2 @ A )
            = ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_natural ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_3561_is__unitE,axiom,
    ! [A: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [B2: int] :
              ( ( B2 != zero_zero_int )
             => ( ( dvd_dvd_int @ B2 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A )
                    = B2 )
                 => ( ( ( divide_divide_int @ one_one_int @ B2 )
                      = A )
                   => ( ( ( times_times_int @ A @ B2 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C @ A )
                       != ( times_times_int @ C @ B2 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_3562_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [B2: nat] :
              ( ( B2 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A )
                    = B2 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B2 )
                      = A )
                   => ( ( ( times_times_nat @ A @ B2 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C @ A )
                       != ( times_times_nat @ C @ B2 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_3563_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B2: code_integer] :
              ( ( B2 != zero_z3403309356797280102nteger )
             => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
               => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
                    = B2 )
                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 )
                      = A )
                   => ( ( ( times_3573771949741848930nteger @ A @ B2 )
                        = one_one_Code_integer )
                     => ( ( divide6298287555418463151nteger @ C @ A )
                       != ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_3564_is__unitE,axiom,
    ! [A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ~ ( ( A != zero_z2226904508553997617atural )
         => ! [B2: code_natural] :
              ( ( B2 != zero_z2226904508553997617atural )
             => ( ( dvd_dvd_Code_natural @ B2 @ one_one_Code_natural )
               => ( ( ( divide5121882707175180666atural @ one_one_Code_natural @ A )
                    = B2 )
                 => ( ( ( divide5121882707175180666atural @ one_one_Code_natural @ B2 )
                      = A )
                   => ( ( ( times_2397367101498566445atural @ A @ B2 )
                        = one_one_Code_natural )
                     => ( ( divide5121882707175180666atural @ C @ A )
                       != ( times_2397367101498566445atural @ C @ B2 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_3565_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
          = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_3566_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
          = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_3567_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_3568_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
       => ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ A @ B ) )
          = ( divide5121882707175180666atural @ one_one_Code_natural @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_3569_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
          = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_3570_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
          = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_3571_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_3572_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
       => ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ B @ A ) )
          = ( divide5121882707175180666atural @ one_one_Code_natural @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_3573_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_3574_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_3575_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_3576_odd__even__add,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( plus_p4538020629002901425atural @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_3577_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_3578_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_3579_odd__one,axiom,
    ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).

% odd_one
thf(fact_3580_odd__one,axiom,
    ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ one_one_Code_natural ) ).

% odd_one
thf(fact_3581_dvd__power__iff,axiom,
    ! [X: code_integer,M2: nat,N2: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M2 ) @ ( power_8256067586552552935nteger @ X @ N2 ) )
        = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_3582_dvd__power__iff,axiom,
    ! [X: nat,M2: nat,N2: nat] :
      ( ( X != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M2 ) @ ( power_power_nat @ X @ N2 ) )
        = ( ( dvd_dvd_nat @ X @ one_one_nat )
          | ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_3583_dvd__power__iff,axiom,
    ! [X: int,M2: nat,N2: nat] :
      ( ( X != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X @ M2 ) @ ( power_power_int @ X @ N2 ) )
        = ( ( dvd_dvd_int @ X @ one_one_int )
          | ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_3584_dvd__power,axiom,
    ! [N2: nat,X: code_integer] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_Code_integer ) )
     => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_3585_dvd__power,axiom,
    ! [N2: nat,X: rat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_rat ) )
     => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_3586_dvd__power,axiom,
    ! [N2: nat,X: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_nat ) )
     => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_3587_dvd__power,axiom,
    ! [N2: nat,X: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X = one_one_int ) )
     => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N2 ) ) ) ).

% dvd_power
thf(fact_3588_Suc3__eq__add__3,axiom,
    ! [N2: nat] :
      ( ( suc @ ( suc @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ).

% Suc3_eq_add_3
thf(fact_3589_dvd__mult__cancel2,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N2 @ M2 ) @ M2 )
        = ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_3590_dvd__mult__cancel1,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M2 @ N2 ) @ M2 )
        = ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_3591_dvd__minus__add,axiom,
    ! [Q3: nat,N2: nat,R2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ Q3 @ N2 )
     => ( ( ord_less_eq_nat @ Q3 @ ( times_times_nat @ R2 @ M2 ) )
       => ( ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N2 @ Q3 ) )
          = ( dvd_dvd_nat @ M2 @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M2 ) @ Q3 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_3592_power__dvd__imp__le,axiom,
    ! [I: nat,M2: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N2 ) )
     => ( ( ord_less_nat @ one_one_nat @ I )
       => ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).

% power_dvd_imp_le
thf(fact_3593_aset_I10_J,axiom,
    ! [D2: int,D3: int,A2: set_int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D3 )
     => ! [X8: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A2 )
                 => ( X8
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X8 @ T ) )
           => ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( plus_plus_int @ X8 @ D3 ) @ T ) ) ) ) ) ).

% aset(10)
thf(fact_3594_aset_I9_J,axiom,
    ! [D2: int,D3: int,A2: set_int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D3 )
     => ! [X8: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A2 )
                 => ( X8
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X8 @ T ) )
           => ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( plus_plus_int @ X8 @ D3 ) @ T ) ) ) ) ) ).

% aset(9)
thf(fact_3595_bset_I10_J,axiom,
    ! [D2: int,D3: int,B5: set_int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D3 )
     => ! [X8: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B5 )
                 => ( X8
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X8 @ T ) )
           => ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X8 @ D3 ) @ T ) ) ) ) ) ).

% bset(10)
thf(fact_3596_bset_I9_J,axiom,
    ! [D2: int,D3: int,B5: set_int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D3 )
     => ! [X8: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B5 )
                 => ( X8
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X8 @ T ) )
           => ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X8 @ D3 ) @ T ) ) ) ) ) ).

% bset(9)
thf(fact_3597_num_Osize__gen_I3_J,axiom,
    ! [X32: num] :
      ( ( size_num @ ( bit1 @ X32 ) )
      = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(3)
thf(fact_3598_num_Osize_I6_J,axiom,
    ! [X32: num] :
      ( ( size_size_num @ ( bit1 @ X32 ) )
      = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(6)
thf(fact_3599_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_3600_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_3601_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_3602_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_natural] :
      ( ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = one_one_Code_natural ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_3603_Suc__mod__eq__add3__mod,axiom,
    ! [M2: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ N2 ) ) ).

% Suc_mod_eq_add3_mod
thf(fact_3604_Suc__div__eq__add3__div,axiom,
    ! [M2: nat,N2: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N2 )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ N2 ) ) ).

% Suc_div_eq_add3_div
thf(fact_3605_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).

% even_diff_iff
thf(fact_3606_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).

% even_abs_add_iff
thf(fact_3607_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).

% even_add_abs_iff
thf(fact_3608_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N4: nat,K4: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ ( plus_plus_int @ K4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_3609_take__bit__incr__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
       != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) )
     => ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ K @ one_one_int ) )
        = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ) ).

% take_bit_incr_eq
thf(fact_3610_oddE,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: nat] :
            ( A
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_3611_oddE,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: int] :
            ( A
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_3612_oddE,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: code_integer] :
            ( A
           != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) @ one_one_Code_integer ) ) ) ).

% oddE
thf(fact_3613_oddE,axiom,
    ! [A: code_natural] :
      ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ~ ! [B2: code_natural] :
            ( A
           != ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B2 ) @ one_one_Code_natural ) ) ) ).

% oddE
thf(fact_3614_parity__cases,axiom,
    ! [A: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
         != zero_zero_int ) )
     => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
           != one_one_int ) ) ) ).

% parity_cases
thf(fact_3615_parity__cases,axiom,
    ! [A: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
         != zero_zero_nat ) )
     => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
           != one_one_nat ) ) ) ).

% parity_cases
thf(fact_3616_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
         != zero_z3403309356797280102nteger ) )
     => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
           != one_one_Code_integer ) ) ) ).

% parity_cases
thf(fact_3617_parity__cases,axiom,
    ! [A: code_natural] :
      ( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
       => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
         != zero_z2226904508553997617atural ) )
     => ~ ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
         => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
           != one_one_Code_natural ) ) ) ).

% parity_cases
thf(fact_3618_mod2__eq__if,axiom,
    ! [A: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = zero_zero_int ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = one_one_int ) ) ) ).

% mod2_eq_if
thf(fact_3619_mod2__eq__if,axiom,
    ! [A: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = zero_zero_nat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = one_one_nat ) ) ) ).

% mod2_eq_if
thf(fact_3620_mod2__eq__if,axiom,
    ! [A: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = one_one_Code_integer ) ) ) ).

% mod2_eq_if
thf(fact_3621_mod2__eq__if,axiom,
    ! [A: code_natural] :
      ( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
       => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
          = zero_z2226904508553997617atural ) )
      & ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
       => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
          = one_one_Code_natural ) ) ) ).

% mod2_eq_if
thf(fact_3622_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
          = one_one_int ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% minus_one_power_iff
thf(fact_3623_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
          = one_one_Code_integer ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% minus_one_power_iff
thf(fact_3624_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
          = one_one_rat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% minus_one_power_iff
thf(fact_3625_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 @ one ) ) @ N2 ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_int_greater_eq
thf(fact_3626_take__bit__Suc__minus__1__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ one_one_Code_integer ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_3627_take__bit__Suc__minus__1__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_3628_take__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_3629_take__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_3630_take__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ A )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_3631_take__bit__Suc,axiom,
    ! [N2: nat,A: code_natural] :
      ( ( bit_se569199155075624693atural @ ( suc @ N2 ) @ A )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N2 @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_3632_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_Code_integer ) ) ).

% take_bit_numeral_minus_1_eq
thf(fact_3633_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_1_eq
thf(fact_3634_even__mask__div__iff_H,axiom,
    ! [M2: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_3635_even__mask__div__iff_H,axiom,
    ! [M2: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_3636_even__mask__div__iff_H,axiom,
    ! [M2: nat,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_3637_even__mask__div__iff_H,axiom,
    ! [M2: nat,N2: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) @ one_one_Code_natural ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_3638_even__mask__div__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_int )
        | ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_3639_even__mask__div__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_nat )
        | ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_3640_even__mask__div__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
          = zero_z3403309356797280102nteger )
        | ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_3641_even__mask__div__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) @ one_one_Code_natural ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 )
          = zero_z2226904508553997617atural )
        | ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_3642_Code__Numeral_Onegative__def,axiom,
    ( code_negative
    = ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).

% Code_Numeral.negative_def
thf(fact_3643_take__bit__numeral__minus__numeral__int,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( case_option_int_num @ zero_zero_int
        @ ^ [Q5: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M2 ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_int @ Q5 ) ) )
        @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M2 ) @ N2 ) ) ) ).

% take_bit_numeral_minus_numeral_int
thf(fact_3644_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_minus_bit1
thf(fact_3645_execute__change,axiom,
    ! [F: array_a > array_a,R2: ref_array_a,H3: heap_e7401611519738050253t_unit] :
      ( ( heap_T5591663578999710211rray_a @ ( ref_change_array_a @ F @ R2 ) @ H3 )
      = ( some_P6105628898608307625it_nat @ ( produc1733996804261680252it_nat @ ( F @ ( ref_get_array_a @ H3 @ R2 ) ) @ ( produc584006145561248582it_nat @ ( ref_set_array_a @ R2 @ ( F @ ( ref_get_array_a @ H3 @ R2 ) ) @ H3 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ) ).

% execute_change
thf(fact_3646_execute__change,axiom,
    ! [F: product_unit > product_unit,R2: ref_Product_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( heap_T875086893843062177t_unit @ ( ref_ch7259622376331601608t_unit @ F @ R2 ) @ H3 )
      = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ ( F @ ( ref_get_Product_unit @ H3 @ R2 ) ) @ ( produc584006145561248582it_nat @ ( ref_set_Product_unit @ R2 @ ( F @ ( ref_get_Product_unit @ H3 @ R2 ) ) @ H3 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ) ).

% execute_change
thf(fact_3647_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_bit1
thf(fact_3648_take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_bit1
thf(fact_3649_divmod__BitM__2__eq,axiom,
    ! [M2: num] :
      ( ( unique5052692396658037445od_int @ ( bitM @ M2 ) @ ( bit0 @ one ) )
      = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ one_one_int ) ) ).

% divmod_BitM_2_eq
thf(fact_3650_flip__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
      = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_3651_flip__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
      = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_3652_flip__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
      = ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_3653_flip__bit__0,axiom,
    ! [A: code_natural] :
      ( ( bit_se168947363167071951atural @ zero_zero_nat @ A )
      = ( plus_p4538020629002901425atural @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_3654_of__bool__eq_I2_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $true )
    = one_one_rat ) ).

% of_bool_eq(2)
thf(fact_3655_of__bool__eq_I2_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $true )
    = one_one_nat ) ).

% of_bool_eq(2)
thf(fact_3656_of__bool__eq_I2_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $true )
    = one_one_int ) ).

% of_bool_eq(2)
thf(fact_3657_of__bool__eq_I2_J,axiom,
    ( ( zero_n356916108424825756nteger @ $true )
    = one_one_Code_integer ) ).

% of_bool_eq(2)
thf(fact_3658_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = one_one_rat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_3659_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = one_one_nat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_3660_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = one_one_int )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_3661_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = one_one_Code_integer )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_3662_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_3663_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_3664_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_3665_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_3666_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat @ ~ P )
      = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).

% of_bool_not_iff
thf(fact_3667_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2684676970156552555ol_int @ ~ P )
      = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).

% of_bool_not_iff
thf(fact_3668_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger @ ~ P )
      = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).

% of_bool_not_iff
thf(fact_3669_take__bit__num__simps_I3_J,axiom,
    ! [N2: nat,M2: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M2 ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
        @ ( bit_take_bit_num @ N2 @ M2 ) ) ) ).

% take_bit_num_simps(3)
thf(fact_3670_take__bit__num__simps_I4_J,axiom,
    ! [N2: nat,M2: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit1 @ M2 ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M2 ) ) ) ) ).

% take_bit_num_simps(4)
thf(fact_3671_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M2: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M2 ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
        @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M2 ) ) ) ).

% take_bit_num_simps(6)
thf(fact_3672_take__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se1745604003318907178nteger @ N2 @ one_one_Code_integer )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_1
thf(fact_3673_take__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ N2 @ one_one_int )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_1
thf(fact_3674_take__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ one_one_nat )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_1
thf(fact_3675_take__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se569199155075624693atural @ N2 @ one_one_Code_natural )
      = ( zero_n8403883297036319079atural @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_1
thf(fact_3676_add__neg__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M2 ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_3677_add__neg__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M2 ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_3678_add__neg__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M2 ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_3679_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N2 ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_3680_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N2 ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_3681_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N2 ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_3682_diff__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N2 ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_3683_diff__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N2 ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_3684_diff__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N2 ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N2 ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_3685_diff__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( inc @ M2 ) ) ) ).

% diff_numeral_special(6)
thf(fact_3686_diff__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( numera6620942414471956472nteger @ ( inc @ M2 ) ) ) ).

% diff_numeral_special(6)
thf(fact_3687_diff__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( numeral_numeral_rat @ ( inc @ M2 ) ) ) ).

% diff_numeral_special(6)
thf(fact_3688_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M2: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M2 ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M2 ) ) ) ) ).

% take_bit_num_simps(7)
thf(fact_3689_take__bit__numeral__numeral,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( case_o356765784539232260er_num @ zero_z3403309356797280102nteger @ numera6620942414471956472nteger @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M2 ) @ N2 ) ) ) ).

% take_bit_numeral_numeral
thf(fact_3690_take__bit__numeral__numeral,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_int @ N2 ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M2 ) @ N2 ) ) ) ).

% take_bit_numeral_numeral
thf(fact_3691_take__bit__numeral__numeral,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
      = ( case_option_nat_num @ zero_zero_nat @ numeral_numeral_nat @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M2 ) @ N2 ) ) ) ).

% take_bit_numeral_numeral
thf(fact_3692_take__bit__numeral__numeral,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se569199155075624693atural @ ( numeral_numeral_nat @ M2 ) @ ( numera5444537566228673987atural @ N2 ) )
      = ( case_o5621594795226839503al_num @ zero_z2226904508553997617atural @ numera5444537566228673987atural @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M2 ) @ N2 ) ) ) ).

% take_bit_numeral_numeral
thf(fact_3693_bits__1__div__exp,axiom,
    ! [N2: nat] :
      ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n8403883297036319079atural @ ( N2 = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_3694_bits__1__div__exp,axiom,
    ! [N2: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_3695_bits__1__div__exp,axiom,
    ! [N2: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_3696_bits__1__div__exp,axiom,
    ! [N2: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n356916108424825756nteger @ ( N2 = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_3697_one__div__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n8403883297036319079atural @ ( N2 = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_3698_one__div__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_3699_one__div__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_3700_one__div__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n356916108424825756nteger @ ( N2 = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_3701_one__mod__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n8403883297036319079atural @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% one_mod_2_pow_eq
thf(fact_3702_one__mod__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% one_mod_2_pow_eq
thf(fact_3703_one__mod__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% one_mod_2_pow_eq
thf(fact_3704_one__mod__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% one_mod_2_pow_eq
thf(fact_3705_mult__inc,axiom,
    ! [X: num,Y: num] :
      ( ( times_times_num @ X @ ( inc @ Y ) )
      = ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).

% mult_inc
thf(fact_3706_add__inc,axiom,
    ! [X: num,Y: num] :
      ( ( plus_plus_num @ X @ ( inc @ Y ) )
      = ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).

% add_inc
thf(fact_3707_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).

% of_bool_less_eq_one
thf(fact_3708_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).

% of_bool_less_eq_one
thf(fact_3709_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).

% of_bool_less_eq_one
thf(fact_3710_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).

% of_bool_less_eq_one
thf(fact_3711_of__bool__def,axiom,
    ( zero_n2052037380579107095ol_rat
    = ( ^ [P6: $o] : ( if_rat @ P6 @ one_one_rat @ zero_zero_rat ) ) ) ).

% of_bool_def
thf(fact_3712_of__bool__def,axiom,
    ( zero_n2687167440665602831ol_nat
    = ( ^ [P6: $o] : ( if_nat @ P6 @ one_one_nat @ zero_zero_nat ) ) ) ).

% of_bool_def
thf(fact_3713_of__bool__def,axiom,
    ( zero_n2684676970156552555ol_int
    = ( ^ [P6: $o] : ( if_int @ P6 @ one_one_int @ zero_zero_int ) ) ) ).

% of_bool_def
thf(fact_3714_of__bool__def,axiom,
    ( zero_n356916108424825756nteger
    = ( ^ [P6: $o] : ( if_Code_integer @ P6 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).

% of_bool_def
thf(fact_3715_split__of__bool,axiom,
    ! [P: rat > $o,P4: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
      = ( ( P4
         => ( P @ one_one_rat ) )
        & ( ~ P4
         => ( P @ zero_zero_rat ) ) ) ) ).

% split_of_bool
thf(fact_3716_split__of__bool,axiom,
    ! [P: nat > $o,P4: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
      = ( ( P4
         => ( P @ one_one_nat ) )
        & ( ~ P4
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_of_bool
thf(fact_3717_split__of__bool,axiom,
    ! [P: int > $o,P4: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
      = ( ( P4
         => ( P @ one_one_int ) )
        & ( ~ P4
         => ( P @ zero_zero_int ) ) ) ) ).

% split_of_bool
thf(fact_3718_split__of__bool,axiom,
    ! [P: code_integer > $o,P4: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P4 ) )
      = ( ( P4
         => ( P @ one_one_Code_integer ) )
        & ( ~ P4
         => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).

% split_of_bool
thf(fact_3719_split__of__bool__asm,axiom,
    ! [P: rat > $o,P4: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_rat ) )
            | ( ~ P4
              & ~ ( P @ zero_zero_rat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_3720_split__of__bool__asm,axiom,
    ! [P: nat > $o,P4: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_nat ) )
            | ( ~ P4
              & ~ ( P @ zero_zero_nat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_3721_split__of__bool__asm,axiom,
    ! [P: int > $o,P4: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_int ) )
            | ( ~ P4
              & ~ ( P @ zero_zero_int ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_3722_split__of__bool__asm,axiom,
    ! [P: code_integer > $o,P4: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_Code_integer ) )
            | ( ~ P4
              & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_3723_add__One,axiom,
    ! [X: num] :
      ( ( plus_plus_num @ X @ one )
      = ( inc @ X ) ) ).

% add_One
thf(fact_3724_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K4: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K4 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_3725_BitM__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ ( bitM @ N2 ) @ one )
      = ( bit0 @ N2 ) ) ).

% BitM_plus_one
thf(fact_3726_one__plus__BitM,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N2 ) )
      = ( bit0 @ N2 ) ) ).

% one_plus_BitM
thf(fact_3727_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_nat @ ( inc @ X ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).

% numeral_inc
thf(fact_3728_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_int @ ( inc @ X ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).

% numeral_inc
thf(fact_3729_numeral__inc,axiom,
    ! [X: num] :
      ( ( numera6620942414471956472nteger @ ( inc @ X ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer ) ) ).

% numeral_inc
thf(fact_3730_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_rat @ ( inc @ X ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).

% numeral_inc
thf(fact_3731_numeral__inc,axiom,
    ! [X: num] :
      ( ( numera5444537566228673987atural @ ( inc @ X ) )
      = ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ X ) @ one_one_Code_natural ) ) ).

% numeral_inc
thf(fact_3732_numeral__BitM,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bitM @ N2 ) )
      = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ one_one_int ) ) ).

% numeral_BitM
thf(fact_3733_numeral__BitM,axiom,
    ! [N2: num] :
      ( ( numera6620942414471956472nteger @ ( bitM @ N2 ) )
      = ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ one_one_Code_integer ) ) ).

% numeral_BitM
thf(fact_3734_numeral__BitM,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bitM @ N2 ) )
      = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N2 ) ) @ one_one_rat ) ) ).

% numeral_BitM
thf(fact_3735_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_bit1
thf(fact_3736_bits__induct,axiom,
    ! [P: code_natural > $o,A: code_natural] :
      ( ! [A3: code_natural] :
          ( ( ( divide5121882707175180666atural @ A3 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
            = A3 )
         => ( P @ A3 ) )
     => ( ! [A3: code_natural,B2: $o] :
            ( ( P @ A3 )
           => ( ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( zero_n8403883297036319079atural @ B2 ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A3 ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
                = A3 )
             => ( P @ ( plus_p4538020629002901425atural @ ( zero_n8403883297036319079atural @ B2 ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_3737_bits__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [A3: nat] :
          ( ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = A3 )
         => ( P @ A3 ) )
     => ( ! [A3: nat,B2: $o] :
            ( ( P @ A3 )
           => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                = A3 )
             => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_3738_bits__induct,axiom,
    ! [P: int > $o,A: int] :
      ( ! [A3: int] :
          ( ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = A3 )
         => ( P @ A3 ) )
     => ( ! [A3: int,B2: $o] :
            ( ( P @ A3 )
           => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = A3 )
             => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_3739_bits__induct,axiom,
    ! [P: code_integer > $o,A: code_integer] :
      ( ! [A3: code_integer] :
          ( ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = A3 )
         => ( P @ A3 ) )
     => ( ! [A3: code_integer,B2: $o] :
            ( ( P @ A3 )
           => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B2 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = A3 )
             => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B2 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_3740_divide__int__unfold,axiom,
    ! [L: int,K: int,N2: nat,M2: 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 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_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 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M2 @ N2 ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M2 @ N2 )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N2 @ M2 ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_3741_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ L ) @ ( numera6620942414471956472nteger @ ( bit1 @ K ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( pred_numeral @ L ) @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ).

% take_bit_numeral_bit1
thf(fact_3742_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_bit1
thf(fact_3743_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_numeral_bit1
thf(fact_3744_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se569199155075624693atural @ ( numeral_numeral_nat @ L ) @ ( numera5444537566228673987atural @ ( bit1 @ K ) ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( pred_numeral @ L ) @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ one_one_Code_natural ) ) ).

% take_bit_numeral_bit1
thf(fact_3745_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R2: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q3 @ R2 ) )
      = ( plus_plus_int @ Q3 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_3746_mask__numeral,axiom,
    ! [N2: num] :
      ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N2 ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N2 ) ) ) ) ) ).

% mask_numeral
thf(fact_3747_mask__numeral,axiom,
    ! [N2: num] :
      ( ( bit_se2119862282449309892nteger @ ( numeral_numeral_nat @ N2 ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ ( pred_numeral @ N2 ) ) ) ) ) ).

% mask_numeral
thf(fact_3748_mask__numeral,axiom,
    ! [N2: num] :
      ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N2 ) ) ) ) ) ).

% mask_numeral
thf(fact_3749_mask__numeral,axiom,
    ! [N2: num] :
      ( ( bit_se943457434206027407atural @ ( numeral_numeral_nat @ N2 ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se943457434206027407atural @ ( pred_numeral @ N2 ) ) ) ) ) ).

% mask_numeral
thf(fact_3750_and__int__unfold,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K4: int,L3: int] :
          ( if_int
          @ ( ( K4 = zero_zero_int )
            | ( L3 = zero_zero_int ) )
          @ zero_zero_int
          @ ( if_int
            @ ( K4
              = ( uminus_uminus_int @ one_one_int ) )
            @ L3
            @ ( if_int
              @ ( L3
                = ( uminus_uminus_int @ one_one_int ) )
              @ K4
              @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% and_int_unfold
thf(fact_3751_the__res__def,axiom,
    ( time_the_res_array_a
    = ( ^ [M: heap_T6362127782908069795rray_a,H: heap_e7401611519738050253t_unit] :
          ( case_o8314599424437789472it_nat @ undefined_array_a
          @ ( produc8767961711439910440rray_a
            @ ^ [R3: array_a] :
                ( produc2025300630104146186rray_a
                @ ^ [X3: heap_e7401611519738050253t_unit,Xa4: nat] : R3 ) )
          @ ( heap_T5591663578999710211rray_a @ M @ H ) ) ) ) ).

% the_res_def
thf(fact_3752_the__res__def,axiom,
    ( time_t2745601220510131320t_unit
    = ( ^ [M: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit] :
          ( case_o9196786658626276042it_nat @ undefi2974082611360736373t_unit
          @ ( produc5075907435489727646t_unit
            @ ^ [R3: product_unit] :
                ( produc1863717115126956122t_unit
                @ ^ [X3: heap_e7401611519738050253t_unit,Xa4: nat] : R3 ) )
          @ ( heap_T875086893843062177t_unit @ M @ H ) ) ) ) ).

% the_res_def
thf(fact_3753_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
        = ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).

% take_bit_eq_mask_iff_exp_dvd
thf(fact_3754_Code__Numeral_Opositive__def,axiom,
    code_positive = numera6620942414471956472nteger ).

% Code_Numeral.positive_def
thf(fact_3755_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K4: int,L3: int] :
          ( if_int @ ( L3 = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K4 )
              = ( sgn_sgn_int @ L3 ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K4 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K4 ) ) @ ( nat2 @ ( abs_abs_int @ L3 ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L3 @ K4 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_3756_sum__gp0,axiom,
    ! [X: rat,N2: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( semiri681578069525770553at_rat @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp0
thf(fact_3757_atMost__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( set_ord_atMost_nat @ X )
        = ( set_ord_atMost_nat @ Y ) )
      = ( X = Y ) ) ).

% atMost_eq_iff
thf(fact_3758_atMost__iff,axiom,
    ! [I: $o,K: $o] :
      ( ( member_o @ I @ ( set_ord_atMost_o @ K ) )
      = ( ord_less_eq_o @ I @ K ) ) ).

% atMost_iff
thf(fact_3759_atMost__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K ) )
      = ( ord_less_eq_set_nat @ I @ K ) ) ).

% atMost_iff
thf(fact_3760_atMost__iff,axiom,
    ! [I: rat,K: rat] :
      ( ( member_rat @ I @ ( set_ord_atMost_rat @ K ) )
      = ( ord_less_eq_rat @ I @ K ) ) ).

% atMost_iff
thf(fact_3761_atMost__iff,axiom,
    ! [I: num,K: num] :
      ( ( member_num @ I @ ( set_ord_atMost_num @ K ) )
      = ( ord_less_eq_num @ I @ K ) ) ).

% atMost_iff
thf(fact_3762_atMost__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_atMost_int @ K ) )
      = ( ord_less_eq_int @ I @ K ) ) ).

% atMost_iff
thf(fact_3763_atMost__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
      = ( ord_less_eq_nat @ I @ K ) ) ).

% atMost_iff
thf(fact_3764_atMost__subset__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X ) @ ( set_or4236626031148496127et_nat @ Y ) )
      = ( ord_less_eq_set_nat @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_3765_atMost__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X ) @ ( set_ord_atMost_rat @ Y ) )
      = ( ord_less_eq_rat @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_3766_atMost__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X ) @ ( set_ord_atMost_num @ Y ) )
      = ( ord_less_eq_num @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_3767_atMost__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_3768_atMost__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_3769_bit_Oconj__one__right,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ X @ ( uminus_uminus_int @ one_one_int ) )
      = X ) ).

% bit.conj_one_right
thf(fact_3770_bit_Oconj__one__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se3949692690581998587nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = X ) ).

% bit.conj_one_right
thf(fact_3771_and_Oright__neutral,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = A ) ).

% and.right_neutral
thf(fact_3772_and_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = A ) ).

% and.right_neutral
thf(fact_3773_and_Oleft__neutral,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
      = A ) ).

% and.left_neutral
thf(fact_3774_and_Oleft__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
      = A ) ).

% and.left_neutral
thf(fact_3775_and__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = one_one_int ) ).

% and_numerals(2)
thf(fact_3776_and__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = one_one_nat ) ).

% and_numerals(2)
thf(fact_3777_and__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = one_one_Code_integer ) ).

% and_numerals(2)
thf(fact_3778_and__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se2773287842338716102atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = one_one_Code_natural ) ).

% and_numerals(2)
thf(fact_3779_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
      = one_one_int ) ).

% and_numerals(8)
thf(fact_3780_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
      = one_one_nat ) ).

% and_numerals(8)
thf(fact_3781_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% and_numerals(8)
thf(fact_3782_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ one_one_Code_natural )
      = one_one_Code_natural ) ).

% and_numerals(8)
thf(fact_3783_Icc__subset__Iic__iff,axiom,
    ! [L: set_nat,H3: set_nat,H5: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ L @ H3 ) @ ( set_or4236626031148496127et_nat @ H5 ) )
      = ( ~ ( ord_less_eq_set_nat @ L @ H3 )
        | ( ord_less_eq_set_nat @ H3 @ H5 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_3784_Icc__subset__Iic__iff,axiom,
    ! [L: rat,H3: rat,H5: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H3 ) @ ( set_ord_atMost_rat @ H5 ) )
      = ( ~ ( ord_less_eq_rat @ L @ H3 )
        | ( ord_less_eq_rat @ H3 @ H5 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_3785_Icc__subset__Iic__iff,axiom,
    ! [L: num,H3: num,H5: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L @ H3 ) @ ( set_ord_atMost_num @ H5 ) )
      = ( ~ ( ord_less_eq_num @ L @ H3 )
        | ( ord_less_eq_num @ H3 @ H5 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_3786_Icc__subset__Iic__iff,axiom,
    ! [L: int,H3: int,H5: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H3 ) @ ( set_ord_atMost_int @ H5 ) )
      = ( ~ ( ord_less_eq_int @ L @ H3 )
        | ( ord_less_eq_int @ H3 @ H5 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_3787_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H3: nat,H5: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H3 ) @ ( set_ord_atMost_nat @ H5 ) )
      = ( ~ ( ord_less_eq_nat @ L @ H3 )
        | ( ord_less_eq_nat @ H3 @ H5 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_3788_Icc__subset__Iic__iff,axiom,
    ! [L: code_integer,H3: code_integer,H5: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or189985376899183464nteger @ L @ H3 ) @ ( set_or9101266186257409494nteger @ H5 ) )
      = ( ~ ( ord_le3102999989581377725nteger @ L @ H3 )
        | ( ord_le3102999989581377725nteger @ H3 @ H5 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_3789_mask__Suc__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% mask_Suc_0
thf(fact_3790_mask__Suc__0,axiom,
    ( ( bit_se2119862282449309892nteger @ ( suc @ zero_zero_nat ) )
    = one_one_Code_integer ) ).

% mask_Suc_0
thf(fact_3791_mask__Suc__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% mask_Suc_0
thf(fact_3792_mask__Suc__0,axiom,
    ( ( bit_se943457434206027407atural @ ( suc @ zero_zero_nat ) )
    = one_one_Code_natural ) ).

% mask_Suc_0
thf(fact_3793_sum_OatMost__Suc,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_3794_sum_OatMost__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_3795_sum_OatMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_3796_sum_OatMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_3797_take__bit__minus__one__eq__mask,axiom,
    ! [N2: nat] :
      ( ( bit_se1745604003318907178nteger @ N2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( bit_se2119862282449309892nteger @ N2 ) ) ).

% take_bit_minus_one_eq_mask
thf(fact_3798_take__bit__minus__one__eq__mask,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( bit_se2000444600071755411sk_int @ N2 ) ) ).

% take_bit_minus_one_eq_mask
thf(fact_3799_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = zero_zero_int ) ).

% and_numerals(1)
thf(fact_3800_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = zero_zero_nat ) ).

% and_numerals(1)
thf(fact_3801_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = zero_z3403309356797280102nteger ) ).

% and_numerals(1)
thf(fact_3802_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se2773287842338716102atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = zero_z2226904508553997617atural ) ).

% and_numerals(1)
thf(fact_3803_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_numerals(5)
thf(fact_3804_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
      = zero_zero_nat ) ).

% and_numerals(5)
thf(fact_3805_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% and_numerals(5)
thf(fact_3806_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ one_one_Code_natural )
      = zero_z2226904508553997617atural ) ).

% and_numerals(5)
thf(fact_3807_nat__numeral__diff__1,axiom,
    ! [V: num] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
      = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).

% nat_numeral_diff_1
thf(fact_3808_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_3809_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_3810_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_3811_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_3812_mask__integer_Oabs__eq,axiom,
    ( bit_se2119862282449309892nteger
    = ( ^ [X3: nat] : ( code_integer_of_int @ ( bit_se2000444600071755411sk_int @ X3 ) ) ) ) ).

% mask_integer.abs_eq
thf(fact_3813_not__Iic__eq__Icc,axiom,
    ! [H5: int,L: int,H3: int] :
      ( ( set_ord_atMost_int @ H5 )
     != ( set_or1266510415728281911st_int @ L @ H3 ) ) ).

% not_Iic_eq_Icc
thf(fact_3814_atMost__def,axiom,
    ( set_or4236626031148496127et_nat
    = ( ^ [U2: set_nat] :
          ( collect_set_nat
          @ ^ [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_3815_atMost__def,axiom,
    ( set_ord_atMost_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X3: rat] : ( ord_less_eq_rat @ X3 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_3816_atMost__def,axiom,
    ( set_ord_atMost_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X3: num] : ( ord_less_eq_num @ X3 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_3817_atMost__def,axiom,
    ( set_ord_atMost_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X3: int] : ( ord_less_eq_int @ X3 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_3818_atMost__def,axiom,
    ( set_ord_atMost_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X3: nat] : ( ord_less_eq_nat @ X3 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_3819_and__eq__minus__1__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( bit_se725231765392027082nd_int @ A @ B )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( ( A
          = ( uminus_uminus_int @ one_one_int ) )
        & ( B
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% and_eq_minus_1_iff
thf(fact_3820_and__eq__minus__1__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( bit_se3949692690581998587nteger @ A @ B )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( ( A
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
        & ( B
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% and_eq_minus_1_iff
thf(fact_3821_atMost__atLeast0,axiom,
    ( set_ord_atMost_nat
    = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).

% atMost_atLeast0
thf(fact_3822_nat__one__as__int,axiom,
    ( one_one_nat
    = ( nat2 @ one_one_int ) ) ).

% nat_one_as_int
thf(fact_3823_not__Iic__le__Icc,axiom,
    ! [H3: int,L4: int,H5: int] :
      ~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H3 ) @ ( set_or1266510415728281911st_int @ L4 @ H5 ) ) ).

% not_Iic_le_Icc
thf(fact_3824_nat__int__add,axiom,
    ! [A: nat,B: nat] :
      ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
      = ( plus_plus_nat @ A @ B ) ) ).

% nat_int_add
thf(fact_3825_option_Othe__def,axiom,
    ( the_num
    = ( case_option_num_num @ undefined_num
      @ ^ [X23: num] : X23 ) ) ).

% option.the_def
thf(fact_3826_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_3827_nat__add__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
       => ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
          = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_3828_Suc__as__int,axiom,
    ( suc
    = ( ^ [A5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_3829_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 ) ) ) ) ).

% nat_abs_triangle_ineq
thf(fact_3830_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_p7104986032573967614at_nat @ ( G @ zero_zero_nat )
        @ ( groups6857163185585827899at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_3831_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_3832_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_3833_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_3834_sum__telescope,axiom,
    ! [F: nat > code_integer,I: nat] :
      ( ( groups7501900531339628137nteger
        @ ^ [I3: nat] : ( minus_8373710615458151222nteger @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_8373710615458151222nteger @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_3835_sum__telescope,axiom,
    ! [F: nat > rat,I: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_3836_sum__telescope,axiom,
    ! [F: nat > int,I: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( minus_minus_int @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).

% sum_telescope
thf(fact_3837_the__heap__def,axiom,
    ( time_t490086488869527696rray_a
    = ( ^ [M: heap_T6362127782908069795rray_a,H: heap_e7401611519738050253t_unit] :
          ( case_o8180226181140434498it_nat @ undefi8805113437130903861t_unit
          @ ( produc7016084256669755260t_unit
            @ ^ [X3: array_a] :
                ( produc9092781844693468442t_unit
                @ ^ [H6: heap_e7401611519738050253t_unit,Xa4: nat] : H6 ) )
          @ ( heap_T5591663578999710211rray_a @ M @ H ) ) ) ) ).

% the_heap_def
thf(fact_3838_the__heap__def,axiom,
    ( time_t3350536682685678164t_unit
    = ( ^ [M: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit] :
          ( case_o2566993838309857162it_nat @ undefi8805113437130903861t_unit
          @ ( produc6568152436032731486t_unit
            @ ^ [X3: product_unit] :
                ( produc9092781844693468442t_unit
                @ ^ [H6: heap_e7401611519738050253t_unit,Xa4: nat] : H6 ) )
          @ ( heap_T875086893843062177t_unit @ M @ H ) ) ) ) ).

% the_heap_def
thf(fact_3839_and__one__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ one_one_int )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_3840_and__one__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ one_one_nat )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_3841_and__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_3842_and__one__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se2773287842338716102atural @ A @ one_one_Code_natural )
      = ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_3843_one__and__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ A )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_3844_one__and__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ A )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_3845_one__and__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_3846_one__and__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se2773287842338716102atural @ one_one_Code_natural @ A )
      = ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_3847_take__bit__eq__mask__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
        = ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ K @ one_one_int ) )
        = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
thf(fact_3848_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( suc @ ( nat2 @ Z ) )
        = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).

% Suc_nat_eq_nat_zadd1
thf(fact_3849_Divides_Oadjust__div__def,axiom,
    ( adjust_div
    = ( produc8211389475949308722nt_int
      @ ^ [Q5: int,R3: int] : ( plus_plus_int @ Q5 @ ( zero_n2684676970156552555ol_int @ ( R3 != zero_zero_int ) ) ) ) ) ).

% Divides.adjust_div_def
thf(fact_3850_sum__gp__basic,axiom,
    ! [X: code_integer,N2: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ X @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_3851_sum__gp__basic,axiom,
    ! [X: rat,N2: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_3852_sum__gp__basic,axiom,
    ! [X: int,N2: nat] :
      ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_3853_sum__power__shift,axiom,
    ! [M2: nat,N2: nat,X: rat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( times_times_rat @ ( power_power_rat @ X @ M2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_3854_sum__power__shift,axiom,
    ! [M2: nat,N2: nat,X: int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( times_times_int @ ( power_power_int @ X @ M2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_3855_sum_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J2 ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K4: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K4 @ I3 ) )
            @ ( set_ord_atMost_nat @ K4 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.triangle_reindex_eq
thf(fact_3856_and__int__rec,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K4: int,L3: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K4 )
              & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_int_rec
thf(fact_3857_mask__half__int,axiom,
    ! [N2: nat] :
      ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).

% mask_half_int
thf(fact_3858_sum_Oin__pairs__0,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6857163185585827899at_nat
        @ ^ [I3: nat] : ( plus_p7104986032573967614at_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_3859_sum_Oin__pairs__0,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_3860_sum_Oin__pairs__0,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_3861_sum_Oin__pairs__0,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_3862_mask__nat__def,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N4: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) ).

% mask_nat_def
thf(fact_3863_mask__eq__exp__minus__1,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N4: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) @ one_one_int ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_3864_mask__eq__exp__minus__1,axiom,
    ( bit_se2119862282449309892nteger
    = ( ^ [N4: nat] : ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N4 ) @ one_one_Code_integer ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_3865_mask__eq__exp__minus__1,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N4: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_3866_mask__eq__exp__minus__1,axiom,
    ( bit_se943457434206027407atural
    = ( ^ [N4: nat] : ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N4 ) @ one_one_Code_natural ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_3867_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > code_integer,H3: nat > code_integer] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups7501900531339628137nteger
            @ ^ [J2: nat] : ( if_Code_integer @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_Code_integer @ ( J2 = K ) @ zero_z3403309356797280102nteger @ ( H3 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups7501900531339628137nteger
            @ ^ [J2: nat] : ( if_Code_integer @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H3 @ J2 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_3868_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > multis2468970476368604999at_nat,H3: nat > multis2468970476368604999at_nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups6857163185585827899at_nat
            @ ^ [J2: nat] : ( if_mul8430962117462786573at_nat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_mul8430962117462786573at_nat @ ( J2 = K ) @ zero_z1048942125864253310at_nat @ ( H3 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups6857163185585827899at_nat
            @ ^ [J2: nat] : ( if_mul8430962117462786573at_nat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H3 @ J2 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_3869_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > rat,H3: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [J2: nat] : ( if_rat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_rat @ ( J2 = K ) @ zero_zero_rat @ ( H3 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups2906978787729119204at_rat
            @ ^ [J2: nat] : ( if_rat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H3 @ J2 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_3870_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > int,H3: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups3539618377306564664at_int
            @ ^ [J2: nat] : ( if_int @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_int @ ( J2 = K ) @ zero_zero_int @ ( H3 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups3539618377306564664at_int
            @ ^ [J2: nat] : ( if_int @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H3 @ J2 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_3871_sum_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > nat,H3: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_nat @ ( J2 = K ) @ zero_zero_nat @ ( H3 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups3542108847815614940at_nat
            @ ^ [J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H3 @ J2 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_3872_and__minus__numerals_I3_J,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ ( bitM @ N2 ) ) ) ) ).

% and_minus_numerals(3)
thf(fact_3873_and__minus__numerals_I7_J,axiom,
    ! [N2: num,M2: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ ( numeral_numeral_int @ M2 ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ ( bitM @ N2 ) ) ) ) ).

% and_minus_numerals(7)
thf(fact_3874_and__minus__numerals_I4_J,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ ( bit0 @ N2 ) ) ) ) ).

% and_minus_numerals(4)
thf(fact_3875_and__minus__numerals_I8_J,axiom,
    ! [N2: num,M2: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ ( numeral_numeral_int @ M2 ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ ( bit0 @ N2 ) ) ) ) ).

% and_minus_numerals(8)
thf(fact_3876_gbinomial__partial__row__sum,axiom,
    ! [A: rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K4: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K4 ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K4 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ one_one_rat ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ A @ ( plus_plus_nat @ M2 @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_3877_gbinomial__r__part__sum,axiom,
    ! [M2: nat] :
      ( ( groups2906978787729119204at_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M2 ) ) @ one_one_rat ) ) @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% gbinomial_r_part_sum
thf(fact_3878_and__num_Oelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa = one )
           => ( Y
             != ( some_num @ one ) ) ) )
       => ( ( ( X = one )
           => ( ? [N: num] :
                  ( Xa
                  = ( bit0 @ N ) )
             => ( Y != none_num ) ) )
         => ( ( ( X = one )
             => ( ? [N: num] :
                    ( Xa
                    = ( bit1 @ N ) )
               => ( Y
                 != ( some_num @ one ) ) ) )
           => ( ( ? [M5: num] :
                    ( X
                    = ( bit0 @ M5 ) )
               => ( ( Xa = one )
                 => ( Y != none_num ) ) )
             => ( ! [M5: num] :
                    ( ( X
                      = ( bit0 @ M5 ) )
                   => ! [N: num] :
                        ( ( Xa
                          = ( bit0 @ N ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X
                        = ( bit0 @ M5 ) )
                     => ! [N: num] :
                          ( ( Xa
                            = ( bit1 @ N ) )
                         => ( Y
                           != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N ) ) ) ) )
                 => ( ( ? [M5: num] :
                          ( X
                          = ( bit1 @ M5 ) )
                     => ( ( Xa = one )
                       => ( Y
                         != ( some_num @ one ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X
                            = ( bit1 @ M5 ) )
                         => ! [N: num] :
                              ( ( Xa
                                = ( bit0 @ N ) )
                             => ( Y
                               != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N ) ) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X
                              = ( bit1 @ M5 ) )
                           => ! [N: num] :
                                ( ( Xa
                                  = ( bit1 @ N ) )
                               => ( Y
                                 != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                    @ ^ [N8: num] : ( some_num @ ( bit1 @ N8 ) )
                                    @ ( bit_un7362597486090784418nd_num @ M5 @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
thf(fact_3879_take__bit__Suc__from__most,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K )
      = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_Suc_from_most
thf(fact_3880_gbinomial__0_I1_J,axiom,
    ! [A: code_integer] :
      ( ( gbinom8545251970709558553nteger @ A @ zero_zero_nat )
      = one_one_Code_integer ) ).

% gbinomial_0(1)
thf(fact_3881_gbinomial__0_I1_J,axiom,
    ! [A: rat] :
      ( ( gbinomial_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% gbinomial_0(1)
thf(fact_3882_gbinomial__0_I1_J,axiom,
    ! [A: nat] :
      ( ( gbinomial_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% gbinomial_0(1)
thf(fact_3883_gbinomial__0_I1_J,axiom,
    ! [A: int] :
      ( ( gbinomial_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% gbinomial_0(1)
thf(fact_3884_and__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = one_one_nat ) ).

% and_nat_numerals(2)
thf(fact_3885_and__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% and_nat_numerals(4)
thf(fact_3886_bit__disjunctive__add__iff,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ! [N: nat] :
          ( ~ ( bit_se1146084159140164899it_int @ A @ N )
          | ~ ( bit_se1146084159140164899it_int @ B @ N ) )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B ) @ N2 )
        = ( ( bit_se1146084159140164899it_int @ A @ N2 )
          | ( bit_se1146084159140164899it_int @ B @ N2 ) ) ) ) ).

% bit_disjunctive_add_iff
thf(fact_3887_bit__disjunctive__add__iff,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] :
      ( ! [N: nat] :
          ( ~ ( bit_se9216721137139052372nteger @ A @ N )
          | ~ ( bit_se9216721137139052372nteger @ B @ N ) )
     => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ N2 )
        = ( ( bit_se9216721137139052372nteger @ A @ N2 )
          | ( bit_se9216721137139052372nteger @ B @ N2 ) ) ) ) ).

% bit_disjunctive_add_iff
thf(fact_3888_bit__disjunctive__add__iff,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ! [N: nat] :
          ( ~ ( bit_se1148574629649215175it_nat @ A @ N )
          | ~ ( bit_se1148574629649215175it_nat @ B @ N ) )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
        = ( ( bit_se1148574629649215175it_nat @ A @ N2 )
          | ( bit_se1148574629649215175it_nat @ B @ N2 ) ) ) ) ).

% bit_disjunctive_add_iff
thf(fact_3889_bit__disjunctive__add__iff,axiom,
    ! [A: code_natural,B: code_natural,N2: nat] :
      ( ! [N: nat] :
          ( ~ ( bit_se8040316288895769887atural @ A @ N )
          | ~ ( bit_se8040316288895769887atural @ B @ N ) )
     => ( ( bit_se8040316288895769887atural @ ( plus_p4538020629002901425atural @ A @ B ) @ N2 )
        = ( ( bit_se8040316288895769887atural @ A @ N2 )
          | ( bit_se8040316288895769887atural @ B @ N2 ) ) ) ) ).

% bit_disjunctive_add_iff
thf(fact_3890_bit__1__iff,axiom,
    ! [N2: nat] :
      ( ( bit_se1146084159140164899it_int @ one_one_int @ N2 )
      = ( N2 = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_3891_bit__1__iff,axiom,
    ! [N2: nat] :
      ( ( bit_se9216721137139052372nteger @ one_one_Code_integer @ N2 )
      = ( N2 = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_3892_bit__1__iff,axiom,
    ! [N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N2 )
      = ( N2 = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_3893_bit__1__iff,axiom,
    ! [N2: nat] :
      ( ( bit_se8040316288895769887atural @ one_one_Code_natural @ N2 )
      = ( N2 = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_3894_not__bit__1__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N2 ) ) ).

% not_bit_1_Suc
thf(fact_3895_not__bit__1__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se9216721137139052372nteger @ one_one_Code_integer @ ( suc @ N2 ) ) ).

% not_bit_1_Suc
thf(fact_3896_not__bit__1__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N2 ) ) ).

% not_bit_1_Suc
thf(fact_3897_not__bit__1__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se8040316288895769887atural @ one_one_Code_natural @ ( suc @ N2 ) ) ).

% not_bit_1_Suc
thf(fact_3898_bit__numeral__simps_I1_J,axiom,
    ! [N2: num] :
      ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( numeral_numeral_nat @ N2 ) ) ).

% bit_numeral_simps(1)
thf(fact_3899_bit__numeral__simps_I1_J,axiom,
    ! [N2: num] :
      ~ ( bit_se9216721137139052372nteger @ one_one_Code_integer @ ( numeral_numeral_nat @ N2 ) ) ).

% bit_numeral_simps(1)
thf(fact_3900_bit__numeral__simps_I1_J,axiom,
    ! [N2: num] :
      ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% bit_numeral_simps(1)
thf(fact_3901_bit__numeral__simps_I1_J,axiom,
    ! [N2: num] :
      ~ ( bit_se8040316288895769887atural @ one_one_Code_natural @ ( numeral_numeral_nat @ N2 ) ) ).

% bit_numeral_simps(1)
thf(fact_3902_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_3903_and__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( ( bit_se3949692690581998587nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se725231765392027082nd_int @ Xa @ X ) ) ) ).

% and_integer.abs_eq
thf(fact_3904_gbinomial__addition__formula,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_3905_gbinomial__absorb__comp,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_3906_gbinomial__mult__1_H,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_3907_gbinomial__mult__1,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_3908_Suc__times__gbinomial,axiom,
    ! [K: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_3909_gbinomial__absorption,axiom,
    ! [K: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_3910_gbinomial__parallel__sum,axiom,
    ! [A: rat,N2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K4: nat] : ( gbinomial_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K4 ) ) @ K4 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( gbinomial_rat @ ( plus_plus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N2 ) ) @ one_one_rat ) @ N2 ) ) ).

% gbinomial_parallel_sum
thf(fact_3911_gbinomial__rec,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_3912_gbinomial__factors,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_3913_gbinomial__negated__upper,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K4: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K4 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K4 ) @ A5 ) @ one_one_rat ) @ K4 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_3914_gbinomial__index__swap,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ one_one_rat ) @ K ) )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ N2 ) ) ) ).

% gbinomial_index_swap
thf(fact_3915_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M6: nat] :
              ( ( ord_less_eq_nat @ N @ M6 )
             => ( ( bit_se1146084159140164899it_int @ K @ M6 )
                = ( bit_se1146084159140164899it_int @ K @ N ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N )
             => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ) ) ).

% int_bit_bound
thf(fact_3916_gbinomial__minus,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_3917_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A @ K )
        = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_3918_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K4: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K4 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K4 ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ M2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ M2 ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_3919_numeral__and__num,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) ) ).

% numeral_and_num
thf(fact_3920_numeral__and__num,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
      = ( case_option_nat_num @ zero_zero_nat @ numeral_numeral_nat @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) ) ).

% numeral_and_num
thf(fact_3921_numeral__and__num,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( case_o356765784539232260er_num @ zero_z3403309356797280102nteger @ numera6620942414471956472nteger @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) ) ).

% numeral_and_num
thf(fact_3922_numeral__and__num,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ M2 ) @ ( numera5444537566228673987atural @ N2 ) )
      = ( case_o5621594795226839503al_num @ zero_z2226904508553997617atural @ numera5444537566228673987atural @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) ) ).

% numeral_and_num
thf(fact_3923_gbinomial__sum__up__index,axiom,
    ! [K: nat,N2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [J2: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J2 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_3924_gbinomial__partial__sum__poly,axiom,
    ! [M2: nat,A: rat,X: rat,Y: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K4: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ A ) @ K4 ) @ ( power_power_rat @ X @ K4 ) ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ M2 @ K4 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( groups2906978787729119204at_rat
        @ ^ [K4: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K4 ) @ ( power_power_rat @ ( uminus_uminus_rat @ X ) @ K4 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_nat @ M2 @ K4 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_3925_even__bit__succ__iff,axiom,
    ! [A: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N2 )
        = ( ( bit_se1146084159140164899it_int @ A @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_3926_even__bit__succ__iff,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ N2 )
        = ( ( bit_se9216721137139052372nteger @ A @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_3927_even__bit__succ__iff,axiom,
    ! [A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N2 )
        = ( ( bit_se1148574629649215175it_nat @ A @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_3928_even__bit__succ__iff,axiom,
    ! [A: code_natural,N2: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se8040316288895769887atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ N2 )
        = ( ( bit_se8040316288895769887atural @ A @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_3929_odd__bit__iff__bit__pred,axiom,
    ! [A: int,N2: nat] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1146084159140164899it_int @ A @ N2 )
        = ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A @ one_one_int ) @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_3930_odd__bit__iff__bit__pred,axiom,
    ! [A: code_integer,N2: nat] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se9216721137139052372nteger @ A @ N2 )
        = ( ( bit_se9216721137139052372nteger @ ( minus_8373710615458151222nteger @ A @ one_one_Code_integer ) @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_3931_odd__bit__iff__bit__pred,axiom,
    ! [A: nat,N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1148574629649215175it_nat @ A @ N2 )
        = ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A @ one_one_nat ) @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_3932_odd__bit__iff__bit__pred,axiom,
    ! [A: code_natural,N2: nat] :
      ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se8040316288895769887atural @ A @ N2 )
        = ( ( bit_se8040316288895769887atural @ ( minus_7197305767214868737atural @ A @ one_one_Code_natural ) @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_3933_and__num_Osimps_I9_J,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N8: num] : ( some_num @ ( bit1 @ N8 ) )
        @ ( bit_un7362597486090784418nd_num @ M2 @ N2 ) ) ) ).

% and_num.simps(9)
thf(fact_3934_gbinomial__absorption_H,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A @ K )
        = ( times_times_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_3935_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M2: nat,A: rat,X: rat,Y: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K4: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ A ) @ K4 ) @ ( power_power_rat @ X @ K4 ) ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ M2 @ K4 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( groups2906978787729119204at_rat
        @ ^ [K4: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ K4 ) @ A ) @ one_one_rat ) @ K4 ) @ ( power_power_rat @ X @ K4 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_nat @ M2 @ K4 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_3936_gbinomial__sum__nat__pow2,axiom,
    ! [M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K4: nat] : ( divide_divide_rat @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M2 @ K4 ) ) @ K4 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ K4 ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ M2 ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_3937_bit__sum__mult__2__cases,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ! [J3: nat] :
          ~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J3 ) )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ N2 )
        = ( ( ( N2 = zero_zero_nat )
           => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
          & ( ( N2 != zero_zero_nat )
           => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_3938_bit__sum__mult__2__cases,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] :
      ( ! [J3: nat] :
          ~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J3 ) )
     => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ N2 )
        = ( ( ( N2 = zero_zero_nat )
           => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
          & ( ( N2 != zero_zero_nat )
           => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_3939_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ! [J3: nat] :
          ~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J3 ) )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ N2 )
        = ( ( ( N2 = zero_zero_nat )
           => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
          & ( ( N2 != zero_zero_nat )
           => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_3940_bit__sum__mult__2__cases,axiom,
    ! [A: code_natural,B: code_natural,N2: nat] :
      ( ! [J3: nat] :
          ~ ( bit_se8040316288895769887atural @ A @ ( suc @ J3 ) )
     => ( ( bit_se8040316288895769887atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B ) ) @ N2 )
        = ( ( ( N2 = zero_zero_nat )
           => ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) )
          & ( ( N2 != zero_zero_nat )
           => ( bit_se8040316288895769887atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_3941_and__not__num_Osimps_I8_J,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_and_not_num @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N8: num] : ( some_num @ ( bit1 @ N8 ) )
        @ ( bit_and_not_num @ M2 @ N2 ) ) ) ).

% and_not_num.simps(8)
thf(fact_3942_bit__rec,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [A5: int,N4: nat] :
          ( ( ( N4 = zero_zero_nat )
           => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) )
          & ( ( N4 != zero_zero_nat )
           => ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_3943_bit__rec,axiom,
    ( bit_se9216721137139052372nteger
    = ( ^ [A5: code_integer,N4: nat] :
          ( ( ( N4 = zero_zero_nat )
           => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) )
          & ( ( N4 != zero_zero_nat )
           => ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_3944_bit__rec,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [A5: nat,N4: nat] :
          ( ( ( N4 = zero_zero_nat )
           => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) )
          & ( ( N4 != zero_zero_nat )
           => ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_3945_bit__rec,axiom,
    ( bit_se8040316288895769887atural
    = ( ^ [A5: code_natural,N4: nat] :
          ( ( ( N4 = zero_zero_nat )
           => ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A5 ) )
          & ( ( N4 != zero_zero_nat )
           => ( bit_se8040316288895769887atural @ ( divide5121882707175180666atural @ A5 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_3946_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M: nat,N4: nat] :
          ( if_nat
          @ ( ( M = zero_zero_nat )
            | ( N4 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_3947_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M: nat,N4: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_3948_set__bit__eq,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N4: nat,K4: int] :
          ( plus_plus_int @ K4
          @ ( times_times_int
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( bit_se1146084159140164899it_int @ K4 @ N4 ) )
            @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).

% set_bit_eq
thf(fact_3949_and__not__num_Oelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( ( bit_and_not_num @ X @ Xa )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ( ? [N: num] :
                  ( Xa
                  = ( bit0 @ N ) )
             => ( Y
               != ( some_num @ one ) ) ) )
         => ( ( ( X = one )
             => ( ? [N: num] :
                    ( Xa
                    = ( bit1 @ N ) )
               => ( Y != none_num ) ) )
           => ( ! [M5: num] :
                  ( ( X
                    = ( bit0 @ M5 ) )
                 => ( ( Xa = one )
                   => ( Y
                     != ( some_num @ ( bit0 @ M5 ) ) ) ) )
             => ( ! [M5: num] :
                    ( ( X
                      = ( bit0 @ M5 ) )
                   => ! [N: num] :
                        ( ( Xa
                          = ( bit0 @ N ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X
                        = ( bit0 @ M5 ) )
                     => ! [N: num] :
                          ( ( Xa
                            = ( bit1 @ N ) )
                         => ( Y
                           != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N ) ) ) ) )
                 => ( ! [M5: num] :
                        ( ( X
                          = ( bit1 @ M5 ) )
                       => ( ( Xa = one )
                         => ( Y
                           != ( some_num @ ( bit0 @ M5 ) ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X
                            = ( bit1 @ M5 ) )
                         => ! [N: num] :
                              ( ( Xa
                                = ( bit0 @ N ) )
                             => ( Y
                               != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                  @ ^ [N8: num] : ( some_num @ ( bit1 @ N8 ) )
                                  @ ( bit_and_not_num @ M5 @ N ) ) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X
                              = ( bit1 @ M5 ) )
                           => ! [N: num] :
                                ( ( Xa
                                  = ( bit1 @ N ) )
                               => ( Y
                                 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
thf(fact_3950_and__not__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( ( bit_and_not_num @ X @ Xa )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X @ Xa ) )
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = none_num )
               => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X = one )
             => ! [N: num] :
                  ( ( Xa
                    = ( bit0 @ N ) )
                 => ( ( Y
                      = ( some_num @ one ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N ) ) ) ) ) )
           => ( ( ( X = one )
               => ! [N: num] :
                    ( ( Xa
                      = ( bit1 @ N ) )
                   => ( ( Y = none_num )
                     => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N ) ) ) ) ) )
             => ( ! [M5: num] :
                    ( ( X
                      = ( bit0 @ M5 ) )
                   => ( ( Xa = one )
                     => ( ( Y
                          = ( some_num @ ( bit0 @ M5 ) ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X
                        = ( bit0 @ M5 ) )
                     => ! [N: num] :
                          ( ( Xa
                            = ( bit0 @ N ) )
                         => ( ( Y
                              = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N ) ) ) ) ) )
                 => ( ! [M5: num] :
                        ( ( X
                          = ( bit0 @ M5 ) )
                       => ! [N: num] :
                            ( ( Xa
                              = ( bit1 @ N ) )
                           => ( ( Y
                                = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N ) ) ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X
                            = ( bit1 @ M5 ) )
                         => ( ( Xa = one )
                           => ( ( Y
                                = ( some_num @ ( bit0 @ M5 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
                     => ( ! [M5: num] :
                            ( ( X
                              = ( bit1 @ M5 ) )
                           => ! [N: num] :
                                ( ( Xa
                                  = ( bit0 @ N ) )
                               => ( ( Y
                                    = ( case_o6005452278849405969um_num @ ( some_num @ one )
                                      @ ^ [N8: num] : ( some_num @ ( bit1 @ N8 ) )
                                      @ ( bit_and_not_num @ M5 @ N ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N ) ) ) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X
                                = ( bit1 @ M5 ) )
                             => ! [N: num] :
                                  ( ( Xa
                                    = ( bit1 @ N ) )
                                 => ( ( Y
                                      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
thf(fact_3951_and__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X @ Xa ) )
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y
                  = ( some_num @ one ) )
               => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X = one )
             => ! [N: num] :
                  ( ( Xa
                    = ( bit0 @ N ) )
                 => ( ( Y = none_num )
                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N ) ) ) ) ) )
           => ( ( ( X = one )
               => ! [N: num] :
                    ( ( Xa
                      = ( bit1 @ N ) )
                   => ( ( Y
                        = ( some_num @ one ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N ) ) ) ) ) )
             => ( ! [M5: num] :
                    ( ( X
                      = ( bit0 @ M5 ) )
                   => ( ( Xa = one )
                     => ( ( Y = none_num )
                       => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X
                        = ( bit0 @ M5 ) )
                     => ! [N: num] :
                          ( ( Xa
                            = ( bit0 @ N ) )
                         => ( ( Y
                              = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N ) ) ) ) ) )
                 => ( ! [M5: num] :
                        ( ( X
                          = ( bit0 @ M5 ) )
                       => ! [N: num] :
                            ( ( Xa
                              = ( bit1 @ N ) )
                           => ( ( Y
                                = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N ) ) ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X
                            = ( bit1 @ M5 ) )
                         => ( ( Xa = one )
                           => ( ( Y
                                = ( some_num @ one ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
                     => ( ! [M5: num] :
                            ( ( X
                              = ( bit1 @ M5 ) )
                           => ! [N: num] :
                                ( ( Xa
                                  = ( bit0 @ N ) )
                               => ( ( Y
                                    = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N ) ) ) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X
                                = ( bit1 @ M5 ) )
                             => ! [N: num] :
                                  ( ( Xa
                                    = ( bit1 @ N ) )
                                 => ( ( Y
                                      = ( case_o6005452278849405969um_num @ ( some_num @ one )
                                        @ ^ [N8: num] : ( some_num @ ( bit1 @ N8 ) )
                                        @ ( bit_un7362597486090784418nd_num @ M5 @ N ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
thf(fact_3952_gbinomial__code,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K4: nat] :
          ( if_rat @ ( K4 = zero_zero_nat ) @ one_one_rat
          @ ( divide_divide_rat
            @ ( set_fo1949268297981939178at_rat
              @ ^ [L3: nat] : ( times_times_rat @ ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ L3 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K4 @ one_one_nat )
              @ one_one_rat )
            @ ( semiri773545260158071498ct_rat @ K4 ) ) ) ) ) ).

% gbinomial_code
thf(fact_3953_xor__num_Oelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ! [N: num] :
                ( ( Xa
                  = ( bit0 @ N ) )
               => ( Y
                 != ( some_num @ ( bit1 @ N ) ) ) ) )
         => ( ( ( X = one )
             => ! [N: num] :
                  ( ( Xa
                    = ( bit1 @ N ) )
                 => ( Y
                   != ( some_num @ ( bit0 @ N ) ) ) ) )
           => ( ! [M5: num] :
                  ( ( X
                    = ( bit0 @ M5 ) )
                 => ( ( Xa = one )
                   => ( Y
                     != ( some_num @ ( bit1 @ M5 ) ) ) ) )
             => ( ! [M5: num] :
                    ( ( X
                      = ( bit0 @ M5 ) )
                   => ! [N: num] :
                        ( ( Xa
                          = ( bit0 @ N ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X
                        = ( bit0 @ M5 ) )
                     => ! [N: num] :
                          ( ( Xa
                            = ( bit1 @ N ) )
                         => ( Y
                           != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N ) ) ) ) ) )
                 => ( ! [M5: num] :
                        ( ( X
                          = ( bit1 @ M5 ) )
                       => ( ( Xa = one )
                         => ( Y
                           != ( some_num @ ( bit0 @ M5 ) ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X
                            = ( bit1 @ M5 ) )
                         => ! [N: num] :
                              ( ( Xa
                                = ( bit0 @ N ) )
                             => ( Y
                               != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N ) ) ) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X
                              = ( bit1 @ M5 ) )
                           => ! [N: num] :
                                ( ( Xa
                                  = ( bit1 @ N ) )
                               => ( Y
                                 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
thf(fact_3954_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I3 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_rat ) ) ).

% choose_alternating_sum
thf(fact_3955_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int ) ) ).

% choose_alternating_sum
thf(fact_3956_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I3 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
thf(fact_3957_choose__linear__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( times_times_nat @ I3 @ ( binomial @ N2 @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% choose_linear_sum
thf(fact_3958_binomial__r__part__sum,axiom,
    ! [M2: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% binomial_r_part_sum
thf(fact_3959_sum_Otriangle__reindex,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J2: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K4: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K4 @ I3 ) )
            @ ( set_ord_atMost_nat @ K4 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.triangle_reindex
thf(fact_3960_lessThan__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( set_ord_lessThan_nat @ X )
        = ( set_ord_lessThan_nat @ Y ) )
      = ( X = Y ) ) ).

% lessThan_eq_iff
thf(fact_3961_lessThan__iff,axiom,
    ! [I: $o,K: $o] :
      ( ( member_o @ I @ ( set_ord_lessThan_o @ K ) )
      = ( ord_less_o @ I @ K ) ) ).

% lessThan_iff
thf(fact_3962_lessThan__iff,axiom,
    ! [I: code_integer,K: code_integer] :
      ( ( member_Code_integer @ I @ ( set_or5754767410780653050nteger @ K ) )
      = ( ord_le6747313008572928689nteger @ I @ K ) ) ).

% lessThan_iff
thf(fact_3963_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
      = ( ord_less_rat @ I @ K ) ) ).

% lessThan_iff
thf(fact_3964_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
      = ( ord_less_num @ I @ K ) ) ).

% lessThan_iff
thf(fact_3965_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
      = ( ord_less_int @ I @ K ) ) ).

% lessThan_iff
thf(fact_3966_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
      = ( ord_less_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_3967_binomial__n__n,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ N2 )
      = one_one_nat ) ).

% binomial_n_n
thf(fact_3968_lessThan__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X ) @ ( set_ord_lessThan_rat @ Y ) )
      = ( ord_less_eq_rat @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_3969_lessThan__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y ) )
      = ( ord_less_eq_num @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_3970_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_3971_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_3972_binomial__Suc__Suc,axiom,
    ! [N2: nat,K: nat] :
      ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
      = ( plus_plus_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).

% binomial_Suc_Suc
thf(fact_3973_binomial__n__0,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_3974_sum_OlessThan__Suc,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_3975_sum_OlessThan__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_3976_sum_OlessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_3977_sum_OlessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_3978_choose__one,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ one_one_nat )
      = N2 ) ).

% choose_one
thf(fact_3979_lessThan__def,axiom,
    ( set_or5754767410780653050nteger
    = ( ^ [U2: code_integer] :
          ( collect_Code_integer
          @ ^ [X3: code_integer] : ( ord_le6747313008572928689nteger @ X3 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_3980_lessThan__def,axiom,
    ( set_ord_lessThan_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X3: rat] : ( ord_less_rat @ X3 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_3981_lessThan__def,axiom,
    ( set_ord_lessThan_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X3: num] : ( ord_less_num @ X3 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_3982_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X3: int] : ( ord_less_int @ X3 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_3983_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X3: nat] : ( ord_less_nat @ X3 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_3984_choose__mult__lemma,axiom,
    ! [M2: nat,R2: nat,K: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M2 @ R2 ) @ K ) @ ( plus_plus_nat @ M2 @ K ) ) @ ( binomial @ ( plus_plus_nat @ M2 @ K ) @ K ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M2 @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M2 @ R2 ) @ M2 ) ) ) ).

% choose_mult_lemma
thf(fact_3985_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_3986_bit__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( bit_se9216721137139052372nteger @ ( code_integer_of_int @ X ) )
      = ( bit_se1146084159140164899it_int @ X ) ) ).

% bit_integer.abs_eq
thf(fact_3987_lessThan__strict__subset__iff,axiom,
    ! [M2: code_integer,N2: code_integer] :
      ( ( ord_le1307284697595431911nteger @ ( set_or5754767410780653050nteger @ M2 ) @ ( set_or5754767410780653050nteger @ N2 ) )
      = ( ord_le6747313008572928689nteger @ M2 @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_3988_lessThan__strict__subset__iff,axiom,
    ! [M2: rat,N2: rat] :
      ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M2 ) @ ( set_ord_lessThan_rat @ N2 ) )
      = ( ord_less_rat @ M2 @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_3989_lessThan__strict__subset__iff,axiom,
    ! [M2: num,N2: num] :
      ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M2 ) @ ( set_ord_lessThan_num @ N2 ) )
      = ( ord_less_num @ M2 @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_3990_lessThan__strict__subset__iff,axiom,
    ! [M2: int,N2: int] :
      ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M2 ) @ ( set_ord_lessThan_int @ N2 ) )
      = ( ord_less_int @ M2 @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_3991_lessThan__strict__subset__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M2 ) @ ( set_ord_lessThan_nat @ N2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_3992_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] :
      ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
      = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).

% Suc_times_binomial_add
thf(fact_3993_binomial__absorb__comp,axiom,
    ! [N2: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ N2 @ K ) @ ( binomial @ N2 @ K ) )
      = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).

% binomial_absorb_comp
thf(fact_3994_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N2 ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_3995_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N2 ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_3996_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N2 ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_3997_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_3998_sum_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nat_diff_reindex
thf(fact_3999_Iic__subset__Iio__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or9101266186257409494nteger @ A ) @ ( set_or5754767410780653050nteger @ B ) )
      = ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_4000_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A ) @ ( set_ord_lessThan_rat @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_4001_Iic__subset__Iio__iff,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ A ) @ ( set_ord_lessThan_num @ B ) )
      = ( ord_less_num @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_4002_Iic__subset__Iio__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_4003_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_4004_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 ) ) ) ).

% binomial_absorption
thf(fact_4005_sum__diff__distrib,axiom,
    ! [Q: nat > nat,P: nat > nat,N2: nat] :
      ( ! [X4: nat] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
     => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N2 ) ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X3: nat] : ( minus_minus_nat @ ( P @ X3 ) @ ( Q @ X3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_4006_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_p7104986032573967614at_nat @ ( G @ zero_zero_nat )
        @ ( groups6857163185585827899at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_4007_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_4008_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_4009_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_4010_sum__lessThan__telescope,axiom,
    ! [F: nat > code_integer,M2: nat] :
      ( ( groups7501900531339628137nteger
        @ ^ [N4: nat] : ( minus_8373710615458151222nteger @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_8373710615458151222nteger @ ( F @ M2 ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_4011_sum__lessThan__telescope,axiom,
    ! [F: nat > rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N4: nat] : ( minus_minus_rat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_rat @ ( F @ M2 ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_4012_sum__lessThan__telescope,axiom,
    ! [F: nat > int,M2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N4: nat] : ( minus_minus_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_int @ ( F @ M2 ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_4013_sum__lessThan__telescope_H,axiom,
    ! [F: nat > code_integer,M2: nat] :
      ( ( groups7501900531339628137nteger
        @ ^ [N4: nat] : ( minus_8373710615458151222nteger @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_8373710615458151222nteger @ ( F @ zero_zero_nat ) @ ( F @ M2 ) ) ) ).

% sum_lessThan_telescope'
thf(fact_4014_sum__lessThan__telescope_H,axiom,
    ! [F: nat > rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N4: nat] : ( minus_minus_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M2 ) ) ) ).

% sum_lessThan_telescope'
thf(fact_4015_sum__lessThan__telescope_H,axiom,
    ! [F: nat > int,M2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N4: nat] : ( minus_minus_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M2 ) ) ) ).

% sum_lessThan_telescope'
thf(fact_4016_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K4: nat] : ( G @ ( suc @ K4 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_4017_sum__bounds__lt__plus1,axiom,
    ! [F: nat > nat,Mm: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K4: nat] : ( F @ ( suc @ K4 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_4018_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [J2: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( A @ I3 @ J2 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J2 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nested_swap'
thf(fact_4019_sum__choose__lower,axiom,
    ! [R2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K4: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K4 ) @ K4 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N2 ) ) @ N2 ) ) ).

% sum_choose_lower
thf(fact_4020_choose__rising__sum_I2_J,axiom,
    ! [N2: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J2: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J2 ) @ N2 )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ one_one_nat ) @ M2 ) ) ).

% choose_rising_sum(2)
thf(fact_4021_choose__rising__sum_I1_J,axiom,
    ! [N2: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J2: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J2 ) @ N2 )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ one_one_nat ) @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_4022_binomial__code,axiom,
    ( binomial
    = ( ^ [N4: nat,K4: nat] : ( if_nat @ ( ord_less_nat @ N4 @ K4 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K4 ) ) @ ( binomial @ N4 @ ( minus_minus_nat @ N4 @ K4 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N4 @ K4 ) @ one_one_nat ) @ N4 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K4 ) ) ) ) ) ) ).

% binomial_code
thf(fact_4023_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 ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_4024_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 ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_4025_one__diff__power__eq,axiom,
    ! [X: code_integer,N2: nat] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ X @ N2 ) )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_4026_one__diff__power__eq,axiom,
    ! [X: rat,N2: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_4027_one__diff__power__eq,axiom,
    ! [X: int,N2: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_4028_power__diff__1__eq,axiom,
    ! [X: code_integer,N2: nat] :
      ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ N2 ) @ one_one_Code_integer )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ X @ one_one_Code_integer ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_4029_power__diff__1__eq,axiom,
    ! [X: rat,N2: nat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ one_one_rat )
      = ( times_times_rat @ ( minus_minus_rat @ X @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_4030_power__diff__1__eq,axiom,
    ! [X: int,N2: nat] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N2 ) @ one_one_int )
      = ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_4031_geometric__sum,axiom,
    ! [X: rat,N2: nat] :
      ( ( X != one_one_rat )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ) ).

% geometric_sum
thf(fact_4032_sum_OatMost__shift,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_p7104986032573967614at_nat @ ( G @ zero_zero_nat )
        @ ( groups6857163185585827899at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_4033_sum_OatMost__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_4034_sum_OatMost__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_4035_sum_OatMost__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_4036_vandermonde,axiom,
    ! [M2: nat,N2: nat,R2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K4: nat] : ( times_times_nat @ ( binomial @ M2 @ K4 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R2 @ K4 ) ) )
        @ ( set_ord_atMost_nat @ R2 ) )
      = ( binomial @ ( plus_plus_nat @ M2 @ N2 ) @ R2 ) ) ).

% vandermonde
thf(fact_4037_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 ) ) ) ) ).

% binomial_addition_formula
thf(fact_4038_power__diff__sumr2,axiom,
    ! [X: code_integer,N2: nat,Y: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ N2 ) @ ( power_8256067586552552935nteger @ Y @ N2 ) )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ X @ Y )
        @ ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_8256067586552552935nteger @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_4039_power__diff__sumr2,axiom,
    ! [X: rat,N2: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ Y @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_rat @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_4040_power__diff__sumr2,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_int @ X @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_4041_diff__power__eq__sum,axiom,
    ! [X: code_integer,N2: nat,Y: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ ( suc @ N2 ) ) @ ( power_8256067586552552935nteger @ Y @ ( suc @ N2 ) ) )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ X @ Y )
        @ ( groups7501900531339628137nteger
          @ ^ [P6: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ P6 ) @ ( power_8256067586552552935nteger @ Y @ ( minus_minus_nat @ N2 @ P6 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_4042_diff__power__eq__sum,axiom,
    ! [X: rat,N2: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) @ ( power_power_rat @ Y @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [P6: nat] : ( times_times_rat @ ( power_power_rat @ X @ P6 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ N2 @ P6 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_4043_diff__power__eq__sum,axiom,
    ! [X: int,N2: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N2 ) ) @ ( power_power_int @ Y @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [P6: nat] : ( times_times_int @ ( power_power_int @ X @ P6 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ P6 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_4044_sum__gp__strict,axiom,
    ! [X: rat,N2: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( semiri681578069525770553at_rat @ N2 ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_4045_binomial,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
      = ( groups3542108847815614940at_nat
        @ ^ [K4: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K4 ) ) @ ( power_power_nat @ A @ K4 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K4 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial
thf(fact_4046_choose__two,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% choose_two
thf(fact_4047_one__diff__power__eq_H,axiom,
    ! [X: code_integer,N2: nat] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ X @ N2 ) )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X )
        @ ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( power_8256067586552552935nteger @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_4048_one__diff__power__eq_H,axiom,
    ! [X: rat,N2: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_4049_one__diff__power__eq_H,axiom,
    ! [X: int,N2: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_4050_binomial__ring,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( power_power_rat @ ( plus_plus_rat @ A @ B ) @ N2 )
      = ( groups2906978787729119204at_rat
        @ ^ [K4: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K4 ) ) @ ( power_power_rat @ A @ K4 ) ) @ ( power_power_rat @ B @ ( minus_minus_nat @ N2 @ K4 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_4051_binomial__ring,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( power_power_int @ ( plus_plus_int @ A @ B ) @ N2 )
      = ( groups3539618377306564664at_int
        @ ^ [K4: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ K4 ) ) @ ( power_power_int @ A @ K4 ) ) @ ( power_power_int @ B @ ( minus_minus_nat @ N2 @ K4 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_4052_binomial__ring,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ N2 )
      = ( groups7501900531339628137nteger
        @ ^ [K4: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ K4 ) ) @ ( power_8256067586552552935nteger @ A @ K4 ) ) @ ( power_8256067586552552935nteger @ B @ ( minus_minus_nat @ N2 @ K4 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_4053_binomial__ring,axiom,
    ! [A: code_natural,B: code_natural,N2: nat] :
      ( ( power_7079662738309270450atural @ ( plus_p4538020629002901425atural @ A @ B ) @ N2 )
      = ( groups6325495683096345652atural
        @ ^ [K4: nat] : ( times_2397367101498566445atural @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ ( binomial @ N2 @ K4 ) ) @ ( power_7079662738309270450atural @ A @ K4 ) ) @ ( power_7079662738309270450atural @ B @ ( minus_minus_nat @ N2 @ K4 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_4054_binomial__ring,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
      = ( groups3542108847815614940at_nat
        @ ^ [K4: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K4 ) ) @ ( power_power_nat @ A @ K4 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K4 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_4055_xor__num_Osimps_I6_J,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) ) ) ).

% xor_num.simps(6)
thf(fact_4056_xor__num_Osimps_I8_J,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) ) ) ).

% xor_num.simps(8)
thf(fact_4057_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I3 ) @ ( semiri681578069525770553at_rat @ I3 ) ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
thf(fact_4058_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ I3 ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int ) ) ).

% choose_alternating_linear_sum
thf(fact_4059_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I3 ) @ ( semiri4939895301339042750nteger @ I3 ) ) @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
thf(fact_4060_fact__Suc__0,axiom,
    ( ( semiri3624122377584611663nteger @ ( suc @ zero_zero_nat ) )
    = one_one_Code_integer ) ).

% fact_Suc_0
thf(fact_4061_fact__Suc__0,axiom,
    ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
    = one_one_rat ) ).

% fact_Suc_0
thf(fact_4062_fact__Suc__0,axiom,
    ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% fact_Suc_0
thf(fact_4063_fact__Suc__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% fact_Suc_0
thf(fact_4064_fact__1,axiom,
    ( ( semiri3624122377584611663nteger @ one_one_nat )
    = one_one_Code_integer ) ).

% fact_1
thf(fact_4065_fact__1,axiom,
    ( ( semiri773545260158071498ct_rat @ one_one_nat )
    = one_one_rat ) ).

% fact_1
thf(fact_4066_fact__1,axiom,
    ( ( semiri1406184849735516958ct_int @ one_one_nat )
    = one_one_int ) ).

% fact_1
thf(fact_4067_fact__1,axiom,
    ( ( semiri1408675320244567234ct_nat @ one_one_nat )
    = one_one_nat ) ).

% fact_1
thf(fact_4068_fact__0,axiom,
    ( ( semiri3624122377584611663nteger @ zero_zero_nat )
    = one_one_Code_integer ) ).

% fact_0
thf(fact_4069_fact__0,axiom,
    ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
    = one_one_rat ) ).

% fact_0
thf(fact_4070_fact__0,axiom,
    ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
    = one_one_int ) ).

% fact_0
thf(fact_4071_fact__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
    = one_one_nat ) ).

% fact_0
thf(fact_4072_fact__code,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N4: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_4073_fact__code,axiom,
    ( semiri3624122377584611663nteger
    = ( ^ [N4: nat] : ( semiri4939895301339042750nteger @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_4074_fact__code,axiom,
    ( semiri2447717529341329178atural
    = ( ^ [N4: nat] : ( semiri3763490453095760265atural @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_4075_fact__code,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N4: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_4076_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri773545260158071498ct_rat @ N2 )
        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_4077_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri1406184849735516958ct_int @ N2 )
        = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_4078_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri3624122377584611663nteger @ N2 )
        = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_4079_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri2447717529341329178atural @ N2 )
        = ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ N2 ) @ ( semiri2447717529341329178atural @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_4080_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri1408675320244567234ct_nat @ N2 )
        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_4081_fact__num__eq__if,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [M: nat] : ( if_rat @ ( M = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_4082_fact__num__eq__if,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [M: nat] : ( if_int @ ( M = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_4083_fact__num__eq__if,axiom,
    ( semiri3624122377584611663nteger
    = ( ^ [M: nat] : ( if_Code_integer @ ( M = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_4084_fact__num__eq__if,axiom,
    ( semiri2447717529341329178atural
    = ( ^ [M: nat] : ( if_Code_natural @ ( M = zero_zero_nat ) @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M ) @ ( semiri2447717529341329178atural @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_4085_fact__num__eq__if,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [M: nat] : ( if_nat @ ( M = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_4086_xor__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option_num] :
      ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X @ Xa ) )
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = none_num )
               => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X = one )
             => ! [N: num] :
                  ( ( Xa
                    = ( bit0 @ N ) )
                 => ( ( Y
                      = ( some_num @ ( bit1 @ N ) ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N ) ) ) ) ) )
           => ( ( ( X = one )
               => ! [N: num] :
                    ( ( Xa
                      = ( bit1 @ N ) )
                   => ( ( Y
                        = ( some_num @ ( bit0 @ N ) ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N ) ) ) ) ) )
             => ( ! [M5: num] :
                    ( ( X
                      = ( bit0 @ M5 ) )
                   => ( ( Xa = one )
                     => ( ( Y
                          = ( some_num @ ( bit1 @ M5 ) ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X
                        = ( bit0 @ M5 ) )
                     => ! [N: num] :
                          ( ( Xa
                            = ( bit0 @ N ) )
                         => ( ( Y
                              = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N ) ) ) ) ) )
                 => ( ! [M5: num] :
                        ( ( X
                          = ( bit0 @ M5 ) )
                       => ! [N: num] :
                            ( ( Xa
                              = ( bit1 @ N ) )
                           => ( ( Y
                                = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N ) ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N ) ) ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X
                            = ( bit1 @ M5 ) )
                         => ( ( Xa = one )
                           => ( ( Y
                                = ( some_num @ ( bit0 @ M5 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
                     => ( ! [M5: num] :
                            ( ( X
                              = ( bit1 @ M5 ) )
                           => ! [N: num] :
                                ( ( Xa
                                  = ( bit0 @ N ) )
                               => ( ( Y
                                    = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N ) ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N ) ) ) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X
                                = ( bit1 @ M5 ) )
                             => ! [N: num] :
                                  ( ( Xa
                                    = ( bit1 @ N ) )
                                 => ( ( Y
                                      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
thf(fact_4087_dvd__fact,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ N2 )
       => ( dvd_dvd_nat @ M2 @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% dvd_fact
thf(fact_4088_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( semiri3624122377584611663nteger @ N2 ) ) ).

% fact_ge_1
thf(fact_4089_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_ge_1
thf(fact_4090_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_ge_1
thf(fact_4091_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_1
thf(fact_4092_pochhammer__double,axiom,
    ! [Z: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_4093_fact__double,axiom,
    ! [N2: nat] :
      ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% fact_double
thf(fact_4094_pochhammer__code,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A5: rat,N4: nat] :
          ( if_rat @ ( N4 = zero_zero_nat ) @ one_one_rat
          @ ( set_fo1949268297981939178at_rat
            @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_rat ) ) ) ) ).

% pochhammer_code
thf(fact_4095_pochhammer__code,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A5: int,N4: nat] :
          ( if_int @ ( N4 = zero_zero_nat ) @ one_one_int
          @ ( set_fo2581907887559384638at_int
            @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_int ) ) ) ) ).

% pochhammer_code
thf(fact_4096_pochhammer__code,axiom,
    ( comm_s8582702949713902594nteger
    = ( ^ [A5: code_integer,N4: nat] :
          ( if_Code_integer @ ( N4 = zero_zero_nat ) @ one_one_Code_integer
          @ ( set_fo1084959871951514735nteger
            @ ^ [O: nat] : ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A5 @ ( semiri4939895301339042750nteger @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_Code_integer ) ) ) ) ).

% pochhammer_code
thf(fact_4097_pochhammer__code,axiom,
    ( comm_s7406298101470620109atural
    = ( ^ [A5: code_natural,N4: nat] :
          ( if_Code_natural @ ( N4 = zero_zero_nat ) @ one_one_Code_natural
          @ ( set_fo9131927060563008058atural
            @ ^ [O: nat] : ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ A5 @ ( semiri3763490453095760265atural @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_Code_natural ) ) ) ) ).

% pochhammer_code
thf(fact_4098_pochhammer__code,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A5: nat,N4: nat] :
          ( if_nat @ ( N4 = zero_zero_nat ) @ one_one_nat
          @ ( set_fo2584398358068434914at_nat
            @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_nat ) ) ) ) ).

% pochhammer_code
thf(fact_4099_xor__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se6528837805403552850or_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% xor_Suc_0_eq
thf(fact_4100_Suc__0__xor__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% Suc_0_xor_eq
thf(fact_4101_push__bit__numeral__minus__1,axiom,
    ! [N2: num] :
      ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ N2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ).

% push_bit_numeral_minus_1
thf(fact_4102_push__bit__numeral__minus__1,axiom,
    ! [N2: num] :
      ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ N2 ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ).

% push_bit_numeral_minus_1
thf(fact_4103_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K4: code_integer] :
          ( if_num @ ( ord_le3102999989581377725nteger @ K4 @ one_one_Code_integer ) @ one
          @ ( produc7336495610019696514er_num
            @ ^ [L3: code_integer,J2: code_integer] : ( if_num @ ( J2 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ one ) )
            @ ( code_divmod_integer @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_4104_Sum__Ico__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M2 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_4105_ivl__diff,axiom,
    ! [I: rat,N2: rat,M2: rat] :
      ( ( ord_less_eq_rat @ I @ N2 )
     => ( ( minus_minus_set_rat @ ( set_or4029947393144176647an_rat @ I @ M2 ) @ ( set_or4029947393144176647an_rat @ I @ N2 ) )
        = ( set_or4029947393144176647an_rat @ N2 @ M2 ) ) ) ).

% ivl_diff
thf(fact_4106_ivl__diff,axiom,
    ! [I: num,N2: num,M2: num] :
      ( ( ord_less_eq_num @ I @ N2 )
     => ( ( minus_minus_set_num @ ( set_or1222409239386451017an_num @ I @ M2 ) @ ( set_or1222409239386451017an_num @ I @ N2 ) )
        = ( set_or1222409239386451017an_num @ N2 @ M2 ) ) ) ).

% ivl_diff
thf(fact_4107_ivl__diff,axiom,
    ! [I: nat,N2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I @ N2 )
     => ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M2 ) @ ( set_or4665077453230672383an_nat @ I @ N2 ) )
        = ( set_or4665077453230672383an_nat @ N2 @ M2 ) ) ) ).

% ivl_diff
thf(fact_4108_ivl__diff,axiom,
    ! [I: code_integer,N2: code_integer,M2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ I @ N2 )
     => ( ( minus_2355218937544613996nteger @ ( set_or8404916559141939852nteger @ I @ M2 ) @ ( set_or8404916559141939852nteger @ I @ N2 ) )
        = ( set_or8404916559141939852nteger @ N2 @ M2 ) ) ) ).

% ivl_diff
thf(fact_4109_ivl__diff,axiom,
    ! [I: int,N2: int,M2: int] :
      ( ( ord_less_eq_int @ I @ N2 )
     => ( ( minus_minus_set_int @ ( set_or4662586982721622107an_int @ I @ M2 ) @ ( set_or4662586982721622107an_int @ I @ N2 ) )
        = ( set_or4662586982721622107an_int @ N2 @ M2 ) ) ) ).

% ivl_diff
thf(fact_4110_push__bit__push__bit,axiom,
    ! [M2: nat,N2: nat,A: nat] :
      ( ( bit_se547839408752420682it_nat @ M2 @ ( bit_se547839408752420682it_nat @ N2 @ A ) )
      = ( bit_se547839408752420682it_nat @ ( plus_plus_nat @ M2 @ N2 ) @ A ) ) ).

% push_bit_push_bit
thf(fact_4111_push__bit__push__bit,axiom,
    ! [M2: nat,N2: nat,A: code_integer] :
      ( ( bit_se7788150548672797655nteger @ M2 @ ( bit_se7788150548672797655nteger @ N2 @ A ) )
      = ( bit_se7788150548672797655nteger @ ( plus_plus_nat @ M2 @ N2 ) @ A ) ) ).

% push_bit_push_bit
thf(fact_4112_push__bit__push__bit,axiom,
    ! [M2: nat,N2: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ M2 @ ( bit_se545348938243370406it_int @ N2 @ A ) )
      = ( bit_se545348938243370406it_int @ ( plus_plus_nat @ M2 @ N2 ) @ A ) ) ).

% push_bit_push_bit
thf(fact_4113_push__bit__push__bit,axiom,
    ! [M2: nat,N2: nat,A: code_natural] :
      ( ( bit_se6611745700429515170atural @ M2 @ ( bit_se6611745700429515170atural @ N2 @ A ) )
      = ( bit_se6611745700429515170atural @ ( plus_plus_nat @ M2 @ N2 ) @ A ) ) ).

% push_bit_push_bit
thf(fact_4114_lessThan__minus__lessThan,axiom,
    ! [N2: nat,M2: nat] :
      ( ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( set_ord_lessThan_nat @ M2 ) )
      = ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ).

% lessThan_minus_lessThan
thf(fact_4115_lessThan__minus__lessThan,axiom,
    ! [N2: code_integer,M2: code_integer] :
      ( ( minus_2355218937544613996nteger @ ( set_or5754767410780653050nteger @ N2 ) @ ( set_or5754767410780653050nteger @ M2 ) )
      = ( set_or8404916559141939852nteger @ M2 @ N2 ) ) ).

% lessThan_minus_lessThan
thf(fact_4116_lessThan__minus__lessThan,axiom,
    ! [N2: int,M2: int] :
      ( ( minus_minus_set_int @ ( set_ord_lessThan_int @ N2 ) @ ( set_ord_lessThan_int @ M2 ) )
      = ( set_or4662586982721622107an_int @ M2 @ N2 ) ) ).

% lessThan_minus_lessThan
thf(fact_4117_atLeastLessThan__iff,axiom,
    ! [I: $o,L: $o,U: $o] :
      ( ( member_o @ I @ ( set_or7139685690850216873Than_o @ L @ U ) )
      = ( ( ord_less_eq_o @ L @ I )
        & ( ord_less_o @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_4118_atLeastLessThan__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( ( member_set_nat @ I @ ( set_or3540276404033026485et_nat @ L @ U ) )
      = ( ( ord_less_eq_set_nat @ L @ I )
        & ( ord_less_set_nat @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_4119_atLeastLessThan__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( ( member_rat @ I @ ( set_or4029947393144176647an_rat @ L @ U ) )
      = ( ( ord_less_eq_rat @ L @ I )
        & ( ord_less_rat @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_4120_atLeastLessThan__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( ( member_num @ I @ ( set_or1222409239386451017an_num @ L @ U ) )
      = ( ( ord_less_eq_num @ L @ I )
        & ( ord_less_num @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_4121_atLeastLessThan__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
      = ( ( ord_less_eq_nat @ L @ I )
        & ( ord_less_nat @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_4122_atLeastLessThan__iff,axiom,
    ! [I: code_integer,L: code_integer,U: code_integer] :
      ( ( member_Code_integer @ I @ ( set_or8404916559141939852nteger @ L @ U ) )
      = ( ( ord_le3102999989581377725nteger @ L @ I )
        & ( ord_le6747313008572928689nteger @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_4123_atLeastLessThan__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( ( member_int @ I @ ( set_or4662586982721622107an_int @ L @ U ) )
      = ( ( ord_less_eq_int @ L @ I )
        & ( ord_less_int @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_4124_ivl__subset,axiom,
    ! [I: rat,J: rat,M2: rat,N2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ I @ J ) @ ( set_or4029947393144176647an_rat @ M2 @ N2 ) )
      = ( ( ord_less_eq_rat @ J @ I )
        | ( ( ord_less_eq_rat @ M2 @ I )
          & ( ord_less_eq_rat @ J @ N2 ) ) ) ) ).

% ivl_subset
thf(fact_4125_ivl__subset,axiom,
    ! [I: num,J: num,M2: num,N2: num] :
      ( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ I @ J ) @ ( set_or1222409239386451017an_num @ M2 @ N2 ) )
      = ( ( ord_less_eq_num @ J @ I )
        | ( ( ord_less_eq_num @ M2 @ I )
          & ( ord_less_eq_num @ J @ N2 ) ) ) ) ).

% ivl_subset
thf(fact_4126_ivl__subset,axiom,
    ! [I: nat,J: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
      = ( ( ord_less_eq_nat @ J @ I )
        | ( ( ord_less_eq_nat @ M2 @ I )
          & ( ord_less_eq_nat @ J @ N2 ) ) ) ) ).

% ivl_subset
thf(fact_4127_ivl__subset,axiom,
    ! [I: code_integer,J: code_integer,M2: code_integer,N2: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or8404916559141939852nteger @ I @ J ) @ ( set_or8404916559141939852nteger @ M2 @ N2 ) )
      = ( ( ord_le3102999989581377725nteger @ J @ I )
        | ( ( ord_le3102999989581377725nteger @ M2 @ I )
          & ( ord_le3102999989581377725nteger @ J @ N2 ) ) ) ) ).

% ivl_subset
thf(fact_4128_ivl__subset,axiom,
    ! [I: int,J: int,M2: int,N2: int] :
      ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I @ J ) @ ( set_or4662586982721622107an_int @ M2 @ N2 ) )
      = ( ( ord_less_eq_int @ J @ I )
        | ( ( ord_less_eq_int @ M2 @ I )
          & ( ord_less_eq_int @ J @ N2 ) ) ) ) ).

% ivl_subset
thf(fact_4129_pochhammer__0,axiom,
    ! [A: code_integer] :
      ( ( comm_s8582702949713902594nteger @ A @ zero_zero_nat )
      = one_one_Code_integer ) ).

% pochhammer_0
thf(fact_4130_pochhammer__0,axiom,
    ! [A: rat] :
      ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% pochhammer_0
thf(fact_4131_pochhammer__0,axiom,
    ! [A: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% pochhammer_0
thf(fact_4132_pochhammer__0,axiom,
    ! [A: int] :
      ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% pochhammer_0
thf(fact_4133_xor__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% xor_numerals(1)
thf(fact_4134_xor__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) ) ).

% xor_numerals(1)
thf(fact_4135_xor__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).

% xor_numerals(1)
thf(fact_4136_xor__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se2046307713759805098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = ( numera5444537566228673987atural @ ( bit1 @ Y ) ) ) ).

% xor_numerals(1)
thf(fact_4137_xor__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).

% xor_numerals(2)
thf(fact_4138_xor__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) ) ).

% xor_numerals(2)
thf(fact_4139_xor__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_int @ ( bit0 @ Y ) ) ) ).

% xor_numerals(2)
thf(fact_4140_xor__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se2046307713759805098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = ( numera5444537566228673987atural @ ( bit0 @ Y ) ) ) ).

% xor_numerals(2)
thf(fact_4141_xor__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% xor_numerals(5)
thf(fact_4142_xor__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( bit1 @ X ) ) ) ).

% xor_numerals(5)
thf(fact_4143_xor__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).

% xor_numerals(5)
thf(fact_4144_xor__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( bit1 @ X ) ) ) ).

% xor_numerals(5)
thf(fact_4145_xor__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).

% xor_numerals(8)
thf(fact_4146_xor__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( bit0 @ X ) ) ) ).

% xor_numerals(8)
thf(fact_4147_xor__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit0 @ X ) ) ) ).

% xor_numerals(8)
thf(fact_4148_xor__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( bit0 @ X ) ) ) ).

% xor_numerals(8)
thf(fact_4149_push__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se547839408752420682it_nat @ N2 @ one_one_nat )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% push_bit_of_1
thf(fact_4150_push__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se7788150548672797655nteger @ N2 @ one_one_Code_integer )
      = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ).

% push_bit_of_1
thf(fact_4151_push__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se545348938243370406it_int @ N2 @ one_one_int )
      = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% push_bit_of_1
thf(fact_4152_push__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se6611745700429515170atural @ N2 @ one_one_Code_natural )
      = ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) ) ).

% push_bit_of_1
thf(fact_4153_sum_Oop__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > code_integer] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups7501900531339628137nteger @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups7501900531339628137nteger @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = ( plus_p5714425477246183910nteger @ ( groups7501900531339628137nteger @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_4154_sum_Oop__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = zero_z1048942125864253310at_nat ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_4155_sum_Oop__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_4156_sum_Oop__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > int] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_4157_sum_Oop__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.op_ivl_Suc
thf(fact_4158_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_4159_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_4160_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_4161_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_4162_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_4163_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_4164_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_4165_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_4166_flip__bit__nat__def,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [M: nat,N4: nat] : ( bit_se6528837805403552850or_nat @ N4 @ ( bit_se547839408752420682it_nat @ M @ one_one_nat ) ) ) ) ).

% flip_bit_nat_def
thf(fact_4167_push__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,X: int] :
      ( ( bit_se7788150548672797655nteger @ Xa @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se545348938243370406it_int @ Xa @ X ) ) ) ).

% push_bit_integer.abs_eq
thf(fact_4168_atLeastLessThan__inj_I2_J,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ( set_or4029947393144176647an_rat @ A @ B )
        = ( set_or4029947393144176647an_rat @ C @ D2 ) )
     => ( ( ord_less_rat @ A @ B )
       => ( ( ord_less_rat @ C @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_4169_atLeastLessThan__inj_I2_J,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ( set_or1222409239386451017an_num @ A @ B )
        = ( set_or1222409239386451017an_num @ C @ D2 ) )
     => ( ( ord_less_num @ A @ B )
       => ( ( ord_less_num @ C @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_4170_atLeastLessThan__inj_I2_J,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = ( set_or4665077453230672383an_nat @ C @ D2 ) )
     => ( ( ord_less_nat @ A @ B )
       => ( ( ord_less_nat @ C @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_4171_atLeastLessThan__inj_I2_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A @ B )
        = ( set_or8404916559141939852nteger @ C @ D2 ) )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( ord_le6747313008572928689nteger @ C @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_4172_atLeastLessThan__inj_I2_J,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = ( set_or4662586982721622107an_int @ C @ D2 ) )
     => ( ( ord_less_int @ A @ B )
       => ( ( ord_less_int @ C @ D2 )
         => ( B = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_4173_atLeastLessThan__inj_I1_J,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ( set_or4029947393144176647an_rat @ A @ B )
        = ( set_or4029947393144176647an_rat @ C @ D2 ) )
     => ( ( ord_less_rat @ A @ B )
       => ( ( ord_less_rat @ C @ D2 )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_4174_atLeastLessThan__inj_I1_J,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ( set_or1222409239386451017an_num @ A @ B )
        = ( set_or1222409239386451017an_num @ C @ D2 ) )
     => ( ( ord_less_num @ A @ B )
       => ( ( ord_less_num @ C @ D2 )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_4175_atLeastLessThan__inj_I1_J,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = ( set_or4665077453230672383an_nat @ C @ D2 ) )
     => ( ( ord_less_nat @ A @ B )
       => ( ( ord_less_nat @ C @ D2 )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_4176_atLeastLessThan__inj_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A @ B )
        = ( set_or8404916559141939852nteger @ C @ D2 ) )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( ord_le6747313008572928689nteger @ C @ D2 )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_4177_atLeastLessThan__inj_I1_J,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = ( set_or4662586982721622107an_int @ C @ D2 ) )
     => ( ( ord_less_int @ A @ B )
       => ( ( ord_less_int @ C @ D2 )
         => ( A = C ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_4178_atLeastLessThan__eq__iff,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ( ( set_or4029947393144176647an_rat @ A @ B )
            = ( set_or4029947393144176647an_rat @ C @ D2 ) )
          = ( ( A = C )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_4179_atLeastLessThan__eq__iff,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_num @ C @ D2 )
       => ( ( ( set_or1222409239386451017an_num @ A @ B )
            = ( set_or1222409239386451017an_num @ C @ D2 ) )
          = ( ( A = C )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_4180_atLeastLessThan__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( ( set_or4665077453230672383an_nat @ A @ B )
            = ( set_or4665077453230672383an_nat @ C @ D2 ) )
          = ( ( A = C )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_4181_atLeastLessThan__eq__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ( ( set_or8404916559141939852nteger @ A @ B )
            = ( set_or8404916559141939852nteger @ C @ D2 ) )
          = ( ( A = C )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_4182_atLeastLessThan__eq__iff,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( ( set_or4662586982721622107an_int @ A @ B )
            = ( set_or4662586982721622107an_int @ C @ D2 ) )
          = ( ( A = C )
            & ( B = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_4183_flip__bit__eq__xor,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [N4: nat,A5: nat] : ( bit_se6528837805403552850or_nat @ A5 @ ( bit_se547839408752420682it_nat @ N4 @ one_one_nat ) ) ) ) ).

% flip_bit_eq_xor
thf(fact_4184_flip__bit__eq__xor,axiom,
    ( bit_se1345352211410354436nteger
    = ( ^ [N4: nat,A5: code_integer] : ( bit_se3222712562003087583nteger @ A5 @ ( bit_se7788150548672797655nteger @ N4 @ one_one_Code_integer ) ) ) ) ).

% flip_bit_eq_xor
thf(fact_4185_flip__bit__eq__xor,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N4: nat,A5: int] : ( bit_se6526347334894502574or_int @ A5 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ).

% flip_bit_eq_xor
thf(fact_4186_flip__bit__eq__xor,axiom,
    ( bit_se168947363167071951atural
    = ( ^ [N4: nat,A5: code_natural] : ( bit_se2046307713759805098atural @ A5 @ ( bit_se6611745700429515170atural @ N4 @ one_one_Code_natural ) ) ) ) ).

% flip_bit_eq_xor
thf(fact_4187_push__bit__add,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( bit_se547839408752420682it_nat @ N2 @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N2 @ A ) @ ( bit_se547839408752420682it_nat @ N2 @ B ) ) ) ).

% push_bit_add
thf(fact_4188_push__bit__add,axiom,
    ! [N2: nat,A: code_integer,B: code_integer] :
      ( ( bit_se7788150548672797655nteger @ N2 @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( bit_se7788150548672797655nteger @ N2 @ A ) @ ( bit_se7788150548672797655nteger @ N2 @ B ) ) ) ).

% push_bit_add
thf(fact_4189_push__bit__add,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( bit_se545348938243370406it_int @ N2 @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( bit_se545348938243370406it_int @ N2 @ A ) @ ( bit_se545348938243370406it_int @ N2 @ B ) ) ) ).

% push_bit_add
thf(fact_4190_push__bit__add,axiom,
    ! [N2: nat,A: code_natural,B: code_natural] :
      ( ( bit_se6611745700429515170atural @ N2 @ ( plus_p4538020629002901425atural @ A @ B ) )
      = ( plus_p4538020629002901425atural @ ( bit_se6611745700429515170atural @ N2 @ A ) @ ( bit_se6611745700429515170atural @ N2 @ B ) ) ) ).

% push_bit_add
thf(fact_4191_atLeastLessThan__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ A @ B ) @ ( set_or4029947393144176647an_rat @ C @ D2 ) )
     => ( ( ord_less_eq_rat @ B @ A )
        | ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_4192_atLeastLessThan__subset__iff,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ A @ B ) @ ( set_or1222409239386451017an_num @ C @ D2 ) )
     => ( ( ord_less_eq_num @ B @ A )
        | ( ( ord_less_eq_num @ C @ A )
          & ( ord_less_eq_num @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_4193_atLeastLessThan__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D2 ) )
     => ( ( ord_less_eq_nat @ B @ A )
        | ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_eq_nat @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_4194_atLeastLessThan__subset__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or8404916559141939852nteger @ A @ B ) @ ( set_or8404916559141939852nteger @ C @ D2 ) )
     => ( ( ord_le3102999989581377725nteger @ B @ A )
        | ( ( ord_le3102999989581377725nteger @ C @ A )
          & ( ord_le3102999989581377725nteger @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_4195_atLeastLessThan__subset__iff,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D2 ) )
     => ( ( ord_less_eq_int @ B @ A )
        | ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_eq_int @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_4196_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_4197_lessThan__atLeast0,axiom,
    ( set_ord_lessThan_nat
    = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).

% lessThan_atLeast0
thf(fact_4198_push__bit__take__bit,axiom,
    ! [M2: nat,N2: nat,A: code_integer] :
      ( ( bit_se7788150548672797655nteger @ M2 @ ( bit_se1745604003318907178nteger @ N2 @ A ) )
      = ( bit_se1745604003318907178nteger @ ( plus_plus_nat @ M2 @ N2 ) @ ( bit_se7788150548672797655nteger @ M2 @ A ) ) ) ).

% push_bit_take_bit
thf(fact_4199_push__bit__take__bit,axiom,
    ! [M2: nat,N2: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ M2 @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
      = ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M2 @ N2 ) @ ( bit_se545348938243370406it_int @ M2 @ A ) ) ) ).

% push_bit_take_bit
thf(fact_4200_push__bit__take__bit,axiom,
    ! [M2: nat,N2: nat,A: nat] :
      ( ( bit_se547839408752420682it_nat @ M2 @ ( bit_se2925701944663578781it_nat @ N2 @ A ) )
      = ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M2 @ N2 ) @ ( bit_se547839408752420682it_nat @ M2 @ A ) ) ) ).

% push_bit_take_bit
thf(fact_4201_push__bit__take__bit,axiom,
    ! [M2: nat,N2: nat,A: code_natural] :
      ( ( bit_se6611745700429515170atural @ M2 @ ( bit_se569199155075624693atural @ N2 @ A ) )
      = ( bit_se569199155075624693atural @ ( plus_plus_nat @ M2 @ N2 ) @ ( bit_se6611745700429515170atural @ M2 @ A ) ) ) ).

% push_bit_take_bit
thf(fact_4202_pochhammer__fact,axiom,
    ( semiri3624122377584611663nteger
    = ( comm_s8582702949713902594nteger @ one_one_Code_integer ) ) ).

% pochhammer_fact
thf(fact_4203_pochhammer__fact,axiom,
    ( semiri773545260158071498ct_rat
    = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).

% pochhammer_fact
thf(fact_4204_pochhammer__fact,axiom,
    ( semiri1406184849735516958ct_int
    = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).

% pochhammer_fact
thf(fact_4205_pochhammer__fact,axiom,
    ( semiri1408675320244567234ct_nat
    = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).

% pochhammer_fact
thf(fact_4206_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ).

% sum.shift_bounds_Suc_ivl
thf(fact_4207_sum_Oshift__bounds__nat__ivl,axiom,
    ! [G: nat > nat,M2: nat,K: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ).

% sum.shift_bounds_nat_ivl
thf(fact_4208_sum_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat,G: nat > nat,H3: nat > nat] :
      ( ( A = C )
     => ( ( B = D2 )
       => ( ! [X4: nat] :
              ( ( ord_less_eq_nat @ C @ X4 )
             => ( ( ord_less_nat @ X4 @ D2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) ) )
         => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
            = ( groups3542108847815614940at_nat @ H3 @ ( set_or4665077453230672383an_nat @ C @ D2 ) ) ) ) ) ) ).

% sum.ivl_cong
thf(fact_4209_sum_Oivl__cong,axiom,
    ! [A: int,C: int,B: int,D2: int,G: int > int,H3: int > int] :
      ( ( A = C )
     => ( ( B = D2 )
       => ( ! [X4: int] :
              ( ( ord_less_eq_int @ C @ X4 )
             => ( ( ord_less_int @ X4 @ D2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) ) )
         => ( ( groups4538972089207619220nt_int @ G @ ( set_or4662586982721622107an_int @ A @ B ) )
            = ( groups4538972089207619220nt_int @ H3 @ ( set_or4662586982721622107an_int @ C @ D2 ) ) ) ) ) ) ).

% sum.ivl_cong
thf(fact_4210_sum_OatLeastLessThan__concat,axiom,
    ! [M2: nat,N2: nat,P4: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) )
          = ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ P4 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_4211_sum_OatLeastLessThan__concat,axiom,
    ! [M2: nat,N2: nat,P4: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) )
          = ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ P4 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_4212_sum_OatLeastLessThan__concat,axiom,
    ! [M2: nat,N2: nat,P4: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) )
          = ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ P4 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_4213_sum_OatLeastLessThan__concat,axiom,
    ! [M2: nat,N2: nat,P4: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) )
          = ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ P4 ) ) ) ) ) ).

% sum.atLeastLessThan_concat
thf(fact_4214_sum__diff__nat__ivl,axiom,
    ! [M2: nat,N2: nat,P4: nat,F: nat > code_integer] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( minus_8373710615458151222nteger @ ( groups7501900531339628137nteger @ F @ ( set_or4665077453230672383an_nat @ M2 @ P4 ) ) @ ( groups7501900531339628137nteger @ F @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) )
          = ( groups7501900531339628137nteger @ F @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) ) ) ) ).

% sum_diff_nat_ivl
thf(fact_4215_sum__diff__nat__ivl,axiom,
    ! [M2: nat,N2: nat,P4: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_or4665077453230672383an_nat @ M2 @ P4 ) ) @ ( groups2906978787729119204at_rat @ F @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) )
          = ( groups2906978787729119204at_rat @ F @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) ) ) ) ).

% sum_diff_nat_ivl
thf(fact_4216_sum__diff__nat__ivl,axiom,
    ! [M2: nat,N2: nat,P4: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ M2 @ P4 ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) )
          = ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) ) ) ) ).

% sum_diff_nat_ivl
thf(fact_4217_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s8582702949713902594nteger @ zero_z3403309356797280102nteger @ N2 )
          = one_one_Code_integer ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s8582702949713902594nteger @ zero_z3403309356797280102nteger @ N2 )
          = zero_z3403309356797280102nteger ) ) ) ).

% pochhammer_0_left
thf(fact_4218_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
          = one_one_rat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
          = zero_zero_rat ) ) ) ).

% pochhammer_0_left
thf(fact_4219_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
          = one_one_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
          = zero_zero_nat ) ) ) ).

% pochhammer_0_left
thf(fact_4220_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
          = one_one_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
          = zero_zero_int ) ) ) ).

% pochhammer_0_left
thf(fact_4221_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G @ suc ) @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ).

% sum.atLeast_Suc_lessThan_Suc_shift
thf(fact_4222_sum_OatLeastLessThan__shift__bounds,axiom,
    ! [G: nat > nat,M2: nat,K: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G @ ( plus_plus_nat @ K ) ) @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ).

% sum.atLeastLessThan_shift_bounds
thf(fact_4223_concat__bit__eq,axiom,
    ( bit_concat_bit
    = ( ^ [N4: nat,K4: int,L3: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N4 @ K4 ) @ ( bit_se545348938243370406it_int @ N4 @ L3 ) ) ) ) ).

% concat_bit_eq
thf(fact_4224_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D2: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or3540276404033026485et_nat @ C @ D2 ) )
      = ( ( ord_less_eq_set_nat @ A @ B )
       => ( ( ord_less_eq_set_nat @ C @ A )
          & ( ord_less_set_nat @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4225_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or4029947393144176647an_rat @ C @ D2 ) )
      = ( ( ord_less_eq_rat @ A @ B )
       => ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_rat @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4226_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: num,B: num,C: num,D2: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or1222409239386451017an_num @ C @ D2 ) )
      = ( ( ord_less_eq_num @ A @ B )
       => ( ( ord_less_eq_num @ C @ A )
          & ( ord_less_num @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4227_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D2 ) )
      = ( ( ord_less_eq_nat @ A @ B )
       => ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_nat @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4228_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or189985376899183464nteger @ A @ B ) @ ( set_or8404916559141939852nteger @ C @ D2 ) )
      = ( ( ord_le3102999989581377725nteger @ A @ B )
       => ( ( ord_le3102999989581377725nteger @ C @ A )
          & ( ord_le6747313008572928689nteger @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4229_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D2 ) )
      = ( ( ord_less_eq_int @ A @ B )
       => ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_int @ B @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4230_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D2 ) )
      = ( ( ord_less_rat @ A @ B )
       => ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B @ D2 ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_4231_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > code_integer,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_z3403309356797280102nteger )
     => ( ( groups7501900531339628137nteger @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups7501900531339628137nteger @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_4232_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > multis2468970476368604999at_nat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_z1048942125864253310at_nat )
     => ( ( groups6857163185585827899at_nat @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups6857163185585827899at_nat @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_4233_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > rat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_rat )
     => ( ( groups2906978787729119204at_rat @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups2906978787729119204at_rat @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_4234_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > int,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_int )
     => ( ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3539618377306564664at_int @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_4235_sum__shift__lb__Suc0__0__upt,axiom,
    ! [F: nat > nat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_nat )
     => ( ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
thf(fact_4236_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_4237_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_4238_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_4239_sum_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.atLeast0_lessThan_Suc
thf(fact_4240_sum_OatLeast__Suc__lessThan,axiom,
    ! [M2: nat,N2: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
        = ( plus_p7104986032573967614at_nat @ ( G @ M2 ) @ ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_4241_sum_OatLeast__Suc__lessThan,axiom,
    ! [M2: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
        = ( plus_plus_rat @ ( G @ M2 ) @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_4242_sum_OatLeast__Suc__lessThan,axiom,
    ! [M2: nat,N2: nat,G: nat > int] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
        = ( plus_plus_int @ ( G @ M2 ) @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_4243_sum_OatLeast__Suc__lessThan,axiom,
    ! [M2: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
        = ( plus_plus_nat @ ( G @ M2 ) @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_lessThan
thf(fact_4244_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_4245_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_4246_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_4247_sum_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% sum.atLeastLessThan_Suc
thf(fact_4248_sum_Olast__plus,axiom,
    ! [M2: nat,N2: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( plus_p7104986032573967614at_nat @ ( G @ N2 ) @ ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.last_plus
thf(fact_4249_sum_Olast__plus,axiom,
    ! [M2: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( plus_plus_rat @ ( G @ N2 ) @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.last_plus
thf(fact_4250_sum_Olast__plus,axiom,
    ! [M2: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( plus_plus_int @ ( G @ N2 ) @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.last_plus
thf(fact_4251_sum_Olast__plus,axiom,
    ! [M2: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( plus_plus_nat @ ( G @ N2 ) @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ) ) ).

% sum.last_plus
thf(fact_4252_pochhammer__rec,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( comm_s8582702949713902594nteger @ A @ ( suc @ N2 ) )
      = ( times_3573771949741848930nteger @ A @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_4253_pochhammer__rec,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_4254_pochhammer__rec,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_4255_pochhammer__rec,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_4256_pochhammer__Suc,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N2 ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_4257_pochhammer__Suc,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N2 ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_4258_pochhammer__Suc,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N2 ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_4259_pochhammer__Suc,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( comm_s8582702949713902594nteger @ A @ ( suc @ N2 ) )
      = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ A @ N2 ) @ ( plus_p5714425477246183910nteger @ A @ ( semiri4939895301339042750nteger @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_4260_pochhammer__Suc,axiom,
    ! [A: code_natural,N2: nat] :
      ( ( comm_s7406298101470620109atural @ A @ ( suc @ N2 ) )
      = ( times_2397367101498566445atural @ ( comm_s7406298101470620109atural @ A @ N2 ) @ ( plus_p4538020629002901425atural @ A @ ( semiri3763490453095760265atural @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_4261_pochhammer__rec_H,axiom,
    ! [Z: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N2 ) )
      = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_4262_pochhammer__rec_H,axiom,
    ! [Z: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N2 ) )
      = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( comm_s4660882817536571857er_int @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_4263_pochhammer__rec_H,axiom,
    ! [Z: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N2 ) )
      = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N2 ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_4264_pochhammer__rec_H,axiom,
    ! [Z: code_integer,N2: nat] :
      ( ( comm_s8582702949713902594nteger @ Z @ ( suc @ N2 ) )
      = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ N2 ) ) @ ( comm_s8582702949713902594nteger @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_4265_pochhammer__rec_H,axiom,
    ! [Z: code_natural,N2: nat] :
      ( ( comm_s7406298101470620109atural @ Z @ ( suc @ N2 ) )
      = ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ Z @ ( semiri3763490453095760265atural @ N2 ) ) @ ( comm_s7406298101470620109atural @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_4266_pochhammer__product_H,axiom,
    ! [Z: rat,N2: nat,M2: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N2 @ M2 ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N2 ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_4267_pochhammer__product_H,axiom,
    ! [Z: int,N2: nat,M2: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N2 @ M2 ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N2 ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_4268_pochhammer__product_H,axiom,
    ! [Z: nat,N2: nat,M2: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N2 @ M2 ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N2 ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_4269_pochhammer__product_H,axiom,
    ! [Z: code_integer,N2: nat,M2: nat] :
      ( ( comm_s8582702949713902594nteger @ Z @ ( plus_plus_nat @ N2 @ M2 ) )
      = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z @ N2 ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ N2 ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_4270_pochhammer__product_H,axiom,
    ! [Z: code_natural,N2: nat,M2: nat] :
      ( ( comm_s7406298101470620109atural @ Z @ ( plus_plus_nat @ N2 @ M2 ) )
      = ( times_2397367101498566445atural @ ( comm_s7406298101470620109atural @ Z @ N2 ) @ ( comm_s7406298101470620109atural @ ( plus_p4538020629002901425atural @ Z @ ( semiri3763490453095760265atural @ N2 ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_4271_bit__iff__and__push__bit__not__eq__0,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [A5: nat,N4: nat] :
          ( ( bit_se727722235901077358nd_nat @ A5 @ ( bit_se547839408752420682it_nat @ N4 @ one_one_nat ) )
         != zero_zero_nat ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
thf(fact_4272_bit__iff__and__push__bit__not__eq__0,axiom,
    ( bit_se9216721137139052372nteger
    = ( ^ [A5: code_integer,N4: nat] :
          ( ( bit_se3949692690581998587nteger @ A5 @ ( bit_se7788150548672797655nteger @ N4 @ one_one_Code_integer ) )
         != zero_z3403309356797280102nteger ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
thf(fact_4273_bit__iff__and__push__bit__not__eq__0,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [A5: int,N4: nat] :
          ( ( bit_se725231765392027082nd_int @ A5 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) )
         != zero_zero_int ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
thf(fact_4274_bit__iff__and__push__bit__not__eq__0,axiom,
    ( bit_se8040316288895769887atural
    = ( ^ [A5: code_natural,N4: nat] :
          ( ( bit_se2773287842338716102atural @ A5 @ ( bit_se6611745700429515170atural @ N4 @ one_one_Code_natural ) )
         != zero_z2226904508553997617atural ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
thf(fact_4275_sum__Suc__diff_H,axiom,
    ! [M2: nat,N2: nat,F: nat > code_integer] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( minus_8373710615458151222nteger @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
        = ( minus_8373710615458151222nteger @ ( F @ N2 ) @ ( F @ M2 ) ) ) ) ).

% sum_Suc_diff'
thf(fact_4276_sum__Suc__diff_H,axiom,
    ! [M2: nat,N2: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
        = ( minus_minus_rat @ ( F @ N2 ) @ ( F @ M2 ) ) ) ) ).

% sum_Suc_diff'
thf(fact_4277_sum__Suc__diff_H,axiom,
    ! [M2: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
        = ( minus_minus_int @ ( F @ N2 ) @ ( F @ M2 ) ) ) ) ).

% sum_Suc_diff'
thf(fact_4278_sum_OatLeastLessThan__rev,axiom,
    ! [G: nat > nat,N2: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ M2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ ( suc @ I3 ) ) )
        @ ( set_or4665077453230672383an_nat @ N2 @ M2 ) ) ) ).

% sum.atLeastLessThan_rev
thf(fact_4279_sum_Onested__swap,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( A @ I3 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [J2: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( A @ I3 @ J2 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J2 ) @ N2 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% sum.nested_swap
thf(fact_4280_sum_Onat__group,axiom,
    ! [G: nat > nat,K: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [M: nat] : ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M @ K ) @ ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ K ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( times_times_nat @ N2 @ K ) ) ) ) ).

% sum.nat_group
thf(fact_4281_numeral__xor__num,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
      = ( case_option_nat_num @ zero_zero_nat @ numeral_numeral_nat @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) ) ).

% numeral_xor_num
thf(fact_4282_numeral__xor__num,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( case_o356765784539232260er_num @ zero_z3403309356797280102nteger @ numera6620942414471956472nteger @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) ) ).

% numeral_xor_num
thf(fact_4283_numeral__xor__num,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) ) ).

% numeral_xor_num
thf(fact_4284_numeral__xor__num,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ M2 ) @ ( numera5444537566228673987atural @ N2 ) )
      = ( case_o5621594795226839503al_num @ zero_z2226904508553997617atural @ numera5444537566228673987atural @ ( bit_un2480387367778600638or_num @ M2 @ N2 ) ) ) ).

% numeral_xor_num
thf(fact_4285_sum_Ohead__if,axiom,
    ! [N2: nat,M2: nat,G: nat > code_integer] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups7501900531339628137nteger @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups7501900531339628137nteger @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = ( plus_p5714425477246183910nteger @ ( groups7501900531339628137nteger @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.head_if
thf(fact_4286_sum_Ohead__if,axiom,
    ! [N2: nat,M2: nat,G: nat > multis2468970476368604999at_nat] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = zero_z1048942125864253310at_nat ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups6857163185585827899at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = ( plus_p7104986032573967614at_nat @ ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.head_if
thf(fact_4287_sum_Ohead__if,axiom,
    ! [N2: nat,M2: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.head_if
thf(fact_4288_sum_Ohead__if,axiom,
    ! [N2: nat,M2: nat,G: nat > int] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.head_if
thf(fact_4289_sum_Ohead__if,axiom,
    ! [N2: nat,M2: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% sum.head_if
thf(fact_4290_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [G: nat > multis2468970476368604999at_nat,N2: nat] :
      ( ( groups6857163185585827899at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_p7104986032573967614at_nat @ ( G @ zero_zero_nat ) @ ( groups6857163185585827899at_nat @ ( comp_n8698576032424989604at_nat @ G @ suc ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% sum.atLeast0_lessThan_Suc_shift
thf(fact_4291_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat ) @ ( groups2906978787729119204at_rat @ ( comp_nat_rat_nat @ G @ suc ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% sum.atLeast0_lessThan_Suc_shift
thf(fact_4292_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat ) @ ( groups3539618377306564664at_int @ ( comp_nat_int_nat @ G @ suc ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% sum.atLeast0_lessThan_Suc_shift
thf(fact_4293_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat ) @ ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G @ suc ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% sum.atLeast0_lessThan_Suc_shift
thf(fact_4294_sum_OatLeastLessThan__shift__0,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
      = ( groups3542108847815614940at_nat @ ( comp_nat_nat_nat @ G @ ( plus_plus_nat @ M2 ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ).

% sum.atLeastLessThan_shift_0
thf(fact_4295_pochhammer__product,axiom,
    ! [M2: nat,N2: nat,Z: rat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( comm_s4028243227959126397er_rat @ Z @ N2 )
        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M2 ) ) @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_4296_pochhammer__product,axiom,
    ! [M2: nat,N2: nat,Z: int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( comm_s4660882817536571857er_int @ Z @ N2 )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_4297_pochhammer__product,axiom,
    ! [M2: nat,N2: nat,Z: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( comm_s4663373288045622133er_nat @ Z @ N2 )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_4298_pochhammer__product,axiom,
    ! [M2: nat,N2: nat,Z: code_integer] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( comm_s8582702949713902594nteger @ Z @ N2 )
        = ( times_3573771949741848930nteger @ ( comm_s8582702949713902594nteger @ Z @ M2 ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ Z @ ( semiri4939895301339042750nteger @ M2 ) ) @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_4299_pochhammer__product,axiom,
    ! [M2: nat,N2: nat,Z: code_natural] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( comm_s7406298101470620109atural @ Z @ N2 )
        = ( times_2397367101498566445atural @ ( comm_s7406298101470620109atural @ Z @ M2 ) @ ( comm_s7406298101470620109atural @ ( plus_p4538020629002901425atural @ Z @ ( semiri3763490453095760265atural @ M2 ) ) @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_4300_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: nat > nat,N2: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ M2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M2 ) ) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_4301_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ( comp_nat_nat_nat @ G
          @ ^ [N4: nat] : ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) )
        @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ).

% sum.atLeast_lessThan_pred_shift
thf(fact_4302_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K ) )
      = ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_4303_pochhammer__absorb__comp,axiom,
    ! [R2: int,K: nat] :
      ( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K ) )
      = ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_4304_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K ) )
      = ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_4305_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ N2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% pochhammer_same
thf(fact_4306_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ N2 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% pochhammer_same
thf(fact_4307_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ N2 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( semiri3624122377584611663nteger @ N2 ) ) ) ).

% pochhammer_same
thf(fact_4308_pochhammer__minus,axiom,
    ! [B: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_4309_pochhammer__minus,axiom,
    ! [B: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_4310_pochhammer__minus,axiom,
    ! [B: code_integer,K: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_4311_pochhammer__minus_H,axiom,
    ! [B: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_4312_pochhammer__minus_H,axiom,
    ! [B: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_4313_pochhammer__minus_H,axiom,
    ! [B: code_integer,K: nat] :
      ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_4314_gbinomial__pochhammer_H,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K4: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ K4 ) ) @ one_one_rat ) @ K4 ) @ ( semiri773545260158071498ct_rat @ K4 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_4315_gbinomial__pochhammer,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K4 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A5 ) @ K4 ) ) @ ( semiri773545260158071498ct_rat @ K4 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_4316_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M: nat,N4: nat] : ( if_nat @ ( M = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_4317_num__of__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( code_num_of_integer @ ( code_integer_of_int @ X ) )
      = ( num_of_nat @ ( nat2 @ X ) ) ) ).

% num_of_integer.abs_eq
thf(fact_4318_signed__take__bit__code,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N4: nat,A5: code_integer] : ( if_Code_integer @ ( bit_se9216721137139052372nteger @ ( bit_se1745604003318907178nteger @ ( suc @ N4 ) @ A5 ) @ N4 ) @ ( plus_p5714425477246183910nteger @ ( bit_se1745604003318907178nteger @ ( suc @ N4 ) @ A5 ) @ ( bit_se7788150548672797655nteger @ ( suc @ N4 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) @ ( bit_se1745604003318907178nteger @ ( suc @ N4 ) @ A5 ) ) ) ) ).

% signed_take_bit_code
thf(fact_4319_signed__take__bit__code,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N4: nat,A5: int] : ( if_int @ ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ A5 ) @ N4 ) @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ A5 ) @ ( bit_se545348938243370406it_int @ ( suc @ N4 ) @ ( uminus_uminus_int @ one_one_int ) ) ) @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ A5 ) ) ) ) ).

% signed_take_bit_code
thf(fact_4320_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M: nat,N4: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_4321_xor__one__eq,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ A @ one_one_nat )
      = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% xor_one_eq
thf(fact_4322_xor__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3222712562003087583nteger @ A @ one_one_Code_integer )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n356916108424825756nteger
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% xor_one_eq
thf(fact_4323_xor__one__eq,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ A @ one_one_int )
      = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2684676970156552555ol_int
          @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% xor_one_eq
thf(fact_4324_xor__one__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se2046307713759805098atural @ A @ one_one_Code_natural )
      = ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ A @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n8403883297036319079atural
          @ ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% xor_one_eq
thf(fact_4325_one__xor__eq,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ one_one_nat @ A )
      = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_xor_eq
thf(fact_4326_one__xor__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ A )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n356916108424825756nteger
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_xor_eq
thf(fact_4327_one__xor__eq,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ one_one_int @ A )
      = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2684676970156552555ol_int
          @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_xor_eq
thf(fact_4328_one__xor__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se2046307713759805098atural @ one_one_Code_natural @ A )
      = ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ A @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n8403883297036319079atural
          @ ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_xor_eq
thf(fact_4329_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ B ) @ N2 )
      = ( groups2906978787729119204at_rat
        @ ^ [K4: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K4 ) ) @ ( comm_s4028243227959126397er_rat @ A @ K4 ) ) @ ( comm_s4028243227959126397er_rat @ B @ ( minus_minus_nat @ N2 @ K4 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_4330_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B ) @ N2 )
      = ( groups3539618377306564664at_int
        @ ^ [K4: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ K4 ) ) @ ( comm_s4660882817536571857er_int @ A @ K4 ) ) @ ( comm_s4660882817536571857er_int @ B @ ( minus_minus_nat @ N2 @ K4 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_4331_pochhammer__binomial__sum,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] :
      ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ N2 )
      = ( groups7501900531339628137nteger
        @ ^ [K4: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ K4 ) ) @ ( comm_s8582702949713902594nteger @ A @ K4 ) ) @ ( comm_s8582702949713902594nteger @ B @ ( minus_minus_nat @ N2 @ K4 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_4332_divmod__integer__def,axiom,
    ( code_divmod_integer
    = ( ^ [K4: code_integer,L3: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K4 @ L3 ) @ ( modulo364778990260209775nteger @ K4 @ L3 ) ) ) ) ).

% divmod_integer_def
thf(fact_4333_sum__power2,axiom,
    ! [K: nat] :
      ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
      = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).

% sum_power2
thf(fact_4334_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K4: code_integer] :
          ( if_nat @ ( ord_le3102999989581377725nteger @ K4 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
          @ ( produc1555791787009142072er_nat
            @ ^ [L3: code_integer,J2: code_integer] : ( if_nat @ ( J2 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ one_one_nat ) )
            @ ( code_divmod_integer @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_4335_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,N2: nat] :
      ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [K4: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K4 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_4336_int__of__integer__code,axiom,
    ( code_int_of_integer
    = ( ^ [K4: code_integer] :
          ( if_int @ ( ord_le6747313008572928689nteger @ K4 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K4 ) ) )
          @ ( if_int @ ( K4 = zero_z3403309356797280102nteger ) @ zero_zero_int
            @ ( produc1553301316500091796er_int
              @ ^ [L3: code_integer,J2: code_integer] : ( if_int @ ( J2 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ one_one_int ) )
              @ ( code_divmod_integer @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_4337_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K4: code_integer,L3: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( L3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K4 )
            @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L3
              @ ( if_Pro6119634080678213985nteger
                @ ( ( sgn_sgn_Code_integer @ K4 )
                  = ( sgn_sgn_Code_integer @ L3 ) )
                @ ( code_divmod_abs @ K4 @ L3 )
                @ ( produc6916734918728496179nteger
                  @ ^ [R3: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R3 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L3 ) @ S4 ) ) )
                  @ ( code_divmod_abs @ K4 @ L3 ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_4338_time__def,axiom,
    ( time_time_array_a
    = ( ^ [C2: heap_T6362127782908069795rray_a,H: heap_e7401611519738050253t_unit] :
          ( case_o1134687144775446439it_nat @ zero_zero_nat
          @ ( produc6028340861379781217at_nat
            @ ^ [X3: array_a] :
                ( produc4278983685114131267at_nat
                @ ^ [Xa4: heap_e7401611519738050253t_unit,T2: nat] : T2 ) )
          @ ( heap_T5591663578999710211rray_a @ C2 @ H ) ) ) ) ).

% time_def
thf(fact_4339_time__def,axiom,
    ( time_t4224138285095624986t_unit
    = ( ^ [C2: heap_T5738788834812785303t_unit,H: heap_e7401611519738050253t_unit] :
          ( case_o5424514124273913701it_nat @ zero_zero_nat
          @ ( produc5629734405777326591at_nat
            @ ^ [X3: product_unit] :
                ( produc4278983685114131267at_nat
                @ ^ [Xa4: heap_e7401611519738050253t_unit,T2: nat] : T2 ) )
          @ ( heap_T875086893843062177t_unit @ C2 @ H ) ) ) ) ).

% time_def
thf(fact_4340_bit__cut__integer__def,axiom,
    ( code_bit_cut_integer
    = ( ^ [K4: code_integer] :
          ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K4 ) ) ) ) ).

% bit_cut_integer_def
thf(fact_4341_integer__of__num_I3_J,axiom,
    ! [N2: num] :
      ( ( code_integer_of_num @ ( bit1 @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N2 ) @ ( code_integer_of_num @ N2 ) ) @ one_one_Code_integer ) ) ).

% integer_of_num(3)
thf(fact_4342_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X3: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y3: nat,Z4: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y3 @ V2 ) @ ( plus_plus_nat @ U2 @ Z4 ) ) )
          @ ( rep_Integ @ X3 )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_4343_nat__of__integer__of__nat,axiom,
    ! [N2: nat] :
      ( ( code_nat_of_integer @ ( semiri4939895301339042750nteger @ N2 ) )
      = N2 ) ).

% nat_of_integer_of_nat
thf(fact_4344_int__of__integer__of__nat,axiom,
    ! [N2: nat] :
      ( ( code_int_of_integer @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( semiri1314217659103216013at_int @ N2 ) ) ).

% int_of_integer_of_nat
thf(fact_4345_integer__of__int__int__of__integer,axiom,
    ! [K: code_integer] :
      ( ( code_integer_of_int @ ( code_int_of_integer @ K ) )
      = K ) ).

% integer_of_int_int_of_integer
thf(fact_4346_int__of__integer__integer__of__int,axiom,
    ! [K: int] :
      ( ( code_int_of_integer @ ( code_integer_of_int @ K ) )
      = K ) ).

% int_of_integer_integer_of_int
thf(fact_4347_int__of__integer__inverse,axiom,
    ! [X: code_integer] :
      ( ( code_integer_of_int @ ( code_int_of_integer @ X ) )
      = X ) ).

% int_of_integer_inverse
thf(fact_4348_apsnd__conv,axiom,
    ! [F: produc8923325533196201883nteger > produc8923325533196201883nteger,X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
      ( ( produc717174245515237944e_term @ F @ ( produc8603105652947943368nteger @ X @ Y ) )
      = ( produc8603105652947943368nteger @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_4349_apsnd__conv,axiom,
    ! [F: produc3658429121746597890et_nat > produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
      ( ( produc8750854537940449737_nat_o @ F @ ( produc5001842942810119800et_nat @ X @ Y ) )
      = ( produc5001842942810119800et_nat @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_4350_apsnd__conv,axiom,
    ! [F: produc3658429121746597890et_nat > produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
      ( ( produc7714581247149323085_nat_o @ F @ ( produc5001842942810119800et_nat @ X @ Y ) )
      = ( produc2245416461498447860et_nat @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_4351_apsnd__conv,axiom,
    ! [F: produc3925858234332021118et_nat > produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
      ( ( produc1515462096303866701_nat_o @ F @ ( produc2245416461498447860et_nat @ X @ Y ) )
      = ( produc5001842942810119800et_nat @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_4352_apsnd__conv,axiom,
    ! [F: produc3925858234332021118et_nat > produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
      ( ( produc969530845752564945_nat_o @ F @ ( produc2245416461498447860et_nat @ X @ Y ) )
      = ( produc2245416461498447860et_nat @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_4353_apsnd__conv,axiom,
    ! [F: product_prod_int_int > product_prod_int_int,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( produc4973430039190721449e_term @ F @ ( produc5700946648718959541nt_int @ X @ Y ) )
      = ( produc5700946648718959541nt_int @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_4354_apsnd__conv,axiom,
    ! [F: product_prod_int_int > product_prod_int_int,X: int > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( produc7011684061226097695e_term @ F @ ( produc4305682042979456191nt_int @ X @ Y ) )
      = ( produc4305682042979456191nt_int @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_4355_apsnd__conv,axiom,
    ! [F: code_integer > code_integer,X: code_integer,Y: code_integer] :
      ( ( produc6499014454317279255nteger @ F @ ( produc1086072967326762835nteger @ X @ Y ) )
      = ( produc1086072967326762835nteger @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_4356_prod_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [Uu: nat] : one_one_nat
        @ A2 )
      = one_one_nat ) ).

% prod.neutral_const
thf(fact_4357_prod_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [Uu: nat] : one_one_int
        @ A2 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_4358_prod_Oneutral__const,axiom,
    ! [A2: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [Uu: int] : one_one_int
        @ A2 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_4359_of__int__integer__of,axiom,
    ! [K: code_integer] :
      ( ( ring_18347121197199848620nteger @ ( code_int_of_integer @ K ) )
      = K ) ).

% of_int_integer_of
thf(fact_4360_int__of__integer__of__int,axiom,
    ! [K: int] :
      ( ( code_int_of_integer @ ( ring_18347121197199848620nteger @ K ) )
      = K ) ).

% int_of_integer_of_int
thf(fact_4361_zero__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ zero_z3403309356797280102nteger )
    = zero_zero_int ) ).

% zero_integer.rep_eq
thf(fact_4362_int__of__integer__numeral,axiom,
    ! [K: num] :
      ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% int_of_integer_numeral
thf(fact_4363_plus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X @ Xa ) )
      = ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).

% plus_integer.rep_eq
thf(fact_4364_one__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ one_one_Code_integer )
    = one_one_int ) ).

% one_integer.rep_eq
thf(fact_4365_uminus__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( uminus1351360451143612070nteger @ X ) )
      = ( uminus_uminus_int @ ( code_int_of_integer @ X ) ) ) ).

% uminus_integer.rep_eq
thf(fact_4366_times__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X @ Xa ) )
      = ( times_times_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).

% times_integer.rep_eq
thf(fact_4367_minus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X @ Xa ) )
      = ( minus_minus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).

% minus_integer.rep_eq
thf(fact_4368_abs__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( abs_abs_Code_integer @ X ) )
      = ( abs_abs_int @ ( code_int_of_integer @ X ) ) ) ).

% abs_integer.rep_eq
thf(fact_4369_divide__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X @ Xa ) )
      = ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).

% divide_integer.rep_eq
thf(fact_4370_modulo__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( modulo364778990260209775nteger @ X @ Xa ) )
      = ( modulo_modulo_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).

% modulo_integer.rep_eq
thf(fact_4371_sgn__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( sgn_sgn_Code_integer @ X ) )
      = ( sgn_sgn_int @ ( code_int_of_integer @ X ) ) ) ).

% sgn_integer.rep_eq
thf(fact_4372_prod_OatMost__Suc,axiom,
    ! [G: nat > assn,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_4373_prod_OatMost__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_4374_prod_OatMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_4375_prod_OatMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_4376_prod_OlessThan__Suc,axiom,
    ! [G: nat > assn,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_4377_prod_OlessThan__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_4378_prod_OlessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_4379_prod_OlessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_4380_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
     => ( ( code_nat_of_integer @ K )
        = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
thf(fact_4381_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > code_integer] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = one_one_Code_integer ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_4382_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > assn] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = one_one_assn ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_4383_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_4384_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_4385_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M2 )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_4386_prod_Oop__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > code_integer] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = one_one_Code_integer ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_4387_prod_Oop__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > assn] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = one_one_assn ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_4388_prod_Oop__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_4389_prod_Oop__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_4390_prod_Oop__ivl__Suc,axiom,
    ! [N2: nat,M2: nat,G: nat > int] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.op_ivl_Suc
thf(fact_4391_xor__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( bit_se3222712562003087583nteger @ X @ Xa ) )
      = ( bit_se6526347334894502574or_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).

% xor_integer.rep_eq
thf(fact_4392_push__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa: code_integer] :
      ( ( code_int_of_integer @ ( bit_se7788150548672797655nteger @ X @ Xa ) )
      = ( bit_se545348938243370406it_int @ X @ ( code_int_of_integer @ Xa ) ) ) ).

% push_bit_integer.rep_eq
thf(fact_4393_xor__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( ( bit_se3222712562003087583nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se6526347334894502574or_int @ Xa @ X ) ) ) ).

% xor_integer.abs_eq
thf(fact_4394_nat__of__integer_Orep__eq,axiom,
    ( code_nat_of_integer
    = ( ^ [X3: code_integer] : ( nat2 @ ( code_int_of_integer @ X3 ) ) ) ) ).

% nat_of_integer.rep_eq
thf(fact_4395_integer__eq__iff,axiom,
    ( ( ^ [Y4: code_integer,Z2: code_integer] : ( Y4 = Z2 ) )
    = ( ^ [K4: code_integer,L3: code_integer] :
          ( ( code_int_of_integer @ K4 )
          = ( code_int_of_integer @ L3 ) ) ) ) ).

% integer_eq_iff
thf(fact_4396_integer__eqI,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( code_int_of_integer @ K )
        = ( code_int_of_integer @ L ) )
     => ( K = L ) ) ).

% integer_eqI
thf(fact_4397_int__of__integer__inject,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( code_int_of_integer @ X )
        = ( code_int_of_integer @ Y ) )
      = ( X = Y ) ) ).

% int_of_integer_inject
thf(fact_4398_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: $o > code_integer,A2: set_o] :
      ( ( ( groups7694694392188491536nteger @ G @ A2 )
       != one_one_Code_integer )
     => ~ ! [A3: $o] :
            ( ( member_o @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_Code_integer ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_4399_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > code_integer,A2: set_nat] :
      ( ( ( groups3455450783089532116nteger @ G @ A2 )
       != one_one_Code_integer )
     => ~ ! [A3: nat] :
            ( ( member_nat @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_Code_integer ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_4400_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > code_integer,A2: set_int] :
      ( ( ( groups3827104343326376752nteger @ G @ A2 )
       != one_one_Code_integer )
     => ~ ! [A3: int] :
            ( ( member_int @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_Code_integer ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_4401_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: $o > assn,A2: set_o] :
      ( ( ( groups5301882518646026715o_assn @ G @ A2 )
       != one_one_assn )
     => ~ ! [A3: $o] :
            ( ( member_o @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_assn ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_4402_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > assn,A2: set_nat] :
      ( ( ( groups6906906614972039071t_assn @ G @ A2 )
       != one_one_assn )
     => ~ ! [A3: nat] :
            ( ( member_nat @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_assn ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_4403_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > assn,A2: set_int] :
      ( ( ( groups7882442080178216443t_assn @ G @ A2 )
       != one_one_assn )
     => ~ ! [A3: int] :
            ( ( member_int @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_assn ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_4404_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: $o > rat,A2: set_o] :
      ( ( ( groups2869687844427037835_o_rat @ G @ A2 )
       != one_one_rat )
     => ~ ! [A3: $o] :
            ( ( member_o @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_4405_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > rat,A2: set_nat] :
      ( ( ( groups73079841787564623at_rat @ G @ A2 )
       != one_one_rat )
     => ~ ! [A3: nat] :
            ( ( member_nat @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_4406_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > rat,A2: set_int] :
      ( ( ( groups1072433553688619179nt_rat @ G @ A2 )
       != one_one_rat )
     => ~ ! [A3: int] :
            ( ( member_int @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_4407_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: $o > nat,A2: set_o] :
      ( ( ( groups3504817904513533571_o_nat @ G @ A2 )
       != one_one_nat )
     => ~ ! [A3: $o] :
            ( ( member_o @ A3 @ A2 )
           => ( ( G @ A3 )
              = one_one_nat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_4408_prod_Oneutral,axiom,
    ! [A2: set_nat,G: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ( G @ X4 )
            = one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G @ A2 )
        = one_one_nat ) ) ).

% prod.neutral
thf(fact_4409_prod_Oneutral,axiom,
    ! [A2: set_nat,G: nat > int] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ( G @ X4 )
            = one_one_int ) )
     => ( ( groups705719431365010083at_int @ G @ A2 )
        = one_one_int ) ) ).

% prod.neutral
thf(fact_4410_prod_Oneutral,axiom,
    ! [A2: set_int,G: int > int] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ( G @ X4 )
            = one_one_int ) )
     => ( ( groups1705073143266064639nt_int @ G @ A2 )
        = one_one_int ) ) ).

% prod.neutral
thf(fact_4411_apsnd__compose,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,X: produc8923325533196201883nteger] :
      ( ( produc9020447175693601247nteger @ F @ ( produc6499014454317279255nteger @ G @ X ) )
      = ( produc9020447175693601247nteger @ ( comp_C1593894019821074884nteger @ F @ G ) @ X ) ) ).

% apsnd_compose
thf(fact_4412_apsnd__compose,axiom,
    ! [F: code_integer > code_integer,G: num > code_integer,X: produc7142191362829087448er_num] :
      ( ( produc6499014454317279255nteger @ F @ ( produc7418200839687411156nteger @ G @ X ) )
      = ( produc7418200839687411156nteger @ ( comp_C3531382070062128313er_num @ F @ G ) @ X ) ) ).

% apsnd_compose
thf(fact_4413_apsnd__compose,axiom,
    ! [F: code_integer > code_integer,G: code_integer > code_integer,X: produc8923325533196201883nteger] :
      ( ( produc6499014454317279255nteger @ F @ ( produc6499014454317279255nteger @ G @ X ) )
      = ( produc6499014454317279255nteger @ ( comp_C7449957260575251196nteger @ F @ G ) @ X ) ) ).

% apsnd_compose
thf(fact_4414_prod__ge__1,axiom,
    ! [A2: set_o,F: $o > code_integer] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X4 ) ) )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups7694694392188491536nteger @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_4415_prod__ge__1,axiom,
    ! [A2: set_nat,F: nat > code_integer] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X4 ) ) )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_4416_prod__ge__1,axiom,
    ! [A2: set_int,F: int > code_integer] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ X4 ) ) )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_4417_prod__ge__1,axiom,
    ! [A2: set_o,F: $o > rat] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups2869687844427037835_o_rat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_4418_prod__ge__1,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_4419_prod__ge__1,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_4420_prod__ge__1,axiom,
    ! [A2: set_o,F: $o > nat] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups3504817904513533571_o_nat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_4421_prod__ge__1,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_4422_prod__ge__1,axiom,
    ! [A2: set_o,F: $o > int] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ord_less_eq_int @ one_one_int @ ( F @ X4 ) ) )
     => ( ord_less_eq_int @ one_one_int @ ( groups3502327434004483295_o_int @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_4423_prod__ge__1,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
     => ( ord_less_eq_nat @ one_one_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).

% prod_ge_1
thf(fact_4424_int__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Y ) )
      = ( ord_le6747313008572928689nteger @ X @ Y ) ) ).

% int_of_integer_less_iff
thf(fact_4425_integer__less__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [K4: code_integer,L3: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K4 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).

% integer_less_iff
thf(fact_4426_less__integer_Orep__eq,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X3: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_integer.rep_eq
thf(fact_4427_less__eq__integer_Orep__eq,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [X3: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_eq_integer.rep_eq
thf(fact_4428_integer__less__eq__iff,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [K4: code_integer,L3: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K4 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).

% integer_less_eq_iff
thf(fact_4429_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_4430_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > int,M2: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_4431_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ).

% prod.shift_bounds_Suc_ivl
thf(fact_4432_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [G: nat > int,M2: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ).

% prod.shift_bounds_Suc_ivl
thf(fact_4433_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M2: nat,K: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_4434_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > int,M2: nat,K: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_4435_prod_Oshift__bounds__nat__ivl,axiom,
    ! [G: nat > nat,M2: nat,K: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ).

% prod.shift_bounds_nat_ivl
thf(fact_4436_prod_Oshift__bounds__nat__ivl,axiom,
    ! [G: nat > int,M2: nat,K: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ).

% prod.shift_bounds_nat_ivl
thf(fact_4437_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( set_or8404916559141939852nteger @ L @ ( plus_p5714425477246183910nteger @ U @ one_one_Code_integer ) )
      = ( set_or189985376899183464nteger @ L @ U ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
thf(fact_4438_prod__le__1,axiom,
    ! [A2: set_o,F: $o > code_integer] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X4 ) )
            & ( ord_le3102999989581377725nteger @ ( F @ X4 ) @ one_one_Code_integer ) ) )
     => ( ord_le3102999989581377725nteger @ ( groups7694694392188491536nteger @ F @ A2 ) @ one_one_Code_integer ) ) ).

% prod_le_1
thf(fact_4439_prod__le__1,axiom,
    ! [A2: set_nat,F: nat > code_integer] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X4 ) )
            & ( ord_le3102999989581377725nteger @ ( F @ X4 ) @ one_one_Code_integer ) ) )
     => ( ord_le3102999989581377725nteger @ ( groups3455450783089532116nteger @ F @ A2 ) @ one_one_Code_integer ) ) ).

% prod_le_1
thf(fact_4440_prod__le__1,axiom,
    ! [A2: set_int,F: int > code_integer] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ X4 ) )
            & ( ord_le3102999989581377725nteger @ ( F @ X4 ) @ one_one_Code_integer ) ) )
     => ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F @ A2 ) @ one_one_Code_integer ) ) ).

% prod_le_1
thf(fact_4441_prod__le__1,axiom,
    ! [A2: set_o,F: $o > rat] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
            & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups2869687844427037835_o_rat @ F @ A2 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_4442_prod__le__1,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
            & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_4443_prod__le__1,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
            & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_4444_prod__le__1,axiom,
    ! [A2: set_o,F: $o > nat] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
            & ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups3504817904513533571_o_nat @ F @ A2 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_4445_prod__le__1,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
            & ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_4446_prod__le__1,axiom,
    ! [A2: set_o,F: $o > int] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) )
            & ( ord_less_eq_int @ ( F @ X4 ) @ one_one_int ) ) )
     => ( ord_less_eq_int @ ( groups3502327434004483295_o_int @ F @ A2 ) @ one_one_int ) ) ).

% prod_le_1
thf(fact_4447_prod__le__1,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
            & ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
     => ( ord_less_eq_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ one_one_nat ) ) ).

% prod_le_1
thf(fact_4448_prod_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat,G: nat > nat,H3: nat > nat] :
      ( ( A = C )
     => ( ( B = D2 )
       => ( ! [X4: nat] :
              ( ( ord_less_eq_nat @ C @ X4 )
             => ( ( ord_less_nat @ X4 @ D2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) ) )
         => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
            = ( groups708209901874060359at_nat @ H3 @ ( set_or4665077453230672383an_nat @ C @ D2 ) ) ) ) ) ) ).

% prod.ivl_cong
thf(fact_4449_prod_Oivl__cong,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat,G: nat > int,H3: nat > int] :
      ( ( A = C )
     => ( ( B = D2 )
       => ( ! [X4: nat] :
              ( ( ord_less_eq_nat @ C @ X4 )
             => ( ( ord_less_nat @ X4 @ D2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) ) )
         => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ A @ B ) )
            = ( groups705719431365010083at_int @ H3 @ ( set_or4665077453230672383an_nat @ C @ D2 ) ) ) ) ) ) ).

% prod.ivl_cong
thf(fact_4450_prod_Oivl__cong,axiom,
    ! [A: int,C: int,B: int,D2: int,G: int > int,H3: int > int] :
      ( ( A = C )
     => ( ( B = D2 )
       => ( ! [X4: int] :
              ( ( ord_less_eq_int @ C @ X4 )
             => ( ( ord_less_int @ X4 @ D2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) ) )
         => ( ( groups1705073143266064639nt_int @ G @ ( set_or4662586982721622107an_int @ A @ B ) )
            = ( groups1705073143266064639nt_int @ H3 @ ( set_or4662586982721622107an_int @ C @ D2 ) ) ) ) ) ) ).

% prod.ivl_cong
thf(fact_4451_take__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa: code_integer] :
      ( ( code_int_of_integer @ ( bit_se1745604003318907178nteger @ X @ Xa ) )
      = ( bit_se2923211474154528505it_int @ X @ ( code_int_of_integer @ Xa ) ) ) ).

% take_bit_integer.rep_eq
thf(fact_4452_and__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( bit_se3949692690581998587nteger @ X @ Xa ) )
      = ( bit_se725231765392027082nd_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).

% and_integer.rep_eq
thf(fact_4453_bit__integer_Orep__eq,axiom,
    ( bit_se9216721137139052372nteger
    = ( ^ [X3: code_integer] : ( bit_se1146084159140164899it_int @ ( code_int_of_integer @ X3 ) ) ) ) ).

% bit_integer.rep_eq
thf(fact_4454_prod_OatLeastLessThan__concat,axiom,
    ! [M2: nat,N2: nat,P4: nat,G: nat > assn] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) )
          = ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M2 @ P4 ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_4455_prod_OatLeastLessThan__concat,axiom,
    ! [M2: nat,N2: nat,P4: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) )
          = ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ P4 ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_4456_prod_OatLeastLessThan__concat,axiom,
    ! [M2: nat,N2: nat,P4: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) )
          = ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ P4 ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_4457_prod_OatLeastLessThan__concat,axiom,
    ! [M2: nat,N2: nat,P4: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ P4 )
       => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ N2 @ P4 ) ) )
          = ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ P4 ) ) ) ) ) ).

% prod.atLeastLessThan_concat
thf(fact_4458_mask__integer_Orep__eq,axiom,
    ! [X: nat] :
      ( ( code_int_of_integer @ ( bit_se2119862282449309892nteger @ X ) )
      = ( bit_se2000444600071755411sk_int @ X ) ) ).

% mask_integer.rep_eq
thf(fact_4459_integer__of__num__def,axiom,
    code_integer_of_num = numera6620942414471956472nteger ).

% integer_of_num_def
thf(fact_4460_unset__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa: code_integer] :
      ( ( code_int_of_integer @ ( bit_se8260200283734997820nteger @ X @ Xa ) )
      = ( bit_se4203085406695923979it_int @ X @ ( code_int_of_integer @ Xa ) ) ) ).

% unset_bit_integer.rep_eq
thf(fact_4461_set__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa: code_integer] :
      ( ( code_int_of_integer @ ( bit_se2793503036327961859nteger @ X @ Xa ) )
      = ( bit_se7879613467334960850it_int @ X @ ( code_int_of_integer @ Xa ) ) ) ).

% set_bit_integer.rep_eq
thf(fact_4462_flip__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa: code_integer] :
      ( ( code_int_of_integer @ ( bit_se1345352211410354436nteger @ X @ Xa ) )
      = ( bit_se2159334234014336723it_int @ X @ ( code_int_of_integer @ Xa ) ) ) ).

% flip_bit_integer.rep_eq
thf(fact_4463_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G @ suc ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.atLeast_Suc_atMost_Suc_shift
thf(fact_4464_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [G: nat > int,M2: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups705719431365010083at_int @ ( comp_nat_int_nat @ G @ suc ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.atLeast_Suc_atMost_Suc_shift
thf(fact_4465_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G @ suc ) @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ).

% prod.atLeast_Suc_lessThan_Suc_shift
thf(fact_4466_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [G: nat > int,M2: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups705719431365010083at_int @ ( comp_nat_int_nat @ G @ suc ) @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ).

% prod.atLeast_Suc_lessThan_Suc_shift
thf(fact_4467_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
      = ( set_or1266510415728281911st_int @ L @ U ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_4468_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [G: nat > nat,M2: nat,K: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G @ ( plus_plus_nat @ K ) ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.atLeastAtMost_shift_bounds
thf(fact_4469_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [G: nat > int,M2: nat,K: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups705719431365010083at_int @ ( comp_nat_int_nat @ G @ ( plus_plus_nat @ K ) ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.atLeastAtMost_shift_bounds
thf(fact_4470_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [G: nat > nat,M2: nat,K: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G @ ( plus_plus_nat @ K ) ) @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ).

% prod.atLeastLessThan_shift_bounds
thf(fact_4471_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [G: nat > int,M2: nat,K: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups705719431365010083at_int @ ( comp_nat_int_nat @ G @ ( plus_plus_nat @ K ) ) @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ).

% prod.atLeastLessThan_shift_bounds
thf(fact_4472_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N2: nat,M2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M2 ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_4473_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > int,N2: nat,M2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N2 @ M2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M2 ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_4474_prod_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nat_diff_reindex
thf(fact_4475_prod_Onat__diff__reindex,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nat_diff_reindex
thf(fact_4476_nat__of__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( code_nat_of_integer @ ( code_integer_of_int @ X ) )
      = ( nat2 @ X ) ) ).

% nat_of_integer.abs_eq
thf(fact_4477_num__of__integer_Orep__eq,axiom,
    ( code_num_of_integer
    = ( ^ [X3: code_integer] : ( num_of_nat @ ( nat2 @ ( code_int_of_integer @ X3 ) ) ) ) ) ).

% num_of_integer.rep_eq
thf(fact_4478_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > assn,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_4479_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_4480_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_4481_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_4482_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > assn,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_4483_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_4484_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_4485_prod_OatLeast0__lessThan__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.atLeast0_lessThan_Suc
thf(fact_4486_prod_OatLeast__Suc__lessThan,axiom,
    ! [M2: nat,N2: nat,G: nat > assn] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
        = ( times_times_assn @ ( G @ M2 ) @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_4487_prod_OatLeast__Suc__lessThan,axiom,
    ! [M2: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
        = ( times_times_rat @ ( G @ M2 ) @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_4488_prod_OatLeast__Suc__lessThan,axiom,
    ! [M2: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
        = ( times_times_nat @ ( G @ M2 ) @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_4489_prod_OatLeast__Suc__lessThan,axiom,
    ! [M2: nat,N2: nat,G: nat > int] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
        = ( times_times_int @ ( G @ M2 ) @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_lessThan
thf(fact_4490_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N2: nat,G: nat > assn] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
        = ( times_times_assn @ ( G @ ( suc @ N2 ) ) @ ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_4491_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
        = ( times_times_rat @ ( G @ ( suc @ N2 ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_4492_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
        = ( times_times_nat @ ( G @ ( suc @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_4493_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) ) )
        = ( times_times_int @ ( G @ ( suc @ N2 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_4494_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N2: nat,G: nat > assn] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( times_times_assn @ ( G @ M2 ) @ ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_4495_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( times_times_rat @ ( G @ M2 ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_4496_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( times_times_nat @ ( G @ M2 ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_4497_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( times_times_int @ ( G @ M2 ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_4498_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > assn] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_4499_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_4500_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_4501_prod_OatLeastLessThan__Suc,axiom,
    ! [A: nat,B: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ A @ ( suc @ B ) ) )
        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ A @ B ) ) @ ( G @ B ) ) ) ) ).

% prod.atLeastLessThan_Suc
thf(fact_4502_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).

% divmod_abs_code(6)
thf(fact_4503_prod_Olast__plus,axiom,
    ! [M2: nat,N2: nat,G: nat > assn] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( times_times_assn @ ( G @ N2 ) @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.last_plus
thf(fact_4504_prod_Olast__plus,axiom,
    ! [M2: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( times_times_rat @ ( G @ N2 ) @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.last_plus
thf(fact_4505_prod_Olast__plus,axiom,
    ! [M2: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( times_times_nat @ ( G @ N2 ) @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.last_plus
thf(fact_4506_prod_Olast__plus,axiom,
    ! [M2: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( times_times_int @ ( G @ N2 ) @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.last_plus
thf(fact_4507_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N2: nat,G: nat > assn] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_assn @ ( G @ M2 )
          @ ( groups6906906614972039071t_assn
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_4508_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_rat @ ( G @ M2 )
          @ ( groups73079841787564623at_rat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_4509_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_nat @ ( G @ M2 )
          @ ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_4510_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( times_times_int @ ( G @ M2 )
          @ ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_4511_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > assn,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_assn @ ( G @ zero_zero_nat )
        @ ( groups6906906614972039071t_assn
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_4512_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( G @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_4513_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_4514_prod_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_4515_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > assn,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_assn @ ( G @ zero_zero_nat )
        @ ( groups6906906614972039071t_assn
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_4516_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( G @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_4517_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_4518_prod_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_4519_prod_OatLeastLessThan__rev,axiom,
    ! [G: nat > nat,N2: nat,M2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ M2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ ( suc @ I3 ) ) )
        @ ( set_or4665077453230672383an_nat @ N2 @ M2 ) ) ) ).

% prod.atLeastLessThan_rev
thf(fact_4520_prod_OatLeastLessThan__rev,axiom,
    ! [G: nat > int,N2: nat,M2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ N2 @ M2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ ( suc @ I3 ) ) )
        @ ( set_or4665077453230672383an_nat @ N2 @ M2 ) ) ) ).

% prod.atLeastLessThan_rev
thf(fact_4521_prod_Onested__swap,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( groups708209901874060359at_nat @ ( A @ I3 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [J2: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( A @ I3 @ J2 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J2 ) @ N2 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% prod.nested_swap
thf(fact_4522_prod_Onested__swap,axiom,
    ! [A: nat > nat > int,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( groups705719431365010083at_int @ ( A @ I3 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [J2: nat] :
            ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( A @ I3 @ J2 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J2 ) @ N2 ) )
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% prod.nested_swap
thf(fact_4523_prod_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [K4: nat] : ( G @ ( suc @ K4 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_4524_prod_OatLeast1__atMost__eq,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [K4: nat] : ( G @ ( suc @ K4 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_4525_prod_Onat__group,axiom,
    ! [G: nat > nat,K: nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [M: nat] : ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M @ K ) @ ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ K ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( times_times_nat @ N2 @ K ) ) ) ) ).

% prod.nat_group
thf(fact_4526_prod_Onat__group,axiom,
    ! [G: nat > int,K: nat,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [M: nat] : ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ ( times_times_nat @ M @ K ) @ ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ K ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( times_times_nat @ N2 @ K ) ) ) ) ).

% prod.nat_group
thf(fact_4527_fact__prod,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N4: nat] :
          ( semiri1314217659103216013at_int
          @ ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ) ).

% fact_prod
thf(fact_4528_fact__prod,axiom,
    ( semiri3624122377584611663nteger
    = ( ^ [N4: nat] :
          ( semiri4939895301339042750nteger
          @ ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ) ).

% fact_prod
thf(fact_4529_fact__prod,axiom,
    ( semiri2447717529341329178atural
    = ( ^ [N4: nat] :
          ( semiri3763490453095760265atural
          @ ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ) ).

% fact_prod
thf(fact_4530_fact__prod,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N4: nat] :
          ( semiri1316708129612266289at_nat
          @ ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ) ).

% fact_prod
thf(fact_4531_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( groups708209901874060359at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [J2: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( A @ I3 @ J2 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J2 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nested_swap'
thf(fact_4532_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > int,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( groups705719431365010083at_int @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [J2: nat] :
            ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( A @ I3 @ J2 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J2 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.nested_swap'
thf(fact_4533_prod__atLeastAtMost__code,axiom,
    ! [F: nat > code_integer,A: nat,B: nat] :
      ( ( groups3455450783089532116nteger @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1084959871951514735nteger
        @ ^ [A5: nat] : ( times_3573771949741848930nteger @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_Code_integer ) ) ).

% prod_atLeastAtMost_code
thf(fact_4534_prod__atLeastAtMost__code,axiom,
    ! [F: nat > assn,A: nat,B: nat] :
      ( ( groups6906906614972039071t_assn @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1959793692361082170t_assn
        @ ^ [A5: nat] : ( times_times_assn @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_assn ) ) ).

% prod_atLeastAtMost_code
thf(fact_4535_prod__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B: nat] :
      ( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A5: nat] : ( times_times_rat @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_rat ) ) ).

% prod_atLeastAtMost_code
thf(fact_4536_prod__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B: nat] :
      ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A5: nat] : ( times_times_nat @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_nat ) ) ).

% prod_atLeastAtMost_code
thf(fact_4537_prod__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B: nat] :
      ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A5: nat] : ( times_times_int @ ( F @ A5 ) )
        @ A
        @ B
        @ one_one_int ) ) ).

% prod_atLeastAtMost_code
thf(fact_4538_prod_Ohead__if,axiom,
    ! [N2: nat,M2: nat,G: nat > code_integer] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = one_one_Code_integer ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups3455450783089532116nteger @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.head_if
thf(fact_4539_prod_Ohead__if,axiom,
    ! [N2: nat,M2: nat,G: nat > assn] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = one_one_assn ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.head_if
thf(fact_4540_prod_Ohead__if,axiom,
    ! [N2: nat,M2: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.head_if
thf(fact_4541_prod_Ohead__if,axiom,
    ! [N2: nat,M2: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.head_if
thf(fact_4542_prod_Ohead__if,axiom,
    ! [N2: nat,M2: nat,G: nat > int] :
      ( ( ( ord_less_nat @ N2 @ M2 )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ N2 @ M2 )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) @ ( G @ N2 ) ) ) ) ) ).

% prod.head_if
thf(fact_4543_prod_Oub__add__nat,axiom,
    ! [M2: nat,N2: nat,G: nat > assn,P4: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( times_times_assn @ ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_4544_prod_Oub__add__nat,axiom,
    ! [M2: nat,N2: nat,G: nat > rat,P4: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_4545_prod_Oub__add__nat,axiom,
    ! [M2: nat,N2: nat,G: nat > nat,P4: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_4546_prod_Oub__add__nat,axiom,
    ! [M2: nat,N2: nat,G: nat > int,P4: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_4547_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [G: nat > assn,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_assn @ ( G @ zero_zero_nat ) @ ( groups6906906614972039071t_assn @ ( comp_nat_assn_nat @ G @ suc ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc_shift
thf(fact_4548_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( G @ zero_zero_nat ) @ ( groups73079841787564623at_rat @ ( comp_nat_rat_nat @ G @ suc ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc_shift
thf(fact_4549_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat ) @ ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G @ suc ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc_shift
thf(fact_4550_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat ) @ ( groups705719431365010083at_int @ ( comp_nat_int_nat @ G @ suc ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% prod.atLeast0_atMost_Suc_shift
thf(fact_4551_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [G: nat > assn,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_assn @ ( G @ zero_zero_nat ) @ ( groups6906906614972039071t_assn @ ( comp_nat_assn_nat @ G @ suc ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% prod.atLeast0_lessThan_Suc_shift
thf(fact_4552_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( G @ zero_zero_nat ) @ ( groups73079841787564623at_rat @ ( comp_nat_rat_nat @ G @ suc ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% prod.atLeast0_lessThan_Suc_shift
thf(fact_4553_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_nat @ ( G @ zero_zero_nat ) @ ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G @ suc ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% prod.atLeast0_lessThan_Suc_shift
thf(fact_4554_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( G @ zero_zero_nat ) @ ( groups705719431365010083at_int @ ( comp_nat_int_nat @ G @ suc ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% prod.atLeast0_lessThan_Suc_shift
thf(fact_4555_prod_OatLeastLessThan__shift__0,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
      = ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G @ ( plus_plus_nat @ M2 ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ).

% prod.atLeastLessThan_shift_0
thf(fact_4556_prod_OatLeastLessThan__shift__0,axiom,
    ! [G: nat > int,M2: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) )
      = ( groups705719431365010083at_int @ ( comp_nat_int_nat @ G @ ( plus_plus_nat @ M2 ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ).

% prod.atLeastLessThan_shift_0
thf(fact_4557_integer__of__num__triv_I1_J,axiom,
    ( ( code_integer_of_num @ one )
    = one_one_Code_integer ) ).

% integer_of_num_triv(1)
thf(fact_4558_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: nat > nat,N2: nat,M2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ N2 @ M2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M2 ) ) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_4559_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [G: nat > int,N2: nat,M2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ N2 @ M2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M2 ) ) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_4560_prod_OatMost__shift,axiom,
    ! [G: nat > assn,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_assn @ ( G @ zero_zero_nat )
        @ ( groups6906906614972039071t_assn
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_4561_prod_OatMost__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_rat @ ( G @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_4562_prod_OatMost__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_nat @ ( G @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_4563_prod_OatMost__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_int @ ( G @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% prod.atMost_shift
thf(fact_4564_pochhammer__prod,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A5: rat,N4: nat] :
          ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ I3 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) ) ) ).

% pochhammer_prod
thf(fact_4565_pochhammer__prod,axiom,
    ( comm_s8582702949713902594nteger
    = ( ^ [A5: code_integer,N4: nat] :
          ( groups3455450783089532116nteger
          @ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A5 @ ( semiri4939895301339042750nteger @ I3 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) ) ) ).

% pochhammer_prod
thf(fact_4566_pochhammer__prod,axiom,
    ( comm_s7406298101470620109atural
    = ( ^ [A5: code_natural,N4: nat] :
          ( groups2279045934846249631atural
          @ ^ [I3: nat] : ( plus_p4538020629002901425atural @ A5 @ ( semiri3763490453095760265atural @ I3 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) ) ) ).

% pochhammer_prod
thf(fact_4567_pochhammer__prod,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A5: nat,N4: nat] :
          ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ I3 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) ) ) ).

% pochhammer_prod
thf(fact_4568_pochhammer__prod,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A5: int,N4: nat] :
          ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ I3 ) )
          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) ) ) ).

% pochhammer_prod
thf(fact_4569_prod_OatLeast__atMost__pred__shift,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ( comp_nat_nat_nat @ G
          @ ^ [N4: nat] : ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.atLeast_atMost_pred_shift
thf(fact_4570_prod_OatLeast__atMost__pred__shift,axiom,
    ! [G: nat > int,M2: nat,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ( comp_nat_int_nat @ G
          @ ^ [N4: nat] : ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) )
        @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.atLeast_atMost_pred_shift
thf(fact_4571_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups708209901874060359at_nat
        @ ( comp_nat_nat_nat @ G
          @ ^ [N4: nat] : ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) )
        @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups708209901874060359at_nat @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ).

% prod.atLeast_lessThan_pred_shift
thf(fact_4572_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [G: nat > int,M2: nat,N2: nat] :
      ( ( groups705719431365010083at_int
        @ ( comp_nat_int_nat @ G
          @ ^ [N4: nat] : ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) )
        @ ( set_or4665077453230672383an_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( groups705719431365010083at_int @ G @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ).

% prod.atLeast_lessThan_pred_shift
thf(fact_4573_integer__of__num_I2_J,axiom,
    ! [N2: num] :
      ( ( code_integer_of_num @ ( bit0 @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N2 ) @ ( code_integer_of_num @ N2 ) ) ) ).

% integer_of_num(2)
thf(fact_4574_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).

% divmod_abs_code(5)
thf(fact_4575_prod_OatLeastAtMost__shift__0,axiom,
    ! [M2: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( groups708209901874060359at_nat @ ( comp_nat_nat_nat @ G @ ( plus_plus_nat @ M2 ) ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ).

% prod.atLeastAtMost_shift_0
thf(fact_4576_prod_OatLeastAtMost__shift__0,axiom,
    ! [M2: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) )
        = ( groups705719431365010083at_int @ ( comp_nat_int_nat @ G @ ( plus_plus_nat @ M2 ) ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) ).

% prod.atLeastAtMost_shift_0
thf(fact_4577_sum_OatLeast__int__lessThan__int__shift,axiom,
    ! [G: int > nat,M2: nat,N2: nat] :
      ( ( groups4541462559716669496nt_nat @ G @ ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( groups3542108847815614940at_nat @ ( comp_int_nat_nat @ G @ semiri1314217659103216013at_int ) @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ).

% sum.atLeast_int_lessThan_int_shift
thf(fact_4578_sum_OatLeast__int__lessThan__int__shift,axiom,
    ! [G: int > int,M2: nat,N2: nat] :
      ( ( groups4538972089207619220nt_int @ G @ ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( groups3539618377306564664at_int @ ( comp_int_int_nat @ G @ semiri1314217659103216013at_int ) @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ).

% sum.atLeast_int_lessThan_int_shift
thf(fact_4579_pochhammer__Suc__prod,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( groups73079841787564623at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_4580_pochhammer__Suc__prod,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( comm_s8582702949713902594nteger @ A @ ( suc @ N2 ) )
      = ( groups3455450783089532116nteger
        @ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A @ ( semiri4939895301339042750nteger @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_4581_pochhammer__Suc__prod,axiom,
    ! [A: code_natural,N2: nat] :
      ( ( comm_s7406298101470620109atural @ A @ ( suc @ N2 ) )
      = ( groups2279045934846249631atural
        @ ^ [I3: nat] : ( plus_p4538020629002901425atural @ A @ ( semiri3763490453095760265atural @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_4582_pochhammer__Suc__prod,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_4583_pochhammer__Suc__prod,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod
thf(fact_4584_pochhammer__prod__rev,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A5: rat,N4: nat] :
          ( groups73079841787564623at_rat
          @ ^ [I3: nat] : ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N4 @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_4585_pochhammer__prod__rev,axiom,
    ( comm_s8582702949713902594nteger
    = ( ^ [A5: code_integer,N4: nat] :
          ( groups3455450783089532116nteger
          @ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A5 @ ( semiri4939895301339042750nteger @ ( minus_minus_nat @ N4 @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_4586_pochhammer__prod__rev,axiom,
    ( comm_s7406298101470620109atural
    = ( ^ [A5: code_natural,N4: nat] :
          ( groups2279045934846249631atural
          @ ^ [I3: nat] : ( plus_p4538020629002901425atural @ A5 @ ( semiri3763490453095760265atural @ ( minus_minus_nat @ N4 @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_4587_pochhammer__prod__rev,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A5: nat,N4: nat] :
          ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N4 @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_4588_pochhammer__prod__rev,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A5: int,N4: nat] :
          ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N4 @ I3 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N4 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_4589_fact__div__fact,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N2 @ M2 )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N2 ) )
        = ( groups708209901874060359at_nat
          @ ^ [X3: nat] : X3
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M2 ) ) ) ) ).

% fact_div_fact
thf(fact_4590_integer__of__num__triv_I2_J,axiom,
    ( ( code_integer_of_num @ ( bit0 @ one ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% integer_of_num_triv(2)
thf(fact_4591_prod_Oin__pairs,axiom,
    ! [G: nat > assn,M2: nat,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6906906614972039071t_assn
        @ ^ [I3: nat] : ( times_times_assn @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_4592_prod_Oin__pairs,axiom,
    ! [G: nat > rat,M2: nat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [I3: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_4593_prod_Oin__pairs,axiom,
    ! [G: nat > nat,M2: nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_4594_prod_Oin__pairs,axiom,
    ! [G: nat > int,M2: nat,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.in_pairs
thf(fact_4595_prod_Oin__pairs__0,axiom,
    ! [G: nat > assn,N2: nat] :
      ( ( groups6906906614972039071t_assn @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6906906614972039071t_assn
        @ ^ [I3: nat] : ( times_times_assn @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_4596_prod_Oin__pairs__0,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [I3: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_4597_prod_Oin__pairs__0,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_4598_prod_Oin__pairs__0,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.in_pairs_0
thf(fact_4599_nat__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ord_less_nat @ ( code_nat_of_integer @ X ) @ ( code_nat_of_integer @ Y ) )
          = ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ) ).

% nat_of_integer_less_iff
thf(fact_4600_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( groups73079841787564623at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N2 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_4601_pochhammer__Suc__prod__rev,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( comm_s8582702949713902594nteger @ A @ ( suc @ N2 ) )
      = ( groups3455450783089532116nteger
        @ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A @ ( semiri4939895301339042750nteger @ ( minus_minus_nat @ N2 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_4602_pochhammer__Suc__prod__rev,axiom,
    ! [A: code_natural,N2: nat] :
      ( ( comm_s7406298101470620109atural @ A @ ( suc @ N2 ) )
      = ( groups2279045934846249631atural
        @ ^ [I3: nat] : ( plus_p4538020629002901425atural @ A @ ( semiri3763490453095760265atural @ ( minus_minus_nat @ N2 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_4603_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N2 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_4604_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I3: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ I3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_4605_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K4: code_integer,L3: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L3 )
            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K4 ) @ ( code_divmod_abs @ K4 @ L3 )
              @ ( produc6916734918728496179nteger
                @ ^ [R3: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R3 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L3 @ S4 ) ) )
                @ ( code_divmod_abs @ K4 @ L3 ) ) )
            @ ( if_Pro6119634080678213985nteger @ ( L3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K4 )
              @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
                @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K4 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K4 @ L3 )
                  @ ( produc6916734918728496179nteger
                    @ ^ [R3: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R3 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L3 ) @ S4 ) ) )
                    @ ( code_divmod_abs @ K4 @ L3 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_4606_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K4: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K4 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R3: code_integer,S4: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K4 ) @ R3 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R3 ) @ S4 ) ) @ ( S4 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_4607_prod_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > code_integer,H3: nat > code_integer] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups3455450783089532116nteger
            @ ^ [J2: nat] : ( if_Code_integer @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_Code_integer @ ( J2 = K ) @ one_one_Code_integer @ ( H3 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups3455450783089532116nteger
            @ ^ [J2: nat] : ( if_Code_integer @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H3 @ J2 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_4608_prod_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > assn,H3: nat > assn] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups6906906614972039071t_assn
            @ ^ [J2: nat] : ( if_assn @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_assn @ ( J2 = K ) @ one_one_assn @ ( H3 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups6906906614972039071t_assn
            @ ^ [J2: nat] : ( if_assn @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H3 @ J2 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_4609_prod_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > rat,H3: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups73079841787564623at_rat
            @ ^ [J2: nat] : ( if_rat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_rat @ ( J2 = K ) @ one_one_rat @ ( H3 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups73079841787564623at_rat
            @ ^ [J2: nat] : ( if_rat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H3 @ J2 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_4610_prod_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > nat,H3: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups708209901874060359at_nat
            @ ^ [J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_nat @ ( J2 = K ) @ one_one_nat @ ( H3 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups708209901874060359at_nat
            @ ^ [J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H3 @ J2 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_4611_prod_Ozero__middle,axiom,
    ! [P4: nat,K: nat,G: nat > int,H3: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P4 )
     => ( ( ord_less_eq_nat @ K @ P4 )
       => ( ( groups705719431365010083at_int
            @ ^ [J2: nat] : ( if_int @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( if_int @ ( J2 = K ) @ one_one_int @ ( H3 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P4 ) )
          = ( groups705719431365010083at_int
            @ ^ [J2: nat] : ( if_int @ ( ord_less_nat @ J2 @ K ) @ ( G @ J2 ) @ ( H3 @ J2 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_4612_xor__int__rec,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K4: int,L3: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K4 ) )
             != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_int_rec
thf(fact_4613_divmod__abs__def,axiom,
    ( code_divmod_abs
    = ( ^ [K4: code_integer,L3: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K4 ) @ ( abs_abs_Code_integer @ L3 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K4 ) @ ( abs_abs_Code_integer @ L3 ) ) ) ) ) ).

% divmod_abs_def
thf(fact_4614_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y3: nat,Z4: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y3 @ V2 ) @ ( plus_plus_nat @ U2 @ Z4 ) ) )
          @ ( rep_Integ @ X3 )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_int.rep_eq
thf(fact_4615_xor__int__unfold,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K4: int,L3: int] :
          ( if_int
          @ ( K4
            = ( uminus_uminus_int @ one_one_int ) )
          @ ( bit_ri7919022796975470100ot_int @ L3 )
          @ ( if_int
            @ ( L3
              = ( uminus_uminus_int @ one_one_int ) )
            @ ( bit_ri7919022796975470100ot_int @ K4 )
            @ ( if_int @ ( K4 = zero_zero_int ) @ L3 @ ( if_int @ ( L3 = zero_zero_int ) @ K4 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_4616_take__bit__num__code,axiom,
    ( bit_take_bit_num
    = ( ^ [N4: nat,M: num] :
          ( produc478579273971653890on_num
          @ ^ [A5: nat,X3: num] :
              ( case_nat_option_num @ none_num
              @ ^ [O: nat] :
                  ( case_num_option_num @ ( some_num @ one )
                  @ ^ [P6: num] :
                      ( case_o6005452278849405969um_num @ none_num
                      @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
                      @ ( bit_take_bit_num @ O @ P6 ) )
                  @ ^ [P6: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) )
                  @ X3 )
              @ A5 )
          @ ( product_Pair_nat_num @ N4 @ M ) ) ) ) ).

% take_bit_num_code
thf(fact_4617_prod_Otriangle__reindex,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J2: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [K4: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K4 @ I3 ) )
            @ ( set_ord_atMost_nat @ K4 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.triangle_reindex
thf(fact_4618_prod_Otriangle__reindex,axiom,
    ! [G: nat > nat > int,N2: nat] :
      ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J2: nat] : ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [K4: nat] :
            ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K4 @ I3 ) )
            @ ( set_ord_atMost_nat @ K4 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% prod.triangle_reindex
thf(fact_4619_case__nat__add__eq__if,axiom,
    ! [A: option_num,F: nat > option_num,V: num,N2: nat] :
      ( ( case_nat_option_num @ A @ F @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
      = ( F @ ( plus_plus_nat @ ( pred_numeral @ V ) @ N2 ) ) ) ).

% case_nat_add_eq_if
thf(fact_4620_bit_Ocompl__zero,axiom,
    ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.compl_zero
thf(fact_4621_bit_Ocompl__zero,axiom,
    ( ( bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.compl_zero
thf(fact_4622_bit_Ocompl__one,axiom,
    ( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ one_one_int ) )
    = zero_zero_int ) ).

% bit.compl_one
thf(fact_4623_bit_Ocompl__one,axiom,
    ( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% bit.compl_one
thf(fact_4624_bit_Oxor__cancel__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se3222712562003087583nteger @ X @ ( bit_ri7632146776885996613nteger @ X ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.xor_cancel_right
thf(fact_4625_bit_Oxor__cancel__right,axiom,
    ! [X: int] :
      ( ( bit_se6526347334894502574or_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.xor_cancel_right
thf(fact_4626_bit_Oxor__cancel__left,axiom,
    ! [X: code_integer] :
      ( ( bit_se3222712562003087583nteger @ ( bit_ri7632146776885996613nteger @ X ) @ X )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.xor_cancel_left
thf(fact_4627_bit_Oxor__cancel__left,axiom,
    ! [X: int] :
      ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.xor_cancel_left
thf(fact_4628_bit_Oxor__one__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se3222712562003087583nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( bit_ri7632146776885996613nteger @ X ) ) ).

% bit.xor_one_right
thf(fact_4629_bit_Oxor__one__right,axiom,
    ! [X: int] :
      ( ( bit_se6526347334894502574or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ X ) ) ).

% bit.xor_one_right
thf(fact_4630_bit_Oxor__one__left,axiom,
    ! [X: code_integer] :
      ( ( bit_se3222712562003087583nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
      = ( bit_ri7632146776885996613nteger @ X ) ) ).

% bit.xor_one_left
thf(fact_4631_bit_Oxor__one__left,axiom,
    ! [X: int] :
      ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
      = ( bit_ri7919022796975470100ot_int @ X ) ) ).

% bit.xor_one_left
thf(fact_4632_push__bit__minus__one__eq__not__mask,axiom,
    ! [N2: nat] :
      ( ( bit_se7788150548672797655nteger @ N2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N2 ) ) ) ).

% push_bit_minus_one_eq_not_mask
thf(fact_4633_push__bit__minus__one__eq__not__mask,axiom,
    ! [N2: nat] :
      ( ( bit_se545348938243370406it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) ) ).

% push_bit_minus_one_eq_not_mask
thf(fact_4634_not__one__eq,axiom,
    ( ( bit_ri7919022796975470100ot_int @ one_one_int )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% not_one_eq
thf(fact_4635_not__one__eq,axiom,
    ( ( bit_ri7632146776885996613nteger @ one_one_Code_integer )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% not_one_eq
thf(fact_4636_not__add__distrib,axiom,
    ! [A: int,B: int] :
      ( ( bit_ri7919022796975470100ot_int @ ( plus_plus_int @ A @ B ) )
      = ( minus_minus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).

% not_add_distrib
thf(fact_4637_not__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bit_ri7632146776885996613nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( minus_8373710615458151222nteger @ ( bit_ri7632146776885996613nteger @ A ) @ B ) ) ).

% not_add_distrib
thf(fact_4638_not__diff__distrib,axiom,
    ! [A: int,B: int] :
      ( ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A @ B ) )
      = ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).

% not_diff_distrib
thf(fact_4639_not__diff__distrib,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bit_ri7632146776885996613nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( bit_ri7632146776885996613nteger @ A ) @ B ) ) ).

% not_diff_distrib
thf(fact_4640_minus__eq__not__plus__1,axiom,
    ( uminus_uminus_int
    = ( ^ [A5: int] : ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A5 ) @ one_one_int ) ) ) ).

% minus_eq_not_plus_1
thf(fact_4641_minus__eq__not__plus__1,axiom,
    ( uminus1351360451143612070nteger
    = ( ^ [A5: code_integer] : ( plus_p5714425477246183910nteger @ ( bit_ri7632146776885996613nteger @ A5 ) @ one_one_Code_integer ) ) ) ).

% minus_eq_not_plus_1
thf(fact_4642_minus__eq__not__minus__1,axiom,
    ( uminus_uminus_int
    = ( ^ [A5: int] : ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A5 @ one_one_int ) ) ) ) ).

% minus_eq_not_minus_1
thf(fact_4643_minus__eq__not__minus__1,axiom,
    ( uminus1351360451143612070nteger
    = ( ^ [A5: code_integer] : ( bit_ri7632146776885996613nteger @ ( minus_8373710615458151222nteger @ A5 @ one_one_Code_integer ) ) ) ) ).

% minus_eq_not_minus_1
thf(fact_4644_not__eq__complement,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [A5: int] : ( minus_minus_int @ ( uminus_uminus_int @ A5 ) @ one_one_int ) ) ) ).

% not_eq_complement
thf(fact_4645_not__eq__complement,axiom,
    ( bit_ri7632146776885996613nteger
    = ( ^ [A5: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A5 ) @ one_one_Code_integer ) ) ) ).

% not_eq_complement
thf(fact_4646_prod__int__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X3: int] : X3
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).

% prod_int_eq
thf(fact_4647_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X3: int] : X3
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_4648_push__bit__mask__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( bit_se7788150548672797655nteger @ M2 @ ( bit_se2119862282449309892nteger @ N2 ) )
      = ( bit_se3949692690581998587nteger @ ( bit_se2119862282449309892nteger @ ( plus_plus_nat @ N2 @ M2 ) ) @ ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ M2 ) ) ) ) ).

% push_bit_mask_eq
thf(fact_4649_push__bit__mask__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( bit_se545348938243370406it_int @ M2 @ ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( bit_se725231765392027082nd_int @ ( bit_se2000444600071755411sk_int @ ( plus_plus_nat @ N2 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ M2 ) ) ) ) ).

% push_bit_mask_eq
thf(fact_4650_prod_OatLeast__int__lessThan__int__shift,axiom,
    ! [G: int > nat,M2: nat,N2: nat] :
      ( ( groups1707563613775114915nt_nat @ G @ ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( groups708209901874060359at_nat @ ( comp_int_nat_nat @ G @ semiri1314217659103216013at_int ) @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ).

% prod.atLeast_int_lessThan_int_shift
thf(fact_4651_prod_OatLeast__int__lessThan__int__shift,axiom,
    ! [G: int > int,M2: nat,N2: nat] :
      ( ( groups1705073143266064639nt_int @ G @ ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( groups705719431365010083at_int @ ( comp_int_int_nat @ G @ semiri1314217659103216013at_int ) @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) ).

% prod.atLeast_int_lessThan_int_shift
thf(fact_4652_unset__bit__eq__and__not,axiom,
    ( bit_se8260200283734997820nteger
    = ( ^ [N4: nat,A5: code_integer] : ( bit_se3949692690581998587nteger @ A5 @ ( bit_ri7632146776885996613nteger @ ( bit_se7788150548672797655nteger @ N4 @ one_one_Code_integer ) ) ) ) ) ).

% unset_bit_eq_and_not
thf(fact_4653_unset__bit__eq__and__not,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N4: nat,A5: int] : ( bit_se725231765392027082nd_int @ A5 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ) ).

% unset_bit_eq_and_not
thf(fact_4654_prod_OatLeast__int__atMost__int__shift,axiom,
    ! [G: int > nat,M2: nat,N2: nat] :
      ( ( groups1707563613775114915nt_nat @ G @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( groups708209901874060359at_nat @ ( comp_int_nat_nat @ G @ semiri1314217659103216013at_int ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.atLeast_int_atMost_int_shift
thf(fact_4655_prod_OatLeast__int__atMost__int__shift,axiom,
    ! [G: int > int,M2: nat,N2: nat] :
      ( ( groups1705073143266064639nt_int @ G @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( groups705719431365010083at_int @ ( comp_int_int_nat @ G @ semiri1314217659103216013at_int ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% prod.atLeast_int_atMost_int_shift
thf(fact_4656_int__numeral__not__and__num,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N2 @ M2 ) ) ) ).

% int_numeral_not_and_num
thf(fact_4657_int__numeral__and__not__num,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ N2 ) ) ) ).

% int_numeral_and_not_num
thf(fact_4658_prod_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > nat,N2: nat] :
      ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J2 ) @ N2 ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [K4: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K4 @ I3 ) )
            @ ( set_ord_atMost_nat @ K4 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.triangle_reindex_eq
thf(fact_4659_prod_Otriangle__reindex__eq,axiom,
    ! [G: nat > nat > int,N2: nat] :
      ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [I3: nat,J2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ J2 ) @ N2 ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [K4: nat] :
            ( groups705719431365010083at_int
            @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus_nat @ K4 @ I3 ) )
            @ ( set_ord_atMost_nat @ K4 ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% prod.triangle_reindex_eq
thf(fact_4660_and__not__numerals_I8_J,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% and_not_numerals(8)
thf(fact_4661_not__int__rec,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K4: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K4 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% not_int_rec
thf(fact_4662_Nitpick_Ocase__nat__unfold,axiom,
    ( case_nat_option_num
    = ( ^ [X3: option_num,F2: nat > option_num,N4: nat] : ( if_option_num @ ( N4 = zero_zero_nat ) @ X3 @ ( F2 @ ( minus_minus_nat @ N4 @ one_one_nat ) ) ) ) ) ).

% Nitpick.case_nat_unfold
thf(fact_4663_or__int__rec,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K4: int,L3: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K4 )
              | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_int_rec
thf(fact_4664_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% or_numerals(4)
thf(fact_4665_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% or_numerals(4)
thf(fact_4666_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ) ).

% or_numerals(4)
thf(fact_4667_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ) ).

% or_numerals(4)
thf(fact_4668_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% or_numerals(6)
thf(fact_4669_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% or_numerals(6)
thf(fact_4670_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ) ).

% or_numerals(6)
thf(fact_4671_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ) ).

% or_numerals(6)
thf(fact_4672_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% or_numerals(7)
thf(fact_4673_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% or_numerals(7)
thf(fact_4674_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ) ).

% or_numerals(7)
thf(fact_4675_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ) ).

% or_numerals(7)
thf(fact_4676_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M: nat,N4: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_4677_or__not__numerals_I9_J,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% or_not_numerals(9)
thf(fact_4678_bit_Odisj__one__left,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.disj_one_left
thf(fact_4679_bit_Odisj__one__left,axiom,
    ! [X: code_integer] :
      ( ( bit_se1080825931792720795nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.disj_one_left
thf(fact_4680_bit_Odisj__one__right,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.disj_one_right
thf(fact_4681_bit_Odisj__one__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se1080825931792720795nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.disj_one_right
thf(fact_4682_or__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).

% or_numerals(8)
thf(fact_4683_or__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_numerals(8)
thf(fact_4684_or__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( bit1 @ X ) ) ) ).

% or_numerals(8)
thf(fact_4685_or__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( bit1 @ X ) ) ) ).

% or_numerals(8)
thf(fact_4686_or__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).

% or_numerals(2)
thf(fact_4687_or__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_numerals(2)
thf(fact_4688_or__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) ) ).

% or_numerals(2)
thf(fact_4689_or__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se9127793120404214118atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = ( numera5444537566228673987atural @ ( bit1 @ Y ) ) ) ).

% or_numerals(2)
thf(fact_4690_bit_Odisj__cancel__left,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ X ) @ X )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.disj_cancel_left
thf(fact_4691_bit_Odisj__cancel__left,axiom,
    ! [X: code_integer] :
      ( ( bit_se1080825931792720795nteger @ ( bit_ri7632146776885996613nteger @ X ) @ X )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.disj_cancel_left
thf(fact_4692_bit_Odisj__cancel__right,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ X @ ( bit_ri7919022796975470100ot_int @ X ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.disj_cancel_right
thf(fact_4693_bit_Odisj__cancel__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se1080825931792720795nteger @ X @ ( bit_ri7632146776885996613nteger @ X ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.disj_cancel_right
thf(fact_4694_or__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).

% or_numerals(5)
thf(fact_4695_or__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_numerals(5)
thf(fact_4696_or__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( bit1 @ X ) ) ) ).

% or_numerals(5)
thf(fact_4697_or__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( bit1 @ X ) ) ) ).

% or_numerals(5)
thf(fact_4698_or__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).

% or_numerals(1)
thf(fact_4699_or__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_numerals(1)
thf(fact_4700_or__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) ) ).

% or_numerals(1)
thf(fact_4701_or__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se9127793120404214118atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = ( numera5444537566228673987atural @ ( bit1 @ Y ) ) ) ).

% or_numerals(1)
thf(fact_4702_or__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( ( bit_se1080825931792720795nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se1409905431419307370or_int @ Xa @ X ) ) ) ).

% or_integer.abs_eq
thf(fact_4703_or__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( bit_se1080825931792720795nteger @ X @ Xa ) )
      = ( bit_se1409905431419307370or_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).

% or_integer.rep_eq
thf(fact_4704_disjunctive__add,axiom,
    ! [A: int,B: int] :
      ( ! [N: nat] :
          ( ~ ( bit_se1146084159140164899it_int @ A @ N )
          | ~ ( bit_se1146084159140164899it_int @ B @ N ) )
     => ( ( plus_plus_int @ A @ B )
        = ( bit_se1409905431419307370or_int @ A @ B ) ) ) ).

% disjunctive_add
thf(fact_4705_disjunctive__add,axiom,
    ! [A: nat,B: nat] :
      ( ! [N: nat] :
          ( ~ ( bit_se1148574629649215175it_nat @ A @ N )
          | ~ ( bit_se1148574629649215175it_nat @ B @ N ) )
     => ( ( plus_plus_nat @ A @ B )
        = ( bit_se1412395901928357646or_nat @ A @ B ) ) ) ).

% disjunctive_add
thf(fact_4706_disjunctive__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ! [N: nat] :
          ( ~ ( bit_se9216721137139052372nteger @ A @ N )
          | ~ ( bit_se9216721137139052372nteger @ B @ N ) )
     => ( ( plus_p5714425477246183910nteger @ A @ B )
        = ( bit_se1080825931792720795nteger @ A @ B ) ) ) ).

% disjunctive_add
thf(fact_4707_disjunctive__add,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ! [N: nat] :
          ( ~ ( bit_se8040316288895769887atural @ A @ N )
          | ~ ( bit_se8040316288895769887atural @ B @ N ) )
     => ( ( plus_p4538020629002901425atural @ A @ B )
        = ( bit_se9127793120404214118atural @ A @ B ) ) ) ).

% disjunctive_add
thf(fact_4708_plus__and__or,axiom,
    ! [X: int,Y: int] :
      ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se1409905431419307370or_int @ X @ Y ) )
      = ( plus_plus_int @ X @ Y ) ) ).

% plus_and_or
thf(fact_4709_not__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( bit_ri7632146776885996613nteger @ X ) )
      = ( bit_ri7919022796975470100ot_int @ ( code_int_of_integer @ X ) ) ) ).

% not_integer.rep_eq
thf(fact_4710_set__bit__nat__def,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [M: nat,N4: nat] : ( bit_se1412395901928357646or_nat @ N4 @ ( bit_se547839408752420682it_nat @ M @ one_one_nat ) ) ) ) ).

% set_bit_nat_def
thf(fact_4711_not__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( bit_ri7632146776885996613nteger @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_ri7919022796975470100ot_int @ X ) ) ) ).

% not_integer.abs_eq
thf(fact_4712_set__bit__eq__or,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [N4: nat,A5: nat] : ( bit_se1412395901928357646or_nat @ A5 @ ( bit_se547839408752420682it_nat @ N4 @ one_one_nat ) ) ) ) ).

% set_bit_eq_or
thf(fact_4713_set__bit__eq__or,axiom,
    ( bit_se2793503036327961859nteger
    = ( ^ [N4: nat,A5: code_integer] : ( bit_se1080825931792720795nteger @ A5 @ ( bit_se7788150548672797655nteger @ N4 @ one_one_Code_integer ) ) ) ) ).

% set_bit_eq_or
thf(fact_4714_set__bit__eq__or,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N4: nat,A5: int] : ( bit_se1409905431419307370or_int @ A5 @ ( bit_se545348938243370406it_int @ N4 @ one_one_int ) ) ) ) ).

% set_bit_eq_or
thf(fact_4715_set__bit__eq__or,axiom,
    ( bit_se1617098188084679374atural
    = ( ^ [N4: nat,A5: code_natural] : ( bit_se9127793120404214118atural @ A5 @ ( bit_se6611745700429515170atural @ N4 @ one_one_Code_natural ) ) ) ) ).

% set_bit_eq_or
thf(fact_4716_bit_Ocomplement__unique,axiom,
    ! [A: int,X: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ A @ X )
        = zero_zero_int )
     => ( ( ( bit_se1409905431419307370or_int @ A @ X )
          = ( uminus_uminus_int @ one_one_int ) )
       => ( ( ( bit_se725231765392027082nd_int @ A @ Y )
            = zero_zero_int )
         => ( ( ( bit_se1409905431419307370or_int @ A @ Y )
              = ( uminus_uminus_int @ one_one_int ) )
           => ( X = Y ) ) ) ) ) ).

% bit.complement_unique
thf(fact_4717_bit_Ocomplement__unique,axiom,
    ! [A: code_integer,X: code_integer,Y: code_integer] :
      ( ( ( bit_se3949692690581998587nteger @ A @ X )
        = zero_z3403309356797280102nteger )
     => ( ( ( bit_se1080825931792720795nteger @ A @ X )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
       => ( ( ( bit_se3949692690581998587nteger @ A @ Y )
            = zero_z3403309356797280102nteger )
         => ( ( ( bit_se1080825931792720795nteger @ A @ Y )
              = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
           => ( X = Y ) ) ) ) ) ).

% bit.complement_unique
thf(fact_4718_bit_Ocompl__unique,axiom,
    ! [X: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Y )
        = zero_zero_int )
     => ( ( ( bit_se1409905431419307370or_int @ X @ Y )
          = ( uminus_uminus_int @ one_one_int ) )
       => ( ( bit_ri7919022796975470100ot_int @ X )
          = Y ) ) ) ).

% bit.compl_unique
thf(fact_4719_bit_Ocompl__unique,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( bit_se3949692690581998587nteger @ X @ Y )
        = zero_z3403309356797280102nteger )
     => ( ( ( bit_se1080825931792720795nteger @ X @ Y )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
       => ( ( bit_ri7632146776885996613nteger @ X )
          = Y ) ) ) ).

% bit.compl_unique
thf(fact_4720_or__one__eq,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ A @ one_one_int )
      = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_4721_or__one__eq,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ A @ one_one_nat )
      = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_4722_or__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se1080825931792720795nteger @ A @ one_one_Code_integer )
      = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_4723_or__one__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se9127793120404214118atural @ A @ one_one_Code_natural )
      = ( plus_p4538020629002901425atural @ A @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_4724_one__or__eq,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ A )
      = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_4725_one__or__eq,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ A )
      = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_4726_one__or__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ A )
      = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_4727_one__or__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se9127793120404214118atural @ one_one_Code_natural @ A )
      = ( plus_p4538020629002901425atural @ A @ ( zero_n8403883297036319079atural @ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_4728_mask__Suc__double,axiom,
    ! [N2: nat] :
      ( ( bit_se2000444600071755411sk_int @ ( suc @ N2 ) )
      = ( bit_se1409905431419307370or_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N2 ) ) ) ) ).

% mask_Suc_double
thf(fact_4729_mask__Suc__double,axiom,
    ! [N2: nat] :
      ( ( bit_se2119862282449309892nteger @ ( suc @ N2 ) )
      = ( bit_se1080825931792720795nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N2 ) ) ) ) ).

% mask_Suc_double
thf(fact_4730_mask__Suc__double,axiom,
    ! [N2: nat] :
      ( ( bit_se2002935070580805687sk_nat @ ( suc @ N2 ) )
      = ( bit_se1412395901928357646or_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ) ).

% mask_Suc_double
thf(fact_4731_mask__Suc__double,axiom,
    ! [N2: nat] :
      ( ( bit_se943457434206027407atural @ ( suc @ N2 ) )
      = ( bit_se9127793120404214118atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se943457434206027407atural @ N2 ) ) ) ) ).

% mask_Suc_double
thf(fact_4732_or__not__numerals_I5_J,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% or_not_numerals(5)
thf(fact_4733_Suc__0__or__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% Suc_0_or_eq
thf(fact_4734_or__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se1412395901928357646or_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% or_Suc_0_eq
thf(fact_4735_or__not__numerals_I8_J,axiom,
    ! [M2: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% or_not_numerals(8)
thf(fact_4736_or__int__unfold,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K4: int,L3: int] :
          ( if_int
          @ ( ( K4
              = ( uminus_uminus_int @ one_one_int ) )
            | ( L3
              = ( uminus_uminus_int @ one_one_int ) ) )
          @ ( uminus_uminus_int @ one_one_int )
          @ ( if_int @ ( K4 = zero_zero_int ) @ L3 @ ( if_int @ ( L3 = zero_zero_int ) @ K4 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_4737_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
    boolea2445317508997433345nteger @ bit_se3949692690581998587nteger @ bit_se1080825931792720795nteger @ bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ bit_se3222712562003087583nteger ).

% bit.abstract_boolean_algebra_sym_diff_axioms
thf(fact_4738_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
    boolea8527374999097803216ff_int @ bit_se725231765392027082nd_int @ bit_se1409905431419307370or_int @ bit_ri7919022796975470100ot_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) @ bit_se6526347334894502574or_int ).

% bit.abstract_boolean_algebra_sym_diff_axioms
thf(fact_4739_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M: nat,N4: nat] : ( if_nat @ ( M = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_4740_drop__bit__rec,axiom,
    ( bit_se3928097537394005634nteger
    = ( ^ [N4: nat,A5: code_integer] : ( if_Code_integer @ ( N4 = zero_zero_nat ) @ A5 @ ( bit_se3928097537394005634nteger @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% drop_bit_rec
thf(fact_4741_drop__bit__rec,axiom,
    ( bit_se8568078237143864401it_int
    = ( ^ [N4: nat,A5: int] : ( if_int @ ( N4 = zero_zero_nat ) @ A5 @ ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% drop_bit_rec
thf(fact_4742_drop__bit__rec,axiom,
    ( bit_se8570568707652914677it_nat
    = ( ^ [N4: nat,A5: nat] : ( if_nat @ ( N4 = zero_zero_nat ) @ A5 @ ( bit_se8570568707652914677it_nat @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% drop_bit_rec
thf(fact_4743_drop__bit__rec,axiom,
    ( bit_se2751692689150723149atural
    = ( ^ [N4: nat,A5: code_natural] : ( if_Code_natural @ ( N4 = zero_zero_nat ) @ A5 @ ( bit_se2751692689150723149atural @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide5121882707175180666atural @ A5 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% drop_bit_rec
thf(fact_4744_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
     => ( ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U )
        = ( image_1215581382706833972nteger @ semiri4939895301339042750nteger @ ( set_ord_lessThan_nat @ ( code_nat_of_integer @ U ) ) ) ) ) ).

% image_atLeastZeroLessThan_integer
thf(fact_4745_uminus__int__def,axiom,
    ( uminus_uminus_int
    = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
      @ ( produc2626176000494625587at_nat
        @ ^ [X3: nat,Y3: nat] : ( product_Pair_nat_nat @ Y3 @ X3 ) ) ) ) ).

% uminus_int_def
thf(fact_4746_bit_Oabstract__boolean__algebra__axioms,axiom,
    boolea5469654029284665142ra_int @ bit_se725231765392027082nd_int @ bit_se1409905431419307370or_int @ bit_ri7919022796975470100ot_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ).

% bit.abstract_boolean_algebra_axioms
thf(fact_4747_bit_Oabstract__boolean__algebra__axioms,axiom,
    boolea5643492148547655015nteger @ bit_se3949692690581998587nteger @ bit_se1080825931792720795nteger @ bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ).

% bit.abstract_boolean_algebra_axioms
thf(fact_4748_map__fun__apply,axiom,
    ( map_fu4826362097070443709at_o_o
    = ( ^ [F2: int > product_prod_nat_nat,G2: $o > $o,H: product_prod_nat_nat > $o,X3: int] : ( G2 @ ( H @ ( F2 @ X3 ) ) ) ) ) ).

% map_fun_apply
thf(fact_4749_map__fun__apply,axiom,
    ( map_fu4960017516451851995nt_int
    = ( ^ [F2: int > product_prod_nat_nat,G2: ( product_prod_nat_nat > product_prod_nat_nat ) > int > int,H: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,X3: int] : ( G2 @ ( H @ ( F2 @ X3 ) ) ) ) ) ).

% map_fun_apply
thf(fact_4750_map__fun__apply,axiom,
    ( map_fu434086159418415080_int_o
    = ( ^ [F2: int > product_prod_nat_nat,G2: ( product_prod_nat_nat > $o ) > int > $o,H: product_prod_nat_nat > product_prod_nat_nat > $o,X3: int] : ( G2 @ ( H @ ( F2 @ X3 ) ) ) ) ) ).

% map_fun_apply
thf(fact_4751_map__fun__apply,axiom,
    ( map_fu3424225382358772769nt_rat
    = ( ^ [F2: int > int,G2: product_prod_int_int > rat,H: int > product_prod_int_int,X3: int] : ( G2 @ ( H @ ( F2 @ X3 ) ) ) ) ) ).

% map_fun_apply
thf(fact_4752_map__fun__apply,axiom,
    ( map_fu7831380289885515383nt_rat
    = ( ^ [F2: int > int,G2: ( int > product_prod_int_int ) > int > rat,H: int > int > product_prod_int_int,X3: int] : ( G2 @ ( H @ ( F2 @ X3 ) ) ) ) ) ).

% map_fun_apply
thf(fact_4753_drop__bit__drop__bit,axiom,
    ! [M2: nat,N2: nat,A: code_integer] :
      ( ( bit_se3928097537394005634nteger @ M2 @ ( bit_se3928097537394005634nteger @ N2 @ A ) )
      = ( bit_se3928097537394005634nteger @ ( plus_plus_nat @ M2 @ N2 ) @ A ) ) ).

% drop_bit_drop_bit
thf(fact_4754_drop__bit__drop__bit,axiom,
    ! [M2: nat,N2: nat,A: int] :
      ( ( bit_se8568078237143864401it_int @ M2 @ ( bit_se8568078237143864401it_int @ N2 @ A ) )
      = ( bit_se8568078237143864401it_int @ ( plus_plus_nat @ M2 @ N2 ) @ A ) ) ).

% drop_bit_drop_bit
thf(fact_4755_drop__bit__drop__bit,axiom,
    ! [M2: nat,N2: nat,A: nat] :
      ( ( bit_se8570568707652914677it_nat @ M2 @ ( bit_se8570568707652914677it_nat @ N2 @ A ) )
      = ( bit_se8570568707652914677it_nat @ ( plus_plus_nat @ M2 @ N2 ) @ A ) ) ).

% drop_bit_drop_bit
thf(fact_4756_drop__bit__drop__bit,axiom,
    ! [M2: nat,N2: nat,A: code_natural] :
      ( ( bit_se2751692689150723149atural @ M2 @ ( bit_se2751692689150723149atural @ N2 @ A ) )
      = ( bit_se2751692689150723149atural @ ( plus_plus_nat @ M2 @ N2 ) @ A ) ) ).

% drop_bit_drop_bit
thf(fact_4757_image__add__atLeastAtMost,axiom,
    ! [K: rat,I: rat,J: rat] :
      ( ( image_rat_rat @ ( plus_plus_rat @ K ) @ ( set_or633870826150836451st_rat @ I @ J ) )
      = ( set_or633870826150836451st_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ K ) ) ) ).

% image_add_atLeastAtMost
thf(fact_4758_image__add__atLeastAtMost,axiom,
    ! [K: int,I: int,J: int] :
      ( ( image_int_int @ ( plus_plus_int @ K ) @ ( set_or1266510415728281911st_int @ I @ J ) )
      = ( set_or1266510415728281911st_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ K ) ) ) ).

% image_add_atLeastAtMost
thf(fact_4759_image__add__atLeastAtMost,axiom,
    ! [K: nat,I: nat,J: nat] :
      ( ( image_nat_nat @ ( plus_plus_nat @ K ) @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% image_add_atLeastAtMost
thf(fact_4760_image__add__atLeastAtMost,axiom,
    ! [K: code_integer,I: code_integer,J: code_integer] :
      ( ( image_4470545334726330049nteger @ ( plus_p5714425477246183910nteger @ K ) @ ( set_or189985376899183464nteger @ I @ J ) )
      = ( set_or189985376899183464nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J @ K ) ) ) ).

% image_add_atLeastAtMost
thf(fact_4761_image__diff__atLeastAtMost,axiom,
    ! [D2: rat,A: rat,B: rat] :
      ( ( image_rat_rat @ ( minus_minus_rat @ D2 ) @ ( set_or633870826150836451st_rat @ A @ B ) )
      = ( set_or633870826150836451st_rat @ ( minus_minus_rat @ D2 @ B ) @ ( minus_minus_rat @ D2 @ A ) ) ) ).

% image_diff_atLeastAtMost
thf(fact_4762_image__diff__atLeastAtMost,axiom,
    ! [D2: int,A: int,B: int] :
      ( ( image_int_int @ ( minus_minus_int @ D2 ) @ ( set_or1266510415728281911st_int @ A @ B ) )
      = ( set_or1266510415728281911st_int @ ( minus_minus_int @ D2 @ B ) @ ( minus_minus_int @ D2 @ A ) ) ) ).

% image_diff_atLeastAtMost
thf(fact_4763_image__diff__atLeastAtMost,axiom,
    ! [D2: code_integer,A: code_integer,B: code_integer] :
      ( ( image_4470545334726330049nteger @ ( minus_8373710615458151222nteger @ D2 ) @ ( set_or189985376899183464nteger @ A @ B ) )
      = ( set_or189985376899183464nteger @ ( minus_8373710615458151222nteger @ D2 @ B ) @ ( minus_8373710615458151222nteger @ D2 @ A ) ) ) ).

% image_diff_atLeastAtMost
thf(fact_4764_image__add__atLeastLessThan,axiom,
    ! [K: rat,I: rat,J: rat] :
      ( ( image_rat_rat @ ( plus_plus_rat @ K ) @ ( set_or4029947393144176647an_rat @ I @ J ) )
      = ( set_or4029947393144176647an_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ K ) ) ) ).

% image_add_atLeastLessThan
thf(fact_4765_image__add__atLeastLessThan,axiom,
    ! [K: nat,I: nat,J: nat] :
      ( ( image_nat_nat @ ( plus_plus_nat @ K ) @ ( set_or4665077453230672383an_nat @ I @ J ) )
      = ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% image_add_atLeastLessThan
thf(fact_4766_image__add__atLeastLessThan,axiom,
    ! [K: code_integer,I: code_integer,J: code_integer] :
      ( ( image_4470545334726330049nteger @ ( plus_p5714425477246183910nteger @ K ) @ ( set_or8404916559141939852nteger @ I @ J ) )
      = ( set_or8404916559141939852nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J @ K ) ) ) ).

% image_add_atLeastLessThan
thf(fact_4767_image__add__atLeastLessThan,axiom,
    ! [K: int,I: int,J: int] :
      ( ( image_int_int @ ( plus_plus_int @ K ) @ ( set_or4662586982721622107an_int @ I @ J ) )
      = ( set_or4662586982721622107an_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ K ) ) ) ).

% image_add_atLeastLessThan
thf(fact_4768_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_4769_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_4770_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_4771_max__0__1_I1_J,axiom,
    ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% max_0_1(1)
thf(fact_4772_max__0__1_I1_J,axiom,
    ( ( ord_max_Code_natural @ zero_z2226904508553997617atural @ one_one_Code_natural )
    = one_one_Code_natural ) ).

% max_0_1(1)
thf(fact_4773_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_4774_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_4775_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_4776_max__0__1_I2_J,axiom,
    ( ( ord_max_Code_integer @ one_one_Code_integer @ zero_z3403309356797280102nteger )
    = one_one_Code_integer ) ).

% max_0_1(2)
thf(fact_4777_max__0__1_I2_J,axiom,
    ( ( ord_max_Code_natural @ one_one_Code_natural @ zero_z2226904508553997617atural )
    = one_one_Code_natural ) ).

% max_0_1(2)
thf(fact_4778_image__uminus__atLeastAtMost,axiom,
    ! [X: rat,Y: rat] :
      ( ( image_rat_rat @ uminus_uminus_rat @ ( set_or633870826150836451st_rat @ X @ Y ) )
      = ( set_or633870826150836451st_rat @ ( uminus_uminus_rat @ Y ) @ ( uminus_uminus_rat @ X ) ) ) ).

% image_uminus_atLeastAtMost
thf(fact_4779_image__uminus__atLeastAtMost,axiom,
    ! [X: int,Y: int] :
      ( ( image_int_int @ uminus_uminus_int @ ( set_or1266510415728281911st_int @ X @ Y ) )
      = ( set_or1266510415728281911st_int @ ( uminus_uminus_int @ Y ) @ ( uminus_uminus_int @ X ) ) ) ).

% image_uminus_atLeastAtMost
thf(fact_4780_image__uminus__atLeastAtMost,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( image_4470545334726330049nteger @ uminus1351360451143612070nteger @ ( set_or189985376899183464nteger @ X @ Y ) )
      = ( set_or189985376899183464nteger @ ( uminus1351360451143612070nteger @ Y ) @ ( uminus1351360451143612070nteger @ X ) ) ) ).

% image_uminus_atLeastAtMost
thf(fact_4781_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(6)
thf(fact_4782_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(6)
thf(fact_4783_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(6)
thf(fact_4784_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(6)
thf(fact_4785_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_natural @ ( numera5444537566228673987atural @ X ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ X ) ) ).

% max_0_1(6)
thf(fact_4786_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(5)
thf(fact_4787_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(5)
thf(fact_4788_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X ) )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(5)
thf(fact_4789_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(5)
thf(fact_4790_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_natural @ one_one_Code_natural @ ( numera5444537566228673987atural @ X ) )
      = ( numera5444537566228673987atural @ X ) ) ).

% max_0_1(5)
thf(fact_4791_image__add__atMost,axiom,
    ! [C: rat,A: rat] :
      ( ( image_rat_rat @ ( plus_plus_rat @ C ) @ ( set_ord_atMost_rat @ A ) )
      = ( set_ord_atMost_rat @ ( plus_plus_rat @ C @ A ) ) ) ).

% image_add_atMost
thf(fact_4792_image__add__atMost,axiom,
    ! [C: int,A: int] :
      ( ( image_int_int @ ( plus_plus_int @ C ) @ ( set_ord_atMost_int @ A ) )
      = ( set_ord_atMost_int @ ( plus_plus_int @ C @ A ) ) ) ).

% image_add_atMost
thf(fact_4793_image__add__atLeastAtMost_H,axiom,
    ! [K: rat,I: rat,J: rat] :
      ( ( image_rat_rat
        @ ^ [N4: rat] : ( plus_plus_rat @ N4 @ K )
        @ ( set_or633870826150836451st_rat @ I @ J ) )
      = ( set_or633870826150836451st_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ K ) ) ) ).

% image_add_atLeastAtMost'
thf(fact_4794_image__add__atLeastAtMost_H,axiom,
    ! [K: int,I: int,J: int] :
      ( ( image_int_int
        @ ^ [N4: int] : ( plus_plus_int @ N4 @ K )
        @ ( set_or1266510415728281911st_int @ I @ J ) )
      = ( set_or1266510415728281911st_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ K ) ) ) ).

% image_add_atLeastAtMost'
thf(fact_4795_image__add__atLeastAtMost_H,axiom,
    ! [K: nat,I: nat,J: nat] :
      ( ( image_nat_nat
        @ ^ [N4: nat] : ( plus_plus_nat @ N4 @ K )
        @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% image_add_atLeastAtMost'
thf(fact_4796_image__add__atLeastAtMost_H,axiom,
    ! [K: code_integer,I: code_integer,J: code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [N4: code_integer] : ( plus_p5714425477246183910nteger @ N4 @ K )
        @ ( set_or189985376899183464nteger @ I @ J ) )
      = ( set_or189985376899183464nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J @ K ) ) ) ).

% image_add_atLeastAtMost'
thf(fact_4797_image__minus__const__atLeastAtMost_H,axiom,
    ! [D2: rat,A: rat,B: rat] :
      ( ( image_rat_rat
        @ ^ [T2: rat] : ( minus_minus_rat @ T2 @ D2 )
        @ ( set_or633870826150836451st_rat @ A @ B ) )
      = ( set_or633870826150836451st_rat @ ( minus_minus_rat @ A @ D2 ) @ ( minus_minus_rat @ B @ D2 ) ) ) ).

% image_minus_const_atLeastAtMost'
thf(fact_4798_image__minus__const__atLeastAtMost_H,axiom,
    ! [D2: int,A: int,B: int] :
      ( ( image_int_int
        @ ^ [T2: int] : ( minus_minus_int @ T2 @ D2 )
        @ ( set_or1266510415728281911st_int @ A @ B ) )
      = ( set_or1266510415728281911st_int @ ( minus_minus_int @ A @ D2 ) @ ( minus_minus_int @ B @ D2 ) ) ) ).

% image_minus_const_atLeastAtMost'
thf(fact_4799_image__minus__const__atLeastAtMost_H,axiom,
    ! [D2: code_integer,A: code_integer,B: code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [T2: code_integer] : ( minus_8373710615458151222nteger @ T2 @ D2 )
        @ ( set_or189985376899183464nteger @ A @ B ) )
      = ( set_or189985376899183464nteger @ ( minus_8373710615458151222nteger @ A @ D2 ) @ ( minus_8373710615458151222nteger @ B @ D2 ) ) ) ).

% image_minus_const_atLeastAtMost'
thf(fact_4800_image__add__atLeastLessThan_H,axiom,
    ! [K: rat,I: rat,J: rat] :
      ( ( image_rat_rat
        @ ^ [N4: rat] : ( plus_plus_rat @ N4 @ K )
        @ ( set_or4029947393144176647an_rat @ I @ J ) )
      = ( set_or4029947393144176647an_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ K ) ) ) ).

% image_add_atLeastLessThan'
thf(fact_4801_image__add__atLeastLessThan_H,axiom,
    ! [K: nat,I: nat,J: nat] :
      ( ( image_nat_nat
        @ ^ [N4: nat] : ( plus_plus_nat @ N4 @ K )
        @ ( set_or4665077453230672383an_nat @ I @ J ) )
      = ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% image_add_atLeastLessThan'
thf(fact_4802_image__add__atLeastLessThan_H,axiom,
    ! [K: code_integer,I: code_integer,J: code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [N4: code_integer] : ( plus_p5714425477246183910nteger @ N4 @ K )
        @ ( set_or8404916559141939852nteger @ I @ J ) )
      = ( set_or8404916559141939852nteger @ ( plus_p5714425477246183910nteger @ I @ K ) @ ( plus_p5714425477246183910nteger @ J @ K ) ) ) ).

% image_add_atLeastLessThan'
thf(fact_4803_image__add__atLeastLessThan_H,axiom,
    ! [K: int,I: int,J: int] :
      ( ( image_int_int
        @ ^ [N4: int] : ( plus_plus_int @ N4 @ K )
        @ ( set_or4662586982721622107an_int @ I @ J ) )
      = ( set_or4662586982721622107an_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ K ) ) ) ).

% image_add_atLeastLessThan'
thf(fact_4804_drop__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se3928097537394005634nteger @ N2 @ one_one_Code_integer )
      = ( zero_n356916108424825756nteger @ ( N2 = zero_zero_nat ) ) ) ).

% drop_bit_of_1
thf(fact_4805_drop__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se8568078237143864401it_int @ N2 @ one_one_int )
      = ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).

% drop_bit_of_1
thf(fact_4806_drop__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se8570568707652914677it_nat @ N2 @ one_one_nat )
      = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).

% drop_bit_of_1
thf(fact_4807_drop__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2751692689150723149atural @ N2 @ one_one_Code_natural )
      = ( zero_n8403883297036319079atural @ ( N2 = zero_zero_nat ) ) ) ).

% drop_bit_of_1
thf(fact_4808_image__mult__atLeastAtMost,axiom,
    ! [D2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D2 )
     => ( ( image_rat_rat @ ( times_times_rat @ D2 ) @ ( set_or633870826150836451st_rat @ A @ B ) )
        = ( set_or633870826150836451st_rat @ ( times_times_rat @ D2 @ A ) @ ( times_times_rat @ D2 @ B ) ) ) ) ).

% image_mult_atLeastAtMost
thf(fact_4809_image__divide__atLeastAtMost,axiom,
    ! [D2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D2 )
     => ( ( image_rat_rat
          @ ^ [C2: rat] : ( divide_divide_rat @ C2 @ D2 )
          @ ( set_or633870826150836451st_rat @ A @ B ) )
        = ( set_or633870826150836451st_rat @ ( divide_divide_rat @ A @ D2 ) @ ( divide_divide_rat @ B @ D2 ) ) ) ) ).

% image_divide_atLeastAtMost
thf(fact_4810_drop__bit__integer_Orep__eq,axiom,
    ! [X: nat,Xa: code_integer] :
      ( ( code_int_of_integer @ ( bit_se3928097537394005634nteger @ X @ Xa ) )
      = ( bit_se8568078237143864401it_int @ X @ ( code_int_of_integer @ Xa ) ) ) ).

% drop_bit_integer.rep_eq
thf(fact_4811_drop__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,X: int] :
      ( ( bit_se3928097537394005634nteger @ Xa @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( bit_se8568078237143864401it_int @ Xa @ X ) ) ) ).

% drop_bit_integer.abs_eq
thf(fact_4812_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_4813_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_4814_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_4815_max__add__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z )
      = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Z ) @ ( plus_p5714425477246183910nteger @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_4816_max__add__distrib__left,axiom,
    ! [X: code_natural,Y: code_natural,Z: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( ord_max_Code_natural @ X @ Y ) @ Z )
      = ( ord_max_Code_natural @ ( plus_p4538020629002901425atural @ X @ Z ) @ ( plus_p4538020629002901425atural @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_4817_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z ) )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_4818_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z ) )
      = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_4819_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_4820_max__add__distrib__right,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( plus_p5714425477246183910nteger @ X @ ( ord_max_Code_integer @ Y @ Z ) )
      = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( plus_p5714425477246183910nteger @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_4821_max__add__distrib__right,axiom,
    ! [X: code_natural,Y: code_natural,Z: code_natural] :
      ( ( plus_p4538020629002901425atural @ X @ ( ord_max_Code_natural @ Y @ Z ) )
      = ( ord_max_Code_natural @ ( plus_p4538020629002901425atural @ X @ Y ) @ ( plus_p4538020629002901425atural @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_4822_max__diff__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z )
      = ( ord_max_Code_integer @ ( minus_8373710615458151222nteger @ X @ Z ) @ ( minus_8373710615458151222nteger @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_4823_max__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ord_max_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_4824_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_4825_translation__Compl,axiom,
    ! [A: rat,T: set_rat] :
      ( ( image_rat_rat @ ( plus_plus_rat @ A ) @ ( uminus2201863774496077783et_rat @ T ) )
      = ( uminus2201863774496077783et_rat @ ( image_rat_rat @ ( plus_plus_rat @ A ) @ T ) ) ) ).

% translation_Compl
thf(fact_4826_translation__Compl,axiom,
    ! [A: int,T: set_int] :
      ( ( image_int_int @ ( plus_plus_int @ A ) @ ( uminus1532241313380277803et_int @ T ) )
      = ( uminus1532241313380277803et_int @ ( image_int_int @ ( plus_plus_int @ A ) @ T ) ) ) ).

% translation_Compl
thf(fact_4827_nat__add__max__right,axiom,
    ! [M2: nat,N2: nat,Q3: nat] :
      ( ( plus_plus_nat @ M2 @ ( ord_max_nat @ N2 @ Q3 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M2 @ N2 ) @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ).

% nat_add_max_right
thf(fact_4828_nat__add__max__left,axiom,
    ! [M2: nat,N2: nat,Q3: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M2 @ N2 ) @ Q3 )
      = ( ord_max_nat @ ( plus_plus_nat @ M2 @ Q3 ) @ ( plus_plus_nat @ N2 @ Q3 ) ) ) ).

% nat_add_max_left
thf(fact_4829_image__eq__imp__comp,axiom,
    ! [F: num > code_integer,A2: set_num,G: num > code_integer,B5: set_num,H3: code_integer > code_integer] :
      ( ( ( image_8379761247532268670nteger @ F @ A2 )
        = ( image_8379761247532268670nteger @ G @ B5 ) )
     => ( ( image_8379761247532268670nteger @ ( comp_C3531382070062128313er_num @ H3 @ F ) @ A2 )
        = ( image_8379761247532268670nteger @ ( comp_C3531382070062128313er_num @ H3 @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_4830_image__eq__imp__comp,axiom,
    ! [F: int > nat,A2: set_int,G: int > nat,B5: set_int,H3: nat > num] :
      ( ( ( image_int_nat @ F @ A2 )
        = ( image_int_nat @ G @ B5 ) )
     => ( ( image_int_num @ ( comp_nat_num_int @ H3 @ F ) @ A2 )
        = ( image_int_num @ ( comp_nat_num_int @ H3 @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_4831_image__eq__imp__comp,axiom,
    ! [F: num > code_integer,A2: set_num,G: nat > code_integer,B5: set_nat,H3: code_integer > code_integer] :
      ( ( ( image_8379761247532268670nteger @ F @ A2 )
        = ( image_1215581382706833972nteger @ G @ B5 ) )
     => ( ( image_8379761247532268670nteger @ ( comp_C3531382070062128313er_num @ H3 @ F ) @ A2 )
        = ( image_1215581382706833972nteger @ ( comp_C6974050283906349679er_nat @ H3 @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_4832_image__eq__imp__comp,axiom,
    ! [F: int > code_integer,A2: set_int,G: nat > code_integer,B5: set_nat,H3: code_integer > int] :
      ( ( ( image_1587234942943678608nteger @ F @ A2 )
        = ( image_1215581382706833972nteger @ G @ B5 ) )
     => ( ( image_int_int @ ( comp_C5046419837055054490nt_int @ H3 @ F ) @ A2 )
        = ( image_nat_int @ ( comp_C5048910307564104766nt_nat @ H3 @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_4833_image__eq__imp__comp,axiom,
    ! [F: int > nat,A2: set_int,G: nat > nat,B5: set_nat,H3: nat > num] :
      ( ( ( image_int_nat @ F @ A2 )
        = ( image_nat_nat @ G @ B5 ) )
     => ( ( image_int_num @ ( comp_nat_num_int @ H3 @ F ) @ A2 )
        = ( image_nat_num @ ( comp_nat_num_nat @ H3 @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_4834_image__eq__imp__comp,axiom,
    ! [F: int > nat,A2: set_int,G: nat > nat,B5: set_nat,H3: nat > int] :
      ( ( ( image_int_nat @ F @ A2 )
        = ( image_nat_nat @ G @ B5 ) )
     => ( ( image_int_int @ ( comp_nat_int_int @ H3 @ F ) @ A2 )
        = ( image_nat_int @ ( comp_nat_int_nat @ H3 @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_4835_image__eq__imp__comp,axiom,
    ! [F: nat > code_integer,A2: set_nat,G: num > code_integer,B5: set_num,H3: code_integer > code_integer] :
      ( ( ( image_1215581382706833972nteger @ F @ A2 )
        = ( image_8379761247532268670nteger @ G @ B5 ) )
     => ( ( image_1215581382706833972nteger @ ( comp_C6974050283906349679er_nat @ H3 @ F ) @ A2 )
        = ( image_8379761247532268670nteger @ ( comp_C3531382070062128313er_num @ H3 @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_4836_image__eq__imp__comp,axiom,
    ! [F: nat > code_integer,A2: set_nat,G: int > code_integer,B5: set_int,H3: code_integer > int] :
      ( ( ( image_1215581382706833972nteger @ F @ A2 )
        = ( image_1587234942943678608nteger @ G @ B5 ) )
     => ( ( image_nat_int @ ( comp_C5048910307564104766nt_nat @ H3 @ F ) @ A2 )
        = ( image_int_int @ ( comp_C5046419837055054490nt_int @ H3 @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_4837_image__eq__imp__comp,axiom,
    ! [F: nat > code_integer,A2: set_nat,G: nat > code_integer,B5: set_nat,H3: code_integer > code_integer] :
      ( ( ( image_1215581382706833972nteger @ F @ A2 )
        = ( image_1215581382706833972nteger @ G @ B5 ) )
     => ( ( image_1215581382706833972nteger @ ( comp_C6974050283906349679er_nat @ H3 @ F ) @ A2 )
        = ( image_1215581382706833972nteger @ ( comp_C6974050283906349679er_nat @ H3 @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_4838_image__eq__imp__comp,axiom,
    ! [F: nat > code_integer,A2: set_nat,G: nat > code_integer,B5: set_nat,H3: code_integer > nat] :
      ( ( ( image_1215581382706833972nteger @ F @ A2 )
        = ( image_1215581382706833972nteger @ G @ B5 ) )
     => ( ( image_nat_nat @ ( comp_C4049556595663050210at_nat @ H3 @ F ) @ A2 )
        = ( image_nat_nat @ ( comp_C4049556595663050210at_nat @ H3 @ G ) @ B5 ) ) ) ).

% image_eq_imp_comp
thf(fact_4839_image__comp,axiom,
    ! [F: code_integer > code_integer,G: num > code_integer,R2: set_num] :
      ( ( image_4470545334726330049nteger @ F @ ( image_8379761247532268670nteger @ G @ R2 ) )
      = ( image_8379761247532268670nteger @ ( comp_C3531382070062128313er_num @ F @ G ) @ R2 ) ) ).

% image_comp
thf(fact_4840_image__comp,axiom,
    ! [F: nat > num,G: int > nat,R2: set_int] :
      ( ( image_nat_num @ F @ ( image_int_nat @ G @ R2 ) )
      = ( image_int_num @ ( comp_nat_num_int @ F @ G ) @ R2 ) ) ).

% image_comp
thf(fact_4841_image__comp,axiom,
    ! [F: code_integer > code_integer,G: nat > code_integer,R2: set_nat] :
      ( ( image_4470545334726330049nteger @ F @ ( image_1215581382706833972nteger @ G @ R2 ) )
      = ( image_1215581382706833972nteger @ ( comp_C6974050283906349679er_nat @ F @ G ) @ R2 ) ) ).

% image_comp
thf(fact_4842_image__comp,axiom,
    ! [F: code_integer > nat,G: nat > code_integer,R2: set_nat] :
      ( ( image_951025933927791156er_nat @ F @ ( image_1215581382706833972nteger @ G @ R2 ) )
      = ( image_nat_nat @ ( comp_C4049556595663050210at_nat @ F @ G ) @ R2 ) ) ).

% image_comp
thf(fact_4843_image__comp,axiom,
    ! [F: code_integer > int,G: nat > code_integer,R2: set_nat] :
      ( ( image_948535463418740880er_int @ F @ ( image_1215581382706833972nteger @ G @ R2 ) )
      = ( image_nat_int @ ( comp_C5048910307564104766nt_nat @ F @ G ) @ R2 ) ) ).

% image_comp
thf(fact_4844_image__comp,axiom,
    ! [F: int > code_integer,G: nat > int,R2: set_nat] :
      ( ( image_1587234942943678608nteger @ F @ ( image_nat_int @ G @ R2 ) )
      = ( image_1215581382706833972nteger @ ( comp_i1588355021709917246er_nat @ F @ G ) @ R2 ) ) ).

% image_comp
thf(fact_4845_image__comp,axiom,
    ! [F: int > nat,G: nat > int,R2: set_nat] :
      ( ( image_int_nat @ F @ ( image_nat_int @ G @ R2 ) )
      = ( image_nat_nat @ ( comp_int_nat_nat @ F @ G ) @ R2 ) ) ).

% image_comp
thf(fact_4846_image__comp,axiom,
    ! [F: nat > code_integer,G: nat > nat,R2: set_nat] :
      ( ( image_1215581382706833972nteger @ F @ ( image_nat_nat @ G @ R2 ) )
      = ( image_1215581382706833972nteger @ ( comp_n3898172953802868194er_nat @ F @ G ) @ R2 ) ) ).

% image_comp
thf(fact_4847_image__comp,axiom,
    ! [F: nat > nat,G: nat > nat,R2: set_nat] :
      ( ( image_nat_nat @ F @ ( image_nat_nat @ G @ R2 ) )
      = ( image_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ R2 ) ) ).

% image_comp
thf(fact_4848_image__comp,axiom,
    ! [F: nat > int,G: int > nat,R2: set_int] :
      ( ( image_nat_int @ F @ ( image_int_nat @ G @ R2 ) )
      = ( image_int_int @ ( comp_nat_int_int @ F @ G ) @ R2 ) ) ).

% image_comp
thf(fact_4849_map__fun__def,axiom,
    ( map_fu2576242048321985878nteger
    = ( ^ [F2: code_integer > code_integer,G2: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,H: code_integer > code_integer] : ( comp_C1593894019821074884nteger @ ( comp_C1593894019821074884nteger @ G2 @ H ) @ F2 ) ) ) ).

% map_fun_def
thf(fact_4850_map__fun__def,axiom,
    ( map_fu6423247683660063826nteger
    = ( ^ [F2: code_integer > code_integer,G2: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,H: code_integer > code_integer > code_integer] : ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ G2 @ H ) @ F2 ) ) ) ).

% map_fun_def
thf(fact_4851_map__fun__def,axiom,
    ( map_fu4826362097070443709at_o_o
    = ( ^ [F2: int > product_prod_nat_nat,G2: $o > $o,H: product_prod_nat_nat > $o] : ( comp_P1289042343629989734_o_int @ ( comp_o8022916681612831582at_nat @ G2 @ H ) @ F2 ) ) ) ).

% map_fun_def
thf(fact_4852_map__fun__def,axiom,
    ( map_fu4960017516451851995nt_int
    = ( ^ [F2: int > product_prod_nat_nat,G2: ( product_prod_nat_nat > product_prod_nat_nat ) > int > int,H: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] : ( comp_P1029605119844863363nt_int @ ( comp_P1872448293794610368at_nat @ G2 @ H ) @ F2 ) ) ) ).

% map_fun_def
thf(fact_4853_map__fun__def,axiom,
    ( map_fu434086159418415080_int_o
    = ( ^ [F2: int > product_prod_nat_nat,G2: ( product_prod_nat_nat > $o ) > int > $o,H: product_prod_nat_nat > product_prod_nat_nat > $o] : ( comp_P7507668834397697393_o_int @ ( comp_P9039549498699531069at_nat @ G2 @ H ) @ F2 ) ) ) ).

% map_fun_def
thf(fact_4854_map__fun__def,axiom,
    ( map_fu3424225382358772769nt_rat
    = ( ^ [F2: int > int,G2: product_prod_int_int > rat,H: int > product_prod_int_int] : ( comp_int_rat_int @ ( comp_P5335556815845735580at_int @ G2 @ H ) @ F2 ) ) ) ).

% map_fun_def
thf(fact_4855_map__fun__def,axiom,
    ( map_fu7831380289885515383nt_rat
    = ( ^ [F2: int > int,G2: ( int > product_prod_int_int ) > int > rat,H: int > int > product_prod_int_int] : ( comp_int_int_rat_int @ ( comp_i1302866070986779890at_int @ G2 @ H ) @ F2 ) ) ) ).

% map_fun_def
thf(fact_4856_map__fun_Ocompositionality,axiom,
    ! [F: num > code_integer,G: code_integer > code_integer,H3: code_integer > code_integer,I: num > code_integer,Fun: code_integer > num] :
      ( ( map_fu5269526691104299083nteger @ F @ G @ ( map_fu603974984455542859nteger @ H3 @ I @ Fun ) )
      = ( map_fu8223014087597717000nteger @ ( comp_C3531382070062128313er_num @ H3 @ F ) @ ( comp_C3531382070062128313er_num @ G @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_4857_map__fun_Ocompositionality,axiom,
    ! [F: num > code_integer,G: nat > num,H3: code_integer > code_integer,I: int > nat,Fun: code_integer > int] :
      ( ( map_fu7479816331989151099at_num @ F @ G @ ( map_fu5423760287386111952nt_nat @ H3 @ I @ Fun ) )
      = ( map_fu8479170043890205655nt_num @ ( comp_C3531382070062128313er_num @ H3 @ F ) @ ( comp_nat_num_int @ G @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_4858_map__fun_Ocompositionality,axiom,
    ! [F: int > nat,G: code_integer > code_integer,H3: nat > num,I: num > code_integer,Fun: num > num] :
      ( ( map_fu1717202910479192016nteger @ F @ G @ ( map_fu1090464068875036027nteger @ H3 @ I @ Fun ) )
      = ( map_fu229184005524548567nteger @ ( comp_nat_num_int @ H3 @ F ) @ ( comp_C3531382070062128313er_num @ G @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_4859_map__fun_Ocompositionality,axiom,
    ! [F: int > nat,G: nat > num,H3: nat > num,I: int > nat,Fun: num > int] :
      ( ( map_fu3669222299741785344at_num @ F @ G @ ( map_fu8361869985550671104nt_nat @ H3 @ I @ Fun ) )
      = ( map_fu2609383839613498790nt_num @ ( comp_nat_num_int @ H3 @ F ) @ ( comp_nat_num_int @ G @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_4860_map__fun_Ocompositionality,axiom,
    ! [F: int > int,G: $o > $o,H3: int > product_prod_nat_nat,I: $o > $o,Fun: product_prod_nat_nat > $o] :
      ( ( map_fun_int_int_o_o @ F @ G @ ( map_fu4826362097070443709at_o_o @ H3 @ I @ Fun ) )
      = ( map_fu4826362097070443709at_o_o @ ( comp_i1108739374719436894at_int @ H3 @ F ) @ ( comp_o_o_o @ G @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_4861_map__fun_Ocompositionality,axiom,
    ! [F: int > int,G: rat > rat,H3: int > int,I: product_prod_int_int > rat,Fun: int > product_prod_int_int] :
      ( ( map_fu6531859369801489698at_rat @ F @ G @ ( map_fu3424225382358772769nt_rat @ H3 @ I @ Fun ) )
      = ( map_fu3424225382358772769nt_rat @ ( comp_int_int_int @ H3 @ F ) @ ( comp_r4727614531642239972nt_int @ G @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_4862_map__fun_Ocompositionality,axiom,
    ! [F: int > product_prod_nat_nat,G: $o > $o,H3: product_prod_nat_nat > product_prod_nat_nat,I: $o > $o,Fun: product_prod_nat_nat > $o] :
      ( ( map_fu4826362097070443709at_o_o @ F @ G @ ( map_fu1707080289169145046at_o_o @ H3 @ I @ Fun ) )
      = ( map_fu4826362097070443709at_o_o @ ( comp_P7428279379633703887at_int @ H3 @ F ) @ ( comp_o_o_o @ G @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_4863_map__fun_Ocompositionality,axiom,
    ! [F: int > int,G: product_prod_int_int > rat,H3: int > int,I: product_prod_int_int > product_prod_int_int,Fun: int > product_prod_int_int] :
      ( ( map_fu3424225382358772769nt_rat @ F @ G @ ( map_fu2180209013632630320nt_int @ H3 @ I @ Fun ) )
      = ( map_fu3424225382358772769nt_rat @ ( comp_int_int_int @ H3 @ F ) @ ( comp_P5648219952011055433nt_int @ G @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_4864_map__fun_Ocompositionality,axiom,
    ! [F: num > code_integer,G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,H3: code_integer > code_integer,I: code_integer > code_integer,Fun: code_integer > code_integer] :
      ( ( map_fu1235460277957925907nteger @ F @ G @ ( map_fu5082455417554595470nteger @ H3 @ I @ Fun ) )
      = ( map_fu1235460277957925907nteger @ ( comp_C3531382070062128313er_num @ H3 @ F ) @ ( comp_C1593894019821074884nteger @ G @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_4865_map__fun_Ocompositionality,axiom,
    ! [F: code_integer > code_integer,G: code_integer > code_integer,H3: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,I: num > code_integer,Fun: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > num] :
      ( ( map_fu5082455417554595470nteger @ F @ G @ ( map_fu1335501423231535891nteger @ H3 @ I @ Fun ) )
      = ( map_fu1335501423231535891nteger @ ( comp_C1593894019821074884nteger @ H3 @ F ) @ ( comp_C3531382070062128313er_num @ G @ I ) @ Fun ) ) ).

% map_fun.compositionality
thf(fact_4866_map__fun_Ocomp,axiom,
    ! [F: num > code_integer,G: code_integer > code_integer,H3: code_integer > code_integer,I: num > code_integer] :
      ( ( comp_C9104104092898552938er_num @ ( map_fu5269526691104299083nteger @ F @ G ) @ ( map_fu603974984455542859nteger @ H3 @ I ) )
      = ( map_fu8223014087597717000nteger @ ( comp_C3531382070062128313er_num @ H3 @ F ) @ ( comp_C3531382070062128313er_num @ G @ I ) ) ) ).

% map_fun.comp
thf(fact_4867_map__fun_Ocomp,axiom,
    ! [F: num > code_integer,G: nat > num,H3: code_integer > code_integer,I: int > nat] :
      ( ( comp_C8127656796141369388er_int @ ( map_fu7479816331989151099at_num @ F @ G ) @ ( map_fu5423760287386111952nt_nat @ H3 @ I ) )
      = ( map_fu8479170043890205655nt_num @ ( comp_C3531382070062128313er_num @ H3 @ F ) @ ( comp_nat_num_int @ G @ I ) ) ) ).

% map_fun.comp
thf(fact_4868_map__fun_Ocomp,axiom,
    ! [F: int > nat,G: code_integer > code_integer,H3: nat > num,I: num > code_integer] :
      ( ( comp_n3323228797031117228um_num @ ( map_fu1717202910479192016nteger @ F @ G ) @ ( map_fu1090464068875036027nteger @ H3 @ I ) )
      = ( map_fu229184005524548567nteger @ ( comp_nat_num_int @ H3 @ F ) @ ( comp_C3531382070062128313er_num @ G @ I ) ) ) ).

% map_fun.comp
thf(fact_4869_map__fun_Ocomp,axiom,
    ! [F: int > nat,G: nat > num,H3: nat > num,I: int > nat] :
      ( ( comp_n4765151935410462062um_int @ ( map_fu3669222299741785344at_num @ F @ G ) @ ( map_fu8361869985550671104nt_nat @ H3 @ I ) )
      = ( map_fu2609383839613498790nt_num @ ( comp_nat_num_int @ H3 @ F ) @ ( comp_nat_num_int @ G @ I ) ) ) ).

% map_fun.comp
thf(fact_4870_map__fun_Ocomp,axiom,
    ! [F: int > int,G: $o > $o,H3: int > product_prod_nat_nat,I: $o > $o] :
      ( ( comp_i6226780894115828703_nat_o @ ( map_fun_int_int_o_o @ F @ G ) @ ( map_fu4826362097070443709at_o_o @ H3 @ I ) )
      = ( map_fu4826362097070443709at_o_o @ ( comp_i1108739374719436894at_int @ H3 @ F ) @ ( comp_o_o_o @ G @ I ) ) ) ).

% map_fun.comp
thf(fact_4871_map__fun_Ocomp,axiom,
    ! [F: int > int,G: rat > rat,H3: int > int,I: product_prod_int_int > rat] :
      ( ( comp_i5271019637457297241nt_int @ ( map_fu6531859369801489698at_rat @ F @ G ) @ ( map_fu3424225382358772769nt_rat @ H3 @ I ) )
      = ( map_fu3424225382358772769nt_rat @ ( comp_int_int_int @ H3 @ F ) @ ( comp_r4727614531642239972nt_int @ G @ I ) ) ) ).

% map_fun.comp
thf(fact_4872_map__fun_Ocomp,axiom,
    ! [F: int > product_prod_nat_nat,G: $o > $o,H3: product_prod_nat_nat > product_prod_nat_nat,I: $o > $o] :
      ( ( comp_P286844350285437056_nat_o @ ( map_fu4826362097070443709at_o_o @ F @ G ) @ ( map_fu1707080289169145046at_o_o @ H3 @ I ) )
      = ( map_fu4826362097070443709at_o_o @ ( comp_P7428279379633703887at_int @ H3 @ F ) @ ( comp_o_o_o @ G @ I ) ) ) ).

% map_fun.comp
thf(fact_4873_map__fun_Ocomp,axiom,
    ! [F: int > int,G: product_prod_int_int > rat,H3: int > int,I: product_prod_int_int > product_prod_int_int] :
      ( ( comp_i118865614814752126nt_int @ ( map_fu3424225382358772769nt_rat @ F @ G ) @ ( map_fu2180209013632630320nt_int @ H3 @ I ) )
      = ( map_fu3424225382358772769nt_rat @ ( comp_int_int_int @ H3 @ F ) @ ( comp_P5648219952011055433nt_int @ G @ I ) ) ) ).

% map_fun.comp
thf(fact_4874_map__fun_Ocomp,axiom,
    ! [F: num > code_integer,G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,H3: code_integer > code_integer,I: code_integer > code_integer] :
      ( ( comp_C5301300457492916725nteger @ ( map_fu1235460277957925907nteger @ F @ G ) @ ( map_fu5082455417554595470nteger @ H3 @ I ) )
      = ( map_fu1235460277957925907nteger @ ( comp_C3531382070062128313er_num @ H3 @ F ) @ ( comp_C1593894019821074884nteger @ G @ I ) ) ) ).

% map_fun.comp
thf(fact_4875_map__fun_Ocomp,axiom,
    ! [F: code_integer > code_integer,G: code_integer > code_integer,H3: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,I: num > code_integer] :
      ( ( comp_C3959375555751086581er_num @ ( map_fu5082455417554595470nteger @ F @ G ) @ ( map_fu1335501423231535891nteger @ H3 @ I ) )
      = ( map_fu1335501423231535891nteger @ ( comp_C1593894019821074884nteger @ H3 @ F ) @ ( comp_C3531382070062128313er_num @ G @ I ) ) ) ).

% map_fun.comp
thf(fact_4876_translation__subtract__Compl,axiom,
    ! [A: code_integer,T: set_Code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [X3: code_integer] : ( minus_8373710615458151222nteger @ X3 @ A )
        @ ( uminus804700908173204444nteger @ T ) )
      = ( uminus804700908173204444nteger
        @ ( image_4470545334726330049nteger
          @ ^ [X3: code_integer] : ( minus_8373710615458151222nteger @ X3 @ A )
          @ T ) ) ) ).

% translation_subtract_Compl
thf(fact_4877_translation__subtract__Compl,axiom,
    ! [A: rat,T: set_rat] :
      ( ( image_rat_rat
        @ ^ [X3: rat] : ( minus_minus_rat @ X3 @ A )
        @ ( uminus2201863774496077783et_rat @ T ) )
      = ( uminus2201863774496077783et_rat
        @ ( image_rat_rat
          @ ^ [X3: rat] : ( minus_minus_rat @ X3 @ A )
          @ T ) ) ) ).

% translation_subtract_Compl
thf(fact_4878_translation__subtract__Compl,axiom,
    ! [A: int,T: set_int] :
      ( ( image_int_int
        @ ^ [X3: int] : ( minus_minus_int @ X3 @ A )
        @ ( uminus1532241313380277803et_int @ T ) )
      = ( uminus1532241313380277803et_int
        @ ( image_int_int
          @ ^ [X3: int] : ( minus_minus_int @ X3 @ A )
          @ T ) ) ) ).

% translation_subtract_Compl
thf(fact_4879_nat__minus__add__max,axiom,
    ! [N2: nat,M2: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N2 @ M2 ) @ M2 )
      = ( ord_max_nat @ N2 @ M2 ) ) ).

% nat_minus_add_max
thf(fact_4880_translation__diff,axiom,
    ! [A: rat,S: set_rat,T: set_rat] :
      ( ( image_rat_rat @ ( plus_plus_rat @ A ) @ ( minus_minus_set_rat @ S @ T ) )
      = ( minus_minus_set_rat @ ( image_rat_rat @ ( plus_plus_rat @ A ) @ S ) @ ( image_rat_rat @ ( plus_plus_rat @ A ) @ T ) ) ) ).

% translation_diff
thf(fact_4881_translation__diff,axiom,
    ! [A: int,S: set_int,T: set_int] :
      ( ( image_int_int @ ( plus_plus_int @ A ) @ ( minus_minus_set_int @ S @ T ) )
      = ( minus_minus_set_int @ ( image_int_int @ ( plus_plus_int @ A ) @ S ) @ ( image_int_int @ ( plus_plus_int @ A ) @ T ) ) ) ).

% translation_diff
thf(fact_4882_translation__subtract__diff,axiom,
    ! [A: code_integer,S: set_Code_integer,T: set_Code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [X3: code_integer] : ( minus_8373710615458151222nteger @ X3 @ A )
        @ ( minus_2355218937544613996nteger @ S @ T ) )
      = ( minus_2355218937544613996nteger
        @ ( image_4470545334726330049nteger
          @ ^ [X3: code_integer] : ( minus_8373710615458151222nteger @ X3 @ A )
          @ S )
        @ ( image_4470545334726330049nteger
          @ ^ [X3: code_integer] : ( minus_8373710615458151222nteger @ X3 @ A )
          @ T ) ) ) ).

% translation_subtract_diff
thf(fact_4883_translation__subtract__diff,axiom,
    ! [A: rat,S: set_rat,T: set_rat] :
      ( ( image_rat_rat
        @ ^ [X3: rat] : ( minus_minus_rat @ X3 @ A )
        @ ( minus_minus_set_rat @ S @ T ) )
      = ( minus_minus_set_rat
        @ ( image_rat_rat
          @ ^ [X3: rat] : ( minus_minus_rat @ X3 @ A )
          @ S )
        @ ( image_rat_rat
          @ ^ [X3: rat] : ( minus_minus_rat @ X3 @ A )
          @ T ) ) ) ).

% translation_subtract_diff
thf(fact_4884_translation__subtract__diff,axiom,
    ! [A: int,S: set_int,T: set_int] :
      ( ( image_int_int
        @ ^ [X3: int] : ( minus_minus_int @ X3 @ A )
        @ ( minus_minus_set_int @ S @ T ) )
      = ( minus_minus_set_int
        @ ( image_int_int
          @ ^ [X3: int] : ( minus_minus_int @ X3 @ A )
          @ S )
        @ ( image_int_int
          @ ^ [X3: int] : ( minus_minus_int @ X3 @ A )
          @ T ) ) ) ).

% translation_subtract_diff
thf(fact_4885_take__bit__drop__bit,axiom,
    ! [M2: nat,N2: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger @ M2 @ ( bit_se3928097537394005634nteger @ N2 @ A ) )
      = ( bit_se3928097537394005634nteger @ N2 @ ( bit_se1745604003318907178nteger @ ( plus_plus_nat @ M2 @ N2 ) @ A ) ) ) ).

% take_bit_drop_bit
thf(fact_4886_take__bit__drop__bit,axiom,
    ! [M2: nat,N2: nat,A: int] :
      ( ( bit_se2923211474154528505it_int @ M2 @ ( bit_se8568078237143864401it_int @ N2 @ A ) )
      = ( bit_se8568078237143864401it_int @ N2 @ ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M2 @ N2 ) @ A ) ) ) ).

% take_bit_drop_bit
thf(fact_4887_take__bit__drop__bit,axiom,
    ! [M2: nat,N2: nat,A: nat] :
      ( ( bit_se2925701944663578781it_nat @ M2 @ ( bit_se8570568707652914677it_nat @ N2 @ A ) )
      = ( bit_se8570568707652914677it_nat @ N2 @ ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M2 @ N2 ) @ A ) ) ) ).

% take_bit_drop_bit
thf(fact_4888_take__bit__drop__bit,axiom,
    ! [M2: nat,N2: nat,A: code_natural] :
      ( ( bit_se569199155075624693atural @ M2 @ ( bit_se2751692689150723149atural @ N2 @ A ) )
      = ( bit_se2751692689150723149atural @ N2 @ ( bit_se569199155075624693atural @ ( plus_plus_nat @ M2 @ N2 ) @ A ) ) ) ).

% take_bit_drop_bit
thf(fact_4889_bit__drop__bit__eq,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se9216721137139052372nteger @ ( bit_se3928097537394005634nteger @ N2 @ A ) )
      = ( comp_nat_o_nat @ ( bit_se9216721137139052372nteger @ A ) @ ( plus_plus_nat @ N2 ) ) ) ).

% bit_drop_bit_eq
thf(fact_4890_bit__drop__bit__eq,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se8568078237143864401it_int @ N2 @ A ) )
      = ( comp_nat_o_nat @ ( bit_se1146084159140164899it_int @ A ) @ ( plus_plus_nat @ N2 ) ) ) ).

% bit_drop_bit_eq
thf(fact_4891_bit__drop__bit__eq,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se8570568707652914677it_nat @ N2 @ A ) )
      = ( comp_nat_o_nat @ ( bit_se1148574629649215175it_nat @ A ) @ ( plus_plus_nat @ N2 ) ) ) ).

% bit_drop_bit_eq
thf(fact_4892_bit__drop__bit__eq,axiom,
    ! [N2: nat,A: code_natural] :
      ( ( bit_se8040316288895769887atural @ ( bit_se2751692689150723149atural @ N2 @ A ) )
      = ( comp_nat_o_nat @ ( bit_se8040316288895769887atural @ A ) @ ( plus_plus_nat @ N2 ) ) ) ).

% bit_drop_bit_eq
thf(fact_4893_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: nat,N2: nat] :
      ( ( divide_divide_nat @ A @ ( bit_se547839408752420682it_nat @ N2 @ one_one_nat ) )
      = ( bit_se8570568707652914677it_nat @ N2 @ A ) ) ).

% div_push_bit_of_1_eq_drop_bit
thf(fact_4894_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( divide6298287555418463151nteger @ A @ ( bit_se7788150548672797655nteger @ N2 @ one_one_Code_integer ) )
      = ( bit_se3928097537394005634nteger @ N2 @ A ) ) ).

% div_push_bit_of_1_eq_drop_bit
thf(fact_4895_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: int,N2: nat] :
      ( ( divide_divide_int @ A @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) )
      = ( bit_se8568078237143864401it_int @ N2 @ A ) ) ).

% div_push_bit_of_1_eq_drop_bit
thf(fact_4896_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: code_natural,N2: nat] :
      ( ( divide5121882707175180666atural @ A @ ( bit_se6611745700429515170atural @ N2 @ one_one_Code_natural ) )
      = ( bit_se2751692689150723149atural @ N2 @ A ) ) ).

% div_push_bit_of_1_eq_drop_bit
thf(fact_4897_bit__iff__and__drop__bit__eq__1,axiom,
    ( bit_se9216721137139052372nteger
    = ( ^ [A5: code_integer,N4: nat] :
          ( ( bit_se3949692690581998587nteger @ ( bit_se3928097537394005634nteger @ N4 @ A5 ) @ one_one_Code_integer )
          = one_one_Code_integer ) ) ) ).

% bit_iff_and_drop_bit_eq_1
thf(fact_4898_bit__iff__and__drop__bit__eq__1,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [A5: int,N4: nat] :
          ( ( bit_se725231765392027082nd_int @ ( bit_se8568078237143864401it_int @ N4 @ A5 ) @ one_one_int )
          = one_one_int ) ) ) ).

% bit_iff_and_drop_bit_eq_1
thf(fact_4899_bit__iff__and__drop__bit__eq__1,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [A5: nat,N4: nat] :
          ( ( bit_se727722235901077358nd_nat @ ( bit_se8570568707652914677it_nat @ N4 @ A5 ) @ one_one_nat )
          = one_one_nat ) ) ) ).

% bit_iff_and_drop_bit_eq_1
thf(fact_4900_bit__iff__and__drop__bit__eq__1,axiom,
    ( bit_se8040316288895769887atural
    = ( ^ [A5: code_natural,N4: nat] :
          ( ( bit_se2773287842338716102atural @ ( bit_se2751692689150723149atural @ N4 @ A5 ) @ one_one_Code_natural )
          = one_one_Code_natural ) ) ) ).

% bit_iff_and_drop_bit_eq_1
thf(fact_4901_bits__ident,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( bit_se7788150548672797655nteger @ N2 @ ( bit_se3928097537394005634nteger @ N2 @ A ) ) @ ( bit_se1745604003318907178nteger @ N2 @ A ) )
      = A ) ).

% bits_ident
thf(fact_4902_bits__ident,axiom,
    ! [N2: nat,A: int] :
      ( ( plus_plus_int @ ( bit_se545348938243370406it_int @ N2 @ ( bit_se8568078237143864401it_int @ N2 @ A ) ) @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
      = A ) ).

% bits_ident
thf(fact_4903_bits__ident,axiom,
    ! [N2: nat,A: nat] :
      ( ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N2 @ ( bit_se8570568707652914677it_nat @ N2 @ A ) ) @ ( bit_se2925701944663578781it_nat @ N2 @ A ) )
      = A ) ).

% bits_ident
thf(fact_4904_bits__ident,axiom,
    ! [N2: nat,A: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( bit_se6611745700429515170atural @ N2 @ ( bit_se2751692689150723149atural @ N2 @ A ) ) @ ( bit_se569199155075624693atural @ N2 @ A ) )
      = A ) ).

% bits_ident
thf(fact_4905_slice__eq__mask,axiom,
    ! [N2: nat,M2: nat,A: code_integer] :
      ( ( bit_se7788150548672797655nteger @ N2 @ ( bit_se1745604003318907178nteger @ M2 @ ( bit_se3928097537394005634nteger @ N2 @ A ) ) )
      = ( bit_se3949692690581998587nteger @ A @ ( bit_se3949692690581998587nteger @ ( bit_se2119862282449309892nteger @ ( plus_plus_nat @ M2 @ N2 ) ) @ ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N2 ) ) ) ) ) ).

% slice_eq_mask
thf(fact_4906_slice__eq__mask,axiom,
    ! [N2: nat,M2: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ N2 @ ( bit_se2923211474154528505it_int @ M2 @ ( bit_se8568078237143864401it_int @ N2 @ A ) ) )
      = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ ( bit_se2000444600071755411sk_int @ ( plus_plus_nat @ M2 @ N2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) ) ) ) ).

% slice_eq_mask
thf(fact_4907_image__add__0,axiom,
    ! [S2: set_Code_integer] :
      ( ( image_4470545334726330049nteger @ ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger ) @ S2 )
      = S2 ) ).

% image_add_0
thf(fact_4908_image__add__0,axiom,
    ! [S2: set_mu2057375006010111271at_nat] :
      ( ( image_2516384619306764229at_nat @ ( plus_p7104986032573967614at_nat @ zero_z1048942125864253310at_nat ) @ S2 )
      = S2 ) ).

% image_add_0
thf(fact_4909_image__add__0,axiom,
    ! [S2: set_rat] :
      ( ( image_rat_rat @ ( plus_plus_rat @ zero_zero_rat ) @ S2 )
      = S2 ) ).

% image_add_0
thf(fact_4910_image__add__0,axiom,
    ! [S2: set_nat] :
      ( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S2 )
      = S2 ) ).

% image_add_0
thf(fact_4911_image__add__0,axiom,
    ! [S2: set_int] :
      ( ( image_int_int @ ( plus_plus_int @ zero_zero_int ) @ S2 )
      = S2 ) ).

% image_add_0
thf(fact_4912_times__int__def,axiom,
    ( times_times_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X3: nat,Y3: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y3 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V2 ) @ ( times_times_nat @ Y3 @ U2 ) ) ) ) ) ) ) ).

% times_int_def
thf(fact_4913_minus__int__def,axiom,
    ( minus_minus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X3: nat,Y3: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ Y3 @ U2 ) ) ) ) ) ) ).

% minus_int_def
thf(fact_4914_plus__int__def,axiom,
    ( plus_plus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X3: nat,Y3: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y3 @ V2 ) ) ) ) ) ) ).

% plus_int_def
thf(fact_4915_rat__minus__code,axiom,
    ! [P4: rat,Q3: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P4 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C2: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A5 @ D4 ) @ ( times_times_int @ B4 @ C2 ) ) @ ( times_times_int @ C2 @ D4 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_minus_code
thf(fact_4916_rat__plus__code,axiom,
    ! [P4: rat,Q3: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P4 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C2: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A5 @ D4 ) @ ( times_times_int @ B4 @ C2 ) ) @ ( times_times_int @ C2 @ D4 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_plus_code
thf(fact_4917_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastAtMost
thf(fact_4918_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_4919_in__inv__image,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( inv_im7023619800666061458at_nat @ R2 @ F ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_4920_in__inv__image,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat,F: set_Pr1261947904930325089at_nat > nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( inv_im6750873321330482877at_nat @ R2 @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_4921_in__inv__image,axiom,
    ! [X: nat,Y: nat,R2: set_Pr4329608150637261639at_nat,F: nat > set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_im4437241714082187487at_nat @ R2 @ F ) )
      = ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_4922_in__inv__image,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( inv_image_nat_nat @ R2 @ F ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ X ) @ ( F @ Y ) ) @ R2 ) ) ).

% in_inv_image
thf(fact_4923_pair__imageI,axiom,
    ! [A: nat,B: nat,A2: set_Pr1261947904930325089at_nat,F: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A2 )
     => ( member_o @ ( F @ A @ B ) @ ( image_3693632289388996572_nat_o @ ( produc6081775807080527818_nat_o @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_4924_pair__imageI,axiom,
    ! [A: nat,B: nat,A2: set_Pr1261947904930325089at_nat,F: nat > nat > int] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A2 )
     => ( member_int @ ( F @ A @ B ) @ ( image_2483585944268220136at_int @ ( produc6840382203811409530at_int @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_4925_pair__imageI,axiom,
    ! [A: nat,B: nat,A2: set_Pr1261947904930325089at_nat,F: nat > nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A2 )
     => ( member_nat @ ( F @ A @ B ) @ ( image_2486076414777270412at_nat @ ( produc6842872674320459806at_nat @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_4926_pair__imageI,axiom,
    ! [A: int,B: int,A2: set_Pr958786334691620121nt_int,F: int > int > int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ A2 )
     => ( member_int @ ( F @ A @ B ) @ ( image_5042161079198086560nt_int @ ( produc8211389475949308722nt_int @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_4927_pair__imageI,axiom,
    ! [A: nat,B: nat,A2: set_Pr1261947904930325089at_nat,F: nat > nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A2 )
     => ( member8440522571783428010at_nat @ ( F @ A @ B ) @ ( image_5168914502847457605at_nat @ ( produc2626176000494625587at_nat @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_4928_pair__imageI,axiom,
    ! [A: int,B: int,A2: set_Pr958786334691620121nt_int,F: int > int > product_prod_int_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ A2 )
     => ( member5262025264175285858nt_int @ ( F @ A @ B ) @ ( image_2653370878348428101nt_int @ ( produc4245557441103728435nt_int @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_4929_pair__imageI,axiom,
    ! [A: nat,B: nat,A2: set_Pr1261947904930325089at_nat,F: nat > nat > product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A2 )
     => ( member2006080708140975699_nat_o @ ( F @ A @ B ) @ ( image_2972642778337070200_nat_o @ ( produc8739625826339149834_nat_o @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_4930_pair__imageI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr4329608150637261639at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ A2 )
     => ( member_o @ ( F @ A @ B ) @ ( image_3500134854233901558_nat_o @ ( produc410239310623530412_nat_o @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_4931_pair__imageI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr4329608150637261639at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ A2 )
     => ( member_nat @ ( F @ A @ B ) @ ( image_3198525501578754290at_nat @ ( produc4001655692221972156at_nat @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_4932_pair__imageI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr4329608150637261639at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > int] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ A2 )
     => ( member_int @ ( F @ A @ B ) @ ( image_3196035031069704014at_int @ ( produc3999165221712921880at_int @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_4933_int__of__integer__max,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( code_int_of_integer @ ( ord_max_Code_integer @ K @ L ) )
      = ( ord_max_int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).

% int_of_integer_max
thf(fact_4934_rat__one__code,axiom,
    ( ( quotient_of @ one_one_rat )
    = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).

% rat_one_code
thf(fact_4935_of__nat__of__integer,axiom,
    ! [K: code_integer] :
      ( ( semiri4939895301339042750nteger @ ( code_nat_of_integer @ K ) )
      = ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ K ) ) ).

% of_nat_of_integer
thf(fact_4936_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_4937_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 ) ) ).

% quotient_of_number(3)
thf(fact_4938_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 ) ) ).

% quotient_of_number(4)
thf(fact_4939_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 ) ) ).

% quotient_of_number(5)
thf(fact_4940_diff__rat__def,axiom,
    ( minus_minus_rat
    = ( ^ [Q5: rat,R3: rat] : ( plus_plus_rat @ Q5 @ ( uminus_uminus_rat @ R3 ) ) ) ) ).

% diff_rat_def
thf(fact_4941_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 ) ) ) ) ).

% quotient_of_div
thf(fact_4942_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).

% zero_notin_Suc_image
thf(fact_4943_None__notin__image__Some,axiom,
    ! [A2: set_Pr5142436192348053284it_nat] :
      ~ ( member3517700364412436275it_nat @ none_P2587165478634041381it_nat @ ( image_4799632500972785611it_nat @ some_P6105628898608307625it_nat @ A2 ) ) ).

% None_notin_image_Some
thf(fact_4944_None__notin__image__Some,axiom,
    ! [A2: set_num] :
      ~ ( member_option_num @ none_num @ ( image_num_option_num @ some_num @ A2 ) ) ).

% None_notin_image_Some
thf(fact_4945_None__notin__image__Some,axiom,
    ! [A2: set_Pr1354866905816374718it_nat] :
      ~ ( member5596548051065438575it_nat @ none_P9117596204409417319it_nat @ ( image_3455408117458022631it_nat @ some_P1914260805536162275it_nat @ A2 ) ) ).

% None_notin_image_Some
thf(fact_4946_plus__integer__def,axiom,
    ( plus_p5714425477246183910nteger
    = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ plus_plus_int ) ) ).

% plus_integer_def
thf(fact_4947_times__integer__def,axiom,
    ( times_3573771949741848930nteger
    = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ times_times_int ) ) ).

% times_integer_def
thf(fact_4948_minus__integer__def,axiom,
    ( minus_8373710615458151222nteger
    = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ minus_minus_int ) ) ).

% minus_integer_def
thf(fact_4949_divide__integer__def,axiom,
    ( divide6298287555418463151nteger
    = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ divide_divide_int ) ) ).

% divide_integer_def
thf(fact_4950_modulo__integer__def,axiom,
    ( modulo364778990260209775nteger
    = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ modulo_modulo_int ) ) ).

% modulo_integer_def
thf(fact_4951_and__integer__def,axiom,
    ( bit_se3949692690581998587nteger
    = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se725231765392027082nd_int ) ) ).

% and_integer_def
thf(fact_4952_or__integer__def,axiom,
    ( bit_se1080825931792720795nteger
    = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se1409905431419307370or_int ) ) ).

% or_integer_def
thf(fact_4953_xor__integer__def,axiom,
    ( bit_se3222712562003087583nteger
    = ( map_fu8272188784021352819nteger @ code_int_of_integer @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se6526347334894502574or_int ) ) ).

% xor_integer_def
thf(fact_4954_image__int__atLeastAtMost,axiom,
    ! [A: nat,B: nat] :
      ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% image_int_atLeastAtMost
thf(fact_4955_image__int__atLeastLessThan,axiom,
    ! [A: nat,B: nat] :
      ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B ) )
      = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% image_int_atLeastLessThan
thf(fact_4956_quotient__of__denom__pos,axiom,
    ! [R2: rat,P4: int,Q3: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ P4 @ Q3 ) )
     => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).

% quotient_of_denom_pos
thf(fact_4957_image__Suc__atMost,axiom,
    ! [N2: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N2 ) ) ) ).

% image_Suc_atMost
thf(fact_4958_image__Suc__lessThan,axiom,
    ! [N2: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ).

% image_Suc_lessThan
thf(fact_4959_rat__uminus__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( uminus_uminus_rat @ P4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A5 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_uminus_code
thf(fact_4960_rat__abs__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( abs_abs_rat @ P4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int] : ( product_Pair_int_int @ ( abs_abs_int @ A5 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_abs_code
thf(fact_4961_uminus__integer__def,axiom,
    ( uminus1351360451143612070nteger
    = ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int @ uminus_uminus_int ) ) ).

% uminus_integer_def
thf(fact_4962_abs__integer__def,axiom,
    ( abs_abs_Code_integer
    = ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int @ abs_abs_int ) ) ).

% abs_integer_def
thf(fact_4963_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] :
      ( ( image_int_int
        @ ^ [X3: int] : ( plus_plus_int @ X3 @ L )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
      = ( set_or4662586982721622107an_int @ L @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_4964_sgn__integer__def,axiom,
    ( sgn_sgn_Code_integer
    = ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int @ sgn_sgn_int ) ) ).

% sgn_integer_def
thf(fact_4965_not__integer__def,axiom,
    ( bit_ri7632146776885996613nteger
    = ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int @ bit_ri7919022796975470100ot_int ) ) ).

% not_integer_def
thf(fact_4966_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [X3: code_integer] : ( plus_p5714425477246183910nteger @ X3 @ L )
        @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ U @ L ) ) )
      = ( set_or8404916559141939852nteger @ L @ U ) ) ).

% image_add_integer_atLeastLessThan
thf(fact_4967_rat__times__code,axiom,
    ! [P4: rat,Q3: rat] :
      ( ( quotient_of @ ( times_times_rat @ P4 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C2: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ B4 ) @ ( times_times_int @ C2 @ D4 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_times_code
thf(fact_4968_rat__divide__code,axiom,
    ! [P4: rat,Q3: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P4 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C2: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D4: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ D4 ) @ ( times_times_int @ C2 @ B4 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_divide_code
thf(fact_4969_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ U )
     => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
        = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).

% image_atLeastZeroLessThan_int
thf(fact_4970_quotient__of__int,axiom,
    ! [A: int] :
      ( ( quotient_of @ ( of_int @ A ) )
      = ( product_Pair_int_int @ A @ one_one_int ) ) ).

% quotient_of_int
thf(fact_4971_rat__inverse__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( A5 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A5 ) @ B4 ) @ ( abs_abs_int @ A5 ) ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_inverse_code
thf(fact_4972_diff__numeral__special_I8_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( neg_numeral_sub_int @ one @ M2 ) ) ).

% diff_numeral_special(8)
thf(fact_4973_diff__numeral__special_I8_J,axiom,
    ! [M2: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( neg_nu5755505904847501662nteger @ one @ M2 ) ) ).

% diff_numeral_special(8)
thf(fact_4974_diff__numeral__special_I8_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( neg_numeral_sub_rat @ one @ M2 ) ) ).

% diff_numeral_special(8)
thf(fact_4975_diff__numeral__special_I7_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( neg_numeral_sub_int @ N2 @ one ) ) ).

% diff_numeral_special(7)
thf(fact_4976_diff__numeral__special_I7_J,axiom,
    ! [N2: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( neg_nu5755505904847501662nteger @ N2 @ one ) ) ).

% diff_numeral_special(7)
thf(fact_4977_diff__numeral__special_I7_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( neg_numeral_sub_rat @ N2 @ one ) ) ).

% diff_numeral_special(7)
thf(fact_4978_minus__sub__one__diff__one,axiom,
    ! [M2: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( neg_numeral_sub_int @ M2 @ one ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).

% minus_sub_one_diff_one
thf(fact_4979_minus__sub__one__diff__one,axiom,
    ! [M2: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( neg_nu5755505904847501662nteger @ M2 @ one ) ) @ one_one_Code_integer )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% minus_sub_one_diff_one
thf(fact_4980_minus__sub__one__diff__one,axiom,
    ! [M2: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( neg_numeral_sub_rat @ M2 @ one ) ) @ one_one_rat )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).

% minus_sub_one_diff_one
thf(fact_4981_add__neg__numeral__special_I4_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N2 ) )
      = ( neg_numeral_sub_int @ N2 @ one ) ) ).

% add_neg_numeral_special(4)
thf(fact_4982_add__neg__numeral__special_I4_J,axiom,
    ! [N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( neg_nu5755505904847501662nteger @ N2 @ one ) ) ).

% add_neg_numeral_special(4)
thf(fact_4983_add__neg__numeral__special_I4_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N2 ) )
      = ( neg_numeral_sub_rat @ N2 @ one ) ) ).

% add_neg_numeral_special(4)
thf(fact_4984_int__of__integer__sub,axiom,
    ! [K: num,L: num] :
      ( ( code_int_of_integer @ ( neg_nu5755505904847501662nteger @ K @ L ) )
      = ( neg_numeral_sub_int @ K @ L ) ) ).

% int_of_integer_sub
thf(fact_4985_inverse__1,axiom,
    ( ( inverse_inverse_rat @ one_one_rat )
    = one_one_rat ) ).

% inverse_1
thf(fact_4986_inverse__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( ( inverse_inverse_rat @ X )
        = one_one_rat )
      = ( X = one_one_rat ) ) ).

% inverse_eq_1_iff
thf(fact_4987_right__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
        = one_one_rat ) ) ).

% right_inverse
thf(fact_4988_left__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
        = one_one_rat ) ) ).

% left_inverse
thf(fact_4989_inverse__eq__divide__numeral,axiom,
    ! [W: num] :
      ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ W ) )
      = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ W ) ) ) ).

% inverse_eq_divide_numeral
thf(fact_4990_add__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( neg_numeral_sub_int @ N2 @ M2 ) ) ).

% add_neg_numeral_simps(2)
thf(fact_4991_add__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( neg_nu5755505904847501662nteger @ N2 @ M2 ) ) ).

% add_neg_numeral_simps(2)
thf(fact_4992_add__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N2 ) )
      = ( neg_numeral_sub_rat @ N2 @ M2 ) ) ).

% add_neg_numeral_simps(2)
thf(fact_4993_add__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( neg_numeral_sub_int @ M2 @ N2 ) ) ).

% add_neg_numeral_simps(1)
thf(fact_4994_add__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( neg_nu5755505904847501662nteger @ M2 @ N2 ) ) ).

% add_neg_numeral_simps(1)
thf(fact_4995_add__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( neg_numeral_sub_rat @ M2 @ N2 ) ) ).

% add_neg_numeral_simps(1)
thf(fact_4996_semiring__norm_I166_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( plus_plus_int @ ( neg_numeral_sub_int @ V @ W ) @ Y ) ) ).

% semiring_norm(166)
thf(fact_4997_semiring__norm_I166_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ V ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( plus_p5714425477246183910nteger @ ( neg_nu5755505904847501662nteger @ V @ W ) @ Y ) ) ).

% semiring_norm(166)
thf(fact_4998_semiring__norm_I166_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( plus_plus_rat @ ( neg_numeral_sub_rat @ V @ W ) @ Y ) ) ).

% semiring_norm(166)
thf(fact_4999_semiring__norm_I167_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Y ) )
      = ( plus_plus_int @ ( neg_numeral_sub_int @ W @ V ) @ Y ) ) ).

% semiring_norm(167)
thf(fact_5000_semiring__norm_I167_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ W ) @ Y ) )
      = ( plus_p5714425477246183910nteger @ ( neg_nu5755505904847501662nteger @ W @ V ) @ Y ) ) ).

% semiring_norm(167)
thf(fact_5001_semiring__norm_I167_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
      = ( plus_plus_rat @ ( neg_numeral_sub_rat @ W @ V ) @ Y ) ) ).

% semiring_norm(167)
thf(fact_5002_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] :
      ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( divide_divide_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).

% inverse_eq_divide_neg_numeral
thf(fact_5003_diff__numeral__special_I2_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int )
      = ( neg_numeral_sub_int @ M2 @ one ) ) ).

% diff_numeral_special(2)
thf(fact_5004_diff__numeral__special_I2_J,axiom,
    ! [M2: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M2 ) @ one_one_Code_integer )
      = ( neg_nu5755505904847501662nteger @ M2 @ one ) ) ).

% diff_numeral_special(2)
thf(fact_5005_diff__numeral__special_I2_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M2 ) @ one_one_rat )
      = ( neg_numeral_sub_rat @ M2 @ one ) ) ).

% diff_numeral_special(2)
thf(fact_5006_diff__numeral__special_I1_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
      = ( neg_numeral_sub_int @ one @ N2 ) ) ).

% diff_numeral_special(1)
thf(fact_5007_diff__numeral__special_I1_J,axiom,
    ! [N2: num] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N2 ) )
      = ( neg_nu5755505904847501662nteger @ one @ N2 ) ) ).

% diff_numeral_special(1)
thf(fact_5008_diff__numeral__special_I1_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( neg_numeral_sub_rat @ one @ N2 ) ) ).

% diff_numeral_special(1)
thf(fact_5009_add__neg__numeral__special_I1_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) )
      = ( neg_numeral_sub_int @ one @ M2 ) ) ).

% add_neg_numeral_special(1)
thf(fact_5010_add__neg__numeral__special_I1_J,axiom,
    ! [M2: num] :
      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) )
      = ( neg_nu5755505904847501662nteger @ one @ M2 ) ) ).

% add_neg_numeral_special(1)
thf(fact_5011_add__neg__numeral__special_I1_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) )
      = ( neg_numeral_sub_rat @ one @ M2 ) ) ).

% add_neg_numeral_special(1)
thf(fact_5012_add__neg__numeral__special_I2_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int )
      = ( neg_numeral_sub_int @ one @ M2 ) ) ).

% add_neg_numeral_special(2)
thf(fact_5013_add__neg__numeral__special_I2_J,axiom,
    ! [M2: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer )
      = ( neg_nu5755505904847501662nteger @ one @ M2 ) ) ).

% add_neg_numeral_special(2)
thf(fact_5014_add__neg__numeral__special_I2_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat )
      = ( neg_numeral_sub_rat @ one @ M2 ) ) ).

% add_neg_numeral_special(2)
thf(fact_5015_add__neg__numeral__special_I3_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( neg_numeral_sub_int @ M2 @ one ) ) ).

% add_neg_numeral_special(3)
thf(fact_5016_add__neg__numeral__special_I3_J,axiom,
    ! [M2: num] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( neg_nu5755505904847501662nteger @ M2 @ one ) ) ).

% add_neg_numeral_special(3)
thf(fact_5017_add__neg__numeral__special_I3_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( neg_numeral_sub_rat @ M2 @ one ) ) ).

% add_neg_numeral_special(3)
thf(fact_5018_inverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = one_one_rat )
     => ( ( inverse_inverse_rat @ A )
        = B ) ) ).

% inverse_unique
thf(fact_5019_inverse__eq__divide,axiom,
    ( inverse_inverse_rat
    = ( divide_divide_rat @ one_one_rat ) ) ).

% inverse_eq_divide
thf(fact_5020_inverse__le__1__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
      = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
        | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).

% inverse_le_1_iff
thf(fact_5021_one__less__inverse__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
      = ( ( ord_less_rat @ zero_zero_rat @ X )
        & ( ord_less_rat @ X @ one_one_rat ) ) ) ).

% one_less_inverse_iff
thf(fact_5022_one__less__inverse,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).

% one_less_inverse
thf(fact_5023_division__ring__inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( plus_plus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).

% division_ring_inverse_add
thf(fact_5024_inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( times_times_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( inverse_inverse_rat @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).

% inverse_add
thf(fact_5025_field__class_Ofield__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
        = one_one_rat ) ) ).

% field_class.field_inverse
thf(fact_5026_nonzero__inverse__eq__divide,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( inverse_inverse_rat @ A )
        = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).

% nonzero_inverse_eq_divide
thf(fact_5027_one__le__inverse__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
      = ( ( ord_less_rat @ zero_zero_rat @ X )
        & ( ord_less_eq_rat @ X @ one_one_rat ) ) ) ).

% one_le_inverse_iff
thf(fact_5028_inverse__less__1__iff,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
      = ( ( ord_less_eq_rat @ X @ zero_zero_rat )
        | ( ord_less_rat @ one_one_rat @ X ) ) ) ).

% inverse_less_1_iff
thf(fact_5029_one__le__inverse,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).

% one_le_inverse
thf(fact_5030_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 ) ) ) ).

% Frct_code_post(5)
thf(fact_5031_Code__Numeral_Odup__def,axiom,
    ( code_dup
    = ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int
      @ ^ [K4: int] : ( plus_plus_int @ K4 @ K4 ) ) ) ).

% Code_Numeral.dup_def
thf(fact_5032_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 ) ) ) ).

% Frct_code_post(6)
thf(fact_5033_Frct__code__post_I4_J,axiom,
    ! [K: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
      = ( numeral_numeral_rat @ K ) ) ).

% Frct_code_post(4)
thf(fact_5034_eq__numeral__iff__iszero_I8_J,axiom,
    ! [Y: num] :
      ( ( one_one_int
        = ( uminus_uminus_int @ ( numeral_numeral_int @ Y ) ) )
      = ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ one @ Y ) ) ) ) ).

% eq_numeral_iff_iszero(8)
thf(fact_5035_eq__numeral__iff__iszero_I8_J,axiom,
    ! [Y: num] :
      ( ( one_one_Code_integer
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ Y ) ) )
      = ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ Y ) ) ) ) ).

% eq_numeral_iff_iszero(8)
thf(fact_5036_eq__numeral__iff__iszero_I8_J,axiom,
    ! [Y: num] :
      ( ( one_one_rat
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ Y ) ) )
      = ( ring_1_iszero_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ one @ Y ) ) ) ) ).

% eq_numeral_iff_iszero(8)
thf(fact_5037_eq__numeral__iff__iszero_I7_J,axiom,
    ! [X: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
        = one_one_int )
      = ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X @ one ) ) ) ) ).

% eq_numeral_iff_iszero(7)
thf(fact_5038_eq__numeral__iff__iszero_I7_J,axiom,
    ! [X: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) )
        = one_one_Code_integer )
      = ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ X @ one ) ) ) ) ).

% eq_numeral_iff_iszero(7)
thf(fact_5039_eq__numeral__iff__iszero_I7_J,axiom,
    ! [X: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) )
        = one_one_rat )
      = ( ring_1_iszero_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ X @ one ) ) ) ) ).

% eq_numeral_iff_iszero(7)
thf(fact_5040_Inf_OINF__image,axiom,
    ! [Inf: set_Code_integer > code_integer,G: code_integer > code_integer,F: num > code_integer,A2: set_num] :
      ( ( Inf @ ( image_4470545334726330049nteger @ G @ ( image_8379761247532268670nteger @ F @ A2 ) ) )
      = ( Inf @ ( image_8379761247532268670nteger @ ( comp_C3531382070062128313er_num @ G @ F ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_5041_Inf_OINF__image,axiom,
    ! [Inf: set_num > num,G: nat > num,F: int > nat,A2: set_int] :
      ( ( Inf @ ( image_nat_num @ G @ ( image_int_nat @ F @ A2 ) ) )
      = ( Inf @ ( image_int_num @ ( comp_nat_num_int @ G @ F ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_5042_Inf_OINF__image,axiom,
    ! [Inf: set_Code_integer > code_integer,G: code_integer > code_integer,F: nat > code_integer,A2: set_nat] :
      ( ( Inf @ ( image_4470545334726330049nteger @ G @ ( image_1215581382706833972nteger @ F @ A2 ) ) )
      = ( Inf @ ( image_1215581382706833972nteger @ ( comp_C6974050283906349679er_nat @ G @ F ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_5043_Inf_OINF__image,axiom,
    ! [Inf: set_nat > nat,G: code_integer > nat,F: nat > code_integer,A2: set_nat] :
      ( ( Inf @ ( image_951025933927791156er_nat @ G @ ( image_1215581382706833972nteger @ F @ A2 ) ) )
      = ( Inf @ ( image_nat_nat @ ( comp_C4049556595663050210at_nat @ G @ F ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_5044_Inf_OINF__image,axiom,
    ! [Inf: set_int > int,G: code_integer > int,F: nat > code_integer,A2: set_nat] :
      ( ( Inf @ ( image_948535463418740880er_int @ G @ ( image_1215581382706833972nteger @ F @ A2 ) ) )
      = ( Inf @ ( image_nat_int @ ( comp_C5048910307564104766nt_nat @ G @ F ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_5045_Inf_OINF__image,axiom,
    ! [Inf: set_Code_integer > code_integer,G: int > code_integer,F: nat > int,A2: set_nat] :
      ( ( Inf @ ( image_1587234942943678608nteger @ G @ ( image_nat_int @ F @ A2 ) ) )
      = ( Inf @ ( image_1215581382706833972nteger @ ( comp_i1588355021709917246er_nat @ G @ F ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_5046_Inf_OINF__image,axiom,
    ! [Inf: set_nat > nat,G: int > nat,F: nat > int,A2: set_nat] :
      ( ( Inf @ ( image_int_nat @ G @ ( image_nat_int @ F @ A2 ) ) )
      = ( Inf @ ( image_nat_nat @ ( comp_int_nat_nat @ G @ F ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_5047_Inf_OINF__image,axiom,
    ! [Inf: set_Code_integer > code_integer,G: nat > code_integer,F: nat > nat,A2: set_nat] :
      ( ( Inf @ ( image_1215581382706833972nteger @ G @ ( image_nat_nat @ F @ A2 ) ) )
      = ( Inf @ ( image_1215581382706833972nteger @ ( comp_n3898172953802868194er_nat @ G @ F ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_5048_Inf_OINF__image,axiom,
    ! [Inf: set_nat > nat,G: nat > nat,F: nat > nat,A2: set_nat] :
      ( ( Inf @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A2 ) ) )
      = ( Inf @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_5049_Inf_OINF__image,axiom,
    ! [Inf: set_int > int,G: nat > int,F: int > nat,A2: set_int] :
      ( ( Inf @ ( image_nat_int @ G @ ( image_int_nat @ F @ A2 ) ) )
      = ( Inf @ ( image_int_int @ ( comp_nat_int_int @ G @ F ) @ A2 ) ) ) ).

% Inf.INF_image
thf(fact_5050_not__iszero__1,axiom,
    ~ ( ring_16219924574208605041nteger @ one_one_Code_integer ) ).

% not_iszero_1
thf(fact_5051_not__iszero__1,axiom,
    ~ ( ring_1_iszero_rat @ one_one_rat ) ).

% not_iszero_1
thf(fact_5052_not__iszero__1,axiom,
    ~ ( ring_1_iszero_int @ one_one_int ) ).

% not_iszero_1
thf(fact_5053_Code__Numeral_Odup__code_I1_J,axiom,
    ( ( code_dup @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% Code_Numeral.dup_code(1)
thf(fact_5054_not__iszero__neg__1,axiom,
    ~ ( ring_1_iszero_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_iszero_neg_1
thf(fact_5055_not__iszero__neg__1,axiom,
    ~ ( ring_16219924574208605041nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_iszero_neg_1
thf(fact_5056_not__iszero__neg__1,axiom,
    ~ ( ring_1_iszero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_iszero_neg_1
thf(fact_5057_dup_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( code_dup @ X ) )
      = ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ X ) ) ) ).

% dup.rep_eq
thf(fact_5058_dup_Oabs__eq,axiom,
    ! [X: int] :
      ( ( code_dup @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( plus_plus_int @ X @ X ) ) ) ).

% dup.abs_eq
thf(fact_5059_eq__numeral__iff__iszero_I2_J,axiom,
    ! [X: num,Y: num] :
      ( ( ( numeral_numeral_int @ X )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ Y ) ) )
      = ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X @ Y ) ) ) ) ).

% eq_numeral_iff_iszero(2)
thf(fact_5060_eq__numeral__iff__iszero_I2_J,axiom,
    ! [X: num,Y: num] :
      ( ( ( numera6620942414471956472nteger @ X )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ Y ) ) )
      = ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ X @ Y ) ) ) ) ).

% eq_numeral_iff_iszero(2)
thf(fact_5061_eq__numeral__iff__iszero_I2_J,axiom,
    ! [X: num,Y: num] :
      ( ( ( numeral_numeral_rat @ X )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ Y ) ) )
      = ( ring_1_iszero_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ X @ Y ) ) ) ) ).

% eq_numeral_iff_iszero(2)
thf(fact_5062_eq__numeral__iff__iszero_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ X ) )
        = ( numeral_numeral_int @ Y ) )
      = ( ring_1_iszero_int @ ( numeral_numeral_int @ ( plus_plus_num @ X @ Y ) ) ) ) ).

% eq_numeral_iff_iszero(3)
thf(fact_5063_eq__numeral__iff__iszero_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) )
        = ( numera6620942414471956472nteger @ Y ) )
      = ( ring_16219924574208605041nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ X @ Y ) ) ) ) ).

% eq_numeral_iff_iszero(3)
thf(fact_5064_eq__numeral__iff__iszero_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) )
        = ( numeral_numeral_rat @ Y ) )
      = ( ring_1_iszero_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ X @ Y ) ) ) ) ).

% eq_numeral_iff_iszero(3)
thf(fact_5065_Frct__code__post_I2_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
      = zero_zero_rat ) ).

% Frct_code_post(2)
thf(fact_5066_Frct__code__post_I1_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
      = zero_zero_rat ) ).

% Frct_code_post(1)
thf(fact_5067_Frct__code__post_I3_J,axiom,
    ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
    = one_one_rat ) ).

% Frct_code_post(3)
thf(fact_5068_eq__numeral__iff__iszero_I6_J,axiom,
    ! [Y: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ Y ) )
      = ( ring_1_iszero_int @ ( neg_numeral_sub_int @ one @ Y ) ) ) ).

% eq_numeral_iff_iszero(6)
thf(fact_5069_eq__numeral__iff__iszero_I6_J,axiom,
    ! [Y: num] :
      ( ( one_one_Code_integer
        = ( numera6620942414471956472nteger @ Y ) )
      = ( ring_16219924574208605041nteger @ ( neg_nu5755505904847501662nteger @ one @ Y ) ) ) ).

% eq_numeral_iff_iszero(6)
thf(fact_5070_eq__numeral__iff__iszero_I6_J,axiom,
    ! [Y: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ Y ) )
      = ( ring_1_iszero_rat @ ( neg_numeral_sub_rat @ one @ Y ) ) ) ).

% eq_numeral_iff_iszero(6)
thf(fact_5071_eq__numeral__iff__iszero_I5_J,axiom,
    ! [X: num] :
      ( ( ( numeral_numeral_int @ X )
        = one_one_int )
      = ( ring_1_iszero_int @ ( neg_numeral_sub_int @ X @ one ) ) ) ).

% eq_numeral_iff_iszero(5)
thf(fact_5072_eq__numeral__iff__iszero_I5_J,axiom,
    ! [X: num] :
      ( ( ( numera6620942414471956472nteger @ X )
        = one_one_Code_integer )
      = ( ring_16219924574208605041nteger @ ( neg_nu5755505904847501662nteger @ X @ one ) ) ) ).

% eq_numeral_iff_iszero(5)
thf(fact_5073_eq__numeral__iff__iszero_I5_J,axiom,
    ! [X: num] :
      ( ( ( numeral_numeral_rat @ X )
        = one_one_rat )
      = ( ring_1_iszero_rat @ ( neg_numeral_sub_rat @ X @ one ) ) ) ).

% eq_numeral_iff_iszero(5)
thf(fact_5074_Frct__code__post_I7_J,axiom,
    ! [A: int,B: int] :
      ( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% Frct_code_post(7)
thf(fact_5075_Frct__code__post_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B ) ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% Frct_code_post(8)
thf(fact_5076_Sup_OSUP__image,axiom,
    ! [Sup: set_Code_integer > code_integer,G: code_integer > code_integer,F: num > code_integer,A2: set_num] :
      ( ( Sup @ ( image_4470545334726330049nteger @ G @ ( image_8379761247532268670nteger @ F @ A2 ) ) )
      = ( Sup @ ( image_8379761247532268670nteger @ ( comp_C3531382070062128313er_num @ G @ F ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_5077_Sup_OSUP__image,axiom,
    ! [Sup: set_num > num,G: nat > num,F: int > nat,A2: set_int] :
      ( ( Sup @ ( image_nat_num @ G @ ( image_int_nat @ F @ A2 ) ) )
      = ( Sup @ ( image_int_num @ ( comp_nat_num_int @ G @ F ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_5078_Sup_OSUP__image,axiom,
    ! [Sup: set_Code_integer > code_integer,G: code_integer > code_integer,F: nat > code_integer,A2: set_nat] :
      ( ( Sup @ ( image_4470545334726330049nteger @ G @ ( image_1215581382706833972nteger @ F @ A2 ) ) )
      = ( Sup @ ( image_1215581382706833972nteger @ ( comp_C6974050283906349679er_nat @ G @ F ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_5079_Sup_OSUP__image,axiom,
    ! [Sup: set_nat > nat,G: code_integer > nat,F: nat > code_integer,A2: set_nat] :
      ( ( Sup @ ( image_951025933927791156er_nat @ G @ ( image_1215581382706833972nteger @ F @ A2 ) ) )
      = ( Sup @ ( image_nat_nat @ ( comp_C4049556595663050210at_nat @ G @ F ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_5080_Sup_OSUP__image,axiom,
    ! [Sup: set_int > int,G: code_integer > int,F: nat > code_integer,A2: set_nat] :
      ( ( Sup @ ( image_948535463418740880er_int @ G @ ( image_1215581382706833972nteger @ F @ A2 ) ) )
      = ( Sup @ ( image_nat_int @ ( comp_C5048910307564104766nt_nat @ G @ F ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_5081_Sup_OSUP__image,axiom,
    ! [Sup: set_Code_integer > code_integer,G: int > code_integer,F: nat > int,A2: set_nat] :
      ( ( Sup @ ( image_1587234942943678608nteger @ G @ ( image_nat_int @ F @ A2 ) ) )
      = ( Sup @ ( image_1215581382706833972nteger @ ( comp_i1588355021709917246er_nat @ G @ F ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_5082_Sup_OSUP__image,axiom,
    ! [Sup: set_nat > nat,G: int > nat,F: nat > int,A2: set_nat] :
      ( ( Sup @ ( image_int_nat @ G @ ( image_nat_int @ F @ A2 ) ) )
      = ( Sup @ ( image_nat_nat @ ( comp_int_nat_nat @ G @ F ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_5083_Sup_OSUP__image,axiom,
    ! [Sup: set_Code_integer > code_integer,G: nat > code_integer,F: nat > nat,A2: set_nat] :
      ( ( Sup @ ( image_1215581382706833972nteger @ G @ ( image_nat_nat @ F @ A2 ) ) )
      = ( Sup @ ( image_1215581382706833972nteger @ ( comp_n3898172953802868194er_nat @ G @ F ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_5084_Sup_OSUP__image,axiom,
    ! [Sup: set_nat > nat,G: nat > nat,F: nat > nat,A2: set_nat] :
      ( ( Sup @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A2 ) ) )
      = ( Sup @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_5085_Sup_OSUP__image,axiom,
    ! [Sup: set_int > int,G: nat > int,F: int > nat,A2: set_int] :
      ( ( Sup @ ( image_nat_int @ G @ ( image_int_nat @ F @ A2 ) ) )
      = ( Sup @ ( image_int_int @ ( comp_nat_int_int @ G @ F ) @ A2 ) ) ) ).

% Sup.SUP_image
thf(fact_5086_Code__Numeral_Osub__code_I9_J,axiom,
    ! [M2: num,N2: num] :
      ( ( code_sub @ ( bit0 @ M2 ) @ ( bit1 @ N2 ) )
      = ( minus_8373710615458151222nteger @ ( code_dup @ ( code_sub @ M2 @ N2 ) ) @ one_one_Code_integer ) ) ).

% Code_Numeral.sub_code(9)
thf(fact_5087_Code__Numeral_Osub__code_I8_J,axiom,
    ! [M2: num,N2: num] :
      ( ( code_sub @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( code_dup @ ( code_sub @ M2 @ N2 ) ) @ one_one_Code_integer ) ) ).

% Code_Numeral.sub_code(8)
thf(fact_5088_less__eq__int__def,axiom,
    ( ord_less_eq_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X3: nat,Y3: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) ) ) ) ).

% less_eq_int_def
thf(fact_5089_less__int__def,axiom,
    ( ord_less_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X3: nat,Y3: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) ) ) ) ).

% less_int_def
thf(fact_5090_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: rat,B: rat,M2: rat,C: rat] :
      ( ( ( ( set_or633870826150836451st_rat @ A @ B )
          = bot_bot_set_rat )
       => ( ( image_rat_rat
            @ ^ [X3: rat] : ( minus_minus_rat @ ( divide_divide_rat @ X3 @ M2 ) @ C )
            @ ( set_or633870826150836451st_rat @ A @ B ) )
          = bot_bot_set_rat ) )
      & ( ( ( set_or633870826150836451st_rat @ A @ B )
         != bot_bot_set_rat )
       => ( ( ( ord_less_eq_rat @ zero_zero_rat @ M2 )
           => ( ( image_rat_rat
                @ ^ [X3: rat] : ( minus_minus_rat @ ( divide_divide_rat @ X3 @ M2 ) @ C )
                @ ( set_or633870826150836451st_rat @ A @ B ) )
              = ( set_or633870826150836451st_rat @ ( minus_minus_rat @ ( divide_divide_rat @ A @ M2 ) @ C ) @ ( minus_minus_rat @ ( divide_divide_rat @ B @ M2 ) @ C ) ) ) )
          & ( ~ ( ord_less_eq_rat @ zero_zero_rat @ M2 )
           => ( ( image_rat_rat
                @ ^ [X3: rat] : ( minus_minus_rat @ ( divide_divide_rat @ X3 @ M2 ) @ C )
                @ ( set_or633870826150836451st_rat @ A @ B ) )
              = ( set_or633870826150836451st_rat @ ( minus_minus_rat @ ( divide_divide_rat @ B @ M2 ) @ C ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ M2 ) @ C ) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div_diff
thf(fact_5091_image__affinity__atLeastAtMost__div,axiom,
    ! [A: rat,B: rat,M2: rat,C: rat] :
      ( ( ( ( set_or633870826150836451st_rat @ A @ B )
          = bot_bot_set_rat )
       => ( ( image_rat_rat
            @ ^ [X3: rat] : ( plus_plus_rat @ ( divide_divide_rat @ X3 @ M2 ) @ C )
            @ ( set_or633870826150836451st_rat @ A @ B ) )
          = bot_bot_set_rat ) )
      & ( ( ( set_or633870826150836451st_rat @ A @ B )
         != bot_bot_set_rat )
       => ( ( ( ord_less_eq_rat @ zero_zero_rat @ M2 )
           => ( ( image_rat_rat
                @ ^ [X3: rat] : ( plus_plus_rat @ ( divide_divide_rat @ X3 @ M2 ) @ C )
                @ ( set_or633870826150836451st_rat @ A @ B ) )
              = ( set_or633870826150836451st_rat @ ( plus_plus_rat @ ( divide_divide_rat @ A @ M2 ) @ C ) @ ( plus_plus_rat @ ( divide_divide_rat @ B @ M2 ) @ C ) ) ) )
          & ( ~ ( ord_less_eq_rat @ zero_zero_rat @ M2 )
           => ( ( image_rat_rat
                @ ^ [X3: rat] : ( plus_plus_rat @ ( divide_divide_rat @ X3 @ M2 ) @ C )
                @ ( set_or633870826150836451st_rat @ A @ B ) )
              = ( set_or633870826150836451st_rat @ ( plus_plus_rat @ ( divide_divide_rat @ B @ M2 ) @ C ) @ ( plus_plus_rat @ ( divide_divide_rat @ A @ M2 ) @ C ) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div
thf(fact_5092_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: rat,B: rat,M2: rat,C: rat] :
      ( ( ( ( set_or633870826150836451st_rat @ A @ B )
          = bot_bot_set_rat )
       => ( ( image_rat_rat
            @ ^ [X3: rat] : ( minus_minus_rat @ ( times_times_rat @ M2 @ X3 ) @ C )
            @ ( set_or633870826150836451st_rat @ A @ B ) )
          = bot_bot_set_rat ) )
      & ( ( ( set_or633870826150836451st_rat @ A @ B )
         != bot_bot_set_rat )
       => ( ( ( ord_less_eq_rat @ zero_zero_rat @ M2 )
           => ( ( image_rat_rat
                @ ^ [X3: rat] : ( minus_minus_rat @ ( times_times_rat @ M2 @ X3 ) @ C )
                @ ( set_or633870826150836451st_rat @ A @ B ) )
              = ( set_or633870826150836451st_rat @ ( minus_minus_rat @ ( times_times_rat @ M2 @ A ) @ C ) @ ( minus_minus_rat @ ( times_times_rat @ M2 @ B ) @ C ) ) ) )
          & ( ~ ( ord_less_eq_rat @ zero_zero_rat @ M2 )
           => ( ( image_rat_rat
                @ ^ [X3: rat] : ( minus_minus_rat @ ( times_times_rat @ M2 @ X3 ) @ C )
                @ ( set_or633870826150836451st_rat @ A @ B ) )
              = ( set_or633870826150836451st_rat @ ( minus_minus_rat @ ( times_times_rat @ M2 @ B ) @ C ) @ ( minus_minus_rat @ ( times_times_rat @ M2 @ A ) @ C ) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_diff
thf(fact_5093_id__apply,axiom,
    ( id_o
    = ( ^ [X3: $o] : X3 ) ) ).

% id_apply
thf(fact_5094_id__apply,axiom,
    ( id_nat
    = ( ^ [X3: nat] : X3 ) ) ).

% id_apply
thf(fact_5095_id__apply,axiom,
    ( id_nat_o
    = ( ^ [X3: nat > $o] : X3 ) ) ).

% id_apply
thf(fact_5096_id__apply,axiom,
    ( id_num
    = ( ^ [X3: num] : X3 ) ) ).

% id_apply
thf(fact_5097_id__apply,axiom,
    ( id_int
    = ( ^ [X3: int] : X3 ) ) ).

% id_apply
thf(fact_5098_bijective__Empty,axiom,
    biject3388518990857851981at_nat @ bot_bo228742789529271731at_nat ).

% bijective_Empty
thf(fact_5099_bijective__Empty,axiom,
    bijective_nat_nat @ bot_bo2099793752762293965at_nat ).

% bijective_Empty
thf(fact_5100_image__id,axiom,
    ( ( image_o_o @ id_o )
    = id_set_o ) ).

% image_id
thf(fact_5101_image__id,axiom,
    ( ( image_nat_o_nat_o @ id_nat_o )
    = id_set_nat_o ) ).

% image_id
thf(fact_5102_image__id,axiom,
    ( ( image_num_num @ id_num )
    = id_set_num ) ).

% image_id
thf(fact_5103_image__id,axiom,
    ( ( image_nat_nat @ id_nat )
    = id_set_nat ) ).

% image_id
thf(fact_5104_image__id,axiom,
    ( ( image_int_int @ id_int )
    = id_set_int ) ).

% image_id
thf(fact_5105_fun_Omap__id,axiom,
    ! [T: num > code_integer] :
      ( ( comp_C3531382070062128313er_num @ id_Code_integer @ T )
      = T ) ).

% fun.map_id
thf(fact_5106_id__comp,axiom,
    ! [G: num > code_integer] :
      ( ( comp_C3531382070062128313er_num @ id_Code_integer @ G )
      = G ) ).

% id_comp
thf(fact_5107_comp__id,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( comp_C1593894019821074884nteger @ F @ id_Code_integer )
      = F ) ).

% comp_id
thf(fact_5108_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_5109_apsnd__id,axiom,
    ( ( produc6499014454317279255nteger @ id_Code_integer )
    = id_Pro2309048762231606924nteger ) ).

% apsnd_id
thf(fact_5110_set__to__map__empty,axiom,
    ( ( set_to4447731662395764663at_nat @ bot_bo228742789529271731at_nat )
    = ( ^ [X3: set_Pr1261947904930325089at_nat] : none_s625347054029921090at_nat ) ) ).

% set_to_map_empty
thf(fact_5111_set__to__map__empty,axiom,
    ( ( set_to_map_nat_nat @ bot_bo2099793752762293965at_nat )
    = ( ^ [X3: nat] : none_nat ) ) ).

% set_to_map_empty
thf(fact_5112_atLeastatMost__empty__iff2,axiom,
    ! [A: $o,B: $o] :
      ( ( bot_bot_set_o
        = ( set_or8904488021354931149Most_o @ A @ B ) )
      = ( ~ ( ord_less_eq_o @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5113_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_set_nat
        = ( set_or4548717258645045905et_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5114_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat
        = ( set_or633870826150836451st_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5115_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num
        = ( set_or7049704709247886629st_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5116_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int
        = ( set_or1266510415728281911st_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5117_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat
        = ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5118_atLeastatMost__empty__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bot_bo3990330152332043303nteger
        = ( set_or189985376899183464nteger @ A @ B ) )
      = ( ~ ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5119_atLeastatMost__empty__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( ( set_or8904488021354931149Most_o @ A @ B )
        = bot_bot_set_o )
      = ( ~ ( ord_less_eq_o @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5120_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ A @ B )
        = bot_bot_set_set_nat )
      = ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5121_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5122_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5123_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5124_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5125_atLeastatMost__empty__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( set_or189985376899183464nteger @ A @ B )
        = bot_bo3990330152332043303nteger )
      = ( ~ ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5126_atLeastatMost__empty,axiom,
    ! [B: $o,A: $o] :
      ( ( ord_less_o @ B @ A )
     => ( ( set_or8904488021354931149Most_o @ A @ B )
        = bot_bot_set_o ) ) ).

% atLeastatMost_empty
thf(fact_5127_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
thf(fact_5128_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num ) ) ).

% atLeastatMost_empty
thf(fact_5129_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int ) ) ).

% atLeastatMost_empty
thf(fact_5130_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
thf(fact_5131_atLeastatMost__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( set_or189985376899183464nteger @ A @ B )
        = bot_bo3990330152332043303nteger ) ) ).

% atLeastatMost_empty
thf(fact_5132_atLeastLessThan__empty,axiom,
    ! [B: $o,A: $o] :
      ( ( ord_less_eq_o @ B @ A )
     => ( ( set_or7139685690850216873Than_o @ A @ B )
        = bot_bot_set_o ) ) ).

% atLeastLessThan_empty
thf(fact_5133_atLeastLessThan__empty,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( set_or3540276404033026485et_nat @ A @ B )
        = bot_bot_set_set_nat ) ) ).

% atLeastLessThan_empty
thf(fact_5134_atLeastLessThan__empty,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( set_or4029947393144176647an_rat @ A @ B )
        = bot_bot_set_rat ) ) ).

% atLeastLessThan_empty
thf(fact_5135_atLeastLessThan__empty,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( set_or1222409239386451017an_num @ A @ B )
        = bot_bot_set_num ) ) ).

% atLeastLessThan_empty
thf(fact_5136_atLeastLessThan__empty,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( set_or4665077453230672383an_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% atLeastLessThan_empty
thf(fact_5137_atLeastLessThan__empty,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( set_or8404916559141939852nteger @ A @ B )
        = bot_bo3990330152332043303nteger ) ) ).

% atLeastLessThan_empty
thf(fact_5138_atLeastLessThan__empty,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( set_or4662586982721622107an_int @ A @ B )
        = bot_bot_set_int ) ) ).

% atLeastLessThan_empty
thf(fact_5139_atLeastLessThan__empty__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( ( set_or7139685690850216873Than_o @ A @ B )
        = bot_bot_set_o )
      = ( ~ ( ord_less_o @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5140_atLeastLessThan__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( set_or4029947393144176647an_rat @ A @ B )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_rat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5141_atLeastLessThan__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( ( set_or1222409239386451017an_num @ A @ B )
        = bot_bot_set_num )
      = ( ~ ( ord_less_num @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5142_atLeastLessThan__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_nat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5143_atLeastLessThan__empty__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A @ B )
        = bot_bo3990330152332043303nteger )
      = ( ~ ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5144_atLeastLessThan__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( set_or4662586982721622107an_int @ A @ B )
        = bot_bot_set_int )
      = ( ~ ( ord_less_int @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_5145_atLeastLessThan__empty__iff2,axiom,
    ! [A: $o,B: $o] :
      ( ( bot_bot_set_o
        = ( set_or7139685690850216873Than_o @ A @ B ) )
      = ( ~ ( ord_less_o @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5146_atLeastLessThan__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat
        = ( set_or4029947393144176647an_rat @ A @ B ) )
      = ( ~ ( ord_less_rat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5147_atLeastLessThan__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num
        = ( set_or1222409239386451017an_num @ A @ B ) )
      = ( ~ ( ord_less_num @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5148_atLeastLessThan__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat
        = ( set_or4665077453230672383an_nat @ A @ B ) )
      = ( ~ ( ord_less_nat @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5149_atLeastLessThan__empty__iff2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bot_bo3990330152332043303nteger
        = ( set_or8404916559141939852nteger @ A @ B ) )
      = ( ~ ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5150_atLeastLessThan__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int
        = ( set_or4662586982721622107an_int @ A @ B ) )
      = ( ~ ( ord_less_int @ A @ B ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_5151_prod_Oempty,axiom,
    ! [G: $o > code_integer] :
      ( ( groups7694694392188491536nteger @ G @ bot_bot_set_o )
      = one_one_Code_integer ) ).

% prod.empty
thf(fact_5152_prod_Oempty,axiom,
    ! [G: $o > assn] :
      ( ( groups5301882518646026715o_assn @ G @ bot_bot_set_o )
      = one_one_assn ) ).

% prod.empty
thf(fact_5153_prod_Oempty,axiom,
    ! [G: $o > rat] :
      ( ( groups2869687844427037835_o_rat @ G @ bot_bot_set_o )
      = one_one_rat ) ).

% prod.empty
thf(fact_5154_prod_Oempty,axiom,
    ! [G: $o > nat] :
      ( ( groups3504817904513533571_o_nat @ G @ bot_bot_set_o )
      = one_one_nat ) ).

% prod.empty
thf(fact_5155_prod_Oempty,axiom,
    ! [G: $o > int] :
      ( ( groups3502327434004483295_o_int @ G @ bot_bot_set_o )
      = one_one_int ) ).

% prod.empty
thf(fact_5156_prod_Oempty,axiom,
    ! [G: nat > code_integer] :
      ( ( groups3455450783089532116nteger @ G @ bot_bot_set_nat )
      = one_one_Code_integer ) ).

% prod.empty
thf(fact_5157_prod_Oempty,axiom,
    ! [G: nat > assn] :
      ( ( groups6906906614972039071t_assn @ G @ bot_bot_set_nat )
      = one_one_assn ) ).

% prod.empty
thf(fact_5158_prod_Oempty,axiom,
    ! [G: nat > rat] :
      ( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
      = one_one_rat ) ).

% prod.empty
thf(fact_5159_prod_Oempty,axiom,
    ! [G: int > code_integer] :
      ( ( groups3827104343326376752nteger @ G @ bot_bot_set_int )
      = one_one_Code_integer ) ).

% prod.empty
thf(fact_5160_prod_Oempty,axiom,
    ! [G: int > assn] :
      ( ( groups7882442080178216443t_assn @ G @ bot_bot_set_int )
      = one_one_assn ) ).

% prod.empty
thf(fact_5161_group__add__class_Ominus__comp__minus,axiom,
    ( ( comp_int_int_int @ uminus_uminus_int @ uminus_uminus_int )
    = id_int ) ).

% group_add_class.minus_comp_minus
thf(fact_5162_group__add__class_Ominus__comp__minus,axiom,
    ( ( comp_C7449957260575251196nteger @ uminus1351360451143612070nteger @ uminus1351360451143612070nteger )
    = id_Code_integer ) ).

% group_add_class.minus_comp_minus
thf(fact_5163_group__add__class_Ominus__comp__minus,axiom,
    ( ( comp_rat_rat_rat @ uminus_uminus_rat @ uminus_uminus_rat )
    = id_rat ) ).

% group_add_class.minus_comp_minus
thf(fact_5164_boolean__algebra__class_Ominus__comp__minus,axiom,
    ( ( comp_o_o_o @ uminus_uminus_o @ uminus_uminus_o )
    = id_o ) ).

% boolean_algebra_class.minus_comp_minus
thf(fact_5165_boolean__algebra__class_Ominus__comp__minus,axiom,
    ( ( comp_n5656820349209612572_nat_o @ uminus_uminus_nat_o @ uminus_uminus_nat_o )
    = id_nat_o ) ).

% boolean_algebra_class.minus_comp_minus
thf(fact_5166_boolean__algebra__class_Ominus__comp__minus,axiom,
    ( ( comp_P7645380973975430442t_unit @ uminus2952777764628376836t_unit @ uminus2952777764628376836t_unit )
    = id_Product_unit ) ).

% boolean_algebra_class.minus_comp_minus
thf(fact_5167_rel__of__empty,axiom,
    ! [P: produc3843707927480180839at_nat > $o] :
      ( ( rel_of6563097375766832824at_nat
        @ ^ [X3: set_Pr1261947904930325089at_nat] : none_s625347054029921090at_nat
        @ P )
      = bot_bo228742789529271731at_nat ) ).

% rel_of_empty
thf(fact_5168_rel__of__empty,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( rel_of_nat_nat
        @ ^ [X3: nat] : none_nat
        @ P )
      = bot_bo2099793752762293965at_nat ) ).

% rel_of_empty
thf(fact_5169_comp__the__Some,axiom,
    ( ( comp_option_o_o_o @ the_o @ some_o )
    = id_o ) ).

% comp_the_Some
thf(fact_5170_comp__the__Some,axiom,
    ( ( comp_o4663963155070737957at_nat @ the_nat @ some_nat )
    = id_nat ) ).

% comp_the_Some
thf(fact_5171_comp__the__Some,axiom,
    ( ( comp_o8860755393669798242_nat_o @ the_nat_o @ some_nat_o )
    = id_nat_o ) ).

% comp_the_Some
thf(fact_5172_comp__the__Some,axiom,
    ( ( comp_o2446264010603541305nt_int @ the_int2 @ some_int )
    = id_int ) ).

% comp_the_Some
thf(fact_5173_comp__the__Some,axiom,
    ( ( comp_o6342398210560591875um_num @ the_num @ some_num )
    = id_num ) ).

% comp_the_Some
thf(fact_5174_comp__the__Some,axiom,
    ( ( comp_o279924169626513613it_nat @ the_Pr7776125533777466484it_nat @ some_P1914260805536162275it_nat )
    = id_Pro1159712480615372887it_nat ) ).

% comp_the_Some
thf(fact_5175_fun_Omap__id0,axiom,
    ( ( comp_o_o_nat @ id_o )
    = id_nat_o ) ).

% fun.map_id0
thf(fact_5176_fun_Omap__id0,axiom,
    ( ( comp_C3531382070062128313er_num @ id_Code_integer )
    = id_num_Code_integer ) ).

% fun.map_id0
thf(fact_5177_Iio__eq__empty__iff,axiom,
    ! [N2: $o] :
      ( ( ( set_ord_lessThan_o @ N2 )
        = bot_bot_set_o )
      = ( N2 = bot_bot_o ) ) ).

% Iio_eq_empty_iff
thf(fact_5178_Iio__eq__empty__iff,axiom,
    ! [N2: nat] :
      ( ( ( set_ord_lessThan_nat @ N2 )
        = bot_bot_set_nat )
      = ( N2 = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
thf(fact_5179_option_Omap__id0,axiom,
    ( ( map_option_o_o @ id_o )
    = id_option_o ) ).

% option.map_id0
thf(fact_5180_option_Omap__id0,axiom,
    ( ( map_option_nat_nat @ id_nat )
    = id_option_nat ) ).

% option.map_id0
thf(fact_5181_option_Omap__id0,axiom,
    ( ( map_op6424164270877473355_nat_o @ id_nat_o )
    = id_option_nat_o ) ).

% option.map_id0
thf(fact_5182_option_Omap__id0,axiom,
    ( ( map_option_int_int @ id_int )
    = id_option_int ) ).

% option.map_id0
thf(fact_5183_option_Omap__id0,axiom,
    ( ( map_option_num_num @ id_num )
    = id_option_num ) ).

% option.map_id0
thf(fact_5184_set__notEmptyE,axiom,
    ! [S2: set_Pr4329608150637261639at_nat] :
      ( ( S2 != bot_bo228742789529271731at_nat )
     => ~ ! [X4: produc3843707927480180839at_nat] :
            ~ ( member8757157785044589968at_nat @ X4 @ S2 ) ) ).

% set_notEmptyE
thf(fact_5185_set__notEmptyE,axiom,
    ! [S2: set_Pr1261947904930325089at_nat] :
      ( ( S2 != bot_bo2099793752762293965at_nat )
     => ~ ! [X4: product_prod_nat_nat] :
            ~ ( member8440522571783428010at_nat @ X4 @ S2 ) ) ).

% set_notEmptyE
thf(fact_5186_set__notEmptyE,axiom,
    ! [S2: set_o] :
      ( ( S2 != bot_bot_set_o )
     => ~ ! [X4: $o] :
            ~ ( member_o @ X4 @ S2 ) ) ).

% set_notEmptyE
thf(fact_5187_set__notEmptyE,axiom,
    ! [S2: set_nat] :
      ( ( S2 != bot_bot_set_nat )
     => ~ ! [X4: nat] :
            ~ ( member_nat @ X4 @ S2 ) ) ).

% set_notEmptyE
thf(fact_5188_set__notEmptyE,axiom,
    ! [S2: set_int] :
      ( ( S2 != bot_bot_set_int )
     => ~ ! [X4: int] :
            ~ ( member_int @ X4 @ S2 ) ) ).

% set_notEmptyE
thf(fact_5189_memb__imp__not__empty,axiom,
    ! [X: produc3843707927480180839at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ S2 )
     => ( S2 != bot_bo228742789529271731at_nat ) ) ).

% memb_imp_not_empty
thf(fact_5190_memb__imp__not__empty,axiom,
    ! [X: product_prod_nat_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ S2 )
     => ( S2 != bot_bo2099793752762293965at_nat ) ) ).

% memb_imp_not_empty
thf(fact_5191_memb__imp__not__empty,axiom,
    ! [X: $o,S2: set_o] :
      ( ( member_o @ X @ S2 )
     => ( S2 != bot_bot_set_o ) ) ).

% memb_imp_not_empty
thf(fact_5192_memb__imp__not__empty,axiom,
    ! [X: nat,S2: set_nat] :
      ( ( member_nat @ X @ S2 )
     => ( S2 != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
thf(fact_5193_memb__imp__not__empty,axiom,
    ! [X: int,S2: set_int] :
      ( ( member_int @ X @ S2 )
     => ( S2 != bot_bot_set_int ) ) ).

% memb_imp_not_empty
thf(fact_5194_id__def,axiom,
    ( id_o
    = ( ^ [X3: $o] : X3 ) ) ).

% id_def
thf(fact_5195_id__def,axiom,
    ( id_nat
    = ( ^ [X3: nat] : X3 ) ) ).

% id_def
thf(fact_5196_id__def,axiom,
    ( id_nat_o
    = ( ^ [X3: nat > $o] : X3 ) ) ).

% id_def
thf(fact_5197_id__def,axiom,
    ( id_num
    = ( ^ [X3: num] : X3 ) ) ).

% id_def
thf(fact_5198_id__def,axiom,
    ( id_int
    = ( ^ [X3: int] : X3 ) ) ).

% id_def
thf(fact_5199_eq__id__iff,axiom,
    ! [F: $o > $o] :
      ( ( ! [X3: $o] :
            ( ( F @ X3 )
            = X3 ) )
      = ( F = id_o ) ) ).

% eq_id_iff
thf(fact_5200_eq__id__iff,axiom,
    ! [F: nat > nat] :
      ( ( ! [X3: nat] :
            ( ( F @ X3 )
            = X3 ) )
      = ( F = id_nat ) ) ).

% eq_id_iff
thf(fact_5201_eq__id__iff,axiom,
    ! [F: ( nat > $o ) > nat > $o] :
      ( ( ! [X3: nat > $o] :
            ( ( F @ X3 )
            = X3 ) )
      = ( F = id_nat_o ) ) ).

% eq_id_iff
thf(fact_5202_eq__id__iff,axiom,
    ! [F: num > num] :
      ( ( ! [X3: num] :
            ( ( F @ X3 )
            = X3 ) )
      = ( F = id_num ) ) ).

% eq_id_iff
thf(fact_5203_eq__id__iff,axiom,
    ! [F: int > int] :
      ( ( ! [X3: int] :
            ( ( F @ X3 )
            = X3 ) )
      = ( F = id_int ) ) ).

% eq_id_iff
thf(fact_5204_comp__eq__id__dest,axiom,
    ! [A: code_integer > code_integer,B: num > code_integer,C: num > code_integer,V: num] :
      ( ( ( comp_C3531382070062128313er_num @ A @ B )
        = ( comp_C3531382070062128313er_num @ id_Code_integer @ C ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_5205_comp__eq__id__dest,axiom,
    ! [A: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,B: code_integer > code_integer,C: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,V: code_integer] :
      ( ( ( comp_C1593894019821074884nteger @ A @ B )
        = ( comp_P3928908779438215820nteger @ id_Pro496765588584934260nteger @ C ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_5206_comp__eq__id__dest,axiom,
    ! [A: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,B: code_integer > code_integer > code_integer,C: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,V: code_integer] :
      ( ( ( comp_C8797469213163452608nteger @ A @ B )
        = ( comp_P3928908779438215820nteger @ id_Pro496765588584934260nteger @ C ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_5207_comp__eq__id__dest,axiom,
    ! [A: nat > num,B: int > nat,C: int > num,V: int] :
      ( ( ( comp_nat_num_int @ A @ B )
        = ( comp_num_num_int @ id_num @ C ) )
     => ( ( A @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_5208_option_Omap__id,axiom,
    ! [T: option_o] :
      ( ( map_option_o_o @ id_o @ T )
      = T ) ).

% option.map_id
thf(fact_5209_option_Omap__id,axiom,
    ! [T: option_nat] :
      ( ( map_option_nat_nat @ id_nat @ T )
      = T ) ).

% option.map_id
thf(fact_5210_option_Omap__id,axiom,
    ! [T: option_nat_o] :
      ( ( map_op6424164270877473355_nat_o @ id_nat_o @ T )
      = T ) ).

% option.map_id
thf(fact_5211_option_Omap__id,axiom,
    ! [T: option_int] :
      ( ( map_option_int_int @ id_int @ T )
      = T ) ).

% option.map_id
thf(fact_5212_option_Omap__id,axiom,
    ! [T: option_num] :
      ( ( map_option_num_num @ id_num @ T )
      = T ) ).

% option.map_id
thf(fact_5213_not__empty__eq__Iic__eq__empty,axiom,
    ! [H3: $o] :
      ( bot_bot_set_o
     != ( set_ord_atMost_o @ H3 ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_5214_not__empty__eq__Iic__eq__empty,axiom,
    ! [H3: int] :
      ( bot_bot_set_int
     != ( set_ord_atMost_int @ H3 ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_5215_not__empty__eq__Iic__eq__empty,axiom,
    ! [H3: nat] :
      ( bot_bot_set_nat
     != ( set_ord_atMost_nat @ H3 ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_5216_lessThan__non__empty,axiom,
    ! [X: int] :
      ( ( set_ord_lessThan_int @ X )
     != bot_bot_set_int ) ).

% lessThan_non_empty
thf(fact_5217_subset__minus__empty,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ B5 )
     => ( ( minus_3314409938677909166at_nat @ A2 @ B5 )
        = bot_bo228742789529271731at_nat ) ) ).

% subset_minus_empty
thf(fact_5218_subset__minus__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B5 )
     => ( ( minus_1356011639430497352at_nat @ A2 @ B5 )
        = bot_bo2099793752762293965at_nat ) ) ).

% subset_minus_empty
thf(fact_5219_subset__minus__empty,axiom,
    ! [A2: set_o,B5: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ B5 )
     => ( ( minus_minus_set_o @ A2 @ B5 )
        = bot_bot_set_o ) ) ).

% subset_minus_empty
thf(fact_5220_subset__minus__empty,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B5 )
     => ( ( minus_minus_set_int @ A2 @ B5 )
        = bot_bot_set_int ) ) ).

% subset_minus_empty
thf(fact_5221_subset__minus__empty,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B5 )
     => ( ( minus_minus_set_nat @ A2 @ B5 )
        = bot_bot_set_nat ) ) ).

% subset_minus_empty
thf(fact_5222_atLeastLessThan0,axiom,
    ! [M2: nat] :
      ( ( set_or4665077453230672383an_nat @ M2 @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_5223_lessThan__empty__iff,axiom,
    ! [N2: nat] :
      ( ( ( set_ord_lessThan_nat @ N2 )
        = bot_bot_set_nat )
      = ( N2 = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_5224_set__to__map__empty__iff_I1_J,axiom,
    ! [S2: set_Pr4329608150637261639at_nat] :
      ( ( ( set_to4447731662395764663at_nat @ S2 )
        = ( ^ [X3: set_Pr1261947904930325089at_nat] : none_s625347054029921090at_nat ) )
      = ( S2 = bot_bo228742789529271731at_nat ) ) ).

% set_to_map_empty_iff(1)
thf(fact_5225_set__to__map__empty__iff_I1_J,axiom,
    ! [S2: set_Pr1261947904930325089at_nat] :
      ( ( ( set_to_map_nat_nat @ S2 )
        = ( ^ [X3: nat] : none_nat ) )
      = ( S2 = bot_bo2099793752762293965at_nat ) ) ).

% set_to_map_empty_iff(1)
thf(fact_5226_Code__Numeral_Osub__code_I1_J,axiom,
    ( ( code_sub @ one @ one )
    = zero_z3403309356797280102nteger ) ).

% Code_Numeral.sub_code(1)
thf(fact_5227_set__to__map__empty__iff_I2_J,axiom,
    ! [S2: set_Pr4329608150637261639at_nat] :
      ( ( ( ^ [X3: set_Pr1261947904930325089at_nat] : none_s625347054029921090at_nat )
        = ( set_to4447731662395764663at_nat @ S2 ) )
      = ( S2 = bot_bo228742789529271731at_nat ) ) ).

% set_to_map_empty_iff(2)
thf(fact_5228_set__to__map__empty__iff_I2_J,axiom,
    ! [S2: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X3: nat] : none_nat )
        = ( set_to_map_nat_nat @ S2 ) )
      = ( S2 = bot_bo2099793752762293965at_nat ) ) ).

% set_to_map_empty_iff(2)
thf(fact_5229_less__integer__def,axiom,
    ( ord_le6747313008572928689nteger
    = ( map_fu2496120808607739376eger_o @ code_int_of_integer @ ( map_fu6957801986076833569nt_o_o @ code_int_of_integer @ id_o ) @ ord_less_int ) ) ).

% less_integer_def
thf(fact_5230_less__eq__integer__def,axiom,
    ( ord_le3102999989581377725nteger
    = ( map_fu2496120808607739376eger_o @ code_int_of_integer @ ( map_fu6957801986076833569nt_o_o @ code_int_of_integer @ id_o ) @ ord_less_eq_int ) ) ).

% less_eq_integer_def
thf(fact_5231_Code__Numeral_Osub__code_I6_J,axiom,
    ! [M2: num,N2: num] :
      ( ( code_sub @ ( bit0 @ M2 ) @ ( bit0 @ N2 ) )
      = ( code_dup @ ( code_sub @ M2 @ N2 ) ) ) ).

% Code_Numeral.sub_code(6)
thf(fact_5232_Code__Numeral_Osub__code_I7_J,axiom,
    ! [M2: num,N2: num] :
      ( ( code_sub @ ( bit1 @ M2 ) @ ( bit1 @ N2 ) )
      = ( code_dup @ ( code_sub @ M2 @ N2 ) ) ) ).

% Code_Numeral.sub_code(7)
thf(fact_5233_sub_Orep__eq,axiom,
    ! [X: num,Xa: num] :
      ( ( code_int_of_integer @ ( code_sub @ X @ Xa ) )
      = ( minus_minus_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Xa ) ) ) ).

% sub.rep_eq
thf(fact_5234_sub_Oabs__eq,axiom,
    ( code_sub
    = ( ^ [Xa4: num,X3: num] : ( code_integer_of_int @ ( minus_minus_int @ ( numeral_numeral_int @ Xa4 ) @ ( numeral_numeral_int @ X3 ) ) ) ) ) ).

% sub.abs_eq
thf(fact_5235_image__mult__atLeastAtMost__if,axiom,
    ! [C: rat,X: rat,Y: rat] :
      ( ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ( image_rat_rat @ ( times_times_rat @ C ) @ ( set_or633870826150836451st_rat @ X @ Y ) )
          = ( set_or633870826150836451st_rat @ ( times_times_rat @ C @ X ) @ ( times_times_rat @ C @ Y ) ) ) )
      & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
       => ( ( ( ord_less_eq_rat @ X @ Y )
           => ( ( image_rat_rat @ ( times_times_rat @ C ) @ ( set_or633870826150836451st_rat @ X @ Y ) )
              = ( set_or633870826150836451st_rat @ ( times_times_rat @ C @ Y ) @ ( times_times_rat @ C @ X ) ) ) )
          & ( ~ ( ord_less_eq_rat @ X @ Y )
           => ( ( image_rat_rat @ ( times_times_rat @ C ) @ ( set_or633870826150836451st_rat @ X @ Y ) )
              = bot_bot_set_rat ) ) ) ) ) ).

% image_mult_atLeastAtMost_if
thf(fact_5236_image__mult__atLeastAtMost__if_H,axiom,
    ! [X: rat,Y: rat,C: rat] :
      ( ( ( ord_less_eq_rat @ X @ Y )
       => ( ( ( ord_less_rat @ zero_zero_rat @ C )
           => ( ( image_rat_rat
                @ ^ [X3: rat] : ( times_times_rat @ X3 @ C )
                @ ( set_or633870826150836451st_rat @ X @ Y ) )
              = ( set_or633870826150836451st_rat @ ( times_times_rat @ X @ C ) @ ( times_times_rat @ Y @ C ) ) ) )
          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
           => ( ( image_rat_rat
                @ ^ [X3: rat] : ( times_times_rat @ X3 @ C )
                @ ( set_or633870826150836451st_rat @ X @ Y ) )
              = ( set_or633870826150836451st_rat @ ( times_times_rat @ Y @ C ) @ ( times_times_rat @ X @ C ) ) ) ) ) )
      & ( ~ ( ord_less_eq_rat @ X @ Y )
       => ( ( image_rat_rat
            @ ^ [X3: rat] : ( times_times_rat @ X3 @ C )
            @ ( set_or633870826150836451st_rat @ X @ Y ) )
          = bot_bot_set_rat ) ) ) ).

% image_mult_atLeastAtMost_if'
thf(fact_5237_image__affinity__atLeastAtMost,axiom,
    ! [A: rat,B: rat,M2: rat,C: rat] :
      ( ( ( ( set_or633870826150836451st_rat @ A @ B )
          = bot_bot_set_rat )
       => ( ( image_rat_rat
            @ ^ [X3: rat] : ( plus_plus_rat @ ( times_times_rat @ M2 @ X3 ) @ C )
            @ ( set_or633870826150836451st_rat @ A @ B ) )
          = bot_bot_set_rat ) )
      & ( ( ( set_or633870826150836451st_rat @ A @ B )
         != bot_bot_set_rat )
       => ( ( ( ord_less_eq_rat @ zero_zero_rat @ M2 )
           => ( ( image_rat_rat
                @ ^ [X3: rat] : ( plus_plus_rat @ ( times_times_rat @ M2 @ X3 ) @ C )
                @ ( set_or633870826150836451st_rat @ A @ B ) )
              = ( set_or633870826150836451st_rat @ ( plus_plus_rat @ ( times_times_rat @ M2 @ A ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ M2 @ B ) @ C ) ) ) )
          & ( ~ ( ord_less_eq_rat @ zero_zero_rat @ M2 )
           => ( ( image_rat_rat
                @ ^ [X3: rat] : ( plus_plus_rat @ ( times_times_rat @ M2 @ X3 ) @ C )
                @ ( set_or633870826150836451st_rat @ A @ B ) )
              = ( set_or633870826150836451st_rat @ ( plus_plus_rat @ ( times_times_rat @ M2 @ B ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ M2 @ A ) @ C ) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost
thf(fact_5238_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K4: int,L3: int] :
          ( if_int
          @ ( ( member_int @ K4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
          @ ( uminus_uminus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K4 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K4 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L3 ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_5239_and__int_Oelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
        = Y )
     => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
        & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_5240_and__int_Opelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) )
       => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
              & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).

% and_int.pelims
thf(fact_5241_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( ( accp_P1096762738010456898nt_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_se725231765392027082nd_int @ K @ L )
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ 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_se725231765392027082nd_int @ K @ L )
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_5242_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [K5: int,L5: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K5 @ L5 ) )
           => ( ( ~ ( ( member_int @ K5 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ L5 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P @ ( divide_divide_int @ K5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K5 @ L5 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% and_int.pinduct
thf(fact_5243_mask__mod__exp,axiom,
    ! [N2: nat,M2: nat] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ord_min_nat @ M2 @ N2 ) ) @ one_one_int ) ) ).

% mask_mod_exp
thf(fact_5244_mask__mod__exp,axiom,
    ! [N2: nat,M2: nat] :
      ( ( modulo_modulo_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
      = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( ord_min_nat @ M2 @ N2 ) ) @ one_one_nat ) ) ).

% mask_mod_exp
thf(fact_5245_mask__mod__exp,axiom,
    ! [N2: nat,M2: nat] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) )
      = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ord_min_nat @ M2 @ N2 ) ) @ one_one_Code_integer ) ) ).

% mask_mod_exp
thf(fact_5246_mask__mod__exp,axiom,
    ! [N2: nat,M2: nat] :
      ( ( modulo8411746178871703098atural @ ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_natural ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) )
      = ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( ord_min_nat @ M2 @ N2 ) ) @ one_one_Code_natural ) ) ).

% mask_mod_exp
thf(fact_5247_pair__leqI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_5248_min__eq__arg_I1_J,axiom,
    ! [M2: code_integer,N2: code_integer] :
      ( ( ( ord_min_Code_integer @ M2 @ N2 )
        = M2 )
      = ( ord_le3102999989581377725nteger @ M2 @ N2 ) ) ).

% min_eq_arg(1)
thf(fact_5249_min__eq__arg_I1_J,axiom,
    ! [M2: code_natural,N2: code_natural] :
      ( ( ( ord_min_Code_natural @ M2 @ N2 )
        = M2 )
      = ( ord_le1926595141338095240atural @ M2 @ N2 ) ) ).

% min_eq_arg(1)
thf(fact_5250_min__eq__arg_I1_J,axiom,
    ! [M2: rat,N2: rat] :
      ( ( ( ord_min_rat @ M2 @ N2 )
        = M2 )
      = ( ord_less_eq_rat @ M2 @ N2 ) ) ).

% min_eq_arg(1)
thf(fact_5251_min__eq__arg_I1_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_min_num @ M2 @ N2 )
        = M2 )
      = ( ord_less_eq_num @ M2 @ N2 ) ) ).

% min_eq_arg(1)
thf(fact_5252_min__eq__arg_I1_J,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( ord_min_nat @ M2 @ N2 )
        = M2 )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% min_eq_arg(1)
thf(fact_5253_min__eq__arg_I1_J,axiom,
    ! [M2: int,N2: int] :
      ( ( ( ord_min_int @ M2 @ N2 )
        = M2 )
      = ( ord_less_eq_int @ M2 @ N2 ) ) ).

% min_eq_arg(1)
thf(fact_5254_min__eq__arg_I2_J,axiom,
    ! [M2: code_integer,N2: code_integer] :
      ( ( ( ord_min_Code_integer @ M2 @ N2 )
        = N2 )
      = ( ord_le3102999989581377725nteger @ N2 @ M2 ) ) ).

% min_eq_arg(2)
thf(fact_5255_min__eq__arg_I2_J,axiom,
    ! [M2: code_natural,N2: code_natural] :
      ( ( ( ord_min_Code_natural @ M2 @ N2 )
        = N2 )
      = ( ord_le1926595141338095240atural @ N2 @ M2 ) ) ).

% min_eq_arg(2)
thf(fact_5256_min__eq__arg_I2_J,axiom,
    ! [M2: rat,N2: rat] :
      ( ( ( ord_min_rat @ M2 @ N2 )
        = N2 )
      = ( ord_less_eq_rat @ N2 @ M2 ) ) ).

% min_eq_arg(2)
thf(fact_5257_min__eq__arg_I2_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_min_num @ M2 @ N2 )
        = N2 )
      = ( ord_less_eq_num @ N2 @ M2 ) ) ).

% min_eq_arg(2)
thf(fact_5258_min__eq__arg_I2_J,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( ord_min_nat @ M2 @ N2 )
        = N2 )
      = ( ord_less_eq_nat @ N2 @ M2 ) ) ).

% min_eq_arg(2)
thf(fact_5259_min__eq__arg_I2_J,axiom,
    ! [M2: int,N2: int] :
      ( ( ( ord_min_int @ M2 @ N2 )
        = N2 )
      = ( ord_less_eq_int @ N2 @ M2 ) ) ).

% min_eq_arg(2)
thf(fact_5260_min__arg__le_I1_J,axiom,
    ! [N2: code_integer,M2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ N2 @ ( ord_min_Code_integer @ M2 @ N2 ) )
      = ( ( ord_min_Code_integer @ M2 @ N2 )
        = N2 ) ) ).

% min_arg_le(1)
thf(fact_5261_min__arg__le_I1_J,axiom,
    ! [N2: code_natural,M2: code_natural] :
      ( ( ord_le1926595141338095240atural @ N2 @ ( ord_min_Code_natural @ M2 @ N2 ) )
      = ( ( ord_min_Code_natural @ M2 @ N2 )
        = N2 ) ) ).

% min_arg_le(1)
thf(fact_5262_min__arg__le_I1_J,axiom,
    ! [N2: set_nat,M2: set_nat] :
      ( ( ord_less_eq_set_nat @ N2 @ ( ord_min_set_nat @ M2 @ N2 ) )
      = ( ( ord_min_set_nat @ M2 @ N2 )
        = N2 ) ) ).

% min_arg_le(1)
thf(fact_5263_min__arg__le_I1_J,axiom,
    ! [N2: rat,M2: rat] :
      ( ( ord_less_eq_rat @ N2 @ ( ord_min_rat @ M2 @ N2 ) )
      = ( ( ord_min_rat @ M2 @ N2 )
        = N2 ) ) ).

% min_arg_le(1)
thf(fact_5264_min__arg__le_I1_J,axiom,
    ! [N2: num,M2: num] :
      ( ( ord_less_eq_num @ N2 @ ( ord_min_num @ M2 @ N2 ) )
      = ( ( ord_min_num @ M2 @ N2 )
        = N2 ) ) ).

% min_arg_le(1)
thf(fact_5265_min__arg__le_I1_J,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N2 @ ( ord_min_nat @ M2 @ N2 ) )
      = ( ( ord_min_nat @ M2 @ N2 )
        = N2 ) ) ).

% min_arg_le(1)
thf(fact_5266_min__arg__le_I1_J,axiom,
    ! [N2: int,M2: int] :
      ( ( ord_less_eq_int @ N2 @ ( ord_min_int @ M2 @ N2 ) )
      = ( ( ord_min_int @ M2 @ N2 )
        = N2 ) ) ).

% min_arg_le(1)
thf(fact_5267_min__arg__le_I2_J,axiom,
    ! [M2: code_integer,N2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ M2 @ ( ord_min_Code_integer @ M2 @ N2 ) )
      = ( ( ord_min_Code_integer @ M2 @ N2 )
        = M2 ) ) ).

% min_arg_le(2)
thf(fact_5268_min__arg__le_I2_J,axiom,
    ! [M2: code_natural,N2: code_natural] :
      ( ( ord_le1926595141338095240atural @ M2 @ ( ord_min_Code_natural @ M2 @ N2 ) )
      = ( ( ord_min_Code_natural @ M2 @ N2 )
        = M2 ) ) ).

% min_arg_le(2)
thf(fact_5269_min__arg__le_I2_J,axiom,
    ! [M2: set_nat,N2: set_nat] :
      ( ( ord_less_eq_set_nat @ M2 @ ( ord_min_set_nat @ M2 @ N2 ) )
      = ( ( ord_min_set_nat @ M2 @ N2 )
        = M2 ) ) ).

% min_arg_le(2)
thf(fact_5270_min__arg__le_I2_J,axiom,
    ! [M2: rat,N2: rat] :
      ( ( ord_less_eq_rat @ M2 @ ( ord_min_rat @ M2 @ N2 ) )
      = ( ( ord_min_rat @ M2 @ N2 )
        = M2 ) ) ).

% min_arg_le(2)
thf(fact_5271_min__arg__le_I2_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ord_less_eq_num @ M2 @ ( ord_min_num @ M2 @ N2 ) )
      = ( ( ord_min_num @ M2 @ N2 )
        = M2 ) ) ).

% min_arg_le(2)
thf(fact_5272_min__arg__le_I2_J,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( ord_min_nat @ M2 @ N2 ) )
      = ( ( ord_min_nat @ M2 @ N2 )
        = M2 ) ) ).

% min_arg_le(2)
thf(fact_5273_min__arg__le_I2_J,axiom,
    ! [M2: int,N2: int] :
      ( ( ord_less_eq_int @ M2 @ ( ord_min_int @ M2 @ N2 ) )
      = ( ( ord_min_int @ M2 @ N2 )
        = M2 ) ) ).

% min_arg_le(2)
thf(fact_5274_min__arg__not__ge_I1_J,axiom,
    ! [M2: code_natural,N2: code_natural] :
      ( ( ~ ( ord_le5570908160329646204atural @ ( ord_min_Code_natural @ M2 @ N2 ) @ M2 ) )
      = ( ( ord_min_Code_natural @ M2 @ N2 )
        = M2 ) ) ).

% min_arg_not_ge(1)
thf(fact_5275_min__arg__not__ge_I1_J,axiom,
    ! [M2: code_integer,N2: code_integer] :
      ( ( ~ ( ord_le6747313008572928689nteger @ ( ord_min_Code_integer @ M2 @ N2 ) @ M2 ) )
      = ( ( ord_min_Code_integer @ M2 @ N2 )
        = M2 ) ) ).

% min_arg_not_ge(1)
thf(fact_5276_min__arg__not__ge_I1_J,axiom,
    ! [M2: rat,N2: rat] :
      ( ( ~ ( ord_less_rat @ ( ord_min_rat @ M2 @ N2 ) @ M2 ) )
      = ( ( ord_min_rat @ M2 @ N2 )
        = M2 ) ) ).

% min_arg_not_ge(1)
thf(fact_5277_min__arg__not__ge_I1_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ~ ( ord_less_num @ ( ord_min_num @ M2 @ N2 ) @ M2 ) )
      = ( ( ord_min_num @ M2 @ N2 )
        = M2 ) ) ).

% min_arg_not_ge(1)
thf(fact_5278_min__arg__not__ge_I1_J,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ~ ( ord_less_nat @ ( ord_min_nat @ M2 @ N2 ) @ M2 ) )
      = ( ( ord_min_nat @ M2 @ N2 )
        = M2 ) ) ).

% min_arg_not_ge(1)
thf(fact_5279_min__arg__not__ge_I1_J,axiom,
    ! [M2: int,N2: int] :
      ( ( ~ ( ord_less_int @ ( ord_min_int @ M2 @ N2 ) @ M2 ) )
      = ( ( ord_min_int @ M2 @ N2 )
        = M2 ) ) ).

% min_arg_not_ge(1)
thf(fact_5280_min__arg__not__ge_I2_J,axiom,
    ! [M2: code_natural,N2: code_natural] :
      ( ( ~ ( ord_le5570908160329646204atural @ ( ord_min_Code_natural @ M2 @ N2 ) @ N2 ) )
      = ( ( ord_min_Code_natural @ M2 @ N2 )
        = N2 ) ) ).

% min_arg_not_ge(2)
thf(fact_5281_min__arg__not__ge_I2_J,axiom,
    ! [M2: code_integer,N2: code_integer] :
      ( ( ~ ( ord_le6747313008572928689nteger @ ( ord_min_Code_integer @ M2 @ N2 ) @ N2 ) )
      = ( ( ord_min_Code_integer @ M2 @ N2 )
        = N2 ) ) ).

% min_arg_not_ge(2)
thf(fact_5282_min__arg__not__ge_I2_J,axiom,
    ! [M2: rat,N2: rat] :
      ( ( ~ ( ord_less_rat @ ( ord_min_rat @ M2 @ N2 ) @ N2 ) )
      = ( ( ord_min_rat @ M2 @ N2 )
        = N2 ) ) ).

% min_arg_not_ge(2)
thf(fact_5283_min__arg__not__ge_I2_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ~ ( ord_less_num @ ( ord_min_num @ M2 @ N2 ) @ N2 ) )
      = ( ( ord_min_num @ M2 @ N2 )
        = N2 ) ) ).

% min_arg_not_ge(2)
thf(fact_5284_min__arg__not__ge_I2_J,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ~ ( ord_less_nat @ ( ord_min_nat @ M2 @ N2 ) @ N2 ) )
      = ( ( ord_min_nat @ M2 @ N2 )
        = N2 ) ) ).

% min_arg_not_ge(2)
thf(fact_5285_min__arg__not__ge_I2_J,axiom,
    ! [M2: int,N2: int] :
      ( ( ~ ( ord_less_int @ ( ord_min_int @ M2 @ N2 ) @ N2 ) )
      = ( ( ord_min_int @ M2 @ N2 )
        = N2 ) ) ).

% min_arg_not_ge(2)
thf(fact_5286_min__less__self__conv_I1_J,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( ord_min_Code_natural @ A @ B ) @ A )
      = ( ord_le5570908160329646204atural @ B @ A ) ) ).

% min_less_self_conv(1)
thf(fact_5287_min__less__self__conv_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_min_Code_integer @ A @ B ) @ A )
      = ( ord_le6747313008572928689nteger @ B @ A ) ) ).

% min_less_self_conv(1)
thf(fact_5288_min__less__self__conv_I1_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( ord_min_rat @ A @ B ) @ A )
      = ( ord_less_rat @ B @ A ) ) ).

% min_less_self_conv(1)
thf(fact_5289_min__less__self__conv_I1_J,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ ( ord_min_num @ A @ B ) @ A )
      = ( ord_less_num @ B @ A ) ) ).

% min_less_self_conv(1)
thf(fact_5290_min__less__self__conv_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ A )
      = ( ord_less_nat @ B @ A ) ) ).

% min_less_self_conv(1)
thf(fact_5291_min__less__self__conv_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( ord_min_int @ A @ B ) @ A )
      = ( ord_less_int @ B @ A ) ) ).

% min_less_self_conv(1)
thf(fact_5292_min__less__self__conv_I2_J,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( ord_min_Code_natural @ A @ B ) @ B )
      = ( ord_le5570908160329646204atural @ A @ B ) ) ).

% min_less_self_conv(2)
thf(fact_5293_min__less__self__conv_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_min_Code_integer @ A @ B ) @ B )
      = ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% min_less_self_conv(2)
thf(fact_5294_min__less__self__conv_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( ord_min_rat @ A @ B ) @ B )
      = ( ord_less_rat @ A @ B ) ) ).

% min_less_self_conv(2)
thf(fact_5295_min__less__self__conv_I2_J,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ ( ord_min_num @ A @ B ) @ B )
      = ( ord_less_num @ A @ B ) ) ).

% min_less_self_conv(2)
thf(fact_5296_min__less__self__conv_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ B ) ) ).

% min_less_self_conv(2)
thf(fact_5297_min__less__self__conv_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( ord_min_int @ A @ B ) @ B )
      = ( ord_less_int @ A @ B ) ) ).

% min_less_self_conv(2)
thf(fact_5298_min__simps_I1_J,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ B )
     => ( ( ord_min_Code_natural @ A @ B )
        = A ) ) ).

% min_simps(1)
thf(fact_5299_min__simps_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_min_Code_integer @ A @ B )
        = A ) ) ).

% min_simps(1)
thf(fact_5300_min__simps_I1_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_min_rat @ A @ B )
        = A ) ) ).

% min_simps(1)
thf(fact_5301_min__simps_I1_J,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_min_num @ A @ B )
        = A ) ) ).

% min_simps(1)
thf(fact_5302_min__simps_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ A @ B )
        = A ) ) ).

% min_simps(1)
thf(fact_5303_min__simps_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_min_int @ A @ B )
        = A ) ) ).

% min_simps(1)
thf(fact_5304_min__simps_I2_J,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( ord_le5570908160329646204atural @ B @ A )
     => ( ( ord_min_Code_natural @ A @ B )
        = B ) ) ).

% min_simps(2)
thf(fact_5305_min__simps_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_min_Code_integer @ A @ B )
        = B ) ) ).

% min_simps(2)
thf(fact_5306_min__simps_I2_J,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_min_rat @ A @ B )
        = B ) ) ).

% min_simps(2)
thf(fact_5307_min__simps_I2_J,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_min_num @ A @ B )
        = B ) ) ).

% min_simps(2)
thf(fact_5308_min__simps_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_min_nat @ A @ B )
        = B ) ) ).

% min_simps(2)
thf(fact_5309_min__simps_I2_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_min_int @ A @ B )
        = B ) ) ).

% min_simps(2)
thf(fact_5310_min__0__1_I2_J,axiom,
    ( ( ord_min_rat @ one_one_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% min_0_1(2)
thf(fact_5311_min__0__1_I2_J,axiom,
    ( ( ord_min_nat @ one_one_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% min_0_1(2)
thf(fact_5312_min__0__1_I2_J,axiom,
    ( ( ord_min_Code_integer @ one_one_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% min_0_1(2)
thf(fact_5313_min__0__1_I2_J,axiom,
    ( ( ord_min_int @ one_one_int @ zero_zero_int )
    = zero_zero_int ) ).

% min_0_1(2)
thf(fact_5314_min__0__1_I2_J,axiom,
    ( ( ord_min_Code_natural @ one_one_Code_natural @ zero_z2226904508553997617atural )
    = zero_z2226904508553997617atural ) ).

% min_0_1(2)
thf(fact_5315_min__0__1_I1_J,axiom,
    ( ( ord_min_rat @ zero_zero_rat @ one_one_rat )
    = zero_zero_rat ) ).

% min_0_1(1)
thf(fact_5316_min__0__1_I1_J,axiom,
    ( ( ord_min_nat @ zero_zero_nat @ one_one_nat )
    = zero_zero_nat ) ).

% min_0_1(1)
thf(fact_5317_min__0__1_I1_J,axiom,
    ( ( ord_min_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer )
    = zero_z3403309356797280102nteger ) ).

% min_0_1(1)
thf(fact_5318_min__0__1_I1_J,axiom,
    ( ( ord_min_int @ zero_zero_int @ one_one_int )
    = zero_zero_int ) ).

% min_0_1(1)
thf(fact_5319_min__0__1_I1_J,axiom,
    ( ( ord_min_Code_natural @ zero_z2226904508553997617atural @ one_one_Code_natural )
    = zero_z2226904508553997617atural ) ).

% min_0_1(1)
thf(fact_5320_min__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_min_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = one_one_nat ) ).

% min_0_1(5)
thf(fact_5321_min__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_min_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = one_one_int ) ).

% min_0_1(5)
thf(fact_5322_min__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_min_Code_integer @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X ) )
      = one_one_Code_integer ) ).

% min_0_1(5)
thf(fact_5323_min__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_min_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
      = one_one_rat ) ).

% min_0_1(5)
thf(fact_5324_min__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_min_Code_natural @ one_one_Code_natural @ ( numera5444537566228673987atural @ X ) )
      = one_one_Code_natural ) ).

% min_0_1(5)
thf(fact_5325_min__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_min_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
      = one_one_nat ) ).

% min_0_1(6)
thf(fact_5326_min__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_min_int @ ( numeral_numeral_int @ X ) @ one_one_int )
      = one_one_int ) ).

% min_0_1(6)
thf(fact_5327_min__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_min_Code_integer @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% min_0_1(6)
thf(fact_5328_min__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_min_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
      = one_one_rat ) ).

% min_0_1(6)
thf(fact_5329_min__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_min_Code_natural @ ( numera5444537566228673987atural @ X ) @ one_one_Code_natural )
      = one_one_Code_natural ) ).

% min_0_1(6)
thf(fact_5330_atLeastAtMost__singleton__iff,axiom,
    ! [A: $o,B: $o,C: $o] :
      ( ( ( set_or8904488021354931149Most_o @ A @ B )
        = ( insert_o @ C @ bot_bot_set_o ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_5331_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B )
        = ( insert_int @ C @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_5332_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B )
        = ( insert_nat @ C @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_5333_atLeastAtMost__singleton__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ( set_or189985376899183464nteger @ A @ B )
        = ( insert_Code_integer @ C @ bot_bo3990330152332043303nteger ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_5334_atLeastAtMost__singleton,axiom,
    ! [A: $o] :
      ( ( set_or8904488021354931149Most_o @ A @ A )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% atLeastAtMost_singleton
thf(fact_5335_atLeastAtMost__singleton,axiom,
    ! [A: int] :
      ( ( set_or1266510415728281911st_int @ A @ A )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% atLeastAtMost_singleton
thf(fact_5336_atLeastAtMost__singleton,axiom,
    ! [A: nat] :
      ( ( set_or1269000886237332187st_nat @ A @ A )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% atLeastAtMost_singleton
thf(fact_5337_atLeastAtMost__singleton,axiom,
    ! [A: code_integer] :
      ( ( set_or189985376899183464nteger @ A @ A )
      = ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ).

% atLeastAtMost_singleton
thf(fact_5338_min__Suc__gt_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ B @ ( suc @ A ) )
        = ( suc @ A ) ) ) ).

% min_Suc_gt(2)
thf(fact_5339_min__Suc__gt_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ ( suc @ A ) @ B )
        = ( suc @ A ) ) ) ).

% min_Suc_gt(1)
thf(fact_5340_case__prod__Pair,axiom,
    ( ( produc6512950862096126219nteger @ produc8603105652947943368nteger )
    = id_Pro1379270990444177765nteger ) ).

% case_prod_Pair
thf(fact_5341_case__prod__Pair,axiom,
    ( ( produc4058941399401191971et_nat @ produc5001842942810119800et_nat )
    = id_Pro1888964116897856973et_nat ) ).

% case_prod_Pair
thf(fact_5342_case__prod__Pair,axiom,
    ( ( produc2377985495875741467et_nat @ produc2245416461498447860et_nat )
    = id_Pro1297863087923562825et_nat ) ).

% case_prod_Pair
thf(fact_5343_case__prod__Pair,axiom,
    ( ( produc8492565224438309093nt_int @ produc5700946648718959541nt_int )
    = id_Pro5422067691592394322nt_int ) ).

% case_prod_Pair
thf(fact_5344_case__prod__Pair,axiom,
    ( ( produc5122537100556696953nt_int @ produc4305682042979456191nt_int )
    = id_Pro3923183700162330716nt_int ) ).

% case_prod_Pair
thf(fact_5345_case__prod__Pair,axiom,
    ( ( produc4245557441103728435nt_int @ product_Pair_int_int )
    = id_Pro4381231610383771306nt_int ) ).

% case_prod_Pair
thf(fact_5346_single__Diff__lessThan,axiom,
    ! [K: $o] :
      ( ( minus_minus_set_o @ ( insert_o @ K @ bot_bot_set_o ) @ ( set_ord_lessThan_o @ K ) )
      = ( insert_o @ K @ bot_bot_set_o ) ) ).

% single_Diff_lessThan
thf(fact_5347_single__Diff__lessThan,axiom,
    ! [K: int] :
      ( ( minus_minus_set_int @ ( insert_int @ K @ bot_bot_set_int ) @ ( set_ord_lessThan_int @ K ) )
      = ( insert_int @ K @ bot_bot_set_int ) ) ).

% single_Diff_lessThan
thf(fact_5348_single__Diff__lessThan,axiom,
    ! [K: nat] :
      ( ( minus_minus_set_nat @ ( insert_nat @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
      = ( insert_nat @ K @ bot_bot_set_nat ) ) ).

% single_Diff_lessThan
thf(fact_5349_min__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ X @ ( ord_min_rat @ Y @ Z ) )
      = ( ord_min_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z ) ) ) ).

% min_add_distrib_right
thf(fact_5350_min__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ X @ ( ord_min_nat @ Y @ Z ) )
      = ( ord_min_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).

% min_add_distrib_right
thf(fact_5351_min__add__distrib__right,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( plus_p5714425477246183910nteger @ X @ ( ord_min_Code_integer @ Y @ Z ) )
      = ( ord_min_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( plus_p5714425477246183910nteger @ X @ Z ) ) ) ).

% min_add_distrib_right
thf(fact_5352_min__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ X @ ( ord_min_int @ Y @ Z ) )
      = ( ord_min_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).

% min_add_distrib_right
thf(fact_5353_min__add__distrib__right,axiom,
    ! [X: code_natural,Y: code_natural,Z: code_natural] :
      ( ( plus_p4538020629002901425atural @ X @ ( ord_min_Code_natural @ Y @ Z ) )
      = ( ord_min_Code_natural @ ( plus_p4538020629002901425atural @ X @ Y ) @ ( plus_p4538020629002901425atural @ X @ Z ) ) ) ).

% min_add_distrib_right
thf(fact_5354_min__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ ( ord_min_rat @ X @ Y ) @ Z )
      = ( ord_min_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y @ Z ) ) ) ).

% min_add_distrib_left
thf(fact_5355_min__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ ( ord_min_nat @ X @ Y ) @ Z )
      = ( ord_min_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).

% min_add_distrib_left
thf(fact_5356_min__add__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( ord_min_Code_integer @ X @ Y ) @ Z )
      = ( ord_min_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Z ) @ ( plus_p5714425477246183910nteger @ Y @ Z ) ) ) ).

% min_add_distrib_left
thf(fact_5357_min__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ ( ord_min_int @ X @ Y ) @ Z )
      = ( ord_min_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).

% min_add_distrib_left
thf(fact_5358_min__add__distrib__left,axiom,
    ! [X: code_natural,Y: code_natural,Z: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( ord_min_Code_natural @ X @ Y ) @ Z )
      = ( ord_min_Code_natural @ ( plus_p4538020629002901425atural @ X @ Z ) @ ( plus_p4538020629002901425atural @ Y @ Z ) ) ) ).

% min_add_distrib_left
thf(fact_5359_min__diff__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( ord_min_Code_integer @ X @ Y ) @ Z )
      = ( ord_min_Code_integer @ ( minus_8373710615458151222nteger @ X @ Z ) @ ( minus_8373710615458151222nteger @ Y @ Z ) ) ) ).

% min_diff_distrib_left
thf(fact_5360_min__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( minus_minus_rat @ ( ord_min_rat @ X @ Y ) @ Z )
      = ( ord_min_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y @ Z ) ) ) ).

% min_diff_distrib_left
thf(fact_5361_min__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( minus_minus_int @ ( ord_min_int @ X @ Y ) @ Z )
      = ( ord_min_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).

% min_diff_distrib_left
thf(fact_5362_bot__option__def,axiom,
    bot_bot_option_num = none_num ).

% bot_option_def
thf(fact_5363_bot__empty__eq2,axiom,
    ( bot_bo3000040243691356879eger_o
    = ( ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ bot_bo5443222936135328352nteger ) ) ) ).

% bot_empty_eq2
thf(fact_5364_bot__empty__eq2,axiom,
    ( bot_bo5580076615179976505_nat_o
    = ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ bot_bo1481135142794719944et_nat ) ) ) ).

% bot_empty_eq2
thf(fact_5365_bot__empty__eq2,axiom,
    ( bot_bo3790638025767943357_nat_o
    = ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ bot_bo5635537948650799172et_nat ) ) ) ).

% bot_empty_eq2
thf(fact_5366_bot__empty__eq2,axiom,
    ( bot_bo8662317086119403298_int_o
    = ( ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ bot_bo572930865798478029nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_5367_bot__empty__eq2,axiom,
    ( bot_bo1403522918969695512_int_o
    = ( ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ bot_bo4508923176915781079nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_5368_bot__empty__eq2,axiom,
    ( bot_bo394778441745866138_nat_o
    = ( ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ bot_bo228742789529271731at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_5369_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_5370_map__fun_Oidentity,axiom,
    ( ( map_fun_nat_nat_o_o
      @ ^ [X3: nat] : X3
      @ ^ [X3: $o] : X3 )
    = id_nat_o ) ).

% map_fun.identity
thf(fact_5371_map__option_Oidentity,axiom,
    ( ( map_option_num_num
      @ ^ [X3: num] : X3 )
    = id_option_num ) ).

% map_option.identity
thf(fact_5372_minus__max__eq__min,axiom,
    ! [X: int,Y: int] :
      ( ( uminus_uminus_int @ ( ord_max_int @ X @ Y ) )
      = ( ord_min_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ).

% minus_max_eq_min
thf(fact_5373_minus__max__eq__min,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( ord_max_Code_integer @ X @ Y ) )
      = ( ord_min_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ ( uminus1351360451143612070nteger @ Y ) ) ) ).

% minus_max_eq_min
thf(fact_5374_minus__max__eq__min,axiom,
    ! [X: rat,Y: rat] :
      ( ( uminus_uminus_rat @ ( ord_max_rat @ X @ Y ) )
      = ( ord_min_rat @ ( uminus_uminus_rat @ X ) @ ( uminus_uminus_rat @ Y ) ) ) ).

% minus_max_eq_min
thf(fact_5375_minus__min__eq__max,axiom,
    ! [X: int,Y: int] :
      ( ( uminus_uminus_int @ ( ord_min_int @ X @ Y ) )
      = ( ord_max_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ).

% minus_min_eq_max
thf(fact_5376_minus__min__eq__max,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( ord_min_Code_integer @ X @ Y ) )
      = ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ ( uminus1351360451143612070nteger @ Y ) ) ) ).

% minus_min_eq_max
thf(fact_5377_minus__min__eq__max,axiom,
    ! [X: rat,Y: rat] :
      ( ( uminus_uminus_rat @ ( ord_min_rat @ X @ Y ) )
      = ( ord_max_rat @ ( uminus_uminus_rat @ X ) @ ( uminus_uminus_rat @ Y ) ) ) ).

% minus_min_eq_max
thf(fact_5378_insert__minus__eq,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( X != Y )
     => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A2 ) @ ( insert9069300056098147895at_nat @ Y @ bot_bo228742789529271731at_nat ) )
        = ( insert9069300056098147895at_nat @ X @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ Y @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% insert_minus_eq
thf(fact_5379_insert__minus__eq,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( X != Y )
     => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) )
        = ( insert8211810215607154385at_nat @ X @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% insert_minus_eq
thf(fact_5380_insert__minus__eq,axiom,
    ! [X: $o,Y: $o,A2: set_o] :
      ( ( X != Y )
     => ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ ( insert_o @ Y @ bot_bot_set_o ) )
        = ( insert_o @ X @ ( minus_minus_set_o @ A2 @ ( insert_o @ Y @ bot_bot_set_o ) ) ) ) ) ).

% insert_minus_eq
thf(fact_5381_insert__minus__eq,axiom,
    ! [X: int,Y: int,A2: set_int] :
      ( ( X != Y )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ ( insert_int @ Y @ bot_bot_set_int ) )
        = ( insert_int @ X @ ( minus_minus_set_int @ A2 @ ( insert_int @ Y @ bot_bot_set_int ) ) ) ) ) ).

% insert_minus_eq
thf(fact_5382_insert__minus__eq,axiom,
    ! [X: nat,Y: nat,A2: set_nat] :
      ( ( X != Y )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ Y @ bot_bot_set_nat ) )
        = ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) ) ) ).

% insert_minus_eq
thf(fact_5383_set__minus__singleton__eq,axiom,
    ! [X: produc3843707927480180839at_nat,X11: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ X11 )
     => ( ( minus_3314409938677909166at_nat @ X11 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
        = X11 ) ) ).

% set_minus_singleton_eq
thf(fact_5384_set__minus__singleton__eq,axiom,
    ! [X: product_prod_nat_nat,X11: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ X11 )
     => ( ( minus_1356011639430497352at_nat @ X11 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
        = X11 ) ) ).

% set_minus_singleton_eq
thf(fact_5385_set__minus__singleton__eq,axiom,
    ! [X: $o,X11: set_o] :
      ( ~ ( member_o @ X @ X11 )
     => ( ( minus_minus_set_o @ X11 @ ( insert_o @ X @ bot_bot_set_o ) )
        = X11 ) ) ).

% set_minus_singleton_eq
thf(fact_5386_set__minus__singleton__eq,axiom,
    ! [X: int,X11: set_int] :
      ( ~ ( member_int @ X @ X11 )
     => ( ( minus_minus_set_int @ X11 @ ( insert_int @ X @ bot_bot_set_int ) )
        = X11 ) ) ).

% set_minus_singleton_eq
thf(fact_5387_set__minus__singleton__eq,axiom,
    ! [X: nat,X11: set_nat] :
      ( ~ ( member_nat @ X @ X11 )
     => ( ( minus_minus_set_nat @ X11 @ ( insert_nat @ X @ bot_bot_set_nat ) )
        = X11 ) ) ).

% set_minus_singleton_eq
thf(fact_5388_atLeastAtMost__singleton_H,axiom,
    ! [A: $o,B: $o] :
      ( ( A = B )
     => ( ( set_or8904488021354931149Most_o @ A @ B )
        = ( insert_o @ A @ bot_bot_set_o ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_5389_atLeastAtMost__singleton_H,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( ( set_or1266510415728281911st_int @ A @ B )
        = ( insert_int @ A @ bot_bot_set_int ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_5390_atLeastAtMost__singleton_H,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( ( set_or1269000886237332187st_nat @ A @ B )
        = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_5391_atLeastAtMost__singleton_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
     => ( ( set_or189985376899183464nteger @ A @ B )
        = ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_5392_remove__subset,axiom,
    ! [X: produc3843707927480180839at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ S2 )
     => ( ord_le2604355607129572851at_nat @ ( minus_3314409938677909166at_nat @ S2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ S2 ) ) ).

% remove_subset
thf(fact_5393_remove__subset,axiom,
    ! [X: product_prod_nat_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ S2 )
     => ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ S2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ S2 ) ) ).

% remove_subset
thf(fact_5394_remove__subset,axiom,
    ! [X: $o,S2: set_o] :
      ( ( member_o @ X @ S2 )
     => ( ord_less_set_o @ ( minus_minus_set_o @ S2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ S2 ) ) ).

% remove_subset
thf(fact_5395_remove__subset,axiom,
    ! [X: int,S2: set_int] :
      ( ( member_int @ X @ S2 )
     => ( ord_less_set_int @ ( minus_minus_set_int @ S2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ S2 ) ) ).

% remove_subset
thf(fact_5396_remove__subset,axiom,
    ! [X: nat,S2: set_nat] :
      ( ( member_nat @ X @ S2 )
     => ( ord_less_set_nat @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ S2 ) ) ).

% remove_subset
thf(fact_5397_nat__of__integer__def,axiom,
    ( code_nat_of_integer
    = ( map_fu6539832666145259331at_nat @ code_int_of_integer @ id_nat @ nat2 ) ) ).

% nat_of_integer_def
thf(fact_5398_bit__integer__def,axiom,
    ( bit_se9216721137139052372nteger
    = ( map_fu186220595405829473_nat_o @ code_int_of_integer @ id_nat_o @ bit_se1146084159140164899it_int ) ) ).

% bit_integer_def
thf(fact_5399_mask__integer__def,axiom,
    ( bit_se2119862282449309892nteger
    = ( map_fu6290471996055670595nteger @ id_nat @ code_integer_of_int @ bit_se2000444600071755411sk_int ) ) ).

% mask_integer_def
thf(fact_5400_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ( set_or7139685690850216873Than_o
    = ( ^ [A5: $o,B4: $o] : ( minus_minus_set_o @ ( set_or8904488021354931149Most_o @ A5 @ B4 ) @ ( insert_o @ B4 @ bot_bot_set_o ) ) ) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_5401_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [A5: nat,B4: nat] : ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A5 @ B4 ) @ ( insert_nat @ B4 @ bot_bot_set_nat ) ) ) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_5402_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ( set_or8404916559141939852nteger
    = ( ^ [A5: code_integer,B4: code_integer] : ( minus_2355218937544613996nteger @ ( set_or189985376899183464nteger @ A5 @ B4 ) @ ( insert_Code_integer @ B4 @ bot_bo3990330152332043303nteger ) ) ) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_5403_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [A5: int,B4: int] : ( minus_minus_set_int @ ( set_or1266510415728281911st_int @ A5 @ B4 ) @ ( insert_int @ B4 @ bot_bot_set_int ) ) ) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_5404_atLeastAtMostPlus1__int__conv,axiom,
    ! [M2: int,N2: int] :
      ( ( ord_less_eq_int @ M2 @ ( plus_plus_int @ one_one_int @ N2 ) )
     => ( ( set_or1266510415728281911st_int @ M2 @ ( plus_plus_int @ one_one_int @ N2 ) )
        = ( insert_int @ ( plus_plus_int @ one_one_int @ N2 ) @ ( set_or1266510415728281911st_int @ M2 @ N2 ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_5405_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I3: int,J2: int] : ( if_set_int @ ( ord_less_int @ J2 @ I3 ) @ bot_bot_set_int @ ( insert_int @ I3 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) ) ) ) ) ).

% simp_from_to
thf(fact_5406_Code__Numeral_Osub__def,axiom,
    ( code_sub
    = ( map_fu6891787308814931657nteger @ id_num @ ( map_fu8638147718074629079nteger @ id_num @ code_integer_of_int )
      @ ^ [M: num,N4: num] : ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N4 ) ) ) ) ).

% Code_Numeral.sub_def
thf(fact_5407_num__of__integer__def,axiom,
    ( code_num_of_integer
    = ( map_fu1227494855608507351um_num @ code_int_of_integer @ id_num @ ( comp_nat_num_int @ num_of_nat @ nat2 ) ) ) ).

% num_of_integer_def
thf(fact_5408_take__bit__integer__def,axiom,
    ( bit_se1745604003318907178nteger
    = ( map_fu8224359078269604234nteger @ id_nat @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se2923211474154528505it_int ) ) ).

% take_bit_integer_def
thf(fact_5409_drop__bit__integer__def,axiom,
    ( bit_se3928097537394005634nteger
    = ( map_fu8224359078269604234nteger @ id_nat @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se8568078237143864401it_int ) ) ).

% drop_bit_integer_def
thf(fact_5410_push__bit__integer__def,axiom,
    ( bit_se7788150548672797655nteger
    = ( map_fu8224359078269604234nteger @ id_nat @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se545348938243370406it_int ) ) ).

% push_bit_integer_def
thf(fact_5411_unset__bit__integer__def,axiom,
    ( bit_se8260200283734997820nteger
    = ( map_fu8224359078269604234nteger @ id_nat @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se4203085406695923979it_int ) ) ).

% unset_bit_integer_def
thf(fact_5412_set__bit__integer__def,axiom,
    ( bit_se2793503036327961859nteger
    = ( map_fu8224359078269604234nteger @ id_nat @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se7879613467334960850it_int ) ) ).

% set_bit_integer_def
thf(fact_5413_flip__bit__integer__def,axiom,
    ( bit_se1345352211410354436nteger
    = ( map_fu8224359078269604234nteger @ id_nat @ ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int ) @ bit_se2159334234014336723it_int ) ) ).

% flip_bit_integer_def
thf(fact_5414_pair__leqI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_5415_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( ( ord_less_nat @ C @ Y )
       => ( ( image_nat_nat
            @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
            @ ( set_or4665077453230672383an_nat @ X @ Y ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y )
       => ( ( ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_5416_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).

% card_atLeastAtMost_int
thf(fact_5417_range__mod,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( image_nat_nat
          @ ^ [M: nat] : ( modulo_modulo_nat @ M @ N2 )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% range_mod
thf(fact_5418_Some__image__these__eq,axiom,
    ! [A2: set_op3940070208846178154it_nat] :
      ( ( image_4799632500972785611it_nat @ some_P6105628898608307625it_nat @ ( these_6645423997822192641it_nat @ A2 ) )
      = ( collec8141831914148684149it_nat
        @ ^ [X3: option3216581401566546698it_nat] :
            ( ( member3517700364412436275it_nat @ X3 @ A2 )
            & ( X3 != none_P2587165478634041381it_nat ) ) ) ) ).

% Some_image_these_eq
thf(fact_5419_Some__image__these__eq,axiom,
    ! [A2: set_option_num] :
      ( ( image_num_option_num @ some_num @ ( these_num @ A2 ) )
      = ( collect_option_num
        @ ^ [X3: option_num] :
            ( ( member_option_num @ X3 @ A2 )
            & ( X3 != none_num ) ) ) ) ).

% Some_image_these_eq
thf(fact_5420_Some__image__these__eq,axiom,
    ! [A2: set_op239864471688321678it_nat] :
      ( ( image_3455408117458022631it_nat @ some_P1914260805536162275it_nat @ ( these_3407724651694999947it_nat @ A2 ) )
      = ( collec5742531119322951597it_nat
        @ ^ [X3: option8956607266484857688it_nat] :
            ( ( member5596548051065438575it_nat @ X3 @ A2 )
            & ( X3 != none_P9117596204409417319it_nat ) ) ) ) ).

% Some_image_these_eq
thf(fact_5421_prod__mono2,axiom,
    ! [B5: set_o,A2: set_o,F: $o > code_integer] :
      ( ( finite_finite_o @ B5 )
     => ( ( ord_less_eq_set_o @ A2 @ B5 )
       => ( ! [B2: $o] :
              ( ( member_o @ B2 @ ( minus_minus_set_o @ B5 @ A2 ) )
             => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B2 ) ) )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ A2 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A3 ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups7694694392188491536nteger @ F @ A2 ) @ ( groups7694694392188491536nteger @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_5422_prod__mono2,axiom,
    ! [B5: set_int,A2: set_int,F: int > code_integer] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A2 @ B5 )
       => ( ! [B2: int] :
              ( ( member_int @ B2 @ ( minus_minus_set_int @ B5 @ A2 ) )
             => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B2 ) ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ A2 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A3 ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F @ A2 ) @ ( groups3827104343326376752nteger @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_5423_prod__mono2,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( ! [B2: code_integer] :
              ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ B5 @ A2 ) )
             => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B2 ) ) )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ A2 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A3 ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups3674199335183972705nteger @ F @ A2 ) @ ( groups3674199335183972705nteger @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_5424_prod__mono2,axiom,
    ! [B5: set_o,A2: set_o,F: $o > rat] :
      ( ( finite_finite_o @ B5 )
     => ( ( ord_less_eq_set_o @ A2 @ B5 )
       => ( ! [B2: $o] :
              ( ( member_o @ B2 @ ( minus_minus_set_o @ B5 @ A2 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ A2 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups2869687844427037835_o_rat @ F @ A2 ) @ ( groups2869687844427037835_o_rat @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_5425_prod__mono2,axiom,
    ! [B5: set_int,A2: set_int,F: int > rat] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A2 @ B5 )
       => ( ! [B2: int] :
              ( ( member_int @ B2 @ ( minus_minus_set_int @ B5 @ A2 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ A2 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_5426_prod__mono2,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( ! [B2: code_integer] :
              ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ B5 @ A2 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ A2 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups2555765274223993564er_rat @ F @ A2 ) @ ( groups2555765274223993564er_rat @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_5427_prod__mono2,axiom,
    ! [B5: set_o,A2: set_o,F: $o > int] :
      ( ( finite_finite_o @ B5 )
     => ( ( ord_less_eq_set_o @ A2 @ B5 )
       => ( ! [B2: $o] :
              ( ( member_o @ B2 @ ( minus_minus_set_o @ B5 @ A2 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B2 ) ) )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ A2 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
           => ( ord_less_eq_int @ ( groups3502327434004483295_o_int @ F @ A2 ) @ ( groups3502327434004483295_o_int @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_5428_prod__mono2,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ B5 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ B5 )
       => ( ! [B2: code_integer] :
              ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ B5 @ A2 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B2 ) ) )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ A2 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
           => ( ord_less_eq_int @ ( groups3188404863801439024er_int @ F @ A2 ) @ ( groups3188404863801439024er_int @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_5429_prod__mono2,axiom,
    ! [B5: set_nat,A2: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A2 @ B5 )
       => ( ! [B2: nat] :
              ( ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A2 ) )
             => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ B2 ) ) )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ A2 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ A3 ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups3455450783089532116nteger @ F @ A2 ) @ ( groups3455450783089532116nteger @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_5430_prod__mono2,axiom,
    ! [B5: set_nat,A2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A2 @ B5 )
       => ( ! [B2: nat] :
              ( ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A2 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ A2 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ F @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_5431_the__dflt__None__nonempty,axiom,
    ! [S2: set_Pr4329608150637261639at_nat] :
      ( ( S2 != bot_bo228742789529271731at_nat )
     => ( ( dflt_N3208751546879591902at_nat @ S2 )
        = ( some_s5890477192898017836at_nat @ S2 ) ) ) ).

% the_dflt_None_nonempty
thf(fact_5432_the__dflt__None__nonempty,axiom,
    ! [S2: set_Pr1261947904930325089at_nat] :
      ( ( S2 != bot_bo2099793752762293965at_nat )
     => ( ( dflt_N6592383573632408824at_nat @ S2 )
        = ( some_s147305329494351046at_nat @ S2 ) ) ) ).

% the_dflt_None_nonempty
thf(fact_5433_the__dflt__None__nonempty,axiom,
    ! [S2: set_o] :
      ( ( S2 != bot_bot_set_o )
     => ( ( dflt_None_set_o @ S2 )
        = ( some_set_o @ S2 ) ) ) ).

% the_dflt_None_nonempty
thf(fact_5434_the__dflt__None__nonempty,axiom,
    ! [S2: set_nat] :
      ( ( S2 != bot_bot_set_nat )
     => ( ( dflt_None_set_nat @ S2 )
        = ( some_set_nat @ S2 ) ) ) ).

% the_dflt_None_nonempty
thf(fact_5435_the__dflt__None__nonempty,axiom,
    ! [S2: set_int] :
      ( ( S2 != bot_bot_set_int )
     => ( ( dflt_None_set_int @ S2 )
        = ( some_set_int @ S2 ) ) ) ).

% the_dflt_None_nonempty
thf(fact_5436_the__dflt__None__empty,axiom,
    ( ( dflt_N3208751546879591902at_nat @ bot_bo228742789529271731at_nat )
    = none_s6518873173954064040at_nat ) ).

% the_dflt_None_empty
thf(fact_5437_the__dflt__None__empty,axiom,
    ( ( dflt_N6592383573632408824at_nat @ bot_bo2099793752762293965at_nat )
    = none_s625347054029921090at_nat ) ).

% the_dflt_None_empty
thf(fact_5438_the__dflt__None__empty,axiom,
    ( ( dflt_None_set_o @ bot_bot_set_o )
    = none_set_o ) ).

% the_dflt_None_empty
thf(fact_5439_the__dflt__None__empty,axiom,
    ( ( dflt_None_set_nat @ bot_bot_set_nat )
    = none_set_nat ) ).

% the_dflt_None_empty
thf(fact_5440_the__dflt__None__empty,axiom,
    ( ( dflt_None_set_int @ bot_bot_set_int )
    = none_set_int ) ).

% the_dflt_None_empty
thf(fact_5441_atMost__UNIV__triv,axiom,
    ( ( set_or4236626031148496127et_nat @ top_top_set_nat )
    = top_top_set_set_nat ) ).

% atMost_UNIV_triv
thf(fact_5442_atMost__UNIV__triv,axiom,
    ( ( set_or2827140217781692084t_unit @ top_to1996260823553986621t_unit )
    = top_to1767297665138865437t_unit ) ).

% atMost_UNIV_triv
thf(fact_5443_atMost__UNIV__triv,axiom,
    ( ( set_or58775011639299419et_int @ top_top_set_int )
    = top_top_set_set_int ) ).

% atMost_UNIV_triv
thf(fact_5444_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% finite_atLeastAtMost
thf(fact_5445_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).

% finite_atLeastLessThan
thf(fact_5446_int__of__integer__min,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( code_int_of_integer @ ( ord_min_Code_integer @ K @ L ) )
      = ( ord_min_int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).

% int_of_integer_min
thf(fact_5447_finite__atMost,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).

% finite_atMost
thf(fact_5448_finite__lessThan,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).

% finite_lessThan
thf(fact_5449_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L @ U ) ) ).

% finite_atLeastAtMost_int
thf(fact_5450_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).

% finite_atLeastLessThan_int
thf(fact_5451_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ L @ U ) ) ).

% finite_atLeastLessThan_integer
thf(fact_5452_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or189985376899183464nteger @ L @ U ) ) ).

% finite_atLeastAtMost_integer
thf(fact_5453_finite__option__UNIV,axiom,
    ( ( finite6785661671136154180nteger @ top_to5929521628599800467nteger )
    = ( finite6017078050557962740nteger @ top_to4645266643341252675nteger ) ) ).

% finite_option_UNIV
thf(fact_5454_finite__option__UNIV,axiom,
    ( ( finite6732403688824079472at_nat @ top_to3251141154256563319at_nat )
    = ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ).

% finite_option_UNIV
thf(fact_5455_finite__option__UNIV,axiom,
    ( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% finite_option_UNIV
thf(fact_5456_finite__option__UNIV,axiom,
    ( ( finite1445617369574913404t_unit @ top_to2690860209552263555t_unit )
    = ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).

% finite_option_UNIV
thf(fact_5457_finite__option__UNIV,axiom,
    ( ( finite1345302120164226195on_int @ top_to6430115241214627170on_int )
    = ( finite_finite_int @ top_top_set_int ) ) ).

% finite_option_UNIV
thf(fact_5458_card__eq__UNIV,axiom,
    ! [S2: set_Product_unit] :
      ( ( ( finite410649719033368117t_unit @ S2 )
        = ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) )
      = ( S2 = top_to1996260823553986621t_unit ) ) ).

% card_eq_UNIV
thf(fact_5459_card__eq__UNIV2,axiom,
    ! [S2: set_Product_unit] :
      ( ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
        = ( finite410649719033368117t_unit @ S2 ) )
      = ( S2 = top_to1996260823553986621t_unit ) ) ).

% card_eq_UNIV2
thf(fact_5460_these__empty,axiom,
    ( ( these_1872268174106123684at_nat @ bot_bo1471150103859235065at_nat )
    = bot_bo228742789529271731at_nat ) ).

% these_empty
thf(fact_5461_these__empty,axiom,
    ( ( these_8061402112538453438at_nat @ bot_bo3414888551220300179at_nat )
    = bot_bo2099793752762293965at_nat ) ).

% these_empty
thf(fact_5462_these__empty,axiom,
    ( ( these_o @ bot_bot_set_option_o )
    = bot_bot_set_o ) ).

% these_empty
thf(fact_5463_these__empty,axiom,
    ( ( these_nat @ bot_bo5009843511495006442on_nat )
    = bot_bot_set_nat ) ).

% these_empty
thf(fact_5464_these__empty,axiom,
    ( ( these_int @ bot_bo2519760366563279686on_int )
    = bot_bot_set_int ) ).

% these_empty
thf(fact_5465_these__insert__None,axiom,
    ! [A2: set_op3940070208846178154it_nat] :
      ( ( these_6645423997822192641it_nat @ ( insert7468008629046257882it_nat @ none_P2587165478634041381it_nat @ A2 ) )
      = ( these_6645423997822192641it_nat @ A2 ) ) ).

% these_insert_None
thf(fact_5466_these__insert__None,axiom,
    ! [A2: set_option_num] :
      ( ( these_num @ ( insert_option_num @ none_num @ A2 ) )
      = ( these_num @ A2 ) ) ).

% these_insert_None
thf(fact_5467_these__insert__None,axiom,
    ! [A2: set_op239864471688321678it_nat] :
      ( ( these_3407724651694999947it_nat @ ( insert2007652293347934600it_nat @ none_P9117596204409417319it_nat @ A2 ) )
      = ( these_3407724651694999947it_nat @ A2 ) ) ).

% these_insert_None
thf(fact_5468_the__dflt__None__set,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( the_de6830073021511528542at_nat @ bot_bo228742789529271731at_nat @ ( dflt_N3208751546879591902at_nat @ X ) )
      = X ) ).

% the_dflt_None_set
thf(fact_5469_the__dflt__None__set,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( the_de3668382347305825784at_nat @ bot_bo2099793752762293965at_nat @ ( dflt_N6592383573632408824at_nat @ X ) )
      = X ) ).

% the_dflt_None_set
thf(fact_5470_the__dflt__None__set,axiom,
    ! [X: set_o] :
      ( ( the_default_set_o @ bot_bot_set_o @ ( dflt_None_set_o @ X ) )
      = X ) ).

% the_dflt_None_set
thf(fact_5471_the__dflt__None__set,axiom,
    ! [X: set_nat] :
      ( ( the_default_set_nat @ bot_bot_set_nat @ ( dflt_None_set_nat @ X ) )
      = X ) ).

% the_dflt_None_set
thf(fact_5472_the__dflt__None__set,axiom,
    ! [X: set_int] :
      ( ( the_default_set_int @ bot_bot_set_int @ ( dflt_None_set_int @ X ) )
      = X ) ).

% the_dflt_None_set
thf(fact_5473_surj__plus,axiom,
    ! [A: rat] :
      ( ( image_rat_rat @ ( plus_plus_rat @ A ) @ top_top_set_rat )
      = top_top_set_rat ) ).

% surj_plus
thf(fact_5474_surj__plus,axiom,
    ! [A: int] :
      ( ( image_int_int @ ( plus_plus_int @ A ) @ top_top_set_int )
      = top_top_set_int ) ).

% surj_plus
thf(fact_5475_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
      = ( ord_less_rat @ A @ B ) ) ).

% infinite_Icc_iff
thf(fact_5476_infinite__Ico__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or4029947393144176647an_rat @ A @ B ) ) )
      = ( ord_less_rat @ A @ B ) ) ).

% infinite_Ico_iff
thf(fact_5477_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > code_integer] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups3455450783089532116nteger @ G @ A2 )
        = one_one_Code_integer ) ) ).

% prod.infinite
thf(fact_5478_prod_Oinfinite,axiom,
    ! [A2: set_int,G: int > code_integer] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups3827104343326376752nteger @ G @ A2 )
        = one_one_Code_integer ) ) ).

% prod.infinite
thf(fact_5479_prod_Oinfinite,axiom,
    ! [A2: set_Code_integer,G: code_integer > code_integer] :
      ( ~ ( finite6017078050557962740nteger @ A2 )
     => ( ( groups3674199335183972705nteger @ G @ A2 )
        = one_one_Code_integer ) ) ).

% prod.infinite
thf(fact_5480_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > assn] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups6906906614972039071t_assn @ G @ A2 )
        = one_one_assn ) ) ).

% prod.infinite
thf(fact_5481_prod_Oinfinite,axiom,
    ! [A2: set_int,G: int > assn] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups7882442080178216443t_assn @ G @ A2 )
        = one_one_assn ) ) ).

% prod.infinite
thf(fact_5482_prod_Oinfinite,axiom,
    ! [A2: set_Code_integer,G: code_integer > assn] :
      ( ~ ( finite6017078050557962740nteger @ A2 )
     => ( ( groups1304777262505850412r_assn @ G @ A2 )
        = one_one_assn ) ) ).

% prod.infinite
thf(fact_5483_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > rat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups73079841787564623at_rat @ G @ A2 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_5484_prod_Oinfinite,axiom,
    ! [A2: set_int,G: int > rat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups1072433553688619179nt_rat @ G @ A2 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_5485_prod_Oinfinite,axiom,
    ! [A2: set_Code_integer,G: code_integer > rat] :
      ( ~ ( finite6017078050557962740nteger @ A2 )
     => ( ( groups2555765274223993564er_rat @ G @ A2 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_5486_prod_Oinfinite,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups1707563613775114915nt_nat @ G @ A2 )
        = one_one_nat ) ) ).

% prod.infinite
thf(fact_5487_card__ge__UNIV,axiom,
    ! [S2: set_Product_unit] :
      ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) @ ( finite410649719033368117t_unit @ S2 ) )
      = ( S2 = top_to1996260823553986621t_unit ) ) ).

% card_ge_UNIV
thf(fact_5488_surj__diff__right,axiom,
    ! [A: code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [X3: code_integer] : ( minus_8373710615458151222nteger @ X3 @ A )
        @ top_to4645266643341252675nteger )
      = top_to4645266643341252675nteger ) ).

% surj_diff_right
thf(fact_5489_surj__diff__right,axiom,
    ! [A: rat] :
      ( ( image_rat_rat
        @ ^ [X3: rat] : ( minus_minus_rat @ X3 @ A )
        @ top_top_set_rat )
      = top_top_set_rat ) ).

% surj_diff_right
thf(fact_5490_surj__diff__right,axiom,
    ! [A: int] :
      ( ( image_int_int
        @ ^ [X3: int] : ( minus_minus_int @ X3 @ A )
        @ top_top_set_int )
      = top_top_set_int ) ).

% surj_diff_right
thf(fact_5491_prod__eq__1__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups1707563613775114915nt_nat @ F @ A2 )
          = one_one_nat )
        = ( ! [X3: int] :
              ( ( member_int @ X3 @ A2 )
             => ( ( F @ X3 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_5492_prod__eq__1__iff,axiom,
    ! [A2: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( groups3190895334310489300er_nat @ F @ A2 )
          = one_one_nat )
        = ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A2 )
             => ( ( F @ X3 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_5493_prod__eq__1__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( ( groups4077766827762148844at_nat @ F @ A2 )
          = one_one_nat )
        = ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ A2 )
             => ( ( F @ X3 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_5494_prod__eq__1__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups708209901874060359at_nat @ F @ A2 )
          = one_one_nat )
        = ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
             => ( ( F @ X3 )
                = one_one_nat ) ) ) ) ) ).

% prod_eq_1_iff
thf(fact_5495_prod_Odelta,axiom,
    ! [S2: set_o,A: $o,B: $o > code_integer] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups7694694392188491536nteger
              @ ^ [K4: $o] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ one_one_Code_integer )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups7694694392188491536nteger
              @ ^ [K4: $o] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ one_one_Code_integer )
              @ S2 )
            = one_one_Code_integer ) ) ) ) ).

% prod.delta
thf(fact_5496_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: nat > code_integer] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups3455450783089532116nteger
              @ ^ [K4: nat] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ one_one_Code_integer )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups3455450783089532116nteger
              @ ^ [K4: nat] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ one_one_Code_integer )
              @ S2 )
            = one_one_Code_integer ) ) ) ) ).

% prod.delta
thf(fact_5497_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B: int > code_integer] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups3827104343326376752nteger
              @ ^ [K4: int] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ one_one_Code_integer )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups3827104343326376752nteger
              @ ^ [K4: int] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ one_one_Code_integer )
              @ S2 )
            = one_one_Code_integer ) ) ) ) ).

% prod.delta
thf(fact_5498_prod_Odelta,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( ( member_Code_integer @ A @ S2 )
         => ( ( groups3674199335183972705nteger
              @ ^ [K4: code_integer] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ one_one_Code_integer )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S2 )
         => ( ( groups3674199335183972705nteger
              @ ^ [K4: code_integer] : ( if_Code_integer @ ( K4 = A ) @ ( B @ K4 ) @ one_one_Code_integer )
              @ S2 )
            = one_one_Code_integer ) ) ) ) ).

% prod.delta
thf(fact_5499_prod_Odelta,axiom,
    ! [S2: set_o,A: $o,B: $o > assn] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups5301882518646026715o_assn
              @ ^ [K4: $o] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ one_one_assn )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups5301882518646026715o_assn
              @ ^ [K4: $o] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ one_one_assn )
              @ S2 )
            = one_one_assn ) ) ) ) ).

% prod.delta
thf(fact_5500_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: nat > assn] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups6906906614972039071t_assn
              @ ^ [K4: nat] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ one_one_assn )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups6906906614972039071t_assn
              @ ^ [K4: nat] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ one_one_assn )
              @ S2 )
            = one_one_assn ) ) ) ) ).

% prod.delta
thf(fact_5501_prod_Odelta,axiom,
    ! [S2: set_int,A: int,B: int > assn] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups7882442080178216443t_assn
              @ ^ [K4: int] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ one_one_assn )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups7882442080178216443t_assn
              @ ^ [K4: int] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ one_one_assn )
              @ S2 )
            = one_one_assn ) ) ) ) ).

% prod.delta
thf(fact_5502_prod_Odelta,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( ( member_Code_integer @ A @ S2 )
         => ( ( groups1304777262505850412r_assn
              @ ^ [K4: code_integer] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ one_one_assn )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S2 )
         => ( ( groups1304777262505850412r_assn
              @ ^ [K4: code_integer] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ one_one_assn )
              @ S2 )
            = one_one_assn ) ) ) ) ).

% prod.delta
thf(fact_5503_prod_Odelta,axiom,
    ! [S2: set_o,A: $o,B: $o > rat] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups2869687844427037835_o_rat
              @ ^ [K4: $o] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups2869687844427037835_o_rat
              @ ^ [K4: $o] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = one_one_rat ) ) ) ) ).

% prod.delta
thf(fact_5504_prod_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K4: nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K4: nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = one_one_rat ) ) ) ) ).

% prod.delta
thf(fact_5505_prod_Odelta_H,axiom,
    ! [S2: set_o,A: $o,B: $o > code_integer] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups7694694392188491536nteger
              @ ^ [K4: $o] : ( if_Code_integer @ ( A = K4 ) @ ( B @ K4 ) @ one_one_Code_integer )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups7694694392188491536nteger
              @ ^ [K4: $o] : ( if_Code_integer @ ( A = K4 ) @ ( B @ K4 ) @ one_one_Code_integer )
              @ S2 )
            = one_one_Code_integer ) ) ) ) ).

% prod.delta'
thf(fact_5506_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: nat > code_integer] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups3455450783089532116nteger
              @ ^ [K4: nat] : ( if_Code_integer @ ( A = K4 ) @ ( B @ K4 ) @ one_one_Code_integer )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups3455450783089532116nteger
              @ ^ [K4: nat] : ( if_Code_integer @ ( A = K4 ) @ ( B @ K4 ) @ one_one_Code_integer )
              @ S2 )
            = one_one_Code_integer ) ) ) ) ).

% prod.delta'
thf(fact_5507_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: int > code_integer] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups3827104343326376752nteger
              @ ^ [K4: int] : ( if_Code_integer @ ( A = K4 ) @ ( B @ K4 ) @ one_one_Code_integer )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups3827104343326376752nteger
              @ ^ [K4: int] : ( if_Code_integer @ ( A = K4 ) @ ( B @ K4 ) @ one_one_Code_integer )
              @ S2 )
            = one_one_Code_integer ) ) ) ) ).

% prod.delta'
thf(fact_5508_prod_Odelta_H,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( ( member_Code_integer @ A @ S2 )
         => ( ( groups3674199335183972705nteger
              @ ^ [K4: code_integer] : ( if_Code_integer @ ( A = K4 ) @ ( B @ K4 ) @ one_one_Code_integer )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S2 )
         => ( ( groups3674199335183972705nteger
              @ ^ [K4: code_integer] : ( if_Code_integer @ ( A = K4 ) @ ( B @ K4 ) @ one_one_Code_integer )
              @ S2 )
            = one_one_Code_integer ) ) ) ) ).

% prod.delta'
thf(fact_5509_prod_Odelta_H,axiom,
    ! [S2: set_o,A: $o,B: $o > assn] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups5301882518646026715o_assn
              @ ^ [K4: $o] : ( if_assn @ ( A = K4 ) @ ( B @ K4 ) @ one_one_assn )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups5301882518646026715o_assn
              @ ^ [K4: $o] : ( if_assn @ ( A = K4 ) @ ( B @ K4 ) @ one_one_assn )
              @ S2 )
            = one_one_assn ) ) ) ) ).

% prod.delta'
thf(fact_5510_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: nat > assn] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups6906906614972039071t_assn
              @ ^ [K4: nat] : ( if_assn @ ( A = K4 ) @ ( B @ K4 ) @ one_one_assn )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups6906906614972039071t_assn
              @ ^ [K4: nat] : ( if_assn @ ( A = K4 ) @ ( B @ K4 ) @ one_one_assn )
              @ S2 )
            = one_one_assn ) ) ) ) ).

% prod.delta'
thf(fact_5511_prod_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: int > assn] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups7882442080178216443t_assn
              @ ^ [K4: int] : ( if_assn @ ( A = K4 ) @ ( B @ K4 ) @ one_one_assn )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups7882442080178216443t_assn
              @ ^ [K4: int] : ( if_assn @ ( A = K4 ) @ ( B @ K4 ) @ one_one_assn )
              @ S2 )
            = one_one_assn ) ) ) ) ).

% prod.delta'
thf(fact_5512_prod_Odelta_H,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( ( member_Code_integer @ A @ S2 )
         => ( ( groups1304777262505850412r_assn
              @ ^ [K4: code_integer] : ( if_assn @ ( A = K4 ) @ ( B @ K4 ) @ one_one_assn )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_Code_integer @ A @ S2 )
         => ( ( groups1304777262505850412r_assn
              @ ^ [K4: code_integer] : ( if_assn @ ( A = K4 ) @ ( B @ K4 ) @ one_one_assn )
              @ S2 )
            = one_one_assn ) ) ) ) ).

% prod.delta'
thf(fact_5513_prod_Odelta_H,axiom,
    ! [S2: set_o,A: $o,B: $o > rat] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups2869687844427037835_o_rat
              @ ^ [K4: $o] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups2869687844427037835_o_rat
              @ ^ [K4: $o] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = one_one_rat ) ) ) ) ).

% prod.delta'
thf(fact_5514_prod_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K4: nat] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K4: nat] : ( if_rat @ ( A = K4 ) @ ( B @ K4 ) @ one_one_rat )
              @ S2 )
            = one_one_rat ) ) ) ) ).

% prod.delta'
thf(fact_5515_these__insert__Some,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_op989438649584373901at_nat] :
      ( ( these_1872268174106123684at_nat @ ( insert6833468608972679677at_nat @ ( some_P131194445498448716at_nat @ X ) @ A2 ) )
      = ( insert9069300056098147895at_nat @ X @ ( these_1872268174106123684at_nat @ A2 ) ) ) ).

% these_insert_Some
thf(fact_5516_these__insert__Some,axiom,
    ! [X: product_prod_nat_nat,A2: set_op4508134149509766951at_nat] :
      ( ( these_8061402112538453438at_nat @ ( insert6842972033726296599at_nat @ ( some_P7363390416028606310at_nat @ X ) @ A2 ) )
      = ( insert8211810215607154385at_nat @ X @ ( these_8061402112538453438at_nat @ A2 ) ) ) ).

% these_insert_Some
thf(fact_5517_these__insert__Some,axiom,
    ! [X: $o,A2: set_option_o] :
      ( ( these_o @ ( insert_option_o @ ( some_o @ X ) @ A2 ) )
      = ( insert_o @ X @ ( these_o @ A2 ) ) ) ).

% these_insert_Some
thf(fact_5518_these__insert__Some,axiom,
    ! [X: nat,A2: set_option_nat] :
      ( ( these_nat @ ( insert_option_nat @ ( some_nat @ X ) @ A2 ) )
      = ( insert_nat @ X @ ( these_nat @ A2 ) ) ) ).

% these_insert_Some
thf(fact_5519_these__insert__Some,axiom,
    ! [X: int,A2: set_option_int] :
      ( ( these_int @ ( insert_option_int @ ( some_int @ X ) @ A2 ) )
      = ( insert_int @ X @ ( these_int @ A2 ) ) ) ).

% these_insert_Some
thf(fact_5520_these__insert__Some,axiom,
    ! [X: num,A2: set_option_num] :
      ( ( these_num @ ( insert_option_num @ ( some_num @ X ) @ A2 ) )
      = ( insert_num @ X @ ( these_num @ A2 ) ) ) ).

% these_insert_Some
thf(fact_5521_these__insert__Some,axiom,
    ! [X: produc8664842809031399944it_nat,A2: set_op239864471688321678it_nat] :
      ( ( these_3407724651694999947it_nat @ ( insert2007652293347934600it_nat @ ( some_P1914260805536162275it_nat @ X ) @ A2 ) )
      = ( insert5940566491043231800it_nat @ X @ ( these_3407724651694999947it_nat @ A2 ) ) ) ).

% these_insert_Some
thf(fact_5522_these__image__Some__eq,axiom,
    ! [A2: set_num] :
      ( ( these_num @ ( image_num_option_num @ some_num @ A2 ) )
      = A2 ) ).

% these_image_Some_eq
thf(fact_5523_these__image__Some__eq,axiom,
    ! [A2: set_Pr1354866905816374718it_nat] :
      ( ( these_3407724651694999947it_nat @ ( image_3455408117458022631it_nat @ some_P1914260805536162275it_nat @ A2 ) )
      = A2 ) ).

% these_image_Some_eq
thf(fact_5524_sum_Oinsert,axiom,
    ! [A2: set_o,X: $o,G: $o > rat] :
      ( ( finite_finite_o @ A2 )
     => ( ~ ( member_o @ X @ A2 )
       => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5525_sum_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5526_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5527_sum_Oinsert,axiom,
    ! [A2: set_Code_integer,X: code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ~ ( member_Code_integer @ X @ A2 )
       => ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5528_sum_Oinsert,axiom,
    ! [A2: set_o,X: $o,G: $o > nat] :
      ( ( finite_finite_o @ A2 )
     => ( ~ ( member_o @ X @ A2 )
       => ( ( groups8507830703676809646_o_nat @ G @ ( insert_o @ X @ A2 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups8507830703676809646_o_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5529_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A2 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5530_sum_Oinsert,axiom,
    ! [A2: set_Code_integer,X: code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ~ ( member_Code_integer @ X @ A2 )
       => ( ( groups7237345082560585321er_nat @ G @ ( insert_Code_integer @ X @ A2 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups7237345082560585321er_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5531_sum_Oinsert,axiom,
    ! [A2: set_o,X: $o,G: $o > int] :
      ( ( finite_finite_o @ A2 )
     => ( ~ ( member_o @ X @ A2 )
       => ( ( groups8505340233167759370_o_int @ G @ ( insert_o @ X @ A2 ) )
          = ( plus_plus_int @ ( G @ X ) @ ( groups8505340233167759370_o_int @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5532_sum_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( groups3539618377306564664at_int @ G @ ( insert_nat @ X @ A2 ) )
          = ( plus_plus_int @ ( G @ X ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5533_sum_Oinsert,axiom,
    ! [A2: set_Code_integer,X: code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ~ ( member_Code_integer @ X @ A2 )
       => ( ( groups7234854612051535045er_int @ G @ ( insert_Code_integer @ X @ A2 ) )
          = ( plus_plus_int @ ( G @ X ) @ ( groups7234854612051535045er_int @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_5534_atLeastLessThan__singleton,axiom,
    ! [M2: nat] :
      ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ M2 ) )
      = ( insert_nat @ M2 @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_5535_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_5536_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).

% card_atLeastLessThan_int
thf(fact_5537_card__doubleton__eq__2__iff,axiom,
    ! [A: product_unit,B: product_unit] :
      ( ( ( finite410649719033368117t_unit @ ( insert_Product_unit @ A @ ( insert_Product_unit @ B @ bot_bo3957492148770167129t_unit ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( A != B ) ) ).

% card_doubleton_eq_2_iff
thf(fact_5538_card__doubleton__eq__2__iff,axiom,
    ! [A: list_nat,B: list_nat] :
      ( ( ( finite_card_list_nat @ ( insert_list_nat @ A @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( A != B ) ) ).

% card_doubleton_eq_2_iff
thf(fact_5539_card__doubleton__eq__2__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( ( finite3771342082235030671at_nat @ ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( A != B ) ) ).

% card_doubleton_eq_2_iff
thf(fact_5540_card__doubleton__eq__2__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( A != B ) ) ).

% card_doubleton_eq_2_iff
thf(fact_5541_card__doubleton__eq__2__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( ( finite_card_o @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( A != B ) ) ).

% card_doubleton_eq_2_iff
thf(fact_5542_card__doubleton__eq__2__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( finite_card_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( A != B ) ) ).

% card_doubleton_eq_2_iff
thf(fact_5543_card__doubleton__eq__2__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( finite_card_int @ ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( A != B ) ) ).

% card_doubleton_eq_2_iff
thf(fact_5544_atMost__eq__UNIV__iff,axiom,
    ! [X: set_nat] :
      ( ( ( set_or4236626031148496127et_nat @ X )
        = top_top_set_set_nat )
      = ( X = top_top_set_nat ) ) ).

% atMost_eq_UNIV_iff
thf(fact_5545_atMost__eq__UNIV__iff,axiom,
    ! [X: set_Product_unit] :
      ( ( ( set_or2827140217781692084t_unit @ X )
        = top_to1767297665138865437t_unit )
      = ( X = top_to1996260823553986621t_unit ) ) ).

% atMost_eq_UNIV_iff
thf(fact_5546_atMost__eq__UNIV__iff,axiom,
    ! [X: set_int] :
      ( ( ( set_or58775011639299419et_int @ X )
        = top_top_set_set_int )
      = ( X = top_top_set_int ) ) ).

% atMost_eq_UNIV_iff
thf(fact_5547_atMost__eq__UNIV__iff,axiom,
    ! [X: assn] :
      ( ( ( set_ord_atMost_assn @ X )
        = top_top_set_assn )
      = ( X = top_top_assn ) ) ).

% atMost_eq_UNIV_iff
thf(fact_5548_atMost__eq__UNIV__iff,axiom,
    ! [X: product_unit] :
      ( ( ( set_or8621721900777396692t_unit @ X )
        = top_to1996260823553986621t_unit )
      = ( X = top_top_Product_unit ) ) ).

% atMost_eq_UNIV_iff
thf(fact_5549_card__le__if__inj__on__rel,axiom,
    ! [B5: set_o,A2: set_o,R2: $o > $o > $o] :
      ( ( finite_finite_o @ B5 )
     => ( ! [A3: $o] :
            ( ( member_o @ A3 @ A2 )
           => ? [B7: $o] :
                ( ( member_o @ B7 @ B5 )
                & ( R2 @ A3 @ B7 ) ) )
       => ( ! [A13: $o,A24: $o,B2: $o] :
              ( ( member_o @ A13 @ A2 )
             => ( ( member_o @ A24 @ A2 )
               => ( ( member_o @ B2 @ B5 )
                 => ( ( R2 @ A13 @ B2 )
                   => ( ( R2 @ A24 @ B2 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_o @ A2 ) @ ( finite_card_o @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_5550_card__le__if__inj__on__rel,axiom,
    ! [B5: set_Product_unit,A2: set_o,R2: $o > product_unit > $o] :
      ( ( finite4290736615968046902t_unit @ B5 )
     => ( ! [A3: $o] :
            ( ( member_o @ A3 @ A2 )
           => ? [B7: product_unit] :
                ( ( member_Product_unit @ B7 @ B5 )
                & ( R2 @ A3 @ B7 ) ) )
       => ( ! [A13: $o,A24: $o,B2: product_unit] :
              ( ( member_o @ A13 @ A2 )
             => ( ( member_o @ A24 @ A2 )
               => ( ( member_Product_unit @ B2 @ B5 )
                 => ( ( R2 @ A13 @ B2 )
                   => ( ( R2 @ A24 @ B2 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_o @ A2 ) @ ( finite410649719033368117t_unit @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_5551_card__le__if__inj__on__rel,axiom,
    ! [B5: set_o,A2: set_int,R2: int > $o > $o] :
      ( ( finite_finite_o @ B5 )
     => ( ! [A3: int] :
            ( ( member_int @ A3 @ A2 )
           => ? [B7: $o] :
                ( ( member_o @ B7 @ B5 )
                & ( R2 @ A3 @ B7 ) ) )
       => ( ! [A13: int,A24: int,B2: $o] :
              ( ( member_int @ A13 @ A2 )
             => ( ( member_int @ A24 @ A2 )
               => ( ( member_o @ B2 @ B5 )
                 => ( ( R2 @ A13 @ B2 )
                   => ( ( R2 @ A24 @ B2 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ ( finite_card_o @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_5552_card__le__if__inj__on__rel,axiom,
    ! [B5: set_Product_unit,A2: set_int,R2: int > product_unit > $o] :
      ( ( finite4290736615968046902t_unit @ B5 )
     => ( ! [A3: int] :
            ( ( member_int @ A3 @ A2 )
           => ? [B7: product_unit] :
                ( ( member_Product_unit @ B7 @ B5 )
                & ( R2 @ A3 @ B7 ) ) )
       => ( ! [A13: int,A24: int,B2: product_unit] :
              ( ( member_int @ A13 @ A2 )
             => ( ( member_int @ A24 @ A2 )
               => ( ( member_Product_unit @ B2 @ B5 )
                 => ( ( R2 @ A13 @ B2 )
                   => ( ( R2 @ A24 @ B2 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ ( finite410649719033368117t_unit @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_5553_card__le__if__inj__on__rel,axiom,
    ! [B5: set_o,A2: set_Product_unit,R2: product_unit > $o > $o] :
      ( ( finite_finite_o @ B5 )
     => ( ! [A3: product_unit] :
            ( ( member_Product_unit @ A3 @ A2 )
           => ? [B7: $o] :
                ( ( member_o @ B7 @ B5 )
                & ( R2 @ A3 @ B7 ) ) )
       => ( ! [A13: product_unit,A24: product_unit,B2: $o] :
              ( ( member_Product_unit @ A13 @ A2 )
             => ( ( member_Product_unit @ A24 @ A2 )
               => ( ( member_o @ B2 @ B5 )
                 => ( ( R2 @ A13 @ B2 )
                   => ( ( R2 @ A24 @ B2 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A2 ) @ ( finite_card_o @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_5554_card__le__if__inj__on__rel,axiom,
    ! [B5: set_Product_unit,A2: set_Product_unit,R2: product_unit > product_unit > $o] :
      ( ( finite4290736615968046902t_unit @ B5 )
     => ( ! [A3: product_unit] :
            ( ( member_Product_unit @ A3 @ A2 )
           => ? [B7: product_unit] :
                ( ( member_Product_unit @ B7 @ B5 )
                & ( R2 @ A3 @ B7 ) ) )
       => ( ! [A13: product_unit,A24: product_unit,B2: product_unit] :
              ( ( member_Product_unit @ A13 @ A2 )
             => ( ( member_Product_unit @ A24 @ A2 )
               => ( ( member_Product_unit @ B2 @ B5 )
                 => ( ( R2 @ A13 @ B2 )
                   => ( ( R2 @ A24 @ B2 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A2 ) @ ( finite410649719033368117t_unit @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_5555_card__le__if__inj__on__rel,axiom,
    ! [B5: set_o,A2: set_nat,R2: nat > $o > $o] :
      ( ( finite_finite_o @ B5 )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A2 )
           => ? [B7: $o] :
                ( ( member_o @ B7 @ B5 )
                & ( R2 @ A3 @ B7 ) ) )
       => ( ! [A13: nat,A24: nat,B2: $o] :
              ( ( member_nat @ A13 @ A2 )
             => ( ( member_nat @ A24 @ A2 )
               => ( ( member_o @ B2 @ B5 )
                 => ( ( R2 @ A13 @ B2 )
                   => ( ( R2 @ A24 @ B2 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_o @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_5556_card__le__if__inj__on__rel,axiom,
    ! [B5: set_Product_unit,A2: set_nat,R2: nat > product_unit > $o] :
      ( ( finite4290736615968046902t_unit @ B5 )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A2 )
           => ? [B7: product_unit] :
                ( ( member_Product_unit @ B7 @ B5 )
                & ( R2 @ A3 @ B7 ) ) )
       => ( ! [A13: nat,A24: nat,B2: product_unit] :
              ( ( member_nat @ A13 @ A2 )
             => ( ( member_nat @ A24 @ A2 )
               => ( ( member_Product_unit @ B2 @ B5 )
                 => ( ( R2 @ A13 @ B2 )
                   => ( ( R2 @ A24 @ B2 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite410649719033368117t_unit @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_5557_card__le__if__inj__on__rel,axiom,
    ! [B5: set_nat,A2: set_o,R2: $o > nat > $o] :
      ( ( finite_finite_nat @ B5 )
     => ( ! [A3: $o] :
            ( ( member_o @ A3 @ A2 )
           => ? [B7: nat] :
                ( ( member_nat @ B7 @ B5 )
                & ( R2 @ A3 @ B7 ) ) )
       => ( ! [A13: $o,A24: $o,B2: nat] :
              ( ( member_o @ A13 @ A2 )
             => ( ( member_o @ A24 @ A2 )
               => ( ( member_nat @ B2 @ B5 )
                 => ( ( R2 @ A13 @ B2 )
                   => ( ( R2 @ A24 @ B2 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_o @ A2 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_5558_card__le__if__inj__on__rel,axiom,
    ! [B5: set_nat,A2: set_int,R2: int > nat > $o] :
      ( ( finite_finite_nat @ B5 )
     => ( ! [A3: int] :
            ( ( member_int @ A3 @ A2 )
           => ? [B7: nat] :
                ( ( member_nat @ B7 @ B5 )
                & ( R2 @ A3 @ B7 ) ) )
       => ( ! [A13: int,A24: int,B2: nat] :
              ( ( member_int @ A13 @ A2 )
             => ( ( member_int @ A24 @ A2 )
               => ( ( member_nat @ B2 @ B5 )
                 => ( ( R2 @ A13 @ B2 )
                   => ( ( R2 @ A24 @ B2 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_5559_top__option__def,axiom,
    ( top_to4826455019444611206et_nat
    = ( some_set_nat @ top_top_set_nat ) ) ).

% top_option_def
thf(fact_5560_top__option__def,axiom,
    ( top_to7662761140297458691t_unit
    = ( some_s8043159101241848530t_unit @ top_to1996260823553986621t_unit ) ) ).

% top_option_def
thf(fact_5561_top__option__def,axiom,
    ( top_to2336371874512884450et_int
    = ( some_set_int @ top_top_set_int ) ) ).

% top_option_def
thf(fact_5562_top__option__def,axiom,
    ( top_to4322782652514146083t_unit
    = ( some_Product_unit @ top_top_Product_unit ) ) ).

% top_option_def
thf(fact_5563_top__option__def,axiom,
    ( top_top_option_assn
    = ( some_assn @ top_top_assn ) ) ).

% top_option_def
thf(fact_5564_finite__if__eq__beyond__finite,axiom,
    ! [S2: set_int,S3: set_int] :
      ( ( finite_finite_int @ S2 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [S4: set_int] :
              ( ( minus_minus_set_int @ S4 @ S2 )
              = ( minus_minus_set_int @ S3 @ S2 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_5565_finite__if__eq__beyond__finite,axiom,
    ! [S2: set_Code_integer,S3: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( finite6931041176100689706nteger
        @ ( collec574505750873337192nteger
          @ ^ [S4: set_Code_integer] :
              ( ( minus_2355218937544613996nteger @ S4 @ S2 )
              = ( minus_2355218937544613996nteger @ S3 @ S2 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_5566_finite__if__eq__beyond__finite,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ S2 )
     => ( finite9047747110432174090at_nat
        @ ( collec5514110066124741708at_nat
          @ ^ [S4: set_Pr1261947904930325089at_nat] :
              ( ( minus_1356011639430497352at_nat @ S4 @ S2 )
              = ( minus_1356011639430497352at_nat @ S3 @ S2 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_5567_finite__if__eq__beyond__finite,axiom,
    ! [S2: set_nat,S3: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [S4: set_nat] :
              ( ( minus_minus_set_nat @ S4 @ S2 )
              = ( minus_minus_set_nat @ S3 @ S2 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_5568_finite__range__Some,axiom,
    ( ( finite6785661671136154180nteger @ ( image_8960079959263999889nteger @ some_Code_integer @ top_to4645266643341252675nteger ) )
    = ( finite6017078050557962740nteger @ top_to4645266643341252675nteger ) ) ).

% finite_range_Some
thf(fact_5569_finite__range__Some,axiom,
    ( ( finite6732403688824079472at_nat @ ( image_4198897800814241419at_nat @ some_P7363390416028606310at_nat @ top_to4669805908274784177at_nat ) )
    = ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ).

% finite_range_Some
thf(fact_5570_finite__range__Some,axiom,
    ( ( finite6231397076622260609on_num @ ( image_num_option_num @ some_num @ top_top_set_num ) )
    = ( finite_finite_num @ top_top_set_num ) ) ).

% finite_range_Some
thf(fact_5571_finite__range__Some,axiom,
    ( ( finite6292238467249657455it_nat @ ( image_3455408117458022631it_nat @ some_P1914260805536162275it_nat @ top_to4082048134677576814it_nat ) )
    = ( finite6446286893012251551it_nat @ top_to4082048134677576814it_nat ) ) ).

% finite_range_Some
thf(fact_5572_finite__range__Some,axiom,
    ( ( finite5523153139673422903on_nat @ ( image_nat_option_nat @ some_nat @ top_top_set_nat ) )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% finite_range_Some
thf(fact_5573_finite__range__Some,axiom,
    ( ( finite1445617369574913404t_unit @ ( image_7094380055901048587t_unit @ some_Product_unit @ top_to1996260823553986621t_unit ) )
    = ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).

% finite_range_Some
thf(fact_5574_finite__range__Some,axiom,
    ( ( finite1345302120164226195on_int @ ( image_int_option_int @ some_int @ top_top_set_int ) )
    = ( finite_finite_int @ top_top_set_int ) ) ).

% finite_range_Some
thf(fact_5575_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S2: set_Product_unit] :
      ( ( ord_less_eq_nat @ N2 @ ( finite410649719033368117t_unit @ S2 ) )
     => ~ ! [T3: set_Product_unit] :
            ( ( ord_le3507040750410214029t_unit @ T3 @ S2 )
           => ( ( ( finite410649719033368117t_unit @ T3 )
                = N2 )
             => ~ ( finite4290736615968046902t_unit @ T3 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_5576_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S2: set_list_nat] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_list_nat @ S2 ) )
     => ~ ! [T3: set_list_nat] :
            ( ( ord_le6045566169113846134st_nat @ T3 @ S2 )
           => ( ( ( finite_card_list_nat @ T3 )
                = N2 )
             => ~ ( finite8100373058378681591st_nat @ T3 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_5577_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S2: set_int] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_int @ S2 ) )
     => ~ ! [T3: set_int] :
            ( ( ord_less_eq_set_int @ T3 @ S2 )
           => ( ( ( finite_card_int @ T3 )
                = N2 )
             => ~ ( finite_finite_int @ T3 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_5578_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S2: set_Code_integer] :
      ( ( ord_less_eq_nat @ N2 @ ( finite4902975817058060853nteger @ S2 ) )
     => ~ ! [T3: set_Code_integer] :
            ( ( ord_le7084787975880047091nteger @ T3 @ S2 )
           => ( ( ( finite4902975817058060853nteger @ T3 )
                = N2 )
             => ~ ( finite6017078050557962740nteger @ T3 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_5579_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ N2 @ ( finite711546835091564841at_nat @ S2 ) )
     => ~ ! [T3: set_Pr1261947904930325089at_nat] :
            ( ( ord_le3146513528884898305at_nat @ T3 @ S2 )
           => ( ( ( finite711546835091564841at_nat @ T3 )
                = N2 )
             => ~ ( finite6177210948735845034at_nat @ T3 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_5580_obtain__subset__with__card__n,axiom,
    ! [N2: nat,S2: set_nat] :
      ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S2 ) )
     => ~ ! [T3: set_nat] :
            ( ( ord_less_eq_set_nat @ T3 @ S2 )
           => ( ( ( finite_card_nat @ T3 )
                = N2 )
             => ~ ( finite_finite_nat @ T3 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_5581_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( ( set_or4049648398082403068at_nat @ X @ Y )
        = top_to1754375979021635447at_nat )
      = ( ( X = bot_bo228742789529271731at_nat )
        & ( Y = top_to6833984726390702231at_nat ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
thf(fact_5582_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( set_or2784603332384507286at_nat @ X @ Y )
        = top_to7629004291339433233at_nat )
      = ( ( X = bot_bo2099793752762293965at_nat )
        & ( Y = top_to4669805908274784177at_nat ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
thf(fact_5583_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( ( set_or266616469461849773_set_o @ X @ Y )
        = top_top_set_set_o )
      = ( ( X = bot_bot_set_o )
        & ( Y = top_top_set_o ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
thf(fact_5584_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ X @ Y )
        = top_top_set_set_nat )
      = ( ( X = bot_bot_set_nat )
        & ( Y = top_top_set_nat ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
thf(fact_5585_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] :
      ( ( ( set_or269330187037242274t_unit @ X @ Y )
        = top_to1767297665138865437t_unit )
      = ( ( X = bot_bo3957492148770167129t_unit )
        & ( Y = top_to1996260823553986621t_unit ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
thf(fact_5586_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( set_or370866239135849197et_int @ X @ Y )
        = top_top_set_set_int )
      = ( ( X = bot_bot_set_int )
        & ( Y = top_top_set_int ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
thf(fact_5587_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( set_or7959216805967363635t_assn @ X @ Y )
        = top_top_set_assn )
      = ( ( X = bot_bot_assn )
        & ( Y = top_top_assn ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
thf(fact_5588_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( ( set_or5692540242118139586t_unit @ X @ Y )
        = top_to1996260823553986621t_unit )
      = ( ( X = bot_bot_Product_unit )
        & ( Y = top_top_Product_unit ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
thf(fact_5589_UNIV__option__conv,axiom,
    ( top_to7137183367772546234it_nat
    = ( insert7468008629046257882it_nat @ none_P2587165478634041381it_nat @ ( image_4799632500972785611it_nat @ some_P6105628898608307625it_nat @ top_to3995162779041354868it_nat ) ) ) ).

% UNIV_option_conv
thf(fact_5590_UNIV__option__conv,axiom,
    ( top_to4635417975585700688on_num
    = ( insert_option_num @ none_num @ ( image_num_option_num @ some_num @ top_top_set_num ) ) ) ).

% UNIV_option_conv
thf(fact_5591_UNIV__option__conv,axiom,
    ( top_to6632602878615679294it_nat
    = ( insert2007652293347934600it_nat @ none_P9117596204409417319it_nat @ ( image_3455408117458022631it_nat @ some_P1914260805536162275it_nat @ top_to4082048134677576814it_nat ) ) ) ).

% UNIV_option_conv
thf(fact_5592_UNIV__option__conv,axiom,
    ( top_to8920198386146353926on_nat
    = ( insert_option_nat @ none_nat @ ( image_nat_option_nat @ some_nat @ top_top_set_nat ) ) ) ).

% UNIV_option_conv
thf(fact_5593_UNIV__option__conv,axiom,
    ( top_to2690860209552263555t_unit
    = ( insert2866144167138965411t_unit @ none_Product_unit @ ( image_7094380055901048587t_unit @ some_Product_unit @ top_to1996260823553986621t_unit ) ) ) ).

% UNIV_option_conv
thf(fact_5594_UNIV__option__conv,axiom,
    ( top_to6430115241214627170on_int
    = ( insert_option_int @ none_int @ ( image_int_option_int @ some_int @ top_top_set_int ) ) ) ).

% UNIV_option_conv
thf(fact_5595_bounded__nat__set__is__finite,axiom,
    ! [N6: set_nat,N2: nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ N6 )
         => ( ord_less_nat @ X4 @ N2 ) )
     => ( finite_finite_nat @ N6 ) ) ).

% bounded_nat_set_is_finite
thf(fact_5596_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N9: set_nat] :
        ? [M: nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ N9 )
         => ( ord_less_nat @ X3 @ M ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_5597_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N9: set_nat] :
        ? [M: nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ N9 )
         => ( ord_less_eq_nat @ X3 @ M ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_5598_surjD,axiom,
    ! [F: nat > code_integer,Y: code_integer] :
      ( ( ( image_1215581382706833972nteger @ F @ top_top_set_nat )
        = top_to4645266643341252675nteger )
     => ? [X4: nat] :
          ( Y
          = ( F @ X4 ) ) ) ).

% surjD
thf(fact_5599_surjD,axiom,
    ! [F: nat > nat,Y: nat] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ? [X4: nat] :
          ( Y
          = ( F @ X4 ) ) ) ).

% surjD
thf(fact_5600_surjD,axiom,
    ! [F: nat > product_unit,Y: product_unit] :
      ( ( ( image_8730104196221521654t_unit @ F @ top_top_set_nat )
        = top_to1996260823553986621t_unit )
     => ? [X4: nat] :
          ( Y
          = ( F @ X4 ) ) ) ).

% surjD
thf(fact_5601_surjD,axiom,
    ! [F: nat > int,Y: int] :
      ( ( ( image_nat_int @ F @ top_top_set_nat )
        = top_top_set_int )
     => ? [X4: nat] :
          ( Y
          = ( F @ X4 ) ) ) ).

% surjD
thf(fact_5602_surjD,axiom,
    ! [F: product_unit > nat,Y: nat] :
      ( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
        = top_top_set_nat )
     => ? [X4: product_unit] :
          ( Y
          = ( F @ X4 ) ) ) ).

% surjD
thf(fact_5603_surjD,axiom,
    ! [F: product_unit > product_unit,Y: product_unit] :
      ( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
        = top_to1996260823553986621t_unit )
     => ? [X4: product_unit] :
          ( Y
          = ( F @ X4 ) ) ) ).

% surjD
thf(fact_5604_surjD,axiom,
    ! [F: product_unit > int,Y: int] :
      ( ( ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit )
        = top_top_set_int )
     => ? [X4: product_unit] :
          ( Y
          = ( F @ X4 ) ) ) ).

% surjD
thf(fact_5605_surjD,axiom,
    ! [F: int > nat,Y: nat] :
      ( ( ( image_int_nat @ F @ top_top_set_int )
        = top_top_set_nat )
     => ? [X4: int] :
          ( Y
          = ( F @ X4 ) ) ) ).

% surjD
thf(fact_5606_surjD,axiom,
    ! [F: int > product_unit,Y: product_unit] :
      ( ( ( image_5846526143423701914t_unit @ F @ top_top_set_int )
        = top_to1996260823553986621t_unit )
     => ? [X4: int] :
          ( Y
          = ( F @ X4 ) ) ) ).

% surjD
thf(fact_5607_surjD,axiom,
    ! [F: int > int,Y: int] :
      ( ( ( image_int_int @ F @ top_top_set_int )
        = top_top_set_int )
     => ? [X4: int] :
          ( Y
          = ( F @ X4 ) ) ) ).

% surjD
thf(fact_5608_surjE,axiom,
    ! [F: nat > code_integer,Y: code_integer] :
      ( ( ( image_1215581382706833972nteger @ F @ top_top_set_nat )
        = top_to4645266643341252675nteger )
     => ~ ! [X4: nat] :
            ( Y
           != ( F @ X4 ) ) ) ).

% surjE
thf(fact_5609_surjE,axiom,
    ! [F: nat > nat,Y: nat] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ~ ! [X4: nat] :
            ( Y
           != ( F @ X4 ) ) ) ).

% surjE
thf(fact_5610_surjE,axiom,
    ! [F: nat > product_unit,Y: product_unit] :
      ( ( ( image_8730104196221521654t_unit @ F @ top_top_set_nat )
        = top_to1996260823553986621t_unit )
     => ~ ! [X4: nat] :
            ( Y
           != ( F @ X4 ) ) ) ).

% surjE
thf(fact_5611_surjE,axiom,
    ! [F: nat > int,Y: int] :
      ( ( ( image_nat_int @ F @ top_top_set_nat )
        = top_top_set_int )
     => ~ ! [X4: nat] :
            ( Y
           != ( F @ X4 ) ) ) ).

% surjE
thf(fact_5612_surjE,axiom,
    ! [F: product_unit > nat,Y: nat] :
      ( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
        = top_top_set_nat )
     => ~ ! [X4: product_unit] :
            ( Y
           != ( F @ X4 ) ) ) ).

% surjE
thf(fact_5613_surjE,axiom,
    ! [F: product_unit > product_unit,Y: product_unit] :
      ( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
        = top_to1996260823553986621t_unit )
     => ~ ! [X4: product_unit] :
            ( Y
           != ( F @ X4 ) ) ) ).

% surjE
thf(fact_5614_surjE,axiom,
    ! [F: product_unit > int,Y: int] :
      ( ( ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit )
        = top_top_set_int )
     => ~ ! [X4: product_unit] :
            ( Y
           != ( F @ X4 ) ) ) ).

% surjE
thf(fact_5615_surjE,axiom,
    ! [F: int > nat,Y: nat] :
      ( ( ( image_int_nat @ F @ top_top_set_int )
        = top_top_set_nat )
     => ~ ! [X4: int] :
            ( Y
           != ( F @ X4 ) ) ) ).

% surjE
thf(fact_5616_surjE,axiom,
    ! [F: int > product_unit,Y: product_unit] :
      ( ( ( image_5846526143423701914t_unit @ F @ top_top_set_int )
        = top_to1996260823553986621t_unit )
     => ~ ! [X4: int] :
            ( Y
           != ( F @ X4 ) ) ) ).

% surjE
thf(fact_5617_surjE,axiom,
    ! [F: int > int,Y: int] :
      ( ( ( image_int_int @ F @ top_top_set_int )
        = top_top_set_int )
     => ~ ! [X4: int] :
            ( Y
           != ( F @ X4 ) ) ) ).

% surjE
thf(fact_5618_surjI,axiom,
    ! [G: nat > code_integer,F: code_integer > nat] :
      ( ! [X4: code_integer] :
          ( ( G @ ( F @ X4 ) )
          = X4 )
     => ( ( image_1215581382706833972nteger @ G @ top_top_set_nat )
        = top_to4645266643341252675nteger ) ) ).

% surjI
thf(fact_5619_surjI,axiom,
    ! [G: nat > nat,F: nat > nat] :
      ( ! [X4: nat] :
          ( ( G @ ( F @ X4 ) )
          = X4 )
     => ( ( image_nat_nat @ G @ top_top_set_nat )
        = top_top_set_nat ) ) ).

% surjI
thf(fact_5620_surjI,axiom,
    ! [G: nat > product_unit,F: product_unit > nat] :
      ( ! [X4: product_unit] :
          ( ( G @ ( F @ X4 ) )
          = X4 )
     => ( ( image_8730104196221521654t_unit @ G @ top_top_set_nat )
        = top_to1996260823553986621t_unit ) ) ).

% surjI
thf(fact_5621_surjI,axiom,
    ! [G: nat > int,F: int > nat] :
      ( ! [X4: int] :
          ( ( G @ ( F @ X4 ) )
          = X4 )
     => ( ( image_nat_int @ G @ top_top_set_nat )
        = top_top_set_int ) ) ).

% surjI
thf(fact_5622_surjI,axiom,
    ! [G: product_unit > nat,F: nat > product_unit] :
      ( ! [X4: nat] :
          ( ( G @ ( F @ X4 ) )
          = X4 )
     => ( ( image_875570014554754200it_nat @ G @ top_to1996260823553986621t_unit )
        = top_top_set_nat ) ) ).

% surjI
thf(fact_5623_surjI,axiom,
    ! [G: product_unit > product_unit,F: product_unit > product_unit] :
      ( ! [X4: product_unit] :
          ( ( G @ ( F @ X4 ) )
          = X4 )
     => ( ( image_405062704495631173t_unit @ G @ top_to1996260823553986621t_unit )
        = top_to1996260823553986621t_unit ) ) ).

% surjI
thf(fact_5624_surjI,axiom,
    ! [G: product_unit > int,F: int > product_unit] :
      ( ! [X4: int] :
          ( ( G @ ( F @ X4 ) )
          = X4 )
     => ( ( image_873079544045703924it_int @ G @ top_to1996260823553986621t_unit )
        = top_top_set_int ) ) ).

% surjI
thf(fact_5625_surjI,axiom,
    ! [G: int > nat,F: nat > int] :
      ( ! [X4: nat] :
          ( ( G @ ( F @ X4 ) )
          = X4 )
     => ( ( image_int_nat @ G @ top_top_set_int )
        = top_top_set_nat ) ) ).

% surjI
thf(fact_5626_surjI,axiom,
    ! [G: int > product_unit,F: product_unit > int] :
      ( ! [X4: product_unit] :
          ( ( G @ ( F @ X4 ) )
          = X4 )
     => ( ( image_5846526143423701914t_unit @ G @ top_top_set_int )
        = top_to1996260823553986621t_unit ) ) ).

% surjI
thf(fact_5627_surjI,axiom,
    ! [G: int > int,F: int > int] :
      ( ! [X4: int] :
          ( ( G @ ( F @ X4 ) )
          = X4 )
     => ( ( image_int_int @ G @ top_top_set_int )
        = top_top_set_int ) ) ).

% surjI
thf(fact_5628_surj__def,axiom,
    ! [F: nat > code_integer] :
      ( ( ( image_1215581382706833972nteger @ F @ top_top_set_nat )
        = top_to4645266643341252675nteger )
      = ( ! [Y3: code_integer] :
          ? [X3: nat] :
            ( Y3
            = ( F @ X3 ) ) ) ) ).

% surj_def
thf(fact_5629_surj__def,axiom,
    ! [F: nat > nat] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = top_top_set_nat )
      = ( ! [Y3: nat] :
          ? [X3: nat] :
            ( Y3
            = ( F @ X3 ) ) ) ) ).

% surj_def
thf(fact_5630_surj__def,axiom,
    ! [F: nat > product_unit] :
      ( ( ( image_8730104196221521654t_unit @ F @ top_top_set_nat )
        = top_to1996260823553986621t_unit )
      = ( ! [Y3: product_unit] :
          ? [X3: nat] :
            ( Y3
            = ( F @ X3 ) ) ) ) ).

% surj_def
thf(fact_5631_surj__def,axiom,
    ! [F: nat > int] :
      ( ( ( image_nat_int @ F @ top_top_set_nat )
        = top_top_set_int )
      = ( ! [Y3: int] :
          ? [X3: nat] :
            ( Y3
            = ( F @ X3 ) ) ) ) ).

% surj_def
thf(fact_5632_surj__def,axiom,
    ! [F: product_unit > nat] :
      ( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
        = top_top_set_nat )
      = ( ! [Y3: nat] :
          ? [X3: product_unit] :
            ( Y3
            = ( F @ X3 ) ) ) ) ).

% surj_def
thf(fact_5633_surj__def,axiom,
    ! [F: product_unit > product_unit] :
      ( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
        = top_to1996260823553986621t_unit )
      = ( ! [Y3: product_unit] :
          ? [X3: product_unit] :
            ( Y3
            = ( F @ X3 ) ) ) ) ).

% surj_def
thf(fact_5634_surj__def,axiom,
    ! [F: product_unit > int] :
      ( ( ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit )
        = top_top_set_int )
      = ( ! [Y3: int] :
          ? [X3: product_unit] :
            ( Y3
            = ( F @ X3 ) ) ) ) ).

% surj_def
thf(fact_5635_surj__def,axiom,
    ! [F: int > nat] :
      ( ( ( image_int_nat @ F @ top_top_set_int )
        = top_top_set_nat )
      = ( ! [Y3: nat] :
          ? [X3: int] :
            ( Y3
            = ( F @ X3 ) ) ) ) ).

% surj_def
thf(fact_5636_surj__def,axiom,
    ! [F: int > product_unit] :
      ( ( ( image_5846526143423701914t_unit @ F @ top_top_set_int )
        = top_to1996260823553986621t_unit )
      = ( ! [Y3: product_unit] :
          ? [X3: int] :
            ( Y3
            = ( F @ X3 ) ) ) ) ).

% surj_def
thf(fact_5637_surj__def,axiom,
    ! [F: int > int] :
      ( ( ( image_int_int @ F @ top_top_set_int )
        = top_top_set_int )
      = ( ! [Y3: int] :
          ? [X3: int] :
            ( Y3
            = ( F @ X3 ) ) ) ) ).

% surj_def
thf(fact_5638_these__empty__eq,axiom,
    ! [B5: set_op3940070208846178154it_nat] :
      ( ( ( these_6645423997822192641it_nat @ B5 )
        = bot_bo6361524548789211024it_nat )
      = ( ( B5 = bot_bo8185009268211221974it_nat )
        | ( B5
          = ( insert7468008629046257882it_nat @ none_P2587165478634041381it_nat @ bot_bo8185009268211221974it_nat ) ) ) ) ).

% these_empty_eq
thf(fact_5639_these__empty__eq,axiom,
    ! [B5: set_option_num] :
      ( ( ( these_num @ B5 )
        = bot_bot_set_num )
      = ( ( B5 = bot_bo725063100934353204on_num )
        | ( B5
          = ( insert_option_num @ none_num @ bot_bo725063100934353204on_num ) ) ) ) ).

% these_empty_eq
thf(fact_5640_these__empty__eq,axiom,
    ! [B5: set_op239864471688321678it_nat] :
      ( ( ( these_3407724651694999947it_nat @ B5 )
        = bot_bo5696401558416705618it_nat )
      = ( ( B5 = bot_bo1172055076661450530it_nat )
        | ( B5
          = ( insert2007652293347934600it_nat @ none_P9117596204409417319it_nat @ bot_bo1172055076661450530it_nat ) ) ) ) ).

% these_empty_eq
thf(fact_5641_these__empty__eq,axiom,
    ! [B5: set_op989438649584373901at_nat] :
      ( ( ( these_1872268174106123684at_nat @ B5 )
        = bot_bo228742789529271731at_nat )
      = ( ( B5 = bot_bo1471150103859235065at_nat )
        | ( B5
          = ( insert6833468608972679677at_nat @ none_P5811896708078067656at_nat @ bot_bo1471150103859235065at_nat ) ) ) ) ).

% these_empty_eq
thf(fact_5642_these__empty__eq,axiom,
    ! [B5: set_op4508134149509766951at_nat] :
      ( ( ( these_8061402112538453438at_nat @ B5 )
        = bot_bo2099793752762293965at_nat )
      = ( ( B5 = bot_bo3414888551220300179at_nat )
        | ( B5
          = ( insert6842972033726296599at_nat @ none_P5556105721700978146at_nat @ bot_bo3414888551220300179at_nat ) ) ) ) ).

% these_empty_eq
thf(fact_5643_these__empty__eq,axiom,
    ! [B5: set_option_o] :
      ( ( ( these_o @ B5 )
        = bot_bot_set_o )
      = ( ( B5 = bot_bot_set_option_o )
        | ( B5
          = ( insert_option_o @ none_o @ bot_bot_set_option_o ) ) ) ) ).

% these_empty_eq
thf(fact_5644_these__empty__eq,axiom,
    ! [B5: set_option_nat] :
      ( ( ( these_nat @ B5 )
        = bot_bot_set_nat )
      = ( ( B5 = bot_bo5009843511495006442on_nat )
        | ( B5
          = ( insert_option_nat @ none_nat @ bot_bo5009843511495006442on_nat ) ) ) ) ).

% these_empty_eq
thf(fact_5645_these__empty__eq,axiom,
    ! [B5: set_option_int] :
      ( ( ( these_int @ B5 )
        = bot_bot_set_int )
      = ( ( B5 = bot_bo2519760366563279686on_int )
        | ( B5
          = ( insert_option_int @ none_int @ bot_bo2519760366563279686on_int ) ) ) ) ).

% these_empty_eq
thf(fact_5646_these__not__empty__eq,axiom,
    ! [B5: set_op3940070208846178154it_nat] :
      ( ( ( these_6645423997822192641it_nat @ B5 )
       != bot_bo6361524548789211024it_nat )
      = ( ( B5 != bot_bo8185009268211221974it_nat )
        & ( B5
         != ( insert7468008629046257882it_nat @ none_P2587165478634041381it_nat @ bot_bo8185009268211221974it_nat ) ) ) ) ).

% these_not_empty_eq
thf(fact_5647_these__not__empty__eq,axiom,
    ! [B5: set_option_num] :
      ( ( ( these_num @ B5 )
       != bot_bot_set_num )
      = ( ( B5 != bot_bo725063100934353204on_num )
        & ( B5
         != ( insert_option_num @ none_num @ bot_bo725063100934353204on_num ) ) ) ) ).

% these_not_empty_eq
thf(fact_5648_these__not__empty__eq,axiom,
    ! [B5: set_op239864471688321678it_nat] :
      ( ( ( these_3407724651694999947it_nat @ B5 )
       != bot_bo5696401558416705618it_nat )
      = ( ( B5 != bot_bo1172055076661450530it_nat )
        & ( B5
         != ( insert2007652293347934600it_nat @ none_P9117596204409417319it_nat @ bot_bo1172055076661450530it_nat ) ) ) ) ).

% these_not_empty_eq
thf(fact_5649_these__not__empty__eq,axiom,
    ! [B5: set_op989438649584373901at_nat] :
      ( ( ( these_1872268174106123684at_nat @ B5 )
       != bot_bo228742789529271731at_nat )
      = ( ( B5 != bot_bo1471150103859235065at_nat )
        & ( B5
         != ( insert6833468608972679677at_nat @ none_P5811896708078067656at_nat @ bot_bo1471150103859235065at_nat ) ) ) ) ).

% these_not_empty_eq
thf(fact_5650_these__not__empty__eq,axiom,
    ! [B5: set_op4508134149509766951at_nat] :
      ( ( ( these_8061402112538453438at_nat @ B5 )
       != bot_bo2099793752762293965at_nat )
      = ( ( B5 != bot_bo3414888551220300179at_nat )
        & ( B5
         != ( insert6842972033726296599at_nat @ none_P5556105721700978146at_nat @ bot_bo3414888551220300179at_nat ) ) ) ) ).

% these_not_empty_eq
thf(fact_5651_these__not__empty__eq,axiom,
    ! [B5: set_option_o] :
      ( ( ( these_o @ B5 )
       != bot_bot_set_o )
      = ( ( B5 != bot_bot_set_option_o )
        & ( B5
         != ( insert_option_o @ none_o @ bot_bot_set_option_o ) ) ) ) ).

% these_not_empty_eq
thf(fact_5652_these__not__empty__eq,axiom,
    ! [B5: set_option_nat] :
      ( ( ( these_nat @ B5 )
       != bot_bot_set_nat )
      = ( ( B5 != bot_bo5009843511495006442on_nat )
        & ( B5
         != ( insert_option_nat @ none_nat @ bot_bo5009843511495006442on_nat ) ) ) ) ).

% these_not_empty_eq
thf(fact_5653_these__not__empty__eq,axiom,
    ! [B5: set_option_int] :
      ( ( ( these_int @ B5 )
       != bot_bot_set_int )
      = ( ( B5 != bot_bo2519760366563279686on_int )
        & ( B5
         != ( insert_option_int @ none_int @ bot_bo2519760366563279686on_int ) ) ) ) ).

% these_not_empty_eq
thf(fact_5654_not__UNIV__eq__Icc,axiom,
    ! [L4: int,H5: int] :
      ( top_top_set_int
     != ( set_or1266510415728281911st_int @ L4 @ H5 ) ) ).

% not_UNIV_eq_Icc
thf(fact_5655_not__UNIV__eq__Icc,axiom,
    ! [L4: nat,H5: nat] :
      ( top_top_set_nat
     != ( set_or1269000886237332187st_nat @ L4 @ H5 ) ) ).

% not_UNIV_eq_Icc
thf(fact_5656_infinite__Iic,axiom,
    ! [A: int] :
      ~ ( finite_finite_int @ ( set_ord_atMost_int @ A ) ) ).

% infinite_Iic
thf(fact_5657_infinite__Iio,axiom,
    ! [A: int] :
      ~ ( finite_finite_int @ ( set_ord_lessThan_int @ A ) ) ).

% infinite_Iio
thf(fact_5658_not__UNIV__eq__Iic,axiom,
    ! [H5: int] :
      ( top_top_set_int
     != ( set_ord_atMost_int @ H5 ) ) ).

% not_UNIV_eq_Iic
thf(fact_5659_not__UNIV__eq__Iic,axiom,
    ! [H5: nat] :
      ( top_top_set_nat
     != ( set_ord_atMost_nat @ H5 ) ) ).

% not_UNIV_eq_Iic
thf(fact_5660_card__1__singletonI,axiom,
    ! [S2: set_Product_unit,X: product_unit] :
      ( ( finite4290736615968046902t_unit @ S2 )
     => ( ( ( finite410649719033368117t_unit @ S2 )
          = one_one_nat )
       => ( ( member_Product_unit @ X @ S2 )
         => ( S2
            = ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ).

% card_1_singletonI
thf(fact_5661_card__1__singletonI,axiom,
    ! [S2: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ S2 )
     => ( ( ( finite_card_list_nat @ S2 )
          = one_one_nat )
       => ( ( member_list_nat @ X @ S2 )
         => ( S2
            = ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) ) ) ).

% card_1_singletonI
thf(fact_5662_card__1__singletonI,axiom,
    ! [S2: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( ( finite4902975817058060853nteger @ S2 )
          = one_one_nat )
       => ( ( member_Code_integer @ X @ S2 )
         => ( S2
            = ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ).

% card_1_singletonI
thf(fact_5663_card__1__singletonI,axiom,
    ! [S2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( finite4343798906461161616at_nat @ S2 )
     => ( ( ( finite3771342082235030671at_nat @ S2 )
          = one_one_nat )
       => ( ( member8757157785044589968at_nat @ X @ S2 )
         => ( S2
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% card_1_singletonI
thf(fact_5664_card__1__singletonI,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ S2 )
     => ( ( ( finite711546835091564841at_nat @ S2 )
          = one_one_nat )
       => ( ( member8440522571783428010at_nat @ X @ S2 )
         => ( S2
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% card_1_singletonI
thf(fact_5665_card__1__singletonI,axiom,
    ! [S2: set_o,X: $o] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( finite_card_o @ S2 )
          = one_one_nat )
       => ( ( member_o @ X @ S2 )
         => ( S2
            = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).

% card_1_singletonI
thf(fact_5666_card__1__singletonI,axiom,
    ! [S2: set_nat,X: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( finite_card_nat @ S2 )
          = one_one_nat )
       => ( ( member_nat @ X @ S2 )
         => ( S2
            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).

% card_1_singletonI
thf(fact_5667_card__1__singletonI,axiom,
    ! [S2: set_int,X: int] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( finite_card_int @ S2 )
          = one_one_nat )
       => ( ( member_int @ X @ S2 )
         => ( S2
            = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).

% card_1_singletonI
thf(fact_5668_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K4: nat] :
            ( ( P @ K4 )
            & ( ord_less_nat @ K4 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_5669_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ N @ ( F @ N ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_5670_card__insert__disjoint_H,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( finite4343798906461161616at_nat @ A2 )
     => ( ~ ( member8757157785044589968at_nat @ X @ A2 )
       => ( ( minus_minus_nat @ ( finite3771342082235030671at_nat @ ( insert9069300056098147895at_nat @ X @ A2 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite3771342082235030671at_nat @ A2 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_5671_card__insert__disjoint_H,axiom,
    ! [A2: set_o,X: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ~ ( member_o @ X @ A2 )
       => ( ( minus_minus_nat @ ( finite_card_o @ ( insert_o @ X @ A2 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite_card_o @ A2 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_5672_card__insert__disjoint_H,axiom,
    ! [A2: set_Product_unit,X: product_unit] :
      ( ( finite4290736615968046902t_unit @ A2 )
     => ( ~ ( member_Product_unit @ X @ A2 )
       => ( ( minus_minus_nat @ ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A2 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite410649719033368117t_unit @ A2 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_5673_card__insert__disjoint_H,axiom,
    ! [A2: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A2 )
     => ( ~ ( member_list_nat @ X @ A2 )
       => ( ( minus_minus_nat @ ( finite_card_list_nat @ ( insert_list_nat @ X @ A2 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite_card_list_nat @ A2 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_5674_card__insert__disjoint_H,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( minus_minus_nat @ ( finite_card_nat @ ( insert_nat @ X @ A2 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite_card_nat @ A2 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_5675_card__insert__disjoint_H,axiom,
    ! [A2: set_int,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( minus_minus_nat @ ( finite_card_int @ ( insert_int @ X @ A2 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite_card_int @ A2 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_5676_card__insert__disjoint_H,axiom,
    ! [A2: set_Code_integer,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ~ ( member_Code_integer @ X @ A2 )
       => ( ( minus_minus_nat @ ( finite4902975817058060853nteger @ ( insert_Code_integer @ X @ A2 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite4902975817058060853nteger @ A2 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_5677_card__insert__disjoint_H,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ~ ( member8440522571783428010at_nat @ X @ A2 )
       => ( ( minus_minus_nat @ ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite711546835091564841at_nat @ A2 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_5678_in__these__eq,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_op989438649584373901at_nat] :
      ( ( member8757157785044589968at_nat @ X @ ( these_1872268174106123684at_nat @ A2 ) )
      = ( member4288423094740844502at_nat @ ( some_P131194445498448716at_nat @ X ) @ A2 ) ) ).

% in_these_eq
thf(fact_5679_in__these__eq,axiom,
    ! [X: product_prod_nat_nat,A2: set_op4508134149509766951at_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( these_8061402112538453438at_nat @ A2 ) )
      = ( member3954567711264315760at_nat @ ( some_P7363390416028606310at_nat @ X ) @ A2 ) ) ).

% in_these_eq
thf(fact_5680_in__these__eq,axiom,
    ! [X: $o,A2: set_option_o] :
      ( ( member_o @ X @ ( these_o @ A2 ) )
      = ( member_option_o @ ( some_o @ X ) @ A2 ) ) ).

% in_these_eq
thf(fact_5681_in__these__eq,axiom,
    ! [X: nat,A2: set_option_nat] :
      ( ( member_nat @ X @ ( these_nat @ A2 ) )
      = ( member_option_nat @ ( some_nat @ X ) @ A2 ) ) ).

% in_these_eq
thf(fact_5682_in__these__eq,axiom,
    ! [X: int,A2: set_option_int] :
      ( ( member_int @ X @ ( these_int @ A2 ) )
      = ( member_option_int @ ( some_int @ X ) @ A2 ) ) ).

% in_these_eq
thf(fact_5683_in__these__eq,axiom,
    ! [X: num,A2: set_option_num] :
      ( ( member_num @ X @ ( these_num @ A2 ) )
      = ( member_option_num @ ( some_num @ X ) @ A2 ) ) ).

% in_these_eq
thf(fact_5684_in__these__eq,axiom,
    ! [X: produc8664842809031399944it_nat,A2: set_op239864471688321678it_nat] :
      ( ( member3421417852355157663it_nat @ X @ ( these_3407724651694999947it_nat @ A2 ) )
      = ( member5596548051065438575it_nat @ ( some_P1914260805536162275it_nat @ X ) @ A2 ) ) ).

% in_these_eq
thf(fact_5685_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_5686_infinite__Ico,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or4029947393144176647an_rat @ A @ B ) ) ) ).

% infinite_Ico
thf(fact_5687_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N6: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N6 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
     => ( finite_finite_nat @ N6 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_5688_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N6: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
     => ( finite_finite_nat @ N6 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_5689_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_5690_surj__fun__eq,axiom,
    ! [F: num > code_integer,X11: set_num,G1: code_integer > code_integer,G22: code_integer > code_integer] :
      ( ( ( image_8379761247532268670nteger @ F @ X11 )
        = top_to4645266643341252675nteger )
     => ( ! [X4: num] :
            ( ( member_num @ X4 @ X11 )
           => ( ( comp_C3531382070062128313er_num @ G1 @ F @ X4 )
              = ( comp_C3531382070062128313er_num @ G22 @ F @ X4 ) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
thf(fact_5691_surj__fun__eq,axiom,
    ! [F: code_integer > code_integer,X11: set_Code_integer,G1: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G22: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( image_4470545334726330049nteger @ F @ X11 )
        = top_to4645266643341252675nteger )
     => ( ! [X4: code_integer] :
            ( ( member_Code_integer @ X4 @ X11 )
           => ( ( comp_C1593894019821074884nteger @ G1 @ F @ X4 )
              = ( comp_C1593894019821074884nteger @ G22 @ F @ X4 ) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
thf(fact_5692_surj__fun__eq,axiom,
    ! [F: code_integer > code_integer > code_integer,X11: set_Code_integer,G1: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G22: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( ( image_2177946750938214461nteger @ F @ X11 )
        = top_to3804862627464599871nteger )
     => ( ! [X4: code_integer] :
            ( ( member_Code_integer @ X4 @ X11 )
           => ( ( comp_C8797469213163452608nteger @ G1 @ F @ X4 )
              = ( comp_C8797469213163452608nteger @ G22 @ F @ X4 ) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
thf(fact_5693_surj__fun__eq,axiom,
    ! [F: int > nat,X11: set_int,G1: nat > num,G22: nat > num] :
      ( ( ( image_int_nat @ F @ X11 )
        = top_top_set_nat )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ X11 )
           => ( ( comp_nat_num_int @ G1 @ F @ X4 )
              = ( comp_nat_num_int @ G22 @ F @ X4 ) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
thf(fact_5694_comp__surj,axiom,
    ! [F: num > code_integer,G: code_integer > code_integer] :
      ( ( ( image_8379761247532268670nteger @ F @ top_top_set_num )
        = top_to4645266643341252675nteger )
     => ( ( ( image_4470545334726330049nteger @ G @ top_to4645266643341252675nteger )
          = top_to4645266643341252675nteger )
       => ( ( image_8379761247532268670nteger @ ( comp_C3531382070062128313er_num @ G @ F ) @ top_top_set_num )
          = top_to4645266643341252675nteger ) ) ) ).

% comp_surj
thf(fact_5695_comp__surj,axiom,
    ! [F: nat > code_integer,G: code_integer > code_integer] :
      ( ( ( image_1215581382706833972nteger @ F @ top_top_set_nat )
        = top_to4645266643341252675nteger )
     => ( ( ( image_4470545334726330049nteger @ G @ top_to4645266643341252675nteger )
          = top_to4645266643341252675nteger )
       => ( ( image_1215581382706833972nteger @ ( comp_C6974050283906349679er_nat @ G @ F ) @ top_top_set_nat )
          = top_to4645266643341252675nteger ) ) ) ).

% comp_surj
thf(fact_5696_comp__surj,axiom,
    ! [F: nat > code_integer,G: code_integer > nat] :
      ( ( ( image_1215581382706833972nteger @ F @ top_top_set_nat )
        = top_to4645266643341252675nteger )
     => ( ( ( image_951025933927791156er_nat @ G @ top_to4645266643341252675nteger )
          = top_top_set_nat )
       => ( ( image_nat_nat @ ( comp_C4049556595663050210at_nat @ G @ F ) @ top_top_set_nat )
          = top_top_set_nat ) ) ) ).

% comp_surj
thf(fact_5697_comp__surj,axiom,
    ! [F: nat > code_integer,G: code_integer > product_unit] :
      ( ( ( image_1215581382706833972nteger @ F @ top_top_set_nat )
        = top_to4645266643341252675nteger )
     => ( ( ( image_7182656606676592809t_unit @ G @ top_to4645266643341252675nteger )
          = top_to1996260823553986621t_unit )
       => ( ( image_8730104196221521654t_unit @ ( comp_C8086623346886033437it_nat @ G @ F ) @ top_top_set_nat )
          = top_to1996260823553986621t_unit ) ) ) ).

% comp_surj
thf(fact_5698_comp__surj,axiom,
    ! [F: nat > code_integer,G: code_integer > int] :
      ( ( ( image_1215581382706833972nteger @ F @ top_top_set_nat )
        = top_to4645266643341252675nteger )
     => ( ( ( image_948535463418740880er_int @ G @ top_to4645266643341252675nteger )
          = top_top_set_int )
       => ( ( image_nat_int @ ( comp_C5048910307564104766nt_nat @ G @ F ) @ top_top_set_nat )
          = top_top_set_int ) ) ) ).

% comp_surj
thf(fact_5699_comp__surj,axiom,
    ! [F: nat > nat,G: nat > code_integer] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( image_1215581382706833972nteger @ G @ top_top_set_nat )
          = top_to4645266643341252675nteger )
       => ( ( image_1215581382706833972nteger @ ( comp_n3898172953802868194er_nat @ G @ F ) @ top_top_set_nat )
          = top_to4645266643341252675nteger ) ) ) ).

% comp_surj
thf(fact_5700_comp__surj,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( image_nat_nat @ G @ top_top_set_nat )
          = top_top_set_nat )
       => ( ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ top_top_set_nat )
          = top_top_set_nat ) ) ) ).

% comp_surj
thf(fact_5701_comp__surj,axiom,
    ! [F: nat > nat,G: nat > product_unit] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( image_8730104196221521654t_unit @ G @ top_top_set_nat )
          = top_to1996260823553986621t_unit )
       => ( ( image_8730104196221521654t_unit @ ( comp_n9025697279074352746it_nat @ G @ F ) @ top_top_set_nat )
          = top_to1996260823553986621t_unit ) ) ) ).

% comp_surj
thf(fact_5702_comp__surj,axiom,
    ! [F: nat > nat,G: nat > int] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( image_nat_int @ G @ top_top_set_nat )
          = top_top_set_int )
       => ( ( image_nat_int @ ( comp_nat_int_nat @ G @ F ) @ top_top_set_nat )
          = top_top_set_int ) ) ) ).

% comp_surj
thf(fact_5703_comp__surj,axiom,
    ! [F: nat > product_unit,G: product_unit > code_integer] :
      ( ( ( image_8730104196221521654t_unit @ F @ top_top_set_nat )
        = top_to1996260823553986621t_unit )
     => ( ( ( image_3815118110082124069nteger @ G @ top_to1996260823553986621t_unit )
          = top_to4645266643341252675nteger )
       => ( ( image_1215581382706833972nteger @ ( comp_P756492639069434009er_nat @ G @ F ) @ top_top_set_nat )
          = top_to4645266643341252675nteger ) ) ) ).

% comp_surj
thf(fact_5704_fun_Oset__map,axiom,
    ! [F: code_integer > code_integer,V: num > code_integer] :
      ( ( image_8379761247532268670nteger @ ( comp_C3531382070062128313er_num @ F @ V ) @ top_top_set_num )
      = ( image_4470545334726330049nteger @ F @ ( image_8379761247532268670nteger @ V @ top_top_set_num ) ) ) ).

% fun.set_map
thf(fact_5705_fun_Oset__map,axiom,
    ! [F: code_integer > code_integer,V: nat > code_integer] :
      ( ( image_1215581382706833972nteger @ ( comp_C6974050283906349679er_nat @ F @ V ) @ top_top_set_nat )
      = ( image_4470545334726330049nteger @ F @ ( image_1215581382706833972nteger @ V @ top_top_set_nat ) ) ) ).

% fun.set_map
thf(fact_5706_fun_Oset__map,axiom,
    ! [F: int > code_integer,V: nat > int] :
      ( ( image_1215581382706833972nteger @ ( comp_i1588355021709917246er_nat @ F @ V ) @ top_top_set_nat )
      = ( image_1587234942943678608nteger @ F @ ( image_nat_int @ V @ top_top_set_nat ) ) ) ).

% fun.set_map
thf(fact_5707_fun_Oset__map,axiom,
    ! [F: nat > code_integer,V: nat > nat] :
      ( ( image_1215581382706833972nteger @ ( comp_n3898172953802868194er_nat @ F @ V ) @ top_top_set_nat )
      = ( image_1215581382706833972nteger @ F @ ( image_nat_nat @ V @ top_top_set_nat ) ) ) ).

% fun.set_map
thf(fact_5708_fun_Oset__map,axiom,
    ! [F: code_integer > nat,V: nat > code_integer] :
      ( ( image_nat_nat @ ( comp_C4049556595663050210at_nat @ F @ V ) @ top_top_set_nat )
      = ( image_951025933927791156er_nat @ F @ ( image_1215581382706833972nteger @ V @ top_top_set_nat ) ) ) ).

% fun.set_map
thf(fact_5709_fun_Oset__map,axiom,
    ! [F: int > nat,V: nat > int] :
      ( ( image_nat_nat @ ( comp_int_nat_nat @ F @ V ) @ top_top_set_nat )
      = ( image_int_nat @ F @ ( image_nat_int @ V @ top_top_set_nat ) ) ) ).

% fun.set_map
thf(fact_5710_fun_Oset__map,axiom,
    ! [F: nat > nat,V: nat > nat] :
      ( ( image_nat_nat @ ( comp_nat_nat_nat @ F @ V ) @ top_top_set_nat )
      = ( image_nat_nat @ F @ ( image_nat_nat @ V @ top_top_set_nat ) ) ) ).

% fun.set_map
thf(fact_5711_fun_Oset__map,axiom,
    ! [F: code_integer > int,V: nat > code_integer] :
      ( ( image_nat_int @ ( comp_C5048910307564104766nt_nat @ F @ V ) @ top_top_set_nat )
      = ( image_948535463418740880er_int @ F @ ( image_1215581382706833972nteger @ V @ top_top_set_nat ) ) ) ).

% fun.set_map
thf(fact_5712_fun_Oset__map,axiom,
    ! [F: nat > int,V: nat > nat] :
      ( ( image_nat_int @ ( comp_nat_int_nat @ F @ V ) @ top_top_set_nat )
      = ( image_nat_int @ F @ ( image_nat_nat @ V @ top_top_set_nat ) ) ) ).

% fun.set_map
thf(fact_5713_fun_Oset__map,axiom,
    ! [F: int > int,V: nat > int] :
      ( ( image_nat_int @ ( comp_int_int_nat @ F @ V ) @ top_top_set_nat )
      = ( image_int_int @ F @ ( image_nat_int @ V @ top_top_set_nat ) ) ) ).

% fun.set_map
thf(fact_5714_fun_Omap__cong,axiom,
    ! [X: num > code_integer,Ya: num > code_integer,F: code_integer > code_integer,G: code_integer > code_integer] :
      ( ( X = Ya )
     => ( ! [Z3: code_integer] :
            ( ( member_Code_integer @ Z3 @ ( image_8379761247532268670nteger @ Ya @ top_top_set_num ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( comp_C3531382070062128313er_num @ F @ X )
          = ( comp_C3531382070062128313er_num @ G @ Ya ) ) ) ) ).

% fun.map_cong
thf(fact_5715_fun_Omap__cong,axiom,
    ! [X: code_integer > code_integer,Ya: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( X = Ya )
     => ( ! [Z3: code_integer] :
            ( ( member_Code_integer @ Z3 @ ( image_4470545334726330049nteger @ Ya @ top_to4645266643341252675nteger ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( comp_C1593894019821074884nteger @ F @ X )
          = ( comp_C1593894019821074884nteger @ G @ Ya ) ) ) ) ).

% fun.map_cong
thf(fact_5716_fun_Omap__cong,axiom,
    ! [X: code_integer > code_integer > code_integer,Ya: code_integer > code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ( X = Ya )
     => ( ! [Z3: code_integer > code_integer] :
            ( ( member2790140394383616496nteger @ Z3 @ ( image_2177946750938214461nteger @ Ya @ top_to4645266643341252675nteger ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( comp_C8797469213163452608nteger @ F @ X )
          = ( comp_C8797469213163452608nteger @ G @ Ya ) ) ) ) ).

% fun.map_cong
thf(fact_5717_fun_Omap__cong,axiom,
    ! [X: int > nat,Ya: int > nat,F: nat > num,G: nat > num] :
      ( ( X = Ya )
     => ( ! [Z3: nat] :
            ( ( member_nat @ Z3 @ ( image_int_nat @ Ya @ top_top_set_int ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( comp_nat_num_int @ F @ X )
          = ( comp_nat_num_int @ G @ Ya ) ) ) ) ).

% fun.map_cong
thf(fact_5718_fun_Omap__cong0,axiom,
    ! [X: num > code_integer,F: code_integer > code_integer,G: code_integer > code_integer] :
      ( ! [Z3: code_integer] :
          ( ( member_Code_integer @ Z3 @ ( image_8379761247532268670nteger @ X @ top_top_set_num ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( comp_C3531382070062128313er_num @ F @ X )
        = ( comp_C3531382070062128313er_num @ G @ X ) ) ) ).

% fun.map_cong0
thf(fact_5719_fun_Omap__cong0,axiom,
    ! [X: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ! [Z3: code_integer] :
          ( ( member_Code_integer @ Z3 @ ( image_4470545334726330049nteger @ X @ top_to4645266643341252675nteger ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( comp_C1593894019821074884nteger @ F @ X )
        = ( comp_C1593894019821074884nteger @ G @ X ) ) ) ).

% fun.map_cong0
thf(fact_5720_fun_Omap__cong0,axiom,
    ! [X: code_integer > code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ! [Z3: code_integer > code_integer] :
          ( ( member2790140394383616496nteger @ Z3 @ ( image_2177946750938214461nteger @ X @ top_to4645266643341252675nteger ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( comp_C8797469213163452608nteger @ F @ X )
        = ( comp_C8797469213163452608nteger @ G @ X ) ) ) ).

% fun.map_cong0
thf(fact_5721_fun_Omap__cong0,axiom,
    ! [X: int > nat,F: nat > num,G: nat > num] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( image_int_nat @ X @ top_top_set_int ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( comp_nat_num_int @ F @ X )
        = ( comp_nat_num_int @ G @ X ) ) ) ).

% fun.map_cong0
thf(fact_5722_fun_Oinj__map__strong,axiom,
    ! [X: num > code_integer,Xa: num > code_integer,F: code_integer > code_integer,Fa: code_integer > code_integer] :
      ( ! [Z3: code_integer,Za: code_integer] :
          ( ( member_Code_integer @ Z3 @ ( image_8379761247532268670nteger @ X @ top_top_set_num ) )
         => ( ( member_Code_integer @ Za @ ( image_8379761247532268670nteger @ Xa @ top_top_set_num ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( comp_C3531382070062128313er_num @ F @ X )
          = ( comp_C3531382070062128313er_num @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% fun.inj_map_strong
thf(fact_5723_fun_Oinj__map__strong,axiom,
    ! [X: code_integer > code_integer,Xa: code_integer > code_integer,F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,Fa: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ! [Z3: code_integer,Za: code_integer] :
          ( ( member_Code_integer @ Z3 @ ( image_4470545334726330049nteger @ X @ top_to4645266643341252675nteger ) )
         => ( ( member_Code_integer @ Za @ ( image_4470545334726330049nteger @ Xa @ top_to4645266643341252675nteger ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( comp_C1593894019821074884nteger @ F @ X )
          = ( comp_C1593894019821074884nteger @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% fun.inj_map_strong
thf(fact_5724_fun_Oinj__map__strong,axiom,
    ! [X: code_integer > code_integer > code_integer,Xa: code_integer > code_integer > code_integer,F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,Fa: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger] :
      ( ! [Z3: code_integer > code_integer,Za: code_integer > code_integer] :
          ( ( member2790140394383616496nteger @ Z3 @ ( image_2177946750938214461nteger @ X @ top_to4645266643341252675nteger ) )
         => ( ( member2790140394383616496nteger @ Za @ ( image_2177946750938214461nteger @ Xa @ top_to4645266643341252675nteger ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( comp_C8797469213163452608nteger @ F @ X )
          = ( comp_C8797469213163452608nteger @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% fun.inj_map_strong
thf(fact_5725_fun_Oinj__map__strong,axiom,
    ! [X: int > nat,Xa: int > nat,F: nat > num,Fa: nat > num] :
      ( ! [Z3: nat,Za: nat] :
          ( ( member_nat @ Z3 @ ( image_int_nat @ X @ top_top_set_int ) )
         => ( ( member_nat @ Za @ ( image_int_nat @ Xa @ top_top_set_int ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( comp_nat_num_int @ F @ X )
          = ( comp_nat_num_int @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% fun.inj_map_strong
thf(fact_5726_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_5727_not__UNIV__le__Icc,axiom,
    ! [L: int,H3: int] :
      ~ ( ord_less_eq_set_int @ top_top_set_int @ ( set_or1266510415728281911st_int @ L @ H3 ) ) ).

% not_UNIV_le_Icc
thf(fact_5728_not__UNIV__le__Icc,axiom,
    ! [L: nat,H3: nat] :
      ~ ( ord_less_eq_set_nat @ top_top_set_nat @ ( set_or1269000886237332187st_nat @ L @ H3 ) ) ).

% not_UNIV_le_Icc
thf(fact_5729_surj__id,axiom,
    ( ( image_o_o @ id_o @ top_top_set_o )
    = top_top_set_o ) ).

% surj_id
thf(fact_5730_surj__id,axiom,
    ( ( image_nat_o_nat_o @ id_nat_o @ top_top_set_nat_o )
    = top_top_set_nat_o ) ).

% surj_id
thf(fact_5731_surj__id,axiom,
    ( ( image_num_num @ id_num @ top_top_set_num )
    = top_top_set_num ) ).

% surj_id
thf(fact_5732_surj__id,axiom,
    ( ( image_nat_nat @ id_nat @ top_top_set_nat )
    = top_top_set_nat ) ).

% surj_id
thf(fact_5733_surj__id,axiom,
    ( ( image_405062704495631173t_unit @ id_Product_unit @ top_to1996260823553986621t_unit )
    = top_to1996260823553986621t_unit ) ).

% surj_id
thf(fact_5734_surj__id,axiom,
    ( ( image_int_int @ id_int @ top_top_set_int )
    = top_top_set_int ) ).

% surj_id
thf(fact_5735_not__UNIV__le__Iic,axiom,
    ! [H3: int] :
      ~ ( ord_less_eq_set_int @ top_top_set_int @ ( set_ord_atMost_int @ H3 ) ) ).

% not_UNIV_le_Iic
thf(fact_5736_not__UNIV__le__Iic,axiom,
    ! [H3: nat] :
      ~ ( ord_less_eq_set_nat @ top_top_set_nat @ ( set_ord_atMost_nat @ H3 ) ) ).

% not_UNIV_le_Iic
thf(fact_5737_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_o,X: $o > code_integer,Y: $o > code_integer] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I3: $o] :
              ( ( member_o @ I3 @ I4 )
              & ( ( X @ I3 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I3: $o] :
                ( ( member_o @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != zero_z3403309356797280102nteger ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I3: $o] :
                ( ( member_o @ I3 @ I4 )
                & ( ( plus_p5714425477246183910nteger @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5738_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_nat,X: nat > code_integer,Y: nat > code_integer] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I4 )
              & ( ( X @ I3 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != zero_z3403309356797280102nteger ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I4 )
                & ( ( plus_p5714425477246183910nteger @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5739_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_int,X: int > code_integer,Y: int > code_integer] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I4 )
              & ( ( X @ I3 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != zero_z3403309356797280102nteger ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I4 )
                & ( ( plus_p5714425477246183910nteger @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5740_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_Code_integer,X: code_integer > code_integer,Y: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I4 )
              & ( ( X @ I3 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != zero_z3403309356797280102nteger ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I4 )
                & ( ( plus_p5714425477246183910nteger @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_z3403309356797280102nteger ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5741_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_o,X: $o > rat,Y: $o > rat] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I3: $o] :
              ( ( member_o @ I3 @ I4 )
              & ( ( X @ I3 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I3: $o] :
                ( ( member_o @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I3: $o] :
                ( ( member_o @ I3 @ I4 )
                & ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5742_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_nat,X: nat > rat,Y: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I4 )
              & ( ( X @ I3 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I4 )
                & ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5743_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_int,X: int > rat,Y: int > rat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I4 )
              & ( ( X @ I3 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I4 )
                & ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5744_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_Code_integer,X: code_integer > rat,Y: code_integer > rat] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I4 )
              & ( ( X @ I3 )
               != zero_zero_rat ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != zero_zero_rat ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I4 )
                & ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5745_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_o,X: $o > nat,Y: $o > nat] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I3: $o] :
              ( ( member_o @ I3 @ I4 )
              & ( ( X @ I3 )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I3: $o] :
                ( ( member_o @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != zero_zero_nat ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I3: $o] :
                ( ( member_o @ I3 @ I4 )
                & ( ( plus_plus_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_nat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5746_sum_Ofinite__Collect__op,axiom,
    ! [I4: set_nat,X: nat > nat,Y: nat > nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I4 )
              & ( ( X @ I3 )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != zero_zero_nat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I4 )
                & ( ( plus_plus_nat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != zero_zero_nat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5747_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_o,X: $o > code_integer,Y: $o > code_integer] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I3: $o] :
              ( ( member_o @ I3 @ I4 )
              & ( ( X @ I3 )
               != one_one_Code_integer ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I3: $o] :
                ( ( member_o @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != one_one_Code_integer ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I3: $o] :
                ( ( member_o @ I3 @ I4 )
                & ( ( times_3573771949741848930nteger @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_Code_integer ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5748_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_nat,X: nat > code_integer,Y: nat > code_integer] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I4 )
              & ( ( X @ I3 )
               != one_one_Code_integer ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != one_one_Code_integer ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I4 )
                & ( ( times_3573771949741848930nteger @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_Code_integer ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5749_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_int,X: int > code_integer,Y: int > code_integer] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I4 )
              & ( ( X @ I3 )
               != one_one_Code_integer ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != one_one_Code_integer ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I4 )
                & ( ( times_3573771949741848930nteger @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_Code_integer ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5750_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_Code_integer,X: code_integer > code_integer,Y: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I4 )
              & ( ( X @ I3 )
               != one_one_Code_integer ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != one_one_Code_integer ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I4 )
                & ( ( times_3573771949741848930nteger @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_Code_integer ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5751_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_o,X: $o > assn,Y: $o > assn] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I3: $o] :
              ( ( member_o @ I3 @ I4 )
              & ( ( X @ I3 )
               != one_one_assn ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I3: $o] :
                ( ( member_o @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != one_one_assn ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I3: $o] :
                ( ( member_o @ I3 @ I4 )
                & ( ( times_times_assn @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5752_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_nat,X: nat > assn,Y: nat > assn] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I4 )
              & ( ( X @ I3 )
               != one_one_assn ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != one_one_assn ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I4 )
                & ( ( times_times_assn @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5753_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_int,X: int > assn,Y: int > assn] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I3: int] :
              ( ( member_int @ I3 @ I4 )
              & ( ( X @ I3 )
               != one_one_assn ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != one_one_assn ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I3: int] :
                ( ( member_int @ I3 @ I4 )
                & ( ( times_times_assn @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5754_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_Code_integer,X: code_integer > assn,Y: code_integer > assn] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I3: code_integer] :
              ( ( member_Code_integer @ I3 @ I4 )
              & ( ( X @ I3 )
               != one_one_assn ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != one_one_assn ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I3: code_integer] :
                ( ( member_Code_integer @ I3 @ I4 )
                & ( ( times_times_assn @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_assn ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5755_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_o,X: $o > rat,Y: $o > rat] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I3: $o] :
              ( ( member_o @ I3 @ I4 )
              & ( ( X @ I3 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I3: $o] :
                ( ( member_o @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I3: $o] :
                ( ( member_o @ I3 @ I4 )
                & ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5756_prod_Ofinite__Collect__op,axiom,
    ! [I4: set_nat,X: nat > rat,Y: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I3: nat] :
              ( ( member_nat @ I3 @ I4 )
              & ( ( X @ I3 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I4 )
                & ( ( Y @ I3 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I3: nat] :
                ( ( member_nat @ I3 @ I4 )
                & ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5757_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).

% finite_atLeastZeroLessThan_int
thf(fact_5758_card__eq__sum,axiom,
    ( finite_card_int
    = ( groups4541462559716669496nt_nat
      @ ^ [X3: int] : one_one_nat ) ) ).

% card_eq_sum
thf(fact_5759_card__eq__sum,axiom,
    ( finite410649719033368117t_unit
    = ( groups8766797177574957219it_nat
      @ ^ [X3: product_unit] : one_one_nat ) ) ).

% card_eq_sum
thf(fact_5760_card__eq__sum,axiom,
    ( finite_card_list_nat
    = ( groups4396056296759096172at_nat
      @ ^ [X3: list_nat] : one_one_nat ) ) ).

% card_eq_sum
thf(fact_5761_card__eq__sum,axiom,
    ( finite_card_nat
    = ( groups3542108847815614940at_nat
      @ ^ [X3: nat] : one_one_nat ) ) ).

% card_eq_sum
thf(fact_5762_prod_Ointer__filter,axiom,
    ! [A2: set_o,G: $o > code_integer,P: $o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( groups7694694392188491536nteger @ G
          @ ( collect_o
            @ ^ [X3: $o] :
                ( ( member_o @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups7694694392188491536nteger
          @ ^ [X3: $o] : ( if_Code_integer @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_Code_integer )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_5763_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > code_integer,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups3455450783089532116nteger @ G
          @ ( collect_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups3455450783089532116nteger
          @ ^ [X3: nat] : ( if_Code_integer @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_Code_integer )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_5764_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: int > code_integer,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3827104343326376752nteger @ G
          @ ( collect_int
            @ ^ [X3: int] :
                ( ( member_int @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups3827104343326376752nteger
          @ ^ [X3: int] : ( if_Code_integer @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_Code_integer )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_5765_prod_Ointer__filter,axiom,
    ! [A2: set_Code_integer,G: code_integer > code_integer,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups3674199335183972705nteger @ G
          @ ( collect_Code_integer
            @ ^ [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups3674199335183972705nteger
          @ ^ [X3: code_integer] : ( if_Code_integer @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_Code_integer )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_5766_prod_Ointer__filter,axiom,
    ! [A2: set_o,G: $o > assn,P: $o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( groups5301882518646026715o_assn @ G
          @ ( collect_o
            @ ^ [X3: $o] :
                ( ( member_o @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups5301882518646026715o_assn
          @ ^ [X3: $o] : ( if_assn @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_assn )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_5767_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > assn,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups6906906614972039071t_assn @ G
          @ ( collect_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups6906906614972039071t_assn
          @ ^ [X3: nat] : ( if_assn @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_assn )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_5768_prod_Ointer__filter,axiom,
    ! [A2: set_int,G: int > assn,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups7882442080178216443t_assn @ G
          @ ( collect_int
            @ ^ [X3: int] :
                ( ( member_int @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups7882442080178216443t_assn
          @ ^ [X3: int] : ( if_assn @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_assn )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_5769_prod_Ointer__filter,axiom,
    ! [A2: set_Code_integer,G: code_integer > assn,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups1304777262505850412r_assn @ G
          @ ( collect_Code_integer
            @ ^ [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups1304777262505850412r_assn
          @ ^ [X3: code_integer] : ( if_assn @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_assn )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_5770_prod_Ointer__filter,axiom,
    ! [A2: set_o,G: $o > rat,P: $o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( groups2869687844427037835_o_rat @ G
          @ ( collect_o
            @ ^ [X3: $o] :
                ( ( member_o @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups2869687844427037835_o_rat
          @ ^ [X3: $o] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_rat )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_5771_prod_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > rat,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups73079841787564623at_rat @ G
          @ ( collect_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups73079841787564623at_rat
          @ ^ [X3: nat] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ one_one_rat )
          @ A2 ) ) ) ).

% prod.inter_filter
thf(fact_5772_prod__gen__delta,axiom,
    ! [S2: set_o,A: $o,B: $o > assn,C: assn] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups5301882518646026715o_assn
              @ ^ [K4: $o] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( times_times_assn @ ( B @ A ) @ ( power_power_assn @ C @ ( minus_minus_nat @ ( finite_card_o @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups5301882518646026715o_assn
              @ ^ [K4: $o] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( power_power_assn @ C @ ( finite_card_o @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_5773_prod__gen__delta,axiom,
    ! [S2: set_Product_unit,A: product_unit,B: product_unit > assn,C: assn] :
      ( ( finite4290736615968046902t_unit @ S2 )
     => ( ( ( member_Product_unit @ A @ S2 )
         => ( ( groups5360233303959380816t_assn
              @ ^ [K4: product_unit] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( times_times_assn @ ( B @ A ) @ ( power_power_assn @ C @ ( minus_minus_nat @ ( finite410649719033368117t_unit @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_Product_unit @ A @ S2 )
         => ( ( groups5360233303959380816t_assn
              @ ^ [K4: product_unit] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( power_power_assn @ C @ ( finite410649719033368117t_unit @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_5774_prod__gen__delta,axiom,
    ! [S2: set_nat,A: nat,B: nat > assn,C: assn] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups6906906614972039071t_assn
              @ ^ [K4: nat] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( times_times_assn @ ( B @ A ) @ ( power_power_assn @ C @ ( minus_minus_nat @ ( finite_card_nat @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups6906906614972039071t_assn
              @ ^ [K4: nat] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( power_power_assn @ C @ ( finite_card_nat @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_5775_prod__gen__delta,axiom,
    ! [S2: set_int,A: int,B: int > assn,C: assn] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups7882442080178216443t_assn
              @ ^ [K4: int] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( times_times_assn @ ( B @ A ) @ ( power_power_assn @ C @ ( minus_minus_nat @ ( finite_card_int @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups7882442080178216443t_assn
              @ ^ [K4: int] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( power_power_assn @ C @ ( finite_card_int @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_5776_prod__gen__delta,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: code_integer > assn,C: assn] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( ( member_Code_integer @ A @ S2 )
         => ( ( groups1304777262505850412r_assn
              @ ^ [K4: code_integer] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( times_times_assn @ ( B @ A ) @ ( power_power_assn @ C @ ( minus_minus_nat @ ( finite4902975817058060853nteger @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S2 )
         => ( ( groups1304777262505850412r_assn
              @ ^ [K4: code_integer] : ( if_assn @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( power_power_assn @ C @ ( finite4902975817058060853nteger @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_5777_prod__gen__delta,axiom,
    ! [S2: set_o,A: $o,B: $o > rat,C: rat] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups2869687844427037835_o_rat
              @ ^ [K4: $o] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( times_times_rat @ ( B @ A ) @ ( power_power_rat @ C @ ( minus_minus_nat @ ( finite_card_o @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups2869687844427037835_o_rat
              @ ^ [K4: $o] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( power_power_rat @ C @ ( finite_card_o @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_5778_prod__gen__delta,axiom,
    ! [S2: set_Product_unit,A: product_unit,B: product_unit > rat,C: rat] :
      ( ( finite4290736615968046902t_unit @ S2 )
     => ( ( ( member_Product_unit @ A @ S2 )
         => ( ( groups8984663225794451456it_rat
              @ ^ [K4: product_unit] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( times_times_rat @ ( B @ A ) @ ( power_power_rat @ C @ ( minus_minus_nat @ ( finite410649719033368117t_unit @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_Product_unit @ A @ S2 )
         => ( ( groups8984663225794451456it_rat
              @ ^ [K4: product_unit] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( power_power_rat @ C @ ( finite410649719033368117t_unit @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_5779_prod__gen__delta,axiom,
    ! [S2: set_nat,A: nat,B: nat > rat,C: rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K4: nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( times_times_rat @ ( B @ A ) @ ( power_power_rat @ C @ ( minus_minus_nat @ ( finite_card_nat @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K4: nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( power_power_rat @ C @ ( finite_card_nat @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_5780_prod__gen__delta,axiom,
    ! [S2: set_int,A: int,B: int > rat,C: rat] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K4: int] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( times_times_rat @ ( B @ A ) @ ( power_power_rat @ C @ ( minus_minus_nat @ ( finite_card_int @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K4: int] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( power_power_rat @ C @ ( finite_card_int @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_5781_prod__gen__delta,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: code_integer > rat,C: rat] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( ( member_Code_integer @ A @ S2 )
         => ( ( groups2555765274223993564er_rat
              @ ^ [K4: code_integer] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( times_times_rat @ ( B @ A ) @ ( power_power_rat @ C @ ( minus_minus_nat @ ( finite4902975817058060853nteger @ S2 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S2 )
         => ( ( groups2555765274223993564er_rat
              @ ^ [K4: code_integer] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ C )
              @ S2 )
            = ( power_power_rat @ C @ ( finite4902975817058060853nteger @ S2 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_5782_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U ) ) ).

% finite_atLeastZeroLessThan_integer
thf(fact_5783_card__2__iff_H,axiom,
    ! [S2: set_int] :
      ( ( ( finite_card_int @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: int] :
            ( ( member_int @ X3 @ S2 )
            & ? [Y3: int] :
                ( ( member_int @ Y3 @ S2 )
                & ( X3 != Y3 )
                & ! [Z4: int] :
                    ( ( member_int @ Z4 @ S2 )
                   => ( ( Z4 = X3 )
                      | ( Z4 = Y3 ) ) ) ) ) ) ) ).

% card_2_iff'
thf(fact_5784_card__2__iff_H,axiom,
    ! [S2: set_Product_unit] :
      ( ( ( finite410649719033368117t_unit @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: product_unit] :
            ( ( member_Product_unit @ X3 @ S2 )
            & ? [Y3: product_unit] :
                ( ( member_Product_unit @ Y3 @ S2 )
                & ( X3 != Y3 )
                & ! [Z4: product_unit] :
                    ( ( member_Product_unit @ Z4 @ S2 )
                   => ( ( Z4 = X3 )
                      | ( Z4 = Y3 ) ) ) ) ) ) ) ).

% card_2_iff'
thf(fact_5785_card__2__iff_H,axiom,
    ! [S2: set_nat] :
      ( ( ( finite_card_nat @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat @ X3 @ S2 )
            & ? [Y3: nat] :
                ( ( member_nat @ Y3 @ S2 )
                & ( X3 != Y3 )
                & ! [Z4: nat] :
                    ( ( member_nat @ Z4 @ S2 )
                   => ( ( Z4 = X3 )
                      | ( Z4 = Y3 ) ) ) ) ) ) ) ).

% card_2_iff'
thf(fact_5786_card__2__iff_H,axiom,
    ! [S2: set_list_nat] :
      ( ( ( finite_card_list_nat @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: list_nat] :
            ( ( member_list_nat @ X3 @ S2 )
            & ? [Y3: list_nat] :
                ( ( member_list_nat @ Y3 @ S2 )
                & ( X3 != Y3 )
                & ! [Z4: list_nat] :
                    ( ( member_list_nat @ Z4 @ S2 )
                   => ( ( Z4 = X3 )
                      | ( Z4 = Y3 ) ) ) ) ) ) ) ).

% card_2_iff'
thf(fact_5787_sum_Orelated,axiom,
    ! [R4: code_integer > code_integer > $o,S2: set_nat,H3: nat > code_integer,G: nat > code_integer] :
      ( ( R4 @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
     => ( ! [X13: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( plus_p5714425477246183910nteger @ X13 @ Y12 ) @ ( plus_p5714425477246183910nteger @ X22 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups7501900531339628137nteger @ H3 @ S2 ) @ ( groups7501900531339628137nteger @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5788_sum_Orelated,axiom,
    ! [R4: code_integer > code_integer > $o,S2: set_int,H3: int > code_integer,G: int > code_integer] :
      ( ( R4 @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
     => ( ! [X13: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( plus_p5714425477246183910nteger @ X13 @ Y12 ) @ ( plus_p5714425477246183910nteger @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups7873554091576472773nteger @ H3 @ S2 ) @ ( groups7873554091576472773nteger @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5789_sum_Orelated,axiom,
    ! [R4: code_integer > code_integer > $o,S2: set_Code_integer,H3: code_integer > code_integer,G: code_integer > code_integer] :
      ( ( R4 @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
     => ( ! [X13: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( plus_p5714425477246183910nteger @ X13 @ Y12 ) @ ( plus_p5714425477246183910nteger @ X22 @ Y22 ) ) )
       => ( ( finite6017078050557962740nteger @ S2 )
         => ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups879477027807139574nteger @ H3 @ S2 ) @ ( groups879477027807139574nteger @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5790_sum_Orelated,axiom,
    ! [R4: rat > rat > $o,S2: set_nat,H3: nat > rat,G: nat > rat] :
      ( ( R4 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X13: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( plus_plus_rat @ X13 @ Y12 ) @ ( plus_plus_rat @ X22 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups2906978787729119204at_rat @ H3 @ S2 ) @ ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5791_sum_Orelated,axiom,
    ! [R4: rat > rat > $o,S2: set_int,H3: int > rat,G: int > rat] :
      ( ( R4 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X13: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( plus_plus_rat @ X13 @ Y12 ) @ ( plus_plus_rat @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups3906332499630173760nt_rat @ H3 @ S2 ) @ ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5792_sum_Orelated,axiom,
    ! [R4: rat > rat > $o,S2: set_Code_integer,H3: code_integer > rat,G: code_integer > rat] :
      ( ( R4 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X13: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( plus_plus_rat @ X13 @ Y12 ) @ ( plus_plus_rat @ X22 @ Y22 ) ) )
       => ( ( finite6017078050557962740nteger @ S2 )
         => ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups6602215022474089585er_rat @ H3 @ S2 ) @ ( groups6602215022474089585er_rat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5793_sum_Orelated,axiom,
    ! [R4: nat > nat > $o,S2: set_int,H3: int > nat,G: int > nat] :
      ( ( R4 @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X13: nat,Y12: nat,X22: nat,Y22: nat] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( plus_plus_nat @ X13 @ Y12 ) @ ( plus_plus_nat @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups4541462559716669496nt_nat @ H3 @ S2 ) @ ( groups4541462559716669496nt_nat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5794_sum_Orelated,axiom,
    ! [R4: nat > nat > $o,S2: set_Code_integer,H3: code_integer > nat,G: code_integer > nat] :
      ( ( R4 @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X13: nat,Y12: nat,X22: nat,Y22: nat] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( plus_plus_nat @ X13 @ Y12 ) @ ( plus_plus_nat @ X22 @ Y22 ) ) )
       => ( ( finite6017078050557962740nteger @ S2 )
         => ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups7237345082560585321er_nat @ H3 @ S2 ) @ ( groups7237345082560585321er_nat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5795_sum_Orelated,axiom,
    ! [R4: int > int > $o,S2: set_nat,H3: nat > int,G: nat > int] :
      ( ( R4 @ zero_zero_int @ zero_zero_int )
     => ( ! [X13: int,Y12: int,X22: int,Y22: int] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( plus_plus_int @ X13 @ Y12 ) @ ( plus_plus_int @ X22 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups3539618377306564664at_int @ H3 @ S2 ) @ ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5796_sum_Orelated,axiom,
    ! [R4: int > int > $o,S2: set_Code_integer,H3: code_integer > int,G: code_integer > int] :
      ( ( R4 @ zero_zero_int @ zero_zero_int )
     => ( ! [X13: int,Y12: int,X22: int,Y22: int] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( plus_plus_int @ X13 @ Y12 ) @ ( plus_plus_int @ X22 @ Y22 ) ) )
       => ( ( finite6017078050557962740nteger @ S2 )
         => ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups7234854612051535045er_int @ H3 @ S2 ) @ ( groups7234854612051535045er_int @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_5797_prod_Orelated,axiom,
    ! [R4: code_integer > code_integer > $o,S2: set_nat,H3: nat > code_integer,G: nat > code_integer] :
      ( ( R4 @ one_one_Code_integer @ one_one_Code_integer )
     => ( ! [X13: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( times_3573771949741848930nteger @ X13 @ Y12 ) @ ( times_3573771949741848930nteger @ X22 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups3455450783089532116nteger @ H3 @ S2 ) @ ( groups3455450783089532116nteger @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_5798_prod_Orelated,axiom,
    ! [R4: code_integer > code_integer > $o,S2: set_int,H3: int > code_integer,G: int > code_integer] :
      ( ( R4 @ one_one_Code_integer @ one_one_Code_integer )
     => ( ! [X13: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( times_3573771949741848930nteger @ X13 @ Y12 ) @ ( times_3573771949741848930nteger @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups3827104343326376752nteger @ H3 @ S2 ) @ ( groups3827104343326376752nteger @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_5799_prod_Orelated,axiom,
    ! [R4: code_integer > code_integer > $o,S2: set_Code_integer,H3: code_integer > code_integer,G: code_integer > code_integer] :
      ( ( R4 @ one_one_Code_integer @ one_one_Code_integer )
     => ( ! [X13: code_integer,Y12: code_integer,X22: code_integer,Y22: code_integer] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( times_3573771949741848930nteger @ X13 @ Y12 ) @ ( times_3573771949741848930nteger @ X22 @ Y22 ) ) )
       => ( ( finite6017078050557962740nteger @ S2 )
         => ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups3674199335183972705nteger @ H3 @ S2 ) @ ( groups3674199335183972705nteger @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_5800_prod_Orelated,axiom,
    ! [R4: assn > assn > $o,S2: set_nat,H3: nat > assn,G: nat > assn] :
      ( ( R4 @ one_one_assn @ one_one_assn )
     => ( ! [X13: assn,Y12: assn,X22: assn,Y22: assn] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( times_times_assn @ X13 @ Y12 ) @ ( times_times_assn @ X22 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups6906906614972039071t_assn @ H3 @ S2 ) @ ( groups6906906614972039071t_assn @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_5801_prod_Orelated,axiom,
    ! [R4: assn > assn > $o,S2: set_int,H3: int > assn,G: int > assn] :
      ( ( R4 @ one_one_assn @ one_one_assn )
     => ( ! [X13: assn,Y12: assn,X22: assn,Y22: assn] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( times_times_assn @ X13 @ Y12 ) @ ( times_times_assn @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups7882442080178216443t_assn @ H3 @ S2 ) @ ( groups7882442080178216443t_assn @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_5802_prod_Orelated,axiom,
    ! [R4: assn > assn > $o,S2: set_Code_integer,H3: code_integer > assn,G: code_integer > assn] :
      ( ( R4 @ one_one_assn @ one_one_assn )
     => ( ! [X13: assn,Y12: assn,X22: assn,Y22: assn] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( times_times_assn @ X13 @ Y12 ) @ ( times_times_assn @ X22 @ Y22 ) ) )
       => ( ( finite6017078050557962740nteger @ S2 )
         => ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups1304777262505850412r_assn @ H3 @ S2 ) @ ( groups1304777262505850412r_assn @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_5803_prod_Orelated,axiom,
    ! [R4: rat > rat > $o,S2: set_nat,H3: nat > rat,G: nat > rat] :
      ( ( R4 @ one_one_rat @ one_one_rat )
     => ( ! [X13: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( times_times_rat @ X13 @ Y12 ) @ ( times_times_rat @ X22 @ Y22 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups73079841787564623at_rat @ H3 @ S2 ) @ ( groups73079841787564623at_rat @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_5804_prod_Orelated,axiom,
    ! [R4: rat > rat > $o,S2: set_int,H3: int > rat,G: int > rat] :
      ( ( R4 @ one_one_rat @ one_one_rat )
     => ( ! [X13: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( times_times_rat @ X13 @ Y12 ) @ ( times_times_rat @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups1072433553688619179nt_rat @ H3 @ S2 ) @ ( groups1072433553688619179nt_rat @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_5805_prod_Orelated,axiom,
    ! [R4: rat > rat > $o,S2: set_Code_integer,H3: code_integer > rat,G: code_integer > rat] :
      ( ( R4 @ one_one_rat @ one_one_rat )
     => ( ! [X13: rat,Y12: rat,X22: rat,Y22: rat] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( times_times_rat @ X13 @ Y12 ) @ ( times_times_rat @ X22 @ Y22 ) ) )
       => ( ( finite6017078050557962740nteger @ S2 )
         => ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups2555765274223993564er_rat @ H3 @ S2 ) @ ( groups2555765274223993564er_rat @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_5806_prod_Orelated,axiom,
    ! [R4: nat > nat > $o,S2: set_int,H3: int > nat,G: int > nat] :
      ( ( R4 @ one_one_nat @ one_one_nat )
     => ( ! [X13: nat,Y12: nat,X22: nat,Y22: nat] :
            ( ( ( R4 @ X13 @ X22 )
              & ( R4 @ Y12 @ Y22 ) )
           => ( R4 @ ( times_times_nat @ X13 @ Y12 ) @ ( times_times_nat @ X22 @ Y22 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S2 )
               => ( R4 @ ( H3 @ X4 ) @ ( G @ X4 ) ) )
           => ( R4 @ ( groups1707563613775114915nt_nat @ H3 @ S2 ) @ ( groups1707563613775114915nt_nat @ G @ S2 ) ) ) ) ) ) ).

% prod.related
thf(fact_5807_sum_Oinsert__if,axiom,
    ! [A2: set_o,X: $o,G: $o > rat] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( member_o @ X @ A2 )
         => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A2 ) )
            = ( groups7872700643590313910_o_rat @ G @ A2 ) ) )
        & ( ~ ( member_o @ X @ A2 )
         => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5808_sum_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ X @ A2 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
            = ( groups2906978787729119204at_rat @ G @ A2 ) ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5809_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
            = ( groups3906332499630173760nt_rat @ G @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5810_sum_Oinsert__if,axiom,
    ! [A2: set_Code_integer,X: code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( member_Code_integer @ X @ A2 )
         => ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A2 ) )
            = ( groups6602215022474089585er_rat @ G @ A2 ) ) )
        & ( ~ ( member_Code_integer @ X @ A2 )
         => ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5811_sum_Oinsert__if,axiom,
    ! [A2: set_o,X: $o,G: $o > nat] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( member_o @ X @ A2 )
         => ( ( groups8507830703676809646_o_nat @ G @ ( insert_o @ X @ A2 ) )
            = ( groups8507830703676809646_o_nat @ G @ A2 ) ) )
        & ( ~ ( member_o @ X @ A2 )
         => ( ( groups8507830703676809646_o_nat @ G @ ( insert_o @ X @ A2 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups8507830703676809646_o_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5812_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A2 ) )
            = ( groups4541462559716669496nt_nat @ G @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A2 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5813_sum_Oinsert__if,axiom,
    ! [A2: set_Code_integer,X: code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( member_Code_integer @ X @ A2 )
         => ( ( groups7237345082560585321er_nat @ G @ ( insert_Code_integer @ X @ A2 ) )
            = ( groups7237345082560585321er_nat @ G @ A2 ) ) )
        & ( ~ ( member_Code_integer @ X @ A2 )
         => ( ( groups7237345082560585321er_nat @ G @ ( insert_Code_integer @ X @ A2 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups7237345082560585321er_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5814_sum_Oinsert__if,axiom,
    ! [A2: set_o,X: $o,G: $o > int] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( member_o @ X @ A2 )
         => ( ( groups8505340233167759370_o_int @ G @ ( insert_o @ X @ A2 ) )
            = ( groups8505340233167759370_o_int @ G @ A2 ) ) )
        & ( ~ ( member_o @ X @ A2 )
         => ( ( groups8505340233167759370_o_int @ G @ ( insert_o @ X @ A2 ) )
            = ( plus_plus_int @ ( G @ X ) @ ( groups8505340233167759370_o_int @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5815_sum_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ X @ A2 )
         => ( ( groups3539618377306564664at_int @ G @ ( insert_nat @ X @ A2 ) )
            = ( groups3539618377306564664at_int @ G @ A2 ) ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ( groups3539618377306564664at_int @ G @ ( insert_nat @ X @ A2 ) )
            = ( plus_plus_int @ ( G @ X ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5816_sum_Oinsert__if,axiom,
    ! [A2: set_Code_integer,X: code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( member_Code_integer @ X @ A2 )
         => ( ( groups7234854612051535045er_int @ G @ ( insert_Code_integer @ X @ A2 ) )
            = ( groups7234854612051535045er_int @ G @ A2 ) ) )
        & ( ~ ( member_Code_integer @ X @ A2 )
         => ( ( groups7234854612051535045er_int @ G @ ( insert_Code_integer @ X @ A2 ) )
            = ( plus_plus_int @ ( G @ X ) @ ( groups7234854612051535045er_int @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_5817_atLeast0__atMost__Suc,axiom,
    ! [N2: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% atLeast0_atMost_Suc
thf(fact_5818_atLeast0__lessThan__Suc,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% atLeast0_lessThan_Suc
thf(fact_5819_Icc__eq__insert__lb__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( set_or1269000886237332187st_nat @ M2 @ N2 )
        = ( insert_nat @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_5820_atLeastAtMostSuc__conv,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
     => ( ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N2 ) )
        = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_5821_atLeastAtMost__insertL,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( insert_nat @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N2 ) )
        = ( set_or1269000886237332187st_nat @ M2 @ N2 ) ) ) ).

% atLeastAtMost_insertL
thf(fact_5822_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_o,T4: set_o,S2: set_o,I: $o > $o,J: $o > $o,T5: set_o,G: $o > code_integer,H3: $o > code_integer] :
      ( ( finite_finite_o @ S7 )
     => ( ( finite_finite_o @ T4 )
       => ( ! [A3: $o] :
              ( ( member_o @ A3 @ ( minus_minus_set_o @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ S2 @ S7 ) )
               => ( member_o @ ( J @ A3 ) @ ( minus_minus_set_o @ T5 @ T4 ) ) )
           => ( ! [B2: $o] :
                  ( ( member_o @ B2 @ ( minus_minus_set_o @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: $o] :
                    ( ( member_o @ B2 @ ( minus_minus_set_o @ T5 @ T4 ) )
                   => ( member_o @ ( I @ B2 ) @ ( minus_minus_set_o @ S2 @ S7 ) ) )
               => ( ! [A3: $o] :
                      ( ( member_o @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_Code_integer ) )
                 => ( ! [B2: $o] :
                        ( ( member_o @ B2 @ T4 )
                       => ( ( H3 @ B2 )
                          = one_one_Code_integer ) )
                   => ( ! [A3: $o] :
                          ( ( member_o @ A3 @ S2 )
                         => ( ( H3 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups7694694392188491536nteger @ G @ S2 )
                        = ( groups7694694392188491536nteger @ H3 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_5823_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_o,T4: set_int,S2: set_o,I: int > $o,J: $o > int,T5: set_int,G: $o > code_integer,H3: int > code_integer] :
      ( ( finite_finite_o @ S7 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A3: $o] :
              ( ( member_o @ A3 @ ( minus_minus_set_o @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ S2 @ S7 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T5 @ T4 ) ) )
           => ( ! [B2: int] :
                  ( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: int] :
                    ( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T4 ) )
                   => ( member_o @ ( I @ B2 ) @ ( minus_minus_set_o @ S2 @ S7 ) ) )
               => ( ! [A3: $o] :
                      ( ( member_o @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_Code_integer ) )
                 => ( ! [B2: int] :
                        ( ( member_int @ B2 @ T4 )
                       => ( ( H3 @ B2 )
                          = one_one_Code_integer ) )
                   => ( ! [A3: $o] :
                          ( ( member_o @ A3 @ S2 )
                         => ( ( H3 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups7694694392188491536nteger @ G @ S2 )
                        = ( groups3827104343326376752nteger @ H3 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_5824_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_o,T4: set_Code_integer,S2: set_o,I: code_integer > $o,J: $o > code_integer,T5: set_Code_integer,G: $o > code_integer,H3: code_integer > code_integer] :
      ( ( finite_finite_o @ S7 )
     => ( ( finite6017078050557962740nteger @ T4 )
       => ( ! [A3: $o] :
              ( ( member_o @ A3 @ ( minus_minus_set_o @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ S2 @ S7 ) )
               => ( member_Code_integer @ ( J @ A3 ) @ ( minus_2355218937544613996nteger @ T5 @ T4 ) ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: code_integer] :
                    ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T5 @ T4 ) )
                   => ( member_o @ ( I @ B2 ) @ ( minus_minus_set_o @ S2 @ S7 ) ) )
               => ( ! [A3: $o] :
                      ( ( member_o @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_Code_integer ) )
                 => ( ! [B2: code_integer] :
                        ( ( member_Code_integer @ B2 @ T4 )
                       => ( ( H3 @ B2 )
                          = one_one_Code_integer ) )
                   => ( ! [A3: $o] :
                          ( ( member_o @ A3 @ S2 )
                         => ( ( H3 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups7694694392188491536nteger @ G @ S2 )
                        = ( groups3674199335183972705nteger @ H3 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_5825_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_int,T4: set_o,S2: set_int,I: $o > int,J: int > $o,T5: set_o,G: int > code_integer,H3: $o > code_integer] :
      ( ( finite_finite_int @ S7 )
     => ( ( finite_finite_o @ T4 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S7 ) )
               => ( member_o @ ( J @ A3 ) @ ( minus_minus_set_o @ T5 @ T4 ) ) )
           => ( ! [B2: $o] :
                  ( ( member_o @ B2 @ ( minus_minus_set_o @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: $o] :
                    ( ( member_o @ B2 @ ( minus_minus_set_o @ T5 @ T4 ) )
                   => ( member_int @ ( I @ B2 ) @ ( minus_minus_set_int @ S2 @ S7 ) ) )
               => ( ! [A3: int] :
                      ( ( member_int @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_Code_integer ) )
                 => ( ! [B2: $o] :
                        ( ( member_o @ B2 @ T4 )
                       => ( ( H3 @ B2 )
                          = one_one_Code_integer ) )
                   => ( ! [A3: int] :
                          ( ( member_int @ A3 @ S2 )
                         => ( ( H3 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups3827104343326376752nteger @ G @ S2 )
                        = ( groups7694694392188491536nteger @ H3 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_5826_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T5: set_int,G: int > code_integer,H3: int > code_integer] :
      ( ( finite_finite_int @ S7 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S7 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T5 @ T4 ) ) )
           => ( ! [B2: int] :
                  ( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: int] :
                    ( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T4 ) )
                   => ( member_int @ ( I @ B2 ) @ ( minus_minus_set_int @ S2 @ S7 ) ) )
               => ( ! [A3: int] :
                      ( ( member_int @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_Code_integer ) )
                 => ( ! [B2: int] :
                        ( ( member_int @ B2 @ T4 )
                       => ( ( H3 @ B2 )
                          = one_one_Code_integer ) )
                   => ( ! [A3: int] :
                          ( ( member_int @ A3 @ S2 )
                         => ( ( H3 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups3827104343326376752nteger @ G @ S2 )
                        = ( groups3827104343326376752nteger @ H3 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_5827_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_int,T4: set_Code_integer,S2: set_int,I: code_integer > int,J: int > code_integer,T5: set_Code_integer,G: int > code_integer,H3: code_integer > code_integer] :
      ( ( finite_finite_int @ S7 )
     => ( ( finite6017078050557962740nteger @ T4 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ S2 @ S7 ) )
               => ( member_Code_integer @ ( J @ A3 ) @ ( minus_2355218937544613996nteger @ T5 @ T4 ) ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: code_integer] :
                    ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T5 @ T4 ) )
                   => ( member_int @ ( I @ B2 ) @ ( minus_minus_set_int @ S2 @ S7 ) ) )
               => ( ! [A3: int] :
                      ( ( member_int @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_Code_integer ) )
                 => ( ! [B2: code_integer] :
                        ( ( member_Code_integer @ B2 @ T4 )
                       => ( ( H3 @ B2 )
                          = one_one_Code_integer ) )
                   => ( ! [A3: int] :
                          ( ( member_int @ A3 @ S2 )
                         => ( ( H3 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups3827104343326376752nteger @ G @ S2 )
                        = ( groups3674199335183972705nteger @ H3 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_5828_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_Code_integer,T4: set_o,S2: set_Code_integer,I: $o > code_integer,J: code_integer > $o,T5: set_o,G: code_integer > code_integer,H3: $o > code_integer] :
      ( ( finite6017078050557962740nteger @ S7 )
     => ( ( finite_finite_o @ T4 )
       => ( ! [A3: code_integer] :
              ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ S2 @ S7 ) )
               => ( member_o @ ( J @ A3 ) @ ( minus_minus_set_o @ T5 @ T4 ) ) )
           => ( ! [B2: $o] :
                  ( ( member_o @ B2 @ ( minus_minus_set_o @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: $o] :
                    ( ( member_o @ B2 @ ( minus_minus_set_o @ T5 @ T4 ) )
                   => ( member_Code_integer @ ( I @ B2 ) @ ( minus_2355218937544613996nteger @ S2 @ S7 ) ) )
               => ( ! [A3: code_integer] :
                      ( ( member_Code_integer @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_Code_integer ) )
                 => ( ! [B2: $o] :
                        ( ( member_o @ B2 @ T4 )
                       => ( ( H3 @ B2 )
                          = one_one_Code_integer ) )
                   => ( ! [A3: code_integer] :
                          ( ( member_Code_integer @ A3 @ S2 )
                         => ( ( H3 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups3674199335183972705nteger @ G @ S2 )
                        = ( groups7694694392188491536nteger @ H3 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_5829_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_Code_integer,T4: set_int,S2: set_Code_integer,I: int > code_integer,J: code_integer > int,T5: set_int,G: code_integer > code_integer,H3: int > code_integer] :
      ( ( finite6017078050557962740nteger @ S7 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A3: code_integer] :
              ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ S2 @ S7 ) )
               => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T5 @ T4 ) ) )
           => ( ! [B2: int] :
                  ( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: int] :
                    ( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T4 ) )
                   => ( member_Code_integer @ ( I @ B2 ) @ ( minus_2355218937544613996nteger @ S2 @ S7 ) ) )
               => ( ! [A3: code_integer] :
                      ( ( member_Code_integer @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_Code_integer ) )
                 => ( ! [B2: int] :
                        ( ( member_int @ B2 @ T4 )
                       => ( ( H3 @ B2 )
                          = one_one_Code_integer ) )
                   => ( ! [A3: code_integer] :
                          ( ( member_Code_integer @ A3 @ S2 )
                         => ( ( H3 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups3674199335183972705nteger @ G @ S2 )
                        = ( groups3827104343326376752nteger @ H3 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_5830_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_Code_integer,T4: set_Code_integer,S2: set_Code_integer,I: code_integer > code_integer,J: code_integer > code_integer,T5: set_Code_integer,G: code_integer > code_integer,H3: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ S7 )
     => ( ( finite6017078050557962740nteger @ T4 )
       => ( ! [A3: code_integer] :
              ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ S2 @ S7 ) )
               => ( member_Code_integer @ ( J @ A3 ) @ ( minus_2355218937544613996nteger @ T5 @ T4 ) ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: code_integer] :
                    ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ T5 @ T4 ) )
                   => ( member_Code_integer @ ( I @ B2 ) @ ( minus_2355218937544613996nteger @ S2 @ S7 ) ) )
               => ( ! [A3: code_integer] :
                      ( ( member_Code_integer @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_Code_integer ) )
                 => ( ! [B2: code_integer] :
                        ( ( member_Code_integer @ B2 @ T4 )
                       => ( ( H3 @ B2 )
                          = one_one_Code_integer ) )
                   => ( ! [A3: code_integer] :
                          ( ( member_Code_integer @ A3 @ S2 )
                         => ( ( H3 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups3674199335183972705nteger @ G @ S2 )
                        = ( groups3674199335183972705nteger @ H3 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_5831_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_o,T4: set_o,S2: set_o,I: $o > $o,J: $o > $o,T5: set_o,G: $o > assn,H3: $o > assn] :
      ( ( finite_finite_o @ S7 )
     => ( ( finite_finite_o @ T4 )
       => ( ! [A3: $o] :
              ( ( member_o @ A3 @ ( minus_minus_set_o @ S2 @ S7 ) )
             => ( ( I @ ( J @ A3 ) )
                = A3 ) )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ S2 @ S7 ) )
               => ( member_o @ ( J @ A3 ) @ ( minus_minus_set_o @ T5 @ T4 ) ) )
           => ( ! [B2: $o] :
                  ( ( member_o @ B2 @ ( minus_minus_set_o @ T5 @ T4 ) )
                 => ( ( J @ ( I @ B2 ) )
                    = B2 ) )
             => ( ! [B2: $o] :
                    ( ( member_o @ B2 @ ( minus_minus_set_o @ T5 @ T4 ) )
                   => ( member_o @ ( I @ B2 ) @ ( minus_minus_set_o @ S2 @ S7 ) ) )
               => ( ! [A3: $o] :
                      ( ( member_o @ A3 @ S7 )
                     => ( ( G @ A3 )
                        = one_one_assn ) )
                 => ( ! [B2: $o] :
                        ( ( member_o @ B2 @ T4 )
                       => ( ( H3 @ B2 )
                          = one_one_assn ) )
                   => ( ! [A3: $o] :
                          ( ( member_o @ A3 @ S2 )
                         => ( ( H3 @ ( J @ A3 ) )
                            = ( G @ A3 ) ) )
                     => ( ( groups5301882518646026715o_assn @ G @ S2 )
                        = ( groups5301882518646026715o_assn @ H3 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
thf(fact_5832_dflt__None__set__def,axiom,
    ( dflt_N3208751546879591902at_nat
    = ( ^ [S5: set_Pr4329608150637261639at_nat] : ( if_opt5121585740920863443at_nat @ ( S5 = bot_bo228742789529271731at_nat ) @ none_s6518873173954064040at_nat @ ( some_s5890477192898017836at_nat @ S5 ) ) ) ) ).

% dflt_None_set_def
thf(fact_5833_dflt__None__set__def,axiom,
    ( dflt_N6592383573632408824at_nat
    = ( ^ [S5: set_Pr1261947904930325089at_nat] : ( if_opt7704869406773131885at_nat @ ( S5 = bot_bo2099793752762293965at_nat ) @ none_s625347054029921090at_nat @ ( some_s147305329494351046at_nat @ S5 ) ) ) ) ).

% dflt_None_set_def
thf(fact_5834_dflt__None__set__def,axiom,
    ( dflt_None_set_o
    = ( ^ [S5: set_o] : ( if_option_set_o @ ( S5 = bot_bot_set_o ) @ none_set_o @ ( some_set_o @ S5 ) ) ) ) ).

% dflt_None_set_def
thf(fact_5835_dflt__None__set__def,axiom,
    ( dflt_None_set_nat
    = ( ^ [S5: set_nat] : ( if_option_set_nat @ ( S5 = bot_bot_set_nat ) @ none_set_nat @ ( some_set_nat @ S5 ) ) ) ) ).

% dflt_None_set_def
thf(fact_5836_dflt__None__set__def,axiom,
    ( dflt_None_set_int
    = ( ^ [S5: set_int] : ( if_option_set_int @ ( S5 = bot_bot_set_int ) @ none_set_int @ ( some_set_int @ S5 ) ) ) ) ).

% dflt_None_set_def
thf(fact_5837_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X3: int] :
              ( ( member_int @ X3 @ A2 )
              & ( ( F @ X3 )
                = one_one_nat )
              & ! [Y3: int] :
                  ( ( member_int @ Y3 @ A2 )
                 => ( ( X3 != Y3 )
                   => ( ( F @ Y3 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_5838_sum__eq__1__iff,axiom,
    ! [A2: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( ( groups7237345082560585321er_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A2 )
              & ( ( F @ X3 )
                = one_one_nat )
              & ! [Y3: code_integer] :
                  ( ( member_Code_integer @ Y3 @ A2 )
                 => ( ( X3 != Y3 )
                   => ( ( F @ Y3 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_5839_sum__eq__1__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( ( groups977919841031483927at_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ A2 )
              & ( ( F @ X3 )
                = one_one_nat )
              & ! [Y3: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Y3 @ A2 )
                 => ( ( X3 != Y3 )
                   => ( ( F @ Y3 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_5840_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
              & ( ( F @ X3 )
                = one_one_nat )
              & ! [Y3: nat] :
                  ( ( member_nat @ Y3 @ A2 )
                 => ( ( X3 != Y3 )
                   => ( ( F @ Y3 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_5841_surj__Compl__image__subset,axiom,
    ! [F: nat > code_integer,A2: set_nat] :
      ( ( ( image_1215581382706833972nteger @ F @ top_top_set_nat )
        = top_to4645266643341252675nteger )
     => ( ord_le7084787975880047091nteger @ ( uminus804700908173204444nteger @ ( image_1215581382706833972nteger @ F @ A2 ) ) @ ( image_1215581382706833972nteger @ F @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_5842_surj__Compl__image__subset,axiom,
    ! [F: nat > product_unit,A2: set_nat] :
      ( ( ( image_8730104196221521654t_unit @ F @ top_top_set_nat )
        = top_to1996260823553986621t_unit )
     => ( ord_le3507040750410214029t_unit @ ( uminus5944136376168626660t_unit @ ( image_8730104196221521654t_unit @ F @ A2 ) ) @ ( image_8730104196221521654t_unit @ F @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_5843_surj__Compl__image__subset,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ( image_nat_int @ F @ top_top_set_nat )
        = top_top_set_int )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ ( image_nat_int @ F @ A2 ) ) @ ( image_nat_int @ F @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_5844_surj__Compl__image__subset,axiom,
    ! [F: product_unit > product_unit,A2: set_Product_unit] :
      ( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
        = top_to1996260823553986621t_unit )
     => ( ord_le3507040750410214029t_unit @ ( uminus5944136376168626660t_unit @ ( image_405062704495631173t_unit @ F @ A2 ) ) @ ( image_405062704495631173t_unit @ F @ ( uminus5944136376168626660t_unit @ A2 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_5845_surj__Compl__image__subset,axiom,
    ! [F: product_unit > int,A2: set_Product_unit] :
      ( ( ( image_873079544045703924it_int @ F @ top_to1996260823553986621t_unit )
        = top_top_set_int )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ ( image_873079544045703924it_int @ F @ A2 ) ) @ ( image_873079544045703924it_int @ F @ ( uminus5944136376168626660t_unit @ A2 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_5846_surj__Compl__image__subset,axiom,
    ! [F: int > product_unit,A2: set_int] :
      ( ( ( image_5846526143423701914t_unit @ F @ top_top_set_int )
        = top_to1996260823553986621t_unit )
     => ( ord_le3507040750410214029t_unit @ ( uminus5944136376168626660t_unit @ ( image_5846526143423701914t_unit @ F @ A2 ) ) @ ( image_5846526143423701914t_unit @ F @ ( uminus1532241313380277803et_int @ A2 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_5847_surj__Compl__image__subset,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ( image_int_int @ F @ top_top_set_int )
        = top_top_set_int )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ ( image_int_int @ F @ A2 ) ) @ ( image_int_int @ F @ ( uminus1532241313380277803et_int @ A2 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_5848_surj__Compl__image__subset,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A2 ) ) @ ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_5849_surj__Compl__image__subset,axiom,
    ! [F: product_unit > nat,A2: set_Product_unit] :
      ( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
        = top_top_set_nat )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_875570014554754200it_nat @ F @ A2 ) ) @ ( image_875570014554754200it_nat @ F @ ( uminus5944136376168626660t_unit @ A2 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_5850_surj__Compl__image__subset,axiom,
    ! [F: int > nat,A2: set_int] :
      ( ( ( image_int_nat @ F @ top_top_set_int )
        = top_top_set_nat )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_int_nat @ F @ A2 ) ) @ ( image_int_nat @ F @ ( uminus1532241313380277803et_int @ A2 ) ) ) ) ).

% surj_Compl_image_subset
thf(fact_5851_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
      = ( nat2 @ U ) ) ).

% card_atLeastZeroLessThan_int
thf(fact_5852_atMost__nat__numeral,axiom,
    ! [K: num] :
      ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
      = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).

% atMost_nat_numeral
thf(fact_5853_lessThan__nat__numeral,axiom,
    ! [K: num] :
      ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
      = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).

% lessThan_nat_numeral
thf(fact_5854_sum__Suc,axiom,
    ! [F: int > nat,A2: set_int] :
      ( ( groups4541462559716669496nt_nat
        @ ^ [X3: int] : ( suc @ ( F @ X3 ) )
        @ A2 )
      = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( finite_card_int @ A2 ) ) ) ).

% sum_Suc
thf(fact_5855_sum__Suc,axiom,
    ! [F: product_unit > nat,A2: set_Product_unit] :
      ( ( groups8766797177574957219it_nat
        @ ^ [X3: product_unit] : ( suc @ ( F @ X3 ) )
        @ A2 )
      = ( plus_plus_nat @ ( groups8766797177574957219it_nat @ F @ A2 ) @ ( finite410649719033368117t_unit @ A2 ) ) ) ).

% sum_Suc
thf(fact_5856_sum__Suc,axiom,
    ! [F: list_nat > nat,A2: set_list_nat] :
      ( ( groups4396056296759096172at_nat
        @ ^ [X3: list_nat] : ( suc @ ( F @ X3 ) )
        @ A2 )
      = ( plus_plus_nat @ ( groups4396056296759096172at_nat @ F @ A2 ) @ ( finite_card_list_nat @ A2 ) ) ) ).

% sum_Suc
thf(fact_5857_sum__Suc,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : ( suc @ ( F @ X3 ) )
        @ A2 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( finite_card_nat @ A2 ) ) ) ).

% sum_Suc
thf(fact_5858_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3827104343326376752nteger @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G @ X3 )
                  = one_one_Code_integer ) ) ) )
        = ( groups3827104343326376752nteger @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_5859_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups3674199335183972705nteger @ G
          @ ( minus_2355218937544613996nteger @ A2
            @ ( collect_Code_integer
              @ ^ [X3: code_integer] :
                  ( ( G @ X3 )
                  = one_one_Code_integer ) ) ) )
        = ( groups3674199335183972705nteger @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_5860_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > assn] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups7882442080178216443t_assn @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G @ X3 )
                  = one_one_assn ) ) ) )
        = ( groups7882442080178216443t_assn @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_5861_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups1304777262505850412r_assn @ G
          @ ( minus_2355218937544613996nteger @ A2
            @ ( collect_Code_integer
              @ ^ [X3: code_integer] :
                  ( ( G @ X3 )
                  = one_one_assn ) ) ) )
        = ( groups1304777262505850412r_assn @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_5862_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups1072433553688619179nt_rat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G @ X3 )
                  = one_one_rat ) ) ) )
        = ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_5863_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups2555765274223993564er_rat @ G
          @ ( minus_2355218937544613996nteger @ A2
            @ ( collect_Code_integer
              @ ^ [X3: code_integer] :
                  ( ( G @ X3 )
                  = one_one_rat ) ) ) )
        = ( groups2555765274223993564er_rat @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_5864_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups1707563613775114915nt_nat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G @ X3 )
                  = one_one_nat ) ) ) )
        = ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_5865_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups3190895334310489300er_nat @ G
          @ ( minus_2355218937544613996nteger @ A2
            @ ( collect_Code_integer
              @ ^ [X3: code_integer] :
                  ( ( G @ X3 )
                  = one_one_nat ) ) ) )
        = ( groups3190895334310489300er_nat @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_5866_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups3188404863801439024er_int @ G
          @ ( minus_2355218937544613996nteger @ A2
            @ ( collect_Code_integer
              @ ^ [X3: code_integer] :
                  ( ( G @ X3 )
                  = one_one_int ) ) ) )
        = ( groups3188404863801439024er_int @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_5867_prod_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups3455450783089532116nteger @ G
          @ ( minus_minus_set_nat @ A2
            @ ( collect_nat
              @ ^ [X3: nat] :
                  ( ( G @ X3 )
                  = one_one_Code_integer ) ) ) )
        = ( groups3455450783089532116nteger @ G @ A2 ) ) ) ).

% prod.setdiff_irrelevant
thf(fact_5868_notin__range__Some,axiom,
    ! [X: option3216581401566546698it_nat] :
      ( ( ~ ( member3517700364412436275it_nat @ X @ ( image_4799632500972785611it_nat @ some_P6105628898608307625it_nat @ top_to3995162779041354868it_nat ) ) )
      = ( X = none_P2587165478634041381it_nat ) ) ).

% notin_range_Some
thf(fact_5869_notin__range__Some,axiom,
    ! [X: option_num] :
      ( ( ~ ( member_option_num @ X @ ( image_num_option_num @ some_num @ top_top_set_num ) ) )
      = ( X = none_num ) ) ).

% notin_range_Some
thf(fact_5870_notin__range__Some,axiom,
    ! [X: option8956607266484857688it_nat] :
      ( ( ~ ( member5596548051065438575it_nat @ X @ ( image_3455408117458022631it_nat @ some_P1914260805536162275it_nat @ top_to4082048134677576814it_nat ) ) )
      = ( X = none_P9117596204409417319it_nat ) ) ).

% notin_range_Some
thf(fact_5871_notin__range__Some,axiom,
    ! [X: option_nat] :
      ( ( ~ ( member_option_nat @ X @ ( image_nat_option_nat @ some_nat @ top_top_set_nat ) ) )
      = ( X = none_nat ) ) ).

% notin_range_Some
thf(fact_5872_notin__range__Some,axiom,
    ! [X: option_Product_unit] :
      ( ( ~ ( member1378114217278917244t_unit @ X @ ( image_7094380055901048587t_unit @ some_Product_unit @ top_to1996260823553986621t_unit ) ) )
      = ( X = none_Product_unit ) ) ).

% notin_range_Some
thf(fact_5873_notin__range__Some,axiom,
    ! [X: option_int] :
      ( ( ~ ( member_option_int @ X @ ( image_int_option_int @ some_int @ top_top_set_int ) ) )
      = ( X = none_int ) ) ).

% notin_range_Some
thf(fact_5874_less__1__prod2,axiom,
    ! [I4: set_o,I: $o,F: $o > code_integer] :
      ( ( finite_finite_o @ I4 )
     => ( ( member_o @ I @ I4 )
       => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I ) )
         => ( ! [I2: $o] :
                ( ( member_o @ I2 @ I4 )
               => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I2 ) ) )
           => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups7694694392188491536nteger @ F @ I4 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_5875_less__1__prod2,axiom,
    ! [I4: set_nat,I: nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ I4 )
     => ( ( member_nat @ I @ I4 )
       => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I4 )
               => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I2 ) ) )
           => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F @ I4 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_5876_less__1__prod2,axiom,
    ! [I4: set_int,I: int,F: int > code_integer] :
      ( ( finite_finite_int @ I4 )
     => ( ( member_int @ I @ I4 )
       => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I4 )
               => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I2 ) ) )
           => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F @ I4 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_5877_less__1__prod2,axiom,
    ! [I4: set_Code_integer,I: code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ I4 )
     => ( ( member_Code_integer @ I @ I4 )
       => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I ) )
         => ( ! [I2: code_integer] :
                ( ( member_Code_integer @ I2 @ I4 )
               => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F @ I2 ) ) )
           => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3674199335183972705nteger @ F @ I4 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_5878_less__1__prod2,axiom,
    ! [I4: set_o,I: $o,F: $o > rat] :
      ( ( finite_finite_o @ I4 )
     => ( ( member_o @ I @ I4 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: $o] :
                ( ( member_o @ I2 @ I4 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups2869687844427037835_o_rat @ F @ I4 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_5879_less__1__prod2,axiom,
    ! [I4: set_nat,I: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I4 )
     => ( ( member_nat @ I @ I4 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I4 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I4 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_5880_less__1__prod2,axiom,
    ! [I4: set_int,I: int,F: int > rat] :
      ( ( finite_finite_int @ I4 )
     => ( ( member_int @ I @ I4 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I4 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I4 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_5881_less__1__prod2,axiom,
    ! [I4: set_Code_integer,I: code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ I4 )
     => ( ( member_Code_integer @ I @ I4 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I2: code_integer] :
                ( ( member_Code_integer @ I2 @ I4 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups2555765274223993564er_rat @ F @ I4 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_5882_less__1__prod2,axiom,
    ! [I4: set_o,I: $o,F: $o > int] :
      ( ( finite_finite_o @ I4 )
     => ( ( member_o @ I @ I4 )
       => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
         => ( ! [I2: $o] :
                ( ( member_o @ I2 @ I4 )
               => ( ord_less_eq_int @ one_one_int @ ( F @ I2 ) ) )
           => ( ord_less_int @ one_one_int @ ( groups3502327434004483295_o_int @ F @ I4 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_5883_less__1__prod2,axiom,
    ! [I4: set_Code_integer,I: code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ I4 )
     => ( ( member_Code_integer @ I @ I4 )
       => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
         => ( ! [I2: code_integer] :
                ( ( member_Code_integer @ I2 @ I4 )
               => ( ord_less_eq_int @ one_one_int @ ( F @ I2 ) ) )
           => ( ord_less_int @ one_one_int @ ( groups3188404863801439024er_int @ F @ I4 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_5884_less__1__prod,axiom,
    ! [I4: set_Code_integer,F: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ I4 )
     => ( ( I4 != bot_bo3990330152332043303nteger )
       => ( ! [I2: code_integer] :
              ( ( member_Code_integer @ I2 @ I4 )
             => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) ) )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3674199335183972705nteger @ F @ I4 ) ) ) ) ) ).

% less_1_prod
thf(fact_5885_less__1__prod,axiom,
    ! [I4: set_o,F: $o > code_integer] :
      ( ( finite_finite_o @ I4 )
     => ( ( I4 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I4 )
             => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) ) )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups7694694392188491536nteger @ F @ I4 ) ) ) ) ) ).

% less_1_prod
thf(fact_5886_less__1__prod,axiom,
    ! [I4: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ I4 )
     => ( ( I4 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I4 )
             => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) ) )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F @ I4 ) ) ) ) ) ).

% less_1_prod
thf(fact_5887_less__1__prod,axiom,
    ! [I4: set_int,F: int > code_integer] :
      ( ( finite_finite_int @ I4 )
     => ( ( I4 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I4 )
             => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F @ I2 ) ) )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F @ I4 ) ) ) ) ) ).

% less_1_prod
thf(fact_5888_less__1__prod,axiom,
    ! [I4: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ I4 )
     => ( ( I4 != bot_bo3990330152332043303nteger )
       => ( ! [I2: code_integer] :
              ( ( member_Code_integer @ I2 @ I4 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups2555765274223993564er_rat @ F @ I4 ) ) ) ) ) ).

% less_1_prod
thf(fact_5889_less__1__prod,axiom,
    ! [I4: set_o,F: $o > rat] :
      ( ( finite_finite_o @ I4 )
     => ( ( I4 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I4 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups2869687844427037835_o_rat @ F @ I4 ) ) ) ) ) ).

% less_1_prod
thf(fact_5890_less__1__prod,axiom,
    ! [I4: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I4 )
     => ( ( I4 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I4 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I4 ) ) ) ) ) ).

% less_1_prod
thf(fact_5891_less__1__prod,axiom,
    ! [I4: set_int,F: int > rat] :
      ( ( finite_finite_int @ I4 )
     => ( ( I4 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I4 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I4 ) ) ) ) ) ).

% less_1_prod
thf(fact_5892_less__1__prod,axiom,
    ! [I4: set_Code_integer,F: code_integer > int] :
      ( ( finite6017078050557962740nteger @ I4 )
     => ( ( I4 != bot_bo3990330152332043303nteger )
       => ( ! [I2: code_integer] :
              ( ( member_Code_integer @ I2 @ I4 )
             => ( ord_less_int @ one_one_int @ ( F @ I2 ) ) )
         => ( ord_less_int @ one_one_int @ ( groups3188404863801439024er_int @ F @ I4 ) ) ) ) ) ).

% less_1_prod
thf(fact_5893_less__1__prod,axiom,
    ! [I4: set_o,F: $o > int] :
      ( ( finite_finite_o @ I4 )
     => ( ( I4 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I4 )
             => ( ord_less_int @ one_one_int @ ( F @ I2 ) ) )
         => ( ord_less_int @ one_one_int @ ( groups3502327434004483295_o_int @ F @ I4 ) ) ) ) ) ).

% less_1_prod
thf(fact_5894_sum_Oreindex__nontrivial,axiom,
    ! [A2: set_num,H3: num > code_integer,G: code_integer > code_integer] :
      ( ( finite_finite_num @ A2 )
     => ( ! [X4: num,Y5: num] :
            ( ( member_num @ X4 @ A2 )
           => ( ( member_num @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = zero_z3403309356797280102nteger ) ) ) ) )
       => ( ( groups879477027807139574nteger @ G @ ( image_8379761247532268670nteger @ H3 @ A2 ) )
          = ( groups5442708359310287027nteger @ ( comp_C3531382070062128313er_num @ G @ H3 ) @ A2 ) ) ) ) ).

% sum.reindex_nontrivial
thf(fact_5895_sum_Oreindex__nontrivial,axiom,
    ! [A2: set_nat,H3: nat > code_integer,G: code_integer > code_integer] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat,Y5: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ( member_nat @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = zero_z3403309356797280102nteger ) ) ) ) )
       => ( ( groups879477027807139574nteger @ G @ ( image_1215581382706833972nteger @ H3 @ A2 ) )
          = ( groups7501900531339628137nteger @ ( comp_C6974050283906349679er_nat @ G @ H3 ) @ A2 ) ) ) ) ).

% sum.reindex_nontrivial
thf(fact_5896_sum_Oreindex__nontrivial,axiom,
    ! [A2: set_nat,H3: nat > nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat,Y5: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ( member_nat @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = zero_z3403309356797280102nteger ) ) ) ) )
       => ( ( groups7501900531339628137nteger @ G @ ( image_nat_nat @ H3 @ A2 ) )
          = ( groups7501900531339628137nteger @ ( comp_n3898172953802868194er_nat @ G @ H3 ) @ A2 ) ) ) ) ).

% sum.reindex_nontrivial
thf(fact_5897_sum_Oreindex__nontrivial,axiom,
    ! [A2: set_nat,H3: nat > int,G: int > code_integer] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat,Y5: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ( member_nat @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = zero_z3403309356797280102nteger ) ) ) ) )
       => ( ( groups7873554091576472773nteger @ G @ ( image_nat_int @ H3 @ A2 ) )
          = ( groups7501900531339628137nteger @ ( comp_i1588355021709917246er_nat @ G @ H3 ) @ A2 ) ) ) ) ).

% sum.reindex_nontrivial
thf(fact_5898_sum_Oreindex__nontrivial,axiom,
    ! [A2: set_int,H3: int > int,G: int > code_integer] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X4: int,Y5: int] :
            ( ( member_int @ X4 @ A2 )
           => ( ( member_int @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = zero_z3403309356797280102nteger ) ) ) ) )
       => ( ( groups7873554091576472773nteger @ G @ ( image_int_int @ H3 @ A2 ) )
          = ( groups7873554091576472773nteger @ ( comp_i1585864551200866970er_int @ G @ H3 ) @ A2 ) ) ) ) ).

% sum.reindex_nontrivial
thf(fact_5899_sum_Oreindex__nontrivial,axiom,
    ! [A2: set_nat,H3: nat > code_integer,G: code_integer > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat,Y5: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ( member_nat @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = zero_zero_rat ) ) ) ) )
       => ( ( groups6602215022474089585er_rat @ G @ ( image_1215581382706833972nteger @ H3 @ A2 ) )
          = ( groups2906978787729119204at_rat @ ( comp_C3733001951728184298at_nat @ G @ H3 ) @ A2 ) ) ) ) ).

% sum.reindex_nontrivial
thf(fact_5900_sum_Oreindex__nontrivial,axiom,
    ! [A2: set_nat,H3: nat > nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat,Y5: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ( member_nat @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = zero_zero_rat ) ) ) ) )
       => ( ( groups2906978787729119204at_rat @ G @ ( image_nat_nat @ H3 @ A2 ) )
          = ( groups2906978787729119204at_rat @ ( comp_nat_rat_nat @ G @ H3 ) @ A2 ) ) ) ) ).

% sum.reindex_nontrivial
thf(fact_5901_sum_Oreindex__nontrivial,axiom,
    ! [A2: set_nat,H3: nat > int,G: int > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat,Y5: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ( member_nat @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = zero_zero_rat ) ) ) ) )
       => ( ( groups3906332499630173760nt_rat @ G @ ( image_nat_int @ H3 @ A2 ) )
          = ( groups2906978787729119204at_rat @ ( comp_int_rat_nat @ G @ H3 ) @ A2 ) ) ) ) ).

% sum.reindex_nontrivial
thf(fact_5902_sum_Oreindex__nontrivial,axiom,
    ! [A2: set_int,H3: int > int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X4: int,Y5: int] :
            ( ( member_int @ X4 @ A2 )
           => ( ( member_int @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = zero_zero_rat ) ) ) ) )
       => ( ( groups3906332499630173760nt_rat @ G @ ( image_int_int @ H3 @ A2 ) )
          = ( groups3906332499630173760nt_rat @ ( comp_int_rat_int @ G @ H3 ) @ A2 ) ) ) ) ).

% sum.reindex_nontrivial
thf(fact_5903_sum_Oreindex__nontrivial,axiom,
    ! [A2: set_int,H3: int > int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X4: int,Y5: int] :
            ( ( member_int @ X4 @ A2 )
           => ( ( member_int @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = zero_zero_nat ) ) ) ) )
       => ( ( groups4541462559716669496nt_nat @ G @ ( image_int_int @ H3 @ A2 ) )
          = ( groups4541462559716669496nt_nat @ ( comp_int_nat_int @ G @ H3 ) @ A2 ) ) ) ) ).

% sum.reindex_nontrivial
thf(fact_5904_prod_Oreindex__nontrivial,axiom,
    ! [A2: set_num,H3: num > code_integer,G: code_integer > code_integer] :
      ( ( finite_finite_num @ A2 )
     => ( ! [X4: num,Y5: num] :
            ( ( member_num @ X4 @ A2 )
           => ( ( member_num @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = one_one_Code_integer ) ) ) ) )
       => ( ( groups3674199335183972705nteger @ G @ ( image_8379761247532268670nteger @ H3 @ A2 ) )
          = ( groups1396258611060191006nteger @ ( comp_C3531382070062128313er_num @ G @ H3 ) @ A2 ) ) ) ) ).

% prod.reindex_nontrivial
thf(fact_5905_prod_Oreindex__nontrivial,axiom,
    ! [A2: set_nat,H3: nat > code_integer,G: code_integer > code_integer] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat,Y5: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ( member_nat @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = one_one_Code_integer ) ) ) ) )
       => ( ( groups3674199335183972705nteger @ G @ ( image_1215581382706833972nteger @ H3 @ A2 ) )
          = ( groups3455450783089532116nteger @ ( comp_C6974050283906349679er_nat @ G @ H3 ) @ A2 ) ) ) ) ).

% prod.reindex_nontrivial
thf(fact_5906_prod_Oreindex__nontrivial,axiom,
    ! [A2: set_nat,H3: nat > nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat,Y5: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ( member_nat @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = one_one_Code_integer ) ) ) ) )
       => ( ( groups3455450783089532116nteger @ G @ ( image_nat_nat @ H3 @ A2 ) )
          = ( groups3455450783089532116nteger @ ( comp_n3898172953802868194er_nat @ G @ H3 ) @ A2 ) ) ) ) ).

% prod.reindex_nontrivial
thf(fact_5907_prod_Oreindex__nontrivial,axiom,
    ! [A2: set_nat,H3: nat > int,G: int > code_integer] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat,Y5: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ( member_nat @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = one_one_Code_integer ) ) ) ) )
       => ( ( groups3827104343326376752nteger @ G @ ( image_nat_int @ H3 @ A2 ) )
          = ( groups3455450783089532116nteger @ ( comp_i1588355021709917246er_nat @ G @ H3 ) @ A2 ) ) ) ) ).

% prod.reindex_nontrivial
thf(fact_5908_prod_Oreindex__nontrivial,axiom,
    ! [A2: set_int,H3: int > int,G: int > code_integer] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X4: int,Y5: int] :
            ( ( member_int @ X4 @ A2 )
           => ( ( member_int @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = one_one_Code_integer ) ) ) ) )
       => ( ( groups3827104343326376752nteger @ G @ ( image_int_int @ H3 @ A2 ) )
          = ( groups3827104343326376752nteger @ ( comp_i1585864551200866970er_int @ G @ H3 ) @ A2 ) ) ) ) ).

% prod.reindex_nontrivial
thf(fact_5909_prod_Oreindex__nontrivial,axiom,
    ! [A2: set_nat,H3: nat > code_integer,G: code_integer > assn] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat,Y5: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ( member_nat @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = one_one_assn ) ) ) ) )
       => ( ( groups1304777262505850412r_assn @ G @ ( image_1215581382706833972nteger @ H3 @ A2 ) )
          = ( groups6906906614972039071t_assn @ ( comp_C8462447471145125178sn_nat @ G @ H3 ) @ A2 ) ) ) ) ).

% prod.reindex_nontrivial
thf(fact_5910_prod_Oreindex__nontrivial,axiom,
    ! [A2: set_nat,H3: nat > nat,G: nat > assn] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat,Y5: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ( member_nat @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = one_one_assn ) ) ) ) )
       => ( ( groups6906906614972039071t_assn @ G @ ( image_nat_nat @ H3 @ A2 ) )
          = ( groups6906906614972039071t_assn @ ( comp_nat_assn_nat @ G @ H3 ) @ A2 ) ) ) ) ).

% prod.reindex_nontrivial
thf(fact_5911_prod_Oreindex__nontrivial,axiom,
    ! [A2: set_nat,H3: nat > int,G: int > assn] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat,Y5: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ( member_nat @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = one_one_assn ) ) ) ) )
       => ( ( groups7882442080178216443t_assn @ G @ ( image_nat_int @ H3 @ A2 ) )
          = ( groups6906906614972039071t_assn @ ( comp_int_assn_nat @ G @ H3 ) @ A2 ) ) ) ) ).

% prod.reindex_nontrivial
thf(fact_5912_prod_Oreindex__nontrivial,axiom,
    ! [A2: set_int,H3: int > int,G: int > assn] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X4: int,Y5: int] :
            ( ( member_int @ X4 @ A2 )
           => ( ( member_int @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = one_one_assn ) ) ) ) )
       => ( ( groups7882442080178216443t_assn @ G @ ( image_int_int @ H3 @ A2 ) )
          = ( groups7882442080178216443t_assn @ ( comp_int_assn_int @ G @ H3 ) @ A2 ) ) ) ) ).

% prod.reindex_nontrivial
thf(fact_5913_prod_Oreindex__nontrivial,axiom,
    ! [A2: set_nat,H3: nat > code_integer,G: code_integer > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat,Y5: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ( member_nat @ Y5 @ A2 )
             => ( ( X4 != Y5 )
               => ( ( ( H3 @ X4 )
                    = ( H3 @ Y5 ) )
                 => ( ( G @ ( H3 @ X4 ) )
                    = one_one_rat ) ) ) ) )
       => ( ( groups2555765274223993564er_rat @ G @ ( image_1215581382706833972nteger @ H3 @ A2 ) )
          = ( groups73079841787564623at_rat @ ( comp_C3733001951728184298at_nat @ G @ H3 ) @ A2 ) ) ) ) ).

% prod.reindex_nontrivial
thf(fact_5914_sum_Osubset__diff,axiom,
    ! [B5: set_int,A2: set_int,G: int > rat] :
      ( ( ord_less_eq_set_int @ B5 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups3906332499630173760nt_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups3906332499630173760nt_rat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5915_sum_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,G: code_integer > rat] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
     => ( ( finite6017078050557962740nteger @ A2 )
       => ( ( groups6602215022474089585er_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A2 @ B5 ) ) @ ( groups6602215022474089585er_rat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5916_sum_Osubset__diff,axiom,
    ! [B5: set_int,A2: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ B5 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups4541462559716669496nt_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups4541462559716669496nt_nat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5917_sum_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,G: code_integer > nat] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
     => ( ( finite6017078050557962740nteger @ A2 )
       => ( ( groups7237345082560585321er_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A2 @ B5 ) ) @ ( groups7237345082560585321er_nat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5918_sum_Osubset__diff,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer,G: code_integer > int] :
      ( ( ord_le7084787975880047091nteger @ B5 @ A2 )
     => ( ( finite6017078050557962740nteger @ A2 )
       => ( ( groups7234854612051535045er_int @ G @ A2 )
          = ( plus_plus_int @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A2 @ B5 ) ) @ ( groups7234854612051535045er_int @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5919_sum_Osubset__diff,axiom,
    ! [B5: set_nat,A2: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B5 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups2906978787729119204at_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups2906978787729119204at_rat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5920_sum_Osubset__diff,axiom,
    ! [B5: set_nat,A2: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ B5 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups3539618377306564664at_int @ G @ A2 )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups3539618377306564664at_int @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5921_sum_Osubset__diff,axiom,
    ! [B5: set_nat,A2: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ B5 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups3542108847815614940at_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups3542108847815614940at_nat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5922_sum_Osubset__diff,axiom,
    ! [B5: set_int,A2: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ B5 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups4538972089207619220nt_int @ G @ A2 )
          = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups4538972089207619220nt_int @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5923_sum_Osubset__diff,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > rat] :
      ( ( ord_le3146513528884898305at_nat @ B5 @ A2 )
     => ( ( finite6177210948735845034at_nat @ A2 )
       => ( ( groups342789780944988191at_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups342789780944988191at_rat @ G @ ( minus_1356011639430497352at_nat @ A2 @ B5 ) ) @ ( groups342789780944988191at_rat @ G @ B5 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_5924_prod_Osame__carrier,axiom,
    ! [C4: set_o,A2: set_o,B5: set_o,G: $o > code_integer,H3: $o > code_integer] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A2 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_Code_integer ) )
           => ( ! [B2: $o] :
                  ( ( member_o @ B2 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_Code_integer ) )
             => ( ( ( groups7694694392188491536nteger @ G @ A2 )
                  = ( groups7694694392188491536nteger @ H3 @ B5 ) )
                = ( ( groups7694694392188491536nteger @ G @ C4 )
                  = ( groups7694694392188491536nteger @ H3 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_5925_prod_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B5: set_int,G: int > code_integer,H3: int > code_integer] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B5 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_Code_integer ) )
           => ( ! [B2: int] :
                  ( ( member_int @ B2 @ ( minus_minus_set_int @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_Code_integer ) )
             => ( ( ( groups3827104343326376752nteger @ G @ A2 )
                  = ( groups3827104343326376752nteger @ H3 @ B5 ) )
                = ( ( groups3827104343326376752nteger @ G @ C4 )
                  = ( groups3827104343326376752nteger @ H3 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_5926_prod_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B5: set_Code_integer,G: code_integer > code_integer,H3: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_Code_integer ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_Code_integer ) )
             => ( ( ( groups3674199335183972705nteger @ G @ A2 )
                  = ( groups3674199335183972705nteger @ H3 @ B5 ) )
                = ( ( groups3674199335183972705nteger @ G @ C4 )
                  = ( groups3674199335183972705nteger @ H3 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_5927_prod_Osame__carrier,axiom,
    ! [C4: set_o,A2: set_o,B5: set_o,G: $o > assn,H3: $o > assn] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A2 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_assn ) )
           => ( ! [B2: $o] :
                  ( ( member_o @ B2 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_assn ) )
             => ( ( ( groups5301882518646026715o_assn @ G @ A2 )
                  = ( groups5301882518646026715o_assn @ H3 @ B5 ) )
                = ( ( groups5301882518646026715o_assn @ G @ C4 )
                  = ( groups5301882518646026715o_assn @ H3 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_5928_prod_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B5: set_int,G: int > assn,H3: int > assn] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B5 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_assn ) )
           => ( ! [B2: int] :
                  ( ( member_int @ B2 @ ( minus_minus_set_int @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_assn ) )
             => ( ( ( groups7882442080178216443t_assn @ G @ A2 )
                  = ( groups7882442080178216443t_assn @ H3 @ B5 ) )
                = ( ( groups7882442080178216443t_assn @ G @ C4 )
                  = ( groups7882442080178216443t_assn @ H3 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_5929_prod_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B5: set_Code_integer,G: code_integer > assn,H3: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_assn ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_assn ) )
             => ( ( ( groups1304777262505850412r_assn @ G @ A2 )
                  = ( groups1304777262505850412r_assn @ H3 @ B5 ) )
                = ( ( groups1304777262505850412r_assn @ G @ C4 )
                  = ( groups1304777262505850412r_assn @ H3 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_5930_prod_Osame__carrier,axiom,
    ! [C4: set_o,A2: set_o,B5: set_o,G: $o > rat,H3: $o > rat] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A2 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B2: $o] :
                  ( ( member_o @ B2 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_rat ) )
             => ( ( ( groups2869687844427037835_o_rat @ G @ A2 )
                  = ( groups2869687844427037835_o_rat @ H3 @ B5 ) )
                = ( ( groups2869687844427037835_o_rat @ G @ C4 )
                  = ( groups2869687844427037835_o_rat @ H3 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_5931_prod_Osame__carrier,axiom,
    ! [C4: set_int,A2: set_int,B5: set_int,G: int > rat,H3: int > rat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B5 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B2: int] :
                  ( ( member_int @ B2 @ ( minus_minus_set_int @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_rat ) )
             => ( ( ( groups1072433553688619179nt_rat @ G @ A2 )
                  = ( groups1072433553688619179nt_rat @ H3 @ B5 ) )
                = ( ( groups1072433553688619179nt_rat @ G @ C4 )
                  = ( groups1072433553688619179nt_rat @ H3 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_5932_prod_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B5: set_Code_integer,G: code_integer > rat,H3: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_rat ) )
             => ( ( ( groups2555765274223993564er_rat @ G @ A2 )
                  = ( groups2555765274223993564er_rat @ H3 @ B5 ) )
                = ( ( groups2555765274223993564er_rat @ G @ C4 )
                  = ( groups2555765274223993564er_rat @ H3 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_5933_prod_Osame__carrier,axiom,
    ! [C4: set_o,A2: set_o,B5: set_o,G: $o > nat,H3: $o > nat] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A2 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_nat ) )
           => ( ! [B2: $o] :
                  ( ( member_o @ B2 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_nat ) )
             => ( ( ( groups3504817904513533571_o_nat @ G @ A2 )
                  = ( groups3504817904513533571_o_nat @ H3 @ B5 ) )
                = ( ( groups3504817904513533571_o_nat @ G @ C4 )
                  = ( groups3504817904513533571_o_nat @ H3 @ C4 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_5934_prod_Osame__carrierI,axiom,
    ! [C4: set_o,A2: set_o,B5: set_o,G: $o > code_integer,H3: $o > code_integer] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A2 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_Code_integer ) )
           => ( ! [B2: $o] :
                  ( ( member_o @ B2 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_Code_integer ) )
             => ( ( ( groups7694694392188491536nteger @ G @ C4 )
                  = ( groups7694694392188491536nteger @ H3 @ C4 ) )
               => ( ( groups7694694392188491536nteger @ G @ A2 )
                  = ( groups7694694392188491536nteger @ H3 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_5935_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B5: set_int,G: int > code_integer,H3: int > code_integer] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B5 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_Code_integer ) )
           => ( ! [B2: int] :
                  ( ( member_int @ B2 @ ( minus_minus_set_int @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_Code_integer ) )
             => ( ( ( groups3827104343326376752nteger @ G @ C4 )
                  = ( groups3827104343326376752nteger @ H3 @ C4 ) )
               => ( ( groups3827104343326376752nteger @ G @ A2 )
                  = ( groups3827104343326376752nteger @ H3 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_5936_prod_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B5: set_Code_integer,G: code_integer > code_integer,H3: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_Code_integer ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_Code_integer ) )
             => ( ( ( groups3674199335183972705nteger @ G @ C4 )
                  = ( groups3674199335183972705nteger @ H3 @ C4 ) )
               => ( ( groups3674199335183972705nteger @ G @ A2 )
                  = ( groups3674199335183972705nteger @ H3 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_5937_prod_Osame__carrierI,axiom,
    ! [C4: set_o,A2: set_o,B5: set_o,G: $o > assn,H3: $o > assn] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A2 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_assn ) )
           => ( ! [B2: $o] :
                  ( ( member_o @ B2 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_assn ) )
             => ( ( ( groups5301882518646026715o_assn @ G @ C4 )
                  = ( groups5301882518646026715o_assn @ H3 @ C4 ) )
               => ( ( groups5301882518646026715o_assn @ G @ A2 )
                  = ( groups5301882518646026715o_assn @ H3 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_5938_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B5: set_int,G: int > assn,H3: int > assn] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B5 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_assn ) )
           => ( ! [B2: int] :
                  ( ( member_int @ B2 @ ( minus_minus_set_int @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_assn ) )
             => ( ( ( groups7882442080178216443t_assn @ G @ C4 )
                  = ( groups7882442080178216443t_assn @ H3 @ C4 ) )
               => ( ( groups7882442080178216443t_assn @ G @ A2 )
                  = ( groups7882442080178216443t_assn @ H3 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_5939_prod_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B5: set_Code_integer,G: code_integer > assn,H3: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_assn ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_assn ) )
             => ( ( ( groups1304777262505850412r_assn @ G @ C4 )
                  = ( groups1304777262505850412r_assn @ H3 @ C4 ) )
               => ( ( groups1304777262505850412r_assn @ G @ A2 )
                  = ( groups1304777262505850412r_assn @ H3 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_5940_prod_Osame__carrierI,axiom,
    ! [C4: set_o,A2: set_o,B5: set_o,G: $o > rat,H3: $o > rat] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A2 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B2: $o] :
                  ( ( member_o @ B2 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_rat ) )
             => ( ( ( groups2869687844427037835_o_rat @ G @ C4 )
                  = ( groups2869687844427037835_o_rat @ H3 @ C4 ) )
               => ( ( groups2869687844427037835_o_rat @ G @ A2 )
                  = ( groups2869687844427037835_o_rat @ H3 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_5941_prod_Osame__carrierI,axiom,
    ! [C4: set_int,A2: set_int,B5: set_int,G: int > rat,H3: int > rat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A2 @ C4 )
       => ( ( ord_less_eq_set_int @ B5 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B2: int] :
                  ( ( member_int @ B2 @ ( minus_minus_set_int @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_rat ) )
             => ( ( ( groups1072433553688619179nt_rat @ G @ C4 )
                  = ( groups1072433553688619179nt_rat @ H3 @ C4 ) )
               => ( ( groups1072433553688619179nt_rat @ G @ A2 )
                  = ( groups1072433553688619179nt_rat @ H3 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_5942_prod_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A2: set_Code_integer,B5: set_Code_integer,G: code_integer > rat,H3: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A2 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B5 @ C4 )
         => ( ! [A3: code_integer] :
                ( ( member_Code_integer @ A3 @ ( minus_2355218937544613996nteger @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B2: code_integer] :
                  ( ( member_Code_integer @ B2 @ ( minus_2355218937544613996nteger @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_rat ) )
             => ( ( ( groups2555765274223993564er_rat @ G @ C4 )
                  = ( groups2555765274223993564er_rat @ H3 @ C4 ) )
               => ( ( groups2555765274223993564er_rat @ G @ A2 )
                  = ( groups2555765274223993564er_rat @ H3 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_5943_prod_Osame__carrierI,axiom,
    ! [C4: set_o,A2: set_o,B5: set_o,G: $o > nat,H3: $o > nat] :
      ( ( finite_finite_o @ C4 )
     => ( ( ord_less_eq_set_o @ A2 @ C4 )
       => ( ( ord_less_eq_set_o @ B5 @ C4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C4 @ A2 ) )
               => ( ( G @ A3 )
                  = one_one_nat ) )
           => ( ! [B2: $o] :
                  ( ( member_o @ B2 @ ( minus_minus_set_o @ C4 @ B5 ) )
                 => ( ( H3 @ B2 )
                    = one_one_nat ) )
             => ( ( ( groups3504817904513533571_o_nat @ G @ C4 )
                  = ( groups3504817904513533571_o_nat @ H3 @ C4 ) )
               => ( ( groups3504817904513533571_o_nat @ G @ A2 )
                  = ( groups3504817904513533571_o_nat @ H3 @ B5 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_5944_prod_Omono__neutral__left,axiom,
    ! [T5: set_int,S2: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S2 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_Code_integer ) )
         => ( ( groups3827104343326376752nteger @ G @ S2 )
            = ( groups3827104343326376752nteger @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_5945_prod_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_Code_integer ) )
         => ( ( groups3674199335183972705nteger @ G @ S2 )
            = ( groups3674199335183972705nteger @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_5946_prod_Omono__neutral__left,axiom,
    ! [T5: set_int,S2: set_int,G: int > assn] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S2 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_assn ) )
         => ( ( groups7882442080178216443t_assn @ G @ S2 )
            = ( groups7882442080178216443t_assn @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_5947_prod_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_assn ) )
         => ( ( groups1304777262505850412r_assn @ G @ S2 )
            = ( groups1304777262505850412r_assn @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_5948_prod_Omono__neutral__left,axiom,
    ! [T5: set_int,S2: set_int,G: int > rat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S2 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_rat ) )
         => ( ( groups1072433553688619179nt_rat @ G @ S2 )
            = ( groups1072433553688619179nt_rat @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_5949_prod_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_rat ) )
         => ( ( groups2555765274223993564er_rat @ G @ S2 )
            = ( groups2555765274223993564er_rat @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_5950_prod_Omono__neutral__left,axiom,
    ! [T5: set_int,S2: set_int,G: int > nat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S2 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_nat ) )
         => ( ( groups1707563613775114915nt_nat @ G @ S2 )
            = ( groups1707563613775114915nt_nat @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_5951_prod_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_nat ) )
         => ( ( groups3190895334310489300er_nat @ G @ S2 )
            = ( groups3190895334310489300er_nat @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_5952_prod_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_int ) )
         => ( ( groups3188404863801439024er_int @ G @ S2 )
            = ( groups3188404863801439024er_int @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_5953_prod_Omono__neutral__left,axiom,
    ! [T5: set_nat,S2: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ T5 )
     => ( ( ord_less_eq_set_nat @ S2 @ T5 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_Code_integer ) )
         => ( ( groups3455450783089532116nteger @ G @ S2 )
            = ( groups3455450783089532116nteger @ G @ T5 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_5954_prod_Omono__neutral__right,axiom,
    ! [T5: set_int,S2: set_int,G: int > code_integer] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S2 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_Code_integer ) )
         => ( ( groups3827104343326376752nteger @ G @ T5 )
            = ( groups3827104343326376752nteger @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_5955_prod_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_Code_integer ) )
         => ( ( groups3674199335183972705nteger @ G @ T5 )
            = ( groups3674199335183972705nteger @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_5956_prod_Omono__neutral__right,axiom,
    ! [T5: set_int,S2: set_int,G: int > assn] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S2 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_assn ) )
         => ( ( groups7882442080178216443t_assn @ G @ T5 )
            = ( groups7882442080178216443t_assn @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_5957_prod_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_assn ) )
         => ( ( groups1304777262505850412r_assn @ G @ T5 )
            = ( groups1304777262505850412r_assn @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_5958_prod_Omono__neutral__right,axiom,
    ! [T5: set_int,S2: set_int,G: int > rat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S2 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_rat ) )
         => ( ( groups1072433553688619179nt_rat @ G @ T5 )
            = ( groups1072433553688619179nt_rat @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_5959_prod_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_rat ) )
         => ( ( groups2555765274223993564er_rat @ G @ T5 )
            = ( groups2555765274223993564er_rat @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_5960_prod_Omono__neutral__right,axiom,
    ! [T5: set_int,S2: set_int,G: int > nat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S2 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_nat ) )
         => ( ( groups1707563613775114915nt_nat @ G @ T5 )
            = ( groups1707563613775114915nt_nat @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_5961_prod_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_nat ) )
         => ( ( groups3190895334310489300er_nat @ G @ T5 )
            = ( groups3190895334310489300er_nat @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_5962_prod_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_int ) )
         => ( ( groups3188404863801439024er_int @ G @ T5 )
            = ( groups3188404863801439024er_int @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_5963_prod_Omono__neutral__right,axiom,
    ! [T5: set_nat,S2: set_nat,G: nat > code_integer] :
      ( ( finite_finite_nat @ T5 )
     => ( ( ord_less_eq_set_nat @ S2 @ T5 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_Code_integer ) )
         => ( ( groups3455450783089532116nteger @ G @ T5 )
            = ( groups3455450783089532116nteger @ G @ S2 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_5964_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_o,S2: set_o,H3: $o > code_integer,G: $o > code_integer] :
      ( ( finite_finite_o @ T5 )
     => ( ( ord_less_eq_set_o @ S2 @ T5 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ ( minus_minus_set_o @ T5 @ S2 ) )
             => ( ( H3 @ X4 )
                = one_one_Code_integer ) )
         => ( ! [X4: $o] :
                ( ( member_o @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups7694694392188491536nteger @ G @ S2 )
              = ( groups7694694392188491536nteger @ H3 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_5965_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_int,S2: set_int,H3: int > code_integer,G: int > code_integer] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S2 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S2 ) )
             => ( ( H3 @ X4 )
                = one_one_Code_integer ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups3827104343326376752nteger @ G @ S2 )
              = ( groups3827104343326376752nteger @ H3 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_5966_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,H3: code_integer > code_integer,G: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( H3 @ X4 )
                = one_one_Code_integer ) )
         => ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups3674199335183972705nteger @ G @ S2 )
              = ( groups3674199335183972705nteger @ H3 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_5967_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_o,S2: set_o,H3: $o > assn,G: $o > assn] :
      ( ( finite_finite_o @ T5 )
     => ( ( ord_less_eq_set_o @ S2 @ T5 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ ( minus_minus_set_o @ T5 @ S2 ) )
             => ( ( H3 @ X4 )
                = one_one_assn ) )
         => ( ! [X4: $o] :
                ( ( member_o @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups5301882518646026715o_assn @ G @ S2 )
              = ( groups5301882518646026715o_assn @ H3 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_5968_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_int,S2: set_int,H3: int > assn,G: int > assn] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S2 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S2 ) )
             => ( ( H3 @ X4 )
                = one_one_assn ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups7882442080178216443t_assn @ G @ S2 )
              = ( groups7882442080178216443t_assn @ H3 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_5969_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,H3: code_integer > assn,G: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( H3 @ X4 )
                = one_one_assn ) )
         => ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups1304777262505850412r_assn @ G @ S2 )
              = ( groups1304777262505850412r_assn @ H3 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_5970_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_o,S2: set_o,H3: $o > rat,G: $o > rat] :
      ( ( finite_finite_o @ T5 )
     => ( ( ord_less_eq_set_o @ S2 @ T5 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ ( minus_minus_set_o @ T5 @ S2 ) )
             => ( ( H3 @ X4 )
                = one_one_rat ) )
         => ( ! [X4: $o] :
                ( ( member_o @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups2869687844427037835_o_rat @ G @ S2 )
              = ( groups2869687844427037835_o_rat @ H3 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_5971_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_int,S2: set_int,H3: int > rat,G: int > rat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S2 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S2 ) )
             => ( ( H3 @ X4 )
                = one_one_rat ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups1072433553688619179nt_rat @ G @ S2 )
              = ( groups1072433553688619179nt_rat @ H3 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_5972_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,H3: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( H3 @ X4 )
                = one_one_rat ) )
         => ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups2555765274223993564er_rat @ G @ S2 )
              = ( groups2555765274223993564er_rat @ H3 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_5973_prod_Omono__neutral__cong__left,axiom,
    ! [T5: set_o,S2: set_o,H3: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ T5 )
     => ( ( ord_less_eq_set_o @ S2 @ T5 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ ( minus_minus_set_o @ T5 @ S2 ) )
             => ( ( H3 @ X4 )
                = one_one_nat ) )
         => ( ! [X4: $o] :
                ( ( member_o @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups3504817904513533571_o_nat @ G @ S2 )
              = ( groups3504817904513533571_o_nat @ H3 @ T5 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_5974_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_o,S2: set_o,G: $o > code_integer,H3: $o > code_integer] :
      ( ( finite_finite_o @ T5 )
     => ( ( ord_less_eq_set_o @ S2 @ T5 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ ( minus_minus_set_o @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_Code_integer ) )
         => ( ! [X4: $o] :
                ( ( member_o @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups7694694392188491536nteger @ G @ T5 )
              = ( groups7694694392188491536nteger @ H3 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_5975_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_int,S2: set_int,G: int > code_integer,H3: int > code_integer] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S2 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_Code_integer ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups3827104343326376752nteger @ G @ T5 )
              = ( groups3827104343326376752nteger @ H3 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_5976_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > code_integer,H3: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_Code_integer ) )
         => ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups3674199335183972705nteger @ G @ T5 )
              = ( groups3674199335183972705nteger @ H3 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_5977_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_o,S2: set_o,G: $o > assn,H3: $o > assn] :
      ( ( finite_finite_o @ T5 )
     => ( ( ord_less_eq_set_o @ S2 @ T5 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ ( minus_minus_set_o @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_assn ) )
         => ( ! [X4: $o] :
                ( ( member_o @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups5301882518646026715o_assn @ G @ T5 )
              = ( groups5301882518646026715o_assn @ H3 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_5978_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_int,S2: set_int,G: int > assn,H3: int > assn] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S2 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_assn ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups7882442080178216443t_assn @ G @ T5 )
              = ( groups7882442080178216443t_assn @ H3 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_5979_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > assn,H3: code_integer > assn] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_assn ) )
         => ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups1304777262505850412r_assn @ G @ T5 )
              = ( groups1304777262505850412r_assn @ H3 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_5980_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_o,S2: set_o,G: $o > rat,H3: $o > rat] :
      ( ( finite_finite_o @ T5 )
     => ( ( ord_less_eq_set_o @ S2 @ T5 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ ( minus_minus_set_o @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_rat ) )
         => ( ! [X4: $o] :
                ( ( member_o @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups2869687844427037835_o_rat @ G @ T5 )
              = ( groups2869687844427037835_o_rat @ H3 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_5981_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_int,S2: set_int,G: int > rat,H3: int > rat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S2 @ T5 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_rat ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups1072433553688619179nt_rat @ G @ T5 )
              = ( groups1072433553688619179nt_rat @ H3 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_5982_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_Code_integer,S2: set_Code_integer,G: code_integer > rat,H3: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T5 )
     => ( ( ord_le7084787975880047091nteger @ S2 @ T5 )
       => ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ ( minus_2355218937544613996nteger @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_rat ) )
         => ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups2555765274223993564er_rat @ G @ T5 )
              = ( groups2555765274223993564er_rat @ H3 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_5983_prod_Omono__neutral__cong__right,axiom,
    ! [T5: set_o,S2: set_o,G: $o > nat,H3: $o > nat] :
      ( ( finite_finite_o @ T5 )
     => ( ( ord_less_eq_set_o @ S2 @ T5 )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ ( minus_minus_set_o @ T5 @ S2 ) )
             => ( ( G @ X4 )
                = one_one_nat ) )
         => ( ! [X4: $o] :
                ( ( member_o @ X4 @ S2 )
               => ( ( G @ X4 )
                  = ( H3 @ X4 ) ) )
           => ( ( groups3504817904513533571_o_nat @ G @ T5 )
              = ( groups3504817904513533571_o_nat @ H3 @ S2 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_5984_atLeastLessThanSuc,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) )
          = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ M2 @ N2 ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N2 )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N2 ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_5985_Option_Othese__def,axiom,
    ( these_6645423997822192641it_nat
    = ( ^ [A7: set_op3940070208846178154it_nat] :
          ( image_5549452628634654015it_nat @ the_Pr5856008525237892824it_nat
          @ ( collec8141831914148684149it_nat
            @ ^ [X3: option3216581401566546698it_nat] :
                ( ( member3517700364412436275it_nat @ X3 @ A7 )
                & ( X3 != none_P2587165478634041381it_nat ) ) ) ) ) ) ).

% Option.these_def
thf(fact_5986_Option_Othese__def,axiom,
    ( these_num
    = ( ^ [A7: set_option_num] :
          ( image_option_num_num @ the_num
          @ ( collect_option_num
            @ ^ [X3: option_num] :
                ( ( member_option_num @ X3 @ A7 )
                & ( X3 != none_num ) ) ) ) ) ) ).

% Option.these_def
thf(fact_5987_Option_Othese__def,axiom,
    ( these_3407724651694999947it_nat
    = ( ^ [A7: set_op239864471688321678it_nat] :
          ( image_1237920015125996135it_nat @ the_Pr7776125533777466484it_nat
          @ ( collec5742531119322951597it_nat
            @ ^ [X3: option8956607266484857688it_nat] :
                ( ( member5596548051065438575it_nat @ X3 @ A7 )
                & ( X3 != none_P9117596204409417319it_nat ) ) ) ) ) ) ).

% Option.these_def
thf(fact_5988_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% atLeast0_atMost_Suc_eq_insert_0
thf(fact_5989_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_5990_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 ) ) ) ) ).

% atMost_Suc_eq_insert_0
thf(fact_5991_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 ) ) ) ) ).

% lessThan_Suc_eq_insert_0
thf(fact_5992_card__2__iff,axiom,
    ! [S2: set_Product_unit] :
      ( ( ( finite410649719033368117t_unit @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: product_unit,Y3: product_unit] :
            ( ( S2
              = ( insert_Product_unit @ X3 @ ( insert_Product_unit @ Y3 @ bot_bo3957492148770167129t_unit ) ) )
            & ( X3 != Y3 ) ) ) ) ).

% card_2_iff
thf(fact_5993_card__2__iff,axiom,
    ! [S2: set_list_nat] :
      ( ( ( finite_card_list_nat @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: list_nat,Y3: list_nat] :
            ( ( S2
              = ( insert_list_nat @ X3 @ ( insert_list_nat @ Y3 @ bot_bot_set_list_nat ) ) )
            & ( X3 != Y3 ) ) ) ) ).

% card_2_iff
thf(fact_5994_card__2__iff,axiom,
    ! [S2: set_Pr4329608150637261639at_nat] :
      ( ( ( finite3771342082235030671at_nat @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
            ( ( S2
              = ( insert9069300056098147895at_nat @ X3 @ ( insert9069300056098147895at_nat @ Y3 @ bot_bo228742789529271731at_nat ) ) )
            & ( X3 != Y3 ) ) ) ) ).

% card_2_iff
thf(fact_5995_card__2__iff,axiom,
    ! [S2: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( S2
              = ( insert8211810215607154385at_nat @ X3 @ ( insert8211810215607154385at_nat @ Y3 @ bot_bo2099793752762293965at_nat ) ) )
            & ( X3 != Y3 ) ) ) ) ).

% card_2_iff
thf(fact_5996_card__2__iff,axiom,
    ! [S2: set_o] :
      ( ( ( finite_card_o @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: $o,Y3: $o] :
            ( ( S2
              = ( insert_o @ X3 @ ( insert_o @ Y3 @ bot_bot_set_o ) ) )
            & ( X3 != Y3 ) ) ) ) ).

% card_2_iff
thf(fact_5997_card__2__iff,axiom,
    ! [S2: set_nat] :
      ( ( ( finite_card_nat @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: nat,Y3: nat] :
            ( ( S2
              = ( insert_nat @ X3 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) )
            & ( X3 != Y3 ) ) ) ) ).

% card_2_iff
thf(fact_5998_card__2__iff,axiom,
    ! [S2: set_int] :
      ( ( ( finite_card_int @ S2 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X3: int,Y3: int] :
            ( ( S2
              = ( insert_int @ X3 @ ( insert_int @ Y3 @ bot_bot_set_int ) ) )
            & ( X3 != Y3 ) ) ) ) ).

% card_2_iff
thf(fact_5999_card__3__iff,axiom,
    ! [S2: set_Product_unit] :
      ( ( ( finite410649719033368117t_unit @ S2 )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X3: product_unit,Y3: product_unit,Z4: product_unit] :
            ( ( S2
              = ( insert_Product_unit @ X3 @ ( insert_Product_unit @ Y3 @ ( insert_Product_unit @ Z4 @ bot_bo3957492148770167129t_unit ) ) ) )
            & ( X3 != Y3 )
            & ( Y3 != Z4 )
            & ( X3 != Z4 ) ) ) ) ).

% card_3_iff
thf(fact_6000_card__3__iff,axiom,
    ! [S2: set_list_nat] :
      ( ( ( finite_card_list_nat @ S2 )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X3: list_nat,Y3: list_nat,Z4: list_nat] :
            ( ( S2
              = ( insert_list_nat @ X3 @ ( insert_list_nat @ Y3 @ ( insert_list_nat @ Z4 @ bot_bot_set_list_nat ) ) ) )
            & ( X3 != Y3 )
            & ( Y3 != Z4 )
            & ( X3 != Z4 ) ) ) ) ).

% card_3_iff
thf(fact_6001_card__3__iff,axiom,
    ! [S2: set_Pr4329608150637261639at_nat] :
      ( ( ( finite3771342082235030671at_nat @ S2 )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat,Z4: produc3843707927480180839at_nat] :
            ( ( S2
              = ( insert9069300056098147895at_nat @ X3 @ ( insert9069300056098147895at_nat @ Y3 @ ( insert9069300056098147895at_nat @ Z4 @ bot_bo228742789529271731at_nat ) ) ) )
            & ( X3 != Y3 )
            & ( Y3 != Z4 )
            & ( X3 != Z4 ) ) ) ) ).

% card_3_iff
thf(fact_6002_card__3__iff,axiom,
    ! [S2: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ S2 )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,Z4: product_prod_nat_nat] :
            ( ( S2
              = ( insert8211810215607154385at_nat @ X3 @ ( insert8211810215607154385at_nat @ Y3 @ ( insert8211810215607154385at_nat @ Z4 @ bot_bo2099793752762293965at_nat ) ) ) )
            & ( X3 != Y3 )
            & ( Y3 != Z4 )
            & ( X3 != Z4 ) ) ) ) ).

% card_3_iff
thf(fact_6003_card__3__iff,axiom,
    ! [S2: set_o] :
      ( ( ( finite_card_o @ S2 )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X3: $o,Y3: $o,Z4: $o] :
            ( ( S2
              = ( insert_o @ X3 @ ( insert_o @ Y3 @ ( insert_o @ Z4 @ bot_bot_set_o ) ) ) )
            & ( X3 != Y3 )
            & ( Y3 != Z4 )
            & ( X3 != Z4 ) ) ) ) ).

% card_3_iff
thf(fact_6004_card__3__iff,axiom,
    ! [S2: set_nat] :
      ( ( ( finite_card_nat @ S2 )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X3: nat,Y3: nat,Z4: nat] :
            ( ( S2
              = ( insert_nat @ X3 @ ( insert_nat @ Y3 @ ( insert_nat @ Z4 @ bot_bot_set_nat ) ) ) )
            & ( X3 != Y3 )
            & ( Y3 != Z4 )
            & ( X3 != Z4 ) ) ) ) ).

% card_3_iff
thf(fact_6005_card__3__iff,axiom,
    ! [S2: set_int] :
      ( ( ( finite_card_int @ S2 )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X3: int,Y3: int,Z4: int] :
            ( ( S2
              = ( insert_int @ X3 @ ( insert_int @ Y3 @ ( insert_int @ Z4 @ bot_bot_set_int ) ) ) )
            & ( X3 != Y3 )
            & ( Y3 != Z4 )
            & ( X3 != Z4 ) ) ) ) ).

% card_3_iff
thf(fact_6006_sum__image__le,axiom,
    ! [I4: set_num,G: code_integer > code_integer,F: num > code_integer] :
      ( ( finite_finite_num @ I4 )
     => ( ! [I2: num] :
            ( ( member_num @ I2 @ I4 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ ( F @ I2 ) ) ) )
       => ( ord_le3102999989581377725nteger @ ( groups879477027807139574nteger @ G @ ( image_8379761247532268670nteger @ F @ I4 ) ) @ ( groups5442708359310287027nteger @ ( comp_C3531382070062128313er_num @ G @ F ) @ I4 ) ) ) ) ).

% sum_image_le
thf(fact_6007_sum__image__le,axiom,
    ! [I4: set_nat,G: code_integer > code_integer,F: nat > code_integer] :
      ( ( finite_finite_nat @ I4 )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I4 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ ( F @ I2 ) ) ) )
       => ( ord_le3102999989581377725nteger @ ( groups879477027807139574nteger @ G @ ( image_1215581382706833972nteger @ F @ I4 ) ) @ ( groups7501900531339628137nteger @ ( comp_C6974050283906349679er_nat @ G @ F ) @ I4 ) ) ) ) ).

% sum_image_le
thf(fact_6008_sum__image__le,axiom,
    ! [I4: set_nat,G: nat > code_integer,F: nat > nat] :
      ( ( finite_finite_nat @ I4 )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I4 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ ( F @ I2 ) ) ) )
       => ( ord_le3102999989581377725nteger @ ( groups7501900531339628137nteger @ G @ ( image_nat_nat @ F @ I4 ) ) @ ( groups7501900531339628137nteger @ ( comp_n3898172953802868194er_nat @ G @ F ) @ I4 ) ) ) ) ).

% sum_image_le
thf(fact_6009_sum__image__le,axiom,
    ! [I4: set_nat,G: int > code_integer,F: nat > int] :
      ( ( finite_finite_nat @ I4 )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I4 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ ( F @ I2 ) ) ) )
       => ( ord_le3102999989581377725nteger @ ( groups7873554091576472773nteger @ G @ ( image_nat_int @ F @ I4 ) ) @ ( groups7501900531339628137nteger @ ( comp_i1588355021709917246er_nat @ G @ F ) @ I4 ) ) ) ) ).

% sum_image_le
thf(fact_6010_sum__image__le,axiom,
    ! [I4: set_int,G: int > code_integer,F: int > int] :
      ( ( finite_finite_int @ I4 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I4 )
           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( G @ ( F @ I2 ) ) ) )
       => ( ord_le3102999989581377725nteger @ ( groups7873554091576472773nteger @ G @ ( image_int_int @ F @ I4 ) ) @ ( groups7873554091576472773nteger @ ( comp_i1585864551200866970er_int @ G @ F ) @ I4 ) ) ) ) ).

% sum_image_le
thf(fact_6011_sum__image__le,axiom,
    ! [I4: set_nat,G: code_integer > rat,F: nat > code_integer] :
      ( ( finite_finite_nat @ I4 )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ ( F @ I2 ) ) ) )
       => ( ord_less_eq_rat @ ( groups6602215022474089585er_rat @ G @ ( image_1215581382706833972nteger @ F @ I4 ) ) @ ( groups2906978787729119204at_rat @ ( comp_C3733001951728184298at_nat @ G @ F ) @ I4 ) ) ) ) ).

% sum_image_le
thf(fact_6012_sum__image__le,axiom,
    ! [I4: set_nat,G: nat > rat,F: nat > nat] :
      ( ( finite_finite_nat @ I4 )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ ( F @ I2 ) ) ) )
       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ G @ ( image_nat_nat @ F @ I4 ) ) @ ( groups2906978787729119204at_rat @ ( comp_nat_rat_nat @ G @ F ) @ I4 ) ) ) ) ).

% sum_image_le
thf(fact_6013_sum__image__le,axiom,
    ! [I4: set_nat,G: int > rat,F: nat > int] :
      ( ( finite_finite_nat @ I4 )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ ( F @ I2 ) ) ) )
       => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ G @ ( image_nat_int @ F @ I4 ) ) @ ( groups2906978787729119204at_rat @ ( comp_int_rat_nat @ G @ F ) @ I4 ) ) ) ) ).

% sum_image_le
thf(fact_6014_sum__image__le,axiom,
    ! [I4: set_int,G: int > rat,F: int > int] :
      ( ( finite_finite_int @ I4 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ ( F @ I2 ) ) ) )
       => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ G @ ( image_int_int @ F @ I4 ) ) @ ( groups3906332499630173760nt_rat @ ( comp_int_rat_int @ G @ F ) @ I4 ) ) ) ) ).

% sum_image_le
thf(fact_6015_sum__image__le,axiom,
    ! [I4: set_int,G: int > nat,F: int > int] :
      ( ( finite_finite_int @ I4 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I4 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( G @ ( F @ I2 ) ) ) )
       => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ G @ ( image_int_int @ F @ I4 ) ) @ ( groups4541462559716669496nt_nat @ ( comp_int_nat_int @ G @ F ) @ I4 ) ) ) ) ).

% sum_image_le
thf(fact_6016_sum_Oremove,axiom,
    ! [A2: set_Code_integer,X: code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( member_Code_integer @ X @ A2 )
       => ( ( groups6602215022474089585er_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6017_sum_Oremove,axiom,
    ! [A2: set_Code_integer,X: code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( member_Code_integer @ X @ A2 )
       => ( ( groups7237345082560585321er_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6018_sum_Oremove,axiom,
    ! [A2: set_Code_integer,X: code_integer,G: code_integer > int] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( member_Code_integer @ X @ A2 )
       => ( ( groups7234854612051535045er_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6019_sum_Oremove,axiom,
    ! [A2: set_o,X: $o,G: $o > rat] :
      ( ( finite_finite_o @ A2 )
     => ( ( member_o @ X @ A2 )
       => ( ( groups7872700643590313910_o_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6020_sum_Oremove,axiom,
    ! [A2: set_o,X: $o,G: $o > nat] :
      ( ( finite_finite_o @ A2 )
     => ( ( member_o @ X @ A2 )
       => ( ( groups8507830703676809646_o_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups8507830703676809646_o_nat @ G @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6021_sum_Oremove,axiom,
    ! [A2: set_o,X: $o,G: $o > int] :
      ( ( finite_finite_o @ A2 )
     => ( ( member_o @ X @ A2 )
       => ( ( groups8505340233167759370_o_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups8505340233167759370_o_int @ G @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6022_sum_Oremove,axiom,
    ! [A2: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ( groups3906332499630173760nt_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6023_sum_Oremove,axiom,
    ! [A2: set_int,X: int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ( groups4541462559716669496nt_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6024_sum_Oremove,axiom,
    ! [A2: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ X @ A2 )
       => ( ( groups2906978787729119204at_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6025_sum_Oremove,axiom,
    ! [A2: set_nat,X: nat,G: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ X @ A2 )
       => ( ( groups3539618377306564664at_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_6026_sum_Oinsert__remove,axiom,
    ! [A2: set_Code_integer,G: code_integer > rat,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups6602215022474089585er_rat @ G @ ( insert_Code_integer @ X @ A2 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups6602215022474089585er_rat @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6027_sum_Oinsert__remove,axiom,
    ! [A2: set_Code_integer,G: code_integer > nat,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups7237345082560585321er_nat @ G @ ( insert_Code_integer @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups7237345082560585321er_nat @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6028_sum_Oinsert__remove,axiom,
    ! [A2: set_Code_integer,G: code_integer > int,X: code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( groups7234854612051535045er_int @ G @ ( insert_Code_integer @ X @ A2 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups7234854612051535045er_int @ G @ ( minus_2355218937544613996nteger @ A2 @ ( insert_Code_integer @ X @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6029_sum_Oinsert__remove,axiom,
    ! [A2: set_o,G: $o > rat,X: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A2 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6030_sum_Oinsert__remove,axiom,
    ! [A2: set_o,G: $o > nat,X: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( groups8507830703676809646_o_nat @ G @ ( insert_o @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups8507830703676809646_o_nat @ G @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6031_sum_Oinsert__remove,axiom,
    ! [A2: set_o,G: $o > int,X: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( groups8505340233167759370_o_int @ G @ ( insert_o @ X @ A2 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups8505340233167759370_o_int @ G @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6032_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: int > rat,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6033_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: int > nat,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6034_sum_Oinsert__remove,axiom,
    ! [A2: set_nat,G: nat > rat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6035_sum_Oinsert__remove,axiom,
    ! [A2: set_nat,G: nat > int,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups3539618377306564664at_int @ G @ ( insert_nat @ X @ A2 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_6036_atLeast1__atMost__eq__remove0,axiom,
    ! [N2: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_atMost_eq_remove0
thf(fact_6037_atLeast1__lessThan__eq__remove0,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_lessThan_eq_remove0
thf(fact_6038_atLeastLessThan__nat__numeral,axiom,
    ! [M2: nat,K: num] :
      ( ( ( ord_less_eq_nat @ M2 @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( numeral_numeral_nat @ K ) )
          = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M2 @ ( pred_numeral @ K ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( numeral_numeral_nat @ K ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
thf(fact_6039_sum_Odelta__remove,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: code_integer > rat,C: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( ( member_Code_integer @ A @ S2 )
         => ( ( groups6602215022474089585er_rat
              @ ^ [K4: code_integer] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups6602215022474089585er_rat @ C @ ( minus_2355218937544613996nteger @ S2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S2 )
         => ( ( groups6602215022474089585er_rat
              @ ^ [K4: code_integer] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups6602215022474089585er_rat @ C @ ( minus_2355218937544613996nteger @ S2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6040_sum_Odelta__remove,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: code_integer > nat,C: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( ( member_Code_integer @ A @ S2 )
         => ( ( groups7237345082560585321er_nat
              @ ^ [K4: code_integer] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups7237345082560585321er_nat @ C @ ( minus_2355218937544613996nteger @ S2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S2 )
         => ( ( groups7237345082560585321er_nat
              @ ^ [K4: code_integer] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups7237345082560585321er_nat @ C @ ( minus_2355218937544613996nteger @ S2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6041_sum_Odelta__remove,axiom,
    ! [S2: set_Code_integer,A: code_integer,B: code_integer > int,C: code_integer > int] :
      ( ( finite6017078050557962740nteger @ S2 )
     => ( ( ( member_Code_integer @ A @ S2 )
         => ( ( groups7234854612051535045er_int
              @ ^ [K4: code_integer] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_int @ ( B @ A ) @ ( groups7234854612051535045er_int @ C @ ( minus_2355218937544613996nteger @ S2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A @ S2 )
         => ( ( groups7234854612051535045er_int
              @ ^ [K4: code_integer] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups7234854612051535045er_int @ C @ ( minus_2355218937544613996nteger @ S2 @ ( insert_Code_integer @ A @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6042_sum_Odelta__remove,axiom,
    ! [S2: set_o,A: $o,B: $o > rat,C: $o > rat] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K4: $o] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups7872700643590313910_o_rat @ C @ ( minus_minus_set_o @ S2 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K4: $o] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups7872700643590313910_o_rat @ C @ ( minus_minus_set_o @ S2 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6043_sum_Odelta__remove,axiom,
    ! [S2: set_o,A: $o,B: $o > nat,C: $o > nat] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [K4: $o] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups8507830703676809646_o_nat @ C @ ( minus_minus_set_o @ S2 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups8507830703676809646_o_nat
              @ ^ [K4: $o] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups8507830703676809646_o_nat @ C @ ( minus_minus_set_o @ S2 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6044_sum_Odelta__remove,axiom,
    ! [S2: set_o,A: $o,B: $o > int,C: $o > int] :
      ( ( finite_finite_o @ S2 )
     => ( ( ( member_o @ A @ S2 )
         => ( ( groups8505340233167759370_o_int
              @ ^ [K4: $o] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_int @ ( B @ A ) @ ( groups8505340233167759370_o_int @ C @ ( minus_minus_set_o @ S2 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
        & ( ~ ( member_o @ A @ S2 )
         => ( ( groups8505340233167759370_o_int
              @ ^ [K4: $o] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups8505340233167759370_o_int @ C @ ( minus_minus_set_o @ S2 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6045_sum_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: int > rat,C: int > rat] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K4: int] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups3906332499630173760nt_rat @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K4: int] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups3906332499630173760nt_rat @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6046_sum_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: int > nat,C: int > nat] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [K4: int] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups4541462559716669496nt_nat @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [K4: int] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups4541462559716669496nt_nat @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6047_sum_Odelta__remove,axiom,
    ! [S2: set_nat,A: nat,B: nat > rat,C: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K4: nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups2906978787729119204at_rat @ C @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K4: nat] : ( if_rat @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups2906978787729119204at_rat @ C @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6048_sum_Odelta__remove,axiom,
    ! [S2: set_nat,A: nat,B: nat > int,C: nat > int] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups3539618377306564664at_int
              @ ^ [K4: nat] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( plus_plus_int @ ( B @ A ) @ ( groups3539618377306564664at_int @ C @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups3539618377306564664at_int
              @ ^ [K4: nat] : ( if_int @ ( K4 = A ) @ ( B @ K4 ) @ ( C @ K4 ) )
              @ S2 )
            = ( groups3539618377306564664at_int @ C @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_6049_prod__le__power,axiom,
    ! [A2: set_o,F: $o > code_integer,N2: code_integer,K: nat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A2 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I2 ) )
            & ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ N2 ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_o @ A2 ) @ K )
       => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N2 )
         => ( ord_le3102999989581377725nteger @ ( groups7694694392188491536nteger @ F @ A2 ) @ ( power_8256067586552552935nteger @ N2 @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_6050_prod__le__power,axiom,
    ! [A2: set_int,F: int > code_integer,N2: code_integer,K: nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A2 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I2 ) )
            & ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ N2 ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ K )
       => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N2 )
         => ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F @ A2 ) @ ( power_8256067586552552935nteger @ N2 @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_6051_prod__le__power,axiom,
    ! [A2: set_Product_unit,F: product_unit > code_integer,N2: code_integer,K: nat] :
      ( ! [I2: product_unit] :
          ( ( member_Product_unit @ I2 @ A2 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I2 ) )
            & ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ N2 ) ) )
     => ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A2 ) @ K )
       => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N2 )
         => ( ord_le3102999989581377725nteger @ ( groups8185359811944078981nteger @ F @ A2 ) @ ( power_8256067586552552935nteger @ N2 @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_6052_prod__le__power,axiom,
    ! [A2: set_nat,F: nat > code_integer,N2: code_integer,K: nat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A2 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F @ I2 ) )
            & ( ord_le3102999989581377725nteger @ ( F @ I2 ) @ N2 ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ K )
       => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N2 )
         => ( ord_le3102999989581377725nteger @ ( groups3455450783089532116nteger @ F @ A2 ) @ ( power_8256067586552552935nteger @ N2 @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_6053_prod__le__power,axiom,
    ! [A2: set_o,F: $o > rat,N2: rat,K: nat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ N2 ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_o @ A2 ) @ K )
       => ( ( ord_less_eq_rat @ one_one_rat @ N2 )
         => ( ord_less_eq_rat @ ( groups2869687844427037835_o_rat @ F @ A2 ) @ ( power_power_rat @ N2 @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_6054_prod__le__power,axiom,
    ! [A2: set_int,F: int > rat,N2: rat,K: nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ N2 ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ K )
       => ( ( ord_less_eq_rat @ one_one_rat @ N2 )
         => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( power_power_rat @ N2 @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_6055_prod__le__power,axiom,
    ! [A2: set_Product_unit,F: product_unit > rat,N2: rat,K: nat] :
      ( ! [I2: product_unit] :
          ( ( member_Product_unit @ I2 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ N2 ) ) )
     => ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A2 ) @ K )
       => ( ( ord_less_eq_rat @ one_one_rat @ N2 )
         => ( ord_less_eq_rat @ ( groups8984663225794451456it_rat @ F @ A2 ) @ ( power_power_rat @ N2 @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_6056_prod__le__power,axiom,
    ! [A2: set_nat,F: nat > rat,N2: rat,K: nat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
            & ( ord_less_eq_rat @ ( F @ I2 ) @ N2 ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ K )
       => ( ( ord_less_eq_rat @ one_one_rat @ N2 )
         => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( power_power_rat @ N2 @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_6057_prod__le__power,axiom,
    ! [A2: set_o,F: $o > nat,N2: nat,K: nat] :
      ( ! [I2: $o] :
          ( ( member_o @ I2 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
            & ( ord_less_eq_nat @ ( F @ I2 ) @ N2 ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_o @ A2 ) @ K )
       => ( ( ord_less_eq_nat @ one_one_nat @ N2 )
         => ( ord_less_eq_nat @ ( groups3504817904513533571_o_nat @ F @ A2 ) @ ( power_power_nat @ N2 @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_6058_prod__le__power,axiom,
    ! [A2: set_int,F: int > nat,N2: nat,K: nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
            & ( ord_less_eq_nat @ ( F @ I2 ) @ N2 ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_int @ A2 ) @ K )
       => ( ( ord_less_eq_nat @ one_one_nat @ N2 )
         => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( power_power_nat @ N2 @ K ) ) ) ) ) ).

% prod_le_power
thf(fact_6059_card__Diff__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A2 )
     => ( ~ ( member8757157785044589968at_nat @ A @ B5 )
       => ( ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B5 ) ) )
          = ( minus_minus_nat @ ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B5 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_6060_card__Diff__insert,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A2 )
     => ( ~ ( member8440522571783428010at_nat @ A @ B5 )
       => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B5 ) ) )
          = ( minus_minus_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B5 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_6061_card__Diff__insert,axiom,
    ! [A: $o,A2: set_o,B5: set_o] :
      ( ( member_o @ A @ A2 )
     => ( ~ ( member_o @ A @ B5 )
       => ( ( finite_card_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B5 ) ) )
          = ( minus_minus_nat @ ( finite_card_o @ ( minus_minus_set_o @ A2 @ B5 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_6062_card__Diff__insert,axiom,
    ! [A: int,A2: set_int,B5: set_int] :
      ( ( member_int @ A @ A2 )
     => ( ~ ( member_int @ A @ B5 )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B5 ) ) )
          = ( minus_minus_nat @ ( finite_card_int @ ( minus_minus_set_int @ A2 @ B5 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_6063_card__Diff__insert,axiom,
    ! [A: product_unit,A2: set_Product_unit,B5: set_Product_unit] :
      ( ( member_Product_unit @ A @ A2 )
     => ( ~ ( member_Product_unit @ A @ B5 )
       => ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A2 @ ( insert_Product_unit @ A @ B5 ) ) )
          = ( minus_minus_nat @ ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A2 @ B5 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_6064_card__Diff__insert,axiom,
    ! [A: list_nat,A2: set_list_nat,B5: set_list_nat] :
      ( ( member_list_nat @ A @ A2 )
     => ( ~ ( member_list_nat @ A @ B5 )
       => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ A @ B5 ) ) )
          = ( minus_minus_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ B5 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_6065_card__Diff__insert,axiom,
    ! [A: nat,A2: set_nat,B5: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( ~ ( member_nat @ A @ B5 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B5 ) ) )
          = ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ one_one_nat ) ) ) ) ).

% card_Diff_insert
thf(fact_6066_card__insert__le__m1,axiom,
    ! [N2: nat,Y: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ ( finite3771342082235030671at_nat @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite3771342082235030671at_nat @ ( insert9069300056098147895at_nat @ X @ Y ) ) @ N2 ) ) ) ).

% card_insert_le_m1
thf(fact_6067_card__insert__le__m1,axiom,
    ! [N2: nat,Y: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X @ Y ) ) @ N2 ) ) ) ).

% card_insert_le_m1
thf(fact_6068_card__insert__le__m1,axiom,
    ! [N2: nat,Y: set_o,X: $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ ( finite_card_o @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_o @ ( insert_o @ X @ Y ) ) @ N2 ) ) ) ).

% card_insert_le_m1
thf(fact_6069_card__insert__le__m1,axiom,
    ! [N2: nat,Y: set_int,X: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ ( finite_card_int @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_int @ ( insert_int @ X @ Y ) ) @ N2 ) ) ) ).

% card_insert_le_m1
thf(fact_6070_card__insert__le__m1,axiom,
    ! [N2: nat,Y: set_Product_unit,X: product_unit] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ Y ) ) @ N2 ) ) ) ).

% card_insert_le_m1
thf(fact_6071_card__insert__le__m1,axiom,
    ! [N2: nat,Y: set_nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ ( finite_card_nat @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X @ Y ) ) @ N2 ) ) ) ).

% card_insert_le_m1
thf(fact_6072_card__insert__le__m1,axiom,
    ! [N2: nat,Y: set_list_nat,X: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ Y ) @ ( minus_minus_nat @ N2 @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_list_nat @ ( insert_list_nat @ X @ Y ) ) @ N2 ) ) ) ).

% card_insert_le_m1
thf(fact_6073_card__Diff__singleton__if,axiom,
    ! [X: product_unit,A2: set_Product_unit] :
      ( ( ( member_Product_unit @ X @ A2 )
       => ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
          = ( minus_minus_nat @ ( finite410649719033368117t_unit @ A2 ) @ one_one_nat ) ) )
      & ( ~ ( member_Product_unit @ X @ A2 )
       => ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
          = ( finite410649719033368117t_unit @ A2 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_6074_card__Diff__singleton__if,axiom,
    ! [X: list_nat,A2: set_list_nat] :
      ( ( ( member_list_nat @ X @ A2 )
       => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) )
          = ( minus_minus_nat @ ( finite_card_list_nat @ A2 ) @ one_one_nat ) ) )
      & ( ~ ( member_list_nat @ X @ A2 )
       => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) )
          = ( finite_card_list_nat @ A2 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_6075_card__Diff__singleton__if,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ( member8757157785044589968at_nat @ X @ A2 )
       => ( ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
          = ( minus_minus_nat @ ( finite3771342082235030671at_nat @ A2 ) @ one_one_nat ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X @ A2 )
       => ( ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
          = ( finite3771342082235030671at_nat @ A2 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_6076_card__Diff__singleton__if,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ X @ A2 )
       => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
          = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A2 ) @ one_one_nat ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X @ A2 )
       => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
          = ( finite711546835091564841at_nat @ A2 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_6077_card__Diff__singleton__if,axiom,
    ! [X: $o,A2: set_o] :
      ( ( ( member_o @ X @ A2 )
       => ( ( finite_card_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) )
          = ( minus_minus_nat @ ( finite_card_o @ A2 ) @ one_one_nat ) ) )
      & ( ~ ( member_o @ X @ A2 )
       => ( ( finite_card_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) )
          = ( finite_card_o @ A2 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_6078_card__Diff__singleton__if,axiom,
    ! [X: int,A2: set_int] :
      ( ( ( member_int @ X @ A2 )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) )
          = ( minus_minus_nat @ ( finite_card_int @ A2 ) @ one_one_nat ) ) )
      & ( ~ ( member_int @ X @ A2 )
       => ( ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) )
          = ( finite_card_int @ A2 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_6079_card__Diff__singleton__if,axiom,
    ! [X: nat,A2: set_nat] :
      ( ( ( member_nat @ X @ A2 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
          = ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) )
      & ( ~ ( member_nat @ X @ A2 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
          = ( finite_card_nat @ A2 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_6080_card__Diff__singleton,axiom,
    ! [X: product_unit,A2: set_Product_unit] :
      ( ( member_Product_unit @ X @ A2 )
     => ( ( finite410649719033368117t_unit @ ( minus_6452836326544984404t_unit @ A2 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) )
        = ( minus_minus_nat @ ( finite410649719033368117t_unit @ A2 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_6081_card__Diff__singleton,axiom,
    ! [X: list_nat,A2: set_list_nat] :
      ( ( member_list_nat @ X @ A2 )
     => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) )
        = ( minus_minus_nat @ ( finite_card_list_nat @ A2 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_6082_card__Diff__singleton,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ A2 )
     => ( ( finite3771342082235030671at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
        = ( minus_minus_nat @ ( finite3771342082235030671at_nat @ A2 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_6083_card__Diff__singleton,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ A2 )
     => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
        = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A2 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_6084_card__Diff__singleton,axiom,
    ! [X: $o,A2: set_o] :
      ( ( member_o @ X @ A2 )
     => ( ( finite_card_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) )
        = ( minus_minus_nat @ ( finite_card_o @ A2 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_6085_card__Diff__singleton,axiom,
    ! [X: int,A2: set_int] :
      ( ( member_int @ X @ A2 )
     => ( ( finite_card_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) )
        = ( minus_minus_nat @ ( finite_card_int @ A2 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_6086_card__Diff__singleton,axiom,
    ! [X: nat,A2: set_nat] :
      ( ( member_nat @ X @ A2 )
     => ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
        = ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_6087_in__finite__psubset,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A2 @ B5 ) @ finite_psubset_nat )
      = ( ( ord_less_set_nat @ A2 @ B5 )
        & ( finite_finite_nat @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_6088_in__finite__psubset,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A2 @ B5 ) @ finite_psubset_int )
      = ( ( ord_less_set_int @ A2 @ B5 )
        & ( finite_finite_int @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_6089_in__finite__psubset,axiom,
    ! [A2: set_Code_integer,B5: set_Code_integer] :
      ( ( member4307123515891402160nteger @ ( produc7443773368509356479nteger @ A2 @ B5 ) @ finite2416775604798480986nteger )
      = ( ( ord_le1307284697595431911nteger @ A2 @ B5 )
        & ( finite6017078050557962740nteger @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_6090_in__finite__psubset,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B5 ) @ finite469560695537375940at_nat )
      = ( ( ord_le7866589430770878221at_nat @ A2 @ B5 )
        & ( finite6177210948735845034at_nat @ B5 ) ) ) ).

% in_finite_psubset
thf(fact_6091_card__1__singletonE,axiom,
    ! [A2: set_Product_unit] :
      ( ( ( finite410649719033368117t_unit @ A2 )
        = one_one_nat )
     => ~ ! [X4: product_unit] :
            ( A2
           != ( insert_Product_unit @ X4 @ bot_bo3957492148770167129t_unit ) ) ) ).

% card_1_singletonE
thf(fact_6092_card__1__singletonE,axiom,
    ! [A2: set_list_nat] :
      ( ( ( finite_card_list_nat @ A2 )
        = one_one_nat )
     => ~ ! [X4: list_nat] :
            ( A2
           != ( insert_list_nat @ X4 @ bot_bot_set_list_nat ) ) ) ).

% card_1_singletonE
thf(fact_6093_card__1__singletonE,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( ( finite3771342082235030671at_nat @ A2 )
        = one_one_nat )
     => ~ ! [X4: produc3843707927480180839at_nat] :
            ( A2
           != ( insert9069300056098147895at_nat @ X4 @ bot_bo228742789529271731at_nat ) ) ) ).

% card_1_singletonE
thf(fact_6094_card__1__singletonE,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( ( finite711546835091564841at_nat @ A2 )
        = one_one_nat )
     => ~ ! [X4: product_prod_nat_nat] :
            ( A2
           != ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) ).

% card_1_singletonE
thf(fact_6095_card__1__singletonE,axiom,
    ! [A2: set_o] :
      ( ( ( finite_card_o @ A2 )
        = one_one_nat )
     => ~ ! [X4: $o] :
            ( A2
           != ( insert_o @ X4 @ bot_bot_set_o ) ) ) ).

% card_1_singletonE
thf(fact_6096_card__1__singletonE,axiom,
    ! [A2: set_nat] :
      ( ( ( finite_card_nat @ A2 )
        = one_one_nat )
     => ~ ! [X4: nat] :
            ( A2
           != ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ).

% card_1_singletonE
thf(fact_6097_card__1__singletonE,axiom,
    ! [A2: set_int] :
      ( ( ( finite_card_int @ A2 )
        = one_one_nat )
     => ~ ! [X4: int] :
            ( A2
           != ( insert_int @ X4 @ bot_bot_set_int ) ) ) ).

% card_1_singletonE
thf(fact_6098_integer__of__nat__def,axiom,
    ( code_integer_of_nat
    = ( map_fu6290471996055670595nteger @ id_nat @ code_integer_of_int @ semiri1314217659103216013at_int ) ) ).

% integer_of_nat_def
thf(fact_6099_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_6100_card__lessThan,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
      = U ) ).

% card_lessThan
thf(fact_6101_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_atLeastLessThan
thf(fact_6102_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_6103_Collect__const__case__prod,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec6321179662152712658at_nat
            @ ( produc410239310623530412_nat_o
              @ ^ [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : P ) )
          = top_to6833984726390702231at_nat ) )
      & ( ~ P
       => ( ( collec6321179662152712658at_nat
            @ ( produc410239310623530412_nat_o
              @ ^ [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : P ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% Collect_const_case_prod
thf(fact_6104_Collect__const__case__prod,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec3392354462482085612at_nat
            @ ( produc6081775807080527818_nat_o
              @ ^ [A5: nat,B4: nat] : P ) )
          = top_to4669805908274784177at_nat ) )
      & ( ~ P
       => ( ( collec3392354462482085612at_nat
            @ ( produc6081775807080527818_nat_o
              @ ^ [A5: nat,B4: nat] : P ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_const_case_prod
thf(fact_6105_nat__of__integer__integer__of__nat,axiom,
    ! [N2: nat] :
      ( ( code_nat_of_integer @ ( code_integer_of_nat @ N2 ) )
      = N2 ) ).

% nat_of_integer_integer_of_nat
thf(fact_6106_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).

% card_atLeastAtMost
thf(fact_6107_integer__of__nat_Orep__eq,axiom,
    ! [X: nat] :
      ( ( code_int_of_integer @ ( code_integer_of_nat @ X ) )
      = ( semiri1314217659103216013at_int @ X ) ) ).

% integer_of_nat.rep_eq
thf(fact_6108_int__of__integer__integer__of__nat,axiom,
    ! [N2: nat] :
      ( ( code_int_of_integer @ ( code_integer_of_nat @ N2 ) )
      = ( semiri1314217659103216013at_int @ N2 ) ) ).

% int_of_integer_integer_of_nat
thf(fact_6109_top__empty__eq2,axiom,
    ( top_to8903224817164384382_nat_o
    = ( ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ top_to6833984726390702231at_nat ) ) ) ).

% top_empty_eq2
thf(fact_6110_top__empty__eq2,axiom,
    ( top_top_nat_nat_o
    = ( ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ top_to4669805908274784177at_nat ) ) ) ).

% top_empty_eq2
thf(fact_6111_top__empty__eq2,axiom,
    ( top_to8112647782992986859eger_o
    = ( ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ top_to7512759353274530428nteger ) ) ) ).

% top_empty_eq2
thf(fact_6112_top__empty__eq2,axiom,
    ( top_to2428096842796733269_nat_o
    = ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ top_to8753217654552796900et_nat ) ) ) ).

% top_empty_eq2
thf(fact_6113_top__empty__eq2,axiom,
    ( top_to7190503160269336793_nat_o
    = ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ top_to1599102959340997728et_nat ) ) ) ).

% top_empty_eq2
thf(fact_6114_top__empty__eq2,axiom,
    ( top_to6261655714344447806_int_o
    = ( ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ top_to3181862456159035625nt_int ) ) ) ).

% top_empty_eq2
thf(fact_6115_top__empty__eq2,axiom,
    ( top_to6513043852502318900_int_o
    = ( ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ top_to2069137843433766899nt_int ) ) ) ).

% top_empty_eq2
thf(fact_6116_int__of__integer__induct,axiom,
    ! [Y: int,P: int > $o] :
      ( ( member_int @ Y @ top_top_set_int )
     => ( ! [X4: code_integer] : ( P @ ( code_int_of_integer @ X4 ) )
       => ( P @ Y ) ) ) ).

% int_of_integer_induct
thf(fact_6117_int__of__integer__cases,axiom,
    ! [Y: int] :
      ( ( member_int @ Y @ top_top_set_int )
     => ~ ! [X4: code_integer] :
            ( Y
           != ( code_int_of_integer @ X4 ) ) ) ).

% int_of_integer_cases
thf(fact_6118_int__of__integer,axiom,
    ! [X: code_integer] : ( member_int @ ( code_int_of_integer @ X ) @ top_top_set_int ) ).

% int_of_integer
thf(fact_6119_integer__of__int__inject,axiom,
    ! [X: int,Y: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( member_int @ Y @ top_top_set_int )
       => ( ( ( code_integer_of_int @ X )
            = ( code_integer_of_int @ Y ) )
          = ( X = Y ) ) ) ) ).

% integer_of_int_inject
thf(fact_6120_integer__of__int__induct,axiom,
    ! [P: code_integer > $o,X: code_integer] :
      ( ! [Y5: int] :
          ( ( member_int @ Y5 @ top_top_set_int )
         => ( P @ ( code_integer_of_int @ Y5 ) ) )
     => ( P @ X ) ) ).

% integer_of_int_induct
thf(fact_6121_integer__of__int__cases,axiom,
    ! [X: code_integer] :
      ~ ! [Y5: int] :
          ( ( X
            = ( code_integer_of_int @ Y5 ) )
         => ~ ( member_int @ Y5 @ top_top_set_int ) ) ).

% integer_of_int_cases
thf(fact_6122_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
     => ( A2
        = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).

% subset_card_intvl_is_intvl
thf(fact_6123_integer__of__int__inverse,axiom,
    ! [Y: int] :
      ( ( member_int @ Y @ top_top_set_int )
     => ( ( code_int_of_integer @ ( code_integer_of_int @ Y ) )
        = Y ) ) ).

% integer_of_int_inverse
thf(fact_6124_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N6: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N6 ) @ N2 ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_6125_integer__of__nat__eq__of__nat,axiom,
    code_integer_of_nat = semiri4939895301339042750nteger ).

% integer_of_nat_eq_of_nat
thf(fact_6126_card__less__Suc2,axiom,
    ! [M3: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M3 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K4: nat] :
                ( ( member_nat @ ( suc @ K4 ) @ M3 )
                & ( ord_less_nat @ K4 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K4: nat] :
                ( ( member_nat @ K4 @ M3 )
                & ( ord_less_nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_6127_card__less__Suc,axiom,
    ! [M3: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M3 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K4: nat] :
                  ( ( member_nat @ ( suc @ K4 ) @ M3 )
                  & ( ord_less_nat @ K4 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K4: nat] :
                ( ( member_nat @ K4 @ M3 )
                & ( ord_less_nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_6128_card__less,axiom,
    ! [M3: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M3 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K4: nat] :
                ( ( member_nat @ K4 @ M3 )
                & ( ord_less_nat @ K4 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_6129_integer__of__nat__0,axiom,
    ( ( code_integer_of_nat @ zero_zero_nat )
    = zero_z3403309356797280102nteger ) ).

% integer_of_nat_0
thf(fact_6130_integer__of__nat_Oabs__eq,axiom,
    ( code_integer_of_nat
    = ( ^ [X3: nat] : ( code_integer_of_int @ ( semiri1314217659103216013at_int @ X3 ) ) ) ) ).

% integer_of_nat.abs_eq
thf(fact_6131_integer__of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( code_integer_of_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( numera6620942414471956472nteger @ N2 ) ) ).

% integer_of_nat_numeral
thf(fact_6132_integer__of__nat__1,axiom,
    ( ( code_integer_of_nat @ one_one_nat )
    = one_one_Code_integer ) ).

% integer_of_nat_1
thf(fact_6133_card__sum__le__nat__sum,axiom,
    ! [S2: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ S2 ) ) ).

% card_sum_le_nat_sum
thf(fact_6134_set__encode__insert,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ N2 @ A2 )
       => ( ( nat_set_encode @ ( insert_nat @ N2 @ A2 ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).

% set_encode_insert
thf(fact_6135_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).

% card_greaterThanLessThan_int
thf(fact_6136_wmax__insertI,axiom,
    ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ YS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).

% wmax_insertI
thf(fact_6137_wmin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).

% wmin_insertI
thf(fact_6138_smin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).

% smin_insertI
thf(fact_6139_prod_Oinsert_H,axiom,
    ! [I4: set_o,P4: $o > code_integer,I: $o] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != one_one_Code_integer ) ) ) )
     => ( ( ( member_o @ I @ I4 )
         => ( ( groups2836416601086271911nteger @ P4 @ ( insert_o @ I @ I4 ) )
            = ( groups2836416601086271911nteger @ P4 @ I4 ) ) )
        & ( ~ ( member_o @ I @ I4 )
         => ( ( groups2836416601086271911nteger @ P4 @ ( insert_o @ I @ I4 ) )
            = ( times_3573771949741848930nteger @ ( P4 @ I ) @ ( groups2836416601086271911nteger @ P4 @ I4 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_6140_prod_Oinsert_H,axiom,
    ! [I4: set_nat,P4: nat > code_integer,I: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != one_one_Code_integer ) ) ) )
     => ( ( ( member_nat @ I @ I4 )
         => ( ( groups8290090294887179773nteger @ P4 @ ( insert_nat @ I @ I4 ) )
            = ( groups8290090294887179773nteger @ P4 @ I4 ) ) )
        & ( ~ ( member_nat @ I @ I4 )
         => ( ( groups8290090294887179773nteger @ P4 @ ( insert_nat @ I @ I4 ) )
            = ( times_3573771949741848930nteger @ ( P4 @ I ) @ ( groups8290090294887179773nteger @ P4 @ I4 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_6141_prod_Oinsert_H,axiom,
    ! [I4: set_int,P4: int > code_integer,I: int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X3: int] :
              ( ( member_int @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != one_one_Code_integer ) ) ) )
     => ( ( ( member_int @ I @ I4 )
         => ( ( groups8661743855124024409nteger @ P4 @ ( insert_int @ I @ I4 ) )
            = ( groups8661743855124024409nteger @ P4 @ I4 ) ) )
        & ( ~ ( member_int @ I @ I4 )
         => ( ( groups8661743855124024409nteger @ P4 @ ( insert_int @ I @ I4 ) )
            = ( times_3573771949741848930nteger @ ( P4 @ I ) @ ( groups8661743855124024409nteger @ P4 @ I4 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_6142_prod_Oinsert_H,axiom,
    ! [I4: set_Code_integer,P4: code_integer > code_integer,I: code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != one_one_Code_integer ) ) ) )
     => ( ( ( member_Code_integer @ I @ I4 )
         => ( ( groups7219916603568234122nteger @ P4 @ ( insert_Code_integer @ I @ I4 ) )
            = ( groups7219916603568234122nteger @ P4 @ I4 ) ) )
        & ( ~ ( member_Code_integer @ I @ I4 )
         => ( ( groups7219916603568234122nteger @ P4 @ ( insert_Code_integer @ I @ I4 ) )
            = ( times_3573771949741848930nteger @ ( P4 @ I ) @ ( groups7219916603568234122nteger @ P4 @ I4 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_6143_prod_Oinsert_H,axiom,
    ! [I4: set_o,P4: $o > assn,I: $o] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != one_one_assn ) ) ) )
     => ( ( ( member_o @ I @ I4 )
         => ( ( groups12625163056032370o_assn @ P4 @ ( insert_o @ I @ I4 ) )
            = ( groups12625163056032370o_assn @ P4 @ I4 ) ) )
        & ( ~ ( member_o @ I @ I4 )
         => ( ( groups12625163056032370o_assn @ P4 @ ( insert_o @ I @ I4 ) )
            = ( times_times_assn @ ( P4 @ I ) @ ( groups12625163056032370o_assn @ P4 @ I4 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_6144_prod_Oinsert_H,axiom,
    ! [I4: set_nat,P4: nat > assn,I: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != one_one_assn ) ) ) )
     => ( ( ( member_nat @ I @ I4 )
         => ( ( groups3916077543535352520t_assn @ P4 @ ( insert_nat @ I @ I4 ) )
            = ( groups3916077543535352520t_assn @ P4 @ I4 ) ) )
        & ( ~ ( member_nat @ I @ I4 )
         => ( ( groups3916077543535352520t_assn @ P4 @ ( insert_nat @ I @ I4 ) )
            = ( times_times_assn @ ( P4 @ I ) @ ( groups3916077543535352520t_assn @ P4 @ I4 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_6145_prod_Oinsert_H,axiom,
    ! [I4: set_int,P4: int > assn,I: int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X3: int] :
              ( ( member_int @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != one_one_assn ) ) ) )
     => ( ( ( member_int @ I @ I4 )
         => ( ( groups4891613008741529892t_assn @ P4 @ ( insert_int @ I @ I4 ) )
            = ( groups4891613008741529892t_assn @ P4 @ I4 ) ) )
        & ( ~ ( member_int @ I @ I4 )
         => ( ( groups4891613008741529892t_assn @ P4 @ ( insert_int @ I @ I4 ) )
            = ( times_times_assn @ ( P4 @ I ) @ ( groups4891613008741529892t_assn @ P4 @ I4 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_6146_prod_Oinsert_H,axiom,
    ! [I4: set_Code_integer,P4: code_integer > assn,I: code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != one_one_assn ) ) ) )
     => ( ( ( member_Code_integer @ I @ I4 )
         => ( ( groups980104329299799381r_assn @ P4 @ ( insert_Code_integer @ I @ I4 ) )
            = ( groups980104329299799381r_assn @ P4 @ I4 ) ) )
        & ( ~ ( member_Code_integer @ I @ I4 )
         => ( ( groups980104329299799381r_assn @ P4 @ ( insert_Code_integer @ I @ I4 ) )
            = ( times_times_assn @ ( P4 @ I ) @ ( groups980104329299799381r_assn @ P4 @ I4 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_6147_prod_Oinsert_H,axiom,
    ! [I4: set_o,P4: $o > rat,I: $o] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != one_one_rat ) ) ) )
     => ( ( ( member_o @ I @ I4 )
         => ( ( groups4688112016125141794_o_rat @ P4 @ ( insert_o @ I @ I4 ) )
            = ( groups4688112016125141794_o_rat @ P4 @ I4 ) ) )
        & ( ~ ( member_o @ I @ I4 )
         => ( ( groups4688112016125141794_o_rat @ P4 @ ( insert_o @ I @ I4 ) )
            = ( times_times_rat @ ( P4 @ I ) @ ( groups4688112016125141794_o_rat @ P4 @ I4 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_6148_prod_Oinsert_H,axiom,
    ! [I4: set_nat,P4: nat > rat,I: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != one_one_rat ) ) ) )
     => ( ( ( member_nat @ I @ I4 )
         => ( ( groups4112464933559648120at_rat @ P4 @ ( insert_nat @ I @ I4 ) )
            = ( groups4112464933559648120at_rat @ P4 @ I4 ) ) )
        & ( ~ ( member_nat @ I @ I4 )
         => ( ( groups4112464933559648120at_rat @ P4 @ ( insert_nat @ I @ I4 ) )
            = ( times_times_rat @ ( P4 @ I ) @ ( groups4112464933559648120at_rat @ P4 @ I4 ) ) ) ) ) ) ).

% prod.insert'
thf(fact_6149_greaterThanLessThan__iff,axiom,
    ! [I: $o,L: $o,U: $o] :
      ( ( member_o @ I @ ( set_or1716231572884733764Than_o @ L @ U ) )
      = ( ( ord_less_o @ L @ I )
        & ( ord_less_o @ I @ U ) ) ) ).

% greaterThanLessThan_iff
thf(fact_6150_greaterThanLessThan__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( ( member_rat @ I @ ( set_or5199638295745620268an_rat @ L @ U ) )
      = ( ( ord_less_rat @ L @ I )
        & ( ord_less_rat @ I @ U ) ) ) ).

% greaterThanLessThan_iff
thf(fact_6151_greaterThanLessThan__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( ( member_num @ I @ ( set_or2392100141987894638an_num @ L @ U ) )
      = ( ( ord_less_num @ L @ I )
        & ( ord_less_num @ I @ U ) ) ) ).

% greaterThanLessThan_iff
thf(fact_6152_greaterThanLessThan__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( ( member_int @ I @ ( set_or5832277885323065728an_int @ L @ U ) )
      = ( ( ord_less_int @ L @ I )
        & ( ord_less_int @ I @ U ) ) ) ).

% greaterThanLessThan_iff
thf(fact_6153_greaterThanLessThan__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( ( member_nat @ I @ ( set_or5834768355832116004an_nat @ L @ U ) )
      = ( ( ord_less_nat @ L @ I )
        & ( ord_less_nat @ I @ U ) ) ) ).

% greaterThanLessThan_iff
thf(fact_6154_greaterThanLessThan__iff,axiom,
    ! [I: code_integer,L: code_integer,U: code_integer] :
      ( ( member_Code_integer @ I @ ( set_or4266950643985792945nteger @ L @ U ) )
      = ( ( ord_le6747313008572928689nteger @ L @ I )
        & ( ord_le6747313008572928689nteger @ I @ U ) ) ) ).

% greaterThanLessThan_iff
thf(fact_6155_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L @ U ) ) ).

% finite_greaterThanLessThan_int
thf(fact_6156_greaterThanLessThan__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat
        = ( set_or5199638295745620268an_rat @ A @ B ) )
      = ( ord_less_eq_rat @ B @ A ) ) ).

% greaterThanLessThan_empty_iff2
thf(fact_6157_greaterThanLessThan__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( set_or5199638295745620268an_rat @ A @ B )
        = bot_bot_set_rat )
      = ( ord_less_eq_rat @ B @ A ) ) ).

% greaterThanLessThan_empty_iff
thf(fact_6158_greaterThanLessThan__empty,axiom,
    ! [L: $o,K: $o] :
      ( ( ord_less_eq_o @ L @ K )
     => ( ( set_or1716231572884733764Than_o @ K @ L )
        = bot_bot_set_o ) ) ).

% greaterThanLessThan_empty
thf(fact_6159_greaterThanLessThan__empty,axiom,
    ! [L: set_nat,K: set_nat] :
      ( ( ord_less_eq_set_nat @ L @ K )
     => ( ( set_or8625682525731655386et_nat @ K @ L )
        = bot_bot_set_set_nat ) ) ).

% greaterThanLessThan_empty
thf(fact_6160_greaterThanLessThan__empty,axiom,
    ! [L: rat,K: rat] :
      ( ( ord_less_eq_rat @ L @ K )
     => ( ( set_or5199638295745620268an_rat @ K @ L )
        = bot_bot_set_rat ) ) ).

% greaterThanLessThan_empty
thf(fact_6161_greaterThanLessThan__empty,axiom,
    ! [L: num,K: num] :
      ( ( ord_less_eq_num @ L @ K )
     => ( ( set_or2392100141987894638an_num @ K @ L )
        = bot_bot_set_num ) ) ).

% greaterThanLessThan_empty
thf(fact_6162_greaterThanLessThan__empty,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ L @ K )
     => ( ( set_or5832277885323065728an_int @ K @ L )
        = bot_bot_set_int ) ) ).

% greaterThanLessThan_empty
thf(fact_6163_greaterThanLessThan__empty,axiom,
    ! [L: nat,K: nat] :
      ( ( ord_less_eq_nat @ L @ K )
     => ( ( set_or5834768355832116004an_nat @ K @ L )
        = bot_bot_set_nat ) ) ).

% greaterThanLessThan_empty
thf(fact_6164_greaterThanLessThan__empty,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ K )
     => ( ( set_or4266950643985792945nteger @ K @ L )
        = bot_bo3990330152332043303nteger ) ) ).

% greaterThanLessThan_empty
thf(fact_6165_infinite__Ioo__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or5199638295745620268an_rat @ A @ B ) ) )
      = ( ord_less_rat @ A @ B ) ) ).

% infinite_Ioo_iff
thf(fact_6166_image__uminus__greaterThanLessThan,axiom,
    ! [X: rat,Y: rat] :
      ( ( image_rat_rat @ uminus_uminus_rat @ ( set_or5199638295745620268an_rat @ X @ Y ) )
      = ( set_or5199638295745620268an_rat @ ( uminus_uminus_rat @ Y ) @ ( uminus_uminus_rat @ X ) ) ) ).

% image_uminus_greaterThanLessThan
thf(fact_6167_image__uminus__greaterThanLessThan,axiom,
    ! [X: int,Y: int] :
      ( ( image_int_int @ uminus_uminus_int @ ( set_or5832277885323065728an_int @ X @ Y ) )
      = ( set_or5832277885323065728an_int @ ( uminus_uminus_int @ Y ) @ ( uminus_uminus_int @ X ) ) ) ).

% image_uminus_greaterThanLessThan
thf(fact_6168_image__uminus__greaterThanLessThan,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( image_4470545334726330049nteger @ uminus1351360451143612070nteger @ ( set_or4266950643985792945nteger @ X @ Y ) )
      = ( set_or4266950643985792945nteger @ ( uminus1351360451143612070nteger @ Y ) @ ( uminus1351360451143612070nteger @ X ) ) ) ).

% image_uminus_greaterThanLessThan
thf(fact_6169_prod_Oempty_H,axiom,
    ! [P4: $o > code_integer] :
      ( ( groups2836416601086271911nteger @ P4 @ bot_bot_set_o )
      = one_one_Code_integer ) ).

% prod.empty'
thf(fact_6170_prod_Oempty_H,axiom,
    ! [P4: $o > assn] :
      ( ( groups12625163056032370o_assn @ P4 @ bot_bot_set_o )
      = one_one_assn ) ).

% prod.empty'
thf(fact_6171_prod_Oempty_H,axiom,
    ! [P4: $o > rat] :
      ( ( groups4688112016125141794_o_rat @ P4 @ bot_bot_set_o )
      = one_one_rat ) ).

% prod.empty'
thf(fact_6172_prod_Oempty_H,axiom,
    ! [P4: $o > nat] :
      ( ( groups5323242076211637530_o_nat @ P4 @ bot_bot_set_o )
      = one_one_nat ) ).

% prod.empty'
thf(fact_6173_prod_Oempty_H,axiom,
    ! [P4: $o > int] :
      ( ( groups5320751605702587254_o_int @ P4 @ bot_bot_set_o )
      = one_one_int ) ).

% prod.empty'
thf(fact_6174_prod_Oempty_H,axiom,
    ! [P4: nat > code_integer] :
      ( ( groups8290090294887179773nteger @ P4 @ bot_bot_set_nat )
      = one_one_Code_integer ) ).

% prod.empty'
thf(fact_6175_prod_Oempty_H,axiom,
    ! [P4: nat > assn] :
      ( ( groups3916077543535352520t_assn @ P4 @ bot_bot_set_nat )
      = one_one_assn ) ).

% prod.empty'
thf(fact_6176_prod_Oempty_H,axiom,
    ! [P4: nat > rat] :
      ( ( groups4112464933559648120at_rat @ P4 @ bot_bot_set_nat )
      = one_one_rat ) ).

% prod.empty'
thf(fact_6177_prod_Oempty_H,axiom,
    ! [P4: nat > nat] :
      ( ( groups4747594993646143856at_nat @ P4 @ bot_bot_set_nat )
      = one_one_nat ) ).

% prod.empty'
thf(fact_6178_prod_Oempty_H,axiom,
    ! [P4: nat > int] :
      ( ( groups4745104523137093580at_int @ P4 @ bot_bot_set_nat )
      = one_one_int ) ).

% prod.empty'
thf(fact_6179_smin__emptyI,axiom,
    ! [X11: set_Pr1261947904930325089at_nat] :
      ( ( X11 != bot_bo2099793752762293965at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X11 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).

% smin_emptyI
thf(fact_6180_wmax__emptyI,axiom,
    ! [X11: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ X11 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X11 ) @ fun_max_weak ) ) ).

% wmax_emptyI
thf(fact_6181_wmin__emptyI,axiom,
    ! [X11: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X11 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).

% wmin_emptyI
thf(fact_6182_prod_Onon__neutral_H,axiom,
    ! [G: $o > code_integer,I4: set_o] :
      ( ( groups2836416601086271911nteger @ G
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_Code_integer ) ) ) )
      = ( groups2836416601086271911nteger @ G @ I4 ) ) ).

% prod.non_neutral'
thf(fact_6183_prod_Onon__neutral_H,axiom,
    ! [G: int > code_integer,I4: set_int] :
      ( ( groups8661743855124024409nteger @ G
        @ ( collect_int
          @ ^ [X3: int] :
              ( ( member_int @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_Code_integer ) ) ) )
      = ( groups8661743855124024409nteger @ G @ I4 ) ) ).

% prod.non_neutral'
thf(fact_6184_prod_Onon__neutral_H,axiom,
    ! [G: nat > code_integer,I4: set_nat] :
      ( ( groups8290090294887179773nteger @ G
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_Code_integer ) ) ) )
      = ( groups8290090294887179773nteger @ G @ I4 ) ) ).

% prod.non_neutral'
thf(fact_6185_prod_Onon__neutral_H,axiom,
    ! [G: $o > assn,I4: set_o] :
      ( ( groups12625163056032370o_assn @ G
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_assn ) ) ) )
      = ( groups12625163056032370o_assn @ G @ I4 ) ) ).

% prod.non_neutral'
thf(fact_6186_prod_Onon__neutral_H,axiom,
    ! [G: int > assn,I4: set_int] :
      ( ( groups4891613008741529892t_assn @ G
        @ ( collect_int
          @ ^ [X3: int] :
              ( ( member_int @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_assn ) ) ) )
      = ( groups4891613008741529892t_assn @ G @ I4 ) ) ).

% prod.non_neutral'
thf(fact_6187_prod_Onon__neutral_H,axiom,
    ! [G: nat > assn,I4: set_nat] :
      ( ( groups3916077543535352520t_assn @ G
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_assn ) ) ) )
      = ( groups3916077543535352520t_assn @ G @ I4 ) ) ).

% prod.non_neutral'
thf(fact_6188_prod_Onon__neutral_H,axiom,
    ! [G: $o > rat,I4: set_o] :
      ( ( groups4688112016125141794_o_rat @ G
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_rat ) ) ) )
      = ( groups4688112016125141794_o_rat @ G @ I4 ) ) ).

% prod.non_neutral'
thf(fact_6189_prod_Onon__neutral_H,axiom,
    ! [G: int > rat,I4: set_int] :
      ( ( groups5111818645460702676nt_rat @ G
        @ ( collect_int
          @ ^ [X3: int] :
              ( ( member_int @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_rat ) ) ) )
      = ( groups5111818645460702676nt_rat @ G @ I4 ) ) ).

% prod.non_neutral'
thf(fact_6190_prod_Onon__neutral_H,axiom,
    ! [G: nat > rat,I4: set_nat] :
      ( ( groups4112464933559648120at_rat @ G
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_rat ) ) ) )
      = ( groups4112464933559648120at_rat @ G @ I4 ) ) ).

% prod.non_neutral'
thf(fact_6191_prod_Onon__neutral_H,axiom,
    ! [G: $o > nat,I4: set_o] :
      ( ( groups5323242076211637530_o_nat @ G
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_nat ) ) ) )
      = ( groups5323242076211637530_o_nat @ G @ I4 ) ) ).

% prod.non_neutral'
thf(fact_6192_infinite__Ioo,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or5199638295745620268an_rat @ A @ B ) ) ) ).

% infinite_Ioo
thf(fact_6193_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or5199638295745620268an_rat @ A @ B ) @ ( set_or5199638295745620268an_rat @ C @ D2 ) )
      = ( ( ord_less_rat @ A @ B )
       => ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B @ D2 ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_6194_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K5: nat,M5: nat] :
          ( X
         != ( product_Pair_nat_nat @ K5 @ M5 ) ) ).

% prod_decode_aux.cases
thf(fact_6195_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
      = ( set_or5832277885323065728an_int @ L @ U ) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_6196_prod_Omono__neutral__left_H,axiom,
    ! [S2: set_nat,T5: set_nat,G: nat > code_integer] :
      ( ( ord_less_eq_set_nat @ S2 @ T5 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_Code_integer ) )
       => ( ( groups8290090294887179773nteger @ G @ S2 )
          = ( groups8290090294887179773nteger @ G @ T5 ) ) ) ) ).

% prod.mono_neutral_left'
thf(fact_6197_prod_Omono__neutral__left_H,axiom,
    ! [S2: set_nat,T5: set_nat,G: nat > assn] :
      ( ( ord_less_eq_set_nat @ S2 @ T5 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_assn ) )
       => ( ( groups3916077543535352520t_assn @ G @ S2 )
          = ( groups3916077543535352520t_assn @ G @ T5 ) ) ) ) ).

% prod.mono_neutral_left'
thf(fact_6198_prod_Omono__neutral__left_H,axiom,
    ! [S2: set_nat,T5: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ S2 @ T5 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_rat ) )
       => ( ( groups4112464933559648120at_rat @ G @ S2 )
          = ( groups4112464933559648120at_rat @ G @ T5 ) ) ) ) ).

% prod.mono_neutral_left'
thf(fact_6199_prod_Omono__neutral__left_H,axiom,
    ! [S2: set_nat,T5: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ S2 @ T5 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_nat ) )
       => ( ( groups4747594993646143856at_nat @ G @ S2 )
          = ( groups4747594993646143856at_nat @ G @ T5 ) ) ) ) ).

% prod.mono_neutral_left'
thf(fact_6200_prod_Omono__neutral__left_H,axiom,
    ! [S2: set_nat,T5: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ S2 @ T5 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_int ) )
       => ( ( groups4745104523137093580at_int @ G @ S2 )
          = ( groups4745104523137093580at_int @ G @ T5 ) ) ) ) ).

% prod.mono_neutral_left'
thf(fact_6201_prod_Omono__neutral__right_H,axiom,
    ! [S2: set_nat,T5: set_nat,G: nat > code_integer] :
      ( ( ord_less_eq_set_nat @ S2 @ T5 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_Code_integer ) )
       => ( ( groups8290090294887179773nteger @ G @ T5 )
          = ( groups8290090294887179773nteger @ G @ S2 ) ) ) ) ).

% prod.mono_neutral_right'
thf(fact_6202_prod_Omono__neutral__right_H,axiom,
    ! [S2: set_nat,T5: set_nat,G: nat > assn] :
      ( ( ord_less_eq_set_nat @ S2 @ T5 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_assn ) )
       => ( ( groups3916077543535352520t_assn @ G @ T5 )
          = ( groups3916077543535352520t_assn @ G @ S2 ) ) ) ) ).

% prod.mono_neutral_right'
thf(fact_6203_prod_Omono__neutral__right_H,axiom,
    ! [S2: set_nat,T5: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ S2 @ T5 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_rat ) )
       => ( ( groups4112464933559648120at_rat @ G @ T5 )
          = ( groups4112464933559648120at_rat @ G @ S2 ) ) ) ) ).

% prod.mono_neutral_right'
thf(fact_6204_prod_Omono__neutral__right_H,axiom,
    ! [S2: set_nat,T5: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ S2 @ T5 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_nat ) )
       => ( ( groups4747594993646143856at_nat @ G @ T5 )
          = ( groups4747594993646143856at_nat @ G @ S2 ) ) ) ) ).

% prod.mono_neutral_right'
thf(fact_6205_prod_Omono__neutral__right_H,axiom,
    ! [S2: set_nat,T5: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ S2 @ T5 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_int ) )
       => ( ( groups4745104523137093580at_int @ G @ T5 )
          = ( groups4745104523137093580at_int @ G @ S2 ) ) ) ) ).

% prod.mono_neutral_right'
thf(fact_6206_prod_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_o,T5: set_o,H3: $o > code_integer,G: $o > code_integer] :
      ( ( ord_less_eq_set_o @ S2 @ T5 )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ ( minus_minus_set_o @ T5 @ S2 ) )
           => ( ( H3 @ I2 )
              = one_one_Code_integer ) )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups2836416601086271911nteger @ G @ S2 )
            = ( groups2836416601086271911nteger @ H3 @ T5 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_6207_prod_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_int,T5: set_int,H3: int > code_integer,G: int > code_integer] :
      ( ( ord_less_eq_set_int @ S2 @ T5 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ ( minus_minus_set_int @ T5 @ S2 ) )
           => ( ( H3 @ I2 )
              = one_one_Code_integer ) )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups8661743855124024409nteger @ G @ S2 )
            = ( groups8661743855124024409nteger @ H3 @ T5 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_6208_prod_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_o,T5: set_o,H3: $o > assn,G: $o > assn] :
      ( ( ord_less_eq_set_o @ S2 @ T5 )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ ( minus_minus_set_o @ T5 @ S2 ) )
           => ( ( H3 @ I2 )
              = one_one_assn ) )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups12625163056032370o_assn @ G @ S2 )
            = ( groups12625163056032370o_assn @ H3 @ T5 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_6209_prod_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_int,T5: set_int,H3: int > assn,G: int > assn] :
      ( ( ord_less_eq_set_int @ S2 @ T5 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ ( minus_minus_set_int @ T5 @ S2 ) )
           => ( ( H3 @ I2 )
              = one_one_assn ) )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups4891613008741529892t_assn @ G @ S2 )
            = ( groups4891613008741529892t_assn @ H3 @ T5 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_6210_prod_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_o,T5: set_o,H3: $o > rat,G: $o > rat] :
      ( ( ord_less_eq_set_o @ S2 @ T5 )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ ( minus_minus_set_o @ T5 @ S2 ) )
           => ( ( H3 @ I2 )
              = one_one_rat ) )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups4688112016125141794_o_rat @ G @ S2 )
            = ( groups4688112016125141794_o_rat @ H3 @ T5 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_6211_prod_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_int,T5: set_int,H3: int > rat,G: int > rat] :
      ( ( ord_less_eq_set_int @ S2 @ T5 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ ( minus_minus_set_int @ T5 @ S2 ) )
           => ( ( H3 @ I2 )
              = one_one_rat ) )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups5111818645460702676nt_rat @ G @ S2 )
            = ( groups5111818645460702676nt_rat @ H3 @ T5 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_6212_prod_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_o,T5: set_o,H3: $o > nat,G: $o > nat] :
      ( ( ord_less_eq_set_o @ S2 @ T5 )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ ( minus_minus_set_o @ T5 @ S2 ) )
           => ( ( H3 @ I2 )
              = one_one_nat ) )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups5323242076211637530_o_nat @ G @ S2 )
            = ( groups5323242076211637530_o_nat @ H3 @ T5 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_6213_prod_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_int,T5: set_int,H3: int > nat,G: int > nat] :
      ( ( ord_less_eq_set_int @ S2 @ T5 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ ( minus_minus_set_int @ T5 @ S2 ) )
           => ( ( H3 @ I2 )
              = one_one_nat ) )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups5746948705547198412nt_nat @ G @ S2 )
            = ( groups5746948705547198412nt_nat @ H3 @ T5 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_6214_prod_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_o,T5: set_o,H3: $o > int,G: $o > int] :
      ( ( ord_less_eq_set_o @ S2 @ T5 )
     => ( ! [I2: $o] :
            ( ( member_o @ I2 @ ( minus_minus_set_o @ T5 @ S2 ) )
           => ( ( H3 @ I2 )
              = one_one_int ) )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups5320751605702587254_o_int @ G @ S2 )
            = ( groups5320751605702587254_o_int @ H3 @ T5 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_6215_prod_Omono__neutral__cong__left_H,axiom,
    ! [S2: set_int,T5: set_int,H3: int > int,G: int > int] :
      ( ( ord_less_eq_set_int @ S2 @ T5 )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ ( minus_minus_set_int @ T5 @ S2 ) )
           => ( ( H3 @ I2 )
              = one_one_int ) )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups5744458235038148136nt_int @ G @ S2 )
            = ( groups5744458235038148136nt_int @ H3 @ T5 ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
thf(fact_6216_prod_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_o,T5: set_o,G: $o > code_integer,H3: $o > code_integer] :
      ( ( ord_less_eq_set_o @ S2 @ T5 )
     => ( ! [X4: $o] :
            ( ( member_o @ X4 @ ( minus_minus_set_o @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_Code_integer ) )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups2836416601086271911nteger @ G @ T5 )
            = ( groups2836416601086271911nteger @ H3 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_6217_prod_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_int,T5: set_int,G: int > code_integer,H3: int > code_integer] :
      ( ( ord_less_eq_set_int @ S2 @ T5 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_Code_integer ) )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups8661743855124024409nteger @ G @ T5 )
            = ( groups8661743855124024409nteger @ H3 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_6218_prod_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_o,T5: set_o,G: $o > assn,H3: $o > assn] :
      ( ( ord_less_eq_set_o @ S2 @ T5 )
     => ( ! [X4: $o] :
            ( ( member_o @ X4 @ ( minus_minus_set_o @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_assn ) )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups12625163056032370o_assn @ G @ T5 )
            = ( groups12625163056032370o_assn @ H3 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_6219_prod_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_int,T5: set_int,G: int > assn,H3: int > assn] :
      ( ( ord_less_eq_set_int @ S2 @ T5 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_assn ) )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups4891613008741529892t_assn @ G @ T5 )
            = ( groups4891613008741529892t_assn @ H3 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_6220_prod_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_o,T5: set_o,G: $o > rat,H3: $o > rat] :
      ( ( ord_less_eq_set_o @ S2 @ T5 )
     => ( ! [X4: $o] :
            ( ( member_o @ X4 @ ( minus_minus_set_o @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_rat ) )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups4688112016125141794_o_rat @ G @ T5 )
            = ( groups4688112016125141794_o_rat @ H3 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_6221_prod_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_int,T5: set_int,G: int > rat,H3: int > rat] :
      ( ( ord_less_eq_set_int @ S2 @ T5 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_rat ) )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups5111818645460702676nt_rat @ G @ T5 )
            = ( groups5111818645460702676nt_rat @ H3 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_6222_prod_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_o,T5: set_o,G: $o > nat,H3: $o > nat] :
      ( ( ord_less_eq_set_o @ S2 @ T5 )
     => ( ! [X4: $o] :
            ( ( member_o @ X4 @ ( minus_minus_set_o @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_nat ) )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups5323242076211637530_o_nat @ G @ T5 )
            = ( groups5323242076211637530_o_nat @ H3 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_6223_prod_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_int,T5: set_int,G: int > nat,H3: int > nat] :
      ( ( ord_less_eq_set_int @ S2 @ T5 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_nat ) )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups5746948705547198412nt_nat @ G @ T5 )
            = ( groups5746948705547198412nt_nat @ H3 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_6224_prod_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_o,T5: set_o,G: $o > int,H3: $o > int] :
      ( ( ord_less_eq_set_o @ S2 @ T5 )
     => ( ! [X4: $o] :
            ( ( member_o @ X4 @ ( minus_minus_set_o @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_int ) )
       => ( ! [X4: $o] :
              ( ( member_o @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups5320751605702587254_o_int @ G @ T5 )
            = ( groups5320751605702587254_o_int @ H3 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_6225_prod_Omono__neutral__cong__right_H,axiom,
    ! [S2: set_int,T5: set_int,G: int > int,H3: int > int] :
      ( ( ord_less_eq_set_int @ S2 @ T5 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ ( minus_minus_set_int @ T5 @ S2 ) )
           => ( ( G @ X4 )
              = one_one_int ) )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ S2 )
             => ( ( G @ X4 )
                = ( H3 @ X4 ) ) )
         => ( ( groups5744458235038148136nt_int @ G @ T5 )
            = ( groups5744458235038148136nt_int @ H3 @ S2 ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
thf(fact_6226_prod_Odistrib_H,axiom,
    ! [I4: set_o,G: $o > code_integer,H3: $o > code_integer] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_Code_integer ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [X3: $o] :
                ( ( member_o @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != one_one_Code_integer ) ) ) )
       => ( ( groups2836416601086271911nteger
            @ ^ [I3: $o] : ( times_3573771949741848930nteger @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( times_3573771949741848930nteger @ ( groups2836416601086271911nteger @ G @ I4 ) @ ( groups2836416601086271911nteger @ H3 @ I4 ) ) ) ) ) ).

% prod.distrib'
thf(fact_6227_prod_Odistrib_H,axiom,
    ! [I4: set_nat,G: nat > code_integer,H3: nat > code_integer] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_Code_integer ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != one_one_Code_integer ) ) ) )
       => ( ( groups8290090294887179773nteger
            @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( times_3573771949741848930nteger @ ( groups8290090294887179773nteger @ G @ I4 ) @ ( groups8290090294887179773nteger @ H3 @ I4 ) ) ) ) ) ).

% prod.distrib'
thf(fact_6228_prod_Odistrib_H,axiom,
    ! [I4: set_int,G: int > code_integer,H3: int > code_integer] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X3: int] :
              ( ( member_int @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_Code_integer ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [X3: int] :
                ( ( member_int @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != one_one_Code_integer ) ) ) )
       => ( ( groups8661743855124024409nteger
            @ ^ [I3: int] : ( times_3573771949741848930nteger @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( times_3573771949741848930nteger @ ( groups8661743855124024409nteger @ G @ I4 ) @ ( groups8661743855124024409nteger @ H3 @ I4 ) ) ) ) ) ).

% prod.distrib'
thf(fact_6229_prod_Odistrib_H,axiom,
    ! [I4: set_Code_integer,G: code_integer > code_integer,H3: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_Code_integer ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != one_one_Code_integer ) ) ) )
       => ( ( groups7219916603568234122nteger
            @ ^ [I3: code_integer] : ( times_3573771949741848930nteger @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( times_3573771949741848930nteger @ ( groups7219916603568234122nteger @ G @ I4 ) @ ( groups7219916603568234122nteger @ H3 @ I4 ) ) ) ) ) ).

% prod.distrib'
thf(fact_6230_prod_Odistrib_H,axiom,
    ! [I4: set_o,G: $o > assn,H3: $o > assn] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_assn ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [X3: $o] :
                ( ( member_o @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != one_one_assn ) ) ) )
       => ( ( groups12625163056032370o_assn
            @ ^ [I3: $o] : ( times_times_assn @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( times_times_assn @ ( groups12625163056032370o_assn @ G @ I4 ) @ ( groups12625163056032370o_assn @ H3 @ I4 ) ) ) ) ) ).

% prod.distrib'
thf(fact_6231_prod_Odistrib_H,axiom,
    ! [I4: set_nat,G: nat > assn,H3: nat > assn] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_assn ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != one_one_assn ) ) ) )
       => ( ( groups3916077543535352520t_assn
            @ ^ [I3: nat] : ( times_times_assn @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( times_times_assn @ ( groups3916077543535352520t_assn @ G @ I4 ) @ ( groups3916077543535352520t_assn @ H3 @ I4 ) ) ) ) ) ).

% prod.distrib'
thf(fact_6232_prod_Odistrib_H,axiom,
    ! [I4: set_int,G: int > assn,H3: int > assn] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X3: int] :
              ( ( member_int @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_assn ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [X3: int] :
                ( ( member_int @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != one_one_assn ) ) ) )
       => ( ( groups4891613008741529892t_assn
            @ ^ [I3: int] : ( times_times_assn @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( times_times_assn @ ( groups4891613008741529892t_assn @ G @ I4 ) @ ( groups4891613008741529892t_assn @ H3 @ I4 ) ) ) ) ) ).

% prod.distrib'
thf(fact_6233_prod_Odistrib_H,axiom,
    ! [I4: set_Code_integer,G: code_integer > assn,H3: code_integer > assn] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_assn ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != one_one_assn ) ) ) )
       => ( ( groups980104329299799381r_assn
            @ ^ [I3: code_integer] : ( times_times_assn @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( times_times_assn @ ( groups980104329299799381r_assn @ G @ I4 ) @ ( groups980104329299799381r_assn @ H3 @ I4 ) ) ) ) ) ).

% prod.distrib'
thf(fact_6234_prod_Odistrib_H,axiom,
    ! [I4: set_o,G: $o > rat,H3: $o > rat] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [X3: $o] :
                ( ( member_o @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != one_one_rat ) ) ) )
       => ( ( groups4688112016125141794_o_rat
            @ ^ [I3: $o] : ( times_times_rat @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( times_times_rat @ ( groups4688112016125141794_o_rat @ G @ I4 ) @ ( groups4688112016125141794_o_rat @ H3 @ I4 ) ) ) ) ) ).

% prod.distrib'
thf(fact_6235_prod_Odistrib_H,axiom,
    ! [I4: set_nat,G: nat > rat,H3: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ I4 )
              & ( ( G @ X3 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != one_one_rat ) ) ) )
       => ( ( groups4112464933559648120at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( times_times_rat @ ( groups4112464933559648120at_rat @ G @ I4 ) @ ( groups4112464933559648120at_rat @ H3 @ I4 ) ) ) ) ) ).

% prod.distrib'
thf(fact_6236_prod_OG__def,axiom,
    ( groups2836416601086271911nteger
    = ( ^ [P6: $o > code_integer,I5: set_o] :
          ( if_Code_integer
          @ ( finite_finite_o
            @ ( collect_o
              @ ^ [X3: $o] :
                  ( ( member_o @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_Code_integer ) ) ) )
          @ ( groups7694694392188491536nteger @ P6
            @ ( collect_o
              @ ^ [X3: $o] :
                  ( ( member_o @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_Code_integer ) ) ) )
          @ one_one_Code_integer ) ) ) ).

% prod.G_def
thf(fact_6237_prod_OG__def,axiom,
    ( groups8290090294887179773nteger
    = ( ^ [P6: nat > code_integer,I5: set_nat] :
          ( if_Code_integer
          @ ( finite_finite_nat
            @ ( collect_nat
              @ ^ [X3: nat] :
                  ( ( member_nat @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_Code_integer ) ) ) )
          @ ( groups3455450783089532116nteger @ P6
            @ ( collect_nat
              @ ^ [X3: nat] :
                  ( ( member_nat @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_Code_integer ) ) ) )
          @ one_one_Code_integer ) ) ) ).

% prod.G_def
thf(fact_6238_prod_OG__def,axiom,
    ( groups8661743855124024409nteger
    = ( ^ [P6: int > code_integer,I5: set_int] :
          ( if_Code_integer
          @ ( finite_finite_int
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( member_int @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_Code_integer ) ) ) )
          @ ( groups3827104343326376752nteger @ P6
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( member_int @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_Code_integer ) ) ) )
          @ one_one_Code_integer ) ) ) ).

% prod.G_def
thf(fact_6239_prod_OG__def,axiom,
    ( groups7219916603568234122nteger
    = ( ^ [P6: code_integer > code_integer,I5: set_Code_integer] :
          ( if_Code_integer
          @ ( finite6017078050557962740nteger
            @ ( collect_Code_integer
              @ ^ [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_Code_integer ) ) ) )
          @ ( groups3674199335183972705nteger @ P6
            @ ( collect_Code_integer
              @ ^ [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_Code_integer ) ) ) )
          @ one_one_Code_integer ) ) ) ).

% prod.G_def
thf(fact_6240_prod_OG__def,axiom,
    ( groups12625163056032370o_assn
    = ( ^ [P6: $o > assn,I5: set_o] :
          ( if_assn
          @ ( finite_finite_o
            @ ( collect_o
              @ ^ [X3: $o] :
                  ( ( member_o @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_assn ) ) ) )
          @ ( groups5301882518646026715o_assn @ P6
            @ ( collect_o
              @ ^ [X3: $o] :
                  ( ( member_o @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_assn ) ) ) )
          @ one_one_assn ) ) ) ).

% prod.G_def
thf(fact_6241_prod_OG__def,axiom,
    ( groups3916077543535352520t_assn
    = ( ^ [P6: nat > assn,I5: set_nat] :
          ( if_assn
          @ ( finite_finite_nat
            @ ( collect_nat
              @ ^ [X3: nat] :
                  ( ( member_nat @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_assn ) ) ) )
          @ ( groups6906906614972039071t_assn @ P6
            @ ( collect_nat
              @ ^ [X3: nat] :
                  ( ( member_nat @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_assn ) ) ) )
          @ one_one_assn ) ) ) ).

% prod.G_def
thf(fact_6242_prod_OG__def,axiom,
    ( groups4891613008741529892t_assn
    = ( ^ [P6: int > assn,I5: set_int] :
          ( if_assn
          @ ( finite_finite_int
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( member_int @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_assn ) ) ) )
          @ ( groups7882442080178216443t_assn @ P6
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( member_int @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_assn ) ) ) )
          @ one_one_assn ) ) ) ).

% prod.G_def
thf(fact_6243_prod_OG__def,axiom,
    ( groups980104329299799381r_assn
    = ( ^ [P6: code_integer > assn,I5: set_Code_integer] :
          ( if_assn
          @ ( finite6017078050557962740nteger
            @ ( collect_Code_integer
              @ ^ [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_assn ) ) ) )
          @ ( groups1304777262505850412r_assn @ P6
            @ ( collect_Code_integer
              @ ^ [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_assn ) ) ) )
          @ one_one_assn ) ) ) ).

% prod.G_def
thf(fact_6244_prod_OG__def,axiom,
    ( groups4688112016125141794_o_rat
    = ( ^ [P6: $o > rat,I5: set_o] :
          ( if_rat
          @ ( finite_finite_o
            @ ( collect_o
              @ ^ [X3: $o] :
                  ( ( member_o @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_rat ) ) ) )
          @ ( groups2869687844427037835_o_rat @ P6
            @ ( collect_o
              @ ^ [X3: $o] :
                  ( ( member_o @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_rat ) ) ) )
          @ one_one_rat ) ) ) ).

% prod.G_def
thf(fact_6245_prod_OG__def,axiom,
    ( groups4112464933559648120at_rat
    = ( ^ [P6: nat > rat,I5: set_nat] :
          ( if_rat
          @ ( finite_finite_nat
            @ ( collect_nat
              @ ^ [X3: nat] :
                  ( ( member_nat @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_rat ) ) ) )
          @ ( groups73079841787564623at_rat @ P6
            @ ( collect_nat
              @ ^ [X3: nat] :
                  ( ( member_nat @ X3 @ I5 )
                  & ( ( P6 @ X3 )
                   != one_one_rat ) ) ) )
          @ one_one_rat ) ) ) ).

% prod.G_def
thf(fact_6246_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or5199638295745620268an_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D2 ) )
      = ( ( ord_less_rat @ A @ B )
       => ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B @ D2 ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_6247_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or5199638295745620268an_rat @ A @ B ) @ ( set_or4029947393144176647an_rat @ C @ D2 ) )
      = ( ( ord_less_rat @ A @ B )
       => ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B @ D2 ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_6248_atLeastAtMost__diff__ends,axiom,
    ! [A: $o,B: $o] :
      ( ( minus_minus_set_o @ ( set_or8904488021354931149Most_o @ A @ B ) @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) )
      = ( set_or1716231572884733764Than_o @ A @ B ) ) ).

% atLeastAtMost_diff_ends
thf(fact_6249_atLeastAtMost__diff__ends,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) ) )
      = ( set_or5832277885323065728an_int @ A @ B ) ) ).

% atLeastAtMost_diff_ends
thf(fact_6250_atLeastAtMost__diff__ends,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
      = ( set_or5834768355832116004an_nat @ A @ B ) ) ).

% atLeastAtMost_diff_ends
thf(fact_6251_atLeastAtMost__diff__ends,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_2355218937544613996nteger @ ( set_or189985376899183464nteger @ A @ B ) @ ( insert_Code_integer @ A @ ( insert_Code_integer @ B @ bot_bo3990330152332043303nteger ) ) )
      = ( set_or4266950643985792945nteger @ A @ B ) ) ).

% atLeastAtMost_diff_ends
thf(fact_6252_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y9: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X11: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ Y9 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X11 @ Y9 ) @ fun_max_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ X11 ) @ Y9 ) @ fun_max_strict ) ) ) ) ).

% smax_insertI
thf(fact_6253_set__decode__plus__power__2,axiom,
    ! [N2: nat,Z: nat] :
      ( ~ ( member_nat @ N2 @ ( nat_set_decode @ Z ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ Z ) )
        = ( insert_nat @ N2 @ ( nat_set_decode @ Z ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_6254_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K4: nat,M: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M @ K4 ) @ ( product_Pair_nat_nat @ M @ ( minus_minus_nat @ K4 @ M ) ) @ ( nat_prod_decode_aux @ ( suc @ K4 ) @ ( minus_minus_nat @ M @ ( suc @ K4 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_6255_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa )
        = Y )
     => ( ( ( ord_less_eq_nat @ Xa @ X )
         => ( Y
            = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa @ X )
         => ( Y
            = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_6256_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).

% min_rpair_set
thf(fact_6257_triangle__Suc,axiom,
    ! [N2: nat] :
      ( ( nat_triangle @ ( suc @ N2 ) )
      = ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).

% triangle_Suc
thf(fact_6258_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% finite_greaterThanLessThan
thf(fact_6259_finite__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or4266950643985792945nteger @ L @ U ) ) ).

% finite_greaterThanLessThan_integer
thf(fact_6260_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).

% card_greaterThanLessThan
thf(fact_6261_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).

% max_rpair_set
thf(fact_6262_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
      = ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_6263_smax__emptyI,axiom,
    ! [Y9: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ Y9 )
     => ( ( Y9 != bot_bo2099793752762293965at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y9 ) @ fun_max_strict ) ) ) ).

% smax_emptyI
thf(fact_6264_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( set_or8404916559141939852nteger @ ( plus_p5714425477246183910nteger @ L @ one_one_Code_integer ) @ U )
      = ( set_or4266950643985792945nteger @ L @ U ) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
thf(fact_6265_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa @ X )
               => ( Y
                  = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa @ X )
               => ( Y
                  = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_6266_total__on__singleton,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : ( total_7212417256332265445at_nat @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X @ X ) @ bot_bo228742789529271731at_nat ) ) ).

% total_on_singleton
thf(fact_6267_total__on__singleton,axiom,
    ! [X: produc3843707927480180839at_nat] : ( total_1513685709984341995at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ X @ X ) @ bot_bo7193250159860175667at_nat ) ) ).

% total_on_singleton
thf(fact_6268_total__on__singleton,axiom,
    ! [X: product_prod_nat_nat] : ( total_3592101749530773125at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ bot_bo5327735625951526323at_nat ) ) ).

% total_on_singleton
thf(fact_6269_total__on__singleton,axiom,
    ! [X: $o] : ( total_on_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X @ X ) @ bot_bo7073875226086086771od_o_o ) ) ).

% total_on_singleton
thf(fact_6270_total__on__singleton,axiom,
    ! [X: nat] : ( total_on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).

% total_on_singleton
thf(fact_6271_total__on__singleton,axiom,
    ! [X: int] : ( total_on_int @ ( insert_int @ X @ bot_bot_set_int ) @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ X @ X ) @ bot_bo1796632182523588997nt_int ) ) ).

% total_on_singleton
thf(fact_6272_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M2: nat] :
      ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M2 ) )
      = ( plus_plus_nat @ ( nat_triangle @ K ) @ M2 ) ) ).

% prod_encode_prod_decode_aux
thf(fact_6273_sum_Oinsert_H,axiom,
    ! [I4: set_o,P4: $o > code_integer,I: $o] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( ( member_o @ I @ I4 )
         => ( ( groups1402912129352969042nteger @ P4 @ ( insert_o @ I @ I4 ) )
            = ( groups1402912129352969042nteger @ P4 @ I4 ) ) )
        & ( ~ ( member_o @ I @ I4 )
         => ( ( groups1402912129352969042nteger @ P4 @ ( insert_o @ I @ I4 ) )
            = ( plus_p5714425477246183910nteger @ ( P4 @ I ) @ ( groups1402912129352969042nteger @ P4 @ I4 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_6274_sum_Oinsert_H,axiom,
    ! [I4: set_nat,P4: nat > code_integer,I: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( ( member_nat @ I @ I4 )
         => ( ( groups555127423416065298nteger @ P4 @ ( insert_nat @ I @ I4 ) )
            = ( groups555127423416065298nteger @ P4 @ I4 ) ) )
        & ( ~ ( member_nat @ I @ I4 )
         => ( ( groups555127423416065298nteger @ P4 @ ( insert_nat @ I @ I4 ) )
            = ( plus_p5714425477246183910nteger @ ( P4 @ I ) @ ( groups555127423416065298nteger @ P4 @ I4 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_6275_sum_Oinsert_H,axiom,
    ! [I4: set_int,P4: int > code_integer,I: int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X3: int] :
              ( ( member_int @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( ( member_int @ I @ I4 )
         => ( ( groups926780983652909934nteger @ P4 @ ( insert_int @ I @ I4 ) )
            = ( groups926780983652909934nteger @ P4 @ I4 ) ) )
        & ( ~ ( member_int @ I @ I4 )
         => ( ( groups926780983652909934nteger @ P4 @ ( insert_int @ I @ I4 ) )
            = ( plus_p5714425477246183910nteger @ ( P4 @ I ) @ ( groups926780983652909934nteger @ P4 @ I4 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_6276_sum_Oinsert_H,axiom,
    ! [I4: set_Code_integer,P4: code_integer > code_integer,I: code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( ( member_Code_integer @ I @ I4 )
         => ( ( groups910942671188738463nteger @ P4 @ ( insert_Code_integer @ I @ I4 ) )
            = ( groups910942671188738463nteger @ P4 @ I4 ) ) )
        & ( ~ ( member_Code_integer @ I @ I4 )
         => ( ( groups910942671188738463nteger @ P4 @ ( insert_Code_integer @ I @ I4 ) )
            = ( plus_p5714425477246183910nteger @ ( P4 @ I ) @ ( groups910942671188738463nteger @ P4 @ I4 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_6277_sum_Oinsert_H,axiom,
    ! [I4: set_o,P4: $o > rat,I: $o] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != zero_zero_rat ) ) ) )
     => ( ( ( member_o @ I @ I4 )
         => ( ( groups3921277224699582669_o_rat @ P4 @ ( insert_o @ I @ I4 ) )
            = ( groups3921277224699582669_o_rat @ P4 @ I4 ) ) )
        & ( ~ ( member_o @ I @ I4 )
         => ( ( groups3921277224699582669_o_rat @ P4 @ ( insert_o @ I @ I4 ) )
            = ( plus_plus_rat @ ( P4 @ I ) @ ( groups3921277224699582669_o_rat @ P4 @ I4 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_6278_sum_Oinsert_H,axiom,
    ! [I4: set_nat,P4: nat > rat,I: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != zero_zero_rat ) ) ) )
     => ( ( ( member_nat @ I @ I4 )
         => ( ( groups1351286907653491341at_rat @ P4 @ ( insert_nat @ I @ I4 ) )
            = ( groups1351286907653491341at_rat @ P4 @ I4 ) ) )
        & ( ~ ( member_nat @ I @ I4 )
         => ( ( groups1351286907653491341at_rat @ P4 @ ( insert_nat @ I @ I4 ) )
            = ( plus_plus_rat @ ( P4 @ I ) @ ( groups1351286907653491341at_rat @ P4 @ I4 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_6279_sum_Oinsert_H,axiom,
    ! [I4: set_int,P4: int > rat,I: int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X3: int] :
              ( ( member_int @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != zero_zero_rat ) ) ) )
     => ( ( ( member_int @ I @ I4 )
         => ( ( groups2350640619554545897nt_rat @ P4 @ ( insert_int @ I @ I4 ) )
            = ( groups2350640619554545897nt_rat @ P4 @ I4 ) ) )
        & ( ~ ( member_int @ I @ I4 )
         => ( ( groups2350640619554545897nt_rat @ P4 @ ( insert_int @ I @ I4 ) )
            = ( plus_plus_rat @ ( P4 @ I ) @ ( groups2350640619554545897nt_rat @ P4 @ I4 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_6280_sum_Oinsert_H,axiom,
    ! [I4: set_Code_integer,P4: code_integer > rat,I: code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != zero_zero_rat ) ) ) )
     => ( ( ( member_Code_integer @ I @ I4 )
         => ( ( groups8878813951405302554er_rat @ P4 @ ( insert_Code_integer @ I @ I4 ) )
            = ( groups8878813951405302554er_rat @ P4 @ I4 ) ) )
        & ( ~ ( member_Code_integer @ I @ I4 )
         => ( ( groups8878813951405302554er_rat @ P4 @ ( insert_Code_integer @ I @ I4 ) )
            = ( plus_plus_rat @ ( P4 @ I ) @ ( groups8878813951405302554er_rat @ P4 @ I4 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_6281_sum_Oinsert_H,axiom,
    ! [I4: set_o,P4: $o > nat,I: $o] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != zero_zero_nat ) ) ) )
     => ( ( ( member_o @ I @ I4 )
         => ( ( groups4556407284786078405_o_nat @ P4 @ ( insert_o @ I @ I4 ) )
            = ( groups4556407284786078405_o_nat @ P4 @ I4 ) ) )
        & ( ~ ( member_o @ I @ I4 )
         => ( ( groups4556407284786078405_o_nat @ P4 @ ( insert_o @ I @ I4 ) )
            = ( plus_plus_nat @ ( P4 @ I ) @ ( groups4556407284786078405_o_nat @ P4 @ I4 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_6282_sum_Oinsert_H,axiom,
    ! [I4: set_nat,P4: nat > nat,I: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ I4 )
              & ( ( P4 @ X3 )
               != zero_zero_nat ) ) ) )
     => ( ( ( member_nat @ I @ I4 )
         => ( ( groups1986416967739987077at_nat @ P4 @ ( insert_nat @ I @ I4 ) )
            = ( groups1986416967739987077at_nat @ P4 @ I4 ) ) )
        & ( ~ ( member_nat @ I @ I4 )
         => ( ( groups1986416967739987077at_nat @ P4 @ ( insert_nat @ I @ I4 ) )
            = ( plus_plus_nat @ ( P4 @ I ) @ ( groups1986416967739987077at_nat @ P4 @ I4 ) ) ) ) ) ) ).

% sum.insert'
thf(fact_6283_max__ext_Omax__extI,axiom,
    ! [X11: set_se7855581050983116737at_nat,Y9: set_se7855581050983116737at_nat,R4: set_Pr4329608150637261639at_nat] :
      ( ( finite9047747110432174090at_nat @ X11 )
     => ( ( finite9047747110432174090at_nat @ Y9 )
       => ( ( Y9 != bot_bo3083307316010499117at_nat )
         => ( ! [X4: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ X4 @ X11 )
               => ? [Xa2: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ Xa2 @ Y9 )
                    & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Xa2 ) @ R4 ) ) )
           => ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ X11 @ Y9 ) @ ( max_ex4535816423174879388at_nat @ R4 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_6284_max__ext_Omax__extI,axiom,
    ! [X11: set_Code_integer,Y9: set_Code_integer,R4: set_Pr4811707699266497531nteger] :
      ( ( finite6017078050557962740nteger @ X11 )
     => ( ( finite6017078050557962740nteger @ Y9 )
       => ( ( Y9 != bot_bo3990330152332043303nteger )
         => ( ! [X4: code_integer] :
                ( ( member_Code_integer @ X4 @ X11 )
               => ? [Xa2: code_integer] :
                    ( ( member_Code_integer @ Xa2 @ Y9 )
                    & ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X4 @ Xa2 ) @ R4 ) ) )
           => ( member4307123515891402160nteger @ ( produc7443773368509356479nteger @ X11 @ Y9 ) @ ( max_ext_Code_integer @ R4 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_6285_max__ext_Omax__extI,axiom,
    ! [X11: set_Pr4329608150637261639at_nat,Y9: set_Pr4329608150637261639at_nat,R4: set_Pr5564308138774400199at_nat] :
      ( ( finite4343798906461161616at_nat @ X11 )
     => ( ( finite4343798906461161616at_nat @ Y9 )
       => ( ( Y9 != bot_bo228742789529271731at_nat )
         => ( ! [X4: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X4 @ X11 )
               => ? [Xa2: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ Xa2 @ Y9 )
                    & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X4 @ Xa2 ) @ R4 ) ) )
           => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X11 @ Y9 ) @ ( max_ex8833738776070432290at_nat @ R4 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_6286_max__ext_Omax__extI,axiom,
    ! [X11: set_Pr1261947904930325089at_nat,Y9: set_Pr1261947904930325089at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( finite6177210948735845034at_nat @ X11 )
     => ( ( finite6177210948735845034at_nat @ Y9 )
       => ( ( Y9 != bot_bo2099793752762293965at_nat )
         => ( ! [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ X11 )
               => ? [Xa2: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Xa2 @ Y9 )
                    & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Xa2 ) @ R4 ) ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X11 @ Y9 ) @ ( max_ex8135407076693332796at_nat @ R4 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_6287_max__ext_Omax__extI,axiom,
    ! [X11: set_o,Y9: set_o,R4: set_Product_prod_o_o] :
      ( ( finite_finite_o @ X11 )
     => ( ( finite_finite_o @ Y9 )
       => ( ( Y9 != bot_bot_set_o )
         => ( ! [X4: $o] :
                ( ( member_o @ X4 @ X11 )
               => ? [Xa2: $o] :
                    ( ( member_o @ Xa2 @ Y9 )
                    & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Xa2 ) @ R4 ) ) )
           => ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ X11 @ Y9 ) @ ( max_ext_o @ R4 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_6288_max__ext_Omax__extI,axiom,
    ! [X11: set_nat,Y9: set_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ X11 )
     => ( ( finite_finite_nat @ Y9 )
       => ( ( Y9 != bot_bot_set_nat )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ X11 )
               => ? [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ Y9 )
                    & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Xa2 ) @ R4 ) ) )
           => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X11 @ Y9 ) @ ( max_ext_nat @ R4 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_6289_max__ext_Omax__extI,axiom,
    ! [X11: set_int,Y9: set_int,R4: set_Pr958786334691620121nt_int] :
      ( ( finite_finite_int @ X11 )
     => ( ( finite_finite_int @ Y9 )
       => ( ( Y9 != bot_bot_set_int )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ X11 )
               => ? [Xa2: int] :
                    ( ( member_int @ Xa2 @ Y9 )
                    & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Xa2 ) @ R4 ) ) )
           => ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ X11 @ Y9 ) @ ( max_ext_int @ R4 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_6290_total__on__def,axiom,
    ( total_7212417256332265445at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
        ! [X3: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X3 @ A7 )
         => ! [Y3: set_Pr1261947904930325089at_nat] :
              ( ( member2643936169264416010at_nat @ Y3 @ A7 )
             => ( ( X3 != Y3 )
               => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R3 )
                  | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_6291_total__on__def,axiom,
    ( total_on_nat
    = ( ^ [A7: set_nat,R3: set_Pr1261947904930325089at_nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ A7 )
         => ! [Y3: nat] :
              ( ( member_nat @ Y3 @ A7 )
             => ( ( X3 != Y3 )
               => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R3 )
                  | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ) ).

% total_on_def
thf(fact_6292_total__onI,axiom,
    ! [A2: set_se7855581050983116737at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ! [X4: set_Pr1261947904930325089at_nat,Y5: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ X4 @ A2 )
         => ( ( member2643936169264416010at_nat @ Y5 @ A2 )
           => ( ( X4 != Y5 )
             => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y5 ) @ R2 )
                | ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y5 @ X4 ) @ R2 ) ) ) ) )
     => ( total_7212417256332265445at_nat @ A2 @ R2 ) ) ).

% total_onI
thf(fact_6293_total__onI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ! [X4: produc3843707927480180839at_nat,Y5: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X4 @ A2 )
         => ( ( member8757157785044589968at_nat @ Y5 @ A2 )
           => ( ( X4 != Y5 )
             => ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X4 @ Y5 ) @ R2 )
                | ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ Y5 @ X4 ) @ R2 ) ) ) ) )
     => ( total_1513685709984341995at_nat @ A2 @ R2 ) ) ).

% total_onI
thf(fact_6294_total__onI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ! [X4: product_prod_nat_nat,Y5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A2 )
         => ( ( member8440522571783428010at_nat @ Y5 @ A2 )
           => ( ( X4 != Y5 )
             => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y5 ) @ R2 )
                | ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y5 @ X4 ) @ R2 ) ) ) ) )
     => ( total_3592101749530773125at_nat @ A2 @ R2 ) ) ).

% total_onI
thf(fact_6295_total__onI,axiom,
    ! [A2: set_o,R2: set_Product_prod_o_o] :
      ( ! [X4: $o,Y5: $o] :
          ( ( member_o @ X4 @ A2 )
         => ( ( member_o @ Y5 @ A2 )
           => ( ( X4 != Y5 )
             => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y5 ) @ R2 )
                | ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y5 @ X4 ) @ R2 ) ) ) ) )
     => ( total_on_o @ A2 @ R2 ) ) ).

% total_onI
thf(fact_6296_total__onI,axiom,
    ! [A2: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y5: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ( member_nat @ Y5 @ A2 )
           => ( ( X4 != Y5 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R2 )
                | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y5 @ X4 ) @ R2 ) ) ) ) )
     => ( total_on_nat @ A2 @ R2 ) ) ).

% total_onI
thf(fact_6297_total__onI,axiom,
    ! [A2: set_int,R2: set_Pr958786334691620121nt_int] :
      ( ! [X4: int,Y5: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ( member_int @ Y5 @ A2 )
           => ( ( X4 != Y5 )
             => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ R2 )
                | ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y5 @ X4 ) @ R2 ) ) ) ) )
     => ( total_on_int @ A2 @ R2 ) ) ).

% total_onI
thf(fact_6298_sum_Odistrib__triv_H,axiom,
    ! [I4: set_nat,G: nat > rat,H3: nat > rat] :
      ( ( finite_finite_nat @ I4 )
     => ( ( groups1351286907653491341at_rat
          @ ^ [I3: nat] : ( plus_plus_rat @ ( G @ I3 ) @ ( H3 @ I3 ) )
          @ I4 )
        = ( plus_plus_rat @ ( groups1351286907653491341at_rat @ G @ I4 ) @ ( groups1351286907653491341at_rat @ H3 @ I4 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_6299_sum_Odistrib__triv_H,axiom,
    ! [I4: set_int,G: int > rat,H3: int > rat] :
      ( ( finite_finite_int @ I4 )
     => ( ( groups2350640619554545897nt_rat
          @ ^ [I3: int] : ( plus_plus_rat @ ( G @ I3 ) @ ( H3 @ I3 ) )
          @ I4 )
        = ( plus_plus_rat @ ( groups2350640619554545897nt_rat @ G @ I4 ) @ ( groups2350640619554545897nt_rat @ H3 @ I4 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_6300_sum_Odistrib__triv_H,axiom,
    ! [I4: set_Code_integer,G: code_integer > rat,H3: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ I4 )
     => ( ( groups8878813951405302554er_rat
          @ ^ [I3: code_integer] : ( plus_plus_rat @ ( G @ I3 ) @ ( H3 @ I3 ) )
          @ I4 )
        = ( plus_plus_rat @ ( groups8878813951405302554er_rat @ G @ I4 ) @ ( groups8878813951405302554er_rat @ H3 @ I4 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_6301_sum_Odistrib__triv_H,axiom,
    ! [I4: set_nat,G: nat > nat,H3: nat > nat] :
      ( ( finite_finite_nat @ I4 )
     => ( ( groups1986416967739987077at_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ ( G @ I3 ) @ ( H3 @ I3 ) )
          @ I4 )
        = ( plus_plus_nat @ ( groups1986416967739987077at_nat @ G @ I4 ) @ ( groups1986416967739987077at_nat @ H3 @ I4 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_6302_sum_Odistrib__triv_H,axiom,
    ! [I4: set_int,G: int > nat,H3: int > nat] :
      ( ( finite_finite_int @ I4 )
     => ( ( groups2985770679641041633nt_nat
          @ ^ [I3: int] : ( plus_plus_nat @ ( G @ I3 ) @ ( H3 @ I3 ) )
          @ I4 )
        = ( plus_plus_nat @ ( groups2985770679641041633nt_nat @ G @ I4 ) @ ( groups2985770679641041633nt_nat @ H3 @ I4 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_6303_sum_Odistrib__triv_H,axiom,
    ! [I4: set_Code_integer,G: code_integer > nat,H3: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ I4 )
     => ( ( groups290571974637022482er_nat
          @ ^ [I3: code_integer] : ( plus_plus_nat @ ( G @ I3 ) @ ( H3 @ I3 ) )
          @ I4 )
        = ( plus_plus_nat @ ( groups290571974637022482er_nat @ G @ I4 ) @ ( groups290571974637022482er_nat @ H3 @ I4 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_6304_sum_Odistrib__triv_H,axiom,
    ! [I4: set_nat,G: nat > int,H3: nat > int] :
      ( ( finite_finite_nat @ I4 )
     => ( ( groups1983926497230936801at_int
          @ ^ [I3: nat] : ( plus_plus_int @ ( G @ I3 ) @ ( H3 @ I3 ) )
          @ I4 )
        = ( plus_plus_int @ ( groups1983926497230936801at_int @ G @ I4 ) @ ( groups1983926497230936801at_int @ H3 @ I4 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_6305_sum_Odistrib__triv_H,axiom,
    ! [I4: set_int,G: int > int,H3: int > int] :
      ( ( finite_finite_int @ I4 )
     => ( ( groups2983280209131991357nt_int
          @ ^ [I3: int] : ( plus_plus_int @ ( G @ I3 ) @ ( H3 @ I3 ) )
          @ I4 )
        = ( plus_plus_int @ ( groups2983280209131991357nt_int @ G @ I4 ) @ ( groups2983280209131991357nt_int @ H3 @ I4 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_6306_sum_Odistrib__triv_H,axiom,
    ! [I4: set_Code_integer,G: code_integer > int,H3: code_integer > int] :
      ( ( finite6017078050557962740nteger @ I4 )
     => ( ( groups288081504127972206er_int
          @ ^ [I3: code_integer] : ( plus_plus_int @ ( G @ I3 ) @ ( H3 @ I3 ) )
          @ I4 )
        = ( plus_plus_int @ ( groups288081504127972206er_int @ G @ I4 ) @ ( groups288081504127972206er_int @ H3 @ I4 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_6307_sum_Odistrib__triv_H,axiom,
    ! [I4: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > rat,H3: product_prod_nat_nat > rat] :
      ( ( finite6177210948735845034at_nat @ I4 )
     => ( ( groups3328076802468863542at_rat
          @ ^ [I3: product_prod_nat_nat] : ( plus_plus_rat @ ( G @ I3 ) @ ( H3 @ I3 ) )
          @ I4 )
        = ( plus_plus_rat @ ( groups3328076802468863542at_rat @ G @ I4 ) @ ( groups3328076802468863542at_rat @ H3 @ I4 ) ) ) ) ).

% sum.distrib_triv'
thf(fact_6308_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% le_prod_encode_1
thf(fact_6309_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% le_prod_encode_2
thf(fact_6310_sum_Odistrib_H,axiom,
    ! [I4: set_o,G: $o > code_integer,H3: $o > code_integer] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ I4 )
              & ( ( G @ X3 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [X3: $o] :
                ( ( member_o @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != zero_z3403309356797280102nteger ) ) ) )
       => ( ( groups1402912129352969042nteger
            @ ^ [I3: $o] : ( plus_p5714425477246183910nteger @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( plus_p5714425477246183910nteger @ ( groups1402912129352969042nteger @ G @ I4 ) @ ( groups1402912129352969042nteger @ H3 @ I4 ) ) ) ) ) ).

% sum.distrib'
thf(fact_6311_sum_Odistrib_H,axiom,
    ! [I4: set_nat,G: nat > code_integer,H3: nat > code_integer] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ I4 )
              & ( ( G @ X3 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != zero_z3403309356797280102nteger ) ) ) )
       => ( ( groups555127423416065298nteger
            @ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( plus_p5714425477246183910nteger @ ( groups555127423416065298nteger @ G @ I4 ) @ ( groups555127423416065298nteger @ H3 @ I4 ) ) ) ) ) ).

% sum.distrib'
thf(fact_6312_sum_Odistrib_H,axiom,
    ! [I4: set_int,G: int > code_integer,H3: int > code_integer] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X3: int] :
              ( ( member_int @ X3 @ I4 )
              & ( ( G @ X3 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [X3: int] :
                ( ( member_int @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != zero_z3403309356797280102nteger ) ) ) )
       => ( ( groups926780983652909934nteger
            @ ^ [I3: int] : ( plus_p5714425477246183910nteger @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( plus_p5714425477246183910nteger @ ( groups926780983652909934nteger @ G @ I4 ) @ ( groups926780983652909934nteger @ H3 @ I4 ) ) ) ) ) ).

% sum.distrib'
thf(fact_6313_sum_Odistrib_H,axiom,
    ! [I4: set_Code_integer,G: code_integer > code_integer,H3: code_integer > code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ I4 )
              & ( ( G @ X3 )
               != zero_z3403309356797280102nteger ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != zero_z3403309356797280102nteger ) ) ) )
       => ( ( groups910942671188738463nteger
            @ ^ [I3: code_integer] : ( plus_p5714425477246183910nteger @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( plus_p5714425477246183910nteger @ ( groups910942671188738463nteger @ G @ I4 ) @ ( groups910942671188738463nteger @ H3 @ I4 ) ) ) ) ) ).

% sum.distrib'
thf(fact_6314_sum_Odistrib_H,axiom,
    ! [I4: set_o,G: $o > rat,H3: $o > rat] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ I4 )
              & ( ( G @ X3 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [X3: $o] :
                ( ( member_o @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != zero_zero_rat ) ) ) )
       => ( ( groups3921277224699582669_o_rat
            @ ^ [I3: $o] : ( plus_plus_rat @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( plus_plus_rat @ ( groups3921277224699582669_o_rat @ G @ I4 ) @ ( groups3921277224699582669_o_rat @ H3 @ I4 ) ) ) ) ) ).

% sum.distrib'
thf(fact_6315_sum_Odistrib_H,axiom,
    ! [I4: set_nat,G: nat > rat,H3: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ I4 )
              & ( ( G @ X3 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != zero_zero_rat ) ) ) )
       => ( ( groups1351286907653491341at_rat
            @ ^ [I3: nat] : ( plus_plus_rat @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( plus_plus_rat @ ( groups1351286907653491341at_rat @ G @ I4 ) @ ( groups1351286907653491341at_rat @ H3 @ I4 ) ) ) ) ) ).

% sum.distrib'
thf(fact_6316_sum_Odistrib_H,axiom,
    ! [I4: set_int,G: int > rat,H3: int > rat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X3: int] :
              ( ( member_int @ X3 @ I4 )
              & ( ( G @ X3 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [X3: int] :
                ( ( member_int @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != zero_zero_rat ) ) ) )
       => ( ( groups2350640619554545897nt_rat
            @ ^ [I3: int] : ( plus_plus_rat @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( plus_plus_rat @ ( groups2350640619554545897nt_rat @ G @ I4 ) @ ( groups2350640619554545897nt_rat @ H3 @ I4 ) ) ) ) ) ).

% sum.distrib'
thf(fact_6317_sum_Odistrib_H,axiom,
    ! [I4: set_Code_integer,G: code_integer > rat,H3: code_integer > rat] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ I4 )
              & ( ( G @ X3 )
               != zero_zero_rat ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != zero_zero_rat ) ) ) )
       => ( ( groups8878813951405302554er_rat
            @ ^ [I3: code_integer] : ( plus_plus_rat @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( plus_plus_rat @ ( groups8878813951405302554er_rat @ G @ I4 ) @ ( groups8878813951405302554er_rat @ H3 @ I4 ) ) ) ) ) ).

% sum.distrib'
thf(fact_6318_sum_Odistrib_H,axiom,
    ! [I4: set_o,G: $o > nat,H3: $o > nat] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ I4 )
              & ( ( G @ X3 )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [X3: $o] :
                ( ( member_o @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != zero_zero_nat ) ) ) )
       => ( ( groups4556407284786078405_o_nat
            @ ^ [I3: $o] : ( plus_plus_nat @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( plus_plus_nat @ ( groups4556407284786078405_o_nat @ G @ I4 ) @ ( groups4556407284786078405_o_nat @ H3 @ I4 ) ) ) ) ) ).

% sum.distrib'
thf(fact_6319_sum_Odistrib_H,axiom,
    ! [I4: set_nat,G: nat > nat,H3: nat > nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ I4 )
              & ( ( G @ X3 )
               != zero_zero_nat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ I4 )
                & ( ( H3 @ X3 )
                 != zero_zero_nat ) ) ) )
       => ( ( groups1986416967739987077at_nat
            @ ^ [I3: nat] : ( plus_plus_nat @ ( G @ I3 ) @ ( H3 @ I3 ) )
            @ I4 )
          = ( plus_plus_nat @ ( groups1986416967739987077at_nat @ G @ I4 ) @ ( groups1986416967739987077at_nat @ H3 @ I4 ) ) ) ) ) ).

% sum.distrib'
thf(fact_6320_prod__encode__def,axiom,
    ( nat_prod_encode
    = ( produc6842872674320459806at_nat
      @ ^ [M: nat,N4: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M @ N4 ) ) @ M ) ) ) ).

% prod_encode_def
thf(fact_6321_max__ext_Ocases,axiom,
    ! [A1: set_se7855581050983116737at_nat,A23: set_se7855581050983116737at_nat,R4: set_Pr4329608150637261639at_nat] :
      ( ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ A1 @ A23 ) @ ( max_ex4535816423174879388at_nat @ R4 ) )
     => ~ ( ( finite9047747110432174090at_nat @ A1 )
         => ( ( finite9047747110432174090at_nat @ A23 )
           => ( ( A23 != bot_bo3083307316010499117at_nat )
             => ~ ! [X8: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ X8 @ A1 )
                   => ? [Xa3: set_Pr1261947904930325089at_nat] :
                        ( ( member2643936169264416010at_nat @ Xa3 @ A23 )
                        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X8 @ Xa3 ) @ R4 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_6322_max__ext_Ocases,axiom,
    ! [A1: set_Code_integer,A23: set_Code_integer,R4: set_Pr4811707699266497531nteger] :
      ( ( member4307123515891402160nteger @ ( produc7443773368509356479nteger @ A1 @ A23 ) @ ( max_ext_Code_integer @ R4 ) )
     => ~ ( ( finite6017078050557962740nteger @ A1 )
         => ( ( finite6017078050557962740nteger @ A23 )
           => ( ( A23 != bot_bo3990330152332043303nteger )
             => ~ ! [X8: code_integer] :
                    ( ( member_Code_integer @ X8 @ A1 )
                   => ? [Xa3: code_integer] :
                        ( ( member_Code_integer @ Xa3 @ A23 )
                        & ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X8 @ Xa3 ) @ R4 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_6323_max__ext_Ocases,axiom,
    ! [A1: set_Pr4329608150637261639at_nat,A23: set_Pr4329608150637261639at_nat,R4: set_Pr5564308138774400199at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A1 @ A23 ) @ ( max_ex8833738776070432290at_nat @ R4 ) )
     => ~ ( ( finite4343798906461161616at_nat @ A1 )
         => ( ( finite4343798906461161616at_nat @ A23 )
           => ( ( A23 != bot_bo228742789529271731at_nat )
             => ~ ! [X8: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ X8 @ A1 )
                   => ? [Xa3: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ Xa3 @ A23 )
                        & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X8 @ Xa3 ) @ R4 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_6324_max__ext_Ocases,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A23: set_Pr1261947904930325089at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A23 ) @ ( max_ex8135407076693332796at_nat @ R4 ) )
     => ~ ( ( finite6177210948735845034at_nat @ A1 )
         => ( ( finite6177210948735845034at_nat @ A23 )
           => ( ( A23 != bot_bo2099793752762293965at_nat )
             => ~ ! [X8: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X8 @ A1 )
                   => ? [Xa3: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ Xa3 @ A23 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X8 @ Xa3 ) @ R4 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_6325_max__ext_Ocases,axiom,
    ! [A1: set_o,A23: set_o,R4: set_Product_prod_o_o] :
      ( ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A1 @ A23 ) @ ( max_ext_o @ R4 ) )
     => ~ ( ( finite_finite_o @ A1 )
         => ( ( finite_finite_o @ A23 )
           => ( ( A23 != bot_bot_set_o )
             => ~ ! [X8: $o] :
                    ( ( member_o @ X8 @ A1 )
                   => ? [Xa3: $o] :
                        ( ( member_o @ Xa3 @ A23 )
                        & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X8 @ Xa3 ) @ R4 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_6326_max__ext_Ocases,axiom,
    ! [A1: set_nat,A23: set_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A23 ) @ ( max_ext_nat @ R4 ) )
     => ~ ( ( finite_finite_nat @ A1 )
         => ( ( finite_finite_nat @ A23 )
           => ( ( A23 != bot_bot_set_nat )
             => ~ ! [X8: nat] :
                    ( ( member_nat @ X8 @ A1 )
                   => ? [Xa3: nat] :
                        ( ( member_nat @ Xa3 @ A23 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X8 @ Xa3 ) @ R4 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_6327_max__ext_Ocases,axiom,
    ! [A1: set_int,A23: set_int,R4: set_Pr958786334691620121nt_int] :
      ( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A1 @ A23 ) @ ( max_ext_int @ R4 ) )
     => ~ ( ( finite_finite_int @ A1 )
         => ( ( finite_finite_int @ A23 )
           => ( ( A23 != bot_bot_set_int )
             => ~ ! [X8: int] :
                    ( ( member_int @ X8 @ A1 )
                   => ? [Xa3: int] :
                        ( ( member_int @ Xa3 @ A23 )
                        & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X8 @ Xa3 ) @ R4 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_6328_max__ext_Osimps,axiom,
    ! [A1: set_se7855581050983116737at_nat,A23: set_se7855581050983116737at_nat,R4: set_Pr4329608150637261639at_nat] :
      ( ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ A1 @ A23 ) @ ( max_ex4535816423174879388at_nat @ R4 ) )
      = ( ( finite9047747110432174090at_nat @ A1 )
        & ( finite9047747110432174090at_nat @ A23 )
        & ( A23 != bot_bo3083307316010499117at_nat )
        & ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ A1 )
           => ? [Y3: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Y3 @ A23 )
                & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R4 ) ) ) ) ) ).

% max_ext.simps
thf(fact_6329_max__ext_Osimps,axiom,
    ! [A1: set_Code_integer,A23: set_Code_integer,R4: set_Pr4811707699266497531nteger] :
      ( ( member4307123515891402160nteger @ ( produc7443773368509356479nteger @ A1 @ A23 ) @ ( max_ext_Code_integer @ R4 ) )
      = ( ( finite6017078050557962740nteger @ A1 )
        & ( finite6017078050557962740nteger @ A23 )
        & ( A23 != bot_bo3990330152332043303nteger )
        & ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A1 )
           => ? [Y3: code_integer] :
                ( ( member_Code_integer @ Y3 @ A23 )
                & ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X3 @ Y3 ) @ R4 ) ) ) ) ) ).

% max_ext.simps
thf(fact_6330_max__ext_Osimps,axiom,
    ! [A1: set_Pr4329608150637261639at_nat,A23: set_Pr4329608150637261639at_nat,R4: set_Pr5564308138774400199at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A1 @ A23 ) @ ( max_ex8833738776070432290at_nat @ R4 ) )
      = ( ( finite4343798906461161616at_nat @ A1 )
        & ( finite4343798906461161616at_nat @ A23 )
        & ( A23 != bot_bo228742789529271731at_nat )
        & ! [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A1 )
           => ? [Y3: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ Y3 @ A23 )
                & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y3 ) @ R4 ) ) ) ) ) ).

% max_ext.simps
thf(fact_6331_max__ext_Osimps,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A23: set_Pr1261947904930325089at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A23 ) @ ( max_ex8135407076693332796at_nat @ R4 ) )
      = ( ( finite6177210948735845034at_nat @ A1 )
        & ( finite6177210948735845034at_nat @ A23 )
        & ( A23 != bot_bo2099793752762293965at_nat )
        & ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A1 )
           => ? [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ A23 )
                & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R4 ) ) ) ) ) ).

% max_ext.simps
thf(fact_6332_max__ext_Osimps,axiom,
    ! [A1: set_o,A23: set_o,R4: set_Product_prod_o_o] :
      ( ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A1 @ A23 ) @ ( max_ext_o @ R4 ) )
      = ( ( finite_finite_o @ A1 )
        & ( finite_finite_o @ A23 )
        & ( A23 != bot_bot_set_o )
        & ! [X3: $o] :
            ( ( member_o @ X3 @ A1 )
           => ? [Y3: $o] :
                ( ( member_o @ Y3 @ A23 )
                & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y3 ) @ R4 ) ) ) ) ) ).

% max_ext.simps
thf(fact_6333_max__ext_Osimps,axiom,
    ! [A1: set_nat,A23: set_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A23 ) @ ( max_ext_nat @ R4 ) )
      = ( ( finite_finite_nat @ A1 )
        & ( finite_finite_nat @ A23 )
        & ( A23 != bot_bot_set_nat )
        & ! [X3: nat] :
            ( ( member_nat @ X3 @ A1 )
           => ? [Y3: nat] :
                ( ( member_nat @ Y3 @ A23 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R4 ) ) ) ) ) ).

% max_ext.simps
thf(fact_6334_max__ext_Osimps,axiom,
    ! [A1: set_int,A23: set_int,R4: set_Pr958786334691620121nt_int] :
      ( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A1 @ A23 ) @ ( max_ext_int @ R4 ) )
      = ( ( finite_finite_int @ A1 )
        & ( finite_finite_int @ A23 )
        & ( A23 != bot_bot_set_int )
        & ! [X3: int] :
            ( ( member_int @ X3 @ A1 )
           => ? [Y3: int] :
                ( ( member_int @ Y3 @ A23 )
                & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R4 ) ) ) ) ) ).

% max_ext.simps
thf(fact_6335_max__extp__eq,axiom,
    ( max_ex4864111882549613972at_nat
    = ( ^ [R3: product_prod_nat_nat > product_prod_nat_nat > $o,X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ ( max_ex8135407076693332796at_nat @ ( collec7088162979684241874at_nat @ ( produc6590410687421337004_nat_o @ R3 ) ) ) ) ) ) ).

% max_extp_eq
thf(fact_6336_max__extp__max__ext__eq,axiom,
    ! [R4: set_Pr4329608150637261639at_nat] :
      ( ( max_ex4845989503969680116at_nat
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R4 ) )
      = ( ^ [X3: set_se7855581050983116737at_nat,Y3: set_se7855581050983116737at_nat] : ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ X3 @ Y3 ) @ ( max_ex4535816423174879388at_nat @ R4 ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_6337_max__extp__max__ext__eq,axiom,
    ! [R4: set_Pr1261947904930325089at_nat] :
      ( ( max_extp_nat
        @ ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R4 ) )
      = ( ^ [X3: set_nat,Y3: set_nat] : ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X3 @ Y3 ) @ ( max_ext_nat @ R4 ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_6338_max__extp__max__ext__eq,axiom,
    ! [R4: set_Pr8693737435421807431at_nat] :
      ( ( max_ex4864111882549613972at_nat
        @ ^ [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R4 ) )
      = ( ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ ( max_ex8135407076693332796at_nat @ R4 ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_6339_one__div__numeral,axiom,
    ! [N2: num] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N2 ) )
      = ( produc8508995932063986495nteger @ ( unique3479559517661332726nteger @ one @ N2 ) ) ) ).

% one_div_numeral
thf(fact_6340_one__div__numeral,axiom,
    ! [N2: num] :
      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
      = ( product_fst_int_int @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ).

% one_div_numeral
thf(fact_6341_one__div__numeral,axiom,
    ! [N2: num] :
      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ N2 ) ) ) ).

% one_div_numeral
thf(fact_6342_sqr_Osimps_I3_J,axiom,
    ! [N2: num] :
      ( ( sqr @ ( bit1 @ N2 ) )
      = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N2 ) @ N2 ) ) ) ) ).

% sqr.simps(3)
thf(fact_6343_one__mod__numeral,axiom,
    ! [N2: num] :
      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N2 ) )
      = ( produc6174133586879617921nteger @ ( unique3479559517661332726nteger @ one @ N2 ) ) ) ).

% one_mod_numeral
thf(fact_6344_one__mod__numeral,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
      = ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ).

% one_mod_numeral
thf(fact_6345_one__mod__numeral,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ N2 ) ) ) ).

% one_mod_numeral
thf(fact_6346_card__UNION,axiom,
    ! [A2: set_set_Product_unit] :
      ( ( finite1772178364199683094t_unit @ A2 )
     => ( ! [X4: set_Product_unit] :
            ( ( member5877623283571906838t_unit @ X4 @ A2 )
           => ( finite4290736615968046902t_unit @ X4 ) )
       => ( ( finite410649719033368117t_unit @ ( comple3349607820504768346t_unit @ A2 ) )
          = ( nat2
            @ ( groups9212676010950626495it_int
              @ ^ [I5: set_set_Product_unit] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite22775419188893461t_unit @ I5 ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite410649719033368117t_unit @ ( comple7216939928495492468t_unit @ I5 ) ) ) )
              @ ( collec4600646233855244344t_unit
                @ ^ [I5: set_set_Product_unit] :
                    ( ( ord_le3674001228145610605t_unit @ I5 @ A2 )
                    & ( I5 != bot_bo6284674557190688313t_unit ) ) ) ) ) ) ) ) ).

% card_UNION
thf(fact_6347_card__UNION,axiom,
    ! [A2: set_set_list_nat] :
      ( ( finite7047420756378620717st_nat @ A2 )
     => ( ! [X4: set_list_nat] :
            ( ( member_set_list_nat @ X4 @ A2 )
           => ( finite8100373058378681591st_nat @ X4 ) )
       => ( ( finite_card_list_nat @ ( comple8404747032580312297st_nat @ A2 ) )
          = ( nat2
            @ ( groups7004213669654646580at_int
              @ ^ [I5: set_set_list_nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite2364142230527598318st_nat @ I5 ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite_card_list_nat @ ( comple184543376406953807st_nat @ I5 ) ) ) )
              @ ( collec4691811733418234273st_nat
                @ ^ [I5: set_set_list_nat] :
                    ( ( ord_le1068707526560357548st_nat @ I5 @ A2 )
                    & ( I5 != bot_bo3886227569956363488st_nat ) ) ) ) ) ) ) ) ).

% card_UNION
thf(fact_6348_card__UNION,axiom,
    ! [A2: set_set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ! [X4: set_int] :
            ( ( member_set_int @ X4 @ A2 )
           => ( finite_finite_int @ X4 ) )
       => ( ( finite_card_int @ ( comple3221217463730067765et_int @ A2 ) )
          = ( nat2
            @ ( groups1080061135233207040nt_int
              @ ^ [I5: set_set_int] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite_card_set_int @ I5 ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite_card_int @ ( comple3628384868704368283et_int @ I5 ) ) ) )
              @ ( collect_set_set_int
                @ ^ [I5: set_set_int] :
                    ( ( ord_le4403425263959731960et_int @ I5 @ A2 )
                    & ( I5 != bot_bot_set_set_int ) ) ) ) ) ) ) ) ).

% card_UNION
thf(fact_6349_card__UNION,axiom,
    ! [A2: set_set_Code_integer] :
      ( ( finite6931041176100689706nteger @ A2 )
     => ( ! [X4: set_Code_integer] :
            ( ( member1834298987299537450nteger @ X4 @ A2 )
           => ( finite6017078050557962740nteger @ X4 ) )
       => ( ( finite4902975817058060853nteger @ ( comple739944243200306918nteger @ A2 ) )
          = ( nat2
            @ ( groups2416162284031854641er_int
              @ ^ [I5: set_set_Code_integer] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite1340570857867686763nteger @ I5 ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite4902975817058060853nteger @ ( comple1256288197631046732nteger @ I5 ) ) ) )
              @ ( collec9208119312383884190nteger
                @ ^ [I5: set_set_Code_integer] :
                    ( ( ord_le1914454125413604393nteger @ I5 @ A2 )
                    & ( I5 != bot_bo9204850693367297629nteger ) ) ) ) ) ) ) ) ).

% card_UNION
thf(fact_6350_card__UNION,axiom,
    ! [A2: set_se7855581050983116737at_nat] :
      ( ( finite9047747110432174090at_nat @ A2 )
     => ( ! [X4: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X4 @ A2 )
           => ( finite6177210948735845034at_nat @ X4 ) )
       => ( ( finite711546835091564841at_nat @ ( comple5685304695842803022at_nat @ A2 ) )
          = ( nat2
            @ ( groups1759609815103008563at_int
              @ ^ [I5: set_se7855581050983116737at_nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite4356350796350151305at_nat @ I5 ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite711546835091564841at_nat @ ( comple1930127330929219688at_nat @ I5 ) ) ) )
              @ ( collec6698286964019821996at_nat
                @ ^ [I5: set_se7855581050983116737at_nat] :
                    ( ( ord_le2077887516847798113at_nat @ I5 @ A2 )
                    & ( I5 != bot_bo3083307316010499117at_nat ) ) ) ) ) ) ) ) ).

% card_UNION
thf(fact_6351_card__UNION,axiom,
    ! [A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ! [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A2 )
           => ( finite_finite_nat @ X4 ) )
       => ( ( finite_card_nat @ ( comple7399068483239264473et_nat @ A2 ) )
          = ( nat2
            @ ( groups7084729577923612836at_int
              @ ^ [I5: set_set_nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ ( finite_card_set_nat @ I5 ) @ one_one_nat ) ) @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ ( comple7806235888213564991et_nat @ I5 ) ) ) )
              @ ( collect_set_set_nat
                @ ^ [I5: set_set_nat] :
                    ( ( ord_le6893508408891458716et_nat @ I5 @ A2 )
                    & ( I5 != bot_bot_set_set_nat ) ) ) ) ) ) ) ) ).

% card_UNION
thf(fact_6352_Sup__atMost,axiom,
    ! [Y: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( set_or4236626031148496127et_nat @ Y ) )
      = Y ) ).

% Sup_atMost
thf(fact_6353_Sup__atMost,axiom,
    ! [Y: product_unit] :
      ( ( comple4687483117567863418t_unit @ ( set_or8621721900777396692t_unit @ Y ) )
      = Y ) ).

% Sup_atMost
thf(fact_6354_fst__apsnd,axiom,
    ! [F: code_integer > code_integer,X: produc8923325533196201883nteger] :
      ( ( produc8508995932063986495nteger @ ( produc6499014454317279255nteger @ F @ X ) )
      = ( produc8508995932063986495nteger @ X ) ) ).

% fst_apsnd
thf(fact_6355_fst__apsnd,axiom,
    ! [F: int > int,X: product_prod_int_int] :
      ( ( product_fst_int_int @ ( produc4463282112584876420nt_int @ F @ X ) )
      = ( product_fst_int_int @ X ) ) ).

% fst_apsnd
thf(fact_6356_fst__comp__apsnd,axiom,
    ! [F: code_integer > code_integer] :
      ( ( comp_P3428181540658402820nteger @ produc8508995932063986495nteger @ ( produc6499014454317279255nteger @ F ) )
      = produc8508995932063986495nteger ) ).

% fst_comp_apsnd
thf(fact_6357_fst__comp__apsnd,axiom,
    ! [F: int > int] :
      ( ( comp_P5356232565778667509nt_int @ product_fst_int_int @ ( produc4463282112584876420nt_int @ F ) )
      = product_fst_int_int ) ).

% fst_comp_apsnd
thf(fact_6358_apsnd__eq__conv,axiom,
    ! [F: code_integer > code_integer,X: produc8923325533196201883nteger,G: code_integer > code_integer] :
      ( ( ( produc6499014454317279255nteger @ F @ X )
        = ( produc6499014454317279255nteger @ G @ X ) )
      = ( ( F @ ( produc6174133586879617921nteger @ X ) )
        = ( G @ ( produc6174133586879617921nteger @ X ) ) ) ) ).

% apsnd_eq_conv
thf(fact_6359_snd__apsnd,axiom,
    ! [F: code_integer > code_integer,X: produc8923325533196201883nteger] :
      ( ( produc6174133586879617921nteger @ ( produc6499014454317279255nteger @ F @ X ) )
      = ( F @ ( produc6174133586879617921nteger @ X ) ) ) ).

% snd_apsnd
thf(fact_6360_snd__apsnd,axiom,
    ! [F: int > int,X: product_prod_int_int] :
      ( ( product_snd_int_int @ ( produc4463282112584876420nt_int @ F @ X ) )
      = ( F @ ( product_snd_int_int @ X ) ) ) ).

% snd_apsnd
thf(fact_6361_snd__apsnd,axiom,
    ! [F: produc7822875418678951345atural > produc7822875418678951345atural,X: produc5835291356934675326atural] :
      ( ( produc6591912806276919810atural @ ( produc4464772112783624294atural @ F @ X ) )
      = ( F @ ( produc6591912806276919810atural @ X ) ) ) ).

% snd_apsnd
thf(fact_6362_snd__comp__apsnd,axiom,
    ! [F: code_integer > code_integer] :
      ( ( comp_P3428181540658402820nteger @ produc6174133586879617921nteger @ ( produc6499014454317279255nteger @ F ) )
      = ( comp_C1043215000170864188nteger @ F @ produc6174133586879617921nteger ) ) ).

% snd_comp_apsnd
thf(fact_6363_snd__comp__apsnd,axiom,
    ! [F: int > int] :
      ( ( comp_P5356232565778667509nt_int @ product_snd_int_int @ ( produc4463282112584876420nt_int @ F ) )
      = ( comp_i4330622015127193916nt_int @ F @ product_snd_int_int ) ) ).

% snd_comp_apsnd
thf(fact_6364_snd__comp__apsnd,axiom,
    ! [F: produc7822875418678951345atural > produc7822875418678951345atural] :
      ( ( comp_P7610810914149544718atural @ produc6591912806276919810atural @ ( produc4464772112783624294atural @ F ) )
      = ( comp_P3067963646784605345atural @ F @ produc6591912806276919810atural ) ) ).

% snd_comp_apsnd
thf(fact_6365_fst__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( produc8508995932063986495nteger @ ( code_divmod_integer @ K @ L ) )
      = ( divide6298287555418463151nteger @ K @ L ) ) ).

% fst_divmod_integer
thf(fact_6366_snd__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( produc6174133586879617921nteger @ ( code_divmod_integer @ K @ L ) )
      = ( modulo364778990260209775nteger @ K @ L ) ) ).

% snd_divmod_integer
thf(fact_6367_Sup__atLeastAtMost,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( comple7399068483239264473et_nat @ ( set_or4548717258645045905et_nat @ X @ Y ) )
        = Y ) ) ).

% Sup_atLeastAtMost
thf(fact_6368_Sup__atLeastAtMost,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ X @ Y )
     => ( ( comple4687483117567863418t_unit @ ( set_or5692540242118139586t_unit @ X @ Y ) )
        = Y ) ) ).

% Sup_atLeastAtMost
thf(fact_6369_Inf__atLeastAtMost,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( comple7806235888213564991et_nat @ ( set_or4548717258645045905et_nat @ X @ Y ) )
        = X ) ) ).

% Inf_atLeastAtMost
thf(fact_6370_Inf__atLeastAtMost,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ X @ Y )
     => ( ( comple2584293577114468500t_unit @ ( set_or5692540242118139586t_unit @ X @ Y ) )
        = X ) ) ).

% Inf_atLeastAtMost
thf(fact_6371_Inf__atLeastLessThan,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_set_nat @ X @ Y )
     => ( ( comple7806235888213564991et_nat @ ( set_or3540276404033026485et_nat @ X @ Y ) )
        = X ) ) ).

% Inf_atLeastLessThan
thf(fact_6372_Inf__atLeastLessThan,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( ord_le361264281704409273t_unit @ X @ Y )
     => ( ( comple2584293577114468500t_unit @ ( set_or4663484573592751262t_unit @ X @ Y ) )
        = X ) ) ).

% Inf_atLeastLessThan
thf(fact_6373_prod_Ocollapse,axiom,
    ! [Prod: produc1908205239877642774nteger] :
      ( ( produc8603105652947943368nteger @ ( produc7822682618958472924nteger @ Prod ) @ ( produc7856867400915047194nteger @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_6374_prod_Ocollapse,axiom,
    ! [Prod: produc3925858234332021118et_nat] :
      ( ( produc5001842942810119800et_nat @ ( produc995936583742144908et_nat @ Prod ) @ ( produc4011572625026189258et_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_6375_prod_Ocollapse,axiom,
    ! [Prod: produc2732055786443039994et_nat] :
      ( ( produc2245416461498447860et_nat @ ( produc180342877477747464et_nat @ Prod ) @ ( produc5374455773327741254et_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_6376_prod_Ocollapse,axiom,
    ! [Prod: produc2285326912895808259nt_int] :
      ( ( produc5700946648718959541nt_int @ ( produc6816164490631068361nt_int @ Prod ) @ ( produc7328097813583171335nt_int @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_6377_prod_Ocollapse,axiom,
    ! [Prod: produc7773217078559923341nt_int] :
      ( ( produc4305682042979456191nt_int @ ( produc6230002227079971283nt_int @ Prod ) @ ( produc3162348030201620241nt_int @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_6378_prod_Ocollapse,axiom,
    ! [Prod: produc8923325533196201883nteger] :
      ( ( produc1086072967326762835nteger @ ( produc8508995932063986495nteger @ Prod ) @ ( produc6174133586879617921nteger @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_6379_prod_Ocollapse,axiom,
    ! [Prod: product_prod_int_int] :
      ( ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_6380_prod_Ocollapse,axiom,
    ! [Prod: produc5835291356934675326atural] :
      ( ( produc6639722614265839536atural @ ( produc497848011034438852atural @ Prod ) @ ( produc6591912806276919810atural @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_6381_Inf__atMost,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( comple6751905152567503566at_nat @ ( set_or4071486571221009422at_nat @ X ) )
      = bot_bo228742789529271731at_nat ) ).

% Inf_atMost
thf(fact_6382_Inf__atMost,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( comple1930127330929219688at_nat @ ( set_or2762583123196137896at_nat @ X ) )
      = bot_bo2099793752762293965at_nat ) ).

% Inf_atMost
thf(fact_6383_Inf__atMost,axiom,
    ! [X: set_o] :
      ( ( comple3063163877087187839_set_o @ ( set_ord_atMost_set_o @ X ) )
      = bot_bot_set_o ) ).

% Inf_atMost
thf(fact_6384_Inf__atMost,axiom,
    ! [X: set_int] :
      ( ( comple3628384868704368283et_int @ ( set_or58775011639299419et_int @ X ) )
      = bot_bot_set_int ) ).

% Inf_atMost
thf(fact_6385_Inf__atMost,axiom,
    ! [X: set_nat] :
      ( ( comple7806235888213564991et_nat @ ( set_or4236626031148496127et_nat @ X ) )
      = bot_bot_set_nat ) ).

% Inf_atMost
thf(fact_6386_Inf__atMost,axiom,
    ! [X: product_unit] :
      ( ( comple2584293577114468500t_unit @ ( set_or8621721900777396692t_unit @ X ) )
      = bot_bot_Product_unit ) ).

% Inf_atMost
thf(fact_6387_img__fst,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ S2 )
     => ( member2643936169264416010at_nat @ A @ ( image_7979982168039674559at_nat @ produc6011863565407846595at_nat @ S2 ) ) ) ).

% img_fst
thf(fact_6388_img__fst,axiom,
    ! [A: nat,B: nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ S2 )
     => ( member_nat @ A @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ S2 ) ) ) ).

% img_fst
thf(fact_6389_img__fst,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,S2: set_Pr1281608226676607948nteger] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A @ B ) @ S2 )
     => ( member4242434998011752849e_term @ A @ ( image_4436781500471753239e_term @ produc7822682618958472924nteger @ S2 ) ) ) ).

% img_fst
thf(fact_6390_img__fst,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,S2: set_Pr3286484037609594932et_nat] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ S2 )
     => ( member6576561426505652726_nat_o @ A @ ( image_6802380912191114376_nat_o @ produc995936583742144908et_nat @ S2 ) ) ) ).

% img_fst
thf(fact_6391_img__fst,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,S2: set_Pr8536935166611901872et_nat] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ S2 )
     => ( member6576561426505652726_nat_o @ A @ ( image_2517677226577294348_nat_o @ produc180342877477747464et_nat @ S2 ) ) ) ).

% img_fst
thf(fact_6392_img__fst,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,S2: set_Pr9222295170931077689nt_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ S2 )
     => ( member3222579708246209666e_term @ A @ ( image_142517975987106933e_term @ produc6816164490631068361nt_int @ S2 ) ) ) ).

% img_fst
thf(fact_6393_img__fst,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,S2: set_Pr1872883991513573699nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ S2 )
     => ( member8845023287901829240e_term @ A @ ( image_4217645046469565813e_term @ produc6230002227079971283nt_int @ S2 ) ) ) ).

% img_fst
thf(fact_6394_img__fst,axiom,
    ! [A: code_integer,B: code_integer,S2: set_Pr4811707699266497531nteger] :
      ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ A @ B ) @ S2 )
     => ( member_Code_integer @ A @ ( image_6316824473203919795nteger @ produc8508995932063986495nteger @ S2 ) ) ) ).

% img_fst
thf(fact_6395_img__fst,axiom,
    ! [A: int,B: int,S2: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ S2 )
     => ( member_int @ A @ ( image_5042161079198086560nt_int @ product_fst_int_int @ S2 ) ) ) ).

% img_fst
thf(fact_6396_img__snd,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ S2 )
     => ( member2643936169264416010at_nat @ B @ ( image_7979982168039674559at_nat @ produc2937545259423399429at_nat @ S2 ) ) ) ).

% img_snd
thf(fact_6397_img__snd,axiom,
    ! [A: nat,B: nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ S2 )
     => ( member_nat @ B @ ( image_2486076414777270412at_nat @ product_snd_nat_nat @ S2 ) ) ) ).

% img_snd
thf(fact_6398_img__snd,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,S2: set_Pr1281608226676607948nteger] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A @ B ) @ S2 )
     => ( member157494554546826820nteger @ B @ ( image_3141030398089242942nteger @ produc7856867400915047194nteger @ S2 ) ) ) ).

% img_snd
thf(fact_6399_img__snd,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,S2: set_Pr3286484037609594932et_nat] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ S2 )
     => ( member6260224972018164377et_nat @ B @ ( image_4390311411407041927et_nat @ produc4011572625026189258et_nat @ S2 ) ) ) ).

% img_snd
thf(fact_6400_img__snd,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,S2: set_Pr8536935166611901872et_nat] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ S2 )
     => ( member1996754912294343701et_nat @ B @ ( image_4557310310409863807et_nat @ produc5374455773327741254et_nat @ S2 ) ) ) ).

% img_snd
thf(fact_6401_img__snd,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,S2: set_Pr9222295170931077689nt_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ S2 )
     => ( member5262025264175285858nt_int @ B @ ( image_3915291023682402351nt_int @ produc7328097813583171335nt_int @ S2 ) ) ) ).

% img_snd
thf(fact_6402_img__snd,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,S2: set_Pr1872883991513573699nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ S2 )
     => ( member5262025264175285858nt_int @ B @ ( image_3108077152341812517nt_int @ produc3162348030201620241nt_int @ S2 ) ) ) ).

% img_snd
thf(fact_6403_img__snd,axiom,
    ! [A: code_integer,B: code_integer,S2: set_Pr4811707699266497531nteger] :
      ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ A @ B ) @ S2 )
     => ( member_Code_integer @ B @ ( image_6316824473203919795nteger @ produc6174133586879617921nteger @ S2 ) ) ) ).

% img_snd
thf(fact_6404_img__snd,axiom,
    ! [A: int,B: int,S2: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ S2 )
     => ( member_int @ B @ ( image_5042161079198086560nt_int @ product_snd_int_int @ S2 ) ) ) ).

% img_snd
thf(fact_6405_img__snd,axiom,
    ! [A: code_natural,B: produc7822875418678951345atural,S2: set_Pr2352657477744078772atural] :
      ( ( member2108296798382401941atural @ ( produc6639722614265839536atural @ A @ B ) @ S2 )
     => ( member8280416476884352090atural @ B @ ( image_3837677093542026988atural @ produc6591912806276919810atural @ S2 ) ) ) ).

% img_snd
thf(fact_6406_fst__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( produc8508995932063986495nteger @ ( code_divmod_abs @ K @ L ) )
      = ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L ) ) ) ).

% fst_divmod_abs
thf(fact_6407_snd__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( produc6174133586879617921nteger @ ( code_divmod_abs @ K @ L ) )
      = ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L ) ) ) ).

% snd_divmod_abs
thf(fact_6408_Inf__atMostLessThan,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ top_to6833984726390702231at_nat @ X )
     => ( ( comple6751905152567503566at_nat @ ( set_or2137699863726836970at_nat @ X ) )
        = bot_bo228742789529271731at_nat ) ) ).

% Inf_atMostLessThan
thf(fact_6409_Inf__atMostLessThan,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ top_to4669805908274784177at_nat @ X )
     => ( ( comple1930127330929219688at_nat @ ( set_or5243998310507771012at_nat @ X ) )
        = bot_bo2099793752762293965at_nat ) ) ).

% Inf_atMostLessThan
thf(fact_6410_Inf__atMostLessThan,axiom,
    ! [X: set_o] :
      ( ( ord_less_set_o @ top_top_set_o @ X )
     => ( ( comple3063163877087187839_set_o @ ( set_or3489638696441961371_set_o @ X ) )
        = bot_bot_set_o ) ) ).

% Inf_atMostLessThan
thf(fact_6411_Inf__atMostLessThan,axiom,
    ! [X: set_Product_unit] :
      ( ( ord_le8056459307392131481t_unit @ top_to1996260823553986621t_unit @ X )
     => ( ( comple7216939928495492468t_unit @ ( set_or3324826255835560848t_unit @ X ) )
        = bot_bo3957492148770167129t_unit ) ) ).

% Inf_atMostLessThan
thf(fact_6412_Inf__atMostLessThan,axiom,
    ! [X: set_int] :
      ( ( ord_less_set_int @ top_top_set_int @ X )
     => ( ( comple3628384868704368283et_int @ ( set_or5935648273017318783et_int @ X ) )
        = bot_bot_set_int ) ) ).

% Inf_atMostLessThan
thf(fact_6413_Inf__atMostLessThan,axiom,
    ! [X: set_nat] :
      ( ( ord_less_set_nat @ top_top_set_nat @ X )
     => ( ( comple7806235888213564991et_nat @ ( set_or890127255671739683et_nat @ X ) )
        = bot_bot_set_nat ) ) ).

% Inf_atMostLessThan
thf(fact_6414_Inf__atMostLessThan,axiom,
    ! [X: product_unit] :
      ( ( ord_le361264281704409273t_unit @ top_top_Product_unit @ X )
     => ( ( comple2584293577114468500t_unit @ ( set_or8708608758551125680t_unit @ X ) )
        = bot_bot_Product_unit ) ) ).

% Inf_atMostLessThan
thf(fact_6415_case__prod__beta_H,axiom,
    ( produc6842872674320459806at_nat
    = ( ^ [F2: nat > nat > nat,X3: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ X3 ) @ ( product_snd_nat_nat @ X3 ) ) ) ) ).

% case_prod_beta'
thf(fact_6416_case__prod__beta_H,axiom,
    ( produc27273713700761075at_nat
    = ( ^ [F2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X3: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ X3 ) @ ( product_snd_nat_nat @ X3 ) ) ) ) ).

% case_prod_beta'
thf(fact_6417_case__prod__beta_H,axiom,
    ( produc8739625826339149834_nat_o
    = ( ^ [F2: nat > nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ X3 ) @ ( product_snd_nat_nat @ X3 ) ) ) ) ).

% case_prod_beta'
thf(fact_6418_case__prod__beta_H,axiom,
    ( produc4245557441103728435nt_int
    = ( ^ [F2: int > int > product_prod_int_int,X3: product_prod_int_int] : ( F2 @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ).

% case_prod_beta'
thf(fact_6419_case__prod__beta_H,axiom,
    ( produc8211389475949308722nt_int
    = ( ^ [F2: int > int > int,X3: product_prod_int_int] : ( F2 @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ).

% case_prod_beta'
thf(fact_6420_case__prod__unfold,axiom,
    ( produc6842872674320459806at_nat
    = ( ^ [C2: nat > nat > nat,P6: product_prod_nat_nat] : ( C2 @ ( product_fst_nat_nat @ P6 ) @ ( product_snd_nat_nat @ P6 ) ) ) ) ).

% case_prod_unfold
thf(fact_6421_case__prod__unfold,axiom,
    ( produc27273713700761075at_nat
    = ( ^ [C2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,P6: product_prod_nat_nat] : ( C2 @ ( product_fst_nat_nat @ P6 ) @ ( product_snd_nat_nat @ P6 ) ) ) ) ).

% case_prod_unfold
thf(fact_6422_case__prod__unfold,axiom,
    ( produc8739625826339149834_nat_o
    = ( ^ [C2: nat > nat > product_prod_nat_nat > $o,P6: product_prod_nat_nat] : ( C2 @ ( product_fst_nat_nat @ P6 ) @ ( product_snd_nat_nat @ P6 ) ) ) ) ).

% case_prod_unfold
thf(fact_6423_case__prod__unfold,axiom,
    ( produc4245557441103728435nt_int
    = ( ^ [C2: int > int > product_prod_int_int,P6: product_prod_int_int] : ( C2 @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) ).

% case_prod_unfold
thf(fact_6424_case__prod__unfold,axiom,
    ( produc8211389475949308722nt_int
    = ( ^ [C2: int > int > int,P6: product_prod_int_int] : ( C2 @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) ).

% case_prod_unfold
thf(fact_6425_Product__Type_OCollect__case__prodD,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ( member8757157785044589968at_nat @ X @ ( collec6321179662152712658at_nat @ ( produc410239310623530412_nat_o @ A2 ) ) )
     => ( A2 @ ( produc6011863565407846595at_nat @ X ) @ ( produc2937545259423399429at_nat @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_6426_Product__Type_OCollect__case__prodD,axiom,
    ! [X: product_prod_nat_nat,A2: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ X @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A2 ) ) )
     => ( A2 @ ( product_fst_nat_nat @ X ) @ ( product_snd_nat_nat @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_6427_Product__Type_OCollect__case__prodD,axiom,
    ! [X: produc8923325533196201883nteger,A2: code_integer > code_integer > $o] :
      ( ( member157494554546826820nteger @ X @ ( collec7766031446232907142nteger @ ( produc2066375834425727024eger_o @ A2 ) ) )
     => ( A2 @ ( produc8508995932063986495nteger @ X ) @ ( produc6174133586879617921nteger @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_6428_Product__Type_OCollect__case__prodD,axiom,
    ! [X: product_prod_int_int,A2: int > int > $o] :
      ( ( member5262025264175285858nt_int @ X @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A2 ) ) )
     => ( A2 @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_6429_Product__Type_OCollect__case__prodD,axiom,
    ! [X: produc5835291356934675326atural,A2: code_natural > produc7822875418678951345atural > $o] :
      ( ( member2108296798382401941atural @ X @ ( collec1991479681304692691atural @ ( produc7358226366420480123ural_o @ A2 ) ) )
     => ( A2 @ ( produc497848011034438852atural @ X ) @ ( produc6591912806276919810atural @ X ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_6430_case__prod__beta,axiom,
    ( produc6842872674320459806at_nat
    = ( ^ [F2: nat > nat > nat,P6: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ P6 ) @ ( product_snd_nat_nat @ P6 ) ) ) ) ).

% case_prod_beta
thf(fact_6431_case__prod__beta,axiom,
    ( produc27273713700761075at_nat
    = ( ^ [F2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,P6: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ P6 ) @ ( product_snd_nat_nat @ P6 ) ) ) ) ).

% case_prod_beta
thf(fact_6432_case__prod__beta,axiom,
    ( produc8739625826339149834_nat_o
    = ( ^ [F2: nat > nat > product_prod_nat_nat > $o,P6: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ P6 ) @ ( product_snd_nat_nat @ P6 ) ) ) ) ).

% case_prod_beta
thf(fact_6433_case__prod__beta,axiom,
    ( produc4245557441103728435nt_int
    = ( ^ [F2: int > int > product_prod_int_int,P6: product_prod_int_int] : ( F2 @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) ).

% case_prod_beta
thf(fact_6434_case__prod__beta,axiom,
    ( produc8211389475949308722nt_int
    = ( ^ [F2: int > int > int,P6: product_prod_int_int] : ( F2 @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) ).

% case_prod_beta
thf(fact_6435_split__beta,axiom,
    ( produc6842872674320459806at_nat
    = ( ^ [F2: nat > nat > nat,Prod2: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ Prod2 ) @ ( product_snd_nat_nat @ Prod2 ) ) ) ) ).

% split_beta
thf(fact_6436_split__beta,axiom,
    ( produc27273713700761075at_nat
    = ( ^ [F2: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Prod2: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ Prod2 ) @ ( product_snd_nat_nat @ Prod2 ) ) ) ) ).

% split_beta
thf(fact_6437_split__beta,axiom,
    ( produc8739625826339149834_nat_o
    = ( ^ [F2: nat > nat > product_prod_nat_nat > $o,Prod2: product_prod_nat_nat] : ( F2 @ ( product_fst_nat_nat @ Prod2 ) @ ( product_snd_nat_nat @ Prod2 ) ) ) ) ).

% split_beta
thf(fact_6438_split__beta,axiom,
    ( produc4245557441103728435nt_int
    = ( ^ [F2: int > int > product_prod_int_int,Prod2: product_prod_int_int] : ( F2 @ ( product_fst_int_int @ Prod2 ) @ ( product_snd_int_int @ Prod2 ) ) ) ) ).

% split_beta
thf(fact_6439_split__beta,axiom,
    ( produc8211389475949308722nt_int
    = ( ^ [F2: int > int > int,Prod2: product_prod_int_int] : ( F2 @ ( product_fst_int_int @ Prod2 ) @ ( product_snd_int_int @ Prod2 ) ) ) ) ).

% split_beta
thf(fact_6440_effect__guardI,axiom,
    ! [P: heap_e7401611519738050253t_unit > $o,H3: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,F: heap_e7401611519738050253t_unit > produc8664842809031399944it_nat,N2: nat,R2: product_unit] :
      ( ( P @ H3 )
     => ( ( H5
          = ( produc2838810277069100634it_nat @ ( produc8604850587836428116it_nat @ ( F @ H3 ) ) ) )
       => ( ( N2
            = ( produc2594868675584459928it_nat @ ( produc8604850587836428116it_nat @ ( F @ H3 ) ) ) )
         => ( ( R2
              = ( produc4530234641508181270it_nat @ ( F @ H3 ) ) )
           => ( heap_T6553295506729943825t_unit @ ( heap_T8440541562793052209t_unit @ P @ F ) @ H3 @ H5 @ R2 @ N2 ) ) ) ) ) ).

% effect_guardI
thf(fact_6441_effect__guardI,axiom,
    ! [P: heap_e7401611519738050253t_unit > $o,H3: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,F: heap_e7401611519738050253t_unit > produc6714399278180371524it_nat,N2: nat,R2: array_a] :
      ( ( P @ H3 )
     => ( ( H5
          = ( produc2838810277069100634it_nat @ ( produc6991280839920081578it_nat @ ( F @ H3 ) ) ) )
       => ( ( N2
            = ( produc2594868675584459928it_nat @ ( produc6991280839920081578it_nat @ ( F @ H3 ) ) ) )
         => ( ( R2
              = ( produc491454361838572648it_nat @ ( F @ H3 ) ) )
           => ( heap_T5830633491639066771rray_a @ ( heap_T7685628758766326643rray_a @ P @ F ) @ H3 @ H5 @ R2 @ N2 ) ) ) ) ) ).

% effect_guardI
thf(fact_6442_effect__guardE,axiom,
    ! [P: heap_e7401611519738050253t_unit > $o,F: heap_e7401611519738050253t_unit > produc8664842809031399944it_nat,H3: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,R2: product_unit,N2: nat] :
      ( ( heap_T6553295506729943825t_unit @ ( heap_T8440541562793052209t_unit @ P @ F ) @ H3 @ H5 @ R2 @ N2 )
     => ~ ( ( H5
            = ( produc2838810277069100634it_nat @ ( produc8604850587836428116it_nat @ ( F @ H3 ) ) ) )
         => ( ( N2
              = ( produc2594868675584459928it_nat @ ( produc8604850587836428116it_nat @ ( F @ H3 ) ) ) )
           => ( ( R2
                = ( produc4530234641508181270it_nat @ ( F @ H3 ) ) )
             => ~ ( P @ H3 ) ) ) ) ) ).

% effect_guardE
thf(fact_6443_effect__guardE,axiom,
    ! [P: heap_e7401611519738050253t_unit > $o,F: heap_e7401611519738050253t_unit > produc6714399278180371524it_nat,H3: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,R2: array_a,N2: nat] :
      ( ( heap_T5830633491639066771rray_a @ ( heap_T7685628758766326643rray_a @ P @ F ) @ H3 @ H5 @ R2 @ N2 )
     => ~ ( ( H5
            = ( produc2838810277069100634it_nat @ ( produc6991280839920081578it_nat @ ( F @ H3 ) ) ) )
         => ( ( N2
              = ( produc2594868675584459928it_nat @ ( produc6991280839920081578it_nat @ ( F @ H3 ) ) ) )
           => ( ( R2
                = ( produc491454361838572648it_nat @ ( F @ H3 ) ) )
             => ~ ( P @ H3 ) ) ) ) ) ).

% effect_guardE
thf(fact_6444_effectE,axiom,
    ! [C: heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,R2: product_unit,N2: nat] :
      ( ( heap_T6553295506729943825t_unit @ C @ H3 @ H5 @ R2 @ N2 )
     => ~ ( ( R2
            = ( produc4530234641508181270it_nat @ ( the_Pr7776125533777466484it_nat @ ( heap_T875086893843062177t_unit @ C @ H3 ) ) ) )
         => ( ( H5
              = ( produc2838810277069100634it_nat @ ( produc8604850587836428116it_nat @ ( the_Pr7776125533777466484it_nat @ ( heap_T875086893843062177t_unit @ C @ H3 ) ) ) ) )
           => ( ( N2
                = ( produc2594868675584459928it_nat @ ( produc8604850587836428116it_nat @ ( the_Pr7776125533777466484it_nat @ ( heap_T875086893843062177t_unit @ C @ H3 ) ) ) ) )
             => ~ ( heap_T584514906347983379t_unit @ C @ H3 ) ) ) ) ) ).

% effectE
thf(fact_6445_effectE,axiom,
    ! [C: heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,R2: array_a,N2: nat] :
      ( ( heap_T5830633491639066771rray_a @ C @ H3 @ H5 @ R2 @ N2 )
     => ~ ( ( R2
            = ( produc491454361838572648it_nat @ ( the_Pr5856008525237892824it_nat @ ( heap_T5591663578999710211rray_a @ C @ H3 ) ) ) )
         => ( ( H5
              = ( produc2838810277069100634it_nat @ ( produc6991280839920081578it_nat @ ( the_Pr5856008525237892824it_nat @ ( heap_T5591663578999710211rray_a @ C @ H3 ) ) ) ) )
           => ( ( N2
                = ( produc2594868675584459928it_nat @ ( produc6991280839920081578it_nat @ ( the_Pr5856008525237892824it_nat @ ( heap_T5591663578999710211rray_a @ C @ H3 ) ) ) ) )
             => ~ ( heap_T6932021131025641745rray_a @ C @ H3 ) ) ) ) ) ).

% effectE
thf(fact_6446_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc8923325533196201883nteger,Z2: produc8923325533196201883nteger] : ( Y4 = Z2 ) )
    = ( ^ [S4: produc8923325533196201883nteger,T2: produc8923325533196201883nteger] :
          ( ( ( produc8508995932063986495nteger @ S4 )
            = ( produc8508995932063986495nteger @ T2 ) )
          & ( ( produc6174133586879617921nteger @ S4 )
            = ( produc6174133586879617921nteger @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_6447_prod__eq__iff,axiom,
    ( ( ^ [Y4: product_prod_int_int,Z2: product_prod_int_int] : ( Y4 = Z2 ) )
    = ( ^ [S4: product_prod_int_int,T2: product_prod_int_int] :
          ( ( ( product_fst_int_int @ S4 )
            = ( product_fst_int_int @ T2 ) )
          & ( ( product_snd_int_int @ S4 )
            = ( product_snd_int_int @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_6448_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc5835291356934675326atural,Z2: produc5835291356934675326atural] : ( Y4 = Z2 ) )
    = ( ^ [S4: produc5835291356934675326atural,T2: produc5835291356934675326atural] :
          ( ( ( produc497848011034438852atural @ S4 )
            = ( produc497848011034438852atural @ T2 ) )
          & ( ( produc6591912806276919810atural @ S4 )
            = ( produc6591912806276919810atural @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_6449_prod__eqI,axiom,
    ! [P4: produc8923325533196201883nteger,Q3: produc8923325533196201883nteger] :
      ( ( ( produc8508995932063986495nteger @ P4 )
        = ( produc8508995932063986495nteger @ Q3 ) )
     => ( ( ( produc6174133586879617921nteger @ P4 )
          = ( produc6174133586879617921nteger @ Q3 ) )
       => ( P4 = Q3 ) ) ) ).

% prod_eqI
thf(fact_6450_prod__eqI,axiom,
    ! [P4: product_prod_int_int,Q3: product_prod_int_int] :
      ( ( ( product_fst_int_int @ P4 )
        = ( product_fst_int_int @ Q3 ) )
     => ( ( ( product_snd_int_int @ P4 )
          = ( product_snd_int_int @ Q3 ) )
       => ( P4 = Q3 ) ) ) ).

% prod_eqI
thf(fact_6451_prod__eqI,axiom,
    ! [P4: produc5835291356934675326atural,Q3: produc5835291356934675326atural] :
      ( ( ( produc497848011034438852atural @ P4 )
        = ( produc497848011034438852atural @ Q3 ) )
     => ( ( ( produc6591912806276919810atural @ P4 )
          = ( produc6591912806276919810atural @ Q3 ) )
       => ( P4 = Q3 ) ) ) ).

% prod_eqI
thf(fact_6452_prod_Oexpand,axiom,
    ! [Prod: produc8923325533196201883nteger,Prod3: produc8923325533196201883nteger] :
      ( ( ( ( produc8508995932063986495nteger @ Prod )
          = ( produc8508995932063986495nteger @ Prod3 ) )
        & ( ( produc6174133586879617921nteger @ Prod )
          = ( produc6174133586879617921nteger @ Prod3 ) ) )
     => ( Prod = Prod3 ) ) ).

% prod.expand
thf(fact_6453_prod_Oexpand,axiom,
    ! [Prod: product_prod_int_int,Prod3: product_prod_int_int] :
      ( ( ( ( product_fst_int_int @ Prod )
          = ( product_fst_int_int @ Prod3 ) )
        & ( ( product_snd_int_int @ Prod )
          = ( product_snd_int_int @ Prod3 ) ) )
     => ( Prod = Prod3 ) ) ).

% prod.expand
thf(fact_6454_prod_Oexpand,axiom,
    ! [Prod: produc5835291356934675326atural,Prod3: produc5835291356934675326atural] :
      ( ( ( ( produc497848011034438852atural @ Prod )
          = ( produc497848011034438852atural @ Prod3 ) )
        & ( ( produc6591912806276919810atural @ Prod )
          = ( produc6591912806276919810atural @ Prod3 ) ) )
     => ( Prod = Prod3 ) ) ).

% prod.expand
thf(fact_6455_All__prod__contract,axiom,
    ! [P: code_integer > code_integer > $o] :
      ( ( ! [A5: code_integer,X9: code_integer] : ( P @ A5 @ X9 ) )
      = ( ! [Z4: produc8923325533196201883nteger] : ( P @ ( produc8508995932063986495nteger @ Z4 ) @ ( produc6174133586879617921nteger @ Z4 ) ) ) ) ).

% All_prod_contract
thf(fact_6456_All__prod__contract,axiom,
    ! [P: int > int > $o] :
      ( ( ! [A5: int,X9: int] : ( P @ A5 @ X9 ) )
      = ( ! [Z4: product_prod_int_int] : ( P @ ( product_fst_int_int @ Z4 ) @ ( product_snd_int_int @ Z4 ) ) ) ) ).

% All_prod_contract
thf(fact_6457_All__prod__contract,axiom,
    ! [P: code_natural > produc7822875418678951345atural > $o] :
      ( ( ! [A5: code_natural,X9: produc7822875418678951345atural] : ( P @ A5 @ X9 ) )
      = ( ! [Z4: produc5835291356934675326atural] : ( P @ ( produc497848011034438852atural @ Z4 ) @ ( produc6591912806276919810atural @ Z4 ) ) ) ) ).

% All_prod_contract
thf(fact_6458_Ex__prod__contract,axiom,
    ! [P: code_integer > code_integer > $o] :
      ( ( ? [A5: code_integer,X9: code_integer] : ( P @ A5 @ X9 ) )
      = ( ? [Z4: produc8923325533196201883nteger] : ( P @ ( produc8508995932063986495nteger @ Z4 ) @ ( produc6174133586879617921nteger @ Z4 ) ) ) ) ).

% Ex_prod_contract
thf(fact_6459_Ex__prod__contract,axiom,
    ! [P: int > int > $o] :
      ( ( ? [A5: int,X9: int] : ( P @ A5 @ X9 ) )
      = ( ? [Z4: product_prod_int_int] : ( P @ ( product_fst_int_int @ Z4 ) @ ( product_snd_int_int @ Z4 ) ) ) ) ).

% Ex_prod_contract
thf(fact_6460_Ex__prod__contract,axiom,
    ! [P: code_natural > produc7822875418678951345atural > $o] :
      ( ( ? [A5: code_natural,X9: produc7822875418678951345atural] : ( P @ A5 @ X9 ) )
      = ( ? [Z4: produc5835291356934675326atural] : ( P @ ( produc497848011034438852atural @ Z4 ) @ ( produc6591912806276919810atural @ Z4 ) ) ) ) ).

% Ex_prod_contract
thf(fact_6461_prod_Osplit__sel,axiom,
    ! [P: nat > $o,F: nat > nat > nat,Prod: product_prod_nat_nat] :
      ( ( P @ ( produc6842872674320459806at_nat @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
       => ( P @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_6462_prod_Osplit__sel,axiom,
    ! [P: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
      ( ( P @ ( produc27273713700761075at_nat @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
       => ( P @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_6463_prod_Osplit__sel,axiom,
    ! [P: ( product_prod_nat_nat > $o ) > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
      ( ( P @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
       => ( P @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_6464_prod_Osplit__sel,axiom,
    ! [P: product_prod_int_int > $o,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( P @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) )
       => ( P @ ( F @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_6465_prod_Osplit__sel,axiom,
    ! [P: int > $o,F: int > int > int,Prod: product_prod_int_int] :
      ( ( P @ ( produc8211389475949308722nt_int @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) )
       => ( P @ ( F @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_6466_prod_Osplit__sel__asm,axiom,
    ! [P: nat > $o,F: nat > nat > nat,Prod: product_prod_nat_nat] :
      ( ( P @ ( produc6842872674320459806at_nat @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
            & ~ ( P @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_6467_prod_Osplit__sel__asm,axiom,
    ! [P: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
      ( ( P @ ( produc27273713700761075at_nat @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
            & ~ ( P @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_6468_prod_Osplit__sel__asm,axiom,
    ! [P: ( product_prod_nat_nat > $o ) > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
      ( ( P @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
            & ~ ( P @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_6469_prod_Osplit__sel__asm,axiom,
    ! [P: product_prod_int_int > $o,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( P @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) )
            & ~ ( P @ ( F @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_6470_prod_Osplit__sel__asm,axiom,
    ! [P: int > $o,F: int > int > int,Prod: product_prod_int_int] :
      ( ( P @ ( produc8211389475949308722nt_int @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) )
            & ~ ( P @ ( F @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_6471_surjective__pairing,axiom,
    ! [T: produc1908205239877642774nteger] :
      ( T
      = ( produc8603105652947943368nteger @ ( produc7822682618958472924nteger @ T ) @ ( produc7856867400915047194nteger @ T ) ) ) ).

% surjective_pairing
thf(fact_6472_surjective__pairing,axiom,
    ! [T: produc3925858234332021118et_nat] :
      ( T
      = ( produc5001842942810119800et_nat @ ( produc995936583742144908et_nat @ T ) @ ( produc4011572625026189258et_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_6473_surjective__pairing,axiom,
    ! [T: produc2732055786443039994et_nat] :
      ( T
      = ( produc2245416461498447860et_nat @ ( produc180342877477747464et_nat @ T ) @ ( produc5374455773327741254et_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_6474_surjective__pairing,axiom,
    ! [T: produc2285326912895808259nt_int] :
      ( T
      = ( produc5700946648718959541nt_int @ ( produc6816164490631068361nt_int @ T ) @ ( produc7328097813583171335nt_int @ T ) ) ) ).

% surjective_pairing
thf(fact_6475_surjective__pairing,axiom,
    ! [T: produc7773217078559923341nt_int] :
      ( T
      = ( produc4305682042979456191nt_int @ ( produc6230002227079971283nt_int @ T ) @ ( produc3162348030201620241nt_int @ T ) ) ) ).

% surjective_pairing
thf(fact_6476_surjective__pairing,axiom,
    ! [T: produc8923325533196201883nteger] :
      ( T
      = ( produc1086072967326762835nteger @ ( produc8508995932063986495nteger @ T ) @ ( produc6174133586879617921nteger @ T ) ) ) ).

% surjective_pairing
thf(fact_6477_surjective__pairing,axiom,
    ! [T: product_prod_int_int] :
      ( T
      = ( product_Pair_int_int @ ( product_fst_int_int @ T ) @ ( product_snd_int_int @ T ) ) ) ).

% surjective_pairing
thf(fact_6478_surjective__pairing,axiom,
    ! [T: produc5835291356934675326atural] :
      ( T
      = ( produc6639722614265839536atural @ ( produc497848011034438852atural @ T ) @ ( produc6591912806276919810atural @ T ) ) ) ).

% surjective_pairing
thf(fact_6479_prod_Oexhaust__sel,axiom,
    ! [Prod: produc1908205239877642774nteger] :
      ( Prod
      = ( produc8603105652947943368nteger @ ( produc7822682618958472924nteger @ Prod ) @ ( produc7856867400915047194nteger @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_6480_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3925858234332021118et_nat] :
      ( Prod
      = ( produc5001842942810119800et_nat @ ( produc995936583742144908et_nat @ Prod ) @ ( produc4011572625026189258et_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_6481_prod_Oexhaust__sel,axiom,
    ! [Prod: produc2732055786443039994et_nat] :
      ( Prod
      = ( produc2245416461498447860et_nat @ ( produc180342877477747464et_nat @ Prod ) @ ( produc5374455773327741254et_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_6482_prod_Oexhaust__sel,axiom,
    ! [Prod: produc2285326912895808259nt_int] :
      ( Prod
      = ( produc5700946648718959541nt_int @ ( produc6816164490631068361nt_int @ Prod ) @ ( produc7328097813583171335nt_int @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_6483_prod_Oexhaust__sel,axiom,
    ! [Prod: produc7773217078559923341nt_int] :
      ( Prod
      = ( produc4305682042979456191nt_int @ ( produc6230002227079971283nt_int @ Prod ) @ ( produc3162348030201620241nt_int @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_6484_prod_Oexhaust__sel,axiom,
    ! [Prod: produc8923325533196201883nteger] :
      ( Prod
      = ( produc1086072967326762835nteger @ ( produc8508995932063986495nteger @ Prod ) @ ( produc6174133586879617921nteger @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_6485_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_int_int] :
      ( Prod
      = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_6486_prod_Oexhaust__sel,axiom,
    ! [Prod: produc5835291356934675326atural] :
      ( Prod
      = ( produc6639722614265839536atural @ ( produc497848011034438852atural @ Prod ) @ ( produc6591912806276919810atural @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_6487_exI__realizer,axiom,
    ! [P: produc8923325533196201883nteger > ( produc6241069584506657477e_term > option6357759511663192854e_term ) > $o,Y: produc8923325533196201883nteger,X: produc6241069584506657477e_term > option6357759511663192854e_term] :
      ( ( P @ Y @ X )
     => ( P @ ( produc7856867400915047194nteger @ ( produc8603105652947943368nteger @ X @ Y ) ) @ ( produc7822682618958472924nteger @ ( produc8603105652947943368nteger @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_6488_exI__realizer,axiom,
    ! [P: produc3658429121746597890et_nat > ( produc3658429121746597890et_nat > $o ) > $o,Y: produc3658429121746597890et_nat,X: produc3658429121746597890et_nat > $o] :
      ( ( P @ Y @ X )
     => ( P @ ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) ) @ ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_6489_exI__realizer,axiom,
    ! [P: produc3925858234332021118et_nat > ( produc3658429121746597890et_nat > $o ) > $o,Y: produc3925858234332021118et_nat,X: produc3658429121746597890et_nat > $o] :
      ( ( P @ Y @ X )
     => ( P @ ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) ) @ ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_6490_exI__realizer,axiom,
    ! [P: product_prod_int_int > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,Y: product_prod_int_int,X: produc8551481072490612790e_term > option6357759511663192854e_term] :
      ( ( P @ Y @ X )
     => ( P @ ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) ) @ ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_6491_exI__realizer,axiom,
    ! [P: product_prod_int_int > ( int > option6357759511663192854e_term ) > $o,Y: product_prod_int_int,X: int > option6357759511663192854e_term] :
      ( ( P @ Y @ X )
     => ( P @ ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) ) @ ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_6492_exI__realizer,axiom,
    ! [P: code_integer > code_integer > $o,Y: code_integer,X: code_integer] :
      ( ( P @ Y @ X )
     => ( P @ ( produc6174133586879617921nteger @ ( produc1086072967326762835nteger @ X @ Y ) ) @ ( produc8508995932063986495nteger @ ( produc1086072967326762835nteger @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_6493_exI__realizer,axiom,
    ! [P: int > int > $o,Y: int,X: int] :
      ( ( P @ Y @ X )
     => ( P @ ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) ) @ ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_6494_exI__realizer,axiom,
    ! [P: produc7822875418678951345atural > code_natural > $o,Y: produc7822875418678951345atural,X: code_natural] :
      ( ( P @ Y @ X )
     => ( P @ ( produc6591912806276919810atural @ ( produc6639722614265839536atural @ X @ Y ) ) @ ( produc497848011034438852atural @ ( produc6639722614265839536atural @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_6495_conjI__realizer,axiom,
    ! [P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > $o,P4: produc6241069584506657477e_term > option6357759511663192854e_term,Q: produc8923325533196201883nteger > $o,Q3: produc8923325533196201883nteger] :
      ( ( P @ P4 )
     => ( ( Q @ Q3 )
       => ( ( P @ ( produc7822682618958472924nteger @ ( produc8603105652947943368nteger @ P4 @ Q3 ) ) )
          & ( Q @ ( produc7856867400915047194nteger @ ( produc8603105652947943368nteger @ P4 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_6496_conjI__realizer,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o,P4: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o,Q3: produc3658429121746597890et_nat] :
      ( ( P @ P4 )
     => ( ( Q @ Q3 )
       => ( ( P @ ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ P4 @ Q3 ) ) )
          & ( Q @ ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ P4 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_6497_conjI__realizer,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o,P4: produc3658429121746597890et_nat > $o,Q: produc3925858234332021118et_nat > $o,Q3: produc3925858234332021118et_nat] :
      ( ( P @ P4 )
     => ( ( Q @ Q3 )
       => ( ( P @ ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ P4 @ Q3 ) ) )
          & ( Q @ ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ P4 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_6498_conjI__realizer,axiom,
    ! [P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o,P4: produc8551481072490612790e_term > option6357759511663192854e_term,Q: product_prod_int_int > $o,Q3: product_prod_int_int] :
      ( ( P @ P4 )
     => ( ( Q @ Q3 )
       => ( ( P @ ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ P4 @ Q3 ) ) )
          & ( Q @ ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ P4 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_6499_conjI__realizer,axiom,
    ! [P: ( int > option6357759511663192854e_term ) > $o,P4: int > option6357759511663192854e_term,Q: product_prod_int_int > $o,Q3: product_prod_int_int] :
      ( ( P @ P4 )
     => ( ( Q @ Q3 )
       => ( ( P @ ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ P4 @ Q3 ) ) )
          & ( Q @ ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ P4 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_6500_conjI__realizer,axiom,
    ! [P: code_integer > $o,P4: code_integer,Q: code_integer > $o,Q3: code_integer] :
      ( ( P @ P4 )
     => ( ( Q @ Q3 )
       => ( ( P @ ( produc8508995932063986495nteger @ ( produc1086072967326762835nteger @ P4 @ Q3 ) ) )
          & ( Q @ ( produc6174133586879617921nteger @ ( produc1086072967326762835nteger @ P4 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_6501_conjI__realizer,axiom,
    ! [P: int > $o,P4: int,Q: int > $o,Q3: int] :
      ( ( P @ P4 )
     => ( ( Q @ Q3 )
       => ( ( P @ ( product_fst_int_int @ ( product_Pair_int_int @ P4 @ Q3 ) ) )
          & ( Q @ ( product_snd_int_int @ ( product_Pair_int_int @ P4 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_6502_conjI__realizer,axiom,
    ! [P: code_natural > $o,P4: code_natural,Q: produc7822875418678951345atural > $o,Q3: produc7822875418678951345atural] :
      ( ( P @ P4 )
     => ( ( Q @ Q3 )
       => ( ( P @ ( produc497848011034438852atural @ ( produc6639722614265839536atural @ P4 @ Q3 ) ) )
          & ( Q @ ( produc6591912806276919810atural @ ( produc6639722614265839536atural @ P4 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_6503_execute__bind__success,axiom,
    ! [F: heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit,G: product_unit > heap_T6362127782908069795rray_a] :
      ( ( heap_T584514906347983379t_unit @ F @ H3 )
     => ( ( heap_T5591663578999710211rray_a @ ( heap_T7158140910958840029rray_a @ F @ G ) @ H3 )
        = ( heap_T531568236820715630rray_a @ ( produc2594868675584459928it_nat @ ( produc8604850587836428116it_nat @ ( the_Pr7776125533777466484it_nat @ ( heap_T875086893843062177t_unit @ F @ H3 ) ) ) ) @ ( heap_T5591663578999710211rray_a @ ( G @ ( produc4530234641508181270it_nat @ ( the_Pr7776125533777466484it_nat @ ( heap_T875086893843062177t_unit @ F @ H3 ) ) ) ) @ ( produc2838810277069100634it_nat @ ( produc8604850587836428116it_nat @ ( the_Pr7776125533777466484it_nat @ ( heap_T875086893843062177t_unit @ F @ H3 ) ) ) ) ) ) ) ) ).

% execute_bind_success
thf(fact_6504_execute__bind__success,axiom,
    ! [F: heap_T5738788834812785303t_unit,H3: heap_e7401611519738050253t_unit,G: product_unit > heap_T5738788834812785303t_unit] :
      ( ( heap_T584514906347983379t_unit @ F @ H3 )
     => ( ( heap_T875086893843062177t_unit @ ( heap_T2633723481742716231t_unit @ F @ G ) @ H3 )
        = ( heap_T3616969660504097270t_unit @ ( produc2594868675584459928it_nat @ ( produc8604850587836428116it_nat @ ( the_Pr7776125533777466484it_nat @ ( heap_T875086893843062177t_unit @ F @ H3 ) ) ) ) @ ( heap_T875086893843062177t_unit @ ( G @ ( produc4530234641508181270it_nat @ ( the_Pr7776125533777466484it_nat @ ( heap_T875086893843062177t_unit @ F @ H3 ) ) ) ) @ ( produc2838810277069100634it_nat @ ( produc8604850587836428116it_nat @ ( the_Pr7776125533777466484it_nat @ ( heap_T875086893843062177t_unit @ F @ H3 ) ) ) ) ) ) ) ) ).

% execute_bind_success
thf(fact_6505_execute__bind__success,axiom,
    ! [F: heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit,G: array_a > heap_T6362127782908069795rray_a] :
      ( ( heap_T6932021131025641745rray_a @ F @ H3 )
     => ( ( heap_T5591663578999710211rray_a @ ( heap_T2407400398799471159rray_a @ F @ G ) @ H3 )
        = ( heap_T531568236820715630rray_a @ ( produc2594868675584459928it_nat @ ( produc6991280839920081578it_nat @ ( the_Pr5856008525237892824it_nat @ ( heap_T5591663578999710211rray_a @ F @ H3 ) ) ) ) @ ( heap_T5591663578999710211rray_a @ ( G @ ( produc491454361838572648it_nat @ ( the_Pr5856008525237892824it_nat @ ( heap_T5591663578999710211rray_a @ F @ H3 ) ) ) ) @ ( produc2838810277069100634it_nat @ ( produc6991280839920081578it_nat @ ( the_Pr5856008525237892824it_nat @ ( heap_T5591663578999710211rray_a @ F @ H3 ) ) ) ) ) ) ) ) ).

% execute_bind_success
thf(fact_6506_execute__bind__success,axiom,
    ! [F: heap_T6362127782908069795rray_a,H3: heap_e7401611519738050253t_unit,G: array_a > heap_T5738788834812785303t_unit] :
      ( ( heap_T6932021131025641745rray_a @ F @ H3 )
     => ( ( heap_T875086893843062177t_unit @ ( heap_T7324619625323989613t_unit @ F @ G ) @ H3 )
        = ( heap_T3616969660504097270t_unit @ ( produc2594868675584459928it_nat @ ( produc6991280839920081578it_nat @ ( the_Pr5856008525237892824it_nat @ ( heap_T5591663578999710211rray_a @ F @ H3 ) ) ) ) @ ( heap_T875086893843062177t_unit @ ( G @ ( produc491454361838572648it_nat @ ( the_Pr5856008525237892824it_nat @ ( heap_T5591663578999710211rray_a @ F @ H3 ) ) ) ) @ ( produc2838810277069100634it_nat @ ( produc6991280839920081578it_nat @ ( the_Pr5856008525237892824it_nat @ ( heap_T5591663578999710211rray_a @ F @ H3 ) ) ) ) ) ) ) ) ).

% execute_bind_success
thf(fact_6507_sndI,axiom,
    ! [X: produc1908205239877642774nteger,Y: produc6241069584506657477e_term > option6357759511663192854e_term,Z: produc8923325533196201883nteger] :
      ( ( X
        = ( produc8603105652947943368nteger @ Y @ Z ) )
     => ( ( produc7856867400915047194nteger @ X )
        = Z ) ) ).

% sndI
thf(fact_6508_sndI,axiom,
    ! [X: produc3925858234332021118et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3658429121746597890et_nat] :
      ( ( X
        = ( produc5001842942810119800et_nat @ Y @ Z ) )
     => ( ( produc4011572625026189258et_nat @ X )
        = Z ) ) ).

% sndI
thf(fact_6509_sndI,axiom,
    ! [X: produc2732055786443039994et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3925858234332021118et_nat] :
      ( ( X
        = ( produc2245416461498447860et_nat @ Y @ Z ) )
     => ( ( produc5374455773327741254et_nat @ X )
        = Z ) ) ).

% sndI
thf(fact_6510_sndI,axiom,
    ! [X: produc2285326912895808259nt_int,Y: produc8551481072490612790e_term > option6357759511663192854e_term,Z: product_prod_int_int] :
      ( ( X
        = ( produc5700946648718959541nt_int @ Y @ Z ) )
     => ( ( produc7328097813583171335nt_int @ X )
        = Z ) ) ).

% sndI
thf(fact_6511_sndI,axiom,
    ! [X: produc7773217078559923341nt_int,Y: int > option6357759511663192854e_term,Z: product_prod_int_int] :
      ( ( X
        = ( produc4305682042979456191nt_int @ Y @ Z ) )
     => ( ( produc3162348030201620241nt_int @ X )
        = Z ) ) ).

% sndI
thf(fact_6512_sndI,axiom,
    ! [X: produc8923325533196201883nteger,Y: code_integer,Z: code_integer] :
      ( ( X
        = ( produc1086072967326762835nteger @ Y @ Z ) )
     => ( ( produc6174133586879617921nteger @ X )
        = Z ) ) ).

% sndI
thf(fact_6513_sndI,axiom,
    ! [X: product_prod_int_int,Y: int,Z: int] :
      ( ( X
        = ( product_Pair_int_int @ Y @ Z ) )
     => ( ( product_snd_int_int @ X )
        = Z ) ) ).

% sndI
thf(fact_6514_sndI,axiom,
    ! [X: produc5835291356934675326atural,Y: code_natural,Z: produc7822875418678951345atural] :
      ( ( X
        = ( produc6639722614265839536atural @ Y @ Z ) )
     => ( ( produc6591912806276919810atural @ X )
        = Z ) ) ).

% sndI
thf(fact_6515_sndE,axiom,
    ! [X: produc1908205239877642774nteger,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,P: produc8923325533196201883nteger > $o] :
      ( ( X
        = ( produc8603105652947943368nteger @ A @ B ) )
     => ( ( P @ ( produc7856867400915047194nteger @ X ) )
       => ( P @ B ) ) ) ).

% sndE
thf(fact_6516_sndE,axiom,
    ! [X: produc3925858234332021118et_nat,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,P: produc3658429121746597890et_nat > $o] :
      ( ( X
        = ( produc5001842942810119800et_nat @ A @ B ) )
     => ( ( P @ ( produc4011572625026189258et_nat @ X ) )
       => ( P @ B ) ) ) ).

% sndE
thf(fact_6517_sndE,axiom,
    ! [X: produc2732055786443039994et_nat,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,P: produc3925858234332021118et_nat > $o] :
      ( ( X
        = ( produc2245416461498447860et_nat @ A @ B ) )
     => ( ( P @ ( produc5374455773327741254et_nat @ X ) )
       => ( P @ B ) ) ) ).

% sndE
thf(fact_6518_sndE,axiom,
    ! [X: produc2285326912895808259nt_int,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( X
        = ( produc5700946648718959541nt_int @ A @ B ) )
     => ( ( P @ ( produc7328097813583171335nt_int @ X ) )
       => ( P @ B ) ) ) ).

% sndE
thf(fact_6519_sndE,axiom,
    ! [X: produc7773217078559923341nt_int,A: int > option6357759511663192854e_term,B: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( X
        = ( produc4305682042979456191nt_int @ A @ B ) )
     => ( ( P @ ( produc3162348030201620241nt_int @ X ) )
       => ( P @ B ) ) ) ).

% sndE
thf(fact_6520_sndE,axiom,
    ! [X: produc8923325533196201883nteger,A: code_integer,B: code_integer,P: code_integer > $o] :
      ( ( X
        = ( produc1086072967326762835nteger @ A @ B ) )
     => ( ( P @ ( produc6174133586879617921nteger @ X ) )
       => ( P @ B ) ) ) ).

% sndE
thf(fact_6521_sndE,axiom,
    ! [X: product_prod_int_int,A: int,B: int,P: int > $o] :
      ( ( X
        = ( product_Pair_int_int @ A @ B ) )
     => ( ( P @ ( product_snd_int_int @ X ) )
       => ( P @ B ) ) ) ).

% sndE
thf(fact_6522_sndE,axiom,
    ! [X: produc5835291356934675326atural,A: code_natural,B: produc7822875418678951345atural,P: produc7822875418678951345atural > $o] :
      ( ( X
        = ( produc6639722614265839536atural @ A @ B ) )
     => ( ( P @ ( produc6591912806276919810atural @ X ) )
       => ( P @ B ) ) ) ).

% sndE
thf(fact_6523_snd__eqD,axiom,
    ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger,A: produc8923325533196201883nteger] :
      ( ( ( produc7856867400915047194nteger @ ( produc8603105652947943368nteger @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_6524_snd__eqD,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,A: produc3658429121746597890et_nat] :
      ( ( ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_6525_snd__eqD,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,A: produc3925858234332021118et_nat] :
      ( ( ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_6526_snd__eqD,axiom,
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,A: product_prod_int_int] :
      ( ( ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_6527_snd__eqD,axiom,
    ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int,A: product_prod_int_int] :
      ( ( ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_6528_snd__eqD,axiom,
    ! [X: code_integer,Y: code_integer,A: code_integer] :
      ( ( ( produc6174133586879617921nteger @ ( produc1086072967326762835nteger @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_6529_snd__eqD,axiom,
    ! [X: int,Y: int,A: int] :
      ( ( ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_6530_snd__eqD,axiom,
    ! [X: code_natural,Y: produc7822875418678951345atural,A: produc7822875418678951345atural] :
      ( ( ( produc6591912806276919810atural @ ( produc6639722614265839536atural @ X @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_6531_snd__conv,axiom,
    ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X2: produc8923325533196201883nteger] :
      ( ( produc7856867400915047194nteger @ ( produc8603105652947943368nteger @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_6532_snd__conv,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X2: produc3658429121746597890et_nat] :
      ( ( produc4011572625026189258et_nat @ ( produc5001842942810119800et_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_6533_snd__conv,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X2: produc3925858234332021118et_nat] :
      ( ( produc5374455773327741254et_nat @ ( produc2245416461498447860et_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_6534_snd__conv,axiom,
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X2: product_prod_int_int] :
      ( ( produc7328097813583171335nt_int @ ( produc5700946648718959541nt_int @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_6535_snd__conv,axiom,
    ! [X1: int > option6357759511663192854e_term,X2: product_prod_int_int] :
      ( ( produc3162348030201620241nt_int @ ( produc4305682042979456191nt_int @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_6536_snd__conv,axiom,
    ! [X1: code_integer,X2: code_integer] :
      ( ( produc6174133586879617921nteger @ ( produc1086072967326762835nteger @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_6537_snd__conv,axiom,
    ! [X1: int,X2: int] :
      ( ( product_snd_int_int @ ( product_Pair_int_int @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_6538_snd__conv,axiom,
    ! [X1: code_natural,X2: produc7822875418678951345atural] :
      ( ( produc6591912806276919810atural @ ( produc6639722614265839536atural @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_6539_Some__Inf,axiom,
    ! [A2: set_set_nat] :
      ( ( some_set_nat @ ( comple7806235888213564991et_nat @ A2 ) )
      = ( comple3446467376476514063et_nat @ ( image_838307146230912995et_nat @ some_set_nat @ A2 ) ) ) ).

% Some_Inf
thf(fact_6540_Some__Inf,axiom,
    ! [A2: set_Product_unit] :
      ( ( some_Product_unit @ ( comple2584293577114468500t_unit @ A2 ) )
      = ( comple5564859526659248602t_unit @ ( image_7094380055901048587t_unit @ some_Product_unit @ A2 ) ) ) ).

% Some_Inf
thf(fact_6541_fstI,axiom,
    ! [X: produc1908205239877642774nteger,Y: produc6241069584506657477e_term > option6357759511663192854e_term,Z: produc8923325533196201883nteger] :
      ( ( X
        = ( produc8603105652947943368nteger @ Y @ Z ) )
     => ( ( produc7822682618958472924nteger @ X )
        = Y ) ) ).

% fstI
thf(fact_6542_fstI,axiom,
    ! [X: produc3925858234332021118et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3658429121746597890et_nat] :
      ( ( X
        = ( produc5001842942810119800et_nat @ Y @ Z ) )
     => ( ( produc995936583742144908et_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_6543_fstI,axiom,
    ! [X: produc2732055786443039994et_nat,Y: produc3658429121746597890et_nat > $o,Z: produc3925858234332021118et_nat] :
      ( ( X
        = ( produc2245416461498447860et_nat @ Y @ Z ) )
     => ( ( produc180342877477747464et_nat @ X )
        = Y ) ) ).

% fstI
thf(fact_6544_fstI,axiom,
    ! [X: produc2285326912895808259nt_int,Y: produc8551481072490612790e_term > option6357759511663192854e_term,Z: product_prod_int_int] :
      ( ( X
        = ( produc5700946648718959541nt_int @ Y @ Z ) )
     => ( ( produc6816164490631068361nt_int @ X )
        = Y ) ) ).

% fstI
thf(fact_6545_fstI,axiom,
    ! [X: produc7773217078559923341nt_int,Y: int > option6357759511663192854e_term,Z: product_prod_int_int] :
      ( ( X
        = ( produc4305682042979456191nt_int @ Y @ Z ) )
     => ( ( produc6230002227079971283nt_int @ X )
        = Y ) ) ).

% fstI
thf(fact_6546_fstI,axiom,
    ! [X: produc8923325533196201883nteger,Y: code_integer,Z: code_integer] :
      ( ( X
        = ( produc1086072967326762835nteger @ Y @ Z ) )
     => ( ( produc8508995932063986495nteger @ X )
        = Y ) ) ).

% fstI
thf(fact_6547_fstI,axiom,
    ! [X: product_prod_int_int,Y: int,Z: int] :
      ( ( X
        = ( product_Pair_int_int @ Y @ Z ) )
     => ( ( product_fst_int_int @ X )
        = Y ) ) ).

% fstI
thf(fact_6548_fstE,axiom,
    ! [X: produc1908205239877642774nteger,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > $o] :
      ( ( X
        = ( produc8603105652947943368nteger @ A @ B ) )
     => ( ( P @ ( produc7822682618958472924nteger @ X ) )
       => ( P @ A ) ) ) ).

% fstE
thf(fact_6549_fstE,axiom,
    ! [X: produc3925858234332021118et_nat,A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( X
        = ( produc5001842942810119800et_nat @ A @ B ) )
     => ( ( P @ ( produc995936583742144908et_nat @ X ) )
       => ( P @ A ) ) ) ).

% fstE
thf(fact_6550_fstE,axiom,
    ! [X: produc2732055786443039994et_nat,A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( X
        = ( produc2245416461498447860et_nat @ A @ B ) )
     => ( ( P @ ( produc180342877477747464et_nat @ X ) )
       => ( P @ A ) ) ) ).

% fstE
thf(fact_6551_fstE,axiom,
    ! [X: produc2285326912895808259nt_int,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > $o] :
      ( ( X
        = ( produc5700946648718959541nt_int @ A @ B ) )
     => ( ( P @ ( produc6816164490631068361nt_int @ X ) )
       => ( P @ A ) ) ) ).

% fstE
thf(fact_6552_fstE,axiom,
    ! [X: produc7773217078559923341nt_int,A: int > option6357759511663192854e_term,B: product_prod_int_int,P: ( int > option6357759511663192854e_term ) > $o] :
      ( ( X
        = ( produc4305682042979456191nt_int @ A @ B ) )
     => ( ( P @ ( produc6230002227079971283nt_int @ X ) )
       => ( P @ A ) ) ) ).

% fstE
thf(fact_6553_fstE,axiom,
    ! [X: produc8923325533196201883nteger,A: code_integer,B: code_integer,P: code_integer > $o] :
      ( ( X
        = ( produc1086072967326762835nteger @ A @ B ) )
     => ( ( P @ ( produc8508995932063986495nteger @ X ) )
       => ( P @ A ) ) ) ).

% fstE
thf(fact_6554_fstE,axiom,
    ! [X: product_prod_int_int,A: int,B: int,P: int > $o] :
      ( ( X
        = ( product_Pair_int_int @ A @ B ) )
     => ( ( P @ ( product_fst_int_int @ X ) )
       => ( P @ A ) ) ) ).

% fstE
thf(fact_6555_fst__eqD,axiom,
    ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger,A: produc6241069584506657477e_term > option6357759511663192854e_term] :
      ( ( ( produc7822682618958472924nteger @ ( produc8603105652947943368nteger @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_6556_fst__eqD,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,A: produc3658429121746597890et_nat > $o] :
      ( ( ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_6557_fst__eqD,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,A: produc3658429121746597890et_nat > $o] :
      ( ( ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_6558_fst__eqD,axiom,
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,A: produc8551481072490612790e_term > option6357759511663192854e_term] :
      ( ( ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_6559_fst__eqD,axiom,
    ! [X: int > option6357759511663192854e_term,Y: product_prod_int_int,A: int > option6357759511663192854e_term] :
      ( ( ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_6560_fst__eqD,axiom,
    ! [X: code_integer,Y: code_integer,A: code_integer] :
      ( ( ( produc8508995932063986495nteger @ ( produc1086072967326762835nteger @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_6561_fst__eqD,axiom,
    ! [X: int,Y: int,A: int] :
      ( ( ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) )
        = A )
     => ( X = A ) ) ).

% fst_eqD
thf(fact_6562_fst__conv,axiom,
    ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X2: produc8923325533196201883nteger] :
      ( ( produc7822682618958472924nteger @ ( produc8603105652947943368nteger @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_6563_fst__conv,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X2: produc3658429121746597890et_nat] :
      ( ( produc995936583742144908et_nat @ ( produc5001842942810119800et_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_6564_fst__conv,axiom,
    ! [X1: produc3658429121746597890et_nat > $o,X2: produc3925858234332021118et_nat] :
      ( ( produc180342877477747464et_nat @ ( produc2245416461498447860et_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_6565_fst__conv,axiom,
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X2: product_prod_int_int] :
      ( ( produc6816164490631068361nt_int @ ( produc5700946648718959541nt_int @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_6566_fst__conv,axiom,
    ! [X1: int > option6357759511663192854e_term,X2: product_prod_int_int] :
      ( ( produc6230002227079971283nt_int @ ( produc4305682042979456191nt_int @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_6567_fst__conv,axiom,
    ! [X1: code_integer,X2: code_integer] :
      ( ( produc8508995932063986495nteger @ ( produc1086072967326762835nteger @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_6568_fst__conv,axiom,
    ! [X1: int,X2: int] :
      ( ( product_fst_int_int @ ( product_Pair_int_int @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_6569_snd__fst__flip,axiom,
    ( produc1152141638515386128e_term
    = ( comp_P8650103696410638905e_term @ produc7822682618958472924nteger
      @ ( produc5068368601420144385nteger
        @ ^ [X3: produc8923325533196201883nteger,Y3: produc6241069584506657477e_term > option6357759511663192854e_term] : ( produc8603105652947943368nteger @ Y3 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_6570_snd__fst__flip,axiom,
    ( produc1838968430882301508_nat_o
    = ( comp_P1110778729668777324_nat_o @ produc995936583742144908et_nat
      @ ( produc5206072795621629085et_nat
        @ ^ [X3: produc3658429121746597890et_nat,Y3: produc3658429121746597890et_nat > $o] : ( produc5001842942810119800et_nat @ Y3 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_6571_snd__fst__flip,axiom,
    ( produc6803565743942592456_nat_o
    = ( comp_P4355506512671352436_nat_o @ produc180342877477747464et_nat
      @ ( produc9203276081649983901et_nat
        @ ^ [X3: produc3925858234332021118et_nat,Y3: produc3658429121746597890et_nat > $o] : ( produc2245416461498447860et_nat @ Y3 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_6572_snd__fst__flip,axiom,
    ( produc6847057335086834975e_term
    = ( comp_P8577347034251779110e_term @ produc6816164490631068361nt_int
      @ ( produc199769199162113789nt_int
        @ ^ [X3: product_prod_int_int,Y3: produc8551481072490612790e_term > option6357759511663192854e_term] : ( produc5700946648718959541nt_int @ Y3 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_6573_snd__fst__flip,axiom,
    ( produc6373639247523685013e_term
    = ( comp_P1875353012086385820e_term @ produc6230002227079971283nt_int
      @ ( produc3478612561300218621nt_int
        @ ^ [X3: product_prod_int_int,Y3: int > option6357759511663192854e_term] : ( produc4305682042979456191nt_int @ Y3 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_6574_snd__fst__flip,axiom,
    ( produc6174133586879617921nteger
    = ( comp_P3428181540658402820nteger @ produc8508995932063986495nteger
      @ ( produc6916734918728496179nteger
        @ ^ [X3: code_integer,Y3: code_integer] : ( produc1086072967326762835nteger @ Y3 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_6575_snd__fst__flip,axiom,
    ( produc6591912806276919810atural
    = ( comp_P3887058762840424708atural @ produc4353917038236142998atural
      @ ( produc6785097126449100389atural
        @ ^ [X3: code_natural,Y3: produc7822875418678951345atural] : ( produc1272419604612767874atural @ Y3 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_6576_snd__fst__flip,axiom,
    ( product_snd_int_int
    = ( comp_P5356232565778667509nt_int @ product_fst_int_int
      @ ( produc4245557441103728435nt_int
        @ ^ [X3: int,Y3: int] : ( product_Pair_int_int @ Y3 @ X3 ) ) ) ) ).

% snd_fst_flip
thf(fact_6577_fst__snd__flip,axiom,
    ( produc1117956856558811858e_term
    = ( comp_P1663382703091044646e_term @ produc7856867400915047194nteger
      @ ( produc5068368601420144385nteger
        @ ^ [X3: produc8923325533196201883nteger,Y3: produc6241069584506657477e_term > option6357759511663192854e_term] : ( produc8603105652947943368nteger @ Y3 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_6578_fst__snd__flip,axiom,
    ( produc8046704426453032966_nat_o
    = ( comp_P5089495125153697957_nat_o @ produc4011572625026189258et_nat
      @ ( produc5206072795621629085et_nat
        @ ^ [X3: produc3658429121746597890et_nat,Y3: produc3658429121746597890et_nat > $o] : ( produc5001842942810119800et_nat @ Y3 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_6579_fst__snd__flip,axiom,
    ( produc1609452848092598666_nat_o
    = ( comp_P9183373265949749025_nat_o @ produc5374455773327741254et_nat
      @ ( produc9203276081649983901et_nat
        @ ^ [X3: produc3925858234332021118et_nat,Y3: produc3658429121746597890et_nat > $o] : ( produc2245416461498447860et_nat @ Y3 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_6580_fst__snd__flip,axiom,
    ( produc6335124012134732001e_term
    = ( comp_P4057098934159210662e_term @ produc7328097813583171335nt_int
      @ ( produc199769199162113789nt_int
        @ ^ [X3: product_prod_int_int,Y3: produc8551481072490612790e_term > option6357759511663192854e_term] : ( produc5700946648718959541nt_int @ Y3 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_6581_fst__snd__flip,axiom,
    ( produc217921407547260247e_term
    = ( comp_P5363713469694551954e_term @ produc3162348030201620241nt_int
      @ ( produc3478612561300218621nt_int
        @ ^ [X3: product_prod_int_int,Y3: int > option6357759511663192854e_term] : ( produc4305682042979456191nt_int @ Y3 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_6582_fst__snd__flip,axiom,
    ( produc8508995932063986495nteger
    = ( comp_P3428181540658402820nteger @ produc6174133586879617921nteger
      @ ( produc6916734918728496179nteger
        @ ^ [X3: code_integer,Y3: code_integer] : ( produc1086072967326762835nteger @ Y3 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_6583_fst__snd__flip,axiom,
    ( produc4353917038236142998atural
    = ( comp_P5756991293874479000atural @ produc6591912806276919810atural
      @ ( produc1198152457851389869atural
        @ ^ [X3: produc7822875418678951345atural,Y3: code_natural] : ( produc6639722614265839536atural @ Y3 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_6584_fst__snd__flip,axiom,
    ( product_fst_int_int
    = ( comp_P5356232565778667509nt_int @ product_snd_int_int
      @ ( produc4245557441103728435nt_int
        @ ^ [X3: int,Y3: int] : ( product_Pair_int_int @ Y3 @ X3 ) ) ) ) ).

% fst_snd_flip
thf(fact_6585_finite__UNION__then__finite,axiom,
    ! [B5: $o > set_int,A2: set_o,A: $o] :
      ( ( finite_finite_int @ ( comple3221217463730067765et_int @ ( image_o_set_int @ B5 @ A2 ) ) )
     => ( ( member_o @ A @ A2 )
       => ( finite_finite_int @ ( B5 @ A ) ) ) ) ).

% finite_UNION_then_finite
thf(fact_6586_finite__UNION__then__finite,axiom,
    ! [B5: nat > set_int,A2: set_nat,A: nat] :
      ( ( finite_finite_int @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B5 @ A2 ) ) )
     => ( ( member_nat @ A @ A2 )
       => ( finite_finite_int @ ( B5 @ A ) ) ) ) ).

% finite_UNION_then_finite
thf(fact_6587_finite__UNION__then__finite,axiom,
    ! [B5: int > set_int,A2: set_int,A: int] :
      ( ( finite_finite_int @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B5 @ A2 ) ) )
     => ( ( member_int @ A @ A2 )
       => ( finite_finite_int @ ( B5 @ A ) ) ) ) ).

% finite_UNION_then_finite
thf(fact_6588_finite__UNION__then__finite,axiom,
    ! [B5: $o > set_Code_integer,A2: set_o,A: $o] :
      ( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_6491638074533506790nteger @ B5 @ A2 ) ) )
     => ( ( member_o @ A @ A2 )
       => ( finite6017078050557962740nteger @ ( B5 @ A ) ) ) ) ).

% finite_UNION_then_finite
thf(fact_6589_finite__UNION__then__finite,axiom,
    ! [B5: nat > set_Code_integer,A2: set_nat,A: nat] :
      ( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_2385391725447763818nteger @ B5 @ A2 ) ) )
     => ( ( member_nat @ A @ A2 )
       => ( finite6017078050557962740nteger @ ( B5 @ A ) ) ) ) ).

% finite_UNION_then_finite
thf(fact_6590_finite__UNION__then__finite,axiom,
    ! [B5: int > set_Code_integer,A2: set_int,A: int] :
      ( ( finite6017078050557962740nteger @ ( comple739944243200306918nteger @ ( image_75573793354812870nteger @ B5 @ A2 ) ) )
     => ( ( member_int @ A @ A2 )
       => ( finite6017078050557962740nteger @ ( B5 @ A ) ) ) ) ).

% finite_UNION_then_finite
thf(fact_6591_finite__UNION__then__finite,axiom,
    ! [B5: $o > set_nat,A2: set_o,A: $o] :
      ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B5 @ A2 ) ) )
     => ( ( member_o @ A @ A2 )
       => ( finite_finite_nat @ ( B5 @ A ) ) ) ) ).

% finite_UNION_then_finite
thf(fact_6592_finite__UNION__then__finite,axiom,
    ! [B5: nat > set_nat,A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A2 ) ) )
     => ( ( member_nat @ A @ A2 )
       => ( finite_finite_nat @ ( B5 @ A ) ) ) ) ).

% finite_UNION_then_finite
thf(fact_6593_finite__UNION__then__finite,axiom,
    ! [B5: int > set_nat,A2: set_int,A: int] :
      ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B5 @ A2 ) ) )
     => ( ( member_int @ A @ A2 )
       => ( finite_finite_nat @ ( B5 @ A ) ) ) ) ).

% finite_UNION_then_finite
thf(fact_6594_finite__UNION__then__finite,axiom,
    ! [B5: product_prod_nat_nat > set_int,A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( finite_finite_int @ ( comple3221217463730067765et_int @ ( image_5061345727057949854et_int @ B5 @ A2 ) ) )
     => ( ( member8440522571783428010at_nat @ A @ A2 )
       => ( finite_finite_int @ ( B5 @ A ) ) ) ) ).

% finite_UNION_then_finite
thf(fact_6595_case__prod__comp,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,X: produc3346328345660746739nteger] :
      ( ( produc4281629032016041417nteger @ ( comp_C1593894019821074884nteger @ F @ G ) @ X )
      = ( F @ ( G @ ( produc1861665112190112313nteger @ X ) ) @ ( produc7955729907432593271nteger @ X ) ) ) ).

% case_prod_comp
thf(fact_6596_case__prod__comp,axiom,
    ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,X: produc3346328345660746739nteger] :
      ( ( produc4281629032016041417nteger @ ( comp_C8797469213163452608nteger @ F @ G ) @ X )
      = ( F @ ( G @ ( produc1861665112190112313nteger @ X ) ) @ ( produc7955729907432593271nteger @ X ) ) ) ).

% case_prod_comp
thf(fact_6597_fn__snd__conv,axiom,
    ! [F: nat > nat] :
      ( ( ^ [X3: product_prod_nat_nat] : ( F @ ( product_snd_nat_nat @ X3 ) ) )
      = ( produc6842872674320459806at_nat
        @ ^ [Uu: nat] : F ) ) ).

% fn_snd_conv
thf(fact_6598_fn__snd__conv,axiom,
    ! [F: nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( ^ [X3: product_prod_nat_nat] : ( F @ ( product_snd_nat_nat @ X3 ) ) )
      = ( produc27273713700761075at_nat
        @ ^ [Uu: nat] : F ) ) ).

% fn_snd_conv
thf(fact_6599_fn__snd__conv,axiom,
    ! [F: nat > product_prod_nat_nat > $o] :
      ( ( ^ [X3: product_prod_nat_nat] : ( F @ ( product_snd_nat_nat @ X3 ) ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [Uu: nat] : F ) ) ).

% fn_snd_conv
thf(fact_6600_fn__snd__conv,axiom,
    ! [F: int > product_prod_int_int] :
      ( ( ^ [X3: product_prod_int_int] : ( F @ ( product_snd_int_int @ X3 ) ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Uu: int] : F ) ) ).

% fn_snd_conv
thf(fact_6601_fn__snd__conv,axiom,
    ! [F: int > int] :
      ( ( ^ [X3: product_prod_int_int] : ( F @ ( product_snd_int_int @ X3 ) ) )
      = ( produc8211389475949308722nt_int
        @ ^ [Uu: int] : F ) ) ).

% fn_snd_conv
thf(fact_6602_snd__def,axiom,
    ( produc6174133586879617921nteger
    = ( produc7469247390737344965nteger
      @ ^ [X12: code_integer,X23: code_integer] : X23 ) ) ).

% snd_def
thf(fact_6603_snd__def,axiom,
    ( produc6591912806276919810atural
    = ( produc6302656368764539796atural
      @ ^ [X12: code_natural,X23: produc7822875418678951345atural] : X23 ) ) ).

% snd_def
thf(fact_6604_snd__def,axiom,
    ( product_snd_nat_nat
    = ( produc6842872674320459806at_nat
      @ ^ [X12: nat,X23: nat] : X23 ) ) ).

% snd_def
thf(fact_6605_snd__def,axiom,
    ( product_snd_int_int
    = ( produc8211389475949308722nt_int
      @ ^ [X12: int,X23: int] : X23 ) ) ).

% snd_def
thf(fact_6606_fn__fst__conv,axiom,
    ! [F: nat > nat] :
      ( ( ^ [X3: product_prod_nat_nat] : ( F @ ( product_fst_nat_nat @ X3 ) ) )
      = ( produc6842872674320459806at_nat
        @ ^ [A5: nat,Uu: nat] : ( F @ A5 ) ) ) ).

% fn_fst_conv
thf(fact_6607_fn__fst__conv,axiom,
    ! [F: nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( ^ [X3: product_prod_nat_nat] : ( F @ ( product_fst_nat_nat @ X3 ) ) )
      = ( produc27273713700761075at_nat
        @ ^ [A5: nat,Uu: nat] : ( F @ A5 ) ) ) ).

% fn_fst_conv
thf(fact_6608_fn__fst__conv,axiom,
    ! [F: nat > product_prod_nat_nat > $o] :
      ( ( ^ [X3: product_prod_nat_nat] : ( F @ ( product_fst_nat_nat @ X3 ) ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [A5: nat,Uu: nat] : ( F @ A5 ) ) ) ).

% fn_fst_conv
thf(fact_6609_fn__fst__conv,axiom,
    ! [F: int > product_prod_int_int] :
      ( ( ^ [X3: product_prod_int_int] : ( F @ ( product_fst_int_int @ X3 ) ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,Uu: int] : ( F @ A5 ) ) ) ).

% fn_fst_conv
thf(fact_6610_fn__fst__conv,axiom,
    ! [F: int > int] :
      ( ( ^ [X3: product_prod_int_int] : ( F @ ( product_fst_int_int @ X3 ) ) )
      = ( produc8211389475949308722nt_int
        @ ^ [A5: int,Uu: int] : ( F @ A5 ) ) ) ).

% fn_fst_conv
thf(fact_6611_fst__def,axiom,
    ( produc8508995932063986495nteger
    = ( produc7469247390737344965nteger
      @ ^ [X12: code_integer,X23: code_integer] : X12 ) ) ).

% fst_def
thf(fact_6612_fst__def,axiom,
    ( product_fst_nat_nat
    = ( produc6842872674320459806at_nat
      @ ^ [X12: nat,X23: nat] : X12 ) ) ).

% fst_def
thf(fact_6613_fst__def,axiom,
    ( product_fst_int_int
    = ( produc8211389475949308722nt_int
      @ ^ [X12: int,X23: int] : X12 ) ) ).

% fst_def
thf(fact_6614_Collect__split__mono__strong,axiom,
    ! [X11: set_Code_integer,A2: set_Pr4811707699266497531nteger,Y9: set_Code_integer,P: code_integer > code_integer > $o,Q: code_integer > code_integer > $o] :
      ( ( X11
        = ( image_6316824473203919795nteger @ produc8508995932063986495nteger @ A2 ) )
     => ( ( Y9
          = ( image_6316824473203919795nteger @ produc6174133586879617921nteger @ A2 ) )
       => ( ! [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ X11 )
             => ! [Xa3: code_integer] :
                  ( ( member_Code_integer @ Xa3 @ Y9 )
                 => ( ( P @ X4 @ Xa3 )
                   => ( Q @ X4 @ Xa3 ) ) ) )
         => ( ( ord_le3725938330318615451nteger @ A2 @ ( collec7766031446232907142nteger @ ( produc2066375834425727024eger_o @ P ) ) )
           => ( ord_le3725938330318615451nteger @ A2 @ ( collec7766031446232907142nteger @ ( produc2066375834425727024eger_o @ Q ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_6615_Collect__split__mono__strong,axiom,
    ! [X11: set_int,A2: set_Pr958786334691620121nt_int,Y9: set_int,P: int > int > $o,Q: int > int > $o] :
      ( ( X11
        = ( image_5042161079198086560nt_int @ product_fst_int_int @ A2 ) )
     => ( ( Y9
          = ( image_5042161079198086560nt_int @ product_snd_int_int @ A2 ) )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ X11 )
             => ! [Xa3: int] :
                  ( ( member_int @ Xa3 @ Y9 )
                 => ( ( P @ X4 @ Xa3 )
                   => ( Q @ X4 @ Xa3 ) ) ) )
         => ( ( ord_le2843351958646193337nt_int @ A2 @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ P ) ) )
           => ( ord_le2843351958646193337nt_int @ A2 @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ Q ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_6616_Collect__split__mono__strong,axiom,
    ! [X11: set_Code_natural,A2: set_Pr2352657477744078772atural,Y9: set_Pr7564037697178849041atural,P: code_natural > produc7822875418678951345atural > $o,Q: code_natural > produc7822875418678951345atural > $o] :
      ( ( X11
        = ( image_8194678751871465133atural @ produc497848011034438852atural @ A2 ) )
     => ( ( Y9
          = ( image_3837677093542026988atural @ produc6591912806276919810atural @ A2 ) )
       => ( ! [X4: code_natural] :
              ( ( member_Code_natural @ X4 @ X11 )
             => ! [Xa3: produc7822875418678951345atural] :
                  ( ( member8280416476884352090atural @ Xa3 @ Y9 )
                 => ( ( P @ X4 @ Xa3 )
                   => ( Q @ X4 @ Xa3 ) ) ) )
         => ( ( ord_le2202040463148982292atural @ A2 @ ( collec1991479681304692691atural @ ( produc7358226366420480123ural_o @ P ) ) )
           => ( ord_le2202040463148982292atural @ A2 @ ( collec1991479681304692691atural @ ( produc7358226366420480123ural_o @ Q ) ) ) ) ) ) ) ).

% Collect_split_mono_strong
thf(fact_6617_Some__Sup,axiom,
    ! [A2: set_o] :
      ( ( A2 != bot_bot_set_o )
     => ( ( some_o @ ( complete_Sup_Sup_o @ A2 ) )
        = ( comple4490649148004324043tion_o @ ( image_o_option_o @ some_o @ A2 ) ) ) ) ).

% Some_Sup
thf(fact_6618_Some__Sup,axiom,
    ! [A2: set_set_nat] :
      ( ( A2 != bot_bot_set_set_nat )
     => ( ( some_set_nat @ ( comple7399068483239264473et_nat @ A2 ) )
        = ( comple8455683388168444585et_nat @ ( image_838307146230912995et_nat @ some_set_nat @ A2 ) ) ) ) ).

% Some_Sup
thf(fact_6619_Some__Sup,axiom,
    ! [A2: set_Product_unit] :
      ( ( A2 != bot_bo3957492148770167129t_unit )
     => ( ( some_Product_unit @ ( comple4687483117567863418t_unit @ A2 ) )
        = ( comple7481738165189995200t_unit @ ( image_7094380055901048587t_unit @ some_Product_unit @ A2 ) ) ) ) ).

% Some_Sup
thf(fact_6620_Inf__option__def,axiom,
    ( comple3446467376476514063et_nat
    = ( ^ [A7: set_option_set_nat] : ( if_option_set_nat @ ( member8989860449721436141et_nat @ none_set_nat @ A7 ) @ none_set_nat @ ( some_set_nat @ ( comple7806235888213564991et_nat @ ( these_set_nat @ A7 ) ) ) ) ) ) ).

% Inf_option_def
thf(fact_6621_Inf__option__def,axiom,
    ( comple5564859526659248602t_unit
    = ( ^ [A7: set_op3165557761946182707t_unit] : ( if_opt6227717841141446553t_unit @ ( member1378114217278917244t_unit @ none_Product_unit @ A7 ) @ none_Product_unit @ ( some_Product_unit @ ( comple2584293577114468500t_unit @ ( these_Product_unit @ A7 ) ) ) ) ) ) ).

% Inf_option_def
thf(fact_6622_Some__INF,axiom,
    ! [F: nat > set_nat,A2: set_nat] :
      ( ( some_set_nat @ ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ F @ A2 ) ) )
      = ( comple3446467376476514063et_nat
        @ ( image_1085372205585808685et_nat
          @ ^ [X3: nat] : ( some_set_nat @ ( F @ X3 ) )
          @ A2 ) ) ) ).

% Some_INF
thf(fact_6623_INF__image,axiom,
    ! [G: code_integer > nat,F: nat > code_integer,A2: set_nat] :
      ( ( complete_Inf_Inf_nat @ ( image_951025933927791156er_nat @ G @ ( image_1215581382706833972nteger @ F @ A2 ) ) )
      = ( complete_Inf_Inf_nat @ ( image_nat_nat @ ( comp_C4049556595663050210at_nat @ G @ F ) @ A2 ) ) ) ).

% INF_image
thf(fact_6624_INF__image,axiom,
    ! [G: code_integer > int,F: nat > code_integer,A2: set_nat] :
      ( ( complete_Inf_Inf_int @ ( image_948535463418740880er_int @ G @ ( image_1215581382706833972nteger @ F @ A2 ) ) )
      = ( complete_Inf_Inf_int @ ( image_nat_int @ ( comp_C5048910307564104766nt_nat @ G @ F ) @ A2 ) ) ) ).

% INF_image
thf(fact_6625_INF__image,axiom,
    ! [G: int > nat,F: nat > int,A2: set_nat] :
      ( ( complete_Inf_Inf_nat @ ( image_int_nat @ G @ ( image_nat_int @ F @ A2 ) ) )
      = ( complete_Inf_Inf_nat @ ( image_nat_nat @ ( comp_int_nat_nat @ G @ F ) @ A2 ) ) ) ).

% INF_image
thf(fact_6626_INF__image,axiom,
    ! [G: nat > nat,F: nat > nat,A2: set_nat] :
      ( ( complete_Inf_Inf_nat @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A2 ) ) )
      = ( complete_Inf_Inf_nat @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A2 ) ) ) ).

% INF_image
thf(fact_6627_INF__image,axiom,
    ! [G: nat > int,F: int > nat,A2: set_int] :
      ( ( complete_Inf_Inf_int @ ( image_nat_int @ G @ ( image_int_nat @ F @ A2 ) ) )
      = ( complete_Inf_Inf_int @ ( image_int_int @ ( comp_nat_int_int @ G @ F ) @ A2 ) ) ) ).

% INF_image
thf(fact_6628_INF__image,axiom,
    ! [G: nat > int,F: nat > nat,A2: set_nat] :
      ( ( complete_Inf_Inf_int @ ( image_nat_int @ G @ ( image_nat_nat @ F @ A2 ) ) )
      = ( complete_Inf_Inf_int @ ( image_nat_int @ ( comp_nat_int_nat @ G @ F ) @ A2 ) ) ) ).

% INF_image
thf(fact_6629_INF__image,axiom,
    ! [G: int > int,F: nat > int,A2: set_nat] :
      ( ( complete_Inf_Inf_int @ ( image_int_int @ G @ ( image_nat_int @ F @ A2 ) ) )
      = ( complete_Inf_Inf_int @ ( image_nat_int @ ( comp_int_int_nat @ G @ F ) @ A2 ) ) ) ).

% INF_image
thf(fact_6630_INF__image,axiom,
    ! [G: int > int,F: int > int,A2: set_int] :
      ( ( complete_Inf_Inf_int @ ( image_int_int @ G @ ( image_int_int @ F @ A2 ) ) )
      = ( complete_Inf_Inf_int @ ( image_int_int @ ( comp_int_int_int @ G @ F ) @ A2 ) ) ) ).

% INF_image
thf(fact_6631_INF__image,axiom,
    ! [G: code_integer > product_unit,F: nat > code_integer,A2: set_nat] :
      ( ( comple2584293577114468500t_unit @ ( image_7182656606676592809t_unit @ G @ ( image_1215581382706833972nteger @ F @ A2 ) ) )
      = ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ ( comp_C8086623346886033437it_nat @ G @ F ) @ A2 ) ) ) ).

% INF_image
thf(fact_6632_INF__image,axiom,
    ! [G: nat > product_unit,F: nat > nat,A2: set_nat] :
      ( ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ G @ ( image_nat_nat @ F @ A2 ) ) )
      = ( comple2584293577114468500t_unit @ ( image_8730104196221521654t_unit @ ( comp_n9025697279074352746it_nat @ G @ F ) @ A2 ) ) ) ).

% INF_image
thf(fact_6633_SUP__image,axiom,
    ! [G: code_integer > nat,F: nat > code_integer,A2: set_nat] :
      ( ( complete_Sup_Sup_nat @ ( image_951025933927791156er_nat @ G @ ( image_1215581382706833972nteger @ F @ A2 ) ) )
      = ( complete_Sup_Sup_nat @ ( image_nat_nat @ ( comp_C4049556595663050210at_nat @ G @ F ) @ A2 ) ) ) ).

% SUP_image
thf(fact_6634_SUP__image,axiom,
    ! [G: code_integer > int,F: nat > code_integer,A2: set_nat] :
      ( ( complete_Sup_Sup_int @ ( image_948535463418740880er_int @ G @ ( image_1215581382706833972nteger @ F @ A2 ) ) )
      = ( complete_Sup_Sup_int @ ( image_nat_int @ ( comp_C5048910307564104766nt_nat @ G @ F ) @ A2 ) ) ) ).

% SUP_image
thf(fact_6635_SUP__image,axiom,
    ! [G: int > nat,F: nat > int,A2: set_nat] :
      ( ( complete_Sup_Sup_nat @ ( image_int_nat @ G @ ( image_nat_int @ F @ A2 ) ) )
      = ( complete_Sup_Sup_nat @ ( image_nat_nat @ ( comp_int_nat_nat @ G @ F ) @ A2 ) ) ) ).

% SUP_image
thf(fact_6636_SUP__image,axiom,
    ! [G: nat > nat,F: nat > nat,A2: set_nat] :
      ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A2 ) ) )
      = ( complete_Sup_Sup_nat @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A2 ) ) ) ).

% SUP_image
thf(fact_6637_SUP__image,axiom,
    ! [G: nat > int,F: int > nat,A2: set_int] :
      ( ( complete_Sup_Sup_int @ ( image_nat_int @ G @ ( image_int_nat @ F @ A2 ) ) )
      = ( complete_Sup_Sup_int @ ( image_int_int @ ( comp_nat_int_int @ G @ F ) @ A2 ) ) ) ).

% SUP_image
thf(fact_6638_SUP__image,axiom,
    ! [G: nat > int,F: nat > nat,A2: set_nat] :
      ( ( complete_Sup_Sup_int @ ( image_nat_int @ G @ ( image_nat_nat @ F @ A2 ) ) )
      = ( complete_Sup_Sup_int @ ( image_nat_int @ ( comp_nat_int_nat @ G @ F ) @ A2 ) ) ) ).

% SUP_image
thf(fact_6639_SUP__image,axiom,
    ! [G: int > int,F: nat > int,A2: set_nat] :
      ( ( complete_Sup_Sup_int @ ( image_int_int @ G @ ( image_nat_int @ F @ A2 ) ) )
      = ( complete_Sup_Sup_int @ ( image_nat_int @ ( comp_int_int_nat @ G @ F ) @ A2 ) ) ) ).

% SUP_image
thf(fact_6640_SUP__image,axiom,
    ! [G: int > int,F: int > int,A2: set_int] :
      ( ( complete_Sup_Sup_int @ ( image_int_int @ G @ ( image_int_int @ F @ A2 ) ) )
      = ( complete_Sup_Sup_int @ ( image_int_int @ ( comp_int_int_int @ G @ F ) @ A2 ) ) ) ).

% SUP_image
thf(fact_6641_SUP__image,axiom,
    ! [G: code_integer > product_unit,F: nat > code_integer,A2: set_nat] :
      ( ( comple4687483117567863418t_unit @ ( image_7182656606676592809t_unit @ G @ ( image_1215581382706833972nteger @ F @ A2 ) ) )
      = ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ ( comp_C8086623346886033437it_nat @ G @ F ) @ A2 ) ) ) ).

% SUP_image
thf(fact_6642_SUP__image,axiom,
    ! [G: nat > product_unit,F: nat > nat,A2: set_nat] :
      ( ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ G @ ( image_nat_nat @ F @ A2 ) ) )
      = ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ ( comp_n9025697279074352746it_nat @ G @ F ) @ A2 ) ) ) ).

% SUP_image
thf(fact_6643_in__snd__imageE,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ Y @ ( image_7979982168039674559at_nat @ produc2937545259423399429at_nat @ S2 ) )
     => ~ ! [X4: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y ) @ S2 ) ) ).

% in_snd_imageE
thf(fact_6644_in__snd__imageE,axiom,
    ! [Y: nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ Y @ ( image_2486076414777270412at_nat @ product_snd_nat_nat @ S2 ) )
     => ~ ! [X4: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y ) @ S2 ) ) ).

% in_snd_imageE
thf(fact_6645_in__snd__imageE,axiom,
    ! [Y: produc8923325533196201883nteger,S2: set_Pr1281608226676607948nteger] :
      ( ( member157494554546826820nteger @ Y @ ( image_3141030398089242942nteger @ produc7856867400915047194nteger @ S2 ) )
     => ~ ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term] :
            ~ ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y ) @ S2 ) ) ).

% in_snd_imageE
thf(fact_6646_in__snd__imageE,axiom,
    ! [Y: produc3658429121746597890et_nat,S2: set_Pr3286484037609594932et_nat] :
      ( ( member6260224972018164377et_nat @ Y @ ( image_4390311411407041927et_nat @ produc4011572625026189258et_nat @ S2 ) )
     => ~ ! [X4: produc3658429121746597890et_nat > $o] :
            ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X4 @ Y ) @ S2 ) ) ).

% in_snd_imageE
thf(fact_6647_in__snd__imageE,axiom,
    ! [Y: produc3925858234332021118et_nat,S2: set_Pr8536935166611901872et_nat] :
      ( ( member1996754912294343701et_nat @ Y @ ( image_4557310310409863807et_nat @ produc5374455773327741254et_nat @ S2 ) )
     => ~ ! [X4: produc3658429121746597890et_nat > $o] :
            ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X4 @ Y ) @ S2 ) ) ).

% in_snd_imageE
thf(fact_6648_in__snd__imageE,axiom,
    ! [Y: product_prod_int_int,S2: set_Pr9222295170931077689nt_int] :
      ( ( member5262025264175285858nt_int @ Y @ ( image_3915291023682402351nt_int @ produc7328097813583171335nt_int @ S2 ) )
     => ~ ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term] :
            ~ ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y ) @ S2 ) ) ).

% in_snd_imageE
thf(fact_6649_in__snd__imageE,axiom,
    ! [Y: product_prod_int_int,S2: set_Pr1872883991513573699nt_int] :
      ( ( member5262025264175285858nt_int @ Y @ ( image_3108077152341812517nt_int @ produc3162348030201620241nt_int @ S2 ) )
     => ~ ! [X4: int > option6357759511663192854e_term] :
            ~ ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y ) @ S2 ) ) ).

% in_snd_imageE
thf(fact_6650_in__snd__imageE,axiom,
    ! [Y: code_integer,S2: set_Pr4811707699266497531nteger] :
      ( ( member_Code_integer @ Y @ ( image_6316824473203919795nteger @ produc6174133586879617921nteger @ S2 ) )
     => ~ ! [X4: code_integer] :
            ~ ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X4 @ Y ) @ S2 ) ) ).

% in_snd_imageE
thf(fact_6651_in__snd__imageE,axiom,
    ! [Y: int,S2: set_Pr958786334691620121nt_int] :
      ( ( member_int @ Y @ ( image_5042161079198086560nt_int @ product_snd_int_int @ S2 ) )
     => ~ ! [X4: int] :
            ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y ) @ S2 ) ) ).

% in_snd_imageE
thf(fact_6652_in__snd__imageE,axiom,
    ! [Y: produc7822875418678951345atural,S2: set_Pr2352657477744078772atural] :
      ( ( member8280416476884352090atural @ Y @ ( image_3837677093542026988atural @ produc6591912806276919810atural @ S2 ) )
     => ~ ! [X4: code_natural] :
            ~ ( member2108296798382401941atural @ ( produc6639722614265839536atural @ X4 @ Y ) @ S2 ) ) ).

% in_snd_imageE
thf(fact_6653_in__fst__imageE,axiom,
    ! [X: set_Pr1261947904930325089at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ X @ ( image_7979982168039674559at_nat @ produc6011863565407846595at_nat @ S2 ) )
     => ~ ! [Y5: set_Pr1261947904930325089at_nat] :
            ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y5 ) @ S2 ) ) ).

% in_fst_imageE
thf(fact_6654_in__fst__imageE,axiom,
    ! [X: nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ X @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ S2 ) )
     => ~ ! [Y5: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y5 ) @ S2 ) ) ).

% in_fst_imageE
thf(fact_6655_in__fst__imageE,axiom,
    ! [X: produc6241069584506657477e_term > option6357759511663192854e_term,S2: set_Pr1281608226676607948nteger] :
      ( ( member4242434998011752849e_term @ X @ ( image_4436781500471753239e_term @ produc7822682618958472924nteger @ S2 ) )
     => ~ ! [Y5: produc8923325533196201883nteger] :
            ~ ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y5 ) @ S2 ) ) ).

% in_fst_imageE
thf(fact_6656_in__fst__imageE,axiom,
    ! [X: produc3658429121746597890et_nat > $o,S2: set_Pr3286484037609594932et_nat] :
      ( ( member6576561426505652726_nat_o @ X @ ( image_6802380912191114376_nat_o @ produc995936583742144908et_nat @ S2 ) )
     => ~ ! [Y5: produc3658429121746597890et_nat] :
            ~ ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y5 ) @ S2 ) ) ).

% in_fst_imageE
thf(fact_6657_in__fst__imageE,axiom,
    ! [X: produc3658429121746597890et_nat > $o,S2: set_Pr8536935166611901872et_nat] :
      ( ( member6576561426505652726_nat_o @ X @ ( image_2517677226577294348_nat_o @ produc180342877477747464et_nat @ S2 ) )
     => ~ ! [Y5: produc3925858234332021118et_nat] :
            ~ ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y5 ) @ S2 ) ) ).

% in_fst_imageE
thf(fact_6658_in__fst__imageE,axiom,
    ! [X: produc8551481072490612790e_term > option6357759511663192854e_term,S2: set_Pr9222295170931077689nt_int] :
      ( ( member3222579708246209666e_term @ X @ ( image_142517975987106933e_term @ produc6816164490631068361nt_int @ S2 ) )
     => ~ ! [Y5: product_prod_int_int] :
            ~ ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y5 ) @ S2 ) ) ).

% in_fst_imageE
thf(fact_6659_in__fst__imageE,axiom,
    ! [X: int > option6357759511663192854e_term,S2: set_Pr1872883991513573699nt_int] :
      ( ( member8845023287901829240e_term @ X @ ( image_4217645046469565813e_term @ produc6230002227079971283nt_int @ S2 ) )
     => ~ ! [Y5: product_prod_int_int] :
            ~ ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y5 ) @ S2 ) ) ).

% in_fst_imageE
thf(fact_6660_in__fst__imageE,axiom,
    ! [X: code_integer,S2: set_Pr4811707699266497531nteger] :
      ( ( member_Code_integer @ X @ ( image_6316824473203919795nteger @ produc8508995932063986495nteger @ S2 ) )
     => ~ ! [Y5: code_integer] :
            ~ ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y5 ) @ S2 ) ) ).

% in_fst_imageE
thf(fact_6661_in__fst__imageE,axiom,
    ! [X: int,S2: set_Pr958786334691620121nt_int] :
      ( ( member_int @ X @ ( image_5042161079198086560nt_int @ product_fst_int_int @ S2 ) )
     => ~ ! [Y5: int] :
            ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y5 ) @ S2 ) ) ).

% in_fst_imageE
thf(fact_6662_Some__SUP,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > set_nat] :
      ( ( A2 != bot_bo228742789529271731at_nat )
     => ( ( some_set_nat @ ( comple7399068483239264473et_nat @ ( image_3160451284896210088et_nat @ F @ A2 ) ) )
        = ( comple8455683388168444585et_nat
          @ ( image_934083234596377592et_nat
            @ ^ [X3: produc3843707927480180839at_nat] : ( some_set_nat @ ( F @ X3 ) )
            @ A2 ) ) ) ) ).

% Some_SUP
thf(fact_6663_Some__SUP,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > set_nat] :
      ( ( A2 != bot_bo2099793752762293965at_nat )
     => ( ( some_set_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ F @ A2 ) ) )
        = ( comple8455683388168444585et_nat
          @ ( image_7383884694314771602et_nat
            @ ^ [X3: product_prod_nat_nat] : ( some_set_nat @ ( F @ X3 ) )
            @ A2 ) ) ) ) ).

% Some_SUP
thf(fact_6664_Some__SUP,axiom,
    ! [A2: set_o,F: $o > set_nat] :
      ( ( A2 != bot_bot_set_o )
     => ( ( some_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A2 ) ) )
        = ( comple8455683388168444585et_nat
          @ ( image_966425688229763753et_nat
            @ ^ [X3: $o] : ( some_set_nat @ ( F @ X3 ) )
            @ A2 ) ) ) ) ).

% Some_SUP
thf(fact_6665_Some__SUP,axiom,
    ! [A2: set_nat,F: nat > set_nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ( some_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A2 ) ) )
        = ( comple8455683388168444585et_nat
          @ ( image_1085372205585808685et_nat
            @ ^ [X3: nat] : ( some_set_nat @ ( F @ X3 ) )
            @ A2 ) ) ) ) ).

% Some_SUP
thf(fact_6666_Some__SUP,axiom,
    ! [A2: set_int,F: int > set_nat] :
      ( ( A2 != bot_bot_set_int )
     => ( ( some_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A2 ) ) )
        = ( comple8455683388168444585et_nat
          @ ( image_2293489338514188681et_nat
            @ ^ [X3: int] : ( some_set_nat @ ( F @ X3 ) )
            @ A2 ) ) ) ) ).

% Some_SUP
thf(fact_6667_Some__SUP,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > product_unit] :
      ( ( A2 != bot_bo228742789529271731at_nat )
     => ( ( some_Product_unit @ ( comple4687483117567863418t_unit @ ( image_7082456443403096939t_unit @ F @ A2 ) ) )
        = ( comple7481738165189995200t_unit
          @ ( image_8585598761158579761t_unit
            @ ^ [X3: produc3843707927480180839at_nat] : ( some_Product_unit @ ( F @ X3 ) )
            @ A2 ) ) ) ) ).

% Some_SUP
thf(fact_6668_Some__SUP,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_unit] :
      ( ( A2 != bot_bo2099793752762293965at_nat )
     => ( ( some_Product_unit @ ( comple4687483117567863418t_unit @ ( image_6169909464789311569t_unit @ F @ A2 ) ) )
        = ( comple7481738165189995200t_unit
          @ ( image_7548311901266006551t_unit
            @ ^ [X3: product_prod_nat_nat] : ( some_Product_unit @ ( F @ X3 ) )
            @ A2 ) ) ) ) ).

% Some_SUP
thf(fact_6669_Some__SUP,axiom,
    ! [A2: set_o,F: $o > product_unit] :
      ( ( A2 != bot_bot_set_o )
     => ( ( some_Product_unit @ ( comple4687483117567863418t_unit @ ( image_o_Product_unit @ F @ A2 ) ) )
        = ( comple7481738165189995200t_unit
          @ ( image_4415570909310888640t_unit
            @ ^ [X3: $o] : ( some_Product_unit @ ( F @ X3 ) )
            @ A2 ) ) ) ) ).

% Some_SUP
thf(fact_6670_Some__SUP,axiom,
    ! [A2: set_nat,F: nat > product_unit] :
      ( ( A2 != bot_bot_set_nat )
     => ( ( some_Product_unit @ ( comple4687483117567863418t_unit @ ( image_8730104196221521654t_unit @ F @ A2 ) ) )
        = ( comple7481738165189995200t_unit
          @ ( image_1497621718204147004t_unit
            @ ^ [X3: nat] : ( some_Product_unit @ ( F @ X3 ) )
            @ A2 ) ) ) ) ).

% Some_SUP
thf(fact_6671_Some__SUP,axiom,
    ! [A2: set_int,F: int > product_unit] :
      ( ( A2 != bot_bot_set_int )
     => ( ( some_Product_unit @ ( comple4687483117567863418t_unit @ ( image_5846526143423701914t_unit @ F @ A2 ) ) )
        = ( comple7481738165189995200t_unit
          @ ( image_1735198859854722016t_unit
            @ ^ [X3: int] : ( some_Product_unit @ ( F @ X3 ) )
            @ A2 ) ) ) ) ).

% Some_SUP
thf(fact_6672_finite__range__prod,axiom,
    ! [F: nat > product_prod_nat_nat] :
      ( ( finite_finite_nat @ ( image_nat_nat @ ( comp_P1955610211387027840at_nat @ product_fst_nat_nat @ F ) @ top_top_set_nat ) )
     => ( ( finite_finite_nat @ ( image_nat_nat @ ( comp_P1955610211387027840at_nat @ product_snd_nat_nat @ F ) @ top_top_set_nat ) )
       => ( finite6177210948735845034at_nat @ ( image_5846123807819985514at_nat @ F @ top_top_set_nat ) ) ) ) ).

% finite_range_prod
thf(fact_6673_finite__range__prod,axiom,
    ! [F: nat > product_prod_nat_int] :
      ( ( finite_finite_nat @ ( image_nat_nat @ ( comp_P7964419862382602716at_nat @ product_fst_nat_int @ F ) @ top_top_set_nat ) )
     => ( ( finite_finite_int @ ( image_nat_int @ ( comp_P8963773574283657272nt_nat @ product_snd_nat_int @ F ) @ top_top_set_nat ) )
       => ( finite1999359929226648326at_int @ ( image_1668272788310788806at_int @ F @ top_top_set_nat ) ) ) ) ).

% finite_range_prod
thf(fact_6674_finite__range__prod,axiom,
    ! [F: nat > produc2591615631378489102nteger] :
      ( ( finite_finite_nat @ ( image_nat_nat @ ( comp_P9144323761349829517at_nat @ produc8897218493789310642nteger @ F ) @ top_top_set_nat ) )
     => ( ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ ( comp_P5461341465284484122er_nat @ produc1781891006948753140nteger @ F ) @ top_top_set_nat ) )
       => ( finite1154042800025562295nteger @ ( image_8507997478476487799nteger @ F @ top_top_set_nat ) ) ) ) ).

% finite_range_prod
thf(fact_6675_finite__range__prod,axiom,
    ! [F: nat > product_prod_int_nat] :
      ( ( finite_finite_int @ ( image_nat_int @ ( comp_P645145991195131448nt_nat @ product_fst_int_nat @ F ) @ top_top_set_nat ) )
     => ( ( finite_finite_nat @ ( image_nat_nat @ ( comp_P8869164316148852700at_nat @ product_snd_int_nat @ F ) @ top_top_set_nat ) )
       => ( finite7176564660636899590nt_nat @ ( image_6845477519721040070nt_nat @ F @ top_top_set_nat ) ) ) ) ).

% finite_range_prod
thf(fact_6676_finite__range__prod,axiom,
    ! [F: nat > produc2963269191615333738nteger] :
      ( ( finite_finite_int @ ( image_nat_int @ ( comp_P59025373045620805nt_nat @ produc45500017171379470nteger @ F ) @ top_top_set_nat ) )
     => ( ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ ( comp_P6870028373966156406er_nat @ produc2153544567185597776nteger @ F ) @ top_top_set_nat ) )
       => ( finite1525696360262406931nteger @ ( image_8879651038713332435nteger @ F @ top_top_set_nat ) ) ) ) ).

% finite_range_prod
thf(fact_6677_finite__range__prod,axiom,
    ! [F: nat > produc6433947425880249742er_nat] :
      ( ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ ( comp_P5700178199414968986er_nat @ produc8632663045010267826er_nat @ F ) @ top_top_set_nat ) )
     => ( ( finite_finite_nat @ ( image_nat_nat @ ( comp_P1337767177661389325at_nat @ produc1517335558169710324er_nat @ F ) @ top_top_set_nat ) )
       => ( finite4996374594527322935er_nat @ ( image_3126957236123472631er_nat @ F @ top_top_set_nat ) ) ) ) ).

% finite_range_prod
thf(fact_6678_finite__range__prod,axiom,
    ! [F: nat > produc2256096406371053034er_int] :
      ( ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ ( comp_P2475326853202991862er_nat @ produc8630172574501217550er_int @ F ) @ top_top_set_nat ) )
     => ( ( finite_finite_int @ ( image_nat_int @ ( comp_P8345930540558018757nt_nat @ produc1514845087660660048er_int @ F ) @ top_top_set_nat ) )
       => ( finite818523575018126227er_int @ ( image_8172478253469051731er_int @ F @ top_top_set_nat ) ) ) ) ).

% finite_range_prod
thf(fact_6679_finite__range__prod,axiom,
    ! [F: nat > produc8923325533196201883nteger] :
      ( ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ ( comp_P3109732533133679783er_nat @ produc8508995932063986495nteger @ F ) @ top_top_set_nat ) )
     => ( ( finite6017078050557962740nteger @ ( image_1215581382706833972nteger @ ( comp_P3109732533133679783er_nat @ produc6174133586879617921nteger @ F ) @ top_top_set_nat ) )
       => ( finite2154694888287295812nteger @ ( image_4272405876590741764nteger @ F @ top_top_set_nat ) ) ) ) ).

% finite_range_prod
thf(fact_6680_finite__range__prod,axiom,
    ! [F: nat > product_prod_int_int] :
      ( ( finite_finite_int @ ( image_nat_int @ ( comp_P6653955642190706324nt_nat @ product_fst_int_int @ F ) @ top_top_set_nat ) )
     => ( ( finite_finite_int @ ( image_nat_int @ ( comp_P6653955642190706324nt_nat @ product_snd_int_int @ F ) @ top_top_set_nat ) )
       => ( finite2998713641127702882nt_int @ ( image_2667626500211843362nt_int @ F @ top_top_set_nat ) ) ) ) ).

% finite_range_prod
thf(fact_6681_finite__range__prod,axiom,
    ! [F: product_unit > product_prod_nat_nat] :
      ( ( finite_finite_nat @ ( image_875570014554754200it_nat @ ( comp_P2980510830910040157t_unit @ product_fst_nat_nat @ F ) @ top_to1996260823553986621t_unit ) )
     => ( ( finite_finite_nat @ ( image_875570014554754200it_nat @ ( comp_P2980510830910040157t_unit @ product_snd_nat_nat @ F ) @ top_to1996260823553986621t_unit ) )
       => ( finite6177210948735845034at_nat @ ( image_7281959405765017145at_nat @ F @ top_to1996260823553986621t_unit ) ) ) ) ).

% finite_range_prod
thf(fact_6682_UN__atMost__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_atMost_UNIV
thf(fact_6683_UN__lessThan__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_lessThan_UNIV
thf(fact_6684_Sup__option__def,axiom,
    ( comple8455683388168444585et_nat
    = ( ^ [A7: set_option_set_nat] :
          ( if_option_set_nat
          @ ( ( A7 = bot_bo5650944848895983264et_nat )
            | ( A7
              = ( insert6281397740311767046et_nat @ none_set_nat @ bot_bo5650944848895983264et_nat ) ) )
          @ none_set_nat
          @ ( some_set_nat @ ( comple7399068483239264473et_nat @ ( these_set_nat @ A7 ) ) ) ) ) ) ).

% Sup_option_def
thf(fact_6685_Sup__option__def,axiom,
    ( comple7481738165189995200t_unit
    = ( ^ [A7: set_op3165557761946182707t_unit] :
          ( if_opt6227717841141446553t_unit
          @ ( ( A7 = bot_bo2792743115866486943t_unit )
            | ( A7
              = ( insert2866144167138965411t_unit @ none_Product_unit @ bot_bo2792743115866486943t_unit ) ) )
          @ none_Product_unit
          @ ( some_Product_unit @ ( comple4687483117567863418t_unit @ ( these_Product_unit @ A7 ) ) ) ) ) ) ).

% Sup_option_def
thf(fact_6686_snd__image__mp,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B5: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2077887516847798113at_nat @ ( image_7979982168039674559at_nat @ produc2937545259423399429at_nat @ A2 ) @ B5 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ A2 )
       => ( member2643936169264416010at_nat @ Y @ B5 ) ) ) ).

% snd_image_mp
thf(fact_6687_snd__image__mp,axiom,
    ! [A2: set_Pr1281608226676607948nteger,B5: set_Pr4811707699266497531nteger,X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
      ( ( ord_le3725938330318615451nteger @ ( image_3141030398089242942nteger @ produc7856867400915047194nteger @ A2 ) @ B5 )
     => ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ A2 )
       => ( member157494554546826820nteger @ Y @ B5 ) ) ) ).

% snd_image_mp
thf(fact_6688_snd__image__mp,axiom,
    ! [A2: set_Pr3286484037609594932et_nat,B5: set_Pr3948176798113811640et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
      ( ( ord_le8090042543978450200et_nat @ ( image_4390311411407041927et_nat @ produc4011572625026189258et_nat @ A2 ) @ B5 )
     => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ A2 )
       => ( member6260224972018164377et_nat @ Y @ B5 ) ) ) ).

% snd_image_mp
thf(fact_6689_snd__image__mp,axiom,
    ! [A2: set_Pr8536935166611901872et_nat,B5: set_Pr3286484037609594932et_nat,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
      ( ( ord_le5966269811547037844et_nat @ ( image_4557310310409863807et_nat @ produc5374455773327741254et_nat @ A2 ) @ B5 )
     => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ A2 )
       => ( member1996754912294343701et_nat @ Y @ B5 ) ) ) ).

% snd_image_mp
thf(fact_6690_snd__image__mp,axiom,
    ! [A2: set_Pr9222295170931077689nt_int,B5: set_Pr958786334691620121nt_int,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( ord_le2843351958646193337nt_int @ ( image_3915291023682402351nt_int @ produc7328097813583171335nt_int @ A2 ) @ B5 )
     => ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ A2 )
       => ( member5262025264175285858nt_int @ Y @ B5 ) ) ) ).

% snd_image_mp
thf(fact_6691_snd__image__mp,axiom,
    ! [A2: set_Pr1872883991513573699nt_int,B5: set_Pr958786334691620121nt_int,X: int > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( ord_le2843351958646193337nt_int @ ( image_3108077152341812517nt_int @ produc3162348030201620241nt_int @ A2 ) @ B5 )
     => ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ A2 )
       => ( member5262025264175285858nt_int @ Y @ B5 ) ) ) ).

% snd_image_mp
thf(fact_6692_snd__image__mp,axiom,
    ! [A2: set_Pr4811707699266497531nteger,B5: set_Code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( image_6316824473203919795nteger @ produc6174133586879617921nteger @ A2 ) @ B5 )
     => ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y ) @ A2 )
       => ( member_Code_integer @ Y @ B5 ) ) ) ).

% snd_image_mp
thf(fact_6693_snd__image__mp,axiom,
    ! [A2: set_Pr958786334691620121nt_int,B5: set_int,X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ product_snd_int_int @ A2 ) @ B5 )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ A2 )
       => ( member_int @ Y @ B5 ) ) ) ).

% snd_image_mp
thf(fact_6694_snd__image__mp,axiom,
    ! [A2: set_Pr2352657477744078772atural,B5: set_Pr7564037697178849041atural,X: code_natural,Y: produc7822875418678951345atural] :
      ( ( ord_le6478268328230966961atural @ ( image_3837677093542026988atural @ produc6591912806276919810atural @ A2 ) @ B5 )
     => ( ( member2108296798382401941atural @ ( produc6639722614265839536atural @ X @ Y ) @ A2 )
       => ( member8280416476884352090atural @ Y @ B5 ) ) ) ).

% snd_image_mp
thf(fact_6695_snd__image__mp,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B5: set_nat,X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ product_snd_nat_nat @ A2 ) @ B5 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ A2 )
       => ( member_nat @ Y @ B5 ) ) ) ).

% snd_image_mp
thf(fact_6696_fst__image__mp,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B5: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2077887516847798113at_nat @ ( image_7979982168039674559at_nat @ produc6011863565407846595at_nat @ A2 ) @ B5 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ A2 )
       => ( member2643936169264416010at_nat @ X @ B5 ) ) ) ).

% fst_image_mp
thf(fact_6697_fst__image__mp,axiom,
    ! [A2: set_Pr1281608226676607948nteger,B5: set_Pr7604974323444597168e_term,X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
      ( ( ord_le7106919388332126736e_term @ ( image_4436781500471753239e_term @ produc7822682618958472924nteger @ A2 ) @ B5 )
     => ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ A2 )
       => ( member4242434998011752849e_term @ X @ B5 ) ) ) ).

% fst_image_mp
thf(fact_6698_fst__image__mp,axiom,
    ! [A2: set_Pr3286484037609594932et_nat,B5: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
      ( ( ord_le2965882846123202637_nat_o @ ( image_6802380912191114376_nat_o @ produc995936583742144908et_nat @ A2 ) @ B5 )
     => ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ A2 )
       => ( member6576561426505652726_nat_o @ X @ B5 ) ) ) ).

% fst_image_mp
thf(fact_6699_fst__image__mp,axiom,
    ! [A2: set_Pr8536935166611901872et_nat,B5: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
      ( ( ord_le2965882846123202637_nat_o @ ( image_2517677226577294348_nat_o @ produc180342877477747464et_nat @ A2 ) @ B5 )
     => ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ A2 )
       => ( member6576561426505652726_nat_o @ X @ B5 ) ) ) ).

% fst_image_mp
thf(fact_6700_fst__image__mp,axiom,
    ! [A2: set_Pr9222295170931077689nt_int,B5: set_Pr3642885161833720865e_term,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( ord_le4129093240937979009e_term @ ( image_142517975987106933e_term @ produc6816164490631068361nt_int @ A2 ) @ B5 )
     => ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ A2 )
       => ( member3222579708246209666e_term @ X @ B5 ) ) ) ).

% fst_image_mp
thf(fact_6701_fst__image__mp,axiom,
    ! [A2: set_Pr1872883991513573699nt_int,B5: set_in3461395444621081367e_term,X: int > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( ord_le567748327753481079e_term @ ( image_4217645046469565813e_term @ produc6230002227079971283nt_int @ A2 ) @ B5 )
     => ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ A2 )
       => ( member8845023287901829240e_term @ X @ B5 ) ) ) ).

% fst_image_mp
thf(fact_6702_fst__image__mp,axiom,
    ! [A2: set_Pr4811707699266497531nteger,B5: set_Code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( image_6316824473203919795nteger @ produc8508995932063986495nteger @ A2 ) @ B5 )
     => ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y ) @ A2 )
       => ( member_Code_integer @ X @ B5 ) ) ) ).

% fst_image_mp
thf(fact_6703_fst__image__mp,axiom,
    ! [A2: set_Pr958786334691620121nt_int,B5: set_int,X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ product_fst_int_int @ A2 ) @ B5 )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ A2 )
       => ( member_int @ X @ B5 ) ) ) ).

% fst_image_mp
thf(fact_6704_fst__image__mp,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B5: set_nat,X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ A2 ) @ B5 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ A2 )
       => ( member_nat @ X @ B5 ) ) ) ).

% fst_image_mp
thf(fact_6705_prod_OUnion__comp,axiom,
    ! [B5: set_set_o,G: $o > code_integer] :
      ( ! [X4: set_o] :
          ( ( member_set_o @ X4 @ B5 )
         => ( finite_finite_o @ X4 ) )
     => ( ! [A14: set_o] :
            ( ( member_set_o @ A14 @ B5 )
           => ! [A25: set_o] :
                ( ( member_set_o @ A25 @ B5 )
               => ( ( A14 != A25 )
                 => ! [X4: $o] :
                      ( ( member_o @ X4 @ A14 )
                     => ( ( member_o @ X4 @ A25 )
                       => ( ( G @ X4 )
                          = one_one_Code_integer ) ) ) ) ) )
       => ( ( groups7694694392188491536nteger @ G @ ( comple90263536869209701_set_o @ B5 ) )
          = ( comp_s262121120021782499nteger @ groups8987389756002350576nteger @ groups7694694392188491536nteger @ G @ B5 ) ) ) ) ).

% prod.Union_comp
thf(fact_6706_prod_OUnion__comp,axiom,
    ! [B5: set_set_int,G: int > code_integer] :
      ( ! [X4: set_int] :
          ( ( member_set_int @ X4 @ B5 )
         => ( finite_finite_int @ X4 ) )
     => ( ! [A14: set_int] :
            ( ( member_set_int @ A14 @ B5 )
           => ! [A25: set_int] :
                ( ( member_set_int @ A25 @ B5 )
               => ( ( A14 != A25 )
                 => ! [X4: int] :
                      ( ( member_int @ X4 @ A14 )
                     => ( ( member_int @ X4 @ A25 )
                       => ( ( G @ X4 )
                          = one_one_Code_integer ) ) ) ) ) )
       => ( ( groups3827104343326376752nteger @ G @ ( comple3221217463730067765et_int @ B5 ) )
          = ( comp_s7646638485285982975nteger @ groups8770619586034088166nteger @ groups3827104343326376752nteger @ G @ B5 ) ) ) ) ).

% prod.Union_comp
thf(fact_6707_prod_OUnion__comp,axiom,
    ! [B5: set_set_Code_integer,G: code_integer > code_integer] :
      ( ! [X4: set_Code_integer] :
          ( ( member1834298987299537450nteger @ X4 @ B5 )
         => ( finite6017078050557962740nteger @ X4 ) )
     => ( ! [A14: set_Code_integer] :
            ( ( member1834298987299537450nteger @ A14 @ B5 )
           => ! [A25: set_Code_integer] :
                ( ( member1834298987299537450nteger @ A25 @ B5 )
               => ( ( A14 != A25 )
                 => ! [X4: code_integer] :
                      ( ( member_Code_integer @ X4 @ A14 )
                     => ( ( member_Code_integer @ X4 @ A25 )
                       => ( ( G @ X4 )
                          = one_one_Code_integer ) ) ) ) ) )
       => ( ( groups3674199335183972705nteger @ G @ ( comple739944243200306918nteger @ B5 ) )
          = ( comp_s8484400498494363538nteger @ groups1474571732273379991nteger @ groups3674199335183972705nteger @ G @ B5 ) ) ) ) ).

% prod.Union_comp
thf(fact_6708_prod_OUnion__comp,axiom,
    ! [B5: set_set_o,G: $o > assn] :
      ( ! [X4: set_o] :
          ( ( member_set_o @ X4 @ B5 )
         => ( finite_finite_o @ X4 ) )
     => ( ! [A14: set_o] :
            ( ( member_set_o @ A14 @ B5 )
           => ! [A25: set_o] :
                ( ( member_set_o @ A25 @ B5 )
               => ( ( A14 != A25 )
                 => ! [X4: $o] :
                      ( ( member_o @ X4 @ A14 )
                     => ( ( member_o @ X4 @ A25 )
                       => ( ( G @ X4 )
                          = one_one_assn ) ) ) ) ) )
       => ( ( groups5301882518646026715o_assn @ G @ ( comple90263536869209701_set_o @ B5 ) )
          = ( comp_s518380907012905326o_assn @ groups6963733262650904763o_assn @ groups5301882518646026715o_assn @ G @ B5 ) ) ) ) ).

% prod.Union_comp
thf(fact_6709_prod_OUnion__comp,axiom,
    ! [B5: set_set_int,G: int > assn] :
      ( ! [X4: set_int] :
          ( ( member_set_int @ X4 @ B5 )
         => ( finite_finite_int @ X4 ) )
     => ( ! [A14: set_int] :
            ( ( member_set_int @ A14 @ B5 )
           => ! [A25: set_int] :
                ( ( member_set_int @ A25 @ B5 )
               => ( ( A14 != A25 )
                 => ! [X4: int] :
                      ( ( member_int @ X4 @ A14 )
                     => ( ( member_int @ X4 @ A25 )
                       => ( ( G @ X4 )
                          = one_one_assn ) ) ) ) ) )
       => ( ( groups7882442080178216443t_assn @ G @ ( comple3221217463730067765et_int @ B5 ) )
          = ( comp_s4595925057662634336t_assn @ groups5950967880310095793t_assn @ groups7882442080178216443t_assn @ G @ B5 ) ) ) ) ).

% prod.Union_comp
thf(fact_6710_prod_OUnion__comp,axiom,
    ! [B5: set_set_Code_integer,G: code_integer > assn] :
      ( ! [X4: set_Code_integer] :
          ( ( member1834298987299537450nteger @ X4 @ B5 )
         => ( finite6017078050557962740nteger @ X4 ) )
     => ( ! [A14: set_Code_integer] :
            ( ( member1834298987299537450nteger @ A14 @ B5 )
           => ! [A25: set_Code_integer] :
                ( ( member1834298987299537450nteger @ A25 @ B5 )
               => ( ( A14 != A25 )
                 => ! [X4: code_integer] :
                      ( ( member_Code_integer @ X4 @ A14 )
                     => ( ( member_Code_integer @ X4 @ A25 )
                       => ( ( G @ X4 )
                          = one_one_assn ) ) ) ) ) )
       => ( ( groups1304777262505850412r_assn @ G @ ( comple739944243200306918nteger @ B5 ) )
          = ( comp_s3531641609786562035r_assn @ groups4245288774754739042r_assn @ groups1304777262505850412r_assn @ G @ B5 ) ) ) ) ).

% prod.Union_comp
thf(fact_6711_prod_OUnion__comp,axiom,
    ! [B5: set_set_o,G: $o > rat] :
      ( ! [X4: set_o] :
          ( ( member_set_o @ X4 @ B5 )
         => ( finite_finite_o @ X4 ) )
     => ( ! [A14: set_o] :
            ( ( member_set_o @ A14 @ B5 )
           => ! [A25: set_o] :
                ( ( member_set_o @ A25 @ B5 )
               => ( ( A14 != A25 )
                 => ! [X4: $o] :
                      ( ( member_o @ X4 @ A14 )
                     => ( ( member_o @ X4 @ A25 )
                       => ( ( G @ X4 )
                          = one_one_rat ) ) ) ) ) )
       => ( ( groups2869687844427037835_o_rat @ G @ ( comple90263536869209701_set_o @ B5 ) )
          = ( comp_s1265507639253661726_o_rat @ groups4233643617323539307_o_rat @ groups2869687844427037835_o_rat @ G @ B5 ) ) ) ) ).

% prod.Union_comp
thf(fact_6712_prod_OUnion__comp,axiom,
    ! [B5: set_set_int,G: int > rat] :
      ( ! [X4: set_int] :
          ( ( member_set_int @ X4 @ B5 )
         => ( finite_finite_int @ X4 ) )
     => ( ! [A14: set_int] :
            ( ( member_set_int @ A14 @ B5 )
           => ! [A25: set_int] :
                ( ( member_set_int @ A25 @ B5 )
               => ( ( A14 != A25 )
                 => ! [X4: int] :
                      ( ( member_int @ X4 @ A14 )
                     => ( ( member_int @ X4 @ A25 )
                       => ( ( G @ X4 )
                          = one_one_rat ) ) ) ) ) )
       => ( ( groups1072433553688619179nt_rat @ G @ ( comple3221217463730067765et_int @ B5 ) )
          = ( comp_s4088261402702255472nt_rat @ groups5800339274786551393nt_rat @ groups1072433553688619179nt_rat @ G @ B5 ) ) ) ) ).

% prod.Union_comp
thf(fact_6713_prod_OUnion__comp,axiom,
    ! [B5: set_set_Code_integer,G: code_integer > rat] :
      ( ! [X4: set_Code_integer] :
          ( ( member1834298987299537450nteger @ X4 @ B5 )
         => ( finite6017078050557962740nteger @ X4 ) )
     => ( ! [A14: set_Code_integer] :
            ( ( member1834298987299537450nteger @ A14 @ B5 )
           => ! [A25: set_Code_integer] :
                ( ( member1834298987299537450nteger @ A25 @ B5 )
               => ( ( A14 != A25 )
                 => ! [X4: code_integer] :
                      ( ( member_Code_integer @ X4 @ A14 )
                     => ( ( member_Code_integer @ X4 @ A25 )
                       => ( ( G @ X4 )
                          = one_one_rat ) ) ) ) ) )
       => ( ( groups2555765274223993564er_rat @ G @ ( comple739944243200306918nteger @ B5 ) )
          = ( comp_s9045439038509510147er_rat @ groups5409254039008640018er_rat @ groups2555765274223993564er_rat @ G @ B5 ) ) ) ) ).

% prod.Union_comp
thf(fact_6714_prod_OUnion__comp,axiom,
    ! [B5: set_set_o,G: $o > nat] :
      ( ! [X4: set_o] :
          ( ( member_set_o @ X4 @ B5 )
         => ( finite_finite_o @ X4 ) )
     => ( ! [A14: set_o] :
            ( ( member_set_o @ A14 @ B5 )
           => ! [A25: set_o] :
                ( ( member_set_o @ A25 @ B5 )
               => ( ( A14 != A25 )
                 => ! [X4: $o] :
                      ( ( member_o @ X4 @ A14 )
                     => ( ( member_o @ X4 @ A25 )
                       => ( ( G @ X4 )
                          = one_one_nat ) ) ) ) ) )
       => ( ( groups3504817904513533571_o_nat @ G @ ( comple90263536869209701_set_o @ B5 ) )
          = ( comp_s6084864194472624150_o_nat @ groups4868773677410035043_o_nat @ groups3504817904513533571_o_nat @ G @ B5 ) ) ) ) ).

% prod.Union_comp
thf(fact_6715_UN__UN__finite__eq,axiom,
    ! [A2: nat > set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [N4: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) )
          @ top_top_set_nat ) )
      = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ top_top_set_nat ) ) ) ).

% UN_UN_finite_eq
thf(fact_6716_UN__le__add__shift,axiom,
    ! [M3: nat > set_nat,K: nat,N2: nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [I3: nat] : ( M3 @ ( plus_plus_nat @ I3 @ K ) )
          @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M3 @ ( set_or1269000886237332187st_nat @ K @ ( plus_plus_nat @ N2 @ K ) ) ) ) ) ).

% UN_le_add_shift
thf(fact_6717_UN__le__add__shift__strict,axiom,
    ! [M3: nat > set_nat,K: nat,N2: nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [I3: nat] : ( M3 @ ( plus_plus_nat @ I3 @ K ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) )
      = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M3 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ N2 @ K ) ) ) ) ) ).

% UN_le_add_shift_strict
thf(fact_6718_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 ) ) ).

% rat_sgn_code
thf(fact_6719_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Y )
     => ( ( bezw @ X @ Y )
        = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).

% bezw_non_0
thf(fact_6720_UN__finite__subset,axiom,
    ! [A2: nat > set_nat,C4: set_nat] :
      ( ! [N: nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) @ C4 )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ top_top_set_nat ) ) @ C4 ) ) ).

% UN_finite_subset
thf(fact_6721_UN__finite2__eq,axiom,
    ! [A2: nat > set_nat,B5: nat > set_nat,K: nat] :
      ( ! [N: nat] :
          ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
          = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( plus_plus_nat @ N @ K ) ) ) ) )
     => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ top_top_set_nat ) )
        = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ top_top_set_nat ) ) ) ) ).

% UN_finite2_eq
thf(fact_6722_bezw_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa )
        = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
        & ( ( Xa != zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).

% bezw.elims
thf(fact_6723_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X3: nat,Y3: nat] : ( if_Pro3027730157355071871nt_int @ ( Y3 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Y3 ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_6724_max__ext__def,axiom,
    ( max_ex4535816423174879388at_nat
    = ( ^ [R5: set_Pr4329608150637261639at_nat] :
          ( collec2601550354786890194at_nat
          @ ( produc1491100424724016556_nat_o
            @ ( max_ex4845989503969680116at_nat
              @ ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R5 ) ) ) ) ) ) ).

% max_ext_def
thf(fact_6725_max__ext__def,axiom,
    ( max_ext_nat
    = ( ^ [R5: set_Pr1261947904930325089at_nat] :
          ( collec6662362479098859352et_nat
          @ ( produc6247414631856714078_nat_o
            @ ( max_extp_nat
              @ ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R5 ) ) ) ) ) ) ).

% max_ext_def
thf(fact_6726_max__ext__def,axiom,
    ( max_ex8135407076693332796at_nat
    = ( ^ [R5: set_Pr8693737435421807431at_nat] :
          ( collec6321179662152712658at_nat
          @ ( produc410239310623530412_nat_o
            @ ( max_ex4864111882549613972at_nat
              @ ^ [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R5 ) ) ) ) ) ) ).

% max_ext_def
thf(fact_6727_UN__finite2__subset,axiom,
    ! [A2: nat > set_nat,B5: nat > set_nat,K: nat] :
      ( ! [N: nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( plus_plus_nat @ N @ K ) ) ) ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ top_top_set_nat ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ top_top_set_nat ) ) ) ) ).

% UN_finite2_subset
thf(fact_6728_bezw_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
       => ~ ( ( ( ( Xa = zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
              & ( ( Xa != zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa ) ) ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).

% bezw.pelims
thf(fact_6729_snd__diag__id,axiom,
    ! [Z: $o] :
      ( ( comp_P2137996956111167876_o_o_o @ product_snd_o_o
        @ ^ [X3: $o] : ( product_Pair_o_o @ X3 @ X3 )
        @ Z )
      = ( id_o @ Z ) ) ).

% snd_diag_id
thf(fact_6730_snd__diag__id,axiom,
    ! [Z: nat] :
      ( ( comp_P1955610211387027840at_nat @ product_snd_nat_nat
        @ ^ [X3: nat] : ( product_Pair_nat_nat @ X3 @ X3 )
        @ Z )
      = ( id_nat @ Z ) ) ).

% snd_diag_id
thf(fact_6731_snd__diag__id,axiom,
    ! [Z: nat > $o] :
      ( ( comp_P8312345242777556868_nat_o @ produc6795044007072221829_nat_o
        @ ^ [X3: nat > $o] : ( produc5701653931590357335_nat_o @ X3 @ X3 )
        @ Z )
      = ( id_nat_o @ Z ) ) ).

% snd_diag_id
thf(fact_6732_snd__diag__id,axiom,
    ! [Z: num] :
      ( ( comp_P864883071962432936um_num @ product_snd_num_num
        @ ^ [X3: num] : ( product_Pair_num_num @ X3 @ X3 )
        @ Z )
      = ( id_num @ Z ) ) ).

% snd_diag_id
thf(fact_6733_snd__diag__id,axiom,
    ! [Z: code_integer] :
      ( ( comp_P7111796090210262324nteger @ produc6174133586879617921nteger
        @ ^ [X3: code_integer] : ( produc1086072967326762835nteger @ X3 @ X3 )
        @ Z )
      = ( id_Code_integer @ Z ) ) ).

% snd_diag_id
thf(fact_6734_snd__diag__id,axiom,
    ! [Z: int] :
      ( ( comp_P6651465171681656048nt_int @ product_snd_int_int
        @ ^ [X3: int] : ( product_Pair_int_int @ X3 @ X3 )
        @ Z )
      = ( id_int @ Z ) ) ).

% snd_diag_id
thf(fact_6735_fst__diag__id,axiom,
    ! [Z: $o] :
      ( ( comp_P2137996956111167876_o_o_o @ product_fst_o_o
        @ ^ [X3: $o] : ( product_Pair_o_o @ X3 @ X3 )
        @ Z )
      = ( id_o @ Z ) ) ).

% fst_diag_id
thf(fact_6736_fst__diag__id,axiom,
    ! [Z: nat] :
      ( ( comp_P1955610211387027840at_nat @ product_fst_nat_nat
        @ ^ [X3: nat] : ( product_Pair_nat_nat @ X3 @ X3 )
        @ Z )
      = ( id_nat @ Z ) ) ).

% fst_diag_id
thf(fact_6737_fst__diag__id,axiom,
    ! [Z: nat > $o] :
      ( ( comp_P8312345242777556868_nat_o @ produc6972895793295859523_nat_o
        @ ^ [X3: nat > $o] : ( produc5701653931590357335_nat_o @ X3 @ X3 )
        @ Z )
      = ( id_nat_o @ Z ) ) ).

% fst_diag_id
thf(fact_6738_fst__diag__id,axiom,
    ! [Z: num] :
      ( ( comp_P864883071962432936um_num @ product_fst_num_num
        @ ^ [X3: num] : ( product_Pair_num_num @ X3 @ X3 )
        @ Z )
      = ( id_num @ Z ) ) ).

% fst_diag_id
thf(fact_6739_fst__diag__id,axiom,
    ! [Z: code_integer] :
      ( ( comp_P7111796090210262324nteger @ produc8508995932063986495nteger
        @ ^ [X3: code_integer] : ( produc1086072967326762835nteger @ X3 @ X3 )
        @ Z )
      = ( id_Code_integer @ Z ) ) ).

% fst_diag_id
thf(fact_6740_fst__diag__id,axiom,
    ! [Z: int] :
      ( ( comp_P6651465171681656048nt_int @ product_fst_int_int
        @ ^ [X3: int] : ( product_Pair_int_int @ X3 @ X3 )
        @ Z )
      = ( id_int @ Z ) ) ).

% fst_diag_id
thf(fact_6741_size__prod__simp,axiom,
    ( basic_7565498685090299632nteger
    = ( ^ [F2: code_integer > nat,G2: code_integer > nat,P6: produc8923325533196201883nteger] : ( plus_plus_nat @ ( plus_plus_nat @ ( F2 @ ( produc8508995932063986495nteger @ P6 ) ) @ ( G2 @ ( produc6174133586879617921nteger @ P6 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ).

% size_prod_simp
thf(fact_6742_size__prod__simp,axiom,
    ( basic_1872990034501187214nt_int
    = ( ^ [F2: int > nat,G2: int > nat,P6: product_prod_int_int] : ( plus_plus_nat @ ( plus_plus_nat @ ( F2 @ ( product_fst_int_int @ P6 ) ) @ ( G2 @ ( product_snd_int_int @ P6 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ).

% size_prod_simp
thf(fact_6743_size__prod__simp,axiom,
    ( basic_8185638495112405971atural
    = ( ^ [F2: code_natural > nat,G2: produc7822875418678951345atural > nat,P6: produc5835291356934675326atural] : ( plus_plus_nat @ ( plus_plus_nat @ ( F2 @ ( produc497848011034438852atural @ P6 ) ) @ ( G2 @ ( produc6591912806276919810atural @ P6 ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ).

% size_prod_simp
thf(fact_6744_fst__diag__snd,axiom,
    ( ( comp_P3428181540658402820nteger @ produc8508995932063986495nteger
      @ ( comp_C2277306503614464128nteger
        @ ^ [X3: code_integer] : ( produc1086072967326762835nteger @ X3 @ X3 )
        @ produc6174133586879617921nteger ) )
    = produc6174133586879617921nteger ) ).

% fst_diag_snd
thf(fact_6745_fst__diag__snd,axiom,
    ( ( comp_P5356232565778667509nt_int @ product_fst_int_int
      @ ( comp_i5941103828406351439nt_int
        @ ^ [X3: int] : ( product_Pair_int_int @ X3 @ X3 )
        @ product_snd_int_int ) )
    = product_snd_int_int ) ).

% fst_diag_snd
thf(fact_6746_fst__diag__snd,axiom,
    ( ( comp_P230220204662055659atural @ produc1537665297809786051atural
      @ ( comp_P4943825737035310295atural
        @ ^ [X3: produc7822875418678951345atural] : ( produc4480994950612372183atural @ X3 @ X3 )
        @ produc6591912806276919810atural ) )
    = produc6591912806276919810atural ) ).

% fst_diag_snd
thf(fact_6747_snd__diag__fst,axiom,
    ( ( comp_P3428181540658402820nteger @ produc6174133586879617921nteger
      @ ( comp_C2277306503614464128nteger
        @ ^ [X3: code_integer] : ( produc1086072967326762835nteger @ X3 @ X3 )
        @ produc8508995932063986495nteger ) )
    = produc8508995932063986495nteger ) ).

% snd_diag_fst
thf(fact_6748_snd__diag__fst,axiom,
    ( ( comp_P5356232565778667509nt_int @ product_snd_int_int
      @ ( comp_i5941103828406351439nt_int
        @ ^ [X3: int] : ( product_Pair_int_int @ X3 @ X3 )
        @ product_fst_int_int ) )
    = product_fst_int_int ) ).

% snd_diag_fst
thf(fact_6749_INF__Int__eq2,axiom,
    ! [S2: set_se5479201105039950375at_nat] :
      ( ( comple8322679762649017525_nat_o
        @ ( image_6595146019508340652_nat_o
          @ ^ [I3: set_Pr4329608150637261639at_nat,X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ I3 )
          @ S2 ) )
      = ( ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ ( comple6751905152567503566at_nat @ S2 ) ) ) ) ).

% INF_Int_eq2
thf(fact_6750_INF__Int__eq2,axiom,
    ! [S2: set_se7855581050983116737at_nat] :
      ( ( comple1675732732025788397_nat_o
        @ ( image_6923511907955291850_nat_o
          @ ^ [I3: set_Pr1261947904930325089at_nat,X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ I3 )
          @ S2 ) )
      = ( ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( comple1930127330929219688at_nat @ S2 ) ) ) ) ).

% INF_Int_eq2
thf(fact_6751_INF__Int__eq2,axiom,
    ! [S2: set_se124295367203800706nteger] :
      ( ( comple797978290332539892eger_o
        @ ( image_3377504002992083726eger_o
          @ ^ [I3: set_Pr1281608226676607948nteger,X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ I3 )
          @ S2 ) )
      = ( ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ ( comple141797230998674821nteger @ S2 ) ) ) ) ).

% INF_Int_eq2
thf(fact_6752_INF__Int__eq2,axiom,
    ! [S2: set_se8543413830569400298et_nat] :
      ( ( comple4401490224799119198_nat_o
        @ ( image_1071233342602175200_nat_o
          @ ^ [I3: set_Pr3286484037609594932et_nat,X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ I3 )
          @ S2 ) )
      = ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ ( comple8712884055654468845et_nat @ S2 ) ) ) ) ).

% INF_Int_eq2
thf(fact_6753_INF__Int__eq2,axiom,
    ! [S2: set_se5120235452707311462et_nat] :
      ( ( comple1647972170925272290_nat_o
        @ ( image_6856364241397052896_nat_o
          @ ^ [I3: set_Pr8536935166611901872et_nat,X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ I3 )
          @ S2 ) )
      = ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ ( comple8400030738448405097et_nat @ S2 ) ) ) ) ).

% INF_Int_eq2
thf(fact_6754_INF__Int__eq2,axiom,
    ! [S2: set_se2364587433798273647nt_int] :
      ( ( comple2375672059646964807_int_o
        @ ( image_5684251269295653582_int_o
          @ ^ [I3: set_Pr9222295170931077689nt_int,X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ I3 )
          @ S2 ) )
      = ( ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ ( comple6065809805536145394nt_int @ S2 ) ) ) ) ).

% INF_Int_eq2
thf(fact_6755_INF__Int__eq2,axiom,
    ! [S2: set_se7394616247590735609nt_int] :
      ( ( comple7280212863790966717_int_o
        @ ( image_1744818337813142478_int_o
          @ ^ [I3: set_Pr1872883991513573699nt_int,X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ I3 )
          @ S2 ) )
      = ( ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ ( comple2057271477292249724nt_int @ S2 ) ) ) ) ).

% INF_Int_eq2
thf(fact_6756_SUP__Sup__eq2,axiom,
    ! [S2: set_se5479201105039950375at_nat] :
      ( ( comple493620192902636699_nat_o
        @ ( image_6595146019508340652_nat_o
          @ ^ [I3: set_Pr4329608150637261639at_nat,X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ I3 )
          @ S2 ) )
      = ( ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ ( comple4364388969068068532at_nat @ S2 ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_6757_SUP__Sup__eq2,axiom,
    ! [S2: set_se7855581050983116737at_nat] :
      ( ( comple3592611370556534995_nat_o
        @ ( image_6923511907955291850_nat_o
          @ ^ [I3: set_Pr1261947904930325089at_nat,X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ I3 )
          @ S2 ) )
      = ( ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( comple5685304695842803022at_nat @ S2 ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_6758_SUP__Sup__eq2,axiom,
    ! [S2: set_se124295367203800706nteger] :
      ( ( comple6692557002573936782eger_o
        @ ( image_3377504002992083726eger_o
          @ ^ [I3: set_Pr1281608226676607948nteger,X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ I3 )
          @ S2 ) )
      = ( ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ ( comple1324685164916153887nteger @ S2 ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_6759_SUP__Sup__eq2,axiom,
    ! [S2: set_se8543413830569400298et_nat] :
      ( ( comple2736776311324672504_nat_o
        @ ( image_1071233342602175200_nat_o
          @ ^ [I3: set_Pr3286484037609594932et_nat,X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ I3 )
          @ S2 ) )
      = ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ ( comple4849024849677941639et_nat @ S2 ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_6760_SUP__Sup__eq2,axiom,
    ! [S2: set_se5120235452707311462et_nat] :
      ( ( comple5542144468884744060_nat_o
        @ ( image_6856364241397052896_nat_o
          @ ^ [I3: set_Pr8536935166611901872et_nat,X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ I3 )
          @ S2 ) )
      = ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ ( comple5063570841203093763et_nat @ S2 ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_6761_SUP__Sup__eq2,axiom,
    ! [S2: set_se2364587433798273647nt_int] :
      ( ( comple7735192718774405345_int_o
        @ ( image_5684251269295653582_int_o
          @ ^ [I3: set_Pr9222295170931077689nt_int,X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ I3 )
          @ S2 ) )
      = ( ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ ( comple6871990601243273356nt_int @ S2 ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_6762_SUP__Sup__eq2,axiom,
    ! [S2: set_se7394616247590735609nt_int] :
      ( ( comple8902563120744349527_int_o
        @ ( image_1744818337813142478_int_o
          @ ^ [I3: set_Pr1872883991513573699nt_int,X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ I3 )
          @ S2 ) )
      = ( ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ ( comple5772984051336441366nt_int @ S2 ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_6763_Inf__INT__eq2,axiom,
    ( comple8322679762649017525_nat_o
    = ( ^ [S5: set_se9180647916752606350_nat_o,X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ ( comple6751905152567503566at_nat @ ( image_4902389024228248874at_nat @ collec6321179662152712658at_nat @ ( image_6226802281960588479_nat_o @ produc410239310623530412_nat_o @ S5 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_6764_Inf__INT__eq2,axiom,
    ( comple1675732732025788397_nat_o
    = ( ^ [S5: set_nat_nat_o,X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( comple1930127330929219688at_nat @ ( image_7124889717316225246at_nat @ collec3392354462482085612at_nat @ ( image_7429393840292777309_nat_o @ produc6081775807080527818_nat_o @ S5 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_6765_Inf__INT__eq2,axiom,
    ( comple797978290332539892eger_o
    = ( ^ [S5: set_Pr8914822823351560561eger_o,X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ ( comple141797230998674821nteger @ ( image_2691561895000378432nteger @ collec6950949345391930603nteger @ ( image_5214523798439755767eger_o @ produc6253627499356882019eger_o @ S5 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_6766_Inf__INT__eq2,axiom,
    ( comple4401490224799119198_nat_o
    = ( ^ [S5: set_Pr8208987589773510107_nat_o,X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ ( comple8712884055654468845et_nat @ ( image_8138495846740763712et_nat @ collec1402215087704437587et_nat @ ( image_8189751627993580965_nat_o @ produc1437786849005270451_nat_o @ S5 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_6767_Inf__INT__eq2,axiom,
    ( comple1647972170925272290_nat_o
    = ( ^ [S5: set_Pr1457143144315775327_nat_o,X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ ( comple8400030738448405097et_nat @ ( image_3321512208570918720et_nat @ collec5543584681430388431et_nat @ ( image_1033581555836824229_nat_o @ produc838355143741117751_nat_o @ S5 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_6768_Inf__INT__eq2,axiom,
    ( comple2375672059646964807_int_o
    = ( ^ [S5: set_Pr5050529186185961540_int_o,X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ ( comple6065809805536145394nt_int @ ( image_7643917626522560896nt_int @ collec1790188477890212312nt_int @ ( image_7631425643974863415_int_o @ produc1573362020775583542_int_o @ S5 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_6769_Inf__INT__eq2,axiom,
    ( comple7280212863790966717_int_o
    = ( ^ [S5: set_in8474487674441654202_int_o,X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ ( comple2057271477292249724nt_int @ ( image_2770089787918713472nt_int @ collec506566255779805410nt_int @ ( image_7670822187664137143_int_o @ produc2558449545302689196_int_o @ S5 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_6770_Sup__SUP__eq2,axiom,
    ( comple493620192902636699_nat_o
    = ( ^ [S5: set_se9180647916752606350_nat_o,X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ ( comple4364388969068068532at_nat @ ( image_4902389024228248874at_nat @ collec6321179662152712658at_nat @ ( image_6226802281960588479_nat_o @ produc410239310623530412_nat_o @ S5 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_6771_Sup__SUP__eq2,axiom,
    ( comple3592611370556534995_nat_o
    = ( ^ [S5: set_nat_nat_o,X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( comple5685304695842803022at_nat @ ( image_7124889717316225246at_nat @ collec3392354462482085612at_nat @ ( image_7429393840292777309_nat_o @ produc6081775807080527818_nat_o @ S5 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_6772_Sup__SUP__eq2,axiom,
    ( comple6692557002573936782eger_o
    = ( ^ [S5: set_Pr8914822823351560561eger_o,X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ ( comple1324685164916153887nteger @ ( image_2691561895000378432nteger @ collec6950949345391930603nteger @ ( image_5214523798439755767eger_o @ produc6253627499356882019eger_o @ S5 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_6773_Sup__SUP__eq2,axiom,
    ( comple2736776311324672504_nat_o
    = ( ^ [S5: set_Pr8208987589773510107_nat_o,X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ ( comple4849024849677941639et_nat @ ( image_8138495846740763712et_nat @ collec1402215087704437587et_nat @ ( image_8189751627993580965_nat_o @ produc1437786849005270451_nat_o @ S5 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_6774_Sup__SUP__eq2,axiom,
    ( comple5542144468884744060_nat_o
    = ( ^ [S5: set_Pr1457143144315775327_nat_o,X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ ( comple5063570841203093763et_nat @ ( image_3321512208570918720et_nat @ collec5543584681430388431et_nat @ ( image_1033581555836824229_nat_o @ produc838355143741117751_nat_o @ S5 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_6775_Sup__SUP__eq2,axiom,
    ( comple7735192718774405345_int_o
    = ( ^ [S5: set_Pr5050529186185961540_int_o,X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ ( comple6871990601243273356nt_int @ ( image_7643917626522560896nt_int @ collec1790188477890212312nt_int @ ( image_7631425643974863415_int_o @ produc1573362020775583542_int_o @ S5 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_6776_Sup__SUP__eq2,axiom,
    ( comple8902563120744349527_int_o
    = ( ^ [S5: set_in8474487674441654202_int_o,X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ ( comple5772984051336441366nt_int @ ( image_2770089787918713472nt_int @ collec506566255779805410nt_int @ ( image_7670822187664137143_int_o @ produc2558449545302689196_int_o @ S5 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_6777_fst__diag__fst,axiom,
    ( ( comp_P3428181540658402820nteger @ produc8508995932063986495nteger
      @ ( comp_C2277306503614464128nteger
        @ ^ [X3: code_integer] : ( produc1086072967326762835nteger @ X3 @ X3 )
        @ produc8508995932063986495nteger ) )
    = produc8508995932063986495nteger ) ).

% fst_diag_fst
thf(fact_6778_fst__diag__fst,axiom,
    ( ( comp_P5356232565778667509nt_int @ product_fst_int_int
      @ ( comp_i5941103828406351439nt_int
        @ ^ [X3: int] : ( product_Pair_int_int @ X3 @ X3 )
        @ product_fst_int_int ) )
    = product_fst_int_int ) ).

% fst_diag_fst
thf(fact_6779_snd__diag__snd,axiom,
    ( ( comp_P230220204662055659atural @ produc7455300194457530885atural
      @ ( comp_P4943825737035310295atural
        @ ^ [X3: produc7822875418678951345atural] : ( produc4480994950612372183atural @ X3 @ X3 )
        @ produc6591912806276919810atural ) )
    = produc6591912806276919810atural ) ).

% snd_diag_snd
thf(fact_6780_snd__diag__snd,axiom,
    ( ( comp_P3428181540658402820nteger @ produc6174133586879617921nteger
      @ ( comp_C2277306503614464128nteger
        @ ^ [X3: code_integer] : ( produc1086072967326762835nteger @ X3 @ X3 )
        @ produc6174133586879617921nteger ) )
    = produc6174133586879617921nteger ) ).

% snd_diag_snd
thf(fact_6781_snd__diag__snd,axiom,
    ( ( comp_P5356232565778667509nt_int @ product_snd_int_int
      @ ( comp_i5941103828406351439nt_int
        @ ^ [X3: int] : ( product_Pair_int_int @ X3 @ X3 )
        @ product_snd_int_int ) )
    = product_snd_int_int ) ).

% snd_diag_snd
thf(fact_6782_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger,A: produc1908205239877642774nteger] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc8603105652947943368nteger @ X @ Y ) )
       => ( P @ ( produc7822682618958472924nteger @ A ) @ ( produc7856867400915047194nteger @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_6783_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat,A: produc3925858234332021118et_nat] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc5001842942810119800et_nat @ X @ Y ) )
       => ( P @ ( produc995936583742144908et_nat @ A ) @ ( produc4011572625026189258et_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_6784_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > produc3925858234332021118et_nat > $o,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat,A: produc2732055786443039994et_nat] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc2245416461498447860et_nat @ X @ Y ) )
       => ( P @ ( produc180342877477747464et_nat @ A ) @ ( produc5374455773327741254et_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_6785_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int,A: produc2285326912895808259nt_int] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc5700946648718959541nt_int @ X @ Y ) )
       => ( P @ ( produc6816164490631068361nt_int @ A ) @ ( produc7328097813583171335nt_int @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_6786_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,X: int > option6357759511663192854e_term,Y: product_prod_int_int,A: produc7773217078559923341nt_int] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc4305682042979456191nt_int @ X @ Y ) )
       => ( P @ ( produc6230002227079971283nt_int @ A ) @ ( produc3162348030201620241nt_int @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_6787_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: code_integer > code_integer > $o,X: code_integer,Y: code_integer,A: produc8923325533196201883nteger] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc1086072967326762835nteger @ X @ Y ) )
       => ( P @ ( produc8508995932063986495nteger @ A ) @ ( produc6174133586879617921nteger @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_6788_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: int > int > $o,X: int,Y: int,A: product_prod_int_int] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( product_Pair_int_int @ X @ Y ) )
       => ( P @ ( product_fst_int_int @ A ) @ ( product_snd_int_int @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_6789_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: code_natural > produc7822875418678951345atural > $o,X: code_natural,Y: produc7822875418678951345atural,A: produc5835291356934675326atural] :
      ( ( P @ X @ Y )
     => ( ( A
          = ( produc6639722614265839536atural @ X @ Y ) )
       => ( P @ ( produc497848011034438852atural @ A ) @ ( produc6591912806276919810atural @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_6790_normalize__def,axiom,
    ( normalize
    = ( ^ [P6: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_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_Pro3027730157355071871nt_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 ) ) ) ) ) ) ) ) ) ).

% normalize_def
thf(fact_6791_range__prod,axiom,
    ! [F: nat > produc6433947425880249742er_nat] :
      ( ord_le3209295114279298062er_nat @ ( image_3126957236123472631er_nat @ F @ top_top_set_nat )
      @ ( produc4260658192689636599er_nat @ ( image_1215581382706833972nteger @ ( comp_P5700178199414968986er_nat @ produc8632663045010267826er_nat @ F ) @ top_top_set_nat )
        @ ^ [Uu: code_integer] : ( image_nat_nat @ ( comp_P1337767177661389325at_nat @ produc1517335558169710324er_nat @ F ) @ top_top_set_nat ) ) ) ).

% range_prod
thf(fact_6792_range__prod,axiom,
    ! [F: nat > produc2256096406371053034er_int] :
      ( ord_le719211969347571306er_int @ ( image_8172478253469051731er_int @ F @ top_top_set_nat )
      @ ( produc4258167722180586323er_int @ ( image_1215581382706833972nteger @ ( comp_P2475326853202991862er_nat @ produc8630172574501217550er_int @ F ) @ top_top_set_nat )
        @ ^ [Uu: code_integer] : ( image_nat_int @ ( comp_P8345930540558018757nt_nat @ produc1514845087660660048er_int @ F ) @ top_top_set_nat ) ) ) ).

% range_prod
thf(fact_6793_range__prod,axiom,
    ! [F: nat > produc2591615631378489102nteger] :
      ( ord_le5852610332322655630nteger @ ( image_8507997478476487799nteger @ F @ top_top_set_nat )
      @ ( produc4525213641468679415nteger @ ( image_nat_nat @ ( comp_P9144323761349829517at_nat @ produc8897218493789310642nteger @ F ) @ top_top_set_nat )
        @ ^ [Uu: nat] : ( image_1215581382706833972nteger @ ( comp_P5461341465284484122er_nat @ produc1781891006948753140nteger @ F ) @ top_top_set_nat ) ) ) ).

% range_prod
thf(fact_6794_range__prod,axiom,
    ! [F: nat > product_prod_nat_nat] :
      ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ F @ top_top_set_nat )
      @ ( produc457027306803732586at_nat @ ( image_nat_nat @ ( comp_P1955610211387027840at_nat @ product_fst_nat_nat @ F ) @ top_top_set_nat )
        @ ^ [Uu: nat] : ( image_nat_nat @ ( comp_P1955610211387027840at_nat @ product_snd_nat_nat @ F ) @ top_top_set_nat ) ) ) ).

% range_prod
thf(fact_6795_range__prod,axiom,
    ! [F: nat > product_prod_nat_int] :
      ( ord_le656430383953171549at_int @ ( image_1668272788310788806at_int @ F @ top_top_set_nat )
      @ ( produc454536836294682310at_int @ ( image_nat_nat @ ( comp_P7964419862382602716at_nat @ product_fst_nat_int @ F ) @ top_top_set_nat )
        @ ^ [Uu: nat] : ( image_nat_int @ ( comp_P8963773574283657272nt_nat @ product_snd_nat_int @ F ) @ top_top_set_nat ) ) ) ).

% range_prod
thf(fact_6796_range__prod,axiom,
    ! [F: nat > produc2963269191615333738nteger] :
      ( ord_le2638047946463454698nteger @ ( image_8879651038713332435nteger @ F @ top_top_set_nat )
      @ ( produc4896867201705524051nteger @ ( image_nat_int @ ( comp_P59025373045620805nt_nat @ produc45500017171379470nteger @ F ) @ top_top_set_nat )
        @ ^ [Uu: int] : ( image_1215581382706833972nteger @ ( comp_P6870028373966156406er_nat @ produc2153544567185597776nteger @ F ) @ top_top_set_nat ) ) ) ).

% range_prod
thf(fact_6797_range__prod,axiom,
    ! [F: nat > product_prod_int_nat] :
      ( ord_le5333435103577920093nt_nat @ ( image_6845477519721040070nt_nat @ F @ top_top_set_nat )
      @ ( produc1456381018704787142nt_nat @ ( image_nat_int @ ( comp_P645145991195131448nt_nat @ product_fst_int_nat @ F ) @ top_top_set_nat )
        @ ^ [Uu: int] : ( image_nat_nat @ ( comp_P8869164316148852700at_nat @ product_snd_int_nat @ F ) @ top_top_set_nat ) ) ) ).

% range_prod
thf(fact_6798_range__prod,axiom,
    ! [F: nat > produc8923325533196201883nteger] :
      ( ord_le3725938330318615451nteger @ ( image_4272405876590741764nteger @ F @ top_top_set_nat )
      @ ( produc5882431272649516420nteger @ ( image_1215581382706833972nteger @ ( comp_P3109732533133679783er_nat @ produc8508995932063986495nteger @ F ) @ top_top_set_nat )
        @ ^ [Uu: code_integer] : ( image_1215581382706833972nteger @ ( comp_P3109732533133679783er_nat @ produc6174133586879617921nteger @ F ) @ top_top_set_nat ) ) ) ).

% range_prod
thf(fact_6799_range__prod,axiom,
    ! [F: nat > product_prod_int_int] :
      ( ord_le2843351958646193337nt_int @ ( image_2667626500211843362nt_int @ F @ top_top_set_nat )
      @ ( produc1453890548195736866nt_int @ ( image_nat_int @ ( comp_P6653955642190706324nt_nat @ product_fst_int_int @ F ) @ top_top_set_nat )
        @ ^ [Uu: int] : ( image_nat_int @ ( comp_P6653955642190706324nt_nat @ product_snd_int_int @ F ) @ top_top_set_nat ) ) ) ).

% range_prod
thf(fact_6800_range__prod,axiom,
    ! [F: product_unit > produc8923325533196201883nteger] :
      ( ord_le3725938330318615451nteger @ ( image_8089323762101378259nteger @ F @ top_to1996260823553986621t_unit )
      @ ( produc5882431272649516420nteger @ ( image_3815118110082124069nteger @ ( comp_P659088277599099894t_unit @ produc8508995932063986495nteger @ F ) @ top_to1996260823553986621t_unit )
        @ ^ [Uu: code_integer] : ( image_3815118110082124069nteger @ ( comp_P659088277599099894t_unit @ produc6174133586879617921nteger @ F ) @ top_to1996260823553986621t_unit ) ) ) ).

% range_prod
thf(fact_6801_fun_Oin__rel,axiom,
    ! [R4: code_integer > code_integer > $o,A: nat > code_integer,B: nat > code_integer] :
      ( ( bNF_re1236659899165691067nteger
        @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
        @ R4
        @ A
        @ B )
      = ( ? [Z4: nat > produc8923325533196201883nteger] :
            ( ( member1326526067935671339nteger @ Z4
              @ ( collec7299664631084759149nteger
                @ ^ [X3: nat > produc8923325533196201883nteger] : ( ord_le3725938330318615451nteger @ ( image_4272405876590741764nteger @ X3 @ top_top_set_nat ) @ ( collec7766031446232907142nteger @ ( produc2066375834425727024eger_o @ R4 ) ) ) ) )
            & ( ( comp_P3109732533133679783er_nat @ produc8508995932063986495nteger @ Z4 )
              = A )
            & ( ( comp_P3109732533133679783er_nat @ produc6174133586879617921nteger @ Z4 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_6802_fun_Oin__rel,axiom,
    ! [R4: int > int > $o,A: nat > int,B: nat > int] :
      ( ( bNF_re6650684261131312217nt_int
        @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
        @ R4
        @ A
        @ B )
      = ( ? [Z4: nat > product_prod_int_int] :
            ( ( member6440191985751670089nt_int @ Z4
              @ ( collec8414850927317488011nt_int
                @ ^ [X3: nat > product_prod_int_int] : ( ord_le2843351958646193337nt_int @ ( image_2667626500211843362nt_int @ X3 @ top_top_set_nat ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R4 ) ) ) ) )
            & ( ( comp_P6653955642190706324nt_nat @ product_fst_int_int @ Z4 )
              = A )
            & ( ( comp_P6653955642190706324nt_nat @ product_snd_int_int @ Z4 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_6803_fun_Oin__rel,axiom,
    ! [R4: code_integer > code_integer > $o,A: product_unit > code_integer,B: product_unit > code_integer] :
      ( ( bNF_re8417623677625516633nteger
        @ ^ [Y4: product_unit,Z2: product_unit] : ( Y4 = Z2 )
        @ R4
        @ A
        @ B )
      = ( ? [Z4: product_unit > produc8923325533196201883nteger] :
            ( ( member2837767709637065162nteger @ Z4
              @ ( collec492036462967229448nteger
                @ ^ [X3: product_unit > produc8923325533196201883nteger] : ( ord_le3725938330318615451nteger @ ( image_8089323762101378259nteger @ X3 @ top_to1996260823553986621t_unit ) @ ( collec7766031446232907142nteger @ ( produc2066375834425727024eger_o @ R4 ) ) ) ) )
            & ( ( comp_P659088277599099894t_unit @ produc8508995932063986495nteger @ Z4 )
              = A )
            & ( ( comp_P659088277599099894t_unit @ produc6174133586879617921nteger @ Z4 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_6804_fun_Oin__rel,axiom,
    ! [R4: int > int > $o,A: product_unit > int,B: product_unit > int] :
      ( ( bNF_re8422471777688076791nt_int
        @ ^ [Y4: product_unit,Z2: product_unit] : ( Y4 = Z2 )
        @ R4
        @ A
        @ B )
      = ( ? [Z4: product_unit > product_prod_int_int] :
            ( ( member5541948317052402408nt_int @ Z4
              @ ( collec2513708037538135334nt_int
                @ ^ [X3: product_unit > product_prod_int_int] : ( ord_le2843351958646193337nt_int @ ( image_4103462098156874993nt_int @ X3 @ top_to1996260823553986621t_unit ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R4 ) ) ) ) )
            & ( ( comp_P6796073005956588617t_unit @ product_fst_int_int @ Z4 )
              = A )
            & ( ( comp_P6796073005956588617t_unit @ product_snd_int_int @ Z4 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_6805_fun_Oin__rel,axiom,
    ! [R4: code_integer > code_integer > $o,A: int > code_integer,B: int > code_integer] :
      ( ( bNF_re1784066744496875891nteger
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ R4
        @ A
        @ B )
      = ( ? [Z4: int > produc8923325533196201883nteger] :
            ( ( member2248416676132969167nteger @ Z4
              @ ( collec8221555239282056977nteger
                @ ^ [X3: int > produc8923325533196201883nteger] : ( ord_le3725938330318615451nteger @ ( image_5194296484788039592nteger @ X3 @ top_top_set_int ) @ ( collec7766031446232907142nteger @ ( produc2066375834425727024eger_o @ R4 ) ) ) ) )
            & ( ( comp_P3107242062624629507er_int @ produc8508995932063986495nteger @ Z4 )
              = A )
            & ( ( comp_P3107242062624629507er_int @ produc6174133586879617921nteger @ Z4 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_6806_fun_Oin__rel,axiom,
    ! [R4: int > int > $o,A: int > int,B: int > int] :
      ( ( bNF_re4712519889275205905nt_int
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ R4
        @ A
        @ B )
      = ( ? [Z4: int > product_prod_int_int] :
            ( ( member254662033361028077nt_int @ Z4
              @ ( collec2229320974926845999nt_int
                @ ^ [X3: int > product_prod_int_int] : ( ord_le2843351958646193337nt_int @ ( image_5705468584675977158nt_int @ X3 @ top_top_set_int ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R4 ) ) ) ) )
            & ( ( comp_P6651465171681656048nt_int @ product_fst_int_int @ Z4 )
              = A )
            & ( ( comp_P6651465171681656048nt_int @ product_snd_int_int @ Z4 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_6807_fun_Oin__rel,axiom,
    ! [R4: code_natural > produc7822875418678951345atural > $o,A: nat > code_natural,B: nat > produc7822875418678951345atural] :
      ( ( bNF_re6101645755646507080atural
        @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
        @ R4
        @ A
        @ B )
      = ( ? [Z4: nat > produc5835291356934675326atural] :
            ( ( member528921174545443716atural @ Z4
              @ ( collec3197714919839585474atural
                @ ^ [X3: nat > produc5835291356934675326atural] : ( ord_le2202040463148982292atural @ ( image_4072043198136622805atural @ X3 @ top_top_set_nat ) @ ( collec1991479681304692691atural @ ( produc7358226366420480123ural_o @ R4 ) ) ) ) )
            & ( ( comp_P5592315970379621083al_nat @ produc497848011034438852atural @ Z4 )
              = A )
            & ( ( comp_P8477885140431536992al_nat @ produc6591912806276919810atural @ Z4 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_6808_fun_Oin__rel,axiom,
    ! [R4: code_natural > produc7822875418678951345atural > $o,A: product_unit > code_natural,B: product_unit > produc7822875418678951345atural] :
      ( ( bNF_re5641277236161133866atural
        @ ^ [Y4: product_unit,Z2: product_unit] : ( Y4 = Z2 )
        @ R4
        @ A
        @ B )
      = ( ? [Z4: product_unit > produc5835291356934675326atural] :
            ( ( member4946248492122881829atural @ Z4
              @ ( collec3282430933710331751atural
                @ ^ [X3: product_unit > produc5835291356934675326atural] : ( ord_le2202040463148982292atural @ ( image_220848230107539526atural @ X3 @ top_to1996260823553986621t_unit ) @ ( collec1991479681304692691atural @ ( produc7358226366420480123ural_o @ R4 ) ) ) ) )
            & ( ( comp_P210409258367696578t_unit @ produc497848011034438852atural @ Z4 )
              = A )
            & ( ( comp_P1945487127105810045t_unit @ produc6591912806276919810atural @ Z4 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_6809_fun_Oin__rel,axiom,
    ! [R4: code_natural > produc7822875418678951345atural > $o,A: int > code_natural,B: int > produc7822875418678951345atural] :
      ( ( bNF_re6290619412732081552atural
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ R4
        @ A
        @ B )
      = ( ? [Z4: int > produc5835291356934675326atural] :
            ( ( member3292482584629245920atural @ Z4
              @ ( collec5961276329923387678atural
                @ ^ [X3: int > produc5835291356934675326atural] : ( ord_le2202040463148982292atural @ ( image_6835604608220425009atural @ X3 @ top_top_set_int ) @ ( collec1991479681304692691atural @ ( produc7358226366420480123ural_o @ R4 ) ) ) ) )
            & ( ( comp_P5589825499870570807al_int @ produc497848011034438852atural @ Z4 )
              = A )
            & ( ( comp_P8475394669922486716al_int @ produc6591912806276919810atural @ Z4 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_6810_fun_Oin__rel,axiom,
    ! [R4: ( nat > $o ) > ( nat > $o ) > $o,A: int > nat > $o,B: int > nat > $o] :
      ( ( bNF_re3376528473927230327_nat_o
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ R4
        @ A
        @ B )
      = ( ? [Z4: int > produc7360885484439884007_nat_o] :
            ( ( member7306185171444233115_nat_o @ Z4
              @ ( collec2839945811840875229_nat_o
                @ ^ [X3: int > produc7360885484439884007_nat_o] : ( ord_le3745675095661657703_nat_o @ ( image_259982842891744372_nat_o @ X3 @ top_top_set_int ) @ ( collec5110290359514967890_nat_o @ ( produc2845945524848125740at_o_o @ R4 ) ) ) ) )
            & ( ( comp_P1075760313067033319_o_int @ produc6972895793295859523_nat_o @ Z4 )
              = A )
            & ( ( comp_P1075760313067033319_o_int @ produc6795044007072221829_nat_o @ Z4 )
              = B ) ) ) ) ).

% fun.in_rel
thf(fact_6811_apfst__apsnd,axiom,
    ! [F: code_integer > int > option6357759511663192854e_term,G: code_integer > product_prod_int_int,X: produc8923325533196201883nteger] :
      ( ( produc5167107986097539517nt_int @ F @ ( produc3535056655957556987nteger @ G @ X ) )
      = ( produc4305682042979456191nt_int @ ( F @ ( produc8508995932063986495nteger @ X ) ) @ ( G @ ( produc6174133586879617921nteger @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_6812_apfst__apsnd,axiom,
    ! [F: int > int > option6357759511663192854e_term,G: int > product_prod_int_int,X: product_prod_int_int] :
      ( ( produc7862092043137183022nt_int @ F @ ( produc6100211334659168315nt_int @ G @ X ) )
      = ( produc4305682042979456191nt_int @ ( F @ ( product_fst_int_int @ X ) ) @ ( G @ ( product_snd_int_int @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_6813_apfst__apsnd,axiom,
    ! [F: code_integer > produc3658429121746597890et_nat > $o,G: code_integer > produc3658429121746597890et_nat,X: produc8923325533196201883nteger] :
      ( ( produc4644964300586701942et_nat @ F @ ( produc7246245855585345468nteger @ G @ X ) )
      = ( produc5001842942810119800et_nat @ ( F @ ( produc8508995932063986495nteger @ X ) ) @ ( G @ ( produc6174133586879617921nteger @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_6814_apfst__apsnd,axiom,
    ! [F: int > produc3658429121746597890et_nat > $o,G: int > produc3658429121746597890et_nat,X: product_prod_int_int] :
      ( ( produc5259063908904861543et_nat @ F @ ( produc9121118195053341018at_int @ G @ X ) )
      = ( produc5001842942810119800et_nat @ ( F @ ( product_fst_int_int @ X ) ) @ ( G @ ( product_snd_int_int @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_6815_apfst__apsnd,axiom,
    ! [F: code_natural > int > option6357759511663192854e_term,G: produc7822875418678951345atural > product_prod_int_int,X: produc5835291356934675326atural] :
      ( ( produc7347430915296828978nt_int @ F @ ( produc5167862320620071534atural @ G @ X ) )
      = ( produc4305682042979456191nt_int @ ( F @ ( produc497848011034438852atural @ X ) ) @ ( G @ ( produc6591912806276919810atural @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_6816_apfst__apsnd,axiom,
    ! [F: code_integer > produc6241069584506657477e_term > option6357759511663192854e_term,G: code_integer > produc8923325533196201883nteger,X: produc8923325533196201883nteger] :
      ( ( produc5215714788235631558nteger @ F @ ( produc7166930624044334813nteger @ G @ X ) )
      = ( produc8603105652947943368nteger @ ( F @ ( produc8508995932063986495nteger @ X ) ) @ ( G @ ( produc6174133586879617921nteger @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_6817_apfst__apsnd,axiom,
    ! [F: code_integer > produc8551481072490612790e_term > option6357759511663192854e_term,G: code_integer > product_prod_int_int,X: produc8923325533196201883nteger] :
      ( ( produc7725767832589379507nt_int @ F @ ( produc3535056655957556987nteger @ G @ X ) )
      = ( produc5700946648718959541nt_int @ ( F @ ( produc8508995932063986495nteger @ X ) ) @ ( G @ ( produc6174133586879617921nteger @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_6818_apfst__apsnd,axiom,
    ! [F: int > produc6241069584506657477e_term > option6357759511663192854e_term,G: int > produc8923325533196201883nteger,X: product_prod_int_int] :
      ( ( produc5502177047915545527nteger @ F @ ( produc2935421404040714269er_int @ G @ X ) )
      = ( produc8603105652947943368nteger @ ( F @ ( product_fst_int_int @ X ) ) @ ( G @ ( product_snd_int_int @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_6819_apfst__apsnd,axiom,
    ! [F: int > produc8551481072490612790e_term > option6357759511663192854e_term,G: int > product_prod_int_int,X: product_prod_int_int] :
      ( ( produc9204274687749184420nt_int @ F @ ( produc6100211334659168315nt_int @ G @ X ) )
      = ( produc5700946648718959541nt_int @ ( F @ ( product_fst_int_int @ X ) ) @ ( G @ ( product_snd_int_int @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_6820_apfst__apsnd,axiom,
    ! [F: code_natural > produc3658429121746597890et_nat > $o,G: produc7822875418678951345atural > produc3658429121746597890et_nat,X: produc5835291356934675326atural] :
      ( ( produc7618241212158389867et_nat @ F @ ( produc1304257963789341151atural @ G @ X ) )
      = ( produc5001842942810119800et_nat @ ( F @ ( produc497848011034438852atural @ X ) ) @ ( G @ ( produc6591912806276919810atural @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_6821_apsnd__apfst,axiom,
    ! [F: code_integer > code_integer,G: code_integer > code_integer,X: produc8923325533196201883nteger] :
      ( ( produc6499014454317279255nteger @ F @ ( produc3195571438000033749nteger @ G @ X ) )
      = ( produc1086072967326762835nteger @ ( G @ ( produc8508995932063986495nteger @ X ) ) @ ( F @ ( produc6174133586879617921nteger @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_6822_apsnd__apfst,axiom,
    ! [F: code_integer > product_prod_int_int,G: code_integer > int > option6357759511663192854e_term,X: produc8923325533196201883nteger] :
      ( ( produc8992176017387436671e_term @ F @ ( produc3207347198985423705nteger @ G @ X ) )
      = ( produc4305682042979456191nt_int @ ( G @ ( produc8508995932063986495nteger @ X ) ) @ ( F @ ( produc6174133586879617921nteger @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_6823_apsnd__apfst,axiom,
    ! [F: int > product_prod_int_int,G: int > int > option6357759511663192854e_term,X: product_prod_int_int] :
      ( ( produc3260485439724623344e_term @ F @ ( produc2895628470685950711rm_int @ G @ X ) )
      = ( produc4305682042979456191nt_int @ ( G @ ( product_fst_int_int @ X ) ) @ ( F @ ( product_snd_int_int @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_6824_apsnd__apfst,axiom,
    ! [F: code_integer > produc3658429121746597890et_nat,G: code_integer > produc3658429121746597890et_nat > $o,X: produc8923325533196201883nteger] :
      ( ( produc898583841617791278_nat_o @ F @ ( produc7528800376889492689nteger @ G @ X ) )
      = ( produc5001842942810119800et_nat @ ( G @ ( produc8508995932063986495nteger @ X ) ) @ ( F @ ( produc6174133586879617921nteger @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_6825_apsnd__apfst,axiom,
    ! [F: int > produc3658429121746597890et_nat,G: int > produc3658429121746597890et_nat > $o,X: product_prod_int_int] :
      ( ( produc3447581998352395295_nat_o @ F @ ( produc1933289243015553937_o_int @ G @ X ) )
      = ( produc5001842942810119800et_nat @ ( G @ ( product_fst_int_int @ X ) ) @ ( F @ ( product_snd_int_int @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_6826_apsnd__apfst,axiom,
    ! [F: produc7822875418678951345atural > product_prod_int_int,G: code_natural > int > option6357759511663192854e_term,X: produc5835291356934675326atural] :
      ( ( produc7977970570655791143e_term @ F @ ( produc8323928115967317034atural @ G @ X ) )
      = ( produc4305682042979456191nt_int @ ( G @ ( produc497848011034438852atural @ X ) ) @ ( F @ ( produc6591912806276919810atural @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_6827_apsnd__apfst,axiom,
    ! [F: code_integer > produc8923325533196201883nteger,G: code_integer > produc6241069584506657477e_term > option6357759511663192854e_term,X: produc8923325533196201883nteger] :
      ( ( produc4047920161794714618e_term @ F @ ( produc7628915689858001394nteger @ G @ X ) )
      = ( produc8603105652947943368nteger @ ( G @ ( produc8508995932063986495nteger @ X ) ) @ ( F @ ( produc6174133586879617921nteger @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_6828_apsnd__apfst,axiom,
    ! [F: code_integer > product_prod_int_int,G: code_integer > produc8551481072490612790e_term > option6357759511663192854e_term,X: produc8923325533196201883nteger] :
      ( ( produc6067092816631456265e_term @ F @ ( produc7602971101079744739nteger @ G @ X ) )
      = ( produc5700946648718959541nt_int @ ( G @ ( produc8508995932063986495nteger @ X ) ) @ ( F @ ( produc6174133586879617921nteger @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_6829_apsnd__apfst,axiom,
    ! [F: int > produc8923325533196201883nteger,G: int > produc6241069584506657477e_term > option6357759511663192854e_term,X: product_prod_int_int] :
      ( ( produc552091674517523435e_term @ F @ ( produc7344697412356995344rm_int @ G @ X ) )
      = ( produc8603105652947943368nteger @ ( G @ ( product_fst_int_int @ X ) ) @ ( F @ ( product_snd_int_int @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_6830_apsnd__apfst,axiom,
    ! [F: int > product_prod_int_int,G: int > produc8551481072490612790e_term > option6357759511663192854e_term,X: product_prod_int_int] :
      ( ( produc189803804368834042e_term @ F @ ( produc212012705979908097rm_int @ G @ X ) )
      = ( produc5700946648718959541nt_int @ ( G @ ( product_fst_int_int @ X ) ) @ ( F @ ( product_snd_int_int @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_6831_eq__snd__iff,axiom,
    ! [B: produc8923325533196201883nteger,P4: produc1908205239877642774nteger] :
      ( ( B
        = ( produc7856867400915047194nteger @ P4 ) )
      = ( ? [A5: produc6241069584506657477e_term > option6357759511663192854e_term] :
            ( P4
            = ( produc8603105652947943368nteger @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_6832_eq__snd__iff,axiom,
    ! [B: produc3658429121746597890et_nat,P4: produc3925858234332021118et_nat] :
      ( ( B
        = ( produc4011572625026189258et_nat @ P4 ) )
      = ( ? [A5: produc3658429121746597890et_nat > $o] :
            ( P4
            = ( produc5001842942810119800et_nat @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_6833_eq__snd__iff,axiom,
    ! [B: produc3925858234332021118et_nat,P4: produc2732055786443039994et_nat] :
      ( ( B
        = ( produc5374455773327741254et_nat @ P4 ) )
      = ( ? [A5: produc3658429121746597890et_nat > $o] :
            ( P4
            = ( produc2245416461498447860et_nat @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_6834_eq__snd__iff,axiom,
    ! [B: product_prod_int_int,P4: produc2285326912895808259nt_int] :
      ( ( B
        = ( produc7328097813583171335nt_int @ P4 ) )
      = ( ? [A5: produc8551481072490612790e_term > option6357759511663192854e_term] :
            ( P4
            = ( produc5700946648718959541nt_int @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_6835_eq__snd__iff,axiom,
    ! [B: product_prod_int_int,P4: produc7773217078559923341nt_int] :
      ( ( B
        = ( produc3162348030201620241nt_int @ P4 ) )
      = ( ? [A5: int > option6357759511663192854e_term] :
            ( P4
            = ( produc4305682042979456191nt_int @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_6836_eq__snd__iff,axiom,
    ! [B: code_integer,P4: produc8923325533196201883nteger] :
      ( ( B
        = ( produc6174133586879617921nteger @ P4 ) )
      = ( ? [A5: code_integer] :
            ( P4
            = ( produc1086072967326762835nteger @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_6837_eq__snd__iff,axiom,
    ! [B: int,P4: product_prod_int_int] :
      ( ( B
        = ( product_snd_int_int @ P4 ) )
      = ( ? [A5: int] :
            ( P4
            = ( product_Pair_int_int @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_6838_eq__snd__iff,axiom,
    ! [B: produc7822875418678951345atural,P4: produc5835291356934675326atural] :
      ( ( B
        = ( produc6591912806276919810atural @ P4 ) )
      = ( ? [A5: code_natural] :
            ( P4
            = ( produc6639722614265839536atural @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_6839_SigmaI,axiom,
    ! [A: $o,A2: set_o,B: $o,B5: $o > set_o] :
      ( ( member_o @ A @ A2 )
     => ( ( member_o @ B @ ( B5 @ A ) )
       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A2 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_6840_SigmaI,axiom,
    ! [A: $o,A2: set_o,B: nat,B5: $o > set_nat] :
      ( ( member_o @ A @ A2 )
     => ( ( member_nat @ B @ ( B5 @ A ) )
       => ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A @ B ) @ ( product_Sigma_o_nat @ A2 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_6841_SigmaI,axiom,
    ! [A: $o,A2: set_o,B: int,B5: $o > set_int] :
      ( ( member_o @ A @ A2 )
     => ( ( member_int @ B @ ( B5 @ A ) )
       => ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A @ B ) @ ( product_Sigma_o_int @ A2 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_6842_SigmaI,axiom,
    ! [A: nat,A2: set_nat,B: $o,B5: nat > set_o] :
      ( ( member_nat @ A @ A2 )
     => ( ( member_o @ B @ ( B5 @ A ) )
       => ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A @ B ) @ ( product_Sigma_nat_o @ A2 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_6843_SigmaI,axiom,
    ! [A: nat,A2: set_nat,B: nat,B5: nat > set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( ( member_nat @ B @ ( B5 @ A ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A2 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_6844_SigmaI,axiom,
    ! [A: nat,A2: set_nat,B: int,B5: nat > set_int] :
      ( ( member_nat @ A @ A2 )
     => ( ( member_int @ B @ ( B5 @ A ) )
       => ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A @ B ) @ ( produc454536836294682310at_int @ A2 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_6845_SigmaI,axiom,
    ! [A: int,A2: set_int,B: $o,B5: int > set_o] :
      ( ( member_int @ A @ A2 )
     => ( ( member_o @ B @ ( B5 @ A ) )
       => ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A @ B ) @ ( product_Sigma_int_o @ A2 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_6846_SigmaI,axiom,
    ! [A: int,A2: set_int,B: nat,B5: int > set_nat] :
      ( ( member_int @ A @ A2 )
     => ( ( member_nat @ B @ ( B5 @ A ) )
       => ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A @ B ) @ ( produc1456381018704787142nt_nat @ A2 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_6847_SigmaI,axiom,
    ! [A: int,A2: set_int,B: int,B5: int > set_int] :
      ( ( member_int @ A @ A2 )
     => ( ( member_int @ B @ ( B5 @ A ) )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( produc1453890548195736866nt_int @ A2 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_6848_SigmaI,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: $o,B5: product_prod_nat_nat > set_o] :
      ( ( member8440522571783428010at_nat @ A @ A2 )
     => ( ( member_o @ B @ ( B5 @ A ) )
       => ( member2200362271989506215_nat_o @ ( produc855456346497908590_nat_o @ A @ B ) @ ( produc203973861852599583_nat_o @ A2 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_6849_mem__Sigma__iff,axiom,
    ! [A: $o,B: $o,A2: set_o,B5: $o > set_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A2 @ B5 ) )
      = ( ( member_o @ A @ A2 )
        & ( member_o @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_6850_mem__Sigma__iff,axiom,
    ! [A: $o,B: nat,A2: set_o,B5: $o > set_nat] :
      ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A @ B ) @ ( product_Sigma_o_nat @ A2 @ B5 ) )
      = ( ( member_o @ A @ A2 )
        & ( member_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_6851_mem__Sigma__iff,axiom,
    ! [A: $o,B: int,A2: set_o,B5: $o > set_int] :
      ( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A @ B ) @ ( product_Sigma_o_int @ A2 @ B5 ) )
      = ( ( member_o @ A @ A2 )
        & ( member_int @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_6852_mem__Sigma__iff,axiom,
    ! [A: nat,B: $o,A2: set_nat,B5: nat > set_o] :
      ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A @ B ) @ ( product_Sigma_nat_o @ A2 @ B5 ) )
      = ( ( member_nat @ A @ A2 )
        & ( member_o @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_6853_mem__Sigma__iff,axiom,
    ! [A: nat,B: int,A2: set_nat,B5: nat > set_int] :
      ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A @ B ) @ ( produc454536836294682310at_int @ A2 @ B5 ) )
      = ( ( member_nat @ A @ A2 )
        & ( member_int @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_6854_mem__Sigma__iff,axiom,
    ! [A: int,B: $o,A2: set_int,B5: int > set_o] :
      ( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A @ B ) @ ( product_Sigma_int_o @ A2 @ B5 ) )
      = ( ( member_int @ A @ A2 )
        & ( member_o @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_6855_mem__Sigma__iff,axiom,
    ! [A: int,B: nat,A2: set_int,B5: int > set_nat] :
      ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A @ B ) @ ( produc1456381018704787142nt_nat @ A2 @ B5 ) )
      = ( ( member_int @ A @ A2 )
        & ( member_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_6856_mem__Sigma__iff,axiom,
    ! [A: int,B: int,A2: set_int,B5: int > set_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( produc1453890548195736866nt_int @ A2 @ B5 ) )
      = ( ( member_int @ A @ A2 )
        & ( member_int @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_6857_mem__Sigma__iff,axiom,
    ! [A: nat,B: nat,A2: set_nat,B5: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A2 @ B5 ) )
      = ( ( member_nat @ A @ A2 )
        & ( member_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_6858_mem__Sigma__iff,axiom,
    ! [A: product_prod_nat_nat,B: $o,A2: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_o] :
      ( ( member2200362271989506215_nat_o @ ( produc855456346497908590_nat_o @ A @ B ) @ ( produc203973861852599583_nat_o @ A2 @ B5 ) )
      = ( ( member8440522571783428010at_nat @ A @ A2 )
        & ( member_o @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_6859_gcd__add1,axiom,
    ! [M2: nat,N2: nat] :
      ( ( gcd_gcd_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
      = ( gcd_gcd_nat @ M2 @ N2 ) ) ).

% gcd_add1
thf(fact_6860_gcd__add1,axiom,
    ! [M2: int,N2: int] :
      ( ( gcd_gcd_int @ ( plus_plus_int @ M2 @ N2 ) @ N2 )
      = ( gcd_gcd_int @ M2 @ N2 ) ) ).

% gcd_add1
thf(fact_6861_gcd__add2,axiom,
    ! [M2: nat,N2: nat] :
      ( ( gcd_gcd_nat @ M2 @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( gcd_gcd_nat @ M2 @ N2 ) ) ).

% gcd_add2
thf(fact_6862_gcd__add2,axiom,
    ! [M2: int,N2: int] :
      ( ( gcd_gcd_int @ M2 @ ( plus_plus_int @ M2 @ N2 ) )
      = ( gcd_gcd_int @ M2 @ N2 ) ) ).

% gcd_add2
thf(fact_6863_gcd_Obottom__left__bottom,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ one_one_nat @ A )
      = one_one_nat ) ).

% gcd.bottom_left_bottom
thf(fact_6864_gcd_Obottom__left__bottom,axiom,
    ! [A: int] :
      ( ( gcd_gcd_int @ one_one_int @ A )
      = one_one_int ) ).

% gcd.bottom_left_bottom
thf(fact_6865_gcd_Obottom__right__bottom,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ A @ one_one_nat )
      = one_one_nat ) ).

% gcd.bottom_right_bottom
thf(fact_6866_gcd_Obottom__right__bottom,axiom,
    ! [A: int] :
      ( ( gcd_gcd_int @ A @ one_one_int )
      = one_one_int ) ).

% gcd.bottom_right_bottom
thf(fact_6867_apfst__conv,axiom,
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc6241069584506657477e_term > option6357759511663192854e_term,X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] :
      ( ( produc2515316116081422825nteger @ F @ ( produc8603105652947943368nteger @ X @ Y ) )
      = ( produc8603105652947943368nteger @ ( F @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_6868_apfst__conv,axiom,
    ! [F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat] :
      ( ( produc3198230245617193214et_nat @ F @ ( produc5001842942810119800et_nat @ X @ Y ) )
      = ( produc5001842942810119800et_nat @ ( F @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_6869_apfst__conv,axiom,
    ! [F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,X: produc3658429121746597890et_nat > $o,Y: produc3925858234332021118et_nat] :
      ( ( produc1442628905706551930et_nat @ F @ ( produc2245416461498447860et_nat @ X @ Y ) )
      = ( produc2245416461498447860et_nat @ ( F @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_6870_apfst__conv,axiom,
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > produc8551481072490612790e_term > option6357759511663192854e_term,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( produc7277921532047535397nt_int @ F @ ( produc5700946648718959541nt_int @ X @ Y ) )
      = ( produc5700946648718959541nt_int @ ( F @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_6871_apfst__conv,axiom,
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > int > option6357759511663192854e_term,X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( produc6919319362177967663nt_int @ F @ ( produc5700946648718959541nt_int @ X @ Y ) )
      = ( produc4305682042979456191nt_int @ ( F @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_6872_apfst__conv,axiom,
    ! [F: ( int > option6357759511663192854e_term ) > produc8551481072490612790e_term > option6357759511663192854e_term,X: int > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( produc6067946493426971951nt_int @ F @ ( produc4305682042979456191nt_int @ X @ Y ) )
      = ( produc5700946648718959541nt_int @ ( F @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_6873_apfst__conv,axiom,
    ! [F: ( int > option6357759511663192854e_term ) > int > option6357759511663192854e_term,X: int > option6357759511663192854e_term,Y: product_prod_int_int] :
      ( ( produc7131893314732144953nt_int @ F @ ( produc4305682042979456191nt_int @ X @ Y ) )
      = ( produc4305682042979456191nt_int @ ( F @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_6874_fst__apfst,axiom,
    ! [F: code_integer > code_integer,X: produc8923325533196201883nteger] :
      ( ( produc8508995932063986495nteger @ ( produc3195571438000033749nteger @ F @ X ) )
      = ( F @ ( produc8508995932063986495nteger @ X ) ) ) ).

% fst_apfst
thf(fact_6875_fst__apfst,axiom,
    ! [F: int > int,X: product_prod_int_int] :
      ( ( product_fst_int_int @ ( produc2355237562570658114nt_int @ F @ X ) )
      = ( F @ ( product_fst_int_int @ X ) ) ) ).

% fst_apfst
thf(fact_6876_snd__apfst,axiom,
    ! [F: code_integer > code_integer,X: produc8923325533196201883nteger] :
      ( ( produc6174133586879617921nteger @ ( produc3195571438000033749nteger @ F @ X ) )
      = ( produc6174133586879617921nteger @ X ) ) ).

% snd_apfst
thf(fact_6877_snd__apfst,axiom,
    ! [F: int > int,X: product_prod_int_int] :
      ( ( product_snd_int_int @ ( produc2355237562570658114nt_int @ F @ X ) )
      = ( product_snd_int_int @ X ) ) ).

% snd_apfst
thf(fact_6878_snd__apfst,axiom,
    ! [F: code_natural > code_natural,X: produc5835291356934675326atural] :
      ( ( produc6591912806276919810atural @ ( produc4623092060929046819atural @ F @ X ) )
      = ( produc6591912806276919810atural @ X ) ) ).

% snd_apfst
thf(fact_6879_Collect__case__prod,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [A5: nat,B4: nat] :
              ( ( P @ A5 )
              & ( Q @ B4 ) ) ) )
      = ( produc457027306803732586at_nat @ ( collect_nat @ P )
        @ ^ [Uu: nat] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_6880_Collect__case__prod,axiom,
    ! [P: nat > $o,Q: list_nat > $o] :
      ( ( collec1907656820224499068st_nat
        @ ( produc5265810065896090426_nat_o
          @ ^ [A5: nat,B4: list_nat] :
              ( ( P @ A5 )
              & ( Q @ B4 ) ) ) )
      = ( produc5227933103413159546st_nat @ ( collect_nat @ P )
        @ ^ [Uu: nat] : ( collect_list_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_6881_Collect__case__prod,axiom,
    ! [P: list_nat > $o,Q: nat > $o] :
      ( ( collec8096645339561496572at_nat
        @ ( produc7308574044827705018_nat_o
          @ ^ [A5: list_nat,B4: nat] :
              ( ( P @ A5 )
              & ( Q @ B4 ) ) ) )
      = ( produc7466890150575444730at_nat @ ( collect_list_nat @ P )
        @ ^ [Uu: list_nat] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_6882_Collect__case__prod,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( collec1570431334306492044st_nat
        @ ( produc6382697403170970154_nat_o
          @ ^ [A5: list_nat,B4: list_nat] :
              ( ( P @ A5 )
              & ( Q @ B4 ) ) ) )
      = ( produc6850429418872557834st_nat @ ( collect_list_nat @ P )
        @ ^ [Uu: list_nat] : ( collect_list_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_6883_is__unit__gcd__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ one_one_nat )
      = ( ( gcd_gcd_nat @ A @ B )
        = one_one_nat ) ) ).

% is_unit_gcd_iff
thf(fact_6884_is__unit__gcd__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( gcd_gcd_int @ A @ B ) @ one_one_int )
      = ( ( gcd_gcd_int @ A @ B )
        = one_one_int ) ) ).

% is_unit_gcd_iff
thf(fact_6885_Sigma__empty1,axiom,
    ! [B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ bot_bo3083307316010499117at_nat @ B5 )
      = bot_bo228742789529271731at_nat ) ).

% Sigma_empty1
thf(fact_6886_Sigma__empty1,axiom,
    ! [B5: nat > set_nat] :
      ( ( produc457027306803732586at_nat @ bot_bot_set_nat @ B5 )
      = bot_bo2099793752762293965at_nat ) ).

% Sigma_empty1
thf(fact_6887_Sigma__empty2,axiom,
    ! [A2: set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ A2
        @ ^ [Uu: set_Pr1261947904930325089at_nat] : bot_bo3083307316010499117at_nat )
      = bot_bo228742789529271731at_nat ) ).

% Sigma_empty2
thf(fact_6888_Sigma__empty2,axiom,
    ! [A2: set_nat] :
      ( ( produc457027306803732586at_nat @ A2
        @ ^ [Uu: nat] : bot_bot_set_nat )
      = bot_bo2099793752762293965at_nat ) ).

% Sigma_empty2
thf(fact_6889_Times__empty,axiom,
    ! [A2: set_o,B5: set_o] :
      ( ( ( product_Sigma_o_o @ A2
          @ ^ [Uu: $o] : B5 )
        = bot_bo7073875226086086771od_o_o )
      = ( ( A2 = bot_bot_set_o )
        | ( B5 = bot_bot_set_o ) ) ) ).

% Times_empty
thf(fact_6890_Times__empty,axiom,
    ! [A2: set_o,B5: set_nat] :
      ( ( ( product_Sigma_o_nat @ A2
          @ ^ [Uu: $o] : B5 )
        = bot_bo66526543525301729_o_nat )
      = ( ( A2 = bot_bot_set_o )
        | ( B5 = bot_bot_set_nat ) ) ) ).

% Times_empty
thf(fact_6891_Times__empty,axiom,
    ! [A2: set_o,B5: set_int] :
      ( ( ( product_Sigma_o_int @ A2
          @ ^ [Uu: $o] : B5 )
        = bot_bo6799815435448350781_o_int )
      = ( ( A2 = bot_bot_set_o )
        | ( B5 = bot_bot_set_int ) ) ) ).

% Times_empty
thf(fact_6892_Times__empty,axiom,
    ! [A2: set_nat,B5: set_o] :
      ( ( ( product_Sigma_nat_o @ A2
          @ ^ [Uu: nat] : B5 )
        = bot_bo1114129665703605383_nat_o )
      = ( ( A2 = bot_bot_set_nat )
        | ( B5 = bot_bot_set_o ) ) ) ).

% Times_empty
thf(fact_6893_Times__empty,axiom,
    ! [A2: set_nat,B5: set_int] :
      ( ( ( produc454536836294682310at_int @ A2
          @ ^ [Uu: nat] : B5 )
        = bot_bo8833082644685343017at_int )
      = ( ( A2 = bot_bot_set_nat )
        | ( B5 = bot_bot_set_int ) ) ) ).

% Times_empty
thf(fact_6894_Times__empty,axiom,
    ! [A2: set_int,B5: set_o] :
      ( ( ( product_Sigma_int_o @ A2
          @ ^ [Uu: int] : B5 )
        = bot_bo8092356734887935275_int_o )
      = ( ( A2 = bot_bot_set_int )
        | ( B5 = bot_bot_set_o ) ) ) ).

% Times_empty
thf(fact_6895_Times__empty,axiom,
    ! [A2: set_int,B5: set_nat] :
      ( ( ( produc1456381018704787142nt_nat @ A2
          @ ^ [Uu: int] : B5 )
        = bot_bo4286715327455315753nt_nat )
      = ( ( A2 = bot_bot_set_int )
        | ( B5 = bot_bot_set_nat ) ) ) ).

% Times_empty
thf(fact_6896_Times__empty,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ( produc1453890548195736866nt_int @ A2
          @ ^ [Uu: int] : B5 )
        = bot_bo1796632182523588997nt_int )
      = ( ( A2 = bot_bot_set_int )
        | ( B5 = bot_bot_set_int ) ) ) ).

% Times_empty
thf(fact_6897_Times__empty,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( ( produc457027306803732586at_nat @ A2
          @ ^ [Uu: nat] : B5 )
        = bot_bo2099793752762293965at_nat )
      = ( ( A2 = bot_bot_set_nat )
        | ( B5 = bot_bot_set_nat ) ) ) ).

% Times_empty
thf(fact_6898_Times__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B5: set_o] :
      ( ( ( produc203973861852599583_nat_o @ A2
          @ ^ [Uu: product_prod_nat_nat] : B5 )
        = bot_bo5454027612061941066_nat_o )
      = ( ( A2 = bot_bo2099793752762293965at_nat )
        | ( B5 = bot_bot_set_o ) ) ) ).

% Times_empty
thf(fact_6899_Sigma__UNIV__cancel,axiom,
    ! [A2: set_se7855581050983116737at_nat,X11: set_se7855581050983116737at_nat] :
      ( ( minus_3314409938677909166at_nat
        @ ( produc7726050920695093128at_nat @ A2
          @ ^ [Uu: set_Pr1261947904930325089at_nat] : X11 )
        @ ( produc7726050920695093128at_nat @ A2
          @ ^ [Uu: set_Pr1261947904930325089at_nat] : top_to7629004291339433233at_nat ) )
      = bot_bo228742789529271731at_nat ) ).

% Sigma_UNIV_cancel
thf(fact_6900_Sigma__UNIV__cancel,axiom,
    ! [A2: set_nat,X11: set_nat] :
      ( ( minus_1356011639430497352at_nat
        @ ( produc457027306803732586at_nat @ A2
          @ ^ [Uu: nat] : X11 )
        @ ( produc457027306803732586at_nat @ A2
          @ ^ [Uu: nat] : top_top_set_nat ) )
      = bot_bo2099793752762293965at_nat ) ).

% Sigma_UNIV_cancel
thf(fact_6901_snd__comp__apfst,axiom,
    ! [F: code_integer > code_integer] :
      ( ( comp_P3428181540658402820nteger @ produc6174133586879617921nteger @ ( produc3195571438000033749nteger @ F ) )
      = produc6174133586879617921nteger ) ).

% snd_comp_apfst
thf(fact_6902_snd__comp__apfst,axiom,
    ! [F: int > int] :
      ( ( comp_P5356232565778667509nt_int @ product_snd_int_int @ ( produc2355237562570658114nt_int @ F ) )
      = product_snd_int_int ) ).

% snd_comp_apfst
thf(fact_6903_snd__comp__apfst,axiom,
    ! [F: code_natural > code_natural] :
      ( ( comp_P7610810914149544718atural @ produc6591912806276919810atural @ ( produc4623092060929046819atural @ F ) )
      = produc6591912806276919810atural ) ).

% snd_comp_apfst
thf(fact_6904_UNIV__Times__UNIV,axiom,
    ( ( produc457027306803732586at_nat @ top_top_set_nat
      @ ^ [Uu: nat] : top_top_set_nat )
    = top_to4669805908274784177at_nat ) ).

% UNIV_Times_UNIV
thf(fact_6905_UNIV__Times__UNIV,axiom,
    ( ( produc5506539147108004339t_unit @ top_top_set_nat
      @ ^ [Uu: nat] : top_to1996260823553986621t_unit )
    = top_to8544742955230171288t_unit ) ).

% UNIV_Times_UNIV
thf(fact_6906_UNIV__Times__UNIV,axiom,
    ( ( produc454536836294682310at_int @ top_top_set_nat
      @ ^ [Uu: nat] : top_top_set_int )
    = top_to2179722763343057421at_int ) ).

% UNIV_Times_UNIV
thf(fact_6907_UNIV__Times__UNIV,axiom,
    ( ( produc6875377002296012693it_nat @ top_to1996260823553986621t_unit
      @ ^ [Uu: product_unit] : top_top_set_nat )
    = top_to5974110478112770290it_nat ) ).

% UNIV_Times_UNIV
thf(fact_6908_UNIV__Times__UNIV,axiom,
    ( ( produc6121607364029222536t_unit @ top_to1996260823553986621t_unit
      @ ^ [Uu: product_unit] : top_to1996260823553986621t_unit )
    = top_to1835807148980544151t_unit ) ).

% UNIV_Times_UNIV
thf(fact_6909_UNIV__Times__UNIV,axiom,
    ( ( produc6872886531786962417it_int @ top_to1996260823553986621t_unit
      @ ^ [Uu: product_unit] : top_top_set_int )
    = top_to3484027333181043534it_int ) ).

% UNIV_Times_UNIV
thf(fact_6910_UNIV__Times__UNIV,axiom,
    ( ( produc1456381018704787142nt_nat @ top_top_set_int
      @ ^ [Uu: int] : top_top_set_nat )
    = top_to6856727482967805965nt_nat ) ).

% UNIV_Times_UNIV
thf(fact_6911_UNIV__Times__UNIV,axiom,
    ( ( produc2622961094310184599t_unit @ top_top_set_int
      @ ^ [Uu: int] : top_to1996260823553986621t_unit )
    = top_to876157113852465980t_unit ) ).

% UNIV_Times_UNIV
thf(fact_6912_UNIV__Times__UNIV,axiom,
    ( ( produc1453890548195736866nt_int @ top_top_set_int
      @ ^ [Uu: int] : top_top_set_int )
    = top_to4366644338036079209nt_int ) ).

% UNIV_Times_UNIV
thf(fact_6913_pairself__image__cart,axiom,
    ! [F: nat > code_integer,A2: set_nat,B5: set_nat] :
      ( ( image_5669961043262023647nteger @ ( pairse4825235148789309005nteger @ F )
        @ ( produc457027306803732586at_nat @ A2
          @ ^ [Uu: nat] : B5 ) )
      = ( produc5882431272649516420nteger @ ( image_1215581382706833972nteger @ F @ A2 )
        @ ^ [Uu: code_integer] : ( image_1215581382706833972nteger @ F @ B5 ) ) ) ).

% pairself_image_cart
thf(fact_6914_pairself__image__cart,axiom,
    ! [F: nat > set_nat,A2: set_nat,B5: set_nat] :
      ( ( image_4566292076127975857et_nat @ ( pairself_nat_set_nat @ F )
        @ ( produc457027306803732586at_nat @ A2
          @ ^ [Uu: nat] : B5 ) )
      = ( produc105401717395143638et_nat @ ( image_nat_set_nat @ F @ A2 )
        @ ^ [Uu: set_nat] : ( image_nat_set_nat @ F @ B5 ) ) ) ).

% pairself_image_cart
thf(fact_6915_pairself__image__cart,axiom,
    ! [F: nat > nat,A2: set_nat,B5: set_nat] :
      ( ( image_5168914502847457605at_nat @ ( pairself_nat_nat @ F )
        @ ( produc457027306803732586at_nat @ A2
          @ ^ [Uu: nat] : B5 ) )
      = ( produc457027306803732586at_nat @ ( image_nat_nat @ F @ A2 )
        @ ^ [Uu: nat] : ( image_nat_nat @ F @ B5 ) ) ) ).

% pairself_image_cart
thf(fact_6916_pairself__image__cart,axiom,
    ! [F: nat > int,A2: set_nat,B5: set_nat] :
      ( ( image_1990417195239315453nt_int @ ( pairself_nat_int @ F )
        @ ( produc457027306803732586at_nat @ A2
          @ ^ [Uu: nat] : B5 ) )
      = ( produc1453890548195736866nt_int @ ( image_nat_int @ F @ A2 )
        @ ^ [Uu: int] : ( image_nat_int @ F @ B5 ) ) ) ).

% pairself_image_cart
thf(fact_6917_pairself__image__cart,axiom,
    ! [F: int > int,A2: set_int,B5: set_int] :
      ( ( image_2653370878348428101nt_int @ ( pairself_int_int @ F )
        @ ( produc1453890548195736866nt_int @ A2
          @ ^ [Uu: int] : B5 ) )
      = ( produc1453890548195736866nt_int @ ( image_int_int @ F @ A2 )
        @ ^ [Uu: int] : ( image_int_int @ F @ B5 ) ) ) ).

% pairself_image_cart
thf(fact_6918_fst__image__times,axiom,
    ! [B5: set_Code_integer,A2: set_Code_integer] :
      ( ( ( B5 = bot_bo3990330152332043303nteger )
       => ( ( image_6316824473203919795nteger @ produc8508995932063986495nteger
            @ ( produc5882431272649516420nteger @ A2
              @ ^ [Uu: code_integer] : B5 ) )
          = bot_bo3990330152332043303nteger ) )
      & ( ( B5 != bot_bo3990330152332043303nteger )
       => ( ( image_6316824473203919795nteger @ produc8508995932063986495nteger
            @ ( produc5882431272649516420nteger @ A2
              @ ^ [Uu: code_integer] : B5 ) )
          = A2 ) ) ) ).

% fst_image_times
thf(fact_6919_fst__image__times,axiom,
    ! [B5: set_o,A2: set_o] :
      ( ( ( B5 = bot_bot_set_o )
       => ( ( image_7896445794123959606_o_o_o @ product_fst_o_o
            @ ( product_Sigma_o_o @ A2
              @ ^ [Uu: $o] : B5 ) )
          = bot_bot_set_o ) )
      & ( ( B5 != bot_bot_set_o )
       => ( ( image_7896445794123959606_o_o_o @ product_fst_o_o
            @ ( product_Sigma_o_o @ A2
              @ ^ [Uu: $o] : B5 ) )
          = A2 ) ) ) ).

% fst_image_times
thf(fact_6920_fst__image__times,axiom,
    ! [B5: set_o,A2: set_nat] :
      ( ( ( B5 = bot_bot_set_o )
       => ( ( image_8629803608462650132_o_nat @ product_fst_nat_o
            @ ( product_Sigma_nat_o @ A2
              @ ^ [Uu: nat] : B5 ) )
          = bot_bot_set_nat ) )
      & ( ( B5 != bot_bot_set_o )
       => ( ( image_8629803608462650132_o_nat @ product_fst_nat_o
            @ ( product_Sigma_nat_o @ A2
              @ ^ [Uu: nat] : B5 ) )
          = A2 ) ) ) ).

% fst_image_times
thf(fact_6921_fst__image__times,axiom,
    ! [B5: set_o,A2: set_int] :
      ( ( ( B5 = bot_bot_set_o )
       => ( ( image_90517134139374612_o_int @ product_fst_int_o
            @ ( product_Sigma_int_o @ A2
              @ ^ [Uu: int] : B5 ) )
          = bot_bot_set_int ) )
      & ( ( B5 != bot_bot_set_o )
       => ( ( image_90517134139374612_o_int @ product_fst_int_o
            @ ( product_Sigma_int_o @ A2
              @ ^ [Uu: int] : B5 ) )
          = A2 ) ) ) ).

% fst_image_times
thf(fact_6922_fst__image__times,axiom,
    ! [B5: set_nat,A2: set_o] :
      ( ( ( B5 = bot_bot_set_nat )
       => ( ( image_8105710389281276538_nat_o @ product_fst_o_nat
            @ ( product_Sigma_o_nat @ A2
              @ ^ [Uu: $o] : B5 ) )
          = bot_bot_set_o ) )
      & ( ( B5 != bot_bot_set_nat )
       => ( ( image_8105710389281276538_nat_o @ product_fst_o_nat
            @ ( product_Sigma_o_nat @ A2
              @ ^ [Uu: $o] : B5 ) )
          = A2 ) ) ) ).

% fst_image_times
thf(fact_6923_fst__image__times,axiom,
    ! [B5: set_nat,A2: set_nat] :
      ( ( ( B5 = bot_bot_set_nat )
       => ( ( image_2486076414777270412at_nat @ product_fst_nat_nat
            @ ( produc457027306803732586at_nat @ A2
              @ ^ [Uu: nat] : B5 ) )
          = bot_bot_set_nat ) )
      & ( ( B5 != bot_bot_set_nat )
       => ( ( image_2486076414777270412at_nat @ product_fst_nat_nat
            @ ( produc457027306803732586at_nat @ A2
              @ ^ [Uu: nat] : B5 ) )
          = A2 ) ) ) ).

% fst_image_times
thf(fact_6924_fst__image__times,axiom,
    ! [B5: set_nat,A2: set_int] :
      ( ( ( B5 = bot_bot_set_nat )
       => ( ( image_2855239504505064772at_int @ product_fst_int_nat
            @ ( produc1456381018704787142nt_nat @ A2
              @ ^ [Uu: int] : B5 ) )
          = bot_bot_set_int ) )
      & ( ( B5 != bot_bot_set_nat )
       => ( ( image_2855239504505064772at_int @ product_fst_int_nat
            @ ( produc1456381018704787142nt_nat @ A2
              @ ^ [Uu: int] : B5 ) )
          = A2 ) ) ) ).

% fst_image_times
thf(fact_6925_fst__image__times,axiom,
    ! [B5: set_int,A2: set_o] :
      ( ( ( B5 = bot_bot_set_int )
       => ( ( image_5860565421610830622_int_o @ product_fst_o_int
            @ ( product_Sigma_o_int @ A2
              @ ^ [Uu: $o] : B5 ) )
          = bot_bot_set_o ) )
      & ( ( B5 != bot_bot_set_int )
       => ( ( image_5860565421610830622_int_o @ product_fst_o_int
            @ ( product_Sigma_o_int @ A2
              @ ^ [Uu: $o] : B5 ) )
          = A2 ) ) ) ).

% fst_image_times
thf(fact_6926_fst__image__times,axiom,
    ! [B5: set_int,A2: set_nat] :
      ( ( ( B5 = bot_bot_set_int )
       => ( ( image_4672997989470292200nt_nat @ product_fst_nat_int
            @ ( produc454536836294682310at_int @ A2
              @ ^ [Uu: nat] : B5 ) )
          = bot_bot_set_nat ) )
      & ( ( B5 != bot_bot_set_int )
       => ( ( image_4672997989470292200nt_nat @ product_fst_nat_int
            @ ( produc454536836294682310at_int @ A2
              @ ^ [Uu: nat] : B5 ) )
          = A2 ) ) ) ).

% fst_image_times
thf(fact_6927_fst__image__times,axiom,
    ! [B5: set_int,A2: set_int] :
      ( ( ( B5 = bot_bot_set_int )
       => ( ( image_5042161079198086560nt_int @ product_fst_int_int
            @ ( produc1453890548195736866nt_int @ A2
              @ ^ [Uu: int] : B5 ) )
          = bot_bot_set_int ) )
      & ( ( B5 != bot_bot_set_int )
       => ( ( image_5042161079198086560nt_int @ product_fst_int_int
            @ ( produc1453890548195736866nt_int @ A2
              @ ^ [Uu: int] : B5 ) )
          = A2 ) ) ) ).

% fst_image_times
thf(fact_6928_snd__image__times,axiom,
    ! [A2: set_Code_integer,B5: set_Code_integer] :
      ( ( ( A2 = bot_bo3990330152332043303nteger )
       => ( ( image_6316824473203919795nteger @ produc6174133586879617921nteger
            @ ( produc5882431272649516420nteger @ A2
              @ ^ [Uu: code_integer] : B5 ) )
          = bot_bo3990330152332043303nteger ) )
      & ( ( A2 != bot_bo3990330152332043303nteger )
       => ( ( image_6316824473203919795nteger @ produc6174133586879617921nteger
            @ ( produc5882431272649516420nteger @ A2
              @ ^ [Uu: code_integer] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_6929_snd__image__times,axiom,
    ! [A2: set_o,B5: set_o] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( image_7896445794123959606_o_o_o @ product_snd_o_o
            @ ( product_Sigma_o_o @ A2
              @ ^ [Uu: $o] : B5 ) )
          = bot_bot_set_o ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( image_7896445794123959606_o_o_o @ product_snd_o_o
            @ ( product_Sigma_o_o @ A2
              @ ^ [Uu: $o] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_6930_snd__image__times,axiom,
    ! [A2: set_o,B5: set_nat] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( image_4888532181312399726at_nat @ product_snd_o_nat
            @ ( product_Sigma_o_nat @ A2
              @ ^ [Uu: $o] : B5 ) )
          = bot_bot_set_nat ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( image_4888532181312399726at_nat @ product_snd_o_nat
            @ ( product_Sigma_o_nat @ A2
              @ ^ [Uu: $o] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_6931_snd__image__times,axiom,
    ! [A2: set_o,B5: set_int] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( image_7072963285496371238nt_int @ product_snd_o_int
            @ ( product_Sigma_o_int @ A2
              @ ^ [Uu: $o] : B5 ) )
          = bot_bot_set_int ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( image_7072963285496371238nt_int @ product_snd_o_int
            @ ( product_Sigma_o_int @ A2
              @ ^ [Uu: $o] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_6932_snd__image__times,axiom,
    ! [A2: set_nat,B5: set_o] :
      ( ( ( A2 = bot_bot_set_nat )
       => ( ( image_7478392465989118804at_o_o @ product_snd_nat_o
            @ ( product_Sigma_nat_o @ A2
              @ ^ [Uu: nat] : B5 ) )
          = bot_bot_set_o ) )
      & ( ( A2 != bot_bot_set_nat )
       => ( ( image_7478392465989118804at_o_o @ product_snd_nat_o
            @ ( product_Sigma_nat_o @ A2
              @ ^ [Uu: nat] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_6933_snd__image__times,axiom,
    ! [A2: set_nat,B5: set_nat] :
      ( ( ( A2 = bot_bot_set_nat )
       => ( ( image_2486076414777270412at_nat @ product_snd_nat_nat
            @ ( produc457027306803732586at_nat @ A2
              @ ^ [Uu: nat] : B5 ) )
          = bot_bot_set_nat ) )
      & ( ( A2 != bot_bot_set_nat )
       => ( ( image_2486076414777270412at_nat @ product_snd_nat_nat
            @ ( produc457027306803732586at_nat @ A2
              @ ^ [Uu: nat] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_6934_snd__image__times,axiom,
    ! [A2: set_nat,B5: set_int] :
      ( ( ( A2 = bot_bot_set_nat )
       => ( ( image_4670507518961241924nt_int @ product_snd_nat_int
            @ ( produc454536836294682310at_int @ A2
              @ ^ [Uu: nat] : B5 ) )
          = bot_bot_set_int ) )
      & ( ( A2 != bot_bot_set_nat )
       => ( ( image_4670507518961241924nt_int @ product_snd_nat_int
            @ ( produc454536836294682310at_int @ A2
              @ ^ [Uu: nat] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_6935_snd__image__times,axiom,
    ! [A2: set_int,B5: set_o] :
      ( ( ( A2 = bot_bot_set_int )
       => ( ( image_5561579847027142576nt_o_o @ product_snd_int_o
            @ ( product_Sigma_int_o @ A2
              @ ^ [Uu: int] : B5 ) )
          = bot_bot_set_o ) )
      & ( ( A2 != bot_bot_set_int )
       => ( ( image_5561579847027142576nt_o_o @ product_snd_int_o
            @ ( product_Sigma_int_o @ A2
              @ ^ [Uu: int] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_6936_snd__image__times,axiom,
    ! [A2: set_int,B5: set_nat] :
      ( ( ( A2 = bot_bot_set_int )
       => ( ( image_2857729975014115048at_nat @ product_snd_int_nat
            @ ( produc1456381018704787142nt_nat @ A2
              @ ^ [Uu: int] : B5 ) )
          = bot_bot_set_nat ) )
      & ( ( A2 != bot_bot_set_int )
       => ( ( image_2857729975014115048at_nat @ product_snd_int_nat
            @ ( produc1456381018704787142nt_nat @ A2
              @ ^ [Uu: int] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_6937_snd__image__times,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( ( A2 = bot_bot_set_int )
       => ( ( image_5042161079198086560nt_int @ product_snd_int_int
            @ ( produc1453890548195736866nt_int @ A2
              @ ^ [Uu: int] : B5 ) )
          = bot_bot_set_int ) )
      & ( ( A2 != bot_bot_set_int )
       => ( ( image_5042161079198086560nt_int @ product_snd_int_int
            @ ( produc1453890548195736866nt_int @ A2
              @ ^ [Uu: int] : B5 ) )
          = B5 ) ) ) ).

% snd_image_times
thf(fact_6938_fst__comp__apfst,axiom,
    ! [F: code_integer > code_integer] :
      ( ( comp_P3428181540658402820nteger @ produc8508995932063986495nteger @ ( produc3195571438000033749nteger @ F ) )
      = ( comp_C1043215000170864188nteger @ F @ produc8508995932063986495nteger ) ) ).

% fst_comp_apfst
thf(fact_6939_fst__comp__apfst,axiom,
    ! [F: int > int] :
      ( ( comp_P5356232565778667509nt_int @ product_fst_int_int @ ( produc2355237562570658114nt_int @ F ) )
      = ( comp_i4330622015127193916nt_int @ F @ product_fst_int_int ) ) ).

% fst_comp_apfst
thf(fact_6940_fun_Omap__transfer,axiom,
    ! [Rb2: code_integer > code_integer > $o,Sd: code_integer > code_integer > $o] :
      ( bNF_re5979881370344185769nteger @ ( bNF_re1062419521925302741nteger @ Rb2 @ Sd )
      @ ( bNF_re6097285183634142137nteger
        @ ( bNF_re6741283560065411407nteger
          @ ^ [Y4: num,Z2: num] : ( Y4 = Z2 )
          @ Rb2 )
        @ ( bNF_re6741283560065411407nteger
          @ ^ [Y4: num,Z2: num] : ( Y4 = Z2 )
          @ Sd ) )
      @ comp_C3531382070062128313er_num
      @ comp_C3531382070062128313er_num ) ).

% fun.map_transfer
thf(fact_6941_fun_Omap__transfer,axiom,
    ! [Rb2: nat > nat > $o,Sd: num > num > $o] :
      ( bNF_re3846787885862361009nt_num @ ( bNF_re341483209202555957um_num @ Rb2 @ Sd )
      @ ( bNF_re9099949428469764705nt_num
        @ ( bNF_re3715656647883201625at_nat
          @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
          @ Rb2 )
        @ ( bNF_re7626690874201225453um_num
          @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
          @ Sd ) )
      @ comp_nat_num_int
      @ comp_nat_num_int ) ).

% fun.map_transfer
thf(fact_6942_fun_Omap__transfer,axiom,
    ! [Rb2: int > int > $o,Sd: ( nat > $o ) > ( nat > $o ) > $o] :
      ( bNF_re7908124382625962333_nat_o @ ( bNF_re3376528473927230327_nat_o @ Rb2 @ Sd )
      @ ( bNF_re6033114333830760525_nat_o
        @ ( bNF_re4712519889275205905nt_int
          @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
          @ Rb2 )
        @ ( bNF_re3376528473927230327_nat_o
          @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
          @ Sd ) )
      @ comp_int_nat_o_int
      @ comp_int_nat_o_int ) ).

% fun.map_transfer
thf(fact_6943_fun_Omap__transfer,axiom,
    ! [Rb2: int > int > $o,Sd: ( int > $o ) > ( int > $o ) > $o] :
      ( bNF_re1900612664124570461_int_o @ ( bNF_re3403563459893282935_int_o @ Rb2 @ Sd )
      @ ( bNF_re3014639844587090765_int_o
        @ ( bNF_re4712519889275205905nt_int
          @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
          @ Rb2 )
        @ ( bNF_re3403563459893282935_int_o
          @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
          @ Sd ) )
      @ comp_int_int_o_int
      @ comp_int_int_o_int ) ).

% fun.map_transfer
thf(fact_6944_fun_Omap__transfer,axiom,
    ! [Rb2: int > int > $o,Sd: ( int > int ) > ( int > int ) > $o] :
      ( bNF_re4984751527058004117nt_int @ ( bNF_re711492959462206631nt_int @ Rb2 @ Sd )
      @ ( bNF_re724639330072410579nt_int
        @ ( bNF_re4712519889275205905nt_int
          @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
          @ Rb2 )
        @ ( bNF_re711492959462206631nt_int
          @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
          @ Sd ) )
      @ comp_int_int_int_int
      @ comp_int_int_int_int ) ).

% fun.map_transfer
thf(fact_6945_fun_Omap__transfer,axiom,
    ! [Rb2: int > int > $o,Sd: ( int > product_prod_int_int ) > ( int > rat ) > $o] :
      ( bNF_re62019127700022579nt_rat @ ( bNF_re3461391660133120880nt_rat @ Rb2 @ Sd )
      @ ( bNF_re949775918795716016nt_rat
        @ ( bNF_re4712519889275205905nt_int
          @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
          @ Rb2 )
        @ ( bNF_re3461391660133120880nt_rat
          @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
          @ Sd ) )
      @ comp_i87437609017934369nt_int
      @ comp_int_int_rat_int ) ).

% fun.map_transfer
thf(fact_6946_fun_Omap__transfer,axiom,
    ! [Rb2: ( nat > $o ) > ( nat > $o ) > $o,Sd: ( nat > $o ) > ( nat > $o ) > $o] :
      ( bNF_re1550856938216548981_nat_o @ ( bNF_re2697311982885662685_nat_o @ Rb2 @ Sd )
      @ ( bNF_re5673586069474677853_nat_o
        @ ( bNF_re3376528473927230327_nat_o
          @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
          @ Rb2 )
        @ ( bNF_re3376528473927230327_nat_o
          @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
          @ Sd ) )
      @ comp_nat_o_nat_o_int
      @ comp_nat_o_nat_o_int ) ).

% fun.map_transfer
thf(fact_6947_fun_Omap__transfer,axiom,
    ! [Rb2: ( nat > $o ) > ( nat > $o ) > $o,Sd: ( int > $o ) > ( int > $o ) > $o] :
      ( bNF_re3491090157995142405_int_o @ ( bNF_re2724346968851715293_int_o @ Rb2 @ Sd )
      @ ( bNF_re2655111580231008093_int_o
        @ ( bNF_re3376528473927230327_nat_o
          @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
          @ Rb2 )
        @ ( bNF_re3403563459893282935_int_o
          @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
          @ Sd ) )
      @ comp_nat_o_int_o_int
      @ comp_nat_o_int_o_int ) ).

% fun.map_transfer
thf(fact_6948_fun_Omap__transfer,axiom,
    ! [Rb2: ( nat > $o ) > ( nat > $o ) > $o,Sd: ( int > int ) > ( int > int ) > $o] :
      ( bNF_re5539255234643219293nt_int @ ( bNF_re7183634743573646861nt_int @ Rb2 @ Sd )
      @ ( bNF_re7395152722673581539nt_int
        @ ( bNF_re3376528473927230327_nat_o
          @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
          @ Rb2 )
        @ ( bNF_re711492959462206631nt_int
          @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
          @ Sd ) )
      @ comp_n3090169619561918849nt_int
      @ comp_n3090169619561918849nt_int ) ).

% fun.map_transfer
thf(fact_6949_fun_Omap__transfer,axiom,
    ! [Rb2: ( int > $o ) > ( int > $o ) > $o,Sd: ( nat > $o ) > ( nat > $o ) > $o] :
      ( bNF_re615132412071444453_nat_o @ ( bNF_re2535741173671728349_nat_o @ Rb2 @ Sd )
      @ ( bNF_re2446162618891206493_nat_o
        @ ( bNF_re3403563459893282935_int_o
          @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
          @ Rb2 )
        @ ( bNF_re3376528473927230327_nat_o
          @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
          @ Sd ) )
      @ comp_int_o_nat_o_int
      @ comp_int_o_nat_o_int ) ).

% fun.map_transfer
thf(fact_6950_comp__transfer,axiom,
    ! [B5: code_integer > code_integer > $o,C4: code_integer > code_integer > $o,A2: num > num > $o] : ( bNF_re5979881370344185769nteger @ ( bNF_re1062419521925302741nteger @ B5 @ C4 ) @ ( bNF_re6097285183634142137nteger @ ( bNF_re6741283560065411407nteger @ A2 @ B5 ) @ ( bNF_re6741283560065411407nteger @ A2 @ C4 ) ) @ comp_C3531382070062128313er_num @ comp_C3531382070062128313er_num ) ).

% comp_transfer
thf(fact_6951_comp__transfer,axiom,
    ! [B5: code_integer > nat > $o,C4: code_integer > num > $o,A2: num > int > $o] : ( bNF_re5469402333911078061nt_num @ ( bNF_re1758498617148825349er_num @ B5 @ C4 ) @ ( bNF_re1151383970801924621nt_num @ ( bNF_re6324211044263125204er_nat @ A2 @ B5 ) @ ( bNF_re2881542830418903838er_num @ A2 @ C4 ) ) @ comp_C3531382070062128313er_num @ comp_nat_num_int ) ).

% comp_transfer
thf(fact_6952_comp__transfer,axiom,
    ! [B5: nat > code_integer > $o,C4: num > code_integer > $o,A2: int > num > $o] : ( bNF_re2163612987331707565nteger @ ( bNF_re8763372240999575301nteger @ B5 @ C4 ) @ ( bNF_re8894579221946749965nteger @ ( bNF_re151304249936775380nteger @ A2 @ B5 ) @ ( bNF_re7315484114762210078nteger @ A2 @ C4 ) ) @ comp_nat_num_int @ comp_C3531382070062128313er_num ) ).

% comp_transfer
thf(fact_6953_comp__transfer,axiom,
    ! [B5: nat > nat > $o,C4: num > num > $o,A2: int > int > $o] : ( bNF_re3846787885862361009nt_num @ ( bNF_re341483209202555957um_num @ B5 @ C4 ) @ ( bNF_re9099949428469764705nt_num @ ( bNF_re3715656647883201625at_nat @ A2 @ B5 ) @ ( bNF_re7626690874201225453um_num @ A2 @ C4 ) ) @ comp_nat_num_int @ comp_nat_num_int ) ).

% comp_transfer
thf(fact_6954_comp__transfer,axiom,
    ! [B5: int > int > $o,C4: ( nat > $o ) > ( nat > $o ) > $o,A2: int > int > $o] : ( bNF_re7908124382625962333_nat_o @ ( bNF_re3376528473927230327_nat_o @ B5 @ C4 ) @ ( bNF_re6033114333830760525_nat_o @ ( bNF_re4712519889275205905nt_int @ A2 @ B5 ) @ ( bNF_re3376528473927230327_nat_o @ A2 @ C4 ) ) @ comp_int_nat_o_int @ comp_int_nat_o_int ) ).

% comp_transfer
thf(fact_6955_comp__transfer,axiom,
    ! [B5: int > int > $o,C4: ( int > $o ) > ( int > $o ) > $o,A2: int > int > $o] : ( bNF_re1900612664124570461_int_o @ ( bNF_re3403563459893282935_int_o @ B5 @ C4 ) @ ( bNF_re3014639844587090765_int_o @ ( bNF_re4712519889275205905nt_int @ A2 @ B5 ) @ ( bNF_re3403563459893282935_int_o @ A2 @ C4 ) ) @ comp_int_int_o_int @ comp_int_int_o_int ) ).

% comp_transfer
thf(fact_6956_comp__transfer,axiom,
    ! [B5: int > int > $o,C4: ( int > int ) > ( int > int ) > $o,A2: int > int > $o] : ( bNF_re4984751527058004117nt_int @ ( bNF_re711492959462206631nt_int @ B5 @ C4 ) @ ( bNF_re724639330072410579nt_int @ ( bNF_re4712519889275205905nt_int @ A2 @ B5 ) @ ( bNF_re711492959462206631nt_int @ A2 @ C4 ) ) @ comp_int_int_int_int @ comp_int_int_int_int ) ).

% comp_transfer
thf(fact_6957_comp__transfer,axiom,
    ! [B5: code_integer > code_integer > $o,C4: code_integer > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o,A2: num > code_integer > $o] : ( bNF_re8134695817916415679nteger @ ( bNF_re3370092614548195229nteger @ B5 @ C4 ) @ ( bNF_re2439920061672227847nteger @ ( bNF_re5202476307641864082nteger @ A2 @ B5 ) @ ( bNF_re5031869190687302234nteger @ A2 @ C4 ) ) @ comp_C3531382070062128313er_num @ comp_C1593894019821074884nteger ) ).

% comp_transfer
thf(fact_6958_comp__transfer,axiom,
    ! [B5: code_integer > code_integer > $o,C4: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer > $o,A2: code_integer > num > $o] : ( bNF_re4919612610145136063nteger @ ( bNF_re4341081746073172381nteger @ B5 @ C4 ) @ ( bNF_re6724329528997265415nteger @ ( bNF_re7350422212947406738nteger @ A2 @ B5 ) @ ( bNF_re166502229267952730nteger @ A2 @ C4 ) ) @ comp_C1593894019821074884nteger @ comp_C3531382070062128313er_num ) ).

% comp_transfer
thf(fact_6959_comp__transfer,axiom,
    ! [B5: code_integer > nat > $o,C4: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > num > $o,A2: code_integer > int > $o] : ( bNF_re1890273300020150979nt_num @ ( bNF_re7119966052990372301er_num @ B5 @ C4 ) @ ( bNF_re5190287125156074331nt_num @ ( bNF_re2891348898900095767er_nat @ A2 @ B5 ) @ ( bNF_re8127238762813979689er_num @ A2 @ C4 ) ) @ comp_C1593894019821074884nteger @ comp_nat_num_int ) ).

% comp_transfer
thf(fact_6960_power__transfer,axiom,
    ! [R4: code_integer > code_integer > $o] :
      ( ( R4 @ one_one_Code_integer @ one_one_Code_integer )
     => ( ( bNF_re122555440370955213nteger @ R4 @ ( bNF_re1062419521925302741nteger @ R4 @ R4 ) @ times_3573771949741848930nteger @ times_3573771949741848930nteger )
       => ( bNF_re5462749793222359603nteger @ R4
          @ ( bNF_re1236659899165691067nteger
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4 )
          @ power_8256067586552552935nteger
          @ power_8256067586552552935nteger ) ) ) ).

% power_transfer
thf(fact_6961_power__transfer,axiom,
    ! [R4: code_integer > assn > $o] :
      ( ( R4 @ one_one_Code_integer @ one_one_assn )
     => ( ( bNF_re7789648203153440174n_assn @ R4 @ ( bNF_re1475883931861434731r_assn @ R4 @ R4 ) @ times_3573771949741848930nteger @ times_times_assn )
       => ( bNF_re2546461910754330953t_assn @ R4
          @ ( bNF_re2692373401909853574r_assn
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4 )
          @ power_8256067586552552935nteger
          @ power_power_assn ) ) ) ).

% power_transfer
thf(fact_6962_power__transfer,axiom,
    ! [R4: code_integer > rat > $o] :
      ( ( R4 @ one_one_Code_integer @ one_one_rat )
     => ( ( bNF_re2976496150901874110at_rat @ R4 @ ( bNF_re4698098036185532619er_rat @ R4 @ R4 ) @ times_3573771949741848930nteger @ times_times_rat )
       => ( bNF_re1167376069379428009at_rat @ R4
          @ ( bNF_re2882060999336173110er_rat
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4 )
          @ power_8256067586552552935nteger
          @ power_power_rat ) ) ) ).

% power_transfer
thf(fact_6963_power__transfer,axiom,
    ! [R4: code_integer > nat > $o] :
      ( ( R4 @ one_one_Code_integer @ one_one_nat )
     => ( ( bNF_re3981674901896804262at_nat @ R4 @ ( bNF_re5201166830993046715er_nat @ R4 @ R4 ) @ times_3573771949741848930nteger @ times_times_nat )
       => ( bNF_re4269568176766901913at_nat @ R4
          @ ( bNF_re3517191059422668846er_nat
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4 )
          @ power_8256067586552552935nteger
          @ power_power_nat ) ) ) ).

% power_transfer
thf(fact_6964_power__transfer,axiom,
    ! [R4: code_integer > int > $o] :
      ( ( R4 @ one_one_Code_integer @ one_one_int )
     => ( ( bNF_re8304916493157054650nt_int @ R4 @ ( bNF_re2888858428391045491er_int @ R4 @ R4 ) @ times_3573771949741848930nteger @ times_times_int )
       => ( bNF_re6766650671887726929at_int @ R4
          @ ( bNF_re3514700588913618570er_int
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4 )
          @ power_8256067586552552935nteger
          @ power_power_int ) ) ) ).

% power_transfer
thf(fact_6965_power__transfer,axiom,
    ! [R4: assn > code_integer > $o] :
      ( ( R4 @ one_one_assn @ one_one_Code_integer )
     => ( ( bNF_re5822757241383739566nteger @ R4 @ ( bNF_re4134162077946812779nteger @ R4 @ R4 ) @ times_times_assn @ times_3573771949741848930nteger )
       => ( bNF_re2102180915162216137nteger @ R4
          @ ( bNF_re1831131418152741254nteger
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4 )
          @ power_power_assn
          @ power_8256067586552552935nteger ) ) ) ).

% power_transfer
thf(fact_6966_power__transfer,axiom,
    ! [R4: assn > assn > $o] :
      ( ( R4 @ one_one_assn @ one_one_assn )
     => ( ( bNF_re9137223911888007311n_assn @ R4 @ ( bNF_re1143782165119297793n_assn @ R4 @ R4 ) @ times_times_assn @ times_times_assn )
       => ( bNF_re3650450707543776735t_assn @ R4
          @ ( bNF_re4051817429369991249n_assn
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4 )
          @ power_power_assn
          @ power_power_assn ) ) ) ).

% power_transfer
thf(fact_6967_power__transfer,axiom,
    ! [R4: assn > rat > $o] :
      ( ( R4 @ one_one_assn @ one_one_rat )
     => ( ( bNF_re967558844708792991at_rat @ R4 @ ( bNF_re9003240792272343649sn_rat @ R4 @ R4 ) @ times_times_assn @ times_times_rat )
       => ( bNF_re6138836914430330175at_rat @ R4
          @ ( bNF_re4159627890580825857sn_rat
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4 )
          @ power_power_assn
          @ power_power_rat ) ) ) ).

% power_transfer
thf(fact_6968_power__transfer,axiom,
    ! [R4: assn > nat > $o] :
      ( ( R4 @ one_one_assn @ one_one_nat )
     => ( ( bNF_re4386305596031132807at_nat @ R4 @ ( bNF_re6742580585434904145sn_nat @ R4 @ R4 ) @ times_times_assn @ times_times_nat )
       => ( bNF_re8560028565313717551at_nat @ R4
          @ ( bNF_re4794757950667321593sn_nat
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4 )
          @ power_power_assn
          @ power_power_nat ) ) ) ).

% power_transfer
thf(fact_6969_power__transfer,axiom,
    ! [R4: assn > int > $o] :
      ( ( R4 @ one_one_assn @ one_one_int )
     => ( ( bNF_re7882741803053012379nt_int @ R4 @ ( bNF_re2927800259774540553sn_int @ R4 @ R4 ) @ times_times_assn @ times_times_int )
       => ( bNF_re185025794348816359at_int @ R4
          @ ( bNF_re4792267480158271317sn_int
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4 )
          @ power_power_assn
          @ power_power_int ) ) ) ).

% power_transfer
thf(fact_6970_times__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    @ times_times_int
    @ times_times_int ) ).

% times_integer.rsp
thf(fact_6971_times__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    @ times_times_nat
    @ times_times_nat ) ).

% times_natural.rsp
thf(fact_6972_divide__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    @ divide_divide_nat
    @ divide_divide_nat ) ).

% divide_natural.rsp
thf(fact_6973_divide__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    @ divide_divide_int
    @ divide_divide_int ) ).

% divide_integer.rsp
thf(fact_6974_or__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    @ bit_se1412395901928357646or_nat
    @ bit_se1412395901928357646or_nat ) ).

% or_natural.rsp
thf(fact_6975_or__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    @ bit_se1409905431419307370or_int
    @ bit_se1409905431419307370or_int ) ).

% or_integer.rsp
thf(fact_6976_set__bit__integer_Orsp,axiom,
    ( bNF_re4785983289428654063nt_int
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    @ bit_se7879613467334960850it_int
    @ bit_se7879613467334960850it_int ) ).

% set_bit_integer.rsp
thf(fact_6977_set__bit__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    @ bit_se7882103937844011126it_nat
    @ bit_se7882103937844011126it_nat ) ).

% set_bit_natural.rsp
thf(fact_6978_xor__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    @ bit_se6526347334894502574or_int
    @ bit_se6526347334894502574or_int ) ).

% xor_integer.rsp
thf(fact_6979_xor__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    @ bit_se6528837805403552850or_nat
    @ bit_se6528837805403552850or_nat ) ).

% xor_natural.rsp
thf(fact_6980_and__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    @ bit_se725231765392027082nd_int
    @ bit_se725231765392027082nd_int ) ).

% and_integer.rsp
thf(fact_6981_and__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    @ bit_se727722235901077358nd_nat
    @ bit_se727722235901077358nd_nat ) ).

% and_natural.rsp
thf(fact_6982_modulo__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    @ modulo_modulo_int
    @ modulo_modulo_int ) ).

% modulo_integer.rsp
thf(fact_6983_modulo__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    @ modulo_modulo_nat
    @ modulo_modulo_nat ) ).

% modulo_natural.rsp
thf(fact_6984_less__integer_Orsp,axiom,
    ( bNF_re3403563459893282935_int_o
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ( bNF_re5089333283451836215nt_o_o
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
      @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
    @ ord_less_int
    @ ord_less_int ) ).

% less_integer.rsp
thf(fact_6985_less__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
      @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
    @ ord_less_nat
    @ ord_less_nat ) ).

% less_natural.rsp
thf(fact_6986_unset__bit__integer_Orsp,axiom,
    ( bNF_re4785983289428654063nt_int
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    @ bit_se4203085406695923979it_int
    @ bit_se4203085406695923979it_int ) ).

% unset_bit_integer.rsp
thf(fact_6987_take__bit__integer_Orsp,axiom,
    ( bNF_re4785983289428654063nt_int
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    @ bit_se2923211474154528505it_int
    @ bit_se2923211474154528505it_int ) ).

% take_bit_integer.rsp
thf(fact_6988_flip__bit__integer_Orsp,axiom,
    ( bNF_re4785983289428654063nt_int
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    @ bit_se2159334234014336723it_int
    @ bit_se2159334234014336723it_int ) ).

% flip_bit_integer.rsp
thf(fact_6989_flip__bit__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    @ bit_se2161824704523386999it_nat
    @ bit_se2161824704523386999it_nat ) ).

% flip_bit_natural.rsp
thf(fact_6990_push__bit__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    @ bit_se547839408752420682it_nat
    @ bit_se547839408752420682it_nat ) ).

% push_bit_natural.rsp
thf(fact_6991_push__bit__integer_Orsp,axiom,
    ( bNF_re4785983289428654063nt_int
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    @ bit_se545348938243370406it_int
    @ bit_se545348938243370406it_int ) ).

% push_bit_integer.rsp
thf(fact_6992_plus__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    @ plus_plus_int
    @ plus_plus_int ) ).

% plus_integer.rsp
thf(fact_6993_plus__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    @ plus_plus_nat
    @ plus_plus_nat ) ).

% plus_natural.rsp
thf(fact_6994_drop__bit__integer_Orsp,axiom,
    ( bNF_re4785983289428654063nt_int
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    @ bit_se8568078237143864401it_int
    @ bit_se8568078237143864401it_int ) ).

% drop_bit_integer.rsp
thf(fact_6995_minus__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    @ minus_minus_nat
    @ minus_minus_nat ) ).

% minus_natural.rsp
thf(fact_6996_minus__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    @ minus_minus_int
    @ minus_minus_int ) ).

% minus_integer.rsp
thf(fact_6997_less__eq__integer_Orsp,axiom,
    ( bNF_re3403563459893282935_int_o
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ( bNF_re5089333283451836215nt_o_o
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
      @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
    @ ord_less_eq_int
    @ ord_less_eq_int ) ).

% less_eq_integer.rsp
thf(fact_6998_less__eq__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
      @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
    @ ord_less_eq_nat
    @ ord_less_eq_nat ) ).

% less_eq_natural.rsp
thf(fact_6999_SigmaE,axiom,
    ! [C: product_prod_o_o,A2: set_o,B5: $o > set_o] :
      ( ( member7466972457876170832od_o_o @ C @ ( product_Sigma_o_o @ A2 @ B5 ) )
     => ~ ! [X4: $o] :
            ( ( member_o @ X4 @ A2 )
           => ! [Y5: $o] :
                ( ( member_o @ Y5 @ ( B5 @ X4 ) )
               => ( C
                 != ( product_Pair_o_o @ X4 @ Y5 ) ) ) ) ) ).

% SigmaE
thf(fact_7000_SigmaE,axiom,
    ! [C: product_prod_o_nat,A2: set_o,B5: $o > set_nat] :
      ( ( member2802428098988154798_o_nat @ C @ ( product_Sigma_o_nat @ A2 @ B5 ) )
     => ~ ! [X4: $o] :
            ( ( member_o @ X4 @ A2 )
           => ! [Y5: nat] :
                ( ( member_nat @ Y5 @ ( B5 @ X4 ) )
               => ( C
                 != ( product_Pair_o_nat @ X4 @ Y5 ) ) ) ) ) ).

% SigmaE
thf(fact_7001_SigmaE,axiom,
    ! [C: product_prod_o_int,A2: set_o,B5: $o > set_int] :
      ( ( member7847949116333733898_o_int @ C @ ( product_Sigma_o_int @ A2 @ B5 ) )
     => ~ ! [X4: $o] :
            ( ( member_o @ X4 @ A2 )
           => ! [Y5: int] :
                ( ( member_int @ Y5 @ ( B5 @ X4 ) )
               => ( C
                 != ( product_Pair_o_int @ X4 @ Y5 ) ) ) ) ) ).

% SigmaE
thf(fact_7002_SigmaE,axiom,
    ! [C: product_prod_nat_o,A2: set_nat,B5: nat > set_o] :
      ( ( member6310962623043647828_nat_o @ C @ ( product_Sigma_nat_o @ A2 @ B5 ) )
     => ~ ! [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ! [Y5: $o] :
                ( ( member_o @ Y5 @ ( B5 @ X4 ) )
               => ( C
                 != ( product_Pair_nat_o @ X4 @ Y5 ) ) ) ) ) ).

% SigmaE
thf(fact_7003_SigmaE,axiom,
    ! [C: product_prod_nat_int,A2: set_nat,B5: nat > set_int] :
      ( ( member4262671552274231302at_int @ C @ ( produc454536836294682310at_int @ A2 @ B5 ) )
     => ~ ! [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ! [Y5: int] :
                ( ( member_int @ Y5 @ ( B5 @ X4 ) )
               => ( C
                 != ( product_Pair_nat_int @ X4 @ Y5 ) ) ) ) ) ).

% SigmaE
thf(fact_7004_SigmaE,axiom,
    ! [C: product_prod_int_o,A2: set_int,B5: int > set_o] :
      ( ( member4489920277610959864_int_o @ C @ ( product_Sigma_int_o @ A2 @ B5 ) )
     => ~ ! [X4: int] :
            ( ( member_int @ X4 @ A2 )
           => ! [Y5: $o] :
                ( ( member_o @ Y5 @ ( B5 @ X4 ) )
               => ( C
                 != ( product_Pair_int_o @ X4 @ Y5 ) ) ) ) ) ).

% SigmaE
thf(fact_7005_SigmaE,axiom,
    ! [C: product_prod_int_nat,A2: set_int,B5: int > set_nat] :
      ( ( member216504246829706758nt_nat @ C @ ( produc1456381018704787142nt_nat @ A2 @ B5 ) )
     => ~ ! [X4: int] :
            ( ( member_int @ X4 @ A2 )
           => ! [Y5: nat] :
                ( ( member_nat @ Y5 @ ( B5 @ X4 ) )
               => ( C
                 != ( product_Pair_int_nat @ X4 @ Y5 ) ) ) ) ) ).

% SigmaE
thf(fact_7006_SigmaE,axiom,
    ! [C: product_prod_int_int,A2: set_int,B5: int > set_int] :
      ( ( member5262025264175285858nt_int @ C @ ( produc1453890548195736866nt_int @ A2 @ B5 ) )
     => ~ ! [X4: int] :
            ( ( member_int @ X4 @ A2 )
           => ! [Y5: int] :
                ( ( member_int @ Y5 @ ( B5 @ X4 ) )
               => ( C
                 != ( product_Pair_int_int @ X4 @ Y5 ) ) ) ) ) ).

% SigmaE
thf(fact_7007_SigmaE,axiom,
    ! [C: product_prod_nat_nat,A2: set_nat,B5: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( produc457027306803732586at_nat @ A2 @ B5 ) )
     => ~ ! [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ! [Y5: nat] :
                ( ( member_nat @ Y5 @ ( B5 @ X4 ) )
               => ( C
                 != ( product_Pair_nat_nat @ X4 @ Y5 ) ) ) ) ) ).

% SigmaE
thf(fact_7008_SigmaE,axiom,
    ! [C: produc652263959640188542_nat_o,A2: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_o] :
      ( ( member2200362271989506215_nat_o @ C @ ( produc203973861852599583_nat_o @ A2 @ B5 ) )
     => ~ ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A2 )
           => ! [Y5: $o] :
                ( ( member_o @ Y5 @ ( B5 @ X4 ) )
               => ( C
                 != ( produc855456346497908590_nat_o @ X4 @ Y5 ) ) ) ) ) ).

% SigmaE
thf(fact_7009_SigmaD1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_se7855581050983116737at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc7726050920695093128at_nat @ A2 @ B5 ) )
     => ( member2643936169264416010at_nat @ A @ A2 ) ) ).

% SigmaD1
thf(fact_7010_SigmaD1,axiom,
    ! [A: nat,B: nat,A2: set_nat,B5: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A2 @ B5 ) )
     => ( member_nat @ A @ A2 ) ) ).

% SigmaD1
thf(fact_7011_SigmaD1,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A2: set_Pr7604974323444597168e_term,B5: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > set_Pr4811707699266497531nteger] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A @ B ) @ ( produc7256824055837167575nteger @ A2 @ B5 ) )
     => ( member4242434998011752849e_term @ A @ A2 ) ) ).

% SigmaD1
thf(fact_7012_SigmaD1,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A2: set_Pr4532377907799695533_nat_o,B5: ( produc3658429121746597890et_nat > $o ) > set_Pr3948176798113811640et_nat] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ ( produc8052021730408606599et_nat @ A2 @ B5 ) )
     => ( member6576561426505652726_nat_o @ A @ A2 ) ) ).

% SigmaD1
thf(fact_7013_SigmaD1,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A2: set_Pr4532377907799695533_nat_o,B5: ( produc3658429121746597890et_nat > $o ) > set_Pr3286484037609594932et_nat] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ ( produc7016705354093686019et_nat @ A2 @ B5 ) )
     => ( member6576561426505652726_nat_o @ A @ A2 ) ) ).

% SigmaD1
thf(fact_7014_SigmaD1,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A2: set_Pr3642885161833720865e_term,B5: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr958786334691620121nt_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ ( produc5808185664642557892nt_int @ A2 @ B5 ) )
     => ( member3222579708246209666e_term @ A @ A2 ) ) ).

% SigmaD1
thf(fact_7015_SigmaD1,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A2: set_in3461395444621081367e_term,B5: ( int > option6357759511663192854e_term ) > set_Pr958786334691620121nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ ( produc1318478031590598990nt_int @ A2 @ B5 ) )
     => ( member8845023287901829240e_term @ A @ A2 ) ) ).

% SigmaD1
thf(fact_7016_SigmaD2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_se7855581050983116737at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( produc7726050920695093128at_nat @ A2 @ B5 ) )
     => ( member2643936169264416010at_nat @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_7017_SigmaD2,axiom,
    ! [A: nat,B: nat,A2: set_nat,B5: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A2 @ B5 ) )
     => ( member_nat @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_7018_SigmaD2,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A2: set_Pr7604974323444597168e_term,B5: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > set_Pr4811707699266497531nteger] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ A @ B ) @ ( produc7256824055837167575nteger @ A2 @ B5 ) )
     => ( member157494554546826820nteger @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_7019_SigmaD2,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat,A2: set_Pr4532377907799695533_nat_o,B5: ( produc3658429121746597890et_nat > $o ) > set_Pr3948176798113811640et_nat] :
      ( ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ A @ B ) @ ( produc8052021730408606599et_nat @ A2 @ B5 ) )
     => ( member6260224972018164377et_nat @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_7020_SigmaD2,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3925858234332021118et_nat,A2: set_Pr4532377907799695533_nat_o,B5: ( produc3658429121746597890et_nat > $o ) > set_Pr3286484037609594932et_nat] :
      ( ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ A @ B ) @ ( produc7016705354093686019et_nat @ A2 @ B5 ) )
     => ( member1996754912294343701et_nat @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_7021_SigmaD2,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A2: set_Pr3642885161833720865e_term,B5: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr958786334691620121nt_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ A @ B ) @ ( produc5808185664642557892nt_int @ A2 @ B5 ) )
     => ( member5262025264175285858nt_int @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_7022_SigmaD2,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A2: set_in3461395444621081367e_term,B5: ( int > option6357759511663192854e_term ) > set_Pr958786334691620121nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ A @ B ) @ ( produc1318478031590598990nt_int @ A2 @ B5 ) )
     => ( member5262025264175285858nt_int @ B @ ( B5 @ A ) ) ) ).

% SigmaD2
thf(fact_7023_SigmaE2,axiom,
    ! [A: $o,B: $o,A2: set_o,B5: $o > set_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A2 @ B5 ) )
     => ~ ( ( member_o @ A @ A2 )
         => ~ ( member_o @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_7024_SigmaE2,axiom,
    ! [A: $o,B: nat,A2: set_o,B5: $o > set_nat] :
      ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A @ B ) @ ( product_Sigma_o_nat @ A2 @ B5 ) )
     => ~ ( ( member_o @ A @ A2 )
         => ~ ( member_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_7025_SigmaE2,axiom,
    ! [A: $o,B: int,A2: set_o,B5: $o > set_int] :
      ( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ A @ B ) @ ( product_Sigma_o_int @ A2 @ B5 ) )
     => ~ ( ( member_o @ A @ A2 )
         => ~ ( member_int @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_7026_SigmaE2,axiom,
    ! [A: nat,B: $o,A2: set_nat,B5: nat > set_o] :
      ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A @ B ) @ ( product_Sigma_nat_o @ A2 @ B5 ) )
     => ~ ( ( member_nat @ A @ A2 )
         => ~ ( member_o @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_7027_SigmaE2,axiom,
    ! [A: nat,B: int,A2: set_nat,B5: nat > set_int] :
      ( ( member4262671552274231302at_int @ ( product_Pair_nat_int @ A @ B ) @ ( produc454536836294682310at_int @ A2 @ B5 ) )
     => ~ ( ( member_nat @ A @ A2 )
         => ~ ( member_int @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_7028_SigmaE2,axiom,
    ! [A: int,B: $o,A2: set_int,B5: int > set_o] :
      ( ( member4489920277610959864_int_o @ ( product_Pair_int_o @ A @ B ) @ ( product_Sigma_int_o @ A2 @ B5 ) )
     => ~ ( ( member_int @ A @ A2 )
         => ~ ( member_o @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_7029_SigmaE2,axiom,
    ! [A: int,B: nat,A2: set_int,B5: int > set_nat] :
      ( ( member216504246829706758nt_nat @ ( product_Pair_int_nat @ A @ B ) @ ( produc1456381018704787142nt_nat @ A2 @ B5 ) )
     => ~ ( ( member_int @ A @ A2 )
         => ~ ( member_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_7030_SigmaE2,axiom,
    ! [A: int,B: int,A2: set_int,B5: int > set_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( produc1453890548195736866nt_int @ A2 @ B5 ) )
     => ~ ( ( member_int @ A @ A2 )
         => ~ ( member_int @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_7031_SigmaE2,axiom,
    ! [A: nat,B: nat,A2: set_nat,B5: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( produc457027306803732586at_nat @ A2 @ B5 ) )
     => ~ ( ( member_nat @ A @ A2 )
         => ~ ( member_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_7032_SigmaE2,axiom,
    ! [A: product_prod_nat_nat,B: $o,A2: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_o] :
      ( ( member2200362271989506215_nat_o @ ( produc855456346497908590_nat_o @ A @ B ) @ ( produc203973861852599583_nat_o @ A2 @ B5 ) )
     => ~ ( ( member8440522571783428010at_nat @ A @ A2 )
         => ~ ( member_o @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_7033_Suc_Orsp,axiom,
    ( bNF_re5653821019739307937at_nat
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ suc
    @ suc ) ).

% Suc.rsp
thf(fact_7034_sub_Orsp,axiom,
    ( bNF_re8402795839162346335um_int
    @ ^ [Y4: num,Z2: num] : ( Y4 = Z2 )
    @ ( bNF_re1822329894187522285nt_int
      @ ^ [Y4: num,Z2: num] : ( Y4 = Z2 )
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    @ ^ [M: num,N4: num] : ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N4 ) )
    @ ^ [M: num,N4: num] : ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N4 ) ) ) ).

% sub.rsp
thf(fact_7035_transfer__rule__numeral,axiom,
    ! [R4: nat > nat > $o] :
      ( ( R4 @ zero_zero_nat @ zero_zero_nat )
     => ( ( R4 @ one_one_nat @ one_one_nat )
       => ( ( bNF_re1345281282404953727at_nat @ R4 @ ( bNF_re5653821019739307937at_nat @ R4 @ R4 ) @ plus_plus_nat @ plus_plus_nat )
         => ( bNF_re825466652795518005at_nat
            @ ^ [Y4: num,Z2: num] : ( Y4 = Z2 )
            @ R4
            @ numeral_numeral_nat
            @ numeral_numeral_nat ) ) ) ) ).

% transfer_rule_numeral
thf(fact_7036_transfer__rule__numeral,axiom,
    ! [R4: nat > int > $o] :
      ( ( R4 @ zero_zero_nat @ zero_zero_int )
     => ( ( R4 @ one_one_nat @ one_one_int )
       => ( ( bNF_re5864104203909894035nt_int @ R4 @ ( bNF_re6022984109467102297at_int @ R4 @ R4 ) @ plus_plus_nat @ plus_plus_int )
         => ( bNF_re822976182286467729at_int
            @ ^ [Y4: num,Z2: num] : ( Y4 = Z2 )
            @ R4
            @ numeral_numeral_nat
            @ numeral_numeral_int ) ) ) ) ).

% transfer_rule_numeral
thf(fact_7037_transfer__rule__numeral,axiom,
    ! [R4: nat > code_integer > $o] :
      ( ( R4 @ zero_zero_nat @ zero_z3403309356797280102nteger )
     => ( ( R4 @ one_one_nat @ one_one_Code_integer )
       => ( ( bNF_re7367609096550055590nteger @ R4 @ ( bNF_re1599192376174140603nteger @ R4 @ R4 ) @ plus_plus_nat @ plus_p5714425477246183910nteger )
         => ( bNF_re6129422230220670146nteger
            @ ^ [Y4: num,Z2: num] : ( Y4 = Z2 )
            @ R4
            @ numeral_numeral_nat
            @ numera6620942414471956472nteger ) ) ) ) ).

% transfer_rule_numeral
thf(fact_7038_transfer__rule__numeral,axiom,
    ! [R4: nat > rat > $o] :
      ( ( R4 @ zero_zero_nat @ zero_zero_rat )
     => ( ( R4 @ one_one_nat @ one_one_rat )
       => ( ( bNF_re3590448190763046039at_rat @ R4 @ ( bNF_re5790763769214331825at_rat @ R4 @ R4 ) @ plus_plus_nat @ plus_plus_rat )
         => ( bNF_re190336592709022269at_rat
            @ ^ [Y4: num,Z2: num] : ( Y4 = Z2 )
            @ R4
            @ numeral_numeral_nat
            @ numeral_numeral_rat ) ) ) ) ).

% transfer_rule_numeral
thf(fact_7039_transfer__rule__numeral,axiom,
    ! [R4: nat > code_natural > $o] :
      ( ( R4 @ zero_zero_nat @ zero_z2226904508553997617atural )
     => ( ( R4 @ one_one_nat @ one_one_Code_natural )
       => ( ( bNF_re88643428490162567atural @ R4 @ ( bNF_re3704215830270325841atural @ R4 @ R4 ) @ plus_plus_nat @ plus_p4538020629002901425atural )
         => ( bNF_re4953017381977387661atural
            @ ^ [Y4: num,Z2: num] : ( Y4 = Z2 )
            @ R4
            @ numeral_numeral_nat
            @ numera5444537566228673987atural ) ) ) ) ).

% transfer_rule_numeral
thf(fact_7040_transfer__rule__numeral,axiom,
    ! [R4: int > nat > $o] :
      ( ( R4 @ zero_zero_int @ zero_zero_nat )
     => ( ( R4 @ one_one_int @ one_one_nat )
       => ( ( bNF_re5416042074812042131at_nat @ R4 @ ( bNF_re4343356799547411545nt_nat @ R4 @ R4 ) @ plus_plus_int @ plus_plus_nat )
         => ( bNF_re1824820364696572561nt_nat
            @ ^ [Y4: num,Z2: num] : ( Y4 = Z2 )
            @ R4
            @ numeral_numeral_int
            @ numeral_numeral_nat ) ) ) ) ).

% transfer_rule_numeral
thf(fact_7041_transfer__rule__numeral,axiom,
    ! [R4: int > int > $o] :
      ( ( R4 @ zero_zero_int @ zero_zero_int )
     => ( ( R4 @ one_one_int @ one_one_int )
       => ( ( bNF_re711492959462206631nt_int @ R4 @ ( bNF_re4712519889275205905nt_int @ R4 @ R4 ) @ plus_plus_int @ plus_plus_int )
         => ( bNF_re1822329894187522285nt_int
            @ ^ [Y4: num,Z2: num] : ( Y4 = Z2 )
            @ R4
            @ numeral_numeral_int
            @ numeral_numeral_int ) ) ) ) ).

% transfer_rule_numeral
thf(fact_7042_transfer__rule__numeral,axiom,
    ! [R4: int > code_integer > $o] :
      ( ( R4 @ zero_zero_int @ zero_z3403309356797280102nteger )
     => ( ( R4 @ one_one_int @ one_one_Code_integer )
       => ( ( bNF_re398004352372739002nteger @ R4 @ ( bNF_re3379532845092657523nteger @ R4 @ R4 ) @ plus_plus_int @ plus_p5714425477246183910nteger )
         => ( bNF_re6501075790457514782nteger
            @ ^ [Y4: num,Z2: num] : ( Y4 = Z2 )
            @ R4
            @ numeral_numeral_int
            @ numera6620942414471956472nteger ) ) ) ) ).

% transfer_rule_numeral
thf(fact_7043_transfer__rule__numeral,axiom,
    ! [R4: int > rat > $o] :
      ( ( R4 @ zero_zero_int @ zero_zero_rat )
     => ( ( R4 @ one_one_int @ one_one_rat )
       => ( ( bNF_re7661208983170134443at_rat @ R4 @ ( bNF_re4480299549022435433nt_rat @ R4 @ R4 ) @ plus_plus_int @ plus_plus_rat )
         => ( bNF_re1189690304610076825nt_rat
            @ ^ [Y4: num,Z2: num] : ( Y4 = Z2 )
            @ R4
            @ numeral_numeral_int
            @ numeral_numeral_rat ) ) ) ) ).

% transfer_rule_numeral
thf(fact_7044_transfer__rule__numeral,axiom,
    ! [R4: int > code_natural > $o] :
      ( ( R4 @ zero_zero_int @ zero_z2226904508553997617atural )
     => ( ( R4 @ one_one_int @ one_one_Code_natural )
       => ( ( bNF_re2342410721167621787atural @ R4 @ ( bNF_re5484556299188842761atural @ R4 @ R4 ) @ plus_plus_int @ plus_p4538020629002901425atural )
         => ( bNF_re5324670942214232297atural
            @ ^ [Y4: num,Z2: num] : ( Y4 = Z2 )
            @ R4
            @ numeral_numeral_int
            @ numera5444537566228673987atural ) ) ) ) ).

% transfer_rule_numeral
thf(fact_7045_uminus__integer_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ uminus_uminus_int
    @ uminus_uminus_int ) ).

% uminus_integer.rsp
thf(fact_7046_integer__of__natural_Orsp,axiom,
    ( bNF_re6650684261131312217nt_int
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ semiri1314217659103216013at_int
    @ semiri1314217659103216013at_int ) ).

% integer_of_natural.rsp
thf(fact_7047_natural__of__integer_Orsp,axiom,
    ( bNF_re3715656647883201625at_nat
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ nat2
    @ nat2 ) ).

% natural_of_integer.rsp
thf(fact_7048_abs__integer_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ abs_abs_int
    @ abs_abs_int ) ).

% abs_integer.rsp
thf(fact_7049_bit__integer_Orsp,axiom,
    ( bNF_re3376528473927230327_nat_o
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ^ [Y4: nat > $o,Z2: nat > $o] : ( Y4 = Z2 )
    @ bit_se1146084159140164899it_int
    @ bit_se1146084159140164899it_int ) ).

% bit_integer.rsp
thf(fact_7050_not__integer_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ bit_ri7919022796975470100ot_int
    @ bit_ri7919022796975470100ot_int ) ).

% not_integer.rsp
thf(fact_7051_sgn__integer_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ sgn_sgn_int
    @ sgn_sgn_int ) ).

% sgn_integer.rsp
thf(fact_7052_mask__integer_Orsp,axiom,
    ( bNF_re6650684261131312217nt_int
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ bit_se2000444600071755411sk_int
    @ bit_se2000444600071755411sk_int ) ).

% mask_integer.rsp
thf(fact_7053_mask__natural_Orsp,axiom,
    ( bNF_re5653821019739307937at_nat
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ bit_se2002935070580805687sk_nat
    @ bit_se2002935070580805687sk_nat ) ).

% mask_natural.rsp
thf(fact_7054_transfer__rule__of__int,axiom,
    ! [R4: int > int > $o] :
      ( ( R4 @ zero_zero_int @ zero_zero_int )
     => ( ( R4 @ one_one_int @ one_one_int )
       => ( ( bNF_re711492959462206631nt_int @ R4 @ ( bNF_re4712519889275205905nt_int @ R4 @ R4 ) @ plus_plus_int @ plus_plus_int )
         => ( ( bNF_re4712519889275205905nt_int @ R4 @ R4 @ uminus_uminus_int @ uminus_uminus_int )
           => ( bNF_re4712519889275205905nt_int
              @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
              @ R4
              @ ring_1_of_int_int
              @ ring_1_of_int_int ) ) ) ) ) ).

% transfer_rule_of_int
thf(fact_7055_transfer__rule__of__int,axiom,
    ! [R4: int > code_integer > $o] :
      ( ( R4 @ zero_zero_int @ zero_z3403309356797280102nteger )
     => ( ( R4 @ one_one_int @ one_one_Code_integer )
       => ( ( bNF_re398004352372739002nteger @ R4 @ ( bNF_re3379532845092657523nteger @ R4 @ R4 ) @ plus_plus_int @ plus_p5714425477246183910nteger )
         => ( ( bNF_re3379532845092657523nteger @ R4 @ R4 @ uminus_uminus_int @ uminus1351360451143612070nteger )
           => ( bNF_re982302072995117890nteger
              @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
              @ R4
              @ ring_1_of_int_int
              @ ring_18347121197199848620nteger ) ) ) ) ) ).

% transfer_rule_of_int
thf(fact_7056_transfer__rule__of__int,axiom,
    ! [R4: int > rat > $o] :
      ( ( R4 @ zero_zero_int @ zero_zero_rat )
     => ( ( R4 @ one_one_int @ one_one_rat )
       => ( ( bNF_re7661208983170134443at_rat @ R4 @ ( bNF_re4480299549022435433nt_rat @ R4 @ R4 ) @ plus_plus_int @ plus_plus_rat )
         => ( ( bNF_re4480299549022435433nt_rat @ R4 @ R4 @ uminus_uminus_int @ uminus_uminus_rat )
           => ( bNF_re4079880299697760445nt_rat
              @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
              @ R4
              @ ring_1_of_int_int
              @ ring_1_of_int_rat ) ) ) ) ) ).

% transfer_rule_of_int
thf(fact_7057_transfer__rule__of__int,axiom,
    ! [R4: code_integer > int > $o] :
      ( ( R4 @ zero_z3403309356797280102nteger @ zero_zero_int )
     => ( ( R4 @ one_one_Code_integer @ one_one_int )
       => ( ( bNF_re8304916493157054650nt_int @ R4 @ ( bNF_re2888858428391045491er_int @ R4 @ R4 ) @ plus_p5714425477246183910nteger @ plus_plus_int )
         => ( ( bNF_re2888858428391045491er_int @ R4 @ R4 @ uminus1351360451143612070nteger @ uminus_uminus_int )
           => ( bNF_re343602593470180162er_int
              @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
              @ R4
              @ ring_18347121197199848620nteger
              @ ring_1_of_int_int ) ) ) ) ) ).

% transfer_rule_of_int
thf(fact_7058_transfer__rule__of__int,axiom,
    ! [R4: code_integer > code_integer > $o] :
      ( ( R4 @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
     => ( ( R4 @ one_one_Code_integer @ one_one_Code_integer )
       => ( ( bNF_re122555440370955213nteger @ R4 @ ( bNF_re1062419521925302741nteger @ R4 @ R4 ) @ plus_p5714425477246183910nteger @ plus_p5714425477246183910nteger )
         => ( ( bNF_re1062419521925302741nteger @ R4 @ R4 @ uminus1351360451143612070nteger @ uminus1351360451143612070nteger )
           => ( bNF_re1784066744496875891nteger
              @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
              @ R4
              @ ring_18347121197199848620nteger
              @ ring_18347121197199848620nteger ) ) ) ) ) ).

% transfer_rule_of_int
thf(fact_7059_transfer__rule__of__int,axiom,
    ! [R4: code_integer > rat > $o] :
      ( ( R4 @ zero_z3403309356797280102nteger @ zero_zero_rat )
     => ( ( R4 @ one_one_Code_integer @ one_one_rat )
       => ( ( bNF_re2976496150901874110at_rat @ R4 @ ( bNF_re4698098036185532619er_rat @ R4 @ R4 ) @ plus_p5714425477246183910nteger @ plus_plus_rat )
         => ( ( bNF_re4698098036185532619er_rat @ R4 @ R4 @ uminus1351360451143612070nteger @ uminus_uminus_rat )
           => ( bNF_re8934335040747510510er_rat
              @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
              @ R4
              @ ring_18347121197199848620nteger
              @ ring_1_of_int_rat ) ) ) ) ) ).

% transfer_rule_of_int
thf(fact_7060_transfer__rule__of__int,axiom,
    ! [R4: rat > int > $o] :
      ( ( R4 @ zero_zero_rat @ zero_zero_int )
     => ( ( R4 @ one_one_rat @ one_one_int )
       => ( ( bNF_re8876735282041863595nt_int @ R4 @ ( bNF_re5838490730811218025at_int @ R4 @ R4 ) @ plus_plus_rat @ plus_plus_int )
         => ( ( bNF_re5838490730811218025at_int @ R4 @ R4 @ uminus_uminus_rat @ uminus_uminus_int )
           => ( bNF_re3396611533439285437at_int
              @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
              @ R4
              @ ring_1_of_int_rat
              @ ring_1_of_int_int ) ) ) ) ) ).

% transfer_rule_of_int
thf(fact_7061_transfer__rule__of__int,axiom,
    ! [R4: rat > code_integer > $o] :
      ( ( R4 @ zero_zero_rat @ zero_z3403309356797280102nteger )
     => ( ( R4 @ one_one_rat @ one_one_Code_integer )
       => ( ( bNF_re4307939118855507134nteger @ R4 @ ( bNF_re2328245411608324299nteger @ R4 @ R4 ) @ plus_plus_rat @ plus_p5714425477246183910nteger )
         => ( ( bNF_re2328245411608324299nteger @ R4 @ R4 @ uminus_uminus_rat @ uminus1351360451143612070nteger )
           => ( bNF_re1382721322319792878nteger
              @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
              @ R4
              @ ring_1_of_int_rat
              @ ring_18347121197199848620nteger ) ) ) ) ) ).

% transfer_rule_of_int
thf(fact_7062_transfer__rule__of__int,axiom,
    ! [R4: rat > rat > $o] :
      ( ( R4 @ zero_zero_rat @ zero_zero_rat )
     => ( ( R4 @ one_one_rat @ one_one_rat )
       => ( ( bNF_re6603079268895015599at_rat @ R4 @ ( bNF_re5606270390558447553at_rat @ R4 @ R4 ) @ plus_plus_rat @ plus_plus_rat )
         => ( ( bNF_re5606270390558447553at_rat @ R4 @ R4 @ uminus_uminus_rat @ uminus_uminus_rat )
           => ( bNF_re2763971943861839977at_rat
              @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
              @ R4
              @ ring_1_of_int_rat
              @ ring_1_of_int_rat ) ) ) ) ) ).

% transfer_rule_of_int
thf(fact_7063_fun_Orel__map_I1_J,axiom,
    ! [Sb: ( nat > $o ) > ( nat > $o ) > $o,I: ( nat > $o ) > nat > $o,X: int > nat > $o,Y: int > nat > $o] :
      ( ( bNF_re3376528473927230327_nat_o
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ Sb
        @ ( comp_nat_o_nat_o_int @ I @ X )
        @ Y )
      = ( bNF_re3376528473927230327_nat_o
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ ^ [X3: nat > $o] : ( Sb @ ( I @ X3 ) )
        @ X
        @ Y ) ) ).

% fun.rel_map(1)
thf(fact_7064_fun_Orel__map_I1_J,axiom,
    ! [Sb: ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o,I: ( int > product_prod_int_int ) > int > product_prod_int_int,X: int > int > product_prod_int_int,Y: int > int > product_prod_int_int] :
      ( ( bNF_re157797125943740599nt_int
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ Sb
        @ ( comp_i1790862786758963855nt_int @ I @ X )
        @ Y )
      = ( bNF_re157797125943740599nt_int
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ ^ [X3: int > product_prod_int_int] : ( Sb @ ( I @ X3 ) )
        @ X
        @ Y ) ) ).

% fun.rel_map(1)
thf(fact_7065_fun_Orel__map_I1_J,axiom,
    ! [Sb: ( int > product_prod_int_int ) > ( int > rat ) > $o,I: ( int > product_prod_int_int ) > int > product_prod_int_int,X: int > int > product_prod_int_int,Y: int > int > rat] :
      ( ( bNF_re3461391660133120880nt_rat
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ Sb
        @ ( comp_i1790862786758963855nt_int @ I @ X )
        @ Y )
      = ( bNF_re3461391660133120880nt_rat
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ ^ [X3: int > product_prod_int_int] : ( Sb @ ( I @ X3 ) )
        @ X
        @ Y ) ) ).

% fun.rel_map(1)
thf(fact_7066_fun_Orel__map_I1_J,axiom,
    ! [Sb: ( int > $o ) > ( int > $o ) > $o,I: ( int > $o ) > int > $o,X: int > int > $o,Y: int > int > $o] :
      ( ( bNF_re3403563459893282935_int_o
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ Sb
        @ ( comp_int_o_int_o_int @ I @ X )
        @ Y )
      = ( bNF_re3403563459893282935_int_o
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ ^ [X3: int > $o] : ( Sb @ ( I @ X3 ) )
        @ X
        @ Y ) ) ).

% fun.rel_map(1)
thf(fact_7067_fun_Orel__map_I1_J,axiom,
    ! [Sb: ( int > int ) > ( int > int ) > $o,I: ( int > int ) > int > int,X: int > int > int,Y: int > int > int] :
      ( ( bNF_re711492959462206631nt_int
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ Sb
        @ ( comp_i1129928922100200191nt_int @ I @ X )
        @ Y )
      = ( bNF_re711492959462206631nt_int
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ ^ [X3: int > int] : ( Sb @ ( I @ X3 ) )
        @ X
        @ Y ) ) ).

% fun.rel_map(1)
thf(fact_7068_fun_Orel__map_I2_J,axiom,
    ! [Sa: ( nat > $o ) > ( nat > $o ) > $o,X: int > nat > $o,G: ( nat > $o ) > nat > $o,Y: int > nat > $o] :
      ( ( bNF_re3376528473927230327_nat_o
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ Sa
        @ X
        @ ( comp_nat_o_nat_o_int @ G @ Y ) )
      = ( bNF_re3376528473927230327_nat_o
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ ^ [X3: nat > $o,Y3: nat > $o] : ( Sa @ X3 @ ( G @ Y3 ) )
        @ X
        @ Y ) ) ).

% fun.rel_map(2)
thf(fact_7069_fun_Orel__map_I2_J,axiom,
    ! [Sa: ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o,X: int > int > product_prod_int_int,G: ( int > product_prod_int_int ) > int > product_prod_int_int,Y: int > int > product_prod_int_int] :
      ( ( bNF_re157797125943740599nt_int
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ Sa
        @ X
        @ ( comp_i1790862786758963855nt_int @ G @ Y ) )
      = ( bNF_re157797125943740599nt_int
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ ^ [X3: int > product_prod_int_int,Y3: int > product_prod_int_int] : ( Sa @ X3 @ ( G @ Y3 ) )
        @ X
        @ Y ) ) ).

% fun.rel_map(2)
thf(fact_7070_fun_Orel__map_I2_J,axiom,
    ! [Sa: ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o,X: int > int > product_prod_int_int,G: ( int > rat ) > int > product_prod_int_int,Y: int > int > rat] :
      ( ( bNF_re157797125943740599nt_int
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ Sa
        @ X
        @ ( comp_i929186717396404458nt_int @ G @ Y ) )
      = ( bNF_re3461391660133120880nt_rat
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ ^ [X3: int > product_prod_int_int,Y3: int > rat] : ( Sa @ X3 @ ( G @ Y3 ) )
        @ X
        @ Y ) ) ).

% fun.rel_map(2)
thf(fact_7071_fun_Orel__map_I2_J,axiom,
    ! [Sa: ( int > product_prod_int_int ) > ( int > rat ) > $o,X: int > int > product_prod_int_int,G: ( int > product_prod_int_int ) > int > rat,Y: int > int > product_prod_int_int] :
      ( ( bNF_re3461391660133120880nt_rat
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ Sa
        @ X
        @ ( comp_i1302866070986779890at_int @ G @ Y ) )
      = ( bNF_re157797125943740599nt_int
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ ^ [X3: int > product_prod_int_int,Y3: int > product_prod_int_int] : ( Sa @ X3 @ ( G @ Y3 ) )
        @ X
        @ Y ) ) ).

% fun.rel_map(2)
thf(fact_7072_fun_Orel__map_I2_J,axiom,
    ! [Sa: ( int > product_prod_int_int ) > ( int > rat ) > $o,X: int > int > product_prod_int_int,G: ( int > rat ) > int > rat,Y: int > int > rat] :
      ( ( bNF_re3461391660133120880nt_rat
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ Sa
        @ X
        @ ( comp_i8258899369942393943at_int @ G @ Y ) )
      = ( bNF_re3461391660133120880nt_rat
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ ^ [X3: int > product_prod_int_int,Y3: int > rat] : ( Sa @ X3 @ ( G @ Y3 ) )
        @ X
        @ Y ) ) ).

% fun.rel_map(2)
thf(fact_7073_fun_Orel__map_I2_J,axiom,
    ! [Sa: ( int > $o ) > ( int > $o ) > $o,X: int > int > $o,G: ( int > $o ) > int > $o,Y: int > int > $o] :
      ( ( bNF_re3403563459893282935_int_o
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ Sa
        @ X
        @ ( comp_int_o_int_o_int @ G @ Y ) )
      = ( bNF_re3403563459893282935_int_o
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ ^ [X3: int > $o,Y3: int > $o] : ( Sa @ X3 @ ( G @ Y3 ) )
        @ X
        @ Y ) ) ).

% fun.rel_map(2)
thf(fact_7074_fun_Orel__map_I2_J,axiom,
    ! [Sa: ( int > int ) > ( int > int ) > $o,X: int > int > int,G: ( int > int ) > int > int,Y: int > int > int] :
      ( ( bNF_re711492959462206631nt_int
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ Sa
        @ X
        @ ( comp_i1129928922100200191nt_int @ G @ Y ) )
      = ( bNF_re711492959462206631nt_int
        @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
        @ ^ [X3: int > int,Y3: int > int] : ( Sa @ X3 @ ( G @ Y3 ) )
        @ X
        @ Y ) ) ).

% fun.rel_map(2)
thf(fact_7075_times__eq__iff,axiom,
    ! [A2: set_o,B5: set_o,C4: set_o,D3: set_o] :
      ( ( ( product_Sigma_o_o @ A2
          @ ^ [Uu: $o] : B5 )
        = ( product_Sigma_o_o @ C4
          @ ^ [Uu: $o] : D3 ) )
      = ( ( ( A2 = C4 )
          & ( B5 = D3 ) )
        | ( ( ( A2 = bot_bot_set_o )
            | ( B5 = bot_bot_set_o ) )
          & ( ( C4 = bot_bot_set_o )
            | ( D3 = bot_bot_set_o ) ) ) ) ) ).

% times_eq_iff
thf(fact_7076_times__eq__iff,axiom,
    ! [A2: set_o,B5: set_nat,C4: set_o,D3: set_nat] :
      ( ( ( product_Sigma_o_nat @ A2
          @ ^ [Uu: $o] : B5 )
        = ( product_Sigma_o_nat @ C4
          @ ^ [Uu: $o] : D3 ) )
      = ( ( ( A2 = C4 )
          & ( B5 = D3 ) )
        | ( ( ( A2 = bot_bot_set_o )
            | ( B5 = bot_bot_set_nat ) )
          & ( ( C4 = bot_bot_set_o )
            | ( D3 = bot_bot_set_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_7077_times__eq__iff,axiom,
    ! [A2: set_o,B5: set_int,C4: set_o,D3: set_int] :
      ( ( ( product_Sigma_o_int @ A2
          @ ^ [Uu: $o] : B5 )
        = ( product_Sigma_o_int @ C4
          @ ^ [Uu: $o] : D3 ) )
      = ( ( ( A2 = C4 )
          & ( B5 = D3 ) )
        | ( ( ( A2 = bot_bot_set_o )
            | ( B5 = bot_bot_set_int ) )
          & ( ( C4 = bot_bot_set_o )
            | ( D3 = bot_bot_set_int ) ) ) ) ) ).

% times_eq_iff
thf(fact_7078_times__eq__iff,axiom,
    ! [A2: set_nat,B5: set_o,C4: set_nat,D3: set_o] :
      ( ( ( product_Sigma_nat_o @ A2
          @ ^ [Uu: nat] : B5 )
        = ( product_Sigma_nat_o @ C4
          @ ^ [Uu: nat] : D3 ) )
      = ( ( ( A2 = C4 )
          & ( B5 = D3 ) )
        | ( ( ( A2 = bot_bot_set_nat )
            | ( B5 = bot_bot_set_o ) )
          & ( ( C4 = bot_bot_set_nat )
            | ( D3 = bot_bot_set_o ) ) ) ) ) ).

% times_eq_iff
thf(fact_7079_times__eq__iff,axiom,
    ! [A2: set_nat,B5: set_nat,C4: set_nat,D3: set_nat] :
      ( ( ( produc457027306803732586at_nat @ A2
          @ ^ [Uu: nat] : B5 )
        = ( produc457027306803732586at_nat @ C4
          @ ^ [Uu: nat] : D3 ) )
      = ( ( ( A2 = C4 )
          & ( B5 = D3 ) )
        | ( ( ( A2 = bot_bot_set_nat )
            | ( B5 = bot_bot_set_nat ) )
          & ( ( C4 = bot_bot_set_nat )
            | ( D3 = bot_bot_set_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_7080_times__eq__iff,axiom,
    ! [A2: set_nat,B5: set_int,C4: set_nat,D3: set_int] :
      ( ( ( produc454536836294682310at_int @ A2
          @ ^ [Uu: nat] : B5 )
        = ( produc454536836294682310at_int @ C4
          @ ^ [Uu: nat] : D3 ) )
      = ( ( ( A2 = C4 )
          & ( B5 = D3 ) )
        | ( ( ( A2 = bot_bot_set_nat )
            | ( B5 = bot_bot_set_int ) )
          & ( ( C4 = bot_bot_set_nat )
            | ( D3 = bot_bot_set_int ) ) ) ) ) ).

% times_eq_iff
thf(fact_7081_times__eq__iff,axiom,
    ! [A2: set_int,B5: set_o,C4: set_int,D3: set_o] :
      ( ( ( product_Sigma_int_o @ A2
          @ ^ [Uu: int] : B5 )
        = ( product_Sigma_int_o @ C4
          @ ^ [Uu: int] : D3 ) )
      = ( ( ( A2 = C4 )
          & ( B5 = D3 ) )
        | ( ( ( A2 = bot_bot_set_int )
            | ( B5 = bot_bot_set_o ) )
          & ( ( C4 = bot_bot_set_int )
            | ( D3 = bot_bot_set_o ) ) ) ) ) ).

% times_eq_iff
thf(fact_7082_times__eq__iff,axiom,
    ! [A2: set_int,B5: set_nat,C4: set_int,D3: set_nat] :
      ( ( ( produc1456381018704787142nt_nat @ A2
          @ ^ [Uu: int] : B5 )
        = ( produc1456381018704787142nt_nat @ C4
          @ ^ [Uu: int] : D3 ) )
      = ( ( ( A2 = C4 )
          & ( B5 = D3 ) )
        | ( ( ( A2 = bot_bot_set_int )
            | ( B5 = bot_bot_set_nat ) )
          & ( ( C4 = bot_bot_set_int )
            | ( D3 = bot_bot_set_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_7083_times__eq__iff,axiom,
    ! [A2: set_int,B5: set_int,C4: set_int,D3: set_int] :
      ( ( ( produc1453890548195736866nt_int @ A2
          @ ^ [Uu: int] : B5 )
        = ( produc1453890548195736866nt_int @ C4
          @ ^ [Uu: int] : D3 ) )
      = ( ( ( A2 = C4 )
          & ( B5 = D3 ) )
        | ( ( ( A2 = bot_bot_set_int )
            | ( B5 = bot_bot_set_int ) )
          & ( ( C4 = bot_bot_set_int )
            | ( D3 = bot_bot_set_int ) ) ) ) ) ).

% times_eq_iff
thf(fact_7084_times__eq__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B5: set_o,C4: set_Pr1261947904930325089at_nat,D3: set_o] :
      ( ( ( produc203973861852599583_nat_o @ A2
          @ ^ [Uu: product_prod_nat_nat] : B5 )
        = ( produc203973861852599583_nat_o @ C4
          @ ^ [Uu: product_prod_nat_nat] : D3 ) )
      = ( ( ( A2 = C4 )
          & ( B5 = D3 ) )
        | ( ( ( A2 = bot_bo2099793752762293965at_nat )
            | ( B5 = bot_bot_set_o ) )
          & ( ( C4 = bot_bo2099793752762293965at_nat )
            | ( D3 = bot_bot_set_o ) ) ) ) ) ).

% times_eq_iff
thf(fact_7085_dup_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ^ [K4: int] : ( plus_plus_int @ K4 @ K4 )
    @ ^ [K4: int] : ( plus_plus_int @ K4 @ K4 ) ) ).

% dup.rsp
thf(fact_7086_Collect__case__prod__Sigma,axiom,
    ! [P: nat > $o,Q: nat > nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X3: nat,Y3: nat] :
              ( ( P @ X3 )
              & ( Q @ X3 @ Y3 ) ) ) )
      = ( produc457027306803732586at_nat @ ( collect_nat @ P )
        @ ^ [X3: nat] : ( collect_nat @ ( Q @ X3 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_7087_Collect__case__prod__Sigma,axiom,
    ! [P: nat > $o,Q: nat > list_nat > $o] :
      ( ( collec1907656820224499068st_nat
        @ ( produc5265810065896090426_nat_o
          @ ^ [X3: nat,Y3: list_nat] :
              ( ( P @ X3 )
              & ( Q @ X3 @ Y3 ) ) ) )
      = ( produc5227933103413159546st_nat @ ( collect_nat @ P )
        @ ^ [X3: nat] : ( collect_list_nat @ ( Q @ X3 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_7088_Collect__case__prod__Sigma,axiom,
    ! [P: list_nat > $o,Q: list_nat > nat > $o] :
      ( ( collec8096645339561496572at_nat
        @ ( produc7308574044827705018_nat_o
          @ ^ [X3: list_nat,Y3: nat] :
              ( ( P @ X3 )
              & ( Q @ X3 @ Y3 ) ) ) )
      = ( produc7466890150575444730at_nat @ ( collect_list_nat @ P )
        @ ^ [X3: list_nat] : ( collect_nat @ ( Q @ X3 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_7089_Collect__case__prod__Sigma,axiom,
    ! [P: list_nat > $o,Q: list_nat > list_nat > $o] :
      ( ( collec1570431334306492044st_nat
        @ ( produc6382697403170970154_nat_o
          @ ^ [X3: list_nat,Y3: list_nat] :
              ( ( P @ X3 )
              & ( Q @ X3 @ Y3 ) ) ) )
      = ( produc6850429418872557834st_nat @ ( collect_list_nat @ P )
        @ ^ [X3: list_nat] : ( collect_list_nat @ ( Q @ X3 ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_7090_apsnd__apfst__commute,axiom,
    ! [F: code_integer > code_integer,G: code_integer > code_integer,P4: produc8923325533196201883nteger] :
      ( ( produc6499014454317279255nteger @ F @ ( produc3195571438000033749nteger @ G @ P4 ) )
      = ( produc3195571438000033749nteger @ G @ ( produc6499014454317279255nteger @ F @ P4 ) ) ) ).

% apsnd_apfst_commute
thf(fact_7091_gcd__add__mult,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( gcd_gcd_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ N2 ) )
      = ( gcd_gcd_nat @ M2 @ N2 ) ) ).

% gcd_add_mult
thf(fact_7092_gcd__add__mult,axiom,
    ! [M2: int,K: int,N2: int] :
      ( ( gcd_gcd_int @ M2 @ ( plus_plus_int @ ( times_times_int @ K @ M2 ) @ N2 ) )
      = ( gcd_gcd_int @ M2 @ N2 ) ) ).

% gcd_add_mult
thf(fact_7093_Sigma__mono,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat,B5: produc3843707927480180839at_nat > set_nat,D3: produc3843707927480180839at_nat > set_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ C4 )
     => ( ! [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ A2 )
           => ( ord_less_eq_set_nat @ ( B5 @ X4 ) @ ( D3 @ X4 ) ) )
       => ( ord_le2145644191787020796at_nat @ ( produc1095920968030776047at_nat @ A2 @ B5 ) @ ( produc1095920968030776047at_nat @ C4 @ D3 ) ) ) ) ).

% Sigma_mono
thf(fact_7094_Sigma__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat > set_nat,D3: product_prod_nat_nat > set_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ C4 )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A2 )
           => ( ord_less_eq_set_nat @ ( B5 @ X4 ) @ ( D3 @ X4 ) ) )
       => ( ord_le7094820750414311574at_nat @ ( produc7672662199629908489at_nat @ A2 @ B5 ) @ ( produc7672662199629908489at_nat @ C4 @ D3 ) ) ) ) ).

% Sigma_mono
thf(fact_7095_Sigma__mono,axiom,
    ! [A2: set_o,C4: set_o,B5: $o > set_nat,D3: $o > set_nat] :
      ( ( ord_less_eq_set_o @ A2 @ C4 )
     => ( ! [X4: $o] :
            ( ( member_o @ X4 @ A2 )
           => ( ord_less_eq_set_nat @ ( B5 @ X4 ) @ ( D3 @ X4 ) ) )
       => ( ord_le6472378536984071597_o_nat @ ( product_Sigma_o_nat @ A2 @ B5 ) @ ( product_Sigma_o_nat @ C4 @ D3 ) ) ) ) ).

% Sigma_mono
thf(fact_7096_Sigma__mono,axiom,
    ! [A2: set_int,C4: set_int,B5: int > set_nat,D3: int > set_nat] :
      ( ( ord_less_eq_set_int @ A2 @ C4 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ A2 )
           => ( ord_less_eq_set_nat @ ( B5 @ X4 ) @ ( D3 @ X4 ) ) )
       => ( ord_le5333435103577920093nt_nat @ ( produc1456381018704787142nt_nat @ A2 @ B5 ) @ ( produc1456381018704787142nt_nat @ C4 @ D3 ) ) ) ) ).

% Sigma_mono
thf(fact_7097_Sigma__mono,axiom,
    ! [A2: set_nat,C4: set_nat,B5: nat > set_nat,D3: nat > set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C4 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ord_less_eq_set_nat @ ( B5 @ X4 ) @ ( D3 @ X4 ) ) )
       => ( ord_le3146513528884898305at_nat @ ( produc457027306803732586at_nat @ A2 @ B5 ) @ ( produc457027306803732586at_nat @ C4 @ D3 ) ) ) ) ).

% Sigma_mono
thf(fact_7098_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U3: int,V4: int] :
      ( ( plus_plus_int @ ( times_times_int @ U3 @ X ) @ ( times_times_int @ V4 @ Y ) )
      = ( gcd_gcd_int @ X @ Y ) ) ).

% bezout_int
thf(fact_7099_Times__subset__cancel2,axiom,
    ! [X: produc3843707927480180839at_nat,C4: set_Pr4329608150637261639at_nat,A2: set_nat,B5: set_nat] :
      ( ( member8757157785044589968at_nat @ X @ C4 )
     => ( ( ord_le3218026494883419554at_nat
          @ ( produc396409126344608333at_nat @ A2
            @ ^ [Uu: nat] : C4 )
          @ ( produc396409126344608333at_nat @ B5
            @ ^ [Uu: nat] : C4 ) )
        = ( ord_less_eq_set_nat @ A2 @ B5 ) ) ) ).

% Times_subset_cancel2
thf(fact_7100_Times__subset__cancel2,axiom,
    ! [X: product_prod_nat_nat,C4: set_Pr1261947904930325089at_nat,A2: set_nat,B5: set_nat] :
      ( ( member8440522571783428010at_nat @ X @ C4 )
     => ( ( ord_le3050193496395145148at_nat
          @ ( produc1809337555817847783at_nat @ A2
            @ ^ [Uu: nat] : C4 )
          @ ( produc1809337555817847783at_nat @ B5
            @ ^ [Uu: nat] : C4 ) )
        = ( ord_less_eq_set_nat @ A2 @ B5 ) ) ) ).

% Times_subset_cancel2
thf(fact_7101_Times__subset__cancel2,axiom,
    ! [X: $o,C4: set_o,A2: set_nat,B5: set_nat] :
      ( ( member_o @ X @ C4 )
     => ( ( ord_le7519981659162375251_nat_o
          @ ( product_Sigma_nat_o @ A2
            @ ^ [Uu: nat] : C4 )
          @ ( product_Sigma_nat_o @ B5
            @ ^ [Uu: nat] : C4 ) )
        = ( ord_less_eq_set_nat @ A2 @ B5 ) ) ) ).

% Times_subset_cancel2
thf(fact_7102_Times__subset__cancel2,axiom,
    ! [X: nat,C4: set_nat,A2: set_nat,B5: set_nat] :
      ( ( member_nat @ X @ C4 )
     => ( ( ord_le3146513528884898305at_nat
          @ ( produc457027306803732586at_nat @ A2
            @ ^ [Uu: nat] : C4 )
          @ ( produc457027306803732586at_nat @ B5
            @ ^ [Uu: nat] : C4 ) )
        = ( ord_less_eq_set_nat @ A2 @ B5 ) ) ) ).

% Times_subset_cancel2
thf(fact_7103_Times__subset__cancel2,axiom,
    ! [X: int,C4: set_int,A2: set_nat,B5: set_nat] :
      ( ( member_int @ X @ C4 )
     => ( ( ord_le656430383953171549at_int
          @ ( produc454536836294682310at_int @ A2
            @ ^ [Uu: nat] : C4 )
          @ ( produc454536836294682310at_int @ B5
            @ ^ [Uu: nat] : C4 ) )
        = ( ord_less_eq_set_nat @ A2 @ B5 ) ) ) ).

% Times_subset_cancel2
thf(fact_7104_UN__Times__distrib,axiom,
    ! [E3: nat > set_nat,F5: nat > set_nat,A2: set_nat,B5: set_nat] :
      ( ( comple5685304695842803022at_nat
        @ ( image_5423882813909314213at_nat
          @ ( produc8197505143624133779at_nat
            @ ^ [A5: nat,B4: nat] :
                ( produc457027306803732586at_nat @ ( E3 @ A5 )
                @ ^ [Uu: nat] : ( F5 @ B4 ) ) )
          @ ( produc457027306803732586at_nat @ A2
            @ ^ [Uu: nat] : B5 ) ) )
      = ( produc457027306803732586at_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ E3 @ A2 ) )
        @ ^ [Uu: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F5 @ B5 ) ) ) ) ).

% UN_Times_distrib
thf(fact_7105_mem__Times__iff,axiom,
    ! [X: product_prod_o_o,A2: set_o,B5: set_o] :
      ( ( member7466972457876170832od_o_o @ X
        @ ( product_Sigma_o_o @ A2
          @ ^ [Uu: $o] : B5 ) )
      = ( ( member_o @ ( product_fst_o_o @ X ) @ A2 )
        & ( member_o @ ( product_snd_o_o @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_7106_mem__Times__iff,axiom,
    ! [X: product_prod_o_nat,A2: set_o,B5: set_nat] :
      ( ( member2802428098988154798_o_nat @ X
        @ ( product_Sigma_o_nat @ A2
          @ ^ [Uu: $o] : B5 ) )
      = ( ( member_o @ ( product_fst_o_nat @ X ) @ A2 )
        & ( member_nat @ ( product_snd_o_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_7107_mem__Times__iff,axiom,
    ! [X: product_prod_o_int,A2: set_o,B5: set_int] :
      ( ( member7847949116333733898_o_int @ X
        @ ( product_Sigma_o_int @ A2
          @ ^ [Uu: $o] : B5 ) )
      = ( ( member_o @ ( product_fst_o_int @ X ) @ A2 )
        & ( member_int @ ( product_snd_o_int @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_7108_mem__Times__iff,axiom,
    ! [X: product_prod_nat_o,A2: set_nat,B5: set_o] :
      ( ( member6310962623043647828_nat_o @ X
        @ ( product_Sigma_nat_o @ A2
          @ ^ [Uu: nat] : B5 ) )
      = ( ( member_nat @ ( product_fst_nat_o @ X ) @ A2 )
        & ( member_o @ ( product_snd_nat_o @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_7109_mem__Times__iff,axiom,
    ! [X: product_prod_nat_int,A2: set_nat,B5: set_int] :
      ( ( member4262671552274231302at_int @ X
        @ ( produc454536836294682310at_int @ A2
          @ ^ [Uu: nat] : B5 ) )
      = ( ( member_nat @ ( product_fst_nat_int @ X ) @ A2 )
        & ( member_int @ ( product_snd_nat_int @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_7110_mem__Times__iff,axiom,
    ! [X: product_prod_int_o,A2: set_int,B5: set_o] :
      ( ( member4489920277610959864_int_o @ X
        @ ( product_Sigma_int_o @ A2
          @ ^ [Uu: int] : B5 ) )
      = ( ( member_int @ ( product_fst_int_o @ X ) @ A2 )
        & ( member_o @ ( product_snd_int_o @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_7111_mem__Times__iff,axiom,
    ! [X: product_prod_int_nat,A2: set_int,B5: set_nat] :
      ( ( member216504246829706758nt_nat @ X
        @ ( produc1456381018704787142nt_nat @ A2
          @ ^ [Uu: int] : B5 ) )
      = ( ( member_int @ ( product_fst_int_nat @ X ) @ A2 )
        & ( member_nat @ ( product_snd_int_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_7112_mem__Times__iff,axiom,
    ! [X: product_prod_nat_nat,A2: set_nat,B5: set_nat] :
      ( ( member8440522571783428010at_nat @ X
        @ ( produc457027306803732586at_nat @ A2
          @ ^ [Uu: nat] : B5 ) )
      = ( ( member_nat @ ( product_fst_nat_nat @ X ) @ A2 )
        & ( member_nat @ ( product_snd_nat_nat @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_7113_mem__Times__iff,axiom,
    ! [X: produc8923325533196201883nteger,A2: set_Code_integer,B5: set_Code_integer] :
      ( ( member157494554546826820nteger @ X
        @ ( produc5882431272649516420nteger @ A2
          @ ^ [Uu: code_integer] : B5 ) )
      = ( ( member_Code_integer @ ( produc8508995932063986495nteger @ X ) @ A2 )
        & ( member_Code_integer @ ( produc6174133586879617921nteger @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_7114_mem__Times__iff,axiom,
    ! [X: product_prod_int_int,A2: set_int,B5: set_int] :
      ( ( member5262025264175285858nt_int @ X
        @ ( produc1453890548195736866nt_int @ A2
          @ ^ [Uu: int] : B5 ) )
      = ( ( member_int @ ( product_fst_int_int @ X ) @ A2 )
        & ( member_int @ ( product_snd_int_int @ X ) @ B5 ) ) ) ).

% mem_Times_iff
thf(fact_7115_in__prod__fst__sndI,axiom,
    ! [X: product_prod_o_o,A2: set_o,B5: set_o] :
      ( ( member_o @ ( product_fst_o_o @ X ) @ A2 )
     => ( ( member_o @ ( product_snd_o_o @ X ) @ B5 )
       => ( member7466972457876170832od_o_o @ X
          @ ( product_Sigma_o_o @ A2
            @ ^ [Uu: $o] : B5 ) ) ) ) ).

% in_prod_fst_sndI
thf(fact_7116_in__prod__fst__sndI,axiom,
    ! [X: product_prod_o_nat,A2: set_o,B5: set_nat] :
      ( ( member_o @ ( product_fst_o_nat @ X ) @ A2 )
     => ( ( member_nat @ ( product_snd_o_nat @ X ) @ B5 )
       => ( member2802428098988154798_o_nat @ X
          @ ( product_Sigma_o_nat @ A2
            @ ^ [Uu: $o] : B5 ) ) ) ) ).

% in_prod_fst_sndI
thf(fact_7117_in__prod__fst__sndI,axiom,
    ! [X: product_prod_o_int,A2: set_o,B5: set_int] :
      ( ( member_o @ ( product_fst_o_int @ X ) @ A2 )
     => ( ( member_int @ ( product_snd_o_int @ X ) @ B5 )
       => ( member7847949116333733898_o_int @ X
          @ ( product_Sigma_o_int @ A2
            @ ^ [Uu: $o] : B5 ) ) ) ) ).

% in_prod_fst_sndI
thf(fact_7118_in__prod__fst__sndI,axiom,
    ! [X: product_prod_nat_o,A2: set_nat,B5: set_o] :
      ( ( member_nat @ ( product_fst_nat_o @ X ) @ A2 )
     => ( ( member_o @ ( product_snd_nat_o @ X ) @ B5 )
       => ( member6310962623043647828_nat_o @ X
          @ ( product_Sigma_nat_o @ A2
            @ ^ [Uu: nat] : B5 ) ) ) ) ).

% in_prod_fst_sndI
thf(fact_7119_in__prod__fst__sndI,axiom,
    ! [X: product_prod_nat_nat,A2: set_nat,B5: set_nat] :
      ( ( member_nat @ ( product_fst_nat_nat @ X ) @ A2 )
     => ( ( member_nat @ ( product_snd_nat_nat @ X ) @ B5 )
       => ( member8440522571783428010at_nat @ X
          @ ( produc457027306803732586at_nat @ A2
            @ ^ [Uu: nat] : B5 ) ) ) ) ).

% in_prod_fst_sndI
thf(fact_7120_in__prod__fst__sndI,axiom,
    ! [X: product_prod_nat_int,A2: set_nat,B5: set_int] :
      ( ( member_nat @ ( product_fst_nat_int @ X ) @ A2 )
     => ( ( member_int @ ( product_snd_nat_int @ X ) @ B5 )
       => ( member4262671552274231302at_int @ X
          @ ( produc454536836294682310at_int @ A2
            @ ^ [Uu: nat] : B5 ) ) ) ) ).

% in_prod_fst_sndI
thf(fact_7121_in__prod__fst__sndI,axiom,
    ! [X: product_prod_int_o,A2: set_int,B5: set_o] :
      ( ( member_int @ ( product_fst_int_o @ X ) @ A2 )
     => ( ( member_o @ ( product_snd_int_o @ X ) @ B5 )
       => ( member4489920277610959864_int_o @ X
          @ ( product_Sigma_int_o @ A2
            @ ^ [Uu: int] : B5 ) ) ) ) ).

% in_prod_fst_sndI
thf(fact_7122_in__prod__fst__sndI,axiom,
    ! [X: product_prod_int_nat,A2: set_int,B5: set_nat] :
      ( ( member_int @ ( product_fst_int_nat @ X ) @ A2 )
     => ( ( member_nat @ ( product_snd_int_nat @ X ) @ B5 )
       => ( member216504246829706758nt_nat @ X
          @ ( produc1456381018704787142nt_nat @ A2
            @ ^ [Uu: int] : B5 ) ) ) ) ).

% in_prod_fst_sndI
thf(fact_7123_in__prod__fst__sndI,axiom,
    ! [X: produc8923325533196201883nteger,A2: set_Code_integer,B5: set_Code_integer] :
      ( ( member_Code_integer @ ( produc8508995932063986495nteger @ X ) @ A2 )
     => ( ( member_Code_integer @ ( produc6174133586879617921nteger @ X ) @ B5 )
       => ( member157494554546826820nteger @ X
          @ ( produc5882431272649516420nteger @ A2
            @ ^ [Uu: code_integer] : B5 ) ) ) ) ).

% in_prod_fst_sndI
thf(fact_7124_in__prod__fst__sndI,axiom,
    ! [X: product_prod_int_int,A2: set_int,B5: set_int] :
      ( ( member_int @ ( product_fst_int_int @ X ) @ A2 )
     => ( ( member_int @ ( product_snd_int_int @ X ) @ B5 )
       => ( member5262025264175285858nt_int @ X
          @ ( produc1453890548195736866nt_int @ A2
            @ ^ [Uu: int] : B5 ) ) ) ) ).

% in_prod_fst_sndI
thf(fact_7125_swap__product,axiom,
    ! [A2: set_Pr4811707699266497531nteger,B5: set_Pr7604974323444597168e_term] :
      ( ( image_8159590125932948193nteger
        @ ( produc5068368601420144385nteger
          @ ^ [I3: produc8923325533196201883nteger,J2: produc6241069584506657477e_term > option6357759511663192854e_term] : ( produc8603105652947943368nteger @ J2 @ I3 ) )
        @ ( produc552098293437506509e_term @ A2
          @ ^ [Uu: produc8923325533196201883nteger] : B5 ) )
      = ( produc7256824055837167575nteger @ B5
        @ ^ [Uu: produc6241069584506657477e_term > option6357759511663192854e_term] : A2 ) ) ).

% swap_product
thf(fact_7126_swap__product,axiom,
    ! [A2: set_Pr3948176798113811640et_nat,B5: set_Pr4532377907799695533_nat_o] :
      ( ( image_4501728140019976901et_nat
        @ ( produc5206072795621629085et_nat
          @ ^ [I3: produc3658429121746597890et_nat,J2: produc3658429121746597890et_nat > $o] : ( produc5001842942810119800et_nat @ J2 @ I3 ) )
        @ ( produc5879417536264718849_nat_o @ A2
          @ ^ [Uu: produc3658429121746597890et_nat] : B5 ) )
      = ( produc8052021730408606599et_nat @ B5
        @ ^ [Uu: produc3658429121746597890et_nat > $o] : A2 ) ) ).

% swap_product
thf(fact_7127_swap__product,axiom,
    ! [A2: set_Pr3286484037609594932et_nat,B5: set_Pr4532377907799695533_nat_o] :
      ( ( image_7255662321850427333et_nat
        @ ( produc9203276081649983901et_nat
          @ ^ [I3: produc3925858234332021118et_nat,J2: produc3658429121746597890et_nat > $o] : ( produc2245416461498447860et_nat @ J2 @ I3 ) )
        @ ( produc8445815324708537221_nat_o @ A2
          @ ^ [Uu: produc3925858234332021118et_nat] : B5 ) )
      = ( produc7016705354093686019et_nat @ B5
        @ ^ [Uu: produc3658429121746597890et_nat > $o] : A2 ) ) ).

% swap_product
thf(fact_7128_swap__product,axiom,
    ! [A2: set_Pr958786334691620121nt_int,B5: set_Pr3642885161833720865e_term] :
      ( ( image_1640129903684976221nt_int
        @ ( produc199769199162113789nt_int
          @ ^ [I3: product_prod_int_int,J2: produc8551481072490612790e_term > option6357759511663192854e_term] : ( produc5700946648718959541nt_int @ J2 @ I3 ) )
        @ ( produc5327145186146221532e_term @ A2
          @ ^ [Uu: product_prod_int_int] : B5 ) )
      = ( produc5808185664642557892nt_int @ B5
        @ ^ [Uu: produc8551481072490612790e_term > option6357759511663192854e_term] : A2 ) ) ).

% swap_product
thf(fact_7129_swap__product,axiom,
    ! [A2: set_Pr958786334691620121nt_int,B5: set_in3461395444621081367e_term] :
      ( ( image_1292712855239066845nt_int
        @ ( produc3478612561300218621nt_int
          @ ^ [I3: product_prod_int_int,J2: int > option6357759511663192854e_term] : ( produc4305682042979456191nt_int @ J2 @ I3 ) )
        @ ( produc4529769248912663762e_term @ A2
          @ ^ [Uu: product_prod_int_int] : B5 ) )
      = ( produc1318478031590598990nt_int @ B5
        @ ^ [Uu: int > option6357759511663192854e_term] : A2 ) ) ).

% swap_product
thf(fact_7130_swap__product,axiom,
    ! [A2: set_int,B5: set_int] :
      ( ( image_2653370878348428101nt_int
        @ ( produc4245557441103728435nt_int
          @ ^ [I3: int,J2: int] : ( product_Pair_int_int @ J2 @ I3 ) )
        @ ( produc1453890548195736866nt_int @ A2
          @ ^ [Uu: int] : B5 ) )
      = ( produc1453890548195736866nt_int @ B5
        @ ^ [Uu: int] : A2 ) ) ).

% swap_product
thf(fact_7131_Sigma__empty__iff,axiom,
    ! [I4: set_se7855581050983116737at_nat,X11: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( ( produc7726050920695093128at_nat @ I4 @ X11 )
        = bot_bo228742789529271731at_nat )
      = ( ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ I4 )
           => ( ( X11 @ X3 )
              = bot_bo3083307316010499117at_nat ) ) ) ) ).

% Sigma_empty_iff
thf(fact_7132_Sigma__empty__iff,axiom,
    ! [I4: set_nat,X11: nat > set_nat] :
      ( ( ( produc457027306803732586at_nat @ I4 @ X11 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ I4 )
           => ( ( X11 @ X3 )
              = bot_bot_set_nat ) ) ) ) ).

% Sigma_empty_iff
thf(fact_7133_gcd__mult__unit2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( gcd_gcd_nat @ B @ ( times_times_nat @ C @ A ) )
        = ( gcd_gcd_nat @ B @ C ) ) ) ).

% gcd_mult_unit2
thf(fact_7134_gcd__mult__unit2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( gcd_gcd_int @ B @ ( times_times_int @ C @ A ) )
        = ( gcd_gcd_int @ B @ C ) ) ) ).

% gcd_mult_unit2
thf(fact_7135_gcd__mult__unit1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( gcd_gcd_nat @ ( times_times_nat @ B @ A ) @ C )
        = ( gcd_gcd_nat @ B @ C ) ) ) ).

% gcd_mult_unit1
thf(fact_7136_gcd__mult__unit1,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( gcd_gcd_int @ ( times_times_int @ B @ A ) @ C )
        = ( gcd_gcd_int @ B @ C ) ) ) ).

% gcd_mult_unit1
thf(fact_7137_gcd__div__unit1,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( gcd_gcd_int @ ( divide_divide_int @ B @ A ) @ C )
        = ( gcd_gcd_int @ B @ C ) ) ) ).

% gcd_div_unit1
thf(fact_7138_gcd__div__unit1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( gcd_gcd_nat @ ( divide_divide_nat @ B @ A ) @ C )
        = ( gcd_gcd_nat @ B @ C ) ) ) ).

% gcd_div_unit1
thf(fact_7139_gcd__div__unit2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( gcd_gcd_int @ B @ ( divide_divide_int @ C @ A ) )
        = ( gcd_gcd_int @ B @ C ) ) ) ).

% gcd_div_unit2
thf(fact_7140_gcd__div__unit2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( gcd_gcd_nat @ B @ ( divide_divide_nat @ C @ A ) )
        = ( gcd_gcd_nat @ B @ C ) ) ) ).

% gcd_div_unit2
thf(fact_7141_less__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y3: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) )
    @ ord_less_int ) ).

% less_int.transfer
thf(fact_7142_less__eq__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y3: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) )
    @ ord_less_eq_int ) ).

% less_eq_int.transfer
thf(fact_7143_plus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X3: nat,Y3: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y3 @ V2 ) ) ) )
    @ plus_plus_int ) ).

% plus_int.transfer
thf(fact_7144_minus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X3: nat,Y3: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ Y3 @ U2 ) ) ) )
    @ minus_minus_int ) ).

% minus_int.transfer
thf(fact_7145_times__subset__iff,axiom,
    ! [A2: set_o,C4: set_o,B5: set_o,D3: set_o] :
      ( ( ord_le192195778054999463od_o_o
        @ ( product_Sigma_o_o @ A2
          @ ^ [Uu: $o] : C4 )
        @ ( product_Sigma_o_o @ B5
          @ ^ [Uu: $o] : D3 ) )
      = ( ( A2 = bot_bot_set_o )
        | ( C4 = bot_bot_set_o )
        | ( ( ord_less_eq_set_o @ A2 @ B5 )
          & ( ord_less_eq_set_o @ C4 @ D3 ) ) ) ) ).

% times_subset_iff
thf(fact_7146_times__subset__iff,axiom,
    ! [A2: set_o,C4: set_int,B5: set_o,D3: set_int] :
      ( ( ord_le3982295392052344841_o_int
        @ ( product_Sigma_o_int @ A2
          @ ^ [Uu: $o] : C4 )
        @ ( product_Sigma_o_int @ B5
          @ ^ [Uu: $o] : D3 ) )
      = ( ( A2 = bot_bot_set_o )
        | ( C4 = bot_bot_set_int )
        | ( ( ord_less_eq_set_o @ A2 @ B5 )
          & ( ord_less_eq_set_int @ C4 @ D3 ) ) ) ) ).

% times_subset_iff
thf(fact_7147_times__subset__iff,axiom,
    ! [A2: set_int,C4: set_o,B5: set_int,D3: set_o] :
      ( ( ord_le5274836691491929335_int_o
        @ ( product_Sigma_int_o @ A2
          @ ^ [Uu: int] : C4 )
        @ ( product_Sigma_int_o @ B5
          @ ^ [Uu: int] : D3 ) )
      = ( ( A2 = bot_bot_set_int )
        | ( C4 = bot_bot_set_o )
        | ( ( ord_less_eq_set_int @ A2 @ B5 )
          & ( ord_less_eq_set_o @ C4 @ D3 ) ) ) ) ).

% times_subset_iff
thf(fact_7148_times__subset__iff,axiom,
    ! [A2: set_int,C4: set_int,B5: set_int,D3: set_int] :
      ( ( ord_le2843351958646193337nt_int
        @ ( produc1453890548195736866nt_int @ A2
          @ ^ [Uu: int] : C4 )
        @ ( produc1453890548195736866nt_int @ B5
          @ ^ [Uu: int] : D3 ) )
      = ( ( A2 = bot_bot_set_int )
        | ( C4 = bot_bot_set_int )
        | ( ( ord_less_eq_set_int @ A2 @ B5 )
          & ( ord_less_eq_set_int @ C4 @ D3 ) ) ) ) ).

% times_subset_iff
thf(fact_7149_times__subset__iff,axiom,
    ! [A2: set_o,C4: set_nat,B5: set_o,D3: set_nat] :
      ( ( ord_le6472378536984071597_o_nat
        @ ( product_Sigma_o_nat @ A2
          @ ^ [Uu: $o] : C4 )
        @ ( product_Sigma_o_nat @ B5
          @ ^ [Uu: $o] : D3 ) )
      = ( ( A2 = bot_bot_set_o )
        | ( C4 = bot_bot_set_nat )
        | ( ( ord_less_eq_set_o @ A2 @ B5 )
          & ( ord_less_eq_set_nat @ C4 @ D3 ) ) ) ) ).

% times_subset_iff
thf(fact_7150_times__subset__iff,axiom,
    ! [A2: set_int,C4: set_nat,B5: set_int,D3: set_nat] :
      ( ( ord_le5333435103577920093nt_nat
        @ ( produc1456381018704787142nt_nat @ A2
          @ ^ [Uu: int] : C4 )
        @ ( produc1456381018704787142nt_nat @ B5
          @ ^ [Uu: int] : D3 ) )
      = ( ( A2 = bot_bot_set_int )
        | ( C4 = bot_bot_set_nat )
        | ( ( ord_less_eq_set_int @ A2 @ B5 )
          & ( ord_less_eq_set_nat @ C4 @ D3 ) ) ) ) ).

% times_subset_iff
thf(fact_7151_times__subset__iff,axiom,
    ! [A2: set_nat,C4: set_o,B5: set_nat,D3: set_o] :
      ( ( ord_le7519981659162375251_nat_o
        @ ( product_Sigma_nat_o @ A2
          @ ^ [Uu: nat] : C4 )
        @ ( product_Sigma_nat_o @ B5
          @ ^ [Uu: nat] : D3 ) )
      = ( ( A2 = bot_bot_set_nat )
        | ( C4 = bot_bot_set_o )
        | ( ( ord_less_eq_set_nat @ A2 @ B5 )
          & ( ord_less_eq_set_o @ C4 @ D3 ) ) ) ) ).

% times_subset_iff
thf(fact_7152_times__subset__iff,axiom,
    ! [A2: set_nat,C4: set_int,B5: set_nat,D3: set_int] :
      ( ( ord_le656430383953171549at_int
        @ ( produc454536836294682310at_int @ A2
          @ ^ [Uu: nat] : C4 )
        @ ( produc454536836294682310at_int @ B5
          @ ^ [Uu: nat] : D3 ) )
      = ( ( A2 = bot_bot_set_nat )
        | ( C4 = bot_bot_set_int )
        | ( ( ord_less_eq_set_nat @ A2 @ B5 )
          & ( ord_less_eq_set_int @ C4 @ D3 ) ) ) ) ).

% times_subset_iff
thf(fact_7153_times__subset__iff,axiom,
    ! [A2: set_nat,C4: set_nat,B5: set_nat,D3: set_nat] :
      ( ( ord_le3146513528884898305at_nat
        @ ( produc457027306803732586at_nat @ A2
          @ ^ [Uu: nat] : C4 )
        @ ( produc457027306803732586at_nat @ B5
          @ ^ [Uu: nat] : D3 ) )
      = ( ( A2 = bot_bot_set_nat )
        | ( C4 = bot_bot_set_nat )
        | ( ( ord_less_eq_set_nat @ A2 @ B5 )
          & ( ord_less_eq_set_nat @ C4 @ D3 ) ) ) ) ).

% times_subset_iff
thf(fact_7154_times__subset__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C4: set_o,B5: set_Pr1261947904930325089at_nat,D3: set_o] :
      ( ( ord_le5919393503700438654_nat_o
        @ ( produc203973861852599583_nat_o @ A2
          @ ^ [Uu: product_prod_nat_nat] : C4 )
        @ ( produc203973861852599583_nat_o @ B5
          @ ^ [Uu: product_prod_nat_nat] : D3 ) )
      = ( ( A2 = bot_bo2099793752762293965at_nat )
        | ( C4 = bot_bot_set_o )
        | ( ( ord_le3146513528884898305at_nat @ A2 @ B5 )
          & ( ord_less_eq_set_o @ C4 @ D3 ) ) ) ) ).

% times_subset_iff
thf(fact_7155_image__paired__Times,axiom,
    ! [F: nat > code_integer,G: nat > code_integer,A2: set_nat,B5: set_nat] :
      ( ( image_5669961043262023647nteger
        @ ( produc39701218431283661nteger
          @ ^ [X3: nat,Y3: nat] : ( produc1086072967326762835nteger @ ( F @ X3 ) @ ( G @ Y3 ) ) )
        @ ( produc457027306803732586at_nat @ A2
          @ ^ [Uu: nat] : B5 ) )
      = ( produc5882431272649516420nteger @ ( image_1215581382706833972nteger @ F @ A2 )
        @ ^ [Uu: code_integer] : ( image_1215581382706833972nteger @ G @ B5 ) ) ) ).

% image_paired_Times
thf(fact_7156_image__paired__Times,axiom,
    ! [F: nat > code_integer,G: nat > nat,A2: set_nat,B5: set_nat] :
      ( ( image_1372510298004463058er_nat
        @ ( produc4146132627719282624er_nat
          @ ^ [X3: nat,Y3: nat] : ( produc8807591364040884934er_nat @ ( F @ X3 ) @ ( G @ Y3 ) ) )
        @ ( produc457027306803732586at_nat @ A2
          @ ^ [Uu: nat] : B5 ) )
      = ( produc4260658192689636599er_nat @ ( image_1215581382706833972nteger @ F @ A2 )
        @ ^ [Uu: code_integer] : ( image_nat_nat @ G @ B5 ) ) ) ).

% image_paired_Times
thf(fact_7157_image__paired__Times,axiom,
    ! [F: nat > code_integer,G: nat > int,A2: set_nat,B5: set_nat] :
      ( ( image_6418031315350042158er_int
        @ ( produc9191653645064861724er_int
          @ ^ [X3: nat,Y3: nat] : ( produc8805100893531834658er_int @ ( F @ X3 ) @ ( G @ Y3 ) ) )
        @ ( produc457027306803732586at_nat @ A2
          @ ^ [Uu: nat] : B5 ) )
      = ( produc4258167722180586323er_int @ ( image_1215581382706833972nteger @ F @ A2 )
        @ ^ [Uu: code_integer] : ( image_nat_int @ G @ B5 ) ) ) ).

% image_paired_Times
thf(fact_7158_image__paired__Times,axiom,
    ! [F: nat > code_integer,G: int > int,A2: set_nat,B5: set_int] :
      ( ( image_202211252889950930er_int
        @ ( produc1728318271963197120er_int
          @ ^ [X3: nat,Y3: int] : ( produc8805100893531834658er_int @ ( F @ X3 ) @ ( G @ Y3 ) ) )
        @ ( produc454536836294682310at_int @ A2
          @ ^ [Uu: nat] : B5 ) )
      = ( produc4258167722180586323er_int @ ( image_1215581382706833972nteger @ F @ A2 )
        @ ^ [Uu: code_integer] : ( image_int_int @ G @ B5 ) ) ) ).

% image_paired_Times
thf(fact_7159_image__paired__Times,axiom,
    ! [F: nat > nat,G: nat > code_integer,A2: set_nat,B5: set_nat] :
      ( ( image_6753550540357478226nteger
        @ ( produc303800833217521984nteger
          @ ^ [X3: nat,Y3: nat] : ( produc9072146812819927750nteger @ ( F @ X3 ) @ ( G @ Y3 ) ) )
        @ ( produc457027306803732586at_nat @ A2
          @ ^ [Uu: nat] : B5 ) )
      = ( produc4525213641468679415nteger @ ( image_nat_nat @ F @ A2 )
        @ ^ [Uu: nat] : ( image_1215581382706833972nteger @ G @ B5 ) ) ) ).

% image_paired_Times
thf(fact_7160_image__paired__Times,axiom,
    ! [F: nat > nat,G: nat > nat,A2: set_nat,B5: set_nat] :
      ( ( image_5168914502847457605at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [X3: nat,Y3: nat] : ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y3 ) ) )
        @ ( produc457027306803732586at_nat @ A2
          @ ^ [Uu: nat] : B5 ) )
      = ( produc457027306803732586at_nat @ ( image_nat_nat @ F @ A2 )
        @ ^ [Uu: nat] : ( image_nat_nat @ G @ B5 ) ) ) ).

% image_paired_Times
thf(fact_7161_image__paired__Times,axiom,
    ! [F: nat > nat,G: nat > int,A2: set_nat,B5: set_nat] :
      ( ( image_991063483338260897at_int
        @ ( produc7671697017840204687at_int
          @ ^ [X3: nat,Y3: nat] : ( product_Pair_nat_int @ ( F @ X3 ) @ ( G @ Y3 ) ) )
        @ ( produc457027306803732586at_nat @ A2
          @ ^ [Uu: nat] : B5 ) )
      = ( produc454536836294682310at_int @ ( image_nat_nat @ F @ A2 )
        @ ^ [Uu: nat] : ( image_nat_int @ G @ B5 ) ) ) ).

% image_paired_Times
thf(fact_7162_image__paired__Times,axiom,
    ! [F: nat > nat,G: int > int,A2: set_nat,B5: set_int] :
      ( ( image_9117352539549038149at_int
        @ ( produc1486167065449562675at_int
          @ ^ [X3: nat,Y3: int] : ( product_Pair_nat_int @ ( F @ X3 ) @ ( G @ Y3 ) ) )
        @ ( produc454536836294682310at_int @ A2
          @ ^ [Uu: nat] : B5 ) )
      = ( produc454536836294682310at_int @ ( image_nat_nat @ F @ A2 )
        @ ^ [Uu: nat] : ( image_int_int @ G @ B5 ) ) ) ).

% image_paired_Times
thf(fact_7163_image__paired__Times,axiom,
    ! [F: nat > int,G: nat > code_integer,A2: set_nat,B5: set_nat] :
      ( ( image_7125204100594322862nteger
        @ ( produc675454393454366620nteger
          @ ^ [X3: nat,Y3: nat] : ( produc220428336201996578nteger @ ( F @ X3 ) @ ( G @ Y3 ) ) )
        @ ( produc457027306803732586at_nat @ A2
          @ ^ [Uu: nat] : B5 ) )
      = ( produc4896867201705524051nteger @ ( image_nat_int @ F @ A2 )
        @ ^ [Uu: int] : ( image_1215581382706833972nteger @ G @ B5 ) ) ) ).

% image_paired_Times
thf(fact_7164_image__paired__Times,axiom,
    ! [F: nat > int,G: nat > nat,A2: set_nat,B5: set_nat] :
      ( ( image_6168268214748512161nt_nat
        @ ( produc3625529712395680143nt_nat
          @ ^ [X3: nat,Y3: nat] : ( product_Pair_int_nat @ ( F @ X3 ) @ ( G @ Y3 ) ) )
        @ ( produc457027306803732586at_nat @ A2
          @ ^ [Uu: nat] : B5 ) )
      = ( produc1456381018704787142nt_nat @ ( image_nat_int @ F @ A2 )
        @ ^ [Uu: int] : ( image_nat_nat @ G @ B5 ) ) ) ).

% image_paired_Times
thf(fact_7165_fst__image__Sigma,axiom,
    ! [A2: set_Code_integer,B5: code_integer > set_Code_integer] :
      ( ( image_6316824473203919795nteger @ produc8508995932063986495nteger @ ( produc5882431272649516420nteger @ A2 @ B5 ) )
      = ( collect_Code_integer
        @ ^ [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A2 )
            & ( ( B5 @ X3 )
             != bot_bo3990330152332043303nteger ) ) ) ) ).

% fst_image_Sigma
thf(fact_7166_fst__image__Sigma,axiom,
    ! [A2: set_o,B5: $o > set_o] :
      ( ( image_7896445794123959606_o_o_o @ product_fst_o_o @ ( product_Sigma_o_o @ A2 @ B5 ) )
      = ( collect_o
        @ ^ [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ( ( B5 @ X3 )
             != bot_bot_set_o ) ) ) ) ).

% fst_image_Sigma
thf(fact_7167_fst__image__Sigma,axiom,
    ! [A2: set_int,B5: int > set_o] :
      ( ( image_90517134139374612_o_int @ product_fst_int_o @ ( product_Sigma_int_o @ A2 @ B5 ) )
      = ( collect_int
        @ ^ [X3: int] :
            ( ( member_int @ X3 @ A2 )
            & ( ( B5 @ X3 )
             != bot_bot_set_o ) ) ) ) ).

% fst_image_Sigma
thf(fact_7168_fst__image__Sigma,axiom,
    ! [A2: set_nat,B5: nat > set_o] :
      ( ( image_8629803608462650132_o_nat @ product_fst_nat_o @ ( product_Sigma_nat_o @ A2 @ B5 ) )
      = ( collect_nat
        @ ^ [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ( ( B5 @ X3 )
             != bot_bot_set_o ) ) ) ) ).

% fst_image_Sigma
thf(fact_7169_fst__image__Sigma,axiom,
    ! [A2: set_o,B5: $o > set_nat] :
      ( ( image_8105710389281276538_nat_o @ product_fst_o_nat @ ( product_Sigma_o_nat @ A2 @ B5 ) )
      = ( collect_o
        @ ^ [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ( ( B5 @ X3 )
             != bot_bot_set_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_7170_fst__image__Sigma,axiom,
    ! [A2: set_int,B5: int > set_nat] :
      ( ( image_2855239504505064772at_int @ product_fst_int_nat @ ( produc1456381018704787142nt_nat @ A2 @ B5 ) )
      = ( collect_int
        @ ^ [X3: int] :
            ( ( member_int @ X3 @ A2 )
            & ( ( B5 @ X3 )
             != bot_bot_set_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_7171_fst__image__Sigma,axiom,
    ! [A2: set_nat,B5: nat > set_nat] :
      ( ( image_2486076414777270412at_nat @ product_fst_nat_nat @ ( produc457027306803732586at_nat @ A2 @ B5 ) )
      = ( collect_nat
        @ ^ [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ( ( B5 @ X3 )
             != bot_bot_set_nat ) ) ) ) ).

% fst_image_Sigma
thf(fact_7172_fst__image__Sigma,axiom,
    ! [A2: set_o,B5: $o > set_int] :
      ( ( image_5860565421610830622_int_o @ product_fst_o_int @ ( product_Sigma_o_int @ A2 @ B5 ) )
      = ( collect_o
        @ ^ [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ( ( B5 @ X3 )
             != bot_bot_set_int ) ) ) ) ).

% fst_image_Sigma
thf(fact_7173_fst__image__Sigma,axiom,
    ! [A2: set_nat,B5: nat > set_int] :
      ( ( image_4672997989470292200nt_nat @ product_fst_nat_int @ ( produc454536836294682310at_int @ A2 @ B5 ) )
      = ( collect_nat
        @ ^ [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ( ( B5 @ X3 )
             != bot_bot_set_int ) ) ) ) ).

% fst_image_Sigma
thf(fact_7174_fst__image__Sigma,axiom,
    ! [A2: set_int,B5: int > set_int] :
      ( ( image_5042161079198086560nt_int @ product_fst_int_int @ ( produc1453890548195736866nt_int @ A2 @ B5 ) )
      = ( collect_int
        @ ^ [X3: int] :
            ( ( member_int @ X3 @ A2 )
            & ( ( B5 @ X3 )
             != bot_bot_set_int ) ) ) ) ).

% fst_image_Sigma
thf(fact_7175_Sigma__def,axiom,
    ( produc7256824055837167575nteger
    = ( ^ [A7: set_Pr7604974323444597168e_term,B8: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > set_Pr4811707699266497531nteger] :
          ( comple1324685164916153887nteger
          @ ( image_3218922492622835021nteger
            @ ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term] :
                ( comple1324685164916153887nteger
                @ ( image_5715966689554962722nteger
                  @ ^ [Y3: produc8923325533196201883nteger] : ( insert4245756632806152262nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ bot_bo5443222936135328352nteger )
                  @ ( B8 @ X3 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_7176_Sigma__def,axiom,
    ( produc8052021730408606599et_nat
    = ( ^ [A7: set_Pr4532377907799695533_nat_o,B8: ( produc3658429121746597890et_nat > $o ) > set_Pr3948176798113811640et_nat] :
          ( comple4849024849677941639et_nat
          @ ( image_1916054815030244540et_nat
            @ ^ [X3: produc3658429121746597890et_nat > $o] :
                ( comple4849024849677941639et_nat
                @ ( image_15228131989405757et_nat
                  @ ^ [Y3: produc3658429121746597890et_nat] : ( insert3111075074545270702et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ bot_bo1481135142794719944et_nat )
                  @ ( B8 @ X3 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_7177_Sigma__def,axiom,
    ( produc7016705354093686019et_nat
    = ( ^ [A7: set_Pr4532377907799695533_nat_o,B8: ( produc3658429121746597890et_nat > $o ) > set_Pr3286484037609594932et_nat] :
          ( comple5063570841203093763et_nat
          @ ( image_4127253568139551288et_nat
            @ ^ [X3: produc3658429121746597890et_nat > $o] :
                ( comple5063570841203093763et_nat
                @ ( image_8189767104526903093et_nat
                  @ ^ [Y3: produc3925858234332021118et_nat] : ( insert8297122186708525866et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ bot_bo5635537948650799172et_nat )
                  @ ( B8 @ X3 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_7178_Sigma__def,axiom,
    ( produc5808185664642557892nt_int
    = ( ^ [A7: set_Pr3642885161833720865e_term,B8: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr958786334691620121nt_int] :
          ( comple6871990601243273356nt_int
          @ ( image_5111841350319496619nt_int
            @ ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term] :
                ( comple6871990601243273356nt_int
                @ ( image_815654828163180973nt_int
                  @ ^ [Y3: product_prod_int_int] : ( insert984290209421517107nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ bot_bo572930865798478029nt_int )
                  @ ( B8 @ X3 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_7179_Sigma__def,axiom,
    ( produc1318478031590598990nt_int
    = ( ^ [A7: set_in3461395444621081367e_term,B8: ( int > option6357759511663192854e_term ) > set_Pr958786334691620121nt_int] :
          ( comple5772984051336441366nt_int
          @ ( image_7708985414016646955nt_int
            @ ^ [X3: int > option6357759511663192854e_term] :
                ( comple5772984051336441366nt_int
                @ ( image_5963362749323609783nt_int
                  @ ^ [Y3: product_prod_int_int] : ( insert750232675644355261nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ bot_bo4508923176915781079nt_int )
                  @ ( B8 @ X3 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_7180_Sigma__def,axiom,
    ( produc7726050920695093128at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat,B8: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
          ( comple4364388969068068532at_nat
          @ ( image_6898982319499645995at_nat
            @ ^ [X3: set_Pr1261947904930325089at_nat] :
                ( comple4364388969068068532at_nat
                @ ( image_6898982319499645995at_nat
                  @ ^ [Y3: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ bot_bo228742789529271731at_nat )
                  @ ( B8 @ X3 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_7181_Sigma__def,axiom,
    ( produc457027306803732586at_nat
    = ( ^ [A7: set_nat,B8: nat > set_nat] :
          ( comple5685304695842803022at_nat
          @ ( image_7178329752028323786at_nat
            @ ^ [X3: nat] :
                ( comple5685304695842803022at_nat
                @ ( image_7178329752028323786at_nat
                  @ ^ [Y3: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ bot_bo2099793752762293965at_nat )
                  @ ( B8 @ X3 ) ) )
            @ A7 ) ) ) ) ).

% Sigma_def
thf(fact_7182_times__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X3: nat,Y3: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y3 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V2 ) @ ( times_times_nat @ Y3 @ U2 ) ) ) ) )
    @ times_times_int ) ).

% times_int.transfer
thf(fact_7183_num__of__integer_Orsp,axiom,
    ( bNF_re7626690874201225453um_num
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ^ [Y4: num,Z2: num] : ( Y4 = Z2 )
    @ ( comp_nat_num_int @ num_of_nat @ nat2 )
    @ ( comp_nat_num_int @ num_of_nat @ nat2 ) ) ).

% num_of_integer.rsp
thf(fact_7184_snd__image__Sigma,axiom,
    ! [A2: set_Code_integer,B5: code_integer > set_Code_integer] :
      ( ( image_6316824473203919795nteger @ produc6174133586879617921nteger @ ( produc5882431272649516420nteger @ A2 @ B5 ) )
      = ( comple739944243200306918nteger @ ( image_2291390168754192375nteger @ B5 @ A2 ) ) ) ).

% snd_image_Sigma
thf(fact_7185_snd__image__Sigma,axiom,
    ! [A2: set_int,B5: int > set_int] :
      ( ( image_5042161079198086560nt_int @ product_snd_int_int @ ( produc1453890548195736866nt_int @ A2 @ B5 ) )
      = ( comple3221217463730067765et_int @ ( image_int_set_int @ B5 @ A2 ) ) ) ).

% snd_image_Sigma
thf(fact_7186_snd__image__Sigma,axiom,
    ! [A2: set_Code_natural,B5: code_natural > set_Pr7564037697178849041atural] :
      ( ( image_3837677093542026988atural @ produc6591912806276919810atural @ ( produc6231535905537262655atural @ A2 @ B5 ) )
      = ( comple4956303571586142718atural @ ( image_3530315724569248802atural @ B5 @ A2 ) ) ) ).

% snd_image_Sigma
thf(fact_7187_snd__image__Sigma,axiom,
    ! [A2: set_nat,B5: nat > set_nat] :
      ( ( image_2486076414777270412at_nat @ product_snd_nat_nat @ ( produc457027306803732586at_nat @ A2 @ B5 ) )
      = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B5 @ A2 ) ) ) ).

% snd_image_Sigma
thf(fact_7188_subset__fst__snd,axiom,
    ! [A2: set_Pr4811707699266497531nteger] :
      ( ord_le3725938330318615451nteger @ A2
      @ ( produc5882431272649516420nteger @ ( image_6316824473203919795nteger @ produc8508995932063986495nteger @ A2 )
        @ ^ [Uu: code_integer] : ( image_6316824473203919795nteger @ produc6174133586879617921nteger @ A2 ) ) ) ).

% subset_fst_snd
thf(fact_7189_subset__fst__snd,axiom,
    ! [A2: set_Pr958786334691620121nt_int] :
      ( ord_le2843351958646193337nt_int @ A2
      @ ( produc1453890548195736866nt_int @ ( image_5042161079198086560nt_int @ product_fst_int_int @ A2 )
        @ ^ [Uu: int] : ( image_5042161079198086560nt_int @ product_snd_int_int @ A2 ) ) ) ).

% subset_fst_snd
thf(fact_7190_subset__fst__snd,axiom,
    ! [A2: set_Pr2352657477744078772atural] :
      ( ord_le2202040463148982292atural @ A2
      @ ( produc6231535905537262655atural @ ( image_8194678751871465133atural @ produc497848011034438852atural @ A2 )
        @ ^ [Uu: code_natural] : ( image_3837677093542026988atural @ produc6591912806276919810atural @ A2 ) ) ) ).

% subset_fst_snd
thf(fact_7191_int__transfer,axiom,
    ( bNF_re6830278522597306478at_int
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ pcr_int
    @ ^ [N4: nat] : ( product_Pair_nat_nat @ N4 @ zero_zero_nat )
    @ semiri1314217659103216013at_int ) ).

% int_transfer
thf(fact_7192_uminus__int_Otransfer,axiom,
    ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
    @ ( produc2626176000494625587at_nat
      @ ^ [X3: nat,Y3: nat] : ( product_Pair_nat_nat @ Y3 @ X3 ) )
    @ uminus_uminus_int ) ).

% uminus_int.transfer
thf(fact_7193_eq__fst__iff,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,P4: produc1908205239877642774nteger] :
      ( ( A
        = ( produc7822682618958472924nteger @ P4 ) )
      = ( ? [B4: produc8923325533196201883nteger] :
            ( P4
            = ( produc8603105652947943368nteger @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_7194_eq__fst__iff,axiom,
    ! [A: produc3658429121746597890et_nat > $o,P4: produc3925858234332021118et_nat] :
      ( ( A
        = ( produc995936583742144908et_nat @ P4 ) )
      = ( ? [B4: produc3658429121746597890et_nat] :
            ( P4
            = ( produc5001842942810119800et_nat @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_7195_eq__fst__iff,axiom,
    ! [A: produc3658429121746597890et_nat > $o,P4: produc2732055786443039994et_nat] :
      ( ( A
        = ( produc180342877477747464et_nat @ P4 ) )
      = ( ? [B4: produc3925858234332021118et_nat] :
            ( P4
            = ( produc2245416461498447860et_nat @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_7196_eq__fst__iff,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,P4: produc2285326912895808259nt_int] :
      ( ( A
        = ( produc6816164490631068361nt_int @ P4 ) )
      = ( ? [B4: product_prod_int_int] :
            ( P4
            = ( produc5700946648718959541nt_int @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_7197_eq__fst__iff,axiom,
    ! [A: int > option6357759511663192854e_term,P4: produc7773217078559923341nt_int] :
      ( ( A
        = ( produc6230002227079971283nt_int @ P4 ) )
      = ( ? [B4: product_prod_int_int] :
            ( P4
            = ( produc4305682042979456191nt_int @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_7198_eq__fst__iff,axiom,
    ! [A: code_integer,P4: produc8923325533196201883nteger] :
      ( ( A
        = ( produc8508995932063986495nteger @ P4 ) )
      = ( ? [B4: code_integer] :
            ( P4
            = ( produc1086072967326762835nteger @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_7199_eq__fst__iff,axiom,
    ! [A: int,P4: product_prod_int_int] :
      ( ( A
        = ( product_fst_int_int @ P4 ) )
      = ( ? [B4: int] :
            ( P4
            = ( product_Pair_int_int @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_7200_transfer__rule__of__nat,axiom,
    ! [R4: rat > rat > $o] :
      ( ( R4 @ zero_zero_rat @ zero_zero_rat )
     => ( ( R4 @ one_one_rat @ one_one_rat )
       => ( ( bNF_re6603079268895015599at_rat @ R4 @ ( bNF_re5606270390558447553at_rat @ R4 @ R4 ) @ plus_plus_rat @ plus_plus_rat )
         => ( bNF_re4702136315717946289at_rat
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4
            @ semiri681578069525770553at_rat
            @ semiri681578069525770553at_rat ) ) ) ) ).

% transfer_rule_of_nat
thf(fact_7201_transfer__rule__of__nat,axiom,
    ! [R4: rat > int > $o] :
      ( ( R4 @ zero_zero_rat @ zero_zero_int )
     => ( ( R4 @ one_one_rat @ one_one_int )
       => ( ( bNF_re8876735282041863595nt_int @ R4 @ ( bNF_re5838490730811218025at_int @ R4 @ R4 ) @ plus_plus_rat @ plus_plus_int )
         => ( bNF_re5334775905295391749at_int
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4
            @ semiri681578069525770553at_rat
            @ semiri1314217659103216013at_int ) ) ) ) ).

% transfer_rule_of_nat
thf(fact_7202_transfer__rule__of__nat,axiom,
    ! [R4: rat > nat > $o] :
      ( ( R4 @ zero_zero_rat @ zero_zero_nat )
     => ( ( R4 @ one_one_rat @ one_one_nat )
       => ( ( bNF_re4357912360536923287at_nat @ R4 @ ( bNF_re5469327641083423665at_nat @ R4 @ R4 ) @ plus_plus_rat @ plus_plus_nat )
         => ( bNF_re5337266375804442025at_nat
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4
            @ semiri681578069525770553at_rat
            @ semiri1316708129612266289at_nat ) ) ) ) ).

% transfer_rule_of_nat
thf(fact_7203_transfer__rule__of__nat,axiom,
    ! [R4: rat > code_integer > $o] :
      ( ( R4 @ zero_zero_rat @ zero_z3403309356797280102nteger )
     => ( ( R4 @ one_one_rat @ one_one_Code_integer )
       => ( ( bNF_re4307939118855507134nteger @ R4 @ ( bNF_re2328245411608324299nteger @ R4 @ R4 ) @ plus_plus_rat @ plus_p5714425477246183910nteger )
         => ( bNF_re4553819317763231286nteger
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4
            @ semiri681578069525770553at_rat
            @ semiri4939895301339042750nteger ) ) ) ) ).

% transfer_rule_of_nat
thf(fact_7204_transfer__rule__of__nat,axiom,
    ! [R4: rat > code_natural > $o] :
      ( ( R4 @ zero_zero_rat @ zero_z2226904508553997617atural )
     => ( ( R4 @ one_one_rat @ one_one_Code_natural )
       => ( ( bNF_re6252345487650389919atural @ R4 @ ( bNF_re4433268865704509537atural @ R4 @ R4 ) @ plus_plus_rat @ plus_p4538020629002901425atural )
         => ( bNF_re3377414469519948801atural
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4
            @ semiri681578069525770553at_rat
            @ semiri3763490453095760265atural ) ) ) ) ).

% transfer_rule_of_nat
thf(fact_7205_transfer__rule__of__nat,axiom,
    ! [R4: int > rat > $o] :
      ( ( R4 @ zero_zero_int @ zero_zero_rat )
     => ( ( R4 @ one_one_int @ one_one_rat )
       => ( ( bNF_re7661208983170134443at_rat @ R4 @ ( bNF_re4480299549022435433nt_rat @ R4 @ R4 ) @ plus_plus_int @ plus_plus_rat )
         => ( bNF_re6018044671553866757nt_rat
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4
            @ semiri1314217659103216013at_int
            @ semiri681578069525770553at_rat ) ) ) ) ).

% transfer_rule_of_nat
thf(fact_7206_transfer__rule__of__nat,axiom,
    ! [R4: int > int > $o] :
      ( ( R4 @ zero_zero_int @ zero_zero_int )
     => ( ( R4 @ one_one_int @ one_one_int )
       => ( ( bNF_re711492959462206631nt_int @ R4 @ ( bNF_re4712519889275205905nt_int @ R4 @ R4 ) @ plus_plus_int @ plus_plus_int )
         => ( bNF_re6650684261131312217nt_int
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4
            @ semiri1314217659103216013at_int
            @ semiri1314217659103216013at_int ) ) ) ) ).

% transfer_rule_of_nat
thf(fact_7207_transfer__rule__of__nat,axiom,
    ! [R4: int > nat > $o] :
      ( ( R4 @ zero_zero_int @ zero_zero_nat )
     => ( ( R4 @ one_one_int @ one_one_nat )
       => ( ( bNF_re5416042074812042131at_nat @ R4 @ ( bNF_re4343356799547411545nt_nat @ R4 @ R4 ) @ plus_plus_int @ plus_plus_nat )
         => ( bNF_re6653174731640362493nt_nat
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4
            @ semiri1314217659103216013at_int
            @ semiri1316708129612266289at_nat ) ) ) ) ).

% transfer_rule_of_nat
thf(fact_7208_transfer__rule__of__nat,axiom,
    ! [R4: int > code_integer > $o] :
      ( ( R4 @ zero_zero_int @ zero_z3403309356797280102nteger )
     => ( ( R4 @ one_one_int @ one_one_Code_integer )
       => ( ( bNF_re398004352372739002nteger @ R4 @ ( bNF_re3379532845092657523nteger @ R4 @ R4 ) @ plus_plus_int @ plus_p5714425477246183910nteger )
         => ( bNF_re4153400068438556298nteger
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4
            @ semiri1314217659103216013at_int
            @ semiri4939895301339042750nteger ) ) ) ) ).

% transfer_rule_of_nat
thf(fact_7209_transfer__rule__of__nat,axiom,
    ! [R4: int > code_natural > $o] :
      ( ( R4 @ zero_zero_int @ zero_z2226904508553997617atural )
     => ( ( R4 @ one_one_int @ one_one_Code_natural )
       => ( ( bNF_re2342410721167621787atural @ R4 @ ( bNF_re5484556299188842761atural @ R4 @ R4 ) @ plus_plus_int @ plus_p4538020629002901425atural )
         => ( bNF_re2976995220195273813atural
            @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
            @ R4
            @ semiri1314217659103216013at_int
            @ semiri3763490453095760265atural ) ) ) ) ).

% transfer_rule_of_nat
thf(fact_7210_transfer__rule__of__bool,axiom,
    ! [R4: rat > rat > $o] :
      ( ( R4 @ zero_zero_rat @ zero_zero_rat )
     => ( ( R4 @ one_one_rat @ one_one_rat )
       => ( bNF_re8824113547921424783at_rat
          @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
          @ R4
          @ zero_n2052037380579107095ol_rat
          @ zero_n2052037380579107095ol_rat ) ) ) ).

% transfer_rule_of_bool
thf(fact_7211_transfer__rule__of__bool,axiom,
    ! [R4: rat > nat > $o] :
      ( ( R4 @ zero_zero_rat @ zero_zero_nat )
     => ( ( R4 @ one_one_rat @ one_one_nat )
       => ( bNF_re235871571153144711at_nat
          @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
          @ R4
          @ zero_n2052037380579107095ol_rat
          @ zero_n2687167440665602831ol_nat ) ) ) ).

% transfer_rule_of_bool
thf(fact_7212_transfer__rule__of__bool,axiom,
    ! [R4: rat > int > $o] :
      ( ( R4 @ zero_zero_rat @ zero_zero_int )
     => ( ( R4 @ one_one_rat @ one_one_int )
       => ( bNF_re233381100644094435at_int
          @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
          @ R4
          @ zero_n2052037380579107095ol_rat
          @ zero_n2684676970156552555ol_int ) ) ) ).

% transfer_rule_of_bool
thf(fact_7213_transfer__rule__of__bool,axiom,
    ! [R4: rat > code_integer > $o] :
      ( ( R4 @ zero_zero_rat @ zero_z3403309356797280102nteger )
     => ( ( R4 @ one_one_rat @ one_one_Code_integer )
       => ( bNF_re5635613845597506580nteger
          @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
          @ R4
          @ zero_n2052037380579107095ol_rat
          @ zero_n356916108424825756nteger ) ) ) ).

% transfer_rule_of_bool
thf(fact_7214_transfer__rule__of__bool,axiom,
    ! [R4: nat > rat > $o] :
      ( ( R4 @ zero_zero_nat @ zero_zero_rat )
     => ( ( R4 @ one_one_nat @ one_one_rat )
       => ( bNF_re9140668191856290695at_rat
          @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
          @ R4
          @ zero_n2687167440665602831ol_nat
          @ zero_n2052037380579107095ol_rat ) ) ) ).

% transfer_rule_of_bool
thf(fact_7215_transfer__rule__of__bool,axiom,
    ! [R4: nat > nat > $o] :
      ( ( R4 @ zero_zero_nat @ zero_zero_nat )
     => ( ( R4 @ one_one_nat @ one_one_nat )
       => ( bNF_re552426215088010623at_nat
          @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
          @ R4
          @ zero_n2687167440665602831ol_nat
          @ zero_n2687167440665602831ol_nat ) ) ) ).

% transfer_rule_of_bool
thf(fact_7216_transfer__rule__of__bool,axiom,
    ! [R4: nat > int > $o] :
      ( ( R4 @ zero_zero_nat @ zero_zero_int )
     => ( ( R4 @ one_one_nat @ one_one_int )
       => ( bNF_re549935744578960347at_int
          @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
          @ R4
          @ zero_n2687167440665602831ol_nat
          @ zero_n2684676970156552555ol_int ) ) ) ).

% transfer_rule_of_bool
thf(fact_7217_transfer__rule__of__bool,axiom,
    ! [R4: nat > code_integer > $o] :
      ( ( R4 @ zero_zero_nat @ zero_z3403309356797280102nteger )
     => ( ( R4 @ one_one_nat @ one_one_Code_integer )
       => ( bNF_re4863541036035986956nteger
          @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
          @ R4
          @ zero_n2687167440665602831ol_nat
          @ zero_n356916108424825756nteger ) ) ) ).

% transfer_rule_of_bool
thf(fact_7218_transfer__rule__of__bool,axiom,
    ! [R4: int > rat > $o] :
      ( ( R4 @ zero_zero_int @ zero_zero_rat )
     => ( ( R4 @ one_one_int @ one_one_rat )
       => ( bNF_re916649866902569443nt_rat
          @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
          @ R4
          @ zero_n2684676970156552555ol_int
          @ zero_n2052037380579107095ol_rat ) ) ) ).

% transfer_rule_of_bool
thf(fact_7219_transfer__rule__of__bool,axiom,
    ! [R4: int > nat > $o] :
      ( ( R4 @ zero_zero_int @ zero_zero_nat )
     => ( ( R4 @ one_one_int @ one_one_nat )
       => ( bNF_re1551779926989065179nt_nat
          @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 )
          @ R4
          @ zero_n2684676970156552555ol_int
          @ zero_n2687167440665602831ol_nat ) ) ) ).

% transfer_rule_of_bool
thf(fact_7220_plus__rat_Otransfer,axiom,
    ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
    @ ^ [X3: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) )
    @ plus_plus_rat ) ).

% plus_rat.transfer
thf(fact_7221_inverse__rat_Otransfer,axiom,
    ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
    @ ^ [X3: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X3 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
    @ inverse_inverse_rat ) ).

% inverse_rat.transfer
thf(fact_7222_Id__on__def,axiom,
    ( id_on_6713504163309144702at_nat
    = ( ^ [A7: set_se7855581050983116737at_nat] :
          ( comple4364388969068068532at_nat
          @ ( image_6898982319499645995at_nat
            @ ^ [X3: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X3 @ X3 ) @ bot_bo228742789529271731at_nat )
            @ A7 ) ) ) ) ).

% Id_on_def
thf(fact_7223_Id__on__def,axiom,
    ( id_on_nat
    = ( ^ [A7: set_nat] :
          ( comple5685304695842803022at_nat
          @ ( image_7178329752028323786at_nat
            @ ^ [X3: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ bot_bo2099793752762293965at_nat )
            @ A7 ) ) ) ) ).

% Id_on_def
thf(fact_7224_times__rat_Otransfer,axiom,
    ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
    @ ^ [X3: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) )
    @ times_times_rat ) ).

% times_rat.transfer
thf(fact_7225_gcd__1__nat,axiom,
    ! [M2: nat] :
      ( ( gcd_gcd_nat @ M2 @ one_one_nat )
      = one_one_nat ) ).

% gcd_1_nat
thf(fact_7226_Id__onI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A2: set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ A @ A2 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ ( id_on_6713504163309144702at_nat @ A2 ) ) ) ).

% Id_onI
thf(fact_7227_Id__onI,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A2 )
     => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ A ) @ ( id_on_7869290023980668164at_nat @ A2 ) ) ) ).

% Id_onI
thf(fact_7228_Id__onI,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ ( id_on_2554058798563519774at_nat @ A2 ) ) ) ).

% Id_onI
thf(fact_7229_Id__onI,axiom,
    ! [A: $o,A2: set_o] :
      ( ( member_o @ A @ A2 )
     => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ A ) @ ( id_on_o @ A2 ) ) ) ).

% Id_onI
thf(fact_7230_Id__onI,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( id_on_nat @ A2 ) ) ) ).

% Id_onI
thf(fact_7231_Id__onI,axiom,
    ! [A: int,A2: set_int] :
      ( ( member_int @ A @ A2 )
     => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ A ) @ ( id_on_int @ A2 ) ) ) ).

% Id_onI
thf(fact_7232_unset__bit__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    @ bit_se4205575877204974255it_nat
    @ bit_se4205575877204974255it_nat ) ).

% unset_bit_natural.rsp
thf(fact_7233_take__bit__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    @ bit_se2925701944663578781it_nat
    @ bit_se2925701944663578781it_nat ) ).

% take_bit_natural.rsp
thf(fact_7234_bit__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ^ [Y4: nat > $o,Z2: nat > $o] : ( Y4 = Z2 )
    @ bit_se1148574629649215175it_nat
    @ bit_se1148574629649215175it_nat ) ).

% bit_natural.rsp
thf(fact_7235_drop__bit__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    @ bit_se8570568707652914677it_nat
    @ bit_se8570568707652914677it_nat ) ).

% drop_bit_natural.rsp
thf(fact_7236_Id__on__iff,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ ( id_on_7869290023980668164at_nat @ A2 ) )
      = ( ( X = Y )
        & ( member8757157785044589968at_nat @ X @ A2 ) ) ) ).

% Id_on_iff
thf(fact_7237_Id__on__iff,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( id_on_2554058798563519774at_nat @ A2 ) )
      = ( ( X = Y )
        & ( member8440522571783428010at_nat @ X @ A2 ) ) ) ).

% Id_on_iff
thf(fact_7238_Id__on__iff,axiom,
    ! [X: $o,Y: $o,A2: set_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ ( id_on_o @ A2 ) )
      = ( ( X = Y )
        & ( member_o @ X @ A2 ) ) ) ).

% Id_on_iff
thf(fact_7239_Id__on__iff,axiom,
    ! [X: int,Y: int,A2: set_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( id_on_int @ A2 ) )
      = ( ( X = Y )
        & ( member_int @ X @ A2 ) ) ) ).

% Id_on_iff
thf(fact_7240_Id__on__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,A2: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( id_on_6713504163309144702at_nat @ A2 ) )
      = ( ( X = Y )
        & ( member2643936169264416010at_nat @ X @ A2 ) ) ) ).

% Id_on_iff
thf(fact_7241_Id__on__iff,axiom,
    ! [X: nat,Y: nat,A2: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( id_on_nat @ A2 ) )
      = ( ( X = Y )
        & ( member_nat @ X @ A2 ) ) ) ).

% Id_on_iff
thf(fact_7242_Id__on__eqI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_se7855581050983116737at_nat] :
      ( ( A = B )
     => ( ( member2643936169264416010at_nat @ A @ A2 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( id_on_6713504163309144702at_nat @ A2 ) ) ) ) ).

% Id_on_eqI
thf(fact_7243_Id__on__eqI,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( A = B )
     => ( ( member8757157785044589968at_nat @ A @ A2 )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ ( id_on_7869290023980668164at_nat @ A2 ) ) ) ) ).

% Id_on_eqI
thf(fact_7244_Id__on__eqI,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( A = B )
     => ( ( member8440522571783428010at_nat @ A @ A2 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( id_on_2554058798563519774at_nat @ A2 ) ) ) ) ).

% Id_on_eqI
thf(fact_7245_Id__on__eqI,axiom,
    ! [A: $o,B: $o,A2: set_o] :
      ( ( A = B )
     => ( ( member_o @ A @ A2 )
       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( id_on_o @ A2 ) ) ) ) ).

% Id_on_eqI
thf(fact_7246_Id__on__eqI,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( ( A = B )
     => ( ( member_nat @ A @ A2 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( id_on_nat @ A2 ) ) ) ) ).

% Id_on_eqI
thf(fact_7247_Id__on__eqI,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( ( A = B )
     => ( ( member_int @ A @ A2 )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( id_on_int @ A2 ) ) ) ) ).

% Id_on_eqI
thf(fact_7248_Id__onE,axiom,
    ! [C: produc1906917888121415911at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( member6009668390874684176at_nat @ C @ ( id_on_7869290023980668164at_nat @ A2 ) )
     => ~ ! [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ A2 )
           => ( C
             != ( produc2335810615602371415at_nat @ X4 @ X4 ) ) ) ) ).

% Id_onE
thf(fact_7249_Id__onE,axiom,
    ! [C: produc859450856879609959at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ C @ ( id_on_2554058798563519774at_nat @ A2 ) )
     => ~ ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A2 )
           => ( C
             != ( produc6161850002892822231at_nat @ X4 @ X4 ) ) ) ) ).

% Id_onE
thf(fact_7250_Id__onE,axiom,
    ! [C: product_prod_o_o,A2: set_o] :
      ( ( member7466972457876170832od_o_o @ C @ ( id_on_o @ A2 ) )
     => ~ ! [X4: $o] :
            ( ( member_o @ X4 @ A2 )
           => ( C
             != ( product_Pair_o_o @ X4 @ X4 ) ) ) ) ).

% Id_onE
thf(fact_7251_Id__onE,axiom,
    ! [C: product_prod_int_int,A2: set_int] :
      ( ( member5262025264175285858nt_int @ C @ ( id_on_int @ A2 ) )
     => ~ ! [X4: int] :
            ( ( member_int @ X4 @ A2 )
           => ( C
             != ( product_Pair_int_int @ X4 @ X4 ) ) ) ) ).

% Id_onE
thf(fact_7252_Id__onE,axiom,
    ! [C: produc3843707927480180839at_nat,A2: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( id_on_6713504163309144702at_nat @ A2 ) )
     => ~ ! [X4: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X4 @ A2 )
           => ( C
             != ( produc2922128104949294807at_nat @ X4 @ X4 ) ) ) ) ).

% Id_onE
thf(fact_7253_Id__onE,axiom,
    ! [C: product_prod_nat_nat,A2: set_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( id_on_nat @ A2 ) )
     => ~ ! [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( C
             != ( product_Pair_nat_nat @ X4 @ X4 ) ) ) ) ).

% Id_onE
thf(fact_7254_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X4: nat,Y5: nat] :
          ( ( times_times_nat @ A @ X4 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_nat
thf(fact_7255_one__rat_Otransfer,axiom,
    pcr_rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ one_one_rat ).

% one_rat.transfer
thf(fact_7256_zero__rat_Otransfer,axiom,
    pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).

% zero_rat.transfer
thf(fact_7257_bezw__aux,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y ) )
      = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).

% bezw_aux
thf(fact_7258_uminus__rat_Otransfer,axiom,
    ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
    @ ^ [X3: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X3 ) ) @ ( product_snd_int_int @ X3 ) )
    @ uminus_uminus_rat ) ).

% uminus_rat.transfer
thf(fact_7259_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: nat] :
      ( ( ( gcd_gcd_nat @ X @ Xa )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
       => ~ ( ( ( ( Xa = zero_zero_nat )
               => ( Y = X ) )
              & ( ( Xa != zero_zero_nat )
               => ( Y
                  = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X @ Xa ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).

% gcd_nat.pelims
thf(fact_7260_o__rsp_I1_J,axiom,
    ! [R23: code_integer > code_integer > $o,R32: code_integer > code_integer > $o,R12: num > num > $o] : ( bNF_re5979881370344185769nteger @ ( bNF_re1062419521925302741nteger @ R23 @ R32 ) @ ( bNF_re6097285183634142137nteger @ ( bNF_re6741283560065411407nteger @ R12 @ R23 ) @ ( bNF_re6741283560065411407nteger @ R12 @ R32 ) ) @ comp_C3531382070062128313er_num @ comp_C3531382070062128313er_num ) ).

% o_rsp(1)
thf(fact_7261_o__rsp_I1_J,axiom,
    ! [R23: code_integer > nat > $o,R32: code_integer > num > $o,R12: num > int > $o] : ( bNF_re5469402333911078061nt_num @ ( bNF_re1758498617148825349er_num @ R23 @ R32 ) @ ( bNF_re1151383970801924621nt_num @ ( bNF_re6324211044263125204er_nat @ R12 @ R23 ) @ ( bNF_re2881542830418903838er_num @ R12 @ R32 ) ) @ comp_C3531382070062128313er_num @ comp_nat_num_int ) ).

% o_rsp(1)
thf(fact_7262_o__rsp_I1_J,axiom,
    ! [R23: nat > code_integer > $o,R32: num > code_integer > $o,R12: int > num > $o] : ( bNF_re2163612987331707565nteger @ ( bNF_re8763372240999575301nteger @ R23 @ R32 ) @ ( bNF_re8894579221946749965nteger @ ( bNF_re151304249936775380nteger @ R12 @ R23 ) @ ( bNF_re7315484114762210078nteger @ R12 @ R32 ) ) @ comp_nat_num_int @ comp_C3531382070062128313er_num ) ).

% o_rsp(1)
thf(fact_7263_o__rsp_I1_J,axiom,
    ! [R23: nat > nat > $o,R32: num > num > $o,R12: int > int > $o] : ( bNF_re3846787885862361009nt_num @ ( bNF_re341483209202555957um_num @ R23 @ R32 ) @ ( bNF_re9099949428469764705nt_num @ ( bNF_re3715656647883201625at_nat @ R12 @ R23 ) @ ( bNF_re7626690874201225453um_num @ R12 @ R32 ) ) @ comp_nat_num_int @ comp_nat_num_int ) ).

% o_rsp(1)
thf(fact_7264_o__rsp_I1_J,axiom,
    ! [R23: int > int > $o,R32: ( nat > $o ) > ( nat > $o ) > $o,R12: int > int > $o] : ( bNF_re7908124382625962333_nat_o @ ( bNF_re3376528473927230327_nat_o @ R23 @ R32 ) @ ( bNF_re6033114333830760525_nat_o @ ( bNF_re4712519889275205905nt_int @ R12 @ R23 ) @ ( bNF_re3376528473927230327_nat_o @ R12 @ R32 ) ) @ comp_int_nat_o_int @ comp_int_nat_o_int ) ).

% o_rsp(1)
thf(fact_7265_o__rsp_I1_J,axiom,
    ! [R23: int > int > $o,R32: ( int > $o ) > ( int > $o ) > $o,R12: int > int > $o] : ( bNF_re1900612664124570461_int_o @ ( bNF_re3403563459893282935_int_o @ R23 @ R32 ) @ ( bNF_re3014639844587090765_int_o @ ( bNF_re4712519889275205905nt_int @ R12 @ R23 ) @ ( bNF_re3403563459893282935_int_o @ R12 @ R32 ) ) @ comp_int_int_o_int @ comp_int_int_o_int ) ).

% o_rsp(1)
thf(fact_7266_o__rsp_I1_J,axiom,
    ! [R23: int > int > $o,R32: ( int > int ) > ( int > int ) > $o,R12: int > int > $o] : ( bNF_re4984751527058004117nt_int @ ( bNF_re711492959462206631nt_int @ R23 @ R32 ) @ ( bNF_re724639330072410579nt_int @ ( bNF_re4712519889275205905nt_int @ R12 @ R23 ) @ ( bNF_re711492959462206631nt_int @ R12 @ R32 ) ) @ comp_int_int_int_int @ comp_int_int_int_int ) ).

% o_rsp(1)
thf(fact_7267_o__rsp_I1_J,axiom,
    ! [R23: code_integer > code_integer > $o,R32: code_integer > ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > $o,R12: num > code_integer > $o] : ( bNF_re8134695817916415679nteger @ ( bNF_re3370092614548195229nteger @ R23 @ R32 ) @ ( bNF_re2439920061672227847nteger @ ( bNF_re5202476307641864082nteger @ R12 @ R23 ) @ ( bNF_re5031869190687302234nteger @ R12 @ R32 ) ) @ comp_C3531382070062128313er_num @ comp_C1593894019821074884nteger ) ).

% o_rsp(1)
thf(fact_7268_o__rsp_I1_J,axiom,
    ! [R23: code_integer > code_integer > $o,R32: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > code_integer > $o,R12: code_integer > num > $o] : ( bNF_re4919612610145136063nteger @ ( bNF_re4341081746073172381nteger @ R23 @ R32 ) @ ( bNF_re6724329528997265415nteger @ ( bNF_re7350422212947406738nteger @ R12 @ R23 ) @ ( bNF_re166502229267952730nteger @ R12 @ R32 ) ) @ comp_C1593894019821074884nteger @ comp_C3531382070062128313er_num ) ).

% o_rsp(1)
thf(fact_7269_o__rsp_I1_J,axiom,
    ! [R23: code_integer > nat > $o,R32: ( produc8923325533196201883nteger > produc8923325533196201883nteger ) > num > $o,R12: code_integer > int > $o] : ( bNF_re1890273300020150979nt_num @ ( bNF_re7119966052990372301er_num @ R23 @ R32 ) @ ( bNF_re5190287125156074331nt_num @ ( bNF_re2891348898900095767er_nat @ R12 @ R23 ) @ ( bNF_re8127238762813979689er_num @ R12 @ R32 ) ) @ comp_C1593894019821074884nteger @ comp_nat_num_int ) ).

% o_rsp(1)
thf(fact_7270_o__rsp_I2_J,axiom,
    ! [R12: num > num > $o] :
      ( bNF_re5979881370344185769nteger
      @ ^ [Y4: code_integer > code_integer,Z2: code_integer > code_integer] : ( Y4 = Z2 )
      @ ( bNF_re6097285183634142137nteger
        @ ( bNF_re6741283560065411407nteger @ R12
          @ ^ [Y4: code_integer,Z2: code_integer] : ( Y4 = Z2 ) )
        @ ( bNF_re6741283560065411407nteger @ R12
          @ ^ [Y4: code_integer,Z2: code_integer] : ( Y4 = Z2 ) ) )
      @ comp_C3531382070062128313er_num
      @ comp_C3531382070062128313er_num ) ).

% o_rsp(2)
thf(fact_7271_o__rsp_I2_J,axiom,
    ! [R12: int > int > $o] :
      ( bNF_re3846787885862361009nt_num
      @ ^ [Y4: nat > num,Z2: nat > num] : ( Y4 = Z2 )
      @ ( bNF_re9099949428469764705nt_num
        @ ( bNF_re3715656647883201625at_nat @ R12
          @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
        @ ( bNF_re7626690874201225453um_num @ R12
          @ ^ [Y4: num,Z2: num] : ( Y4 = Z2 ) ) )
      @ comp_nat_num_int
      @ comp_nat_num_int ) ).

% o_rsp(2)
thf(fact_7272_o__rsp_I2_J,axiom,
    ! [R12: int > int > $o] :
      ( bNF_re1550856938216548981_nat_o
      @ ^ [Y4: ( nat > $o ) > nat > $o,Z2: ( nat > $o ) > nat > $o] : ( Y4 = Z2 )
      @ ( bNF_re5673586069474677853_nat_o
        @ ( bNF_re3376528473927230327_nat_o @ R12
          @ ^ [Y4: nat > $o,Z2: nat > $o] : ( Y4 = Z2 ) )
        @ ( bNF_re3376528473927230327_nat_o @ R12
          @ ^ [Y4: nat > $o,Z2: nat > $o] : ( Y4 = Z2 ) ) )
      @ comp_nat_o_nat_o_int
      @ comp_nat_o_nat_o_int ) ).

% o_rsp(2)
thf(fact_7273_o__rsp_I2_J,axiom,
    ! [R12: int > int > $o] :
      ( bNF_re3491090157995142405_int_o
      @ ^ [Y4: ( nat > $o ) > int > $o,Z2: ( nat > $o ) > int > $o] : ( Y4 = Z2 )
      @ ( bNF_re2655111580231008093_int_o
        @ ( bNF_re3376528473927230327_nat_o @ R12
          @ ^ [Y4: nat > $o,Z2: nat > $o] : ( Y4 = Z2 ) )
        @ ( bNF_re3403563459893282935_int_o @ R12
          @ ^ [Y4: int > $o,Z2: int > $o] : ( Y4 = Z2 ) ) )
      @ comp_nat_o_int_o_int
      @ comp_nat_o_int_o_int ) ).

% o_rsp(2)
thf(fact_7274_o__rsp_I2_J,axiom,
    ! [R12: int > int > $o] :
      ( bNF_re5539255234643219293nt_int
      @ ^ [Y4: ( nat > $o ) > int > int,Z2: ( nat > $o ) > int > int] : ( Y4 = Z2 )
      @ ( bNF_re7395152722673581539nt_int
        @ ( bNF_re3376528473927230327_nat_o @ R12
          @ ^ [Y4: nat > $o,Z2: nat > $o] : ( Y4 = Z2 ) )
        @ ( bNF_re711492959462206631nt_int @ R12
          @ ^ [Y4: int > int,Z2: int > int] : ( Y4 = Z2 ) ) )
      @ comp_n3090169619561918849nt_int
      @ comp_n3090169619561918849nt_int ) ).

% o_rsp(2)
thf(fact_7275_o__rsp_I2_J,axiom,
    ! [R12: int > int > $o] :
      ( bNF_re615132412071444453_nat_o
      @ ^ [Y4: ( int > $o ) > nat > $o,Z2: ( int > $o ) > nat > $o] : ( Y4 = Z2 )
      @ ( bNF_re2446162618891206493_nat_o
        @ ( bNF_re3403563459893282935_int_o @ R12
          @ ^ [Y4: int > $o,Z2: int > $o] : ( Y4 = Z2 ) )
        @ ( bNF_re3376528473927230327_nat_o @ R12
          @ ^ [Y4: nat > $o,Z2: nat > $o] : ( Y4 = Z2 ) ) )
      @ comp_int_o_nat_o_int
      @ comp_int_o_nat_o_int ) ).

% o_rsp(2)
thf(fact_7276_o__rsp_I2_J,axiom,
    ! [R12: int > int > $o] :
      ( bNF_re2555365631850037877_int_o
      @ ^ [Y4: ( int > $o ) > int > $o,Z2: ( int > $o ) > int > $o] : ( Y4 = Z2 )
      @ ( bNF_re8651060166502312541_int_o
        @ ( bNF_re3403563459893282935_int_o @ R12
          @ ^ [Y4: int > $o,Z2: int > $o] : ( Y4 = Z2 ) )
        @ ( bNF_re3403563459893282935_int_o @ R12
          @ ^ [Y4: int > $o,Z2: int > $o] : ( Y4 = Z2 ) ) )
      @ comp_int_o_int_o_int
      @ comp_int_o_int_o_int ) ).

% o_rsp(2)
thf(fact_7277_o__rsp_I2_J,axiom,
    ! [R12: int > int > $o] :
      ( bNF_re5051145820034318685nt_int
      @ ^ [Y4: ( int > $o ) > int > int,Z2: ( int > $o ) > int > int] : ( Y4 = Z2 )
      @ ( bNF_re7462748251485907171nt_int
        @ ( bNF_re3403563459893282935_int_o @ R12
          @ ^ [Y4: int > $o,Z2: int > $o] : ( Y4 = Z2 ) )
        @ ( bNF_re711492959462206631nt_int @ R12
          @ ^ [Y4: int > int,Z2: int > int] : ( Y4 = Z2 ) ) )
      @ comp_i1993086638917920293nt_int
      @ comp_i1993086638917920293nt_int ) ).

% o_rsp(2)
thf(fact_7278_o__rsp_I2_J,axiom,
    ! [R12: int > int > $o] :
      ( bNF_re4659646225794970589_nat_o
      @ ^ [Y4: ( int > int ) > nat > $o,Z2: ( int > int ) > nat > $o] : ( Y4 = Z2 )
      @ ( bNF_re4332408551980016867_nat_o
        @ ( bNF_re711492959462206631nt_int @ R12
          @ ^ [Y4: int > int,Z2: int > int] : ( Y4 = Z2 ) )
        @ ( bNF_re3376528473927230327_nat_o @ R12
          @ ^ [Y4: nat > $o,Z2: nat > $o] : ( Y4 = Z2 ) ) )
      @ comp_i4411980392712547921_o_int
      @ comp_i4411980392712547921_o_int ) ).

% o_rsp(2)
thf(fact_7279_o__rsp_I2_J,axiom,
    ! [R12: int > int > $o] :
      ( bNF_re3161894838932316637_int_o
      @ ^ [Y4: ( int > int ) > int > $o,Z2: ( int > int ) > int > $o] : ( Y4 = Z2 )
      @ ( bNF_re1313934062736347107_int_o
        @ ( bNF_re711492959462206631nt_int @ R12
          @ ^ [Y4: int > int,Z2: int > int] : ( Y4 = Z2 ) )
        @ ( bNF_re3403563459893282935_int_o @ R12
          @ ^ [Y4: int > $o,Z2: int > $o] : ( Y4 = Z2 ) ) )
      @ comp_i5098556425753098485_o_int
      @ comp_i5098556425753098485_o_int ) ).

% o_rsp(2)
thf(fact_7280_apfst__convE,axiom,
    ! [Q3: produc1908205239877642774nteger,F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc6241069584506657477e_term > option6357759511663192854e_term,P4: produc1908205239877642774nteger] :
      ( ( Q3
        = ( produc2515316116081422825nteger @ F @ P4 ) )
     => ~ ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
            ( ( P4
              = ( produc8603105652947943368nteger @ X4 @ Y5 ) )
           => ( Q3
             != ( produc8603105652947943368nteger @ ( F @ X4 ) @ Y5 ) ) ) ) ).

% apfst_convE
thf(fact_7281_apfst__convE,axiom,
    ! [Q3: produc3925858234332021118et_nat,F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,P4: produc3925858234332021118et_nat] :
      ( ( Q3
        = ( produc3198230245617193214et_nat @ F @ P4 ) )
     => ~ ! [X4: produc3658429121746597890et_nat > $o,Y5: produc3658429121746597890et_nat] :
            ( ( P4
              = ( produc5001842942810119800et_nat @ X4 @ Y5 ) )
           => ( Q3
             != ( produc5001842942810119800et_nat @ ( F @ X4 ) @ Y5 ) ) ) ) ).

% apfst_convE
thf(fact_7282_apfst__convE,axiom,
    ! [Q3: produc2732055786443039994et_nat,F: ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,P4: produc2732055786443039994et_nat] :
      ( ( Q3
        = ( produc1442628905706551930et_nat @ F @ P4 ) )
     => ~ ! [X4: produc3658429121746597890et_nat > $o,Y5: produc3925858234332021118et_nat] :
            ( ( P4
              = ( produc2245416461498447860et_nat @ X4 @ Y5 ) )
           => ( Q3
             != ( produc2245416461498447860et_nat @ ( F @ X4 ) @ Y5 ) ) ) ) ).

% apfst_convE
thf(fact_7283_apfst__convE,axiom,
    ! [Q3: produc2285326912895808259nt_int,F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > produc8551481072490612790e_term > option6357759511663192854e_term,P4: produc2285326912895808259nt_int] :
      ( ( Q3
        = ( produc7277921532047535397nt_int @ F @ P4 ) )
     => ~ ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] :
            ( ( P4
              = ( produc5700946648718959541nt_int @ X4 @ Y5 ) )
           => ( Q3
             != ( produc5700946648718959541nt_int @ ( F @ X4 ) @ Y5 ) ) ) ) ).

% apfst_convE
thf(fact_7284_apfst__convE,axiom,
    ! [Q3: produc7773217078559923341nt_int,F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > int > option6357759511663192854e_term,P4: produc2285326912895808259nt_int] :
      ( ( Q3
        = ( produc6919319362177967663nt_int @ F @ P4 ) )
     => ~ ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] :
            ( ( P4
              = ( produc5700946648718959541nt_int @ X4 @ Y5 ) )
           => ( Q3
             != ( produc4305682042979456191nt_int @ ( F @ X4 ) @ Y5 ) ) ) ) ).

% apfst_convE
thf(fact_7285_apfst__convE,axiom,
    ! [Q3: produc2285326912895808259nt_int,F: ( int > option6357759511663192854e_term ) > produc8551481072490612790e_term > option6357759511663192854e_term,P4: produc7773217078559923341nt_int] :
      ( ( Q3
        = ( produc6067946493426971951nt_int @ F @ P4 ) )
     => ~ ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
            ( ( P4
              = ( produc4305682042979456191nt_int @ X4 @ Y5 ) )
           => ( Q3
             != ( produc5700946648718959541nt_int @ ( F @ X4 ) @ Y5 ) ) ) ) ).

% apfst_convE
thf(fact_7286_apfst__convE,axiom,
    ! [Q3: produc7773217078559923341nt_int,F: ( int > option6357759511663192854e_term ) > int > option6357759511663192854e_term,P4: produc7773217078559923341nt_int] :
      ( ( Q3
        = ( produc7131893314732144953nt_int @ F @ P4 ) )
     => ~ ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
            ( ( P4
              = ( produc4305682042979456191nt_int @ X4 @ Y5 ) )
           => ( Q3
             != ( produc4305682042979456191nt_int @ ( F @ X4 ) @ Y5 ) ) ) ) ).

% apfst_convE
thf(fact_7287_UN__le__eq__Un0,axiom,
    ! [M3: nat > set_Pr4329608150637261639at_nat,N2: nat] :
      ( ( comple4364388969068068532at_nat @ ( image_1087905878993176368at_nat @ M3 @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ ( image_1087905878993176368at_nat @ M3 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) @ ( M3 @ zero_zero_nat ) ) ) ).

% UN_le_eq_Un0
thf(fact_7288_UN__le__eq__Un0,axiom,
    ! [M3: nat > set_nat,N2: nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M3 @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M3 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) @ ( M3 @ zero_zero_nat ) ) ) ).

% UN_le_eq_Un0
thf(fact_7289_times__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X3: nat,Y3: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y3 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V2 ) @ ( times_times_nat @ Y3 @ U2 ) ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X3: nat,Y3: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y3 @ V2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V2 ) @ ( times_times_nat @ Y3 @ U2 ) ) ) ) ) ) ).

% times_int.rsp
thf(fact_7290_sup__Some,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( sup_su3598758113090618626et_nat @ ( some_set_nat @ X ) @ ( some_set_nat @ Y ) )
      = ( some_set_nat @ ( sup_sup_set_nat @ X @ Y ) ) ) ).

% sup_Some
thf(fact_7291_sup__Some,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( sup_su2273273666271716065at_nat @ ( some_s5890477192898017836at_nat @ X ) @ ( some_s5890477192898017836at_nat @ Y ) )
      = ( some_s5890477192898017836at_nat @ ( sup_su5525570899277871387at_nat @ X @ Y ) ) ) ).

% sup_Some
thf(fact_7292_sup__Some,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( sup_su7717485023253700007t_unit @ ( some_Product_unit @ X ) @ ( some_Product_unit @ Y ) )
      = ( some_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) ) ) ).

% sup_Some
thf(fact_7293_sup__Some,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_option_assn @ ( some_assn @ X ) @ ( some_assn @ Y ) )
      = ( some_assn @ ( sup_sup_assn @ X @ Y ) ) ) ).

% sup_Some
thf(fact_7294_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V: nat] :
      ( ( intrel @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ U @ V ) )
      = ( ( plus_plus_nat @ X @ V )
        = ( plus_plus_nat @ U @ Y ) ) ) ).

% intrel_iff
thf(fact_7295_insert__Times__insert,axiom,
    ! [A: $o,A2: set_o,B: $o,B5: set_o] :
      ( ( product_Sigma_o_o @ ( insert_o @ A @ A2 )
        @ ^ [Uu: $o] : ( insert_o @ B @ B5 ) )
      = ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B )
        @ ( sup_su5769328420594410459od_o_o
          @ ( product_Sigma_o_o @ A2
            @ ^ [Uu: $o] : ( insert_o @ B @ B5 ) )
          @ ( product_Sigma_o_o @ ( insert_o @ A @ A2 )
            @ ^ [Uu: $o] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_7296_insert__Times__insert,axiom,
    ! [A: $o,A2: set_o,B: nat,B5: set_nat] :
      ( ( product_Sigma_o_nat @ ( insert_o @ A @ A2 )
        @ ^ [Uu: $o] : ( insert_nat @ B @ B5 ) )
      = ( insert7948214739558299847_o_nat @ ( product_Pair_o_nat @ A @ B )
        @ ( sup_su8679140417860611193_o_nat
          @ ( product_Sigma_o_nat @ A2
            @ ^ [Uu: $o] : ( insert_nat @ B @ B5 ) )
          @ ( product_Sigma_o_nat @ ( insert_o @ A @ A2 )
            @ ^ [Uu: $o] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_7297_insert__Times__insert,axiom,
    ! [A: $o,A2: set_o,B: int,B5: set_int] :
      ( ( product_Sigma_o_int @ ( insert_o @ A @ A2 )
        @ ^ [Uu: $o] : ( insert_int @ B @ B5 ) )
      = ( insert3770363720049103139_o_int @ ( product_Pair_o_int @ A @ B )
        @ ( sup_su6189057272928884437_o_int
          @ ( product_Sigma_o_int @ A2
            @ ^ [Uu: $o] : ( insert_int @ B @ B5 ) )
          @ ( product_Sigma_o_int @ ( insert_o @ A @ A2 )
            @ ^ [Uu: $o] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_7298_insert__Times__insert,axiom,
    ! [A: nat,A2: set_nat,B: $o,B5: set_o] :
      ( ( product_Sigma_nat_o @ ( insert_nat @ A @ A2 )
        @ ^ [Uu: nat] : ( insert_o @ B @ B5 ) )
      = ( insert2233377226759017069_nat_o @ ( product_Pair_nat_o @ A @ B )
        @ ( sup_su503371503184139039_nat_o
          @ ( product_Sigma_nat_o @ A2
            @ ^ [Uu: nat] : ( insert_o @ B @ B5 ) )
          @ ( product_Sigma_nat_o @ ( insert_nat @ A @ A2 )
            @ ^ [Uu: nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_7299_insert__Times__insert,axiom,
    ! [A: nat,A2: set_nat,B: nat,B5: set_nat] :
      ( ( produc457027306803732586at_nat @ ( insert_nat @ A @ A2 )
        @ ^ [Uu: nat] : ( insert_nat @ B @ B5 ) )
      = ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B )
        @ ( sup_su6327502436637775413at_nat
          @ ( produc457027306803732586at_nat @ A2
            @ ^ [Uu: nat] : ( insert_nat @ B @ B5 ) )
          @ ( produc457027306803732586at_nat @ ( insert_nat @ A @ A2 )
            @ ^ [Uu: nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_7300_insert__Times__insert,axiom,
    ! [A: nat,A2: set_nat,B: int,B5: set_int] :
      ( ( produc454536836294682310at_int @ ( insert_nat @ A @ A2 )
        @ ^ [Uu: nat] : ( insert_int @ B @ B5 ) )
      = ( insert4033959196097957677at_int @ ( product_Pair_nat_int @ A @ B )
        @ ( sup_su3837419291706048657at_int
          @ ( produc454536836294682310at_int @ A2
            @ ^ [Uu: nat] : ( insert_int @ B @ B5 ) )
          @ ( produc454536836294682310at_int @ ( insert_nat @ A @ A2 )
            @ ^ [Uu: nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_7301_insert__Times__insert,axiom,
    ! [A: int,A2: set_int,B: $o,B5: set_o] :
      ( ( product_Sigma_int_o @ ( insert_int @ A @ A2 )
        @ ^ [Uu: int] : ( insert_o @ B @ B5 ) )
      = ( insert412334881326329105_int_o @ ( product_Pair_int_o @ A @ B )
        @ ( sup_su7481598572368468931_int_o
          @ ( product_Sigma_int_o @ A2
            @ ^ [Uu: int] : ( insert_o @ B @ B5 ) )
          @ ( product_Sigma_int_o @ ( insert_int @ A @ A2 )
            @ ^ [Uu: int] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_7302_insert__Times__insert,axiom,
    ! [A: int,A2: set_int,B: nat,B5: set_nat] :
      ( ( produc1456381018704787142nt_nat @ ( insert_int @ A @ A2 )
        @ ^ [Uu: int] : ( insert_nat @ B @ B5 ) )
      = ( insert9211163927508208941nt_nat @ ( product_Pair_int_nat @ A @ B )
        @ ( sup_su8514424011330797201nt_nat
          @ ( produc1456381018704787142nt_nat @ A2
            @ ^ [Uu: int] : ( insert_nat @ B @ B5 ) )
          @ ( produc1456381018704787142nt_nat @ ( insert_int @ A @ A2 )
            @ ^ [Uu: int] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_7303_insert__Times__insert,axiom,
    ! [A: int,A2: set_int,B: int,B5: set_int] :
      ( ( produc1453890548195736866nt_int @ ( insert_int @ A @ A2 )
        @ ^ [Uu: int] : ( insert_int @ B @ B5 ) )
      = ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A @ B )
        @ ( sup_su6024340866399070445nt_int
          @ ( produc1453890548195736866nt_int @ A2
            @ ^ [Uu: int] : ( insert_int @ B @ B5 ) )
          @ ( produc1453890548195736866nt_int @ ( insert_int @ A @ A2 )
            @ ^ [Uu: int] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_7304_insert__Times__insert,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: $o,B5: set_o] :
      ( ( produc203973861852599583_nat_o @ ( insert8211810215607154385at_nat @ A @ A2 )
        @ ^ [Uu: product_prod_nat_nat] : ( insert_o @ B @ B5 ) )
      = ( insert2126339615020983374_nat_o @ ( produc855456346497908590_nat_o @ A @ B )
        @ ( sup_su5921324857680904370_nat_o
          @ ( produc203973861852599583_nat_o @ A2
            @ ^ [Uu: product_prod_nat_nat] : ( insert_o @ B @ B5 ) )
          @ ( produc203973861852599583_nat_o @ ( insert8211810215607154385at_nat @ A @ A2 )
            @ ^ [Uu: product_prod_nat_nat] : B5 ) ) ) ) ).

% insert_Times_insert
thf(fact_7305_Sigma__Un__distrib1,axiom,
    ! [I4: set_se7855581050983116737at_nat,J4: set_se7855581050983116737at_nat,C4: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ ( sup_su3642409539654194069at_nat @ I4 @ J4 ) @ C4 )
      = ( sup_su5525570899277871387at_nat @ ( produc7726050920695093128at_nat @ I4 @ C4 ) @ ( produc7726050920695093128at_nat @ J4 @ C4 ) ) ) ).

% Sigma_Un_distrib1
thf(fact_7306_Sigma__Un__distrib2,axiom,
    ! [I4: set_se7855581050983116737at_nat,A2: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat,B5: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ I4
        @ ^ [I3: set_Pr1261947904930325089at_nat] : ( sup_su3642409539654194069at_nat @ ( A2 @ I3 ) @ ( B5 @ I3 ) ) )
      = ( sup_su5525570899277871387at_nat @ ( produc7726050920695093128at_nat @ I4 @ A2 ) @ ( produc7726050920695093128at_nat @ I4 @ B5 ) ) ) ).

% Sigma_Un_distrib2
thf(fact_7307_Times__Un__distrib1,axiom,
    ! [A2: set_se7855581050983116737at_nat,B5: set_se7855581050983116737at_nat,C4: set_se7855581050983116737at_nat] :
      ( ( produc7726050920695093128at_nat @ ( sup_su3642409539654194069at_nat @ A2 @ B5 )
        @ ^ [Uu: set_Pr1261947904930325089at_nat] : C4 )
      = ( sup_su5525570899277871387at_nat
        @ ( produc7726050920695093128at_nat @ A2
          @ ^ [Uu: set_Pr1261947904930325089at_nat] : C4 )
        @ ( produc7726050920695093128at_nat @ B5
          @ ^ [Uu: set_Pr1261947904930325089at_nat] : C4 ) ) ) ).

% Times_Un_distrib1
thf(fact_7308_set__diff__diff__left,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,C4: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B5 ) @ C4 )
      = ( minus_3314409938677909166at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ B5 @ C4 ) ) ) ).

% set_diff_diff_left
thf(fact_7309_set__diff__diff__left,axiom,
    ! [A2: set_nat,B5: set_nat,C4: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B5 ) @ C4 )
      = ( minus_minus_set_nat @ A2 @ ( sup_sup_set_nat @ B5 @ C4 ) ) ) ).

% set_diff_diff_left
thf(fact_7310_sup__option__def,axiom,
    ( sup_su3598758113090618626et_nat
    = ( ^ [X3: option_set_nat,Y3: option_set_nat] :
          ( case_o4054078431260844265et_nat @ Y3
          @ ^ [X10: set_nat] :
              ( case_o4054078431260844265et_nat @ X3
              @ ^ [Z4: set_nat] : ( some_set_nat @ ( sup_sup_set_nat @ X10 @ Z4 ) )
              @ Y3 )
          @ X3 ) ) ) ).

% sup_option_def
thf(fact_7311_sup__option__def,axiom,
    ( sup_su2273273666271716065at_nat
    = ( ^ [X3: option1583680563626158861at_nat,Y3: option1583680563626158861at_nat] :
          ( case_o9080059180780454917at_nat @ Y3
          @ ^ [X10: set_Pr4329608150637261639at_nat] :
              ( case_o9080059180780454917at_nat @ X3
              @ ^ [Z4: set_Pr4329608150637261639at_nat] : ( some_s5890477192898017836at_nat @ ( sup_su5525570899277871387at_nat @ X10 @ Z4 ) )
              @ Y3 )
          @ X3 ) ) ) ).

% sup_option_def
thf(fact_7312_sup__option__def,axiom,
    ( sup_su7717485023253700007t_unit
    = ( ^ [X3: option_Product_unit,Y3: option_Product_unit] :
          ( case_o2985186191577459077t_unit @ Y3
          @ ^ [X10: product_unit] :
              ( case_o2985186191577459077t_unit @ X3
              @ ^ [Z4: product_unit] : ( some_Product_unit @ ( sup_sup_Product_unit @ X10 @ Z4 ) )
              @ Y3 )
          @ X3 ) ) ) ).

% sup_option_def
thf(fact_7313_sup__option__def,axiom,
    ( sup_sup_option_assn
    = ( ^ [X3: option_assn,Y3: option_assn] :
          ( case_o4484465799723439917n_assn @ Y3
          @ ^ [X10: assn] :
              ( case_o4484465799723439917n_assn @ X3
              @ ^ [Z4: assn] : ( some_assn @ ( sup_sup_assn @ X10 @ Z4 ) )
              @ Y3 )
          @ X3 ) ) ) ).

% sup_option_def
thf(fact_7314_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: rat,M2: rat,U: rat] :
      ( ( ord_less_eq_rat @ L @ M2 )
     => ( ( ord_less_eq_rat @ M2 @ U )
       => ( ( sup_sup_set_rat @ ( set_or633870826150836451st_rat @ L @ M2 ) @ ( set_or633870826150836451st_rat @ M2 @ U ) )
          = ( set_or633870826150836451st_rat @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_7315_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: num,M2: num,U: num] :
      ( ( ord_less_eq_num @ L @ M2 )
     => ( ( ord_less_eq_num @ M2 @ U )
       => ( ( sup_sup_set_num @ ( set_or7049704709247886629st_num @ L @ M2 ) @ ( set_or7049704709247886629st_num @ M2 @ U ) )
          = ( set_or7049704709247886629st_num @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_7316_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: int,M2: int,U: int] :
      ( ( ord_less_eq_int @ L @ M2 )
     => ( ( ord_less_eq_int @ M2 @ U )
       => ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ M2 ) @ ( set_or1266510415728281911st_int @ M2 @ U ) )
          = ( set_or1266510415728281911st_int @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_7317_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: nat,M2: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ U )
       => ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M2 ) @ ( set_or1269000886237332187st_nat @ M2 @ U ) )
          = ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_7318_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: code_integer,M2: code_integer,U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ M2 )
     => ( ( ord_le3102999989581377725nteger @ M2 @ U )
       => ( ( sup_su848401254843788991nteger @ ( set_or189985376899183464nteger @ L @ M2 ) @ ( set_or189985376899183464nteger @ M2 @ U ) )
          = ( set_or189985376899183464nteger @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_7319_ivl__disj__un__two_I3_J,axiom,
    ! [L: rat,M2: rat,U: rat] :
      ( ( ord_less_eq_rat @ L @ M2 )
     => ( ( ord_less_eq_rat @ M2 @ U )
       => ( ( sup_sup_set_rat @ ( set_or4029947393144176647an_rat @ L @ M2 ) @ ( set_or4029947393144176647an_rat @ M2 @ U ) )
          = ( set_or4029947393144176647an_rat @ L @ U ) ) ) ) ).

% ivl_disj_un_two(3)
thf(fact_7320_ivl__disj__un__two_I3_J,axiom,
    ! [L: num,M2: num,U: num] :
      ( ( ord_less_eq_num @ L @ M2 )
     => ( ( ord_less_eq_num @ M2 @ U )
       => ( ( sup_sup_set_num @ ( set_or1222409239386451017an_num @ L @ M2 ) @ ( set_or1222409239386451017an_num @ M2 @ U ) )
          = ( set_or1222409239386451017an_num @ L @ U ) ) ) ) ).

% ivl_disj_un_two(3)
thf(fact_7321_ivl__disj__un__two_I3_J,axiom,
    ! [L: nat,M2: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ U )
       => ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M2 ) @ ( set_or4665077453230672383an_nat @ M2 @ U ) )
          = ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).

% ivl_disj_un_two(3)
thf(fact_7322_ivl__disj__un__two_I3_J,axiom,
    ! [L: code_integer,M2: code_integer,U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ M2 )
     => ( ( ord_le3102999989581377725nteger @ M2 @ U )
       => ( ( sup_su848401254843788991nteger @ ( set_or8404916559141939852nteger @ L @ M2 ) @ ( set_or8404916559141939852nteger @ M2 @ U ) )
          = ( set_or8404916559141939852nteger @ L @ U ) ) ) ) ).

% ivl_disj_un_two(3)
thf(fact_7323_ivl__disj__un__two_I3_J,axiom,
    ! [L: int,M2: int,U: int] :
      ( ( ord_less_eq_int @ L @ M2 )
     => ( ( ord_less_eq_int @ M2 @ U )
       => ( ( sup_sup_set_int @ ( set_or4662586982721622107an_int @ L @ M2 ) @ ( set_or4662586982721622107an_int @ M2 @ U ) )
          = ( set_or4662586982721622107an_int @ L @ U ) ) ) ) ).

% ivl_disj_un_two(3)
thf(fact_7324_max__ext__additive,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,R4: set_Pr8693737435421807431at_nat,C4: set_Pr1261947904930325089at_nat,D3: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B5 ) @ ( max_ex8135407076693332796at_nat @ R4 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ C4 @ D3 ) @ ( max_ex8135407076693332796at_nat @ R4 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( sup_su6327502436637775413at_nat @ A2 @ C4 ) @ ( sup_su6327502436637775413at_nat @ B5 @ D3 ) ) @ ( max_ex8135407076693332796at_nat @ R4 ) ) ) ) ).

% max_ext_additive
thf(fact_7325_max__ext__additive,axiom,
    ! [A2: set_nat,B5: set_nat,R4: set_Pr1261947904930325089at_nat,C4: set_nat,D3: set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A2 @ B5 ) @ ( max_ext_nat @ R4 ) )
     => ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ C4 @ D3 ) @ ( max_ext_nat @ R4 ) )
       => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ ( sup_sup_set_nat @ A2 @ C4 ) @ ( sup_sup_set_nat @ B5 @ D3 ) ) @ ( max_ext_nat @ R4 ) ) ) ) ).

% max_ext_additive
thf(fact_7326_max__ext__additive,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat,R4: set_Pr5564308138774400199at_nat,C4: set_Pr4329608150637261639at_nat,D3: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A2 @ B5 ) @ ( max_ex8833738776070432290at_nat @ R4 ) )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ C4 @ D3 ) @ ( max_ex8833738776070432290at_nat @ R4 ) )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ C4 ) @ ( sup_su5525570899277871387at_nat @ B5 @ D3 ) ) @ ( max_ex8833738776070432290at_nat @ R4 ) ) ) ) ).

% max_ext_additive
thf(fact_7327_zero__int_Orsp,axiom,
    intrel @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ).

% zero_int.rsp
thf(fact_7328_ivl__disj__un__two_I7_J,axiom,
    ! [L: rat,M2: rat,U: rat] :
      ( ( ord_less_eq_rat @ L @ M2 )
     => ( ( ord_less_eq_rat @ M2 @ U )
       => ( ( sup_sup_set_rat @ ( set_or4029947393144176647an_rat @ L @ M2 ) @ ( set_or633870826150836451st_rat @ M2 @ U ) )
          = ( set_or633870826150836451st_rat @ L @ U ) ) ) ) ).

% ivl_disj_un_two(7)
thf(fact_7329_ivl__disj__un__two_I7_J,axiom,
    ! [L: num,M2: num,U: num] :
      ( ( ord_less_eq_num @ L @ M2 )
     => ( ( ord_less_eq_num @ M2 @ U )
       => ( ( sup_sup_set_num @ ( set_or1222409239386451017an_num @ L @ M2 ) @ ( set_or7049704709247886629st_num @ M2 @ U ) )
          = ( set_or7049704709247886629st_num @ L @ U ) ) ) ) ).

% ivl_disj_un_two(7)
thf(fact_7330_ivl__disj__un__two_I7_J,axiom,
    ! [L: nat,M2: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ U )
       => ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ M2 ) @ ( set_or1269000886237332187st_nat @ M2 @ U ) )
          = ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).

% ivl_disj_un_two(7)
thf(fact_7331_ivl__disj__un__two_I7_J,axiom,
    ! [L: code_integer,M2: code_integer,U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ M2 )
     => ( ( ord_le3102999989581377725nteger @ M2 @ U )
       => ( ( sup_su848401254843788991nteger @ ( set_or8404916559141939852nteger @ L @ M2 ) @ ( set_or189985376899183464nteger @ M2 @ U ) )
          = ( set_or189985376899183464nteger @ L @ U ) ) ) ) ).

% ivl_disj_un_two(7)
thf(fact_7332_ivl__disj__un__two_I7_J,axiom,
    ! [L: int,M2: int,U: int] :
      ( ( ord_less_eq_int @ L @ M2 )
     => ( ( ord_less_eq_int @ M2 @ U )
       => ( ( sup_sup_set_int @ ( set_or4662586982721622107an_int @ L @ M2 ) @ ( set_or1266510415728281911st_int @ M2 @ U ) )
          = ( set_or1266510415728281911st_int @ L @ U ) ) ) ) ).

% ivl_disj_un_two(7)
thf(fact_7333_card__Un__le,axiom,
    ! [A2: set_int,B5: set_int] : ( ord_less_eq_nat @ ( finite_card_int @ ( sup_sup_set_int @ A2 @ B5 ) ) @ ( plus_plus_nat @ ( finite_card_int @ A2 ) @ ( finite_card_int @ B5 ) ) ) ).

% card_Un_le
thf(fact_7334_card__Un__le,axiom,
    ! [A2: set_Product_unit,B5: set_Product_unit] : ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( sup_su793286257634532545t_unit @ A2 @ B5 ) ) @ ( plus_plus_nat @ ( finite410649719033368117t_unit @ A2 ) @ ( finite410649719033368117t_unit @ B5 ) ) ) ).

% card_Un_le
thf(fact_7335_card__Un__le,axiom,
    ! [A2: set_list_nat,B5: set_list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( sup_sup_set_list_nat @ A2 @ B5 ) ) @ ( plus_plus_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B5 ) ) ) ).

% card_Un_le
thf(fact_7336_card__Un__le,axiom,
    ! [A2: set_nat,B5: set_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A2 @ B5 ) ) @ ( plus_plus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B5 ) ) ) ).

% card_Un_le
thf(fact_7337_card__Un__le,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : ( ord_less_eq_nat @ ( finite3771342082235030671at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B5 ) ) @ ( plus_plus_nat @ ( finite3771342082235030671at_nat @ A2 ) @ ( finite3771342082235030671at_nat @ B5 ) ) ) ).

% card_Un_le
thf(fact_7338_ivl__disj__un__one_I2_J,axiom,
    ! [L: rat,U: rat] :
      ( ( ord_less_eq_rat @ L @ U )
     => ( ( sup_sup_set_rat @ ( set_ord_lessThan_rat @ L ) @ ( set_or4029947393144176647an_rat @ L @ U ) )
        = ( set_ord_lessThan_rat @ U ) ) ) ).

% ivl_disj_un_one(2)
thf(fact_7339_ivl__disj__un__one_I2_J,axiom,
    ! [L: num,U: num] :
      ( ( ord_less_eq_num @ L @ U )
     => ( ( sup_sup_set_num @ ( set_ord_lessThan_num @ L ) @ ( set_or1222409239386451017an_num @ L @ U ) )
        = ( set_ord_lessThan_num @ U ) ) ) ).

% ivl_disj_un_one(2)
thf(fact_7340_ivl__disj__un__one_I2_J,axiom,
    ! [L: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ U )
     => ( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or4665077453230672383an_nat @ L @ U ) )
        = ( set_ord_lessThan_nat @ U ) ) ) ).

% ivl_disj_un_one(2)
thf(fact_7341_ivl__disj__un__one_I2_J,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ U )
     => ( ( sup_su848401254843788991nteger @ ( set_or5754767410780653050nteger @ L ) @ ( set_or8404916559141939852nteger @ L @ U ) )
        = ( set_or5754767410780653050nteger @ U ) ) ) ).

% ivl_disj_un_one(2)
thf(fact_7342_ivl__disj__un__one_I2_J,axiom,
    ! [L: int,U: int] :
      ( ( ord_less_eq_int @ L @ U )
     => ( ( sup_sup_set_int @ ( set_ord_lessThan_int @ L ) @ ( set_or4662586982721622107an_int @ L @ U ) )
        = ( set_ord_lessThan_int @ U ) ) ) ).

% ivl_disj_un_one(2)
thf(fact_7343_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
        = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% atLeastLessThan_add_Un
thf(fact_7344_uminus__int_Orsp,axiom,
    ( bNF_re2241393799969408733at_nat @ intrel @ intrel
    @ ( produc2626176000494625587at_nat
      @ ^ [X3: nat,Y3: nat] : ( product_Pair_nat_nat @ Y3 @ X3 ) )
    @ ( produc2626176000494625587at_nat
      @ ^ [X3: nat,Y3: nat] : ( product_Pair_nat_nat @ Y3 @ X3 ) ) ) ).

% uminus_int.rsp
thf(fact_7345_one__int_Orsp,axiom,
    intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).

% one_int.rsp
thf(fact_7346_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: rat,M2: rat,U: rat] :
      ( ( ord_less_eq_rat @ L @ M2 )
     => ( ( ord_less_rat @ M2 @ U )
       => ( ( sup_sup_set_rat @ ( set_or633870826150836451st_rat @ L @ M2 ) @ ( set_or4029947393144176647an_rat @ M2 @ U ) )
          = ( set_or4029947393144176647an_rat @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(2)
thf(fact_7347_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: num,M2: num,U: num] :
      ( ( ord_less_eq_num @ L @ M2 )
     => ( ( ord_less_num @ M2 @ U )
       => ( ( sup_sup_set_num @ ( set_or7049704709247886629st_num @ L @ M2 ) @ ( set_or1222409239386451017an_num @ M2 @ U ) )
          = ( set_or1222409239386451017an_num @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(2)
thf(fact_7348_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: nat,M2: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ M2 )
     => ( ( ord_less_nat @ M2 @ U )
       => ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M2 ) @ ( set_or4665077453230672383an_nat @ M2 @ U ) )
          = ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(2)
thf(fact_7349_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: code_integer,M2: code_integer,U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ M2 )
     => ( ( ord_le6747313008572928689nteger @ M2 @ U )
       => ( ( sup_su848401254843788991nteger @ ( set_or189985376899183464nteger @ L @ M2 ) @ ( set_or8404916559141939852nteger @ M2 @ U ) )
          = ( set_or8404916559141939852nteger @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(2)
thf(fact_7350_ivl__disj__un__two__touch_I2_J,axiom,
    ! [L: int,M2: int,U: int] :
      ( ( ord_less_eq_int @ L @ M2 )
     => ( ( ord_less_int @ M2 @ U )
       => ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ M2 ) @ ( set_or4662586982721622107an_int @ M2 @ U ) )
          = ( set_or4662586982721622107an_int @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(2)
thf(fact_7351_ivl__disj__un__two_I1_J,axiom,
    ! [L: rat,M2: rat,U: rat] :
      ( ( ord_less_rat @ L @ M2 )
     => ( ( ord_less_eq_rat @ M2 @ U )
       => ( ( sup_sup_set_rat @ ( set_or5199638295745620268an_rat @ L @ M2 ) @ ( set_or4029947393144176647an_rat @ M2 @ U ) )
          = ( set_or5199638295745620268an_rat @ L @ U ) ) ) ) ).

% ivl_disj_un_two(1)
thf(fact_7352_ivl__disj__un__two_I1_J,axiom,
    ! [L: num,M2: num,U: num] :
      ( ( ord_less_num @ L @ M2 )
     => ( ( ord_less_eq_num @ M2 @ U )
       => ( ( sup_sup_set_num @ ( set_or2392100141987894638an_num @ L @ M2 ) @ ( set_or1222409239386451017an_num @ M2 @ U ) )
          = ( set_or2392100141987894638an_num @ L @ U ) ) ) ) ).

% ivl_disj_un_two(1)
thf(fact_7353_ivl__disj__un__two_I1_J,axiom,
    ! [L: int,M2: int,U: int] :
      ( ( ord_less_int @ L @ M2 )
     => ( ( ord_less_eq_int @ M2 @ U )
       => ( ( sup_sup_set_int @ ( set_or5832277885323065728an_int @ L @ M2 ) @ ( set_or4662586982721622107an_int @ M2 @ U ) )
          = ( set_or5832277885323065728an_int @ L @ U ) ) ) ) ).

% ivl_disj_un_two(1)
thf(fact_7354_ivl__disj__un__two_I1_J,axiom,
    ! [L: nat,M2: nat,U: nat] :
      ( ( ord_less_nat @ L @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ U )
       => ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ M2 ) @ ( set_or4665077453230672383an_nat @ M2 @ U ) )
          = ( set_or5834768355832116004an_nat @ L @ U ) ) ) ) ).

% ivl_disj_un_two(1)
thf(fact_7355_ivl__disj__un__two_I1_J,axiom,
    ! [L: code_integer,M2: code_integer,U: code_integer] :
      ( ( ord_le6747313008572928689nteger @ L @ M2 )
     => ( ( ord_le3102999989581377725nteger @ M2 @ U )
       => ( ( sup_su848401254843788991nteger @ ( set_or4266950643985792945nteger @ L @ M2 ) @ ( set_or8404916559141939852nteger @ M2 @ U ) )
          = ( set_or4266950643985792945nteger @ L @ U ) ) ) ) ).

% ivl_disj_un_two(1)
thf(fact_7356_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: $o] :
      ( ( sup_sup_set_o @ ( set_ord_lessThan_o @ U ) @ ( insert_o @ U @ bot_bot_set_o ) )
      = ( set_ord_atMost_o @ U ) ) ).

% ivl_disj_un_singleton(2)
thf(fact_7357_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: int] :
      ( ( sup_sup_set_int @ ( set_ord_lessThan_int @ U ) @ ( insert_int @ U @ bot_bot_set_int ) )
      = ( set_ord_atMost_int @ U ) ) ).

% ivl_disj_un_singleton(2)
thf(fact_7358_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: nat] :
      ( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ U ) @ ( insert_nat @ U @ bot_bot_set_nat ) )
      = ( set_ord_atMost_nat @ U ) ) ).

% ivl_disj_un_singleton(2)
thf(fact_7359_ivl__disj__un__one_I4_J,axiom,
    ! [L: rat,U: rat] :
      ( ( ord_less_eq_rat @ L @ U )
     => ( ( sup_sup_set_rat @ ( set_ord_lessThan_rat @ L ) @ ( set_or633870826150836451st_rat @ L @ U ) )
        = ( set_ord_atMost_rat @ U ) ) ) ).

% ivl_disj_un_one(4)
thf(fact_7360_ivl__disj__un__one_I4_J,axiom,
    ! [L: num,U: num] :
      ( ( ord_less_eq_num @ L @ U )
     => ( ( sup_sup_set_num @ ( set_ord_lessThan_num @ L ) @ ( set_or7049704709247886629st_num @ L @ U ) )
        = ( set_ord_atMost_num @ U ) ) ) ).

% ivl_disj_un_one(4)
thf(fact_7361_ivl__disj__un__one_I4_J,axiom,
    ! [L: int,U: int] :
      ( ( ord_less_eq_int @ L @ U )
     => ( ( sup_sup_set_int @ ( set_ord_lessThan_int @ L ) @ ( set_or1266510415728281911st_int @ L @ U ) )
        = ( set_ord_atMost_int @ U ) ) ) ).

% ivl_disj_un_one(4)
thf(fact_7362_ivl__disj__un__one_I4_J,axiom,
    ! [L: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ U )
     => ( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or1269000886237332187st_nat @ L @ U ) )
        = ( set_ord_atMost_nat @ U ) ) ) ).

% ivl_disj_un_one(4)
thf(fact_7363_ivl__disj__un__one_I4_J,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ U )
     => ( ( sup_su848401254843788991nteger @ ( set_or5754767410780653050nteger @ L ) @ ( set_or189985376899183464nteger @ L @ U ) )
        = ( set_or9101266186257409494nteger @ U ) ) ) ).

% ivl_disj_un_one(4)
thf(fact_7364_ivl__disj__un__one_I1_J,axiom,
    ! [L: rat,U: rat] :
      ( ( ord_less_rat @ L @ U )
     => ( ( sup_sup_set_rat @ ( set_ord_atMost_rat @ L ) @ ( set_or5199638295745620268an_rat @ L @ U ) )
        = ( set_ord_lessThan_rat @ U ) ) ) ).

% ivl_disj_un_one(1)
thf(fact_7365_ivl__disj__un__one_I1_J,axiom,
    ! [L: num,U: num] :
      ( ( ord_less_num @ L @ U )
     => ( ( sup_sup_set_num @ ( set_ord_atMost_num @ L ) @ ( set_or2392100141987894638an_num @ L @ U ) )
        = ( set_ord_lessThan_num @ U ) ) ) ).

% ivl_disj_un_one(1)
thf(fact_7366_ivl__disj__un__one_I1_J,axiom,
    ! [L: int,U: int] :
      ( ( ord_less_int @ L @ U )
     => ( ( sup_sup_set_int @ ( set_ord_atMost_int @ L ) @ ( set_or5832277885323065728an_int @ L @ U ) )
        = ( set_ord_lessThan_int @ U ) ) ) ).

% ivl_disj_un_one(1)
thf(fact_7367_ivl__disj__un__one_I1_J,axiom,
    ! [L: nat,U: nat] :
      ( ( ord_less_nat @ L @ U )
     => ( ( sup_sup_set_nat @ ( set_ord_atMost_nat @ L ) @ ( set_or5834768355832116004an_nat @ L @ U ) )
        = ( set_ord_lessThan_nat @ U ) ) ) ).

% ivl_disj_un_one(1)
thf(fact_7368_ivl__disj__un__one_I1_J,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( ord_le6747313008572928689nteger @ L @ U )
     => ( ( sup_su848401254843788991nteger @ ( set_or9101266186257409494nteger @ L ) @ ( set_or4266950643985792945nteger @ L @ U ) )
        = ( set_or5754767410780653050nteger @ U ) ) ) ).

% ivl_disj_un_one(1)
thf(fact_7369_intrel__def,axiom,
    ( intrel
    = ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y3: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V2: nat] :
              ( ( plus_plus_nat @ X3 @ V2 )
              = ( plus_plus_nat @ U2 @ Y3 ) ) ) ) ) ).

% intrel_def
thf(fact_7370_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: $o,U: $o] :
      ( ( ord_less_eq_o @ L @ U )
     => ( ( sup_sup_set_o @ ( set_or7139685690850216873Than_o @ L @ U ) @ ( insert_o @ U @ bot_bot_set_o ) )
        = ( set_or8904488021354931149Most_o @ L @ U ) ) ) ).

% ivl_disj_un_singleton(6)
thf(fact_7371_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: rat,U: rat] :
      ( ( ord_less_eq_rat @ L @ U )
     => ( ( sup_sup_set_rat @ ( set_or4029947393144176647an_rat @ L @ U ) @ ( insert_rat @ U @ bot_bot_set_rat ) )
        = ( set_or633870826150836451st_rat @ L @ U ) ) ) ).

% ivl_disj_un_singleton(6)
thf(fact_7372_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: num,U: num] :
      ( ( ord_less_eq_num @ L @ U )
     => ( ( sup_sup_set_num @ ( set_or1222409239386451017an_num @ L @ U ) @ ( insert_num @ U @ bot_bot_set_num ) )
        = ( set_or7049704709247886629st_num @ L @ U ) ) ) ).

% ivl_disj_un_singleton(6)
thf(fact_7373_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ U )
     => ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ U ) @ ( insert_nat @ U @ bot_bot_set_nat ) )
        = ( set_or1269000886237332187st_nat @ L @ U ) ) ) ).

% ivl_disj_un_singleton(6)
thf(fact_7374_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ U )
     => ( ( sup_su848401254843788991nteger @ ( set_or8404916559141939852nteger @ L @ U ) @ ( insert_Code_integer @ U @ bot_bo3990330152332043303nteger ) )
        = ( set_or189985376899183464nteger @ L @ U ) ) ) ).

% ivl_disj_un_singleton(6)
thf(fact_7375_ivl__disj__un__singleton_I6_J,axiom,
    ! [L: int,U: int] :
      ( ( ord_less_eq_int @ L @ U )
     => ( ( sup_sup_set_int @ ( set_or4662586982721622107an_int @ L @ U ) @ ( insert_int @ U @ bot_bot_set_int ) )
        = ( set_or1266510415728281911st_int @ L @ U ) ) ) ).

% ivl_disj_un_singleton(6)
thf(fact_7376_ivl__disj__un__singleton_I3_J,axiom,
    ! [L: $o,U: $o] :
      ( ( ord_less_o @ L @ U )
     => ( ( sup_sup_set_o @ ( insert_o @ L @ bot_bot_set_o ) @ ( set_or1716231572884733764Than_o @ L @ U ) )
        = ( set_or7139685690850216873Than_o @ L @ U ) ) ) ).

% ivl_disj_un_singleton(3)
thf(fact_7377_ivl__disj__un__singleton_I3_J,axiom,
    ! [L: rat,U: rat] :
      ( ( ord_less_rat @ L @ U )
     => ( ( sup_sup_set_rat @ ( insert_rat @ L @ bot_bot_set_rat ) @ ( set_or5199638295745620268an_rat @ L @ U ) )
        = ( set_or4029947393144176647an_rat @ L @ U ) ) ) ).

% ivl_disj_un_singleton(3)
thf(fact_7378_ivl__disj__un__singleton_I3_J,axiom,
    ! [L: num,U: num] :
      ( ( ord_less_num @ L @ U )
     => ( ( sup_sup_set_num @ ( insert_num @ L @ bot_bot_set_num ) @ ( set_or2392100141987894638an_num @ L @ U ) )
        = ( set_or1222409239386451017an_num @ L @ U ) ) ) ).

% ivl_disj_un_singleton(3)
thf(fact_7379_ivl__disj__un__singleton_I3_J,axiom,
    ! [L: int,U: int] :
      ( ( ord_less_int @ L @ U )
     => ( ( sup_sup_set_int @ ( insert_int @ L @ bot_bot_set_int ) @ ( set_or5832277885323065728an_int @ L @ U ) )
        = ( set_or4662586982721622107an_int @ L @ U ) ) ) ).

% ivl_disj_un_singleton(3)
thf(fact_7380_ivl__disj__un__singleton_I3_J,axiom,
    ! [L: nat,U: nat] :
      ( ( ord_less_nat @ L @ U )
     => ( ( sup_sup_set_nat @ ( insert_nat @ L @ bot_bot_set_nat ) @ ( set_or5834768355832116004an_nat @ L @ U ) )
        = ( set_or4665077453230672383an_nat @ L @ U ) ) ) ).

% ivl_disj_un_singleton(3)
thf(fact_7381_ivl__disj__un__singleton_I3_J,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( ord_le6747313008572928689nteger @ L @ U )
     => ( ( sup_su848401254843788991nteger @ ( insert_Code_integer @ L @ bot_bo3990330152332043303nteger ) @ ( set_or4266950643985792945nteger @ L @ U ) )
        = ( set_or8404916559141939852nteger @ L @ U ) ) ) ).

% ivl_disj_un_singleton(3)
thf(fact_7382_ivl__disj__un__two_I4_J,axiom,
    ! [L: rat,M2: rat,U: rat] :
      ( ( ord_less_eq_rat @ L @ M2 )
     => ( ( ord_less_rat @ M2 @ U )
       => ( ( sup_sup_set_rat @ ( set_or633870826150836451st_rat @ L @ M2 ) @ ( set_or5199638295745620268an_rat @ M2 @ U ) )
          = ( set_or4029947393144176647an_rat @ L @ U ) ) ) ) ).

% ivl_disj_un_two(4)
thf(fact_7383_ivl__disj__un__two_I4_J,axiom,
    ! [L: num,M2: num,U: num] :
      ( ( ord_less_eq_num @ L @ M2 )
     => ( ( ord_less_num @ M2 @ U )
       => ( ( sup_sup_set_num @ ( set_or7049704709247886629st_num @ L @ M2 ) @ ( set_or2392100141987894638an_num @ M2 @ U ) )
          = ( set_or1222409239386451017an_num @ L @ U ) ) ) ) ).

% ivl_disj_un_two(4)
thf(fact_7384_ivl__disj__un__two_I4_J,axiom,
    ! [L: int,M2: int,U: int] :
      ( ( ord_less_eq_int @ L @ M2 )
     => ( ( ord_less_int @ M2 @ U )
       => ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ M2 ) @ ( set_or5832277885323065728an_int @ M2 @ U ) )
          = ( set_or4662586982721622107an_int @ L @ U ) ) ) ) ).

% ivl_disj_un_two(4)
thf(fact_7385_ivl__disj__un__two_I4_J,axiom,
    ! [L: nat,M2: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ M2 )
     => ( ( ord_less_nat @ M2 @ U )
       => ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M2 ) @ ( set_or5834768355832116004an_nat @ M2 @ U ) )
          = ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).

% ivl_disj_un_two(4)
thf(fact_7386_ivl__disj__un__two_I4_J,axiom,
    ! [L: code_integer,M2: code_integer,U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ L @ M2 )
     => ( ( ord_le6747313008572928689nteger @ M2 @ U )
       => ( ( sup_su848401254843788991nteger @ ( set_or189985376899183464nteger @ L @ M2 ) @ ( set_or4266950643985792945nteger @ M2 @ U ) )
          = ( set_or8404916559141939852nteger @ L @ U ) ) ) ) ).

% ivl_disj_un_two(4)
thf(fact_7387_less__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y3: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y3: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V2: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) ) ) ).

% less_int.rsp
thf(fact_7388_less__eq__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y3: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X3: nat,Y3: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) ) ) ).

% less_eq_int.rsp
thf(fact_7389_plus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X3: nat,Y3: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y3 @ V2 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X3: nat,Y3: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y3 @ V2 ) ) ) ) ) ).

% plus_int.rsp
thf(fact_7390_minus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X3: nat,Y3: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ Y3 @ U2 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X3: nat,Y3: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V2: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V2 ) @ ( plus_plus_nat @ Y3 @ U2 ) ) ) ) ) ).

% minus_int.rsp
thf(fact_7391_max__weak__def,axiom,
    ( fun_max_weak
    = ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% max_weak_def
thf(fact_7392_min__weak__def,axiom,
    ( fun_min_weak
    = ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% min_weak_def
thf(fact_7393_fold__union__pair,axiom,
    ! [B5: set_Pr4811707699266497531nteger,X: produc6241069584506657477e_term > option6357759511663192854e_term,A2: set_Pr1281608226676607948nteger] :
      ( ( finite2154694888287295812nteger @ B5 )
     => ( ( sup_su4591284015454442744nteger
          @ ( comple1324685164916153887nteger
            @ ( image_5715966689554962722nteger
              @ ^ [Y3: produc8923325533196201883nteger] : ( insert4245756632806152262nteger @ ( produc8603105652947943368nteger @ X @ Y3 ) @ bot_bo5443222936135328352nteger )
              @ B5 ) )
          @ A2 )
        = ( finite7641027033760462367nteger
          @ ^ [Y3: produc8923325533196201883nteger] : ( insert4245756632806152262nteger @ ( produc8603105652947943368nteger @ X @ Y3 ) )
          @ A2
          @ B5 ) ) ) ).

% fold_union_pair
thf(fact_7394_fold__union__pair,axiom,
    ! [B5: set_Pr3948176798113811640et_nat,X: produc3658429121746597890et_nat > $o,A2: set_Pr3286484037609594932et_nat] :
      ( ( finite2970523954037356441et_nat @ B5 )
     => ( ( sup_su7128418612487073120et_nat
          @ ( comple4849024849677941639et_nat
            @ ( image_15228131989405757et_nat
              @ ^ [Y3: produc3658429121746597890et_nat] : ( insert3111075074545270702et_nat @ ( produc5001842942810119800et_nat @ X @ Y3 ) @ bot_bo1481135142794719944et_nat )
              @ B5 ) )
          @ A2 )
        = ( finite2494574492299216512et_nat
          @ ^ [Y3: produc3658429121746597890et_nat] : ( insert3111075074545270702et_nat @ ( produc5001842942810119800et_nat @ X @ Y3 ) )
          @ A2
          @ B5 ) ) ) ).

% fold_union_pair
thf(fact_7395_fold__union__pair,axiom,
    ! [B5: set_Pr3286484037609594932et_nat,X: produc3658429121746597890et_nat > $o,A2: set_Pr8536935166611901872et_nat] :
      ( ( finite4151335170723375381et_nat @ B5 )
     => ( ( sup_su8975264963432250076et_nat
          @ ( comple5063570841203093763et_nat
            @ ( image_8189767104526903093et_nat
              @ ^ [Y3: produc3925858234332021118et_nat] : ( insert8297122186708525866et_nat @ ( produc2245416461498447860et_nat @ X @ Y3 ) @ bot_bo5635537948650799172et_nat )
              @ B5 ) )
          @ A2 )
        = ( finite8124994865635982712et_nat
          @ ^ [Y3: produc3925858234332021118et_nat] : ( insert8297122186708525866et_nat @ ( produc2245416461498447860et_nat @ X @ Y3 ) )
          @ A2
          @ B5 ) ) ) ).

% fold_union_pair
thf(fact_7396_fold__union__pair,axiom,
    ! [B5: set_Pr958786334691620121nt_int,X: produc8551481072490612790e_term > option6357759511663192854e_term,A2: set_Pr9222295170931077689nt_int] :
      ( ( finite2998713641127702882nt_int @ B5 )
     => ( ( sup_su3382966977382714213nt_int
          @ ( comple6871990601243273356nt_int
            @ ( image_815654828163180973nt_int
              @ ^ [Y3: product_prod_int_int] : ( insert984290209421517107nt_int @ ( produc5700946648718959541nt_int @ X @ Y3 ) @ bot_bo572930865798478029nt_int )
              @ B5 ) )
          @ A2 )
        = ( finite1318812335397358250nt_int
          @ ^ [Y3: product_prod_int_int] : ( insert984290209421517107nt_int @ ( produc5700946648718959541nt_int @ X @ Y3 ) )
          @ A2
          @ B5 ) ) ) ).

% fold_union_pair
thf(fact_7397_fold__union__pair,axiom,
    ! [B5: set_Pr958786334691620121nt_int,X: int > option6357759511663192854e_term,A2: set_Pr1872883991513573699nt_int] :
      ( ( finite2998713641127702882nt_int @ B5 )
     => ( ( sup_su3298353300217089135nt_int
          @ ( comple5772984051336441366nt_int
            @ ( image_5963362749323609783nt_int
              @ ^ [Y3: product_prod_int_int] : ( insert750232675644355261nt_int @ ( produc4305682042979456191nt_int @ X @ Y3 ) @ bot_bo4508923176915781079nt_int )
              @ B5 ) )
          @ A2 )
        = ( finite5333674244414288180nt_int
          @ ^ [Y3: product_prod_int_int] : ( insert750232675644355261nt_int @ ( produc4305682042979456191nt_int @ X @ Y3 ) )
          @ A2
          @ B5 ) ) ) ).

% fold_union_pair
thf(fact_7398_fold__union__pair,axiom,
    ! [B5: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( finite9047747110432174090at_nat @ B5 )
     => ( ( sup_su5525570899277871387at_nat
          @ ( comple4364388969068068532at_nat
            @ ( image_6898982319499645995at_nat
              @ ^ [Y3: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X @ Y3 ) @ bot_bo228742789529271731at_nat )
              @ B5 ) )
          @ A2 )
        = ( finite3179781904389220718at_nat
          @ ^ [Y3: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X @ Y3 ) )
          @ A2
          @ B5 ) ) ) ).

% fold_union_pair
thf(fact_7399_fold__union__pair,axiom,
    ! [B5: set_nat,X: nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( sup_su6327502436637775413at_nat
          @ ( comple5685304695842803022at_nat
            @ ( image_7178329752028323786at_nat
              @ ^ [Y3: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ bot_bo2099793752762293965at_nat )
              @ B5 ) )
          @ A2 )
        = ( finite3745491028973389255at_nat
          @ ^ [Y3: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) )
          @ A2
          @ B5 ) ) ) ).

% fold_union_pair
thf(fact_7400_Field__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( field_3145993188331922222at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ R2 ) )
      = ( sup_su3642409539654194069at_nat @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B @ bot_bo3083307316010499117at_nat ) ) @ ( field_3145993188331922222at_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_7401_Field__insert,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( field_2862844103579723188at_nat @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ A @ B ) @ R2 ) )
      = ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) @ ( field_2862844103579723188at_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_7402_Field__insert,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( field_4706345496283612622at_nat @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 ) )
      = ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) @ ( field_4706345496283612622at_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_7403_Field__insert,axiom,
    ! [A: $o,B: $o,R2: set_Product_prod_o_o] :
      ( ( field_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A @ B ) @ R2 ) )
      = ( sup_sup_set_o @ ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R2 ) ) ) ).

% Field_insert
thf(fact_7404_Field__insert,axiom,
    ! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 ) )
      = ( sup_sup_set_nat @ ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) @ ( field_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_7405_Field__insert,axiom,
    ! [A: int,B: int,R2: set_Pr958786334691620121nt_int] :
      ( ( field_int @ ( insert5033312907999012233nt_int @ ( product_Pair_int_int @ A @ B ) @ R2 ) )
      = ( sup_sup_set_int @ ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) ) @ ( field_int @ R2 ) ) ) ).

% Field_insert
thf(fact_7406_set__to__map__insert,axiom,
    ! [Kv: product_prod_nat_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member_nat @ ( product_fst_nat_nat @ Kv ) @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ S2 ) )
     => ( ( set_to_map_nat_nat @ ( insert8211810215607154385at_nat @ Kv @ S2 ) )
        = ( fun_up1493157387958331631on_nat @ ( set_to_map_nat_nat @ S2 ) @ ( product_fst_nat_nat @ Kv ) @ ( some_nat @ ( product_snd_nat_nat @ Kv ) ) ) ) ) ).

% set_to_map_insert
thf(fact_7407_set__to__map__insert,axiom,
    ! [Kv: product_prod_o_num,S2: set_Pr7040061329075590551_o_num] :
      ( ~ ( member_o @ ( product_fst_o_num @ Kv ) @ ( image_8863890952250787504_num_o @ product_fst_o_num @ S2 ) )
     => ( ( set_to_map_o_num @ ( insert8656458676507137553_o_num @ Kv @ S2 ) )
        = ( fun_upd_o_option_num @ ( set_to_map_o_num @ S2 ) @ ( product_fst_o_num @ Kv ) @ ( some_num @ ( product_snd_o_num @ Kv ) ) ) ) ) ).

% set_to_map_insert
thf(fact_7408_set__to__map__insert,axiom,
    ! [Kv: product_prod_nat_num,S2: set_Pr6200539531224447659at_num] :
      ( ~ ( member_nat @ ( product_fst_nat_num @ Kv ) @ ( image_4504471508743443926um_nat @ product_fst_nat_num @ S2 ) )
     => ( ( set_to_map_nat_num @ ( insert8920054152555992091at_num @ Kv @ S2 ) )
        = ( fun_up2201401324907169337on_num @ ( set_to_map_nat_num @ S2 ) @ ( product_fst_nat_num @ Kv ) @ ( some_num @ ( product_snd_nat_num @ Kv ) ) ) ) ) ).

% set_to_map_insert
thf(fact_7409_set__to__map__insert,axiom,
    ! [Kv: product_prod_int_num,S2: set_Pr8387461105917469447nt_num] :
      ( ~ ( member_int @ ( product_fst_int_num @ Kv ) @ ( image_4873634598471238286um_int @ product_fst_int_num @ S2 ) )
     => ( ( set_to_map_int_num @ ( insert696035827602270839nt_num @ Kv @ S2 ) )
        = ( fun_up4328768054909231765on_num @ ( set_to_map_int_num @ S2 ) @ ( product_fst_int_num @ Kv ) @ ( some_num @ ( product_snd_int_num @ Kv ) ) ) ) ) ).

% set_to_map_insert
thf(fact_7410_set__to__map__insert,axiom,
    ! [Kv: produc8923325533196201883nteger,S2: set_Pr4811707699266497531nteger] :
      ( ~ ( member_Code_integer @ ( produc8508995932063986495nteger @ Kv ) @ ( image_6316824473203919795nteger @ produc8508995932063986495nteger @ S2 ) )
     => ( ( set_to6264436334439079987nteger @ ( insert4913895101485356395nteger @ Kv @ S2 ) )
        = ( fun_up6566892301432185865nteger @ ( set_to6264436334439079987nteger @ S2 ) @ ( produc8508995932063986495nteger @ Kv ) @ ( some_Code_integer @ ( produc6174133586879617921nteger @ Kv ) ) ) ) ) ).

% set_to_map_insert
thf(fact_7411_set__to__map__insert,axiom,
    ! [Kv: product_prod_int_int,S2: set_Pr958786334691620121nt_int] :
      ( ~ ( member_int @ ( product_fst_int_int @ Kv ) @ ( image_5042161079198086560nt_int @ product_fst_int_int @ S2 ) )
     => ( ( set_to_map_int_int @ ( insert5033312907999012233nt_int @ Kv @ S2 ) )
        = ( fun_up8666045135305973159on_int @ ( set_to_map_int_int @ S2 ) @ ( product_fst_int_int @ Kv ) @ ( some_int @ ( product_snd_int_int @ Kv ) ) ) ) ) ).

% set_to_map_insert
thf(fact_7412_set__to__map__insert,axiom,
    ! [Kv: produc9082142974458947146at_num,S2: set_Pr7477759153910077568at_num] :
      ( ~ ( member8440522571783428010at_nat @ ( produc1179527155093393624at_num @ Kv ) @ ( image_7418169580930627696at_nat @ produc1179527155093393624at_num @ S2 ) )
     => ( ( set_to5081529003106763492at_num @ ( insert2790150385885543546at_num @ Kv @ S2 ) )
        = ( fun_up8025039454679724718on_num @ ( set_to5081529003106763492at_num @ S2 ) @ ( produc1179527155093393624at_num @ Kv ) @ ( some_num @ ( produc8847702614181477910at_num @ Kv ) ) ) ) ) ).

% set_to_map_insert
thf(fact_7413_set__to__map__insert,axiom,
    ! [Kv: produc5835291356934675326atural,S2: set_Pr2352657477744078772atural] :
      ( ~ ( member_Code_natural @ ( produc497848011034438852atural @ Kv ) @ ( image_8194678751871465133atural @ produc497848011034438852atural @ S2 ) )
     => ( ( set_to6039658477269176016atural @ ( insert6505218763252461998atural @ Kv @ S2 ) )
        = ( fun_up2923169431403285008atural @ ( set_to6039658477269176016atural @ S2 ) @ ( produc497848011034438852atural @ Kv ) @ ( some_P5671840033927537686atural @ ( produc6591912806276919810atural @ Kv ) ) ) ) ) ).

% set_to_map_insert
thf(fact_7414_set__to__map__insert,axiom,
    ! [Kv: produc2900562427881843919it_nat,S2: set_Pr6777250647868329861it_nat] :
      ( ~ ( member_o @ ( produc1440270732834495325it_nat @ Kv ) @ ( image_5849890305759347266_nat_o @ produc1440270732834495325it_nat @ S2 ) )
     => ( ( set_to3023635543527487849it_nat @ ( insert6915099157023601407it_nat @ Kv @ S2 ) )
        = ( fun_up1288979561775348147it_nat @ ( set_to3023635543527487849it_nat @ S2 ) @ ( produc1440270732834495325it_nat @ Kv ) @ ( some_P1914260805536162275it_nat @ ( produc5978051280710547099it_nat @ Kv ) ) ) ) ) ).

% set_to_map_insert
thf(fact_7415_set__to__map__insert,axiom,
    ! [Kv: produc3839731585411834681it_nat,S2: set_Pr633603713510679705it_nat] :
      ( ~ ( member_nat @ ( produc5854401122675660893it_nat @ Kv ) @ ( image_5261730836495542980at_nat @ produc5854401122675660893it_nat @ S2 ) )
     => ( ( set_to785951348762482001it_nat @ ( insert2528233410557839625it_nat @ Kv @ S2 ) )
        = ( fun_up4116053422210368295it_nat @ ( set_to785951348762482001it_nat @ S2 ) @ ( produc5854401122675660893it_nat @ Kv ) @ ( some_P1914260805536162275it_nat @ ( produc8417671938598930591it_nat @ Kv ) ) ) ) ) ).

% set_to_map_insert
thf(fact_7416_plus__rat_Orsp,axiom,
    ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
    @ ^ [X3: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) )
    @ ^ [X3: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ).

% plus_rat.rsp
thf(fact_7417_inverse__rat_Orsp,axiom,
    ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
    @ ^ [X3: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X3 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) )
    @ ^ [X3: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X3 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ).

% inverse_rat.rsp
thf(fact_7418_image__update,axiom,
    ! [X: nat,A2: set_nat,F: nat > code_integer,N2: code_integer] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( image_1215581382706833972nteger @ ( fun_up3904115110581602988nteger @ F @ X @ N2 ) @ A2 )
        = ( image_1215581382706833972nteger @ F @ A2 ) ) ) ).

% image_update
thf(fact_7419_image__update,axiom,
    ! [X: nat,A2: set_nat,F: nat > set_nat,N2: set_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( image_nat_set_nat @ ( fun_upd_nat_set_nat @ F @ X @ N2 ) @ A2 )
        = ( image_nat_set_nat @ F @ A2 ) ) ) ).

% image_update
thf(fact_7420_image__update,axiom,
    ! [X: nat,A2: set_nat,F: nat > nat,N2: nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( image_nat_nat @ ( fun_upd_nat_nat @ F @ X @ N2 ) @ A2 )
        = ( image_nat_nat @ F @ A2 ) ) ) ).

% image_update
thf(fact_7421_image__update,axiom,
    ! [X: nat,A2: set_nat,F: nat > int,N2: int] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( image_nat_int @ ( fun_upd_nat_int @ F @ X @ N2 ) @ A2 )
        = ( image_nat_int @ F @ A2 ) ) ) ).

% image_update
thf(fact_7422_image__update,axiom,
    ! [X: int,A2: set_int,F: int > int,N2: int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( image_int_int @ ( fun_upd_int_int @ F @ X @ N2 ) @ A2 )
        = ( image_int_int @ F @ A2 ) ) ) ).

% image_update
thf(fact_7423_finite__Field__eq__finite,axiom,
    ! [R4: set_Product_prod_o_o] :
      ( ( finite_finite_o @ ( field_o @ R4 ) )
      = ( finite6120865539452801872od_o_o @ R4 ) ) ).

% finite_Field_eq_finite
thf(fact_7424_finite__Field__eq__finite,axiom,
    ! [R4: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ ( field_nat @ R4 ) )
      = ( finite6177210948735845034at_nat @ R4 ) ) ).

% finite_Field_eq_finite
thf(fact_7425_finite__Field__eq__finite,axiom,
    ! [R4: set_Pr958786334691620121nt_int] :
      ( ( finite_finite_int @ ( field_int @ R4 ) )
      = ( finite2998713641127702882nt_int @ R4 ) ) ).

% finite_Field_eq_finite
thf(fact_7426_finite__Field__eq__finite,axiom,
    ! [R4: set_Pr4811707699266497531nteger] :
      ( ( finite6017078050557962740nteger @ ( field_Code_integer @ R4 ) )
      = ( finite2154694888287295812nteger @ R4 ) ) ).

% finite_Field_eq_finite
thf(fact_7427_finite__Field__eq__finite,axiom,
    ! [R4: set_Pr8693737435421807431at_nat] :
      ( ( finite6177210948735845034at_nat @ ( field_4706345496283612622at_nat @ R4 ) )
      = ( finite4392333629123659920at_nat @ R4 ) ) ).

% finite_Field_eq_finite
thf(fact_7428_image__map__upd,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,M2: produc3843707927480180839at_nat > option_num,Y: num] :
      ( ~ ( member8757157785044589968at_nat @ X @ A2 )
     => ( ( image_8464016542990802060on_num @ ( fun_up3286109336741039636on_num @ M2 @ X @ ( some_num @ Y ) ) @ A2 )
        = ( image_8464016542990802060on_num @ M2 @ A2 ) ) ) ).

% image_map_upd
thf(fact_7429_image__map__upd,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,M2: product_prod_nat_nat > option_num,Y: num] :
      ( ~ ( member8440522571783428010at_nat @ X @ A2 )
     => ( ( image_1023191801488348454on_num @ ( fun_up8025039454679724718on_num @ M2 @ X @ ( some_num @ Y ) ) @ A2 )
        = ( image_1023191801488348454on_num @ M2 @ A2 ) ) ) ).

% image_map_upd
thf(fact_7430_image__map__upd,axiom,
    ! [X: $o,A2: set_o,M2: $o > option_num,Y: num] :
      ( ~ ( member_o @ X @ A2 )
     => ( ( image_o_option_num @ ( fun_upd_o_option_num @ M2 @ X @ ( some_num @ Y ) ) @ A2 )
        = ( image_o_option_num @ M2 @ A2 ) ) ) ).

% image_map_upd
thf(fact_7431_image__map__upd,axiom,
    ! [X: nat,A2: set_nat,M2: nat > option_num,Y: num] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( image_nat_option_num @ ( fun_up2201401324907169337on_num @ M2 @ X @ ( some_num @ Y ) ) @ A2 )
        = ( image_nat_option_num @ M2 @ A2 ) ) ) ).

% image_map_upd
thf(fact_7432_image__map__upd,axiom,
    ! [X: int,A2: set_int,M2: int > option_num,Y: num] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( image_int_option_num @ ( fun_up4328768054909231765on_num @ M2 @ X @ ( some_num @ Y ) ) @ A2 )
        = ( image_int_option_num @ M2 @ A2 ) ) ) ).

% image_map_upd
thf(fact_7433_image__map__upd,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,M2: produc3843707927480180839at_nat > option8956607266484857688it_nat,Y: produc8664842809031399944it_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ A2 )
     => ( ( image_140351626946063738it_nat @ ( fun_up5549328659429834498it_nat @ M2 @ X @ ( some_P1914260805536162275it_nat @ Y ) ) @ A2 )
        = ( image_140351626946063738it_nat @ M2 @ A2 ) ) ) ).

% image_map_upd
thf(fact_7434_image__map__upd,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,M2: product_prod_nat_nat > option8956607266484857688it_nat,Y: produc8664842809031399944it_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ A2 )
     => ( ( image_5755308894823448084it_nat @ ( fun_up3996838528606387100it_nat @ M2 @ X @ ( some_P1914260805536162275it_nat @ Y ) ) @ A2 )
        = ( image_5755308894823448084it_nat @ M2 @ A2 ) ) ) ).

% image_map_upd
thf(fact_7435_image__map__upd,axiom,
    ! [X: $o,A2: set_o,M2: $o > option8956607266484857688it_nat,Y: produc8664842809031399944it_nat] :
      ( ~ ( member_o @ X @ A2 )
     => ( ( image_5849929686412825643it_nat @ ( fun_up1288979561775348147it_nat @ M2 @ X @ ( some_P1914260805536162275it_nat @ Y ) ) @ A2 )
        = ( image_5849929686412825643it_nat @ M2 @ A2 ) ) ) ).

% image_map_upd
thf(fact_7436_image__map__upd,axiom,
    ! [X: nat,A2: set_nat,M2: nat > option8956607266484857688it_nat,Y: produc8664842809031399944it_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( image_5028743614110135983it_nat @ ( fun_up4116053422210368295it_nat @ M2 @ X @ ( some_P1914260805536162275it_nat @ Y ) ) @ A2 )
        = ( image_5028743614110135983it_nat @ M2 @ A2 ) ) ) ).

% image_map_upd
thf(fact_7437_image__map__upd,axiom,
    ! [X: int,A2: set_int,M2: int > option8956607266484857688it_nat,Y: produc8664842809031399944it_nat] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( image_7905038228067148555it_nat @ ( fun_up6992348036167380867it_nat @ M2 @ X @ ( some_P1914260805536162275it_nat @ Y ) ) @ A2 )
        = ( image_7905038228067148555it_nat @ M2 @ A2 ) ) ) ).

% image_map_upd
thf(fact_7438_FieldI2,axiom,
    ! [I: produc3843707927480180839at_nat,J: produc3843707927480180839at_nat,R4: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J ) @ R4 )
     => ( member8757157785044589968at_nat @ J @ ( field_2862844103579723188at_nat @ R4 ) ) ) ).

% FieldI2
thf(fact_7439_FieldI2,axiom,
    ! [I: product_prod_nat_nat,J: product_prod_nat_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I @ J ) @ R4 )
     => ( member8440522571783428010at_nat @ J @ ( field_4706345496283612622at_nat @ R4 ) ) ) ).

% FieldI2
thf(fact_7440_FieldI2,axiom,
    ! [I: int,J: int,R4: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ R4 )
     => ( member_int @ J @ ( field_int @ R4 ) ) ) ).

% FieldI2
thf(fact_7441_FieldI2,axiom,
    ! [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat,R4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R4 )
     => ( member2643936169264416010at_nat @ J @ ( field_3145993188331922222at_nat @ R4 ) ) ) ).

% FieldI2
thf(fact_7442_FieldI2,axiom,
    ! [I: nat,J: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R4 )
     => ( member_nat @ J @ ( field_nat @ R4 ) ) ) ).

% FieldI2
thf(fact_7443_FieldI2,axiom,
    ! [I: $o,J: $o,R4: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I @ J ) @ R4 )
     => ( member_o @ J @ ( field_o @ R4 ) ) ) ).

% FieldI2
thf(fact_7444_FieldI1,axiom,
    ! [I: produc3843707927480180839at_nat,J: produc3843707927480180839at_nat,R4: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I @ J ) @ R4 )
     => ( member8757157785044589968at_nat @ I @ ( field_2862844103579723188at_nat @ R4 ) ) ) ).

% FieldI1
thf(fact_7445_FieldI1,axiom,
    ! [I: product_prod_nat_nat,J: product_prod_nat_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I @ J ) @ R4 )
     => ( member8440522571783428010at_nat @ I @ ( field_4706345496283612622at_nat @ R4 ) ) ) ).

% FieldI1
thf(fact_7446_FieldI1,axiom,
    ! [I: int,J: int,R4: set_Pr958786334691620121nt_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ R4 )
     => ( member_int @ I @ ( field_int @ R4 ) ) ) ).

% FieldI1
thf(fact_7447_FieldI1,axiom,
    ! [I: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat,R4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I @ J ) @ R4 )
     => ( member2643936169264416010at_nat @ I @ ( field_3145993188331922222at_nat @ R4 ) ) ) ).

% FieldI1
thf(fact_7448_FieldI1,axiom,
    ! [I: nat,J: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R4 )
     => ( member_nat @ I @ ( field_nat @ R4 ) ) ) ).

% FieldI1
thf(fact_7449_FieldI1,axiom,
    ! [I: $o,J: $o,R4: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I @ J ) @ R4 )
     => ( member_o @ I @ ( field_o @ R4 ) ) ) ).

% FieldI1
thf(fact_7450_sup__Un__eq2,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( sup_sup_nat_nat_o
        @ ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R4 )
        @ ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S2 ) )
      = ( ^ [X3: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( sup_su6327502436637775413at_nat @ R4 @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_7451_sup__Un__eq2,axiom,
    ! [R4: set_Pr1281608226676607948nteger,S2: set_Pr1281608226676607948nteger] :
      ( ( sup_su234547053653886311eger_o
        @ ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ R4 )
        @ ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ S2 ) )
      = ( ^ [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ ( sup_su4591284015454442744nteger @ R4 @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_7452_sup__Un__eq2,axiom,
    ! [R4: set_Pr3286484037609594932et_nat,S2: set_Pr3286484037609594932et_nat] :
      ( ( sup_su1630790145277462993_nat_o
        @ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ R4 )
        @ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ S2 ) )
      = ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3658429121746597890et_nat] : ( member1996754912294343701et_nat @ ( produc5001842942810119800et_nat @ X3 @ Y3 ) @ ( sup_su7128418612487073120et_nat @ R4 @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_7453_sup__Un__eq2,axiom,
    ! [R4: set_Pr8536935166611901872et_nat,S2: set_Pr8536935166611901872et_nat] :
      ( ( sup_su6535292691877529429_nat_o
        @ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ R4 )
        @ ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ S2 ) )
      = ( ^ [X3: produc3658429121746597890et_nat > $o,Y3: produc3925858234332021118et_nat] : ( member6124377750444531601et_nat @ ( produc2245416461498447860et_nat @ X3 @ Y3 ) @ ( sup_su8975264963432250076et_nat @ R4 @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_7454_sup__Un__eq2,axiom,
    ! [R4: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
      ( ( sup_su4182031696650224058_int_o
        @ ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ R4 )
        @ ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ S2 ) )
      = ( ^ [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ ( sup_su3382966977382714213nt_int @ R4 @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_7455_sup__Un__eq2,axiom,
    ! [R4: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
      ( ( sup_su600977994968626096_int_o
        @ ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ R4 )
        @ ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ S2 ) )
      = ( ^ [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ ( sup_su3298353300217089135nt_int @ R4 @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_7456_sup__Un__eq2,axiom,
    ! [R4: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su7519161239522478338_nat_o
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ R4 )
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ S2 ) )
      = ( ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y3 ) @ ( sup_su5525570899277871387at_nat @ R4 @ S2 ) ) ) ) ).

% sup_Un_eq2
thf(fact_7457_fun__upd__comp,axiom,
    ! [F: code_integer > code_integer,G: num > code_integer,X: num,Y: code_integer] :
      ( ( comp_C3531382070062128313er_num @ F @ ( fun_up1844922938552261878nteger @ G @ X @ Y ) )
      = ( fun_up1844922938552261878nteger @ ( comp_C3531382070062128313er_num @ F @ G ) @ X @ ( F @ Y ) ) ) ).

% fun_upd_comp
thf(fact_7458_fun__upd__comp,axiom,
    ! [F: code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer,X: code_integer,Y: code_integer] :
      ( ( comp_C1593894019821074884nteger @ F @ ( fun_up8355007323143964473nteger @ G @ X @ Y ) )
      = ( fun_up8139772397987072257nteger @ ( comp_C1593894019821074884nteger @ F @ G ) @ X @ ( F @ Y ) ) ) ).

% fun_upd_comp
thf(fact_7459_fun__upd__comp,axiom,
    ! [F: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger,G: code_integer > code_integer > code_integer,X: code_integer,Y: code_integer > code_integer] :
      ( ( comp_C8797469213163452608nteger @ F @ ( fun_up8056716234270145717nteger @ G @ X @ Y ) )
      = ( fun_up8139772397987072257nteger @ ( comp_C8797469213163452608nteger @ F @ G ) @ X @ ( F @ Y ) ) ) ).

% fun_upd_comp
thf(fact_7460_fun__upd__comp,axiom,
    ! [F: nat > num,G: int > nat,X: int,Y: nat] :
      ( ( comp_nat_num_int @ F @ ( fun_upd_int_nat @ G @ X @ Y ) )
      = ( fun_upd_int_num @ ( comp_nat_num_int @ F @ G ) @ X @ ( F @ Y ) ) ) ).

% fun_upd_comp
thf(fact_7461_one__rat_Orsp,axiom,
    ratrel @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ).

% one_rat.rsp
thf(fact_7462_R__subset__Field,axiom,
    ! [R4: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat @ R4
      @ ( produc457027306803732586at_nat @ ( field_nat @ R4 )
        @ ^ [Uu: nat] : ( field_nat @ R4 ) ) ) ).

% R_subset_Field
thf(fact_7463_R__subset__Field,axiom,
    ! [R4: set_Product_prod_o_o] :
      ( ord_le192195778054999463od_o_o @ R4
      @ ( product_Sigma_o_o @ ( field_o @ R4 )
        @ ^ [Uu: $o] : ( field_o @ R4 ) ) ) ).

% R_subset_Field
thf(fact_7464_fst__in__Field,axiom,
    ! [R4: set_Product_prod_o_o] : ( ord_less_eq_set_o @ ( image_7896445794123959606_o_o_o @ product_fst_o_o @ R4 ) @ ( field_o @ R4 ) ) ).

% fst_in_Field
thf(fact_7465_fst__in__Field,axiom,
    ! [R4: set_Pr4811707699266497531nteger] : ( ord_le7084787975880047091nteger @ ( image_6316824473203919795nteger @ produc8508995932063986495nteger @ R4 ) @ ( field_Code_integer @ R4 ) ) ).

% fst_in_Field
thf(fact_7466_fst__in__Field,axiom,
    ! [R4: set_Pr958786334691620121nt_int] : ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ product_fst_int_int @ R4 ) @ ( field_int @ R4 ) ) ).

% fst_in_Field
thf(fact_7467_fst__in__Field,axiom,
    ! [R4: set_Pr1261947904930325089at_nat] : ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ product_fst_nat_nat @ R4 ) @ ( field_nat @ R4 ) ) ).

% fst_in_Field
thf(fact_7468_snd__in__Field,axiom,
    ! [R4: set_Product_prod_o_o] : ( ord_less_eq_set_o @ ( image_7896445794123959606_o_o_o @ product_snd_o_o @ R4 ) @ ( field_o @ R4 ) ) ).

% snd_in_Field
thf(fact_7469_snd__in__Field,axiom,
    ! [R4: set_Pr4811707699266497531nteger] : ( ord_le7084787975880047091nteger @ ( image_6316824473203919795nteger @ produc6174133586879617921nteger @ R4 ) @ ( field_Code_integer @ R4 ) ) ).

% snd_in_Field
thf(fact_7470_snd__in__Field,axiom,
    ! [R4: set_Pr958786334691620121nt_int] : ( ord_less_eq_set_int @ ( image_5042161079198086560nt_int @ product_snd_int_int @ R4 ) @ ( field_int @ R4 ) ) ).

% snd_in_Field
thf(fact_7471_snd__in__Field,axiom,
    ! [R4: set_Pr1261947904930325089at_nat] : ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ product_snd_nat_nat @ R4 ) @ ( field_nat @ R4 ) ) ).

% snd_in_Field
thf(fact_7472_zero__rat_Orsp,axiom,
    ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).

% zero_rat.rsp
thf(fact_7473_Fract_Orsp,axiom,
    ( bNF_re157797125943740599nt_int
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ( bNF_re6250860962936578807nt_int
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
      @ ratrel )
    @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B4 ) )
    @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B4 ) ) ) ).

% Fract.rsp
thf(fact_7474_sum_Oeq__fold,axiom,
    ( groups3542108847815614940at_nat
    = ( ^ [G2: nat > nat] : ( finite_fold_nat_nat @ ( comp_nat_nat_nat_nat @ plus_plus_nat @ G2 ) @ zero_zero_nat ) ) ) ).

% sum.eq_fold
thf(fact_7475_sum_Oeq__fold,axiom,
    ( groups4538972089207619220nt_int
    = ( ^ [G2: int > int] : ( finite_fold_int_int @ ( comp_int_int_int_int @ plus_plus_int @ G2 ) @ zero_zero_int ) ) ) ).

% sum.eq_fold
thf(fact_7476_prod_Oeq__fold,axiom,
    ( groups708209901874060359at_nat
    = ( ^ [G2: nat > nat] : ( finite_fold_nat_nat @ ( comp_nat_nat_nat_nat @ times_times_nat @ G2 ) @ one_one_nat ) ) ) ).

% prod.eq_fold
thf(fact_7477_prod_Oeq__fold,axiom,
    ( groups705719431365010083at_int
    = ( ^ [G2: nat > int] : ( finite_fold_nat_int @ ( comp_int_int_int_nat @ times_times_int @ G2 ) @ one_one_int ) ) ) ).

% prod.eq_fold
thf(fact_7478_prod_Oeq__fold,axiom,
    ( groups1705073143266064639nt_int
    = ( ^ [G2: int > int] : ( finite_fold_int_int @ ( comp_int_int_int_int @ times_times_int @ G2 ) @ one_one_int ) ) ) ).

% prod.eq_fold
thf(fact_7479_finite__range__updI,axiom,
    ! [F: nat > option_num,A: nat,B: num] :
      ( ( finite6231397076622260609on_num @ ( image_nat_option_num @ F @ top_top_set_nat ) )
     => ( finite6231397076622260609on_num @ ( image_nat_option_num @ ( fun_up2201401324907169337on_num @ F @ A @ ( some_num @ B ) ) @ top_top_set_nat ) ) ) ).

% finite_range_updI
thf(fact_7480_finite__range__updI,axiom,
    ! [F: nat > option8956607266484857688it_nat,A: nat,B: produc8664842809031399944it_nat] :
      ( ( finite6292238467249657455it_nat @ ( image_5028743614110135983it_nat @ F @ top_top_set_nat ) )
     => ( finite6292238467249657455it_nat @ ( image_5028743614110135983it_nat @ ( fun_up4116053422210368295it_nat @ F @ A @ ( some_P1914260805536162275it_nat @ B ) ) @ top_top_set_nat ) ) ) ).

% finite_range_updI
thf(fact_7481_finite__range__updI,axiom,
    ! [F: product_unit > option_num,A: product_unit,B: num] :
      ( ( finite6231397076622260609on_num @ ( image_5965367930834091826on_num @ F @ top_to1996260823553986621t_unit ) )
     => ( finite6231397076622260609on_num @ ( image_5965367930834091826on_num @ ( fun_up4695802795598371002on_num @ F @ A @ ( some_num @ B ) ) @ top_to1996260823553986621t_unit ) ) ) ).

% finite_range_updI
thf(fact_7482_finite__range__updI,axiom,
    ! [F: product_unit > option8956607266484857688it_nat,A: product_unit,B: produc8664842809031399944it_nat] :
      ( ( finite6292238467249657455it_nat @ ( image_5022536681399171104it_nat @ F @ top_to1996260823553986621t_unit ) )
     => ( finite6292238467249657455it_nat @ ( image_5022536681399171104it_nat @ ( fun_up1973317344121680296it_nat @ F @ A @ ( some_P1914260805536162275it_nat @ B ) ) @ top_to1996260823553986621t_unit ) ) ) ).

% finite_range_updI
thf(fact_7483_finite__range__updI,axiom,
    ! [F: int > option_num,A: int,B: num] :
      ( ( finite6231397076622260609on_num @ ( image_int_option_num @ F @ top_top_set_int ) )
     => ( finite6231397076622260609on_num @ ( image_int_option_num @ ( fun_up4328768054909231765on_num @ F @ A @ ( some_num @ B ) ) @ top_top_set_int ) ) ) ).

% finite_range_updI
thf(fact_7484_finite__range__updI,axiom,
    ! [F: int > option8956607266484857688it_nat,A: int,B: produc8664842809031399944it_nat] :
      ( ( finite6292238467249657455it_nat @ ( image_7905038228067148555it_nat @ F @ top_top_set_int ) )
     => ( finite6292238467249657455it_nat @ ( image_7905038228067148555it_nat @ ( fun_up6992348036167380867it_nat @ F @ A @ ( some_P1914260805536162275it_nat @ B ) ) @ top_top_set_int ) ) ) ).

% finite_range_updI
thf(fact_7485_fun__upd__image,axiom,
    ! [X: $o,A2: set_o,F: $o > $o,Y: $o] :
      ( ( ( member_o @ X @ A2 )
       => ( ( image_o_o @ ( fun_upd_o_o @ F @ X @ Y ) @ A2 )
          = ( insert_o @ Y @ ( image_o_o @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
      & ( ~ ( member_o @ X @ A2 )
       => ( ( image_o_o @ ( fun_upd_o_o @ F @ X @ Y ) @ A2 )
          = ( image_o_o @ F @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_7486_fun__upd__image,axiom,
    ! [X: $o,A2: set_o,F: $o > nat,Y: nat] :
      ( ( ( member_o @ X @ A2 )
       => ( ( image_o_nat @ ( fun_upd_o_nat @ F @ X @ Y ) @ A2 )
          = ( insert_nat @ Y @ ( image_o_nat @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
      & ( ~ ( member_o @ X @ A2 )
       => ( ( image_o_nat @ ( fun_upd_o_nat @ F @ X @ Y ) @ A2 )
          = ( image_o_nat @ F @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_7487_fun__upd__image,axiom,
    ! [X: $o,A2: set_o,F: $o > int,Y: int] :
      ( ( ( member_o @ X @ A2 )
       => ( ( image_o_int @ ( fun_upd_o_int @ F @ X @ Y ) @ A2 )
          = ( insert_int @ Y @ ( image_o_int @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
      & ( ~ ( member_o @ X @ A2 )
       => ( ( image_o_int @ ( fun_upd_o_int @ F @ X @ Y ) @ A2 )
          = ( image_o_int @ F @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_7488_fun__upd__image,axiom,
    ! [X: int,A2: set_int,F: int > $o,Y: $o] :
      ( ( ( member_int @ X @ A2 )
       => ( ( image_int_o @ ( fun_upd_int_o @ F @ X @ Y ) @ A2 )
          = ( insert_o @ Y @ ( image_int_o @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) )
      & ( ~ ( member_int @ X @ A2 )
       => ( ( image_int_o @ ( fun_upd_int_o @ F @ X @ Y ) @ A2 )
          = ( image_int_o @ F @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_7489_fun__upd__image,axiom,
    ! [X: int,A2: set_int,F: int > nat,Y: nat] :
      ( ( ( member_int @ X @ A2 )
       => ( ( image_int_nat @ ( fun_upd_int_nat @ F @ X @ Y ) @ A2 )
          = ( insert_nat @ Y @ ( image_int_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) )
      & ( ~ ( member_int @ X @ A2 )
       => ( ( image_int_nat @ ( fun_upd_int_nat @ F @ X @ Y ) @ A2 )
          = ( image_int_nat @ F @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_7490_fun__upd__image,axiom,
    ! [X: int,A2: set_int,F: int > int,Y: int] :
      ( ( ( member_int @ X @ A2 )
       => ( ( image_int_int @ ( fun_upd_int_int @ F @ X @ Y ) @ A2 )
          = ( insert_int @ Y @ ( image_int_int @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) )
      & ( ~ ( member_int @ X @ A2 )
       => ( ( image_int_int @ ( fun_upd_int_int @ F @ X @ Y ) @ A2 )
          = ( image_int_int @ F @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_7491_fun__upd__image,axiom,
    ! [X: nat,A2: set_nat,F: nat > code_integer,Y: code_integer] :
      ( ( ( member_nat @ X @ A2 )
       => ( ( image_1215581382706833972nteger @ ( fun_up3904115110581602988nteger @ F @ X @ Y ) @ A2 )
          = ( insert_Code_integer @ Y @ ( image_1215581382706833972nteger @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) )
      & ( ~ ( member_nat @ X @ A2 )
       => ( ( image_1215581382706833972nteger @ ( fun_up3904115110581602988nteger @ F @ X @ Y ) @ A2 )
          = ( image_1215581382706833972nteger @ F @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_7492_fun__upd__image,axiom,
    ! [X: nat,A2: set_nat,F: nat > $o,Y: $o] :
      ( ( ( member_nat @ X @ A2 )
       => ( ( image_nat_o @ ( fun_upd_nat_o @ F @ X @ Y ) @ A2 )
          = ( insert_o @ Y @ ( image_nat_o @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) )
      & ( ~ ( member_nat @ X @ A2 )
       => ( ( image_nat_o @ ( fun_upd_nat_o @ F @ X @ Y ) @ A2 )
          = ( image_nat_o @ F @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_7493_fun__upd__image,axiom,
    ! [X: nat,A2: set_nat,F: nat > nat,Y: nat] :
      ( ( ( member_nat @ X @ A2 )
       => ( ( image_nat_nat @ ( fun_upd_nat_nat @ F @ X @ Y ) @ A2 )
          = ( insert_nat @ Y @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) )
      & ( ~ ( member_nat @ X @ A2 )
       => ( ( image_nat_nat @ ( fun_upd_nat_nat @ F @ X @ Y ) @ A2 )
          = ( image_nat_nat @ F @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_7494_fun__upd__image,axiom,
    ! [X: nat,A2: set_nat,F: nat > int,Y: int] :
      ( ( ( member_nat @ X @ A2 )
       => ( ( image_nat_int @ ( fun_upd_nat_int @ F @ X @ Y ) @ A2 )
          = ( insert_int @ Y @ ( image_nat_int @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) )
      & ( ~ ( member_nat @ X @ A2 )
       => ( ( image_nat_int @ ( fun_upd_nat_int @ F @ X @ Y ) @ A2 )
          = ( image_nat_int @ F @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_7495_times__rat_Orsp,axiom,
    ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
    @ ^ [X3: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) )
    @ ^ [X3: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ).

% times_rat.rsp
thf(fact_7496_uminus__rat_Orsp,axiom,
    ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
    @ ^ [X3: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X3 ) ) @ ( product_snd_int_int @ X3 ) )
    @ ^ [X3: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X3 ) ) @ ( product_snd_int_int @ X3 ) ) ) ).

% uminus_rat.rsp
thf(fact_7497_plus__rat_Oabs__eq,axiom,
    ! [Xa: product_prod_int_int,X: product_prod_int_int] :
      ( ( ratrel @ Xa @ Xa )
     => ( ( ratrel @ X @ X )
       => ( ( plus_plus_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
          = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).

% plus_rat.abs_eq
thf(fact_7498_inverse__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( ( ratrel @ X @ X )
     => ( ( inverse_inverse_rat @ ( abs_Rat @ X ) )
        = ( abs_Rat
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_fst_int_int @ X )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ) ).

% inverse_rat.abs_eq
thf(fact_7499_times__rat_Oabs__eq,axiom,
    ! [Xa: product_prod_int_int,X: product_prod_int_int] :
      ( ( ratrel @ Xa @ Xa )
     => ( ( ratrel @ X @ X )
       => ( ( times_times_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
          = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa ) @ ( product_fst_int_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).

% times_rat.abs_eq
thf(fact_7500_uminus__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( ( ratrel @ X @ X )
     => ( ( uminus_uminus_rat @ ( abs_Rat @ X ) )
        = ( abs_Rat @ ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) ) ) ) ) ).

% uminus_rat.abs_eq
thf(fact_7501_one__rat__def,axiom,
    ( one_one_rat
    = ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).

% one_rat_def
thf(fact_7502_zero__rat__def,axiom,
    ( zero_zero_rat
    = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).

% zero_rat_def
thf(fact_7503_plus__rat__def,axiom,
    ( plus_plus_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X3: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ) ).

% plus_rat_def
thf(fact_7504_inverse__rat__def,axiom,
    ( inverse_inverse_rat
    = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
      @ ^ [X3: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int
          @ ( ( product_fst_int_int @ X3 )
            = zero_zero_int )
          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
          @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ) ).

% inverse_rat_def
thf(fact_7505_times__rat__def,axiom,
    ( times_times_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X3: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X3 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ) ).

% times_rat_def
thf(fact_7506_uminus__rat__def,axiom,
    ( uminus_uminus_rat
    = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
      @ ^ [X3: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X3 ) ) @ ( product_snd_int_int @ X3 ) ) ) ) ).

% uminus_rat_def
thf(fact_7507_card__le__Suc__Max,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).

% card_le_Suc_Max
thf(fact_7508_greaterThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
      = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).

% greaterThan_Suc
thf(fact_7509_INT__greaterThan__UNIV,axiom,
    ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
    = bot_bot_set_nat ) ).

% INT_greaterThan_UNIV
thf(fact_7510_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_7511_euclidean__size__integer_Orep__eq,axiom,
    ( euclid6377331345833325938nteger
    = ( ^ [X3: code_integer] : ( euclid4774559944035922753ze_int @ ( code_int_of_integer @ X3 ) ) ) ) ).

% euclidean_size_integer.rep_eq
thf(fact_7512_euclidean__size__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( euclid6377331345833325938nteger @ ( code_integer_of_int @ X ) )
      = ( euclid4774559944035922753ze_int @ X ) ) ).

% euclidean_size_integer.abs_eq
thf(fact_7513_euclidean__size__natural_Orsp,axiom,
    ( bNF_re5653821019739307937at_nat
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ euclid4777050414544973029ze_nat
    @ euclid4777050414544973029ze_nat ) ).

% euclidean_size_natural.rsp
thf(fact_7514_euclidean__size__integer__def,axiom,
    ( euclid6377331345833325938nteger
    = ( map_fu6539832666145259331at_nat @ code_int_of_integer @ id_nat @ euclid4774559944035922753ze_int ) ) ).

% euclidean_size_integer_def
thf(fact_7515_euclidean__size__integer_Orsp,axiom,
    ( bNF_re3715656647883201625at_nat
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ euclid4774559944035922753ze_int
    @ euclid4774559944035922753ze_int ) ).

% euclidean_size_integer.rsp
thf(fact_7516_division__segment__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( euclid6289375185220004616nteger @ X ) )
      = ( euclid3395696857347342551nt_int @ ( code_int_of_integer @ X ) ) ) ).

% division_segment_integer.rep_eq
thf(fact_7517_division__segment__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( euclid6289375185220004616nteger @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( euclid3395696857347342551nt_int @ X ) ) ) ).

% division_segment_integer.abs_eq
thf(fact_7518_division__segment__nat__def,axiom,
    ( euclid3398187327856392827nt_nat
    = ( ^ [N4: nat] : one_one_nat ) ) ).

% division_segment_nat_def
thf(fact_7519_division__segment__integer_Orsp,axiom,
    ( bNF_re4712519889275205905nt_int
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ euclid3395696857347342551nt_int
    @ euclid3395696857347342551nt_int ) ).

% division_segment_integer.rsp
thf(fact_7520_division__segment__natural_Orsp,axiom,
    ( bNF_re5653821019739307937at_nat
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ euclid3398187327856392827nt_nat
    @ euclid3398187327856392827nt_nat ) ).

% division_segment_natural.rsp
thf(fact_7521_division__segment__integer__def,axiom,
    ( euclid6289375185220004616nteger
    = ( map_fu2599414010547811884nteger @ code_int_of_integer @ code_integer_of_int @ euclid3395696857347342551nt_int ) ) ).

% division_segment_integer_def
thf(fact_7522_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% finite_greaterThanAtMost
thf(fact_7523_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L @ U ) ) ).

% finite_greaterThanAtMost_int
thf(fact_7524_finite__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or2715278749043346189nteger @ L @ U ) ) ).

% finite_greaterThanAtMost_integer
thf(fact_7525_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_greaterThanAtMost
thf(fact_7526_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).

% card_greaterThanAtMost_int
thf(fact_7527_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
      = ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_7528_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
      = ( set_or6656581121297822940st_int @ L @ U ) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_7529_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] :
      ( ( set_or189985376899183464nteger @ ( plus_p5714425477246183910nteger @ L @ one_one_Code_integer ) @ U )
      = ( set_or2715278749043346189nteger @ L @ U ) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
thf(fact_7530_UN__atLeast__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_atLeast_UNIV
thf(fact_7531_atLeast__0,axiom,
    ( ( set_ord_atLeast_nat @ zero_zero_nat )
    = top_top_set_nat ) ).

% atLeast_0
thf(fact_7532_Inf__unit__def,axiom,
    ( comple2584293577114468500t_unit
    = ( ^ [Uu3: set_Product_unit] : product_Unity ) ) ).

% Inf_unit_def
thf(fact_7533_Sup__unit__def,axiom,
    ( comple4687483117567863418t_unit
    = ( ^ [Uu3: set_Product_unit] : product_Unity ) ) ).

% Sup_unit_def
thf(fact_7534_old_Ounit_Oexhaust,axiom,
    ! [Y: product_unit] : ( Y = product_Unity ) ).

% old.unit.exhaust
thf(fact_7535_top__unit__def,axiom,
    top_top_Product_unit = product_Unity ).

% top_unit_def
thf(fact_7536_uminus__unit__def,axiom,
    ( uminus2952777764628376836t_unit
    = ( ^ [Uu3: product_unit] : product_Unity ) ) ).

% uminus_unit_def
thf(fact_7537_bot__unit__def,axiom,
    bot_bot_Product_unit = product_Unity ).

% bot_unit_def
thf(fact_7538_sup__unit__def,axiom,
    ( sup_sup_Product_unit
    = ( ^ [Uu3: product_unit,Uv: product_unit] : product_Unity ) ) ).

% sup_unit_def
thf(fact_7539_UNIV__unit,axiom,
    ( top_to1996260823553986621t_unit
    = ( insert_Product_unit @ product_Unity @ bot_bo3957492148770167129t_unit ) ) ).

% UNIV_unit
thf(fact_7540_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( set_or1210151606488870762an_nat @ K ) ) ).

% atLeast_Suc_greaterThan
thf(fact_7541_wait__def,axiom,
    ( heap_Time_wait
    = ( ^ [N4: nat] :
          ( heap_T6183433275982383450t_unit
          @ ^ [H: heap_e7401611519738050253t_unit] : ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ product_Unity @ ( produc584006145561248582it_nat @ H @ N4 ) ) ) ) ) ) ).

% wait_def
thf(fact_7542_execute__assert_H,axiom,
    ! [P: $o,H3: heap_e7401611519738050253t_unit] :
      ( ( P
       => ( ( heap_T875086893843062177t_unit @ ( refine_Imp_assert @ P ) @ H3 )
          = ( some_P1914260805536162275it_nat @ ( produc7133225469290080770it_nat @ product_Unity @ ( produc584006145561248582it_nat @ H3 @ zero_zero_nat ) ) ) ) )
      & ( ~ P
       => ( ( heap_T875086893843062177t_unit @ ( refine_Imp_assert @ P ) @ H3 )
          = none_P9117596204409417319it_nat ) ) ) ).

% execute_assert'
thf(fact_7543_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 ) ) ) ).

% atLeast_Suc
thf(fact_7544_default__unit__def,axiom,
    defaul566961228789861419t_unit = product_Unity ).

% default_unit_def
thf(fact_7545_inf__unit__def,axiom,
    ( inf_inf_Product_unit
    = ( ^ [Uu3: product_unit,Uv: product_unit] : product_Unity ) ) ).

% inf_unit_def
thf(fact_7546_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X3: rat] :
          ( the_int
          @ ^ [Z4: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z4 ) @ X3 )
              & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_7547_rat__floor__lemma,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
      & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).

% rat_floor_lemma
thf(fact_7548_num__of__integer_Otransfer,axiom,
    ( bNF_re6718328864250387230um_num @ code_pcr_integer
    @ ^ [Y4: num,Z2: num] : ( Y4 = Z2 )
    @ ( comp_nat_num_int @ num_of_nat @ nat2 )
    @ code_num_of_integer ) ).

% num_of_integer.transfer
thf(fact_7549_prod__decode__triangle__add,axiom,
    ! [K: nat,M2: nat] :
      ( ( nat_prod_decode @ ( plus_plus_nat @ ( nat_triangle @ K ) @ M2 ) )
      = ( nat_prod_decode_aux @ K @ M2 ) ) ).

% prod_decode_triangle_add
thf(fact_7550_add__rat,axiom,
    ! [B: int,D2: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D2 != zero_zero_int )
       => ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D2 ) )
          = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D2 ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ).

% add_rat
thf(fact_7551_integer_Orep__transfer,axiom,
    ( bNF_re3804157879324367682nt_int @ code_pcr_integer
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ^ [X3: int] : X3
    @ code_int_of_integer ) ).

% integer.rep_transfer
thf(fact_7552_integer_Oid__abs__transfer,axiom,
    ( bNF_re982302072995117890nteger
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ code_pcr_integer
    @ ^ [X3: int] : X3
    @ code_integer_of_int ) ).

% integer.id_abs_transfer
thf(fact_7553_zero__integer_Otransfer,axiom,
    code_pcr_integer @ zero_zero_int @ zero_z3403309356797280102nteger ).

% zero_integer.transfer
thf(fact_7554_plus__integer_Otransfer,axiom,
    bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ plus_plus_int @ plus_p5714425477246183910nteger ).

% plus_integer.transfer
thf(fact_7555_quotient__of__eq,axiom,
    ! [A: int,B: int,P4: int,Q3: int] :
      ( ( ( quotient_of @ ( fract @ A @ B ) )
        = ( product_Pair_int_int @ P4 @ Q3 ) )
     => ( ( fract @ P4 @ Q3 )
        = ( fract @ A @ B ) ) ) ).

% quotient_of_eq
thf(fact_7556_less__integer_Otransfer,axiom,
    ( bNF_re6321650412969554871eger_o @ code_pcr_integer
    @ ( bNF_re6574881592172037608er_o_o @ code_pcr_integer
      @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
    @ ord_less_int
    @ ord_le6747313008572928689nteger ) ).

% less_integer.transfer
thf(fact_7557_one__integer_Otransfer,axiom,
    code_pcr_integer @ one_one_int @ one_one_Code_integer ).

% one_integer.transfer
thf(fact_7558_less__eq__integer_Otransfer,axiom,
    ( bNF_re6321650412969554871eger_o @ code_pcr_integer
    @ ( bNF_re6574881592172037608er_o_o @ code_pcr_integer
      @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
    @ ord_less_eq_int
    @ ord_le3102999989581377725nteger ) ).

% less_eq_integer.transfer
thf(fact_7559_uminus__integer_Otransfer,axiom,
    bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer @ uminus_uminus_int @ uminus1351360451143612070nteger ).

% uminus_integer.transfer
thf(fact_7560_times__integer_Otransfer,axiom,
    bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ times_times_int @ times_3573771949741848930nteger ).

% times_integer.transfer
thf(fact_7561_minus__integer_Otransfer,axiom,
    bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ minus_minus_int @ minus_8373710615458151222nteger ).

% minus_integer.transfer
thf(fact_7562_abs__integer_Otransfer,axiom,
    bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer @ abs_abs_int @ abs_abs_Code_integer ).

% abs_integer.transfer
thf(fact_7563_nat__of__integer_Otransfer,axiom,
    ( bNF_re2807294637932363402at_nat @ code_pcr_integer
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ nat2
    @ code_nat_of_integer ) ).

% nat_of_integer.transfer
thf(fact_7564_normalize__eq,axiom,
    ! [A: int,B: int,P4: int,Q3: int] :
      ( ( ( normalize @ ( product_Pair_int_int @ A @ B ) )
        = ( product_Pair_int_int @ P4 @ Q3 ) )
     => ( ( fract @ P4 @ Q3 )
        = ( fract @ A @ B ) ) ) ).

% normalize_eq
thf(fact_7565_integer__of__nat_Otransfer,axiom,
    ( bNF_re4153400068438556298nteger
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ code_pcr_integer
    @ semiri1314217659103216013at_int
    @ code_integer_of_nat ) ).

% integer_of_nat.transfer
thf(fact_7566_divide__integer_Otransfer,axiom,
    bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ divide_divide_int @ divide6298287555418463151nteger ).

% divide_integer.transfer
thf(fact_7567_modulo__integer_Otransfer,axiom,
    bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ modulo_modulo_int @ modulo364778990260209775nteger ).

% modulo_integer.transfer
thf(fact_7568_sgn__integer_Otransfer,axiom,
    bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer @ sgn_sgn_int @ sgn_sgn_Code_integer ).

% sgn_integer.transfer
thf(fact_7569_dup_Otransfer,axiom,
    ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer
    @ ^ [K4: int] : ( plus_plus_int @ K4 @ K4 )
    @ code_dup ) ).

% dup.transfer
thf(fact_7570_take__bit__integer_Otransfer,axiom,
    ( bNF_re4935368626670024657nteger
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
    @ bit_se2923211474154528505it_int
    @ bit_se1745604003318907178nteger ) ).

% take_bit_integer.transfer
thf(fact_7571_not__integer_Otransfer,axiom,
    bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer @ bit_ri7919022796975470100ot_int @ bit_ri7632146776885996613nteger ).

% not_integer.transfer
thf(fact_7572_and__integer_Otransfer,axiom,
    bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ bit_se725231765392027082nd_int @ bit_se3949692690581998587nteger ).

% and_integer.transfer
thf(fact_7573_bit__integer_Otransfer,axiom,
    ( bNF_re4711666741709854504_nat_o @ code_pcr_integer
    @ ^ [Y4: nat > $o,Z2: nat > $o] : ( Y4 = Z2 )
    @ bit_se1146084159140164899it_int
    @ bit_se9216721137139052372nteger ) ).

% bit_integer.transfer
thf(fact_7574_or__integer_Otransfer,axiom,
    bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ bit_se1409905431419307370or_int @ bit_se1080825931792720795nteger ).

% or_integer.transfer
thf(fact_7575_xor__integer_Otransfer,axiom,
    bNF_re398004352372739002nteger @ code_pcr_integer @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer ) @ bit_se6526347334894502574or_int @ bit_se3222712562003087583nteger ).

% xor_integer.transfer
thf(fact_7576_drop__bit__integer_Otransfer,axiom,
    ( bNF_re4935368626670024657nteger
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
    @ bit_se8568078237143864401it_int
    @ bit_se3928097537394005634nteger ) ).

% drop_bit_integer.transfer
thf(fact_7577_push__bit__integer_Otransfer,axiom,
    ( bNF_re4935368626670024657nteger
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
    @ bit_se545348938243370406it_int
    @ bit_se7788150548672797655nteger ) ).

% push_bit_integer.transfer
thf(fact_7578_mask__integer_Otransfer,axiom,
    ( bNF_re4153400068438556298nteger
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ code_pcr_integer
    @ bit_se2000444600071755411sk_int
    @ bit_se2119862282449309892nteger ) ).

% mask_integer.transfer
thf(fact_7579_unset__bit__integer_Otransfer,axiom,
    ( bNF_re4935368626670024657nteger
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
    @ bit_se4203085406695923979it_int
    @ bit_se8260200283734997820nteger ) ).

% unset_bit_integer.transfer
thf(fact_7580_set__bit__integer_Otransfer,axiom,
    ( bNF_re4935368626670024657nteger
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
    @ bit_se7879613467334960850it_int
    @ bit_se2793503036327961859nteger ) ).

% set_bit_integer.transfer
thf(fact_7581_division__segment__integer_Otransfer,axiom,
    bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer @ euclid3395696857347342551nt_int @ euclid6289375185220004616nteger ).

% division_segment_integer.transfer
thf(fact_7582_flip__bit__integer_Otransfer,axiom,
    ( bNF_re4935368626670024657nteger
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re3379532845092657523nteger @ code_pcr_integer @ code_pcr_integer )
    @ bit_se2159334234014336723it_int
    @ bit_se1345352211410354436nteger ) ).

% flip_bit_integer.transfer
thf(fact_7583_euclidean__size__integer_Otransfer,axiom,
    ( bNF_re2807294637932363402at_nat @ code_pcr_integer
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ euclid4774559944035922753ze_int
    @ euclid6377331345833325938nteger ) ).

% euclidean_size_integer.transfer
thf(fact_7584_quotient__of__Fract,axiom,
    ! [A: int,B: int] :
      ( ( quotient_of @ ( fract @ A @ B ) )
      = ( normalize @ ( product_Pair_int_int @ A @ B ) ) ) ).

% quotient_of_Fract
thf(fact_7585_sub_Otransfer,axiom,
    ( bNF_re7876454716742015248nteger
    @ ^ [Y4: num,Z2: num] : ( Y4 = Z2 )
    @ ( bNF_re6501075790457514782nteger
      @ ^ [Y4: num,Z2: num] : ( Y4 = Z2 )
      @ code_pcr_integer )
    @ ^ [M: num,N4: num] : ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N4 ) )
    @ code_sub ) ).

% sub.transfer
thf(fact_7586_Fract_Oabs__eq,axiom,
    ( fract
    = ( ^ [Xa4: int,X3: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X3 ) ) ) ) ) ).

% Fract.abs_eq
thf(fact_7587_Fract_Otransfer,axiom,
    ( bNF_re3461391660133120880nt_rat
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ ( bNF_re2214769303045360666nt_rat
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
      @ pcr_rat )
    @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B4 ) )
    @ fract ) ).

% Fract.transfer
thf(fact_7588_Fract__add__one,axiom,
    ! [N2: int,M2: int] :
      ( ( N2 != zero_zero_int )
     => ( ( fract @ ( plus_plus_int @ M2 @ N2 ) @ N2 )
        = ( plus_plus_rat @ ( fract @ M2 @ N2 ) @ one_one_rat ) ) ) ).

% Fract_add_one
thf(fact_7589_Fract__def,axiom,
    ( fract
    = ( map_fu7831380289885515383nt_rat @ id_int @ ( map_fu3424225382358772769nt_rat @ id_int @ abs_Rat )
      @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B4 ) ) ) ) ).

% Fract_def
thf(fact_7590_list__decode_Opinduct,axiom,
    ! [A0: nat,P: nat > $o] :
      ( ( accp_nat @ nat_list_decode_rel @ A0 )
     => ( ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
         => ( P @ zero_zero_nat ) )
       => ( ! [N: nat] :
              ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N ) )
             => ( ! [X8: nat,Y6: nat] :
                    ( ( ( product_Pair_nat_nat @ X8 @ Y6 )
                      = ( nat_prod_decode @ N ) )
                   => ( P @ Y6 ) )
               => ( P @ ( suc @ N ) ) ) )
         => ( P @ A0 ) ) ) ) ).

% list_decode.pinduct
thf(fact_7591_type__definition__integer,axiom,
    type_d8366093980585677751er_int @ code_int_of_integer @ code_integer_of_int @ top_top_set_int ).

% type_definition_integer
thf(fact_7592_pred__nat__trancl__eq__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N2 ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% pred_nat_trancl_eq_le
thf(fact_7593_less__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N2 ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% less_eq
thf(fact_7594_upt__0__eq__Nil__conv,axiom,
    ! [J: nat] :
      ( ( ( upt @ zero_zero_nat @ J )
        = nil_nat )
      = ( J = zero_zero_nat ) ) ).

% upt_0_eq_Nil_conv
thf(fact_7595_map__add__upt_H,axiom,
    ! [Ofs: nat,A: nat,B: nat] :
      ( ( map_nat_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ Ofs )
        @ ( upt @ A @ B ) )
      = ( upt @ ( plus_plus_nat @ A @ Ofs ) @ ( plus_plus_nat @ B @ Ofs ) ) ) ).

% map_add_upt'
thf(fact_7596_map__add__upt,axiom,
    ! [N2: nat,M2: nat] :
      ( ( map_nat_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ N2 )
        @ ( upt @ zero_zero_nat @ M2 ) )
      = ( upt @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).

% map_add_upt
thf(fact_7597_upt__filter__extend,axiom,
    ! [U: nat,U4: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ U @ U4 )
     => ( ! [I2: nat] :
            ( ( ( ord_less_eq_nat @ U @ I2 )
              & ( ord_less_nat @ I2 @ U4 ) )
           => ~ ( P @ I2 ) )
       => ( ( filter_nat @ P @ ( upt @ zero_zero_nat @ U ) )
          = ( filter_nat @ P @ ( upt @ zero_zero_nat @ U4 ) ) ) ) ) ).

% upt_filter_extend
thf(fact_7598_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 ) ) ) ).

% nth_upt
thf(fact_7599_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( ( upt @ I @ J )
        = ( cons_nat @ X @ Xs ) )
      = ( ( ord_less_nat @ I @ J )
        & ( I = X )
        & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_7600_list__decode_Opelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( ( nat_list_decode @ X )
        = Y )
     => ( ( accp_nat @ nat_list_decode_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = nil_nat )
             => ~ ( accp_nat @ nat_list_decode_rel @ zero_zero_nat ) ) )
         => ~ ! [N: nat] :
                ( ( X
                  = ( suc @ N ) )
               => ( ( Y
                    = ( produc2761476792215241774st_nat
                      @ ^ [X3: nat,Y3: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y3 ) )
                      @ ( nat_prod_decode @ N ) ) )
                 => ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N ) ) ) ) ) ) ) ).

% list_decode.pelims
thf(fact_7601_drop__upt,axiom,
    ! [M2: nat,I: nat,J: nat] :
      ( ( drop_nat @ M2 @ ( upt @ I @ J ) )
      = ( upt @ ( plus_plus_nat @ I @ M2 ) @ J ) ) ).

% drop_upt
thf(fact_7602_natural__zero__minus__one,axiom,
    ( ( minus_7197305767214868737atural @ zero_z2226904508553997617atural @ one_one_Code_natural )
    = zero_z2226904508553997617atural ) ).

% natural_zero_minus_one
thf(fact_7603_list__decode_Opsimps_I1_J,axiom,
    ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
   => ( ( nat_list_decode @ zero_zero_nat )
      = nil_nat ) ) ).

% list_decode.psimps(1)
thf(fact_7604_list__decode_Opsimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N2 ) )
     => ( ( nat_list_decode @ ( suc @ N2 ) )
        = ( produc2761476792215241774st_nat
          @ ^ [X3: nat,Y3: nat] : ( cons_nat @ X3 @ ( nat_list_decode @ Y3 ) )
          @ ( nat_prod_decode @ N2 ) ) ) ) ).

% list_decode.psimps(2)
thf(fact_7605_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X4: nat,Xs2: list_nat] :
              ( ( X
                = ( cons_nat @ X4 @ Xs2 ) )
             => ( Y
               != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).

% list_encode.elims
thf(fact_7606_upt__merge,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ J @ K ) )
     => ( ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ K ) )
        = ( upt @ I @ K ) ) ) ).

% upt_merge
thf(fact_7607_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( upt @ I @ J )
          = ( append_nat @ Xs @ Ys ) )
        = ( ? [K4: nat] :
              ( ( ord_less_eq_nat @ I @ K4 )
              & ( ord_less_eq_nat @ K4 @ J )
              & ( ( upt @ I @ K4 )
                = Xs )
              & ( ( upt @ K4 @ J )
                = Ys ) ) ) ) ) ).

% upt_eq_append_conv
thf(fact_7608_upt__append,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( append_nat @ ( upt @ zero_zero_nat @ I ) @ ( upt @ I @ J ) )
        = ( upt @ zero_zero_nat @ J ) ) ) ).

% upt_append
thf(fact_7609_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 ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_7610_upt__eq__lel__conv,axiom,
    ! [L: nat,H3: nat,Is1: list_nat,I: nat,Is2: list_nat] :
      ( ( ( upt @ L @ H3 )
        = ( append_nat @ Is1 @ ( cons_nat @ I @ Is2 ) ) )
      = ( ( Is1
          = ( upt @ L @ I ) )
        & ( Is2
          = ( upt @ ( suc @ I ) @ H3 ) )
        & ( ord_less_eq_nat @ L @ I )
        & ( ord_less_nat @ I @ H3 ) ) ) ).

% upt_eq_lel_conv
thf(fact_7611_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( nat_list_encode @ ( cons_nat @ X @ Xs ) )
      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).

% list_encode.simps(2)
thf(fact_7612_list__encode_Opelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X )
        = Y )
     => ( ( accp_list_nat @ nat_list_encode_rel @ X )
       => ( ( ( X = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
         => ~ ! [X4: nat,Xs2: list_nat] :
                ( ( X
                  = ( cons_nat @ X4 @ Xs2 ) )
               => ( ( Y
                    = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) )
                 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X4 @ Xs2 ) ) ) ) ) ) ) ).

% list_encode.pelims
thf(fact_7613_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( last_nat @ ( upt @ I @ J ) )
        = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).

% last_upt
thf(fact_7614_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 @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N2 )
        = ( suc @ ( plus_plus_nat @ I @ N2 ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_7615_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N2: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J @ I ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N2 )
        = ( suc @ ( plus_plus_nat @ I @ N2 ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_7616_upto_Opelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( ( upto @ X @ Xa )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) )
       => ~ ( ( ( ( ord_less_eq_int @ X @ Xa )
               => ( Y
                  = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
              & ( ~ ( ord_less_eq_int @ X @ Xa )
               => ( Y = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).

% upto.pelims
thf(fact_7617_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( ( accp_P1096762738010456898nt_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 ) ) ) ) ).

% upto.psimps
thf(fact_7618_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
     => ( ( nth_int @ ( upto @ I @ J ) @ K )
        = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).

% nth_upto
thf(fact_7619_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 ) ) ) ).

% length_upto
thf(fact_7620_upto__rec__numeral_I1_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
          = ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( numeral_numeral_int @ N2 ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_7621_upto__rec__numeral_I2_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_7622_upto__rec__numeral_I3_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N2 ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) @ ( numeral_numeral_int @ N2 ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N2 ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_7623_upto__rec__numeral_I4_J,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_7624_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 ) ) ) ) ) ).

% upto_split2
thf(fact_7625_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 ) ) ) ) ).

% upto_rec1
thf(fact_7626_upto_Osimps,axiom,
    ( upto
    = ( ^ [I3: int,J2: int] : ( if_list_int @ ( ord_less_eq_int @ I3 @ J2 ) @ ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_7627_upto_Oelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( ( upto @ X @ Xa )
        = Y )
     => ( ( ( ord_less_eq_int @ X @ Xa )
         => ( Y
            = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
        & ( ~ ( ord_less_eq_int @ X @ Xa )
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
thf(fact_7628_greaterThanAtMost__upto,axiom,
    ( set_or6656581121297822940st_int
    = ( ^ [I3: int,J2: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) ) ) ) ).

% greaterThanAtMost_upto
thf(fact_7629_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 ) ) ) ) ) ) ).

% upto_split3
thf(fact_7630_greaterThanLessThan__upto,axiom,
    ( set_or5832277885323065728an_int
    = ( ^ [I3: int,J2: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ ( minus_minus_int @ J2 @ one_one_int ) ) ) ) ) ).

% greaterThanLessThan_upto
thf(fact_7631_take__upt,axiom,
    ! [I: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M2 ) @ N2 )
     => ( ( take_nat @ M2 @ ( upt @ I @ N2 ) )
        = ( upt @ I @ ( plus_plus_nat @ I @ M2 ) ) ) ) ).

% take_upt
thf(fact_7632_butlast__upt,axiom,
    ! [M2: nat,N2: nat] :
      ( ( butlast_nat @ ( upt @ M2 @ N2 ) )
      = ( upt @ M2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).

% butlast_upt
thf(fact_7633_next_Osimps,axiom,
    ! [V: code_natural,W: code_natural] :
      ( ( next @ ( produc3574140220909816553atural @ V @ W ) )
      = ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ one_one_Code_natural ) ) @ one_one_Code_natural ) @ ( produc3574140220909816553atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% next.simps
thf(fact_7634_full__exhaustive__natural_H_Ocases,axiom,
    ! [X: produc989692990947075319atural] :
      ~ ! [F3: produc4972180933644002618e_term > option6357759511663192854e_term,D: code_natural,I2: code_natural] :
          ( X
         != ( produc3831813291587773865atural @ F3 @ ( produc3574140220909816553atural @ D @ I2 ) ) ) ).

% full_exhaustive_natural'.cases
thf(fact_7635_exhaustive__natural_H_Ocases,axiom,
    ! [X: produc8731074985263844745atural] :
      ~ ! [F3: code_natural > option6357759511663192854e_term,D: code_natural,I2: code_natural] :
          ( X
         != ( produc2252593628808123835atural @ F3 @ ( produc3574140220909816553atural @ D @ I2 ) ) ) ).

% exhaustive_natural'.cases
thf(fact_7636_log_Ocases,axiom,
    ! [X: produc7822875418678951345atural] :
      ~ ! [B2: code_natural,I2: code_natural] :
          ( X
         != ( produc3574140220909816553atural @ B2 @ I2 ) ) ).

% log.cases
thf(fact_7637_minus__shift__def,axiom,
    ( minus_shift
    = ( ^ [R3: code_natural,K4: code_natural,L3: code_natural] : ( if_Code_natural @ ( ord_le5570908160329646204atural @ K4 @ L3 ) @ ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ R3 @ K4 ) @ L3 ) @ ( minus_7197305767214868737atural @ K4 @ L3 ) ) ) ) ).

% minus_shift_def
thf(fact_7638_split__seed__def,axiom,
    ( split_seed
    = ( ^ [S4: produc7822875418678951345atural] :
          ( produc8282080750456430313atural
          @ ^ [V2: code_natural,W2: code_natural] :
              ( produc8282080750456430313atural
              @ ^ [V5: code_natural,W3: code_natural] : ( produc4480994950612372183atural @ ( produc3574140220909816553atural @ ( inc_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ V2 ) @ W3 ) @ ( produc3574140220909816553atural @ V5 @ ( inc_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ W2 ) ) )
              @ ( produc6591912806276919810atural @ ( next @ S4 ) ) )
          @ S4 ) ) ) ).

% split_seed_def
thf(fact_7639_log_Osimps,axiom,
    ( log
    = ( ^ [B4: code_natural,I3: code_natural] :
          ( if_Code_natural
          @ ( ( ord_le1926595141338095240atural @ B4 @ one_one_Code_natural )
            | ( ord_le5570908160329646204atural @ I3 @ B4 ) )
          @ one_one_Code_natural
          @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ B4 @ ( divide5121882707175180666atural @ I3 @ B4 ) ) ) ) ) ) ).

% log.simps
thf(fact_7640_log_Oelims,axiom,
    ! [X: code_natural,Xa: code_natural,Y: code_natural] :
      ( ( ( log @ X @ Xa )
        = Y )
     => ( ( ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
            | ( ord_le5570908160329646204atural @ Xa @ X ) )
         => ( Y = one_one_Code_natural ) )
        & ( ~ ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
              | ( ord_le5570908160329646204atural @ Xa @ X ) )
         => ( Y
            = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ X @ ( divide5121882707175180666atural @ Xa @ X ) ) ) ) ) ) ) ).

% log.elims
thf(fact_7641_inc__shift__def,axiom,
    ( inc_shift
    = ( ^ [V2: code_natural,K4: code_natural] : ( if_Code_natural @ ( V2 = K4 ) @ one_one_Code_natural @ ( plus_p4538020629002901425atural @ K4 @ one_one_Code_natural ) ) ) ) ).

% inc_shift_def
thf(fact_7642_log_Opelims,axiom,
    ! [X: code_natural,Xa: code_natural,Y: code_natural] :
      ( ( ( log @ X @ Xa )
        = Y )
     => ( ( accp_P8126237942716283194atural @ log_rel @ ( produc3574140220909816553atural @ X @ Xa ) )
       => ~ ( ( ( ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
                  | ( ord_le5570908160329646204atural @ Xa @ X ) )
               => ( Y = one_one_Code_natural ) )
              & ( ~ ( ( ord_le1926595141338095240atural @ X @ one_one_Code_natural )
                    | ( ord_le5570908160329646204atural @ Xa @ X ) )
               => ( Y
                  = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ X @ ( divide5121882707175180666atural @ Xa @ X ) ) ) ) ) )
           => ~ ( accp_P8126237942716283194atural @ log_rel @ ( produc3574140220909816553atural @ X @ Xa ) ) ) ) ) ).

% log.pelims
thf(fact_7643_Random_Orange__def,axiom,
    ( range
    = ( ^ [K4: code_natural] :
          ( produc5538323210962509403atural
          @ ( iterat8892046348760725948atural @ ( log @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K4 )
            @ ^ [L3: code_natural] :
                ( produc5538323210962509403atural @ next
                @ ^ [V2: code_natural] : ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ V2 @ ( times_2397367101498566445atural @ L3 @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
            @ one_one_Code_natural )
          @ ^ [V2: code_natural] : ( produc6639722614265839536atural @ ( modulo8411746178871703098atural @ V2 @ K4 ) ) ) ) ) ).

% Random.range_def
thf(fact_7644_wmsI,axiom,
    ! [A2: multis2468970476368604999at_nat,B5: multis2468970476368604999at_nat,Z7: multis2468970476368604999at_nat] :
      ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A2 ) @ ( set_ms8126754132646256062at_nat @ B5 ) ) @ fun_max_strict )
        | ( ( A2 = zero_z1048942125864253310at_nat )
          & ( B5 = zero_z1048942125864253310at_nat ) ) )
     => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z7 @ A2 ) @ ( plus_p7104986032573967614at_nat @ Z7 @ B5 ) ) @ ms_weak ) ) ).

% wmsI
thf(fact_7645_smsI,axiom,
    ! [A2: multis2468970476368604999at_nat,B5: multis2468970476368604999at_nat,Z7: multis2468970476368604999at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A2 ) @ ( set_ms8126754132646256062at_nat @ B5 ) ) @ fun_max_strict )
     => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z7 @ A2 ) @ ( plus_p7104986032573967614at_nat @ Z7 @ B5 ) ) @ ms_strict ) ) ).

% smsI
thf(fact_7646_ms__reduction__pair,axiom,
    fun_re7357418987779152907at_nat @ ( produc5245064249948416855at_nat @ ms_strict @ ms_weak ) ).

% ms_reduction_pair
thf(fact_7647_ms__strictI,axiom,
    ! [Z7: multis2468970476368604999at_nat,Z8: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,B5: multis2468970476368604999at_nat] :
      ( ( pw_leq @ Z7 @ Z8 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A2 ) @ ( set_ms8126754132646256062at_nat @ B5 ) ) @ fun_max_strict )
       => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z7 @ A2 ) @ ( plus_p7104986032573967614at_nat @ Z8 @ B5 ) ) @ ms_strict ) ) ) ).

% ms_strictI
thf(fact_7648_ms__weakI1,axiom,
    ! [Z7: multis2468970476368604999at_nat,Z8: multis2468970476368604999at_nat,A2: multis2468970476368604999at_nat,B5: multis2468970476368604999at_nat] :
      ( ( pw_leq @ Z7 @ Z8 )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A2 ) @ ( set_ms8126754132646256062at_nat @ B5 ) ) @ fun_max_strict )
       => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z7 @ A2 ) @ ( plus_p7104986032573967614at_nat @ Z8 @ B5 ) ) @ ms_weak ) ) ) ).

% ms_weakI1
thf(fact_7649_pw__leq__split,axiom,
    ! [X11: multis2468970476368604999at_nat,Y9: multis2468970476368604999at_nat] :
      ( ( pw_leq @ X11 @ Y9 )
     => ? [A8: multis2468970476368604999at_nat,B9: multis2468970476368604999at_nat,Z9: multis2468970476368604999at_nat] :
          ( ( X11
            = ( plus_p7104986032573967614at_nat @ A8 @ Z9 ) )
          & ( Y9
            = ( plus_p7104986032573967614at_nat @ B9 @ Z9 ) )
          & ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( set_ms8126754132646256062at_nat @ A8 ) @ ( set_ms8126754132646256062at_nat @ B9 ) ) @ fun_max_strict )
            | ( ( B9 = zero_z1048942125864253310at_nat )
              & ( A8 = zero_z1048942125864253310at_nat ) ) ) ) ) ).

% pw_leq_split
thf(fact_7650_ms__weakI2,axiom,
    ! [Z7: multis2468970476368604999at_nat,Z8: multis2468970476368604999at_nat] :
      ( ( pw_leq @ Z7 @ Z8 )
     => ( member6689249552917799696at_nat @ ( produc4348348721325984599at_nat @ ( plus_p7104986032573967614at_nat @ Z7 @ zero_z1048942125864253310at_nat ) @ ( plus_p7104986032573967614at_nat @ Z8 @ zero_z1048942125864253310at_nat ) ) @ ms_weak ) ) ).

% ms_weakI2
thf(fact_7651_pw__leq__lstep,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
     => ( pw_leq @ ( add_ms2612439473150266591at_nat @ X @ zero_z1048942125864253310at_nat ) @ ( add_ms2612439473150266591at_nat @ Y @ zero_z1048942125864253310at_nat ) ) ) ).

% pw_leq_lstep
thf(fact_7652_pw__leq__step,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,X11: multis2468970476368604999at_nat,Y9: multis2468970476368604999at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
     => ( ( pw_leq @ X11 @ Y9 )
       => ( pw_leq @ ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ X @ zero_z1048942125864253310at_nat ) @ X11 ) @ ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ Y @ zero_z1048942125864253310at_nat ) @ Y9 ) ) ) ) ).

% pw_leq_step
thf(fact_7653_pw__leq_Osimps,axiom,
    ( pw_leq
    = ( ^ [A12: multis2468970476368604999at_nat,A22: multis2468970476368604999at_nat] :
          ( ( ( A12 = zero_z1048942125864253310at_nat )
            & ( A22 = zero_z1048942125864253310at_nat ) )
          | ? [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,X9: multis2468970476368604999at_nat,Y10: multis2468970476368604999at_nat] :
              ( ( A12
                = ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ X3 @ zero_z1048942125864253310at_nat ) @ X9 ) )
              & ( A22
                = ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ Y3 @ zero_z1048942125864253310at_nat ) @ Y10 ) )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ fun_pair_leq )
              & ( pw_leq @ X9 @ Y10 ) ) ) ) ) ).

% pw_leq.simps
thf(fact_7654_pw__leq_Ocases,axiom,
    ! [A1: multis2468970476368604999at_nat,A23: multis2468970476368604999at_nat] :
      ( ( pw_leq @ A1 @ A23 )
     => ( ( ( A1 = zero_z1048942125864253310at_nat )
         => ( A23 != zero_z1048942125864253310at_nat ) )
       => ~ ! [X4: product_prod_nat_nat,Y5: product_prod_nat_nat,X14: multis2468970476368604999at_nat] :
              ( ( A1
                = ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ X4 @ zero_z1048942125864253310at_nat ) @ X14 ) )
             => ! [Y11: multis2468970476368604999at_nat] :
                  ( ( A23
                    = ( plus_p7104986032573967614at_nat @ ( add_ms2612439473150266591at_nat @ Y5 @ zero_z1048942125864253310at_nat ) @ Y11 ) )
                 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y5 ) @ fun_pair_leq )
                   => ~ ( pw_leq @ X14 @ Y11 ) ) ) ) ) ) ).

% pw_leq.cases
thf(fact_7655_Code__Numeral_Osub__code_I4_J,axiom,
    ! [N2: num] :
      ( ( code_sub @ one @ ( bit0 @ N2 ) )
      = ( code_Neg @ ( bitM @ N2 ) ) ) ).

% Code_Numeral.sub_code(4)
thf(fact_7656_Code__Numeral_Odup__code_I3_J,axiom,
    ! [N2: num] :
      ( ( code_dup @ ( code_Neg @ N2 ) )
      = ( code_Neg @ ( bit0 @ N2 ) ) ) ).

% Code_Numeral.dup_code(3)
thf(fact_7657_less__integer__code_I3_J,axiom,
    ! [L: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( code_Neg @ L ) ) ).

% less_integer_code(3)
thf(fact_7658_less__integer__code_I7_J,axiom,
    ! [K: num] : ( ord_le6747313008572928689nteger @ ( code_Neg @ K ) @ zero_z3403309356797280102nteger ) ).

% less_integer_code(7)
thf(fact_7659_plus__integer__code_I6_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( code_Neg @ M2 ) @ ( code_Neg @ N2 ) )
      = ( code_Neg @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% plus_integer_code(6)
thf(fact_7660_less__eq__integer__code_I7_J,axiom,
    ! [K: num] : ( ord_le3102999989581377725nteger @ ( code_Neg @ K ) @ zero_z3403309356797280102nteger ) ).

% less_eq_integer_code(7)
thf(fact_7661_less__eq__integer__code_I3_J,axiom,
    ! [L: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( code_Neg @ L ) ) ).

% less_eq_integer_code(3)
thf(fact_7662_less__integer__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( ( ord_le6747313008572928689nteger @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
      = ( ord_less_num @ L @ K ) ) ).

% less_integer_code(9)
thf(fact_7663_less__eq__integer__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( ( ord_le3102999989581377725nteger @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
      = ( ord_less_eq_num @ L @ K ) ) ).

% less_eq_integer_code(9)
thf(fact_7664_minus__integer__code_I6_J,axiom,
    ! [M2: num,N2: num] :
      ( ( minus_8373710615458151222nteger @ ( code_Neg @ M2 ) @ ( code_Neg @ N2 ) )
      = ( code_sub @ N2 @ M2 ) ) ).

% minus_integer_code(6)
thf(fact_7665_divmod__abs__code_I2_J,axiom,
    ! [K: num,L: num] :
      ( ( code_divmod_abs @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
      = ( unique3479559517661332726nteger @ K @ L ) ) ).

% divmod_abs_code(2)
thf(fact_7666_Code__Numeral_Osub__code_I5_J,axiom,
    ! [N2: num] :
      ( ( code_sub @ one @ ( bit1 @ N2 ) )
      = ( code_Neg @ ( bit0 @ N2 ) ) ) ).

% Code_Numeral.sub_code(5)
thf(fact_7667_card__length__sum__list__rec,axiom,
    ! [M2: nat,N6: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M2 )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [L3: list_nat] :
                ( ( ( size_size_list_nat @ L3 )
                  = M2 )
                & ( ( groups4561878855575611511st_nat @ L3 )
                  = N6 ) ) ) )
        = ( plus_plus_nat
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L3: list_nat] :
                  ( ( ( size_size_list_nat @ L3 )
                    = ( minus_minus_nat @ M2 @ one_one_nat ) )
                  & ( ( groups4561878855575611511st_nat @ L3 )
                    = N6 ) ) ) )
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L3: list_nat] :
                  ( ( ( size_size_list_nat @ L3 )
                    = M2 )
                  & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L3 ) @ one_one_nat )
                    = N6 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_7668_card__length__sum__list,axiom,
    ! [M2: nat,N6: nat] :
      ( ( finite_card_list_nat
        @ ( collect_list_nat
          @ ^ [L3: list_nat] :
              ( ( ( size_size_list_nat @ L3 )
                = M2 )
              & ( ( groups4561878855575611511st_nat @ L3 )
                = N6 ) ) ) )
      = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N6 @ M2 ) @ one_one_nat ) @ N6 ) ) ).

% card_length_sum_list
thf(fact_7669_natural__of__nat__of__nat,axiom,
    code_natural_of_nat = semiri3763490453095760265atural ).

% natural_of_nat_of_nat
thf(fact_7670_division__segment__natural_Oabs__eq,axiom,
    ! [X: nat] :
      ( ( euclid5112970336976722131atural @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( euclid3398187327856392827nt_nat @ X ) ) ) ).

% division_segment_natural.abs_eq
thf(fact_7671_set__bit__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ( bit_se1617098188084679374atural @ Xa @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( bit_se7882103937844011126it_nat @ Xa @ X ) ) ) ).

% set_bit_natural.abs_eq
thf(fact_7672_flip__bit__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ( bit_se168947363167071951atural @ Xa @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( bit_se2161824704523386999it_nat @ Xa @ X ) ) ) ).

% flip_bit_natural.abs_eq
thf(fact_7673_unset__bit__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ( bit_se7083795435491715335atural @ Xa @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( bit_se4205575877204974255it_nat @ Xa @ X ) ) ) ).

% unset_bit_natural.abs_eq
thf(fact_7674_euclidean__size__natural_Oabs__eq,axiom,
    ! [X: nat] :
      ( ( euclid5200926497590043453atural @ ( code_natural_of_nat @ X ) )
      = ( euclid4777050414544973029ze_nat @ X ) ) ).

% euclidean_size_natural.abs_eq
thf(fact_7675_bit__natural_Oabs__eq,axiom,
    ! [X: nat] :
      ( ( bit_se8040316288895769887atural @ ( code_natural_of_nat @ X ) )
      = ( bit_se1148574629649215175it_nat @ X ) ) ).

% bit_natural.abs_eq
thf(fact_7676_natural__of__nat__cases,axiom,
    ! [X: code_natural] :
      ~ ! [Y5: nat] :
          ( ( X
            = ( code_natural_of_nat @ Y5 ) )
         => ~ ( member_nat @ Y5 @ top_top_set_nat ) ) ).

% natural_of_nat_cases
thf(fact_7677_natural__of__nat__induct,axiom,
    ! [P: code_natural > $o,X: code_natural] :
      ( ! [Y5: nat] :
          ( ( member_nat @ Y5 @ top_top_set_nat )
         => ( P @ ( code_natural_of_nat @ Y5 ) ) )
     => ( P @ X ) ) ).

% natural_of_nat_induct
thf(fact_7678_natural__of__nat__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( member_nat @ X @ top_top_set_nat )
     => ( ( member_nat @ Y @ top_top_set_nat )
       => ( ( ( code_natural_of_nat @ X )
            = ( code_natural_of_nat @ Y ) )
          = ( X = Y ) ) ) ) ).

% natural_of_nat_inject
thf(fact_7679_take__bit__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ( bit_se569199155075624693atural @ Xa @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( bit_se2925701944663578781it_nat @ Xa @ X ) ) ) ).

% take_bit_natural.abs_eq
thf(fact_7680_and__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ( bit_se2773287842338716102atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( bit_se727722235901077358nd_nat @ Xa @ X ) ) ) ).

% and_natural.abs_eq
thf(fact_7681_xor__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ( bit_se2046307713759805098atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( bit_se6528837805403552850or_nat @ Xa @ X ) ) ) ).

% xor_natural.abs_eq
thf(fact_7682_or__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ( bit_se9127793120404214118atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( bit_se1412395901928357646or_nat @ Xa @ X ) ) ) ).

% or_natural.abs_eq
thf(fact_7683_mask__natural_Oabs__eq,axiom,
    ( bit_se943457434206027407atural
    = ( ^ [X3: nat] : ( code_natural_of_nat @ ( bit_se2002935070580805687sk_nat @ X3 ) ) ) ) ).

% mask_natural.abs_eq
thf(fact_7684_push__bit__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ( bit_se6611745700429515170atural @ Xa @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( bit_se547839408752420682it_nat @ Xa @ X ) ) ) ).

% push_bit_natural.abs_eq
thf(fact_7685_drop__bit__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ( bit_se2751692689150723149atural @ Xa @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( bit_se8570568707652914677it_nat @ Xa @ X ) ) ) ).

% drop_bit_natural.abs_eq
thf(fact_7686_zero__natural__def,axiom,
    ( zero_z2226904508553997617atural
    = ( code_natural_of_nat @ zero_zero_nat ) ) ).

% zero_natural_def
thf(fact_7687_one__natural__def,axiom,
    ( one_one_Code_natural
    = ( code_natural_of_nat @ one_one_nat ) ) ).

% one_natural_def
thf(fact_7688_less__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ( ord_le5570908160329646204atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
      = ( ord_less_nat @ Xa @ X ) ) ).

% less_natural.abs_eq
thf(fact_7689_plus__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ( plus_p4538020629002901425atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( plus_plus_nat @ Xa @ X ) ) ) ).

% plus_natural.abs_eq
thf(fact_7690_minus__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ( minus_7197305767214868737atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( minus_minus_nat @ Xa @ X ) ) ) ).

% minus_natural.abs_eq
thf(fact_7691_less__eq__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ( ord_le1926595141338095240atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
      = ( ord_less_eq_nat @ Xa @ X ) ) ).

% less_eq_natural.abs_eq
thf(fact_7692_times__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ( times_2397367101498566445atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( times_times_nat @ Xa @ X ) ) ) ).

% times_natural.abs_eq
thf(fact_7693_divide__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ( divide5121882707175180666atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( divide_divide_nat @ Xa @ X ) ) ) ).

% divide_natural.abs_eq
thf(fact_7694_mask__natural__def,axiom,
    ( bit_se943457434206027407atural
    = ( map_fu4742413587575543474atural @ id_nat @ code_natural_of_nat @ bit_se2002935070580805687sk_nat ) ) ).

% mask_natural_def
thf(fact_7695_modulo__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] :
      ( ( modulo8411746178871703098atural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( modulo_modulo_nat @ Xa @ X ) ) ) ).

% modulo_natural.abs_eq
thf(fact_7696_Nat_Oabs__eq,axiom,
    ! [X: int] :
      ( ( code_Nat @ ( code_integer_of_int @ X ) )
      = ( code_natural_of_nat @ ( nat2 @ X ) ) ) ).

% Nat.abs_eq
thf(fact_7697_Nat__def,axiom,
    ( code_Nat
    = ( map_fu1051355602067684763atural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).

% Nat_def
thf(fact_7698_natural__of__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( code_n4118661773612635043nteger @ ( code_integer_of_int @ X ) )
      = ( code_natural_of_nat @ ( nat2 @ X ) ) ) ).

% natural_of_integer.abs_eq
thf(fact_7699_natural__of__nat__of__natural__inverse,axiom,
    ! [N2: code_natural] :
      ( ( code_natural_of_nat @ ( code_nat_of_natural @ N2 ) )
      = N2 ) ).

% natural_of_nat_of_natural_inverse
thf(fact_7700_nat__of__natural__of__nat__inverse,axiom,
    ! [N2: nat] :
      ( ( code_nat_of_natural @ ( code_natural_of_nat @ N2 ) )
      = N2 ) ).

% nat_of_natural_of_nat_inverse
thf(fact_7701_nat__of__natural__inverse,axiom,
    ! [X: code_natural] :
      ( ( code_natural_of_nat @ ( code_nat_of_natural @ X ) )
      = X ) ).

% nat_of_natural_inverse
thf(fact_7702_nat__of__natural__min,axiom,
    ! [K: code_natural,L: code_natural] :
      ( ( code_nat_of_natural @ ( ord_min_Code_natural @ K @ L ) )
      = ( ord_min_nat @ ( code_nat_of_natural @ K ) @ ( code_nat_of_natural @ L ) ) ) ).

% nat_of_natural_min
thf(fact_7703_nat__of__natural__max,axiom,
    ! [K: code_natural,L: code_natural] :
      ( ( code_nat_of_natural @ ( ord_max_Code_natural @ K @ L ) )
      = ( ord_max_nat @ ( code_nat_of_natural @ K ) @ ( code_nat_of_natural @ L ) ) ) ).

% nat_of_natural_max
thf(fact_7704_of__nat__of__natural,axiom,
    ! [N2: code_natural] :
      ( ( semiri3763490453095760265atural @ ( code_nat_of_natural @ N2 ) )
      = N2 ) ).

% of_nat_of_natural
thf(fact_7705_nat__of__natural__of__nat,axiom,
    ! [N2: nat] :
      ( ( code_nat_of_natural @ ( semiri3763490453095760265atural @ N2 ) )
      = N2 ) ).

% nat_of_natural_of_nat
thf(fact_7706_zero__natural_Orep__eq,axiom,
    ( ( code_nat_of_natural @ zero_z2226904508553997617atural )
    = zero_zero_nat ) ).

% zero_natural.rep_eq
thf(fact_7707_one__natural_Orep__eq,axiom,
    ( ( code_nat_of_natural @ one_one_Code_natural )
    = one_one_nat ) ).

% one_natural.rep_eq
thf(fact_7708_plus__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa: code_natural] :
      ( ( code_nat_of_natural @ ( plus_p4538020629002901425atural @ X @ Xa ) )
      = ( plus_plus_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).

% plus_natural.rep_eq
thf(fact_7709_minus__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa: code_natural] :
      ( ( code_nat_of_natural @ ( minus_7197305767214868737atural @ X @ Xa ) )
      = ( minus_minus_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).

% minus_natural.rep_eq
thf(fact_7710_nat__of__natural__numeral,axiom,
    ! [K: num] :
      ( ( code_nat_of_natural @ ( numera5444537566228673987atural @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_of_natural_numeral
thf(fact_7711_times__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa: code_natural] :
      ( ( code_nat_of_natural @ ( times_2397367101498566445atural @ X @ Xa ) )
      = ( times_times_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).

% times_natural.rep_eq
thf(fact_7712_divide__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa: code_natural] :
      ( ( code_nat_of_natural @ ( divide5121882707175180666atural @ X @ Xa ) )
      = ( divide_divide_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).

% divide_natural.rep_eq
thf(fact_7713_modulo__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa: code_natural] :
      ( ( code_nat_of_natural @ ( modulo8411746178871703098atural @ X @ Xa ) )
      = ( modulo_modulo_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).

% modulo_natural.rep_eq
thf(fact_7714_division__segment__natural_Orep__eq,axiom,
    ! [X: code_natural] :
      ( ( code_nat_of_natural @ ( euclid5112970336976722131atural @ X ) )
      = ( euclid3398187327856392827nt_nat @ ( code_nat_of_natural @ X ) ) ) ).

% division_segment_natural.rep_eq
thf(fact_7715_euclidean__size__natural_Orep__eq,axiom,
    ( euclid5200926497590043453atural
    = ( ^ [X3: code_natural] : ( euclid4777050414544973029ze_nat @ ( code_nat_of_natural @ X3 ) ) ) ) ).

% euclidean_size_natural.rep_eq
thf(fact_7716_unset__bit__natural_Orep__eq,axiom,
    ! [X: nat,Xa: code_natural] :
      ( ( code_nat_of_natural @ ( bit_se7083795435491715335atural @ X @ Xa ) )
      = ( bit_se4205575877204974255it_nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).

% unset_bit_natural.rep_eq
thf(fact_7717_flip__bit__natural_Orep__eq,axiom,
    ! [X: nat,Xa: code_natural] :
      ( ( code_nat_of_natural @ ( bit_se168947363167071951atural @ X @ Xa ) )
      = ( bit_se2161824704523386999it_nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).

% flip_bit_natural.rep_eq
thf(fact_7718_set__bit__natural_Orep__eq,axiom,
    ! [X: nat,Xa: code_natural] :
      ( ( code_nat_of_natural @ ( bit_se1617098188084679374atural @ X @ Xa ) )
      = ( bit_se7882103937844011126it_nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).

% set_bit_natural.rep_eq
thf(fact_7719_Nat_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_nat_of_natural @ ( code_Nat @ X ) )
      = ( nat2 @ ( code_int_of_integer @ X ) ) ) ).

% Nat.rep_eq
thf(fact_7720_push__bit__natural_Orep__eq,axiom,
    ! [X: nat,Xa: code_natural] :
      ( ( code_nat_of_natural @ ( bit_se6611745700429515170atural @ X @ Xa ) )
      = ( bit_se547839408752420682it_nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).

% push_bit_natural.rep_eq
thf(fact_7721_mask__natural_Orep__eq,axiom,
    ! [X: nat] :
      ( ( code_nat_of_natural @ ( bit_se943457434206027407atural @ X ) )
      = ( bit_se2002935070580805687sk_nat @ X ) ) ).

% mask_natural.rep_eq
thf(fact_7722_and__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa: code_natural] :
      ( ( code_nat_of_natural @ ( bit_se2773287842338716102atural @ X @ Xa ) )
      = ( bit_se727722235901077358nd_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).

% and_natural.rep_eq
thf(fact_7723_bit__natural_Orep__eq,axiom,
    ( bit_se8040316288895769887atural
    = ( ^ [X3: code_natural] : ( bit_se1148574629649215175it_nat @ ( code_nat_of_natural @ X3 ) ) ) ) ).

% bit_natural.rep_eq
thf(fact_7724_nat__of__natural__inject,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( ( code_nat_of_natural @ X )
        = ( code_nat_of_natural @ Y ) )
      = ( X = Y ) ) ).

% nat_of_natural_inject
thf(fact_7725_natural__eqI,axiom,
    ! [M2: code_natural,N2: code_natural] :
      ( ( ( code_nat_of_natural @ M2 )
        = ( code_nat_of_natural @ N2 ) )
     => ( M2 = N2 ) ) ).

% natural_eqI
thf(fact_7726_natural__eq__iff,axiom,
    ( ( ^ [Y4: code_natural,Z2: code_natural] : ( Y4 = Z2 ) )
    = ( ^ [M: code_natural,N4: code_natural] :
          ( ( code_nat_of_natural @ M )
          = ( code_nat_of_natural @ N4 ) ) ) ) ).

% natural_eq_iff
thf(fact_7727_natural__cases,axiom,
    ! [M2: code_natural] :
      ~ ! [N: nat] :
          ( M2
         != ( semiri3763490453095760265atural @ N ) ) ).

% natural_cases
thf(fact_7728_nat__of__natural,axiom,
    ! [X: code_natural] : ( member_nat @ ( code_nat_of_natural @ X ) @ top_top_set_nat ) ).

% nat_of_natural
thf(fact_7729_nat__of__natural__cases,axiom,
    ! [Y: nat] :
      ( ( member_nat @ Y @ top_top_set_nat )
     => ~ ! [X4: code_natural] :
            ( Y
           != ( code_nat_of_natural @ X4 ) ) ) ).

% nat_of_natural_cases
thf(fact_7730_nat__of__natural__induct,axiom,
    ! [Y: nat,P: nat > $o] :
      ( ( member_nat @ Y @ top_top_set_nat )
     => ( ! [X4: code_natural] : ( P @ ( code_nat_of_natural @ X4 ) )
       => ( P @ Y ) ) ) ).

% nat_of_natural_induct
thf(fact_7731_take__bit__natural_Orep__eq,axiom,
    ! [X: nat,Xa: code_natural] :
      ( ( code_nat_of_natural @ ( bit_se569199155075624693atural @ X @ Xa ) )
      = ( bit_se2925701944663578781it_nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).

% take_bit_natural.rep_eq
thf(fact_7732_or__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa: code_natural] :
      ( ( code_nat_of_natural @ ( bit_se9127793120404214118atural @ X @ Xa ) )
      = ( bit_se1412395901928357646or_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).

% or_natural.rep_eq
thf(fact_7733_xor__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa: code_natural] :
      ( ( code_nat_of_natural @ ( bit_se2046307713759805098atural @ X @ Xa ) )
      = ( bit_se6528837805403552850or_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).

% xor_natural.rep_eq
thf(fact_7734_drop__bit__natural_Orep__eq,axiom,
    ! [X: nat,Xa: code_natural] :
      ( ( code_nat_of_natural @ ( bit_se2751692689150723149atural @ X @ Xa ) )
      = ( bit_se8570568707652914677it_nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).

% drop_bit_natural.rep_eq
thf(fact_7735_natural__of__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_nat_of_natural @ ( code_n4118661773612635043nteger @ X ) )
      = ( nat2 @ ( code_int_of_integer @ X ) ) ) ).

% natural_of_integer.rep_eq
thf(fact_7736_less__natural_Orep__eq,axiom,
    ( ord_le5570908160329646204atural
    = ( ^ [X3: code_natural,Xa4: code_natural] : ( ord_less_nat @ ( code_nat_of_natural @ X3 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).

% less_natural.rep_eq
thf(fact_7737_natural__of__nat__inverse,axiom,
    ! [Y: nat] :
      ( ( member_nat @ Y @ top_top_set_nat )
     => ( ( code_nat_of_natural @ ( code_natural_of_nat @ Y ) )
        = Y ) ) ).

% natural_of_nat_inverse
thf(fact_7738_type__definition__natural,axiom,
    type_d4410041424927559462al_nat @ code_nat_of_natural @ code_natural_of_nat @ top_top_set_nat ).

% type_definition_natural
thf(fact_7739_or__natural__def,axiom,
    ( bit_se9127793120404214118atural
    = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se1412395901928357646or_nat ) ) ).

% or_natural_def
thf(fact_7740_xor__natural__def,axiom,
    ( bit_se2046307713759805098atural
    = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se6528837805403552850or_nat ) ) ).

% xor_natural_def
thf(fact_7741_and__natural__def,axiom,
    ( bit_se2773287842338716102atural
    = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se727722235901077358nd_nat ) ) ).

% and_natural_def
thf(fact_7742_less__eq__natural_Orep__eq,axiom,
    ( ord_le1926595141338095240atural
    = ( ^ [X3: code_natural,Xa4: code_natural] : ( ord_less_eq_nat @ ( code_nat_of_natural @ X3 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).

% less_eq_natural.rep_eq
thf(fact_7743_division__segment__natural__def,axiom,
    ( euclid5112970336976722131atural
    = ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat @ euclid3398187327856392827nt_nat ) ) ).

% division_segment_natural_def
thf(fact_7744_euclidean__size__natural__def,axiom,
    ( euclid5200926497590043453atural
    = ( map_fu226235371393106610at_nat @ code_nat_of_natural @ id_nat @ euclid4777050414544973029ze_nat ) ) ).

% euclidean_size_natural_def
thf(fact_7745_plus__natural__def,axiom,
    ( plus_p4538020629002901425atural
    = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ plus_plus_nat ) ) ).

% plus_natural_def
thf(fact_7746_minus__natural__def,axiom,
    ( minus_7197305767214868737atural
    = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ minus_minus_nat ) ) ).

% minus_natural_def
thf(fact_7747_bit__natural__def,axiom,
    ( bit_se8040316288895769887atural
    = ( map_fu6244533035418998608_nat_o @ code_nat_of_natural @ id_nat_o @ bit_se1148574629649215175it_nat ) ) ).

% bit_natural_def
thf(fact_7748_less__natural__def,axiom,
    ( ord_le5570908160329646204atural
    = ( map_fu6256889081107267320ural_o @ code_nat_of_natural @ ( map_fu4892316939951275536at_o_o @ code_nat_of_natural @ id_o ) @ ord_less_nat ) ) ).

% less_natural_def
thf(fact_7749_times__natural__def,axiom,
    ( times_2397367101498566445atural
    = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ times_times_nat ) ) ).

% times_natural_def
thf(fact_7750_divide__natural__def,axiom,
    ( divide5121882707175180666atural
    = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ divide_divide_nat ) ) ).

% divide_natural_def
thf(fact_7751_take__bit__natural__def,axiom,
    ( bit_se569199155075624693atural
    = ( map_fu6981160461238011112atural @ id_nat @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se2925701944663578781it_nat ) ) ).

% take_bit_natural_def
thf(fact_7752_push__bit__natural__def,axiom,
    ( bit_se6611745700429515170atural
    = ( map_fu6981160461238011112atural @ id_nat @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se547839408752420682it_nat ) ) ).

% push_bit_natural_def
thf(fact_7753_less__eq__natural__def,axiom,
    ( ord_le1926595141338095240atural
    = ( map_fu6256889081107267320ural_o @ code_nat_of_natural @ ( map_fu4892316939951275536at_o_o @ code_nat_of_natural @ id_o ) @ ord_less_eq_nat ) ) ).

% less_eq_natural_def
thf(fact_7754_modulo__natural__def,axiom,
    ( modulo8411746178871703098atural
    = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ modulo_modulo_nat ) ) ).

% modulo_natural_def
thf(fact_7755_drop__bit__natural__def,axiom,
    ( bit_se2751692689150723149atural
    = ( map_fu6981160461238011112atural @ id_nat @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se8570568707652914677it_nat ) ) ).

% drop_bit_natural_def
thf(fact_7756_unset__bit__natural__def,axiom,
    ( bit_se7083795435491715335atural
    = ( map_fu6981160461238011112atural @ id_nat @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se4205575877204974255it_nat ) ) ).

% unset_bit_natural_def
thf(fact_7757_flip__bit__natural__def,axiom,
    ( bit_se168947363167071951atural
    = ( map_fu6981160461238011112atural @ id_nat @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se2161824704523386999it_nat ) ) ).

% flip_bit_natural_def
thf(fact_7758_set__bit__natural__def,axiom,
    ( bit_se1617098188084679374atural
    = ( map_fu6981160461238011112atural @ id_nat @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ bit_se7882103937844011126it_nat ) ) ).

% set_bit_natural_def
thf(fact_7759_natural__decr,axiom,
    ! [N2: code_natural] :
      ( ( N2 != zero_z2226904508553997617atural )
     => ( ord_less_nat @ ( minus_minus_nat @ ( code_nat_of_natural @ N2 ) @ ( suc @ zero_zero_nat ) ) @ ( code_nat_of_natural @ N2 ) ) ) ).

% natural_decr
thf(fact_7760_natural__of__integer__def,axiom,
    ( code_n4118661773612635043nteger
    = ( map_fu1051355602067684763atural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).

% natural_of_integer_def
thf(fact_7761_integer__of__natural__of__integer,axiom,
    ! [K: code_integer] :
      ( ( code_i5400310926305786745atural @ ( code_n4118661773612635043nteger @ K ) )
      = ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ K ) ) ).

% integer_of_natural_of_integer
thf(fact_7762_integer__of__natural_Orep__eq,axiom,
    ! [X: code_natural] :
      ( ( code_int_of_integer @ ( code_i5400310926305786745atural @ X ) )
      = ( semiri1314217659103216013at_int @ ( code_nat_of_natural @ X ) ) ) ).

% integer_of_natural.rep_eq
thf(fact_7763_int__of__integer__of__natural,axiom,
    ! [N2: code_natural] :
      ( ( code_int_of_integer @ ( code_i5400310926305786745atural @ N2 ) )
      = ( semiri1314217659103216013at_int @ ( code_nat_of_natural @ N2 ) ) ) ).

% int_of_integer_of_natural
thf(fact_7764_integer__of__natural__of__nat,axiom,
    ! [N2: nat] :
      ( ( code_i5400310926305786745atural @ ( semiri3763490453095760265atural @ N2 ) )
      = ( semiri4939895301339042750nteger @ N2 ) ) ).

% integer_of_natural_of_nat
thf(fact_7765_natural__of__integer__of__natural,axiom,
    ! [N2: code_natural] :
      ( ( code_n4118661773612635043nteger @ ( code_i5400310926305786745atural @ N2 ) )
      = N2 ) ).

% natural_of_integer_of_natural
thf(fact_7766_integer__of__natural__def,axiom,
    ( code_i5400310926305786745atural
    = ( map_fu2787874002554666395nteger @ code_nat_of_natural @ code_integer_of_int @ semiri1314217659103216013at_int ) ) ).

% integer_of_natural_def
thf(fact_7767_integer__of__natural_Oabs__eq,axiom,
    ! [X: nat] :
      ( ( code_i5400310926305786745atural @ ( code_natural_of_nat @ X ) )
      = ( code_integer_of_int @ ( semiri1314217659103216013at_int @ X ) ) ) ).

% integer_of_natural.abs_eq
thf(fact_7768_Code__Numeral_Osize__nat__def,axiom,
    code_size_nat = code_nat_of_natural ).

% Code_Numeral.size_nat_def
thf(fact_7769_Code__Numeral_OSuc__def,axiom,
    ( code_Suc
    = ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat @ suc ) ) ).

% Code_Numeral.Suc_def
thf(fact_7770_natural_Oinject,axiom,
    ! [Natural: code_natural,Natural2: code_natural] :
      ( ( ( code_Suc @ Natural )
        = ( code_Suc @ Natural2 ) )
      = ( Natural = Natural2 ) ) ).

% natural.inject
thf(fact_7771_Suc_Orep__eq,axiom,
    ! [X: code_natural] :
      ( ( code_nat_of_natural @ ( code_Suc @ X ) )
      = ( suc @ ( code_nat_of_natural @ X ) ) ) ).

% Suc.rep_eq
thf(fact_7772_natural_Oexhaust,axiom,
    ! [Y: code_natural] :
      ( ( Y != zero_z2226904508553997617atural )
     => ~ ! [Natural3: code_natural] :
            ( Y
           != ( code_Suc @ Natural3 ) ) ) ).

% natural.exhaust
thf(fact_7773_natural_Odistinct_I1_J,axiom,
    ! [Natural2: code_natural] :
      ( zero_z2226904508553997617atural
     != ( code_Suc @ Natural2 ) ) ).

% natural.distinct(1)
thf(fact_7774_natural_Odistinct_I2_J,axiom,
    ! [Natural2: code_natural] :
      ( ( code_Suc @ Natural2 )
     != zero_z2226904508553997617atural ) ).

% natural.distinct(2)
thf(fact_7775_Suc_Oabs__eq,axiom,
    ! [X: nat] :
      ( ( code_Suc @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( suc @ X ) ) ) ).

% Suc.abs_eq
thf(fact_7776_Suc__natural__minus__one,axiom,
    ! [N2: code_natural] :
      ( ( minus_7197305767214868737atural @ ( code_Suc @ N2 ) @ one_one_Code_natural )
      = N2 ) ).

% Suc_natural_minus_one
thf(fact_7777_random__aux__set_Ocases,axiom,
    ! [X: produc7822875418678951345atural] :
      ( ! [J3: code_natural] :
          ( X
         != ( produc3574140220909816553atural @ zero_z2226904508553997617atural @ J3 ) )
     => ~ ! [I2: code_natural,J3: code_natural] :
            ( X
           != ( produc3574140220909816553atural @ ( code_Suc @ I2 ) @ J3 ) ) ) ).

% random_aux_set.cases
thf(fact_7778_Code__Numeral_Osub__code_I2_J,axiom,
    ! [M2: num] :
      ( ( code_sub @ ( bit0 @ M2 ) @ one )
      = ( code_Pos @ ( bitM @ M2 ) ) ) ).

% Code_Numeral.sub_code(2)
thf(fact_7779_Code__Numeral_OPos__def,axiom,
    code_Pos = numera6620942414471956472nteger ).

% Code_Numeral.Pos_def
thf(fact_7780_Pos__fold_I2_J,axiom,
    ! [K: num] :
      ( ( numera6620942414471956472nteger @ ( bit0 @ K ) )
      = ( code_Pos @ ( bit0 @ K ) ) ) ).

% Pos_fold(2)
thf(fact_7781_Pos__fold_I1_J,axiom,
    ( ( numera6620942414471956472nteger @ one )
    = ( code_Pos @ one ) ) ).

% Pos_fold(1)
thf(fact_7782_Pos__fold_I3_J,axiom,
    ! [K: num] :
      ( ( numera6620942414471956472nteger @ ( bit1 @ K ) )
      = ( code_Pos @ ( bit1 @ K ) ) ) ).

% Pos_fold(3)
thf(fact_7783_one__integer__code,axiom,
    ( one_one_Code_integer
    = ( code_Pos @ one ) ) ).

% one_integer_code
thf(fact_7784_Code__Numeral_Odup__code_I2_J,axiom,
    ! [N2: num] :
      ( ( code_dup @ ( code_Pos @ N2 ) )
      = ( code_Pos @ ( bit0 @ N2 ) ) ) ).

% Code_Numeral.dup_code(2)
thf(fact_7785_less__integer__code_I4_J,axiom,
    ! [K: num] :
      ~ ( ord_le6747313008572928689nteger @ ( code_Pos @ K ) @ zero_z3403309356797280102nteger ) ).

% less_integer_code(4)
thf(fact_7786_less__integer__code_I2_J,axiom,
    ! [L: num] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( code_Pos @ L ) ) ).

% less_integer_code(2)
thf(fact_7787_plus__integer__code_I3_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( code_Pos @ M2 ) @ ( code_Pos @ N2 ) )
      = ( code_Pos @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% plus_integer_code(3)
thf(fact_7788_less__eq__integer__code_I2_J,axiom,
    ! [L: num] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( code_Pos @ L ) ) ).

% less_eq_integer_code(2)
thf(fact_7789_less__eq__integer__code_I4_J,axiom,
    ! [K: num] :
      ~ ( ord_le3102999989581377725nteger @ ( code_Pos @ K ) @ zero_z3403309356797280102nteger ) ).

% less_eq_integer_code(4)
thf(fact_7790_less__integer__code_I8_J,axiom,
    ! [K: num,L: num] : ( ord_le6747313008572928689nteger @ ( code_Neg @ K ) @ ( code_Pos @ L ) ) ).

% less_integer_code(8)
thf(fact_7791_less__integer__code_I6_J,axiom,
    ! [K: num,L: num] :
      ~ ( ord_le6747313008572928689nteger @ ( code_Pos @ K ) @ ( code_Neg @ L ) ) ).

% less_integer_code(6)
thf(fact_7792_uminus__integer__code_I3_J,axiom,
    ! [M2: num] :
      ( ( uminus1351360451143612070nteger @ ( code_Neg @ M2 ) )
      = ( code_Pos @ M2 ) ) ).

% uminus_integer_code(3)
thf(fact_7793_uminus__integer__code_I2_J,axiom,
    ! [M2: num] :
      ( ( uminus1351360451143612070nteger @ ( code_Pos @ M2 ) )
      = ( code_Neg @ M2 ) ) ).

% uminus_integer_code(2)
thf(fact_7794_Code__Numeral_ONeg__def,axiom,
    ( code_Neg
    = ( ^ [N4: num] : ( uminus1351360451143612070nteger @ ( code_Pos @ N4 ) ) ) ) ).

% Code_Numeral.Neg_def
thf(fact_7795_less__eq__integer__code_I8_J,axiom,
    ! [K: num,L: num] : ( ord_le3102999989581377725nteger @ ( code_Neg @ K ) @ ( code_Pos @ L ) ) ).

% less_eq_integer_code(8)
thf(fact_7796_less__eq__integer__code_I6_J,axiom,
    ! [K: num,L: num] :
      ~ ( ord_le3102999989581377725nteger @ ( code_Pos @ K ) @ ( code_Neg @ L ) ) ).

% less_eq_integer_code(6)
thf(fact_7797_times__integer__code_I3_J,axiom,
    ! [M2: num,N2: num] :
      ( ( times_3573771949741848930nteger @ ( code_Pos @ M2 ) @ ( code_Pos @ N2 ) )
      = ( code_Pos @ ( times_times_num @ M2 @ N2 ) ) ) ).

% times_integer_code(3)
thf(fact_7798_less__integer__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( ( ord_le6747313008572928689nteger @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
      = ( ord_less_num @ K @ L ) ) ).

% less_integer_code(5)
thf(fact_7799_less__eq__integer__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( ( ord_le3102999989581377725nteger @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
      = ( ord_less_eq_num @ K @ L ) ) ).

% less_eq_integer_code(5)
thf(fact_7800_minus__integer__code_I3_J,axiom,
    ! [M2: num,N2: num] :
      ( ( minus_8373710615458151222nteger @ ( code_Pos @ M2 ) @ ( code_Pos @ N2 ) )
      = ( code_sub @ M2 @ N2 ) ) ).

% minus_integer_code(3)
thf(fact_7801_divmod__abs__code_I1_J,axiom,
    ! [K: num,L: num] :
      ( ( code_divmod_abs @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
      = ( unique3479559517661332726nteger @ K @ L ) ) ).

% divmod_abs_code(1)
thf(fact_7802_minus__integer__code_I4_J,axiom,
    ! [M2: num,N2: num] :
      ( ( minus_8373710615458151222nteger @ ( code_Pos @ M2 ) @ ( code_Neg @ N2 ) )
      = ( code_Pos @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% minus_integer_code(4)
thf(fact_7803_minus__integer__code_I5_J,axiom,
    ! [M2: num,N2: num] :
      ( ( minus_8373710615458151222nteger @ ( code_Neg @ M2 ) @ ( code_Pos @ N2 ) )
      = ( code_Neg @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% minus_integer_code(5)
thf(fact_7804_times__integer__code_I4_J,axiom,
    ! [M2: num,N2: num] :
      ( ( times_3573771949741848930nteger @ ( code_Pos @ M2 ) @ ( code_Neg @ N2 ) )
      = ( code_Neg @ ( times_times_num @ M2 @ N2 ) ) ) ).

% times_integer_code(4)
thf(fact_7805_times__integer__code_I5_J,axiom,
    ! [M2: num,N2: num] :
      ( ( times_3573771949741848930nteger @ ( code_Neg @ M2 ) @ ( code_Pos @ N2 ) )
      = ( code_Neg @ ( times_times_num @ M2 @ N2 ) ) ) ).

% times_integer_code(5)
thf(fact_7806_times__integer__code_I6_J,axiom,
    ! [M2: num,N2: num] :
      ( ( times_3573771949741848930nteger @ ( code_Neg @ M2 ) @ ( code_Neg @ N2 ) )
      = ( code_Pos @ ( times_times_num @ M2 @ N2 ) ) ) ).

% times_integer_code(6)
thf(fact_7807_plus__integer__code_I4_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( code_Pos @ M2 ) @ ( code_Neg @ N2 ) )
      = ( code_sub @ M2 @ N2 ) ) ).

% plus_integer_code(4)
thf(fact_7808_plus__integer__code_I5_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( code_Neg @ M2 ) @ ( code_Pos @ N2 ) )
      = ( code_sub @ N2 @ M2 ) ) ).

% plus_integer_code(5)
thf(fact_7809_divmod__abs__code_I3_J,axiom,
    ! [K: num,L: num] :
      ( ( code_divmod_abs @ ( code_Neg @ K ) @ ( code_Pos @ L ) )
      = ( unique3479559517661332726nteger @ K @ L ) ) ).

% divmod_abs_code(3)
thf(fact_7810_divmod__abs__code_I4_J,axiom,
    ! [K: num,L: num] :
      ( ( code_divmod_abs @ ( code_Pos @ K ) @ ( code_Neg @ L ) )
      = ( unique3479559517661332726nteger @ K @ L ) ) ).

% divmod_abs_code(4)
thf(fact_7811_Code__Numeral_Osub__code_I3_J,axiom,
    ! [M2: num] :
      ( ( code_sub @ ( bit1 @ M2 ) @ one )
      = ( code_Pos @ ( bit0 @ M2 ) ) ) ).

% Code_Numeral.sub_code(3)
thf(fact_7812_normalize__stable,axiom,
    ! [Q3: int,P4: int] :
      ( ( ord_less_int @ zero_zero_int @ Q3 )
     => ( ( algebr932160517623751201me_int @ P4 @ Q3 )
       => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q3 ) )
          = ( product_Pair_int_int @ P4 @ Q3 ) ) ) ) ).

% normalize_stable
thf(fact_7813_quotient__of__coprime,axiom,
    ! [R2: rat,P4: int,Q3: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ P4 @ Q3 ) )
     => ( algebr932160517623751201me_int @ P4 @ Q3 ) ) ).

% quotient_of_coprime
thf(fact_7814_normalize__coprime,axiom,
    ! [R2: product_prod_int_int,P4: int,Q3: int] :
      ( ( ( normalize @ R2 )
        = ( product_Pair_int_int @ P4 @ Q3 ) )
     => ( algebr932160517623751201me_int @ P4 @ Q3 ) ) ).

% normalize_coprime
thf(fact_7815_coprime__common__divisor__nat,axiom,
    ! [A: nat,B: nat,X: nat] :
      ( ( algebr934650988132801477me_nat @ A @ B )
     => ( ( dvd_dvd_nat @ X @ A )
       => ( ( dvd_dvd_nat @ X @ B )
         => ( X = one_one_nat ) ) ) ) ).

% coprime_common_divisor_nat
thf(fact_7816_coprime__diff__one__right__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( algebr934650988132801477me_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).

% coprime_diff_one_right_nat
thf(fact_7817_coprime__diff__one__left__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ N2 ) ) ).

% coprime_diff_one_left_nat
thf(fact_7818_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
      ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ B72 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B62 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B52 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B42 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B32 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B22 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B1 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B0 ) ) ) ).

% integer_of_char_code
thf(fact_7819_Suc__funpow,axiom,
    ! [N2: nat] :
      ( ( compow_nat_nat @ N2 @ suc )
      = ( plus_plus_nat @ N2 ) ) ).

% Suc_funpow
thf(fact_7820_or__not__num__neg_Opelims,axiom,
    ! [X: num,Xa: num,Y: num] :
      ( ( ( bit_or_not_num_neg @ X @ Xa )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X @ Xa ) )
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = one )
               => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X = one )
             => ! [M5: num] :
                  ( ( Xa
                    = ( bit0 @ M5 ) )
                 => ( ( Y
                      = ( bit1 @ M5 ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M5 ) ) ) ) ) )
           => ( ( ( X = one )
               => ! [M5: num] :
                    ( ( Xa
                      = ( bit1 @ M5 ) )
                   => ( ( Y
                        = ( bit1 @ M5 ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M5 ) ) ) ) ) )
             => ( ! [N: num] :
                    ( ( X
                      = ( bit0 @ N ) )
                   => ( ( Xa = one )
                     => ( ( Y
                          = ( bit0 @ one ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N ) @ one ) ) ) ) )
               => ( ! [N: num] :
                      ( ( X
                        = ( bit0 @ N ) )
                     => ! [M5: num] :
                          ( ( Xa
                            = ( bit0 @ M5 ) )
                         => ( ( Y
                              = ( bitM @ ( bit_or_not_num_neg @ N @ M5 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N ) @ ( bit0 @ M5 ) ) ) ) ) )
                 => ( ! [N: num] :
                        ( ( X
                          = ( bit0 @ N ) )
                       => ! [M5: num] :
                            ( ( Xa
                              = ( bit1 @ M5 ) )
                           => ( ( Y
                                = ( bit0 @ ( bit_or_not_num_neg @ N @ M5 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N ) @ ( bit1 @ M5 ) ) ) ) ) )
                   => ( ! [N: num] :
                          ( ( X
                            = ( bit1 @ N ) )
                         => ( ( Xa = one )
                           => ( ( Y = one )
                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N ) @ one ) ) ) ) )
                     => ( ! [N: num] :
                            ( ( X
                              = ( bit1 @ N ) )
                           => ! [M5: num] :
                                ( ( Xa
                                  = ( bit0 @ M5 ) )
                               => ( ( Y
                                    = ( bitM @ ( bit_or_not_num_neg @ N @ M5 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N ) @ ( bit0 @ M5 ) ) ) ) ) )
                       => ~ ! [N: num] :
                              ( ( X
                                = ( bit1 @ N ) )
                             => ! [M5: num] :
                                  ( ( Xa
                                    = ( bit1 @ M5 ) )
                                 => ( ( Y
                                      = ( bitM @ ( bit_or_not_num_neg @ N @ M5 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N ) @ ( bit1 @ M5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
thf(fact_7821_cr__natural__def,axiom,
    ( code_cr_natural
    = ( ^ [X3: nat,Y3: code_natural] :
          ( X3
          = ( code_nat_of_natural @ Y3 ) ) ) ) ).

% cr_natural_def
thf(fact_7822_cr__integer__def,axiom,
    ( code_cr_integer
    = ( ^ [X3: int,Y3: code_integer] :
          ( X3
          = ( code_int_of_integer @ Y3 ) ) ) ) ).

% cr_integer_def
thf(fact_7823_integer_Opcr__cr__eq,axiom,
    code_pcr_integer = code_cr_integer ).

% integer.pcr_cr_eq
thf(fact_7824_unit__factor__simps_I2_J,axiom,
    ! [N2: nat] :
      ( ( unit_f2748546683901255202or_nat @ ( suc @ N2 ) )
      = one_one_nat ) ).

% unit_factor_simps(2)
thf(fact_7825_unit__factor__nat__def,axiom,
    ( unit_f2748546683901255202or_nat
    = ( ^ [N4: nat] : ( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat @ one_one_nat ) ) ) ).

% unit_factor_nat_def
thf(fact_7826_Nat_Otransfer,axiom,
    bNF_re1831474436612530402atural @ code_pcr_integer @ code_pcr_natural @ nat2 @ code_Nat ).

% Nat.transfer
thf(fact_7827_pcr__natural__def,axiom,
    ( code_pcr_natural
    = ( relcom1729257228895983867atural
      @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
      @ code_cr_natural ) ) ).

% pcr_natural_def
thf(fact_7828_natural_Orep__transfer,axiom,
    ( bNF_re7333046803109187065at_nat @ code_pcr_natural
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ^ [X3: nat] : X3
    @ code_nat_of_natural ) ).

% natural.rep_transfer
thf(fact_7829_natural_Oid__abs__transfer,axiom,
    ( bNF_re2605341659958429177atural
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ code_pcr_natural
    @ ^ [X3: nat] : X3
    @ code_natural_of_nat ) ).

% natural.id_abs_transfer
thf(fact_7830_natural_Opcr__cr__eq,axiom,
    code_pcr_natural = code_cr_natural ).

% natural.pcr_cr_eq
thf(fact_7831_zero__natural_Otransfer,axiom,
    code_pcr_natural @ zero_zero_nat @ zero_z2226904508553997617atural ).

% zero_natural.transfer
thf(fact_7832_one__natural_Otransfer,axiom,
    code_pcr_natural @ one_one_nat @ one_one_Code_natural ).

% one_natural.transfer
thf(fact_7833_less__natural_Otransfer,axiom,
    ( bNF_re1639080489988575423ural_o @ code_pcr_natural
    @ ( bNF_re2785088596696291543al_o_o @ code_pcr_natural
      @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
    @ ord_less_nat
    @ ord_le5570908160329646204atural ) ).

% less_natural.transfer
thf(fact_7834_plus__natural_Otransfer,axiom,
    bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ plus_plus_nat @ plus_p4538020629002901425atural ).

% plus_natural.transfer
thf(fact_7835_Suc_Otransfer,axiom,
    bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural @ suc @ code_Suc ).

% Suc.transfer
thf(fact_7836_minus__natural_Otransfer,axiom,
    bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ minus_minus_nat @ minus_7197305767214868737atural ).

% minus_natural.transfer
thf(fact_7837_pcr__integer__def,axiom,
    ( code_pcr_integer
    = ( relcom967497705283160040nteger
      @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
      @ code_cr_integer ) ) ).

% pcr_integer_def
thf(fact_7838_less__eq__natural_Otransfer,axiom,
    ( bNF_re1639080489988575423ural_o @ code_pcr_natural
    @ ( bNF_re2785088596696291543al_o_o @ code_pcr_natural
      @ ^ [Y4: $o,Z2: $o] : ( Y4 = Z2 ) )
    @ ord_less_eq_nat
    @ ord_le1926595141338095240atural ) ).

% less_eq_natural.transfer
thf(fact_7839_times__natural_Otransfer,axiom,
    bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ times_times_nat @ times_2397367101498566445atural ).

% times_natural.transfer
thf(fact_7840_divide__natural_Otransfer,axiom,
    bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ divide_divide_nat @ divide5121882707175180666atural ).

% divide_natural.transfer
thf(fact_7841_modulo__natural_Otransfer,axiom,
    bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ modulo_modulo_nat @ modulo8411746178871703098atural ).

% modulo_natural.transfer
thf(fact_7842_take__bit__natural_Otransfer,axiom,
    ( bNF_re3692170009638431535atural
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural )
    @ bit_se2925701944663578781it_nat
    @ bit_se569199155075624693atural ) ).

% take_bit_natural.transfer
thf(fact_7843_xor__natural_Otransfer,axiom,
    bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ bit_se6528837805403552850or_nat @ bit_se2046307713759805098atural ).

% xor_natural.transfer
thf(fact_7844_or__natural_Otransfer,axiom,
    bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ bit_se1412395901928357646or_nat @ bit_se9127793120404214118atural ).

% or_natural.transfer
thf(fact_7845_and__natural_Otransfer,axiom,
    bNF_re88643428490162567atural @ code_pcr_natural @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural ) @ bit_se727722235901077358nd_nat @ bit_se2773287842338716102atural ).

% and_natural.transfer
thf(fact_7846_bit__natural_Otransfer,axiom,
    ( bNF_re2895713572246619415_nat_o @ code_pcr_natural
    @ ^ [Y4: nat > $o,Z2: nat > $o] : ( Y4 = Z2 )
    @ bit_se1148574629649215175it_nat
    @ bit_se8040316288895769887atural ) ).

% bit_natural.transfer
thf(fact_7847_push__bit__natural_Otransfer,axiom,
    ( bNF_re3692170009638431535atural
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural )
    @ bit_se547839408752420682it_nat
    @ bit_se6611745700429515170atural ) ).

% push_bit_natural.transfer
thf(fact_7848_mask__natural_Otransfer,axiom,
    ( bNF_re2605341659958429177atural
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ code_pcr_natural
    @ bit_se2002935070580805687sk_nat
    @ bit_se943457434206027407atural ) ).

% mask_natural.transfer
thf(fact_7849_drop__bit__natural_Otransfer,axiom,
    ( bNF_re3692170009638431535atural
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural )
    @ bit_se8570568707652914677it_nat
    @ bit_se2751692689150723149atural ) ).

% drop_bit_natural.transfer
thf(fact_7850_division__segment__natural_Otransfer,axiom,
    bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural @ euclid3398187327856392827nt_nat @ euclid5112970336976722131atural ).

% division_segment_natural.transfer
thf(fact_7851_flip__bit__natural_Otransfer,axiom,
    ( bNF_re3692170009638431535atural
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural )
    @ bit_se2161824704523386999it_nat
    @ bit_se168947363167071951atural ) ).

% flip_bit_natural.transfer
thf(fact_7852_set__bit__natural_Otransfer,axiom,
    ( bNF_re3692170009638431535atural
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural )
    @ bit_se7882103937844011126it_nat
    @ bit_se1617098188084679374atural ) ).

% set_bit_natural.transfer
thf(fact_7853_unset__bit__natural_Otransfer,axiom,
    ( bNF_re3692170009638431535atural
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ ( bNF_re3704215830270325841atural @ code_pcr_natural @ code_pcr_natural )
    @ bit_se4205575877204974255it_nat
    @ bit_se7083795435491715335atural ) ).

% unset_bit_natural.transfer
thf(fact_7854_euclidean__size__natural_Otransfer,axiom,
    ( bNF_re7333046803109187065at_nat @ code_pcr_natural
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ euclid4777050414544973029ze_nat
    @ euclid5200926497590043453atural ) ).

% euclidean_size_natural.transfer
thf(fact_7855_integer__of__natural_Otransfer,axiom,
    bNF_re5252274238750452962nteger @ code_pcr_natural @ code_pcr_integer @ semiri1314217659103216013at_int @ code_i5400310926305786745atural ).

% integer_of_natural.transfer
thf(fact_7856_natural__of__integer_Otransfer,axiom,
    bNF_re1831474436612530402atural @ code_pcr_integer @ code_pcr_natural @ nat2 @ code_n4118661773612635043nteger ).

% natural_of_integer.transfer
thf(fact_7857_plus__num__def,axiom,
    ( plus_plus_num
    = ( ^ [M: num,N4: num] : ( num_of_nat @ ( plus_plus_nat @ ( nat_of_num @ M ) @ ( nat_of_num @ N4 ) ) ) ) ) ).

% plus_num_def
thf(fact_7858_Lcm__nat__empty,axiom,
    ( ( gcd_Lcm_nat @ bot_bot_set_nat )
    = one_one_nat ) ).

% Lcm_nat_empty
thf(fact_7859_nat__of__num_Osimps_I2_J,axiom,
    ! [X: num] :
      ( ( nat_of_num @ ( bit0 @ X ) )
      = ( plus_plus_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).

% nat_of_num.simps(2)
thf(fact_7860_nat__of__num__code_I1_J,axiom,
    ( ( nat_of_num @ one )
    = one_one_nat ) ).

% nat_of_num_code(1)
thf(fact_7861_nat__of__num__add,axiom,
    ! [X: num,Y: num] :
      ( ( nat_of_num @ ( plus_plus_num @ X @ Y ) )
      = ( plus_plus_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).

% nat_of_num_add
thf(fact_7862_nat__of__num__code_I2_J,axiom,
    ! [N2: num] :
      ( ( nat_of_num @ ( bit0 @ N2 ) )
      = ( plus_plus_nat @ ( nat_of_num @ N2 ) @ ( nat_of_num @ N2 ) ) ) ).

% nat_of_num_code(2)
thf(fact_7863_nat__of__num_Osimps_I3_J,axiom,
    ! [X: num] :
      ( ( nat_of_num @ ( bit1 @ X ) )
      = ( suc @ ( plus_plus_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ) ).

% nat_of_num.simps(3)
thf(fact_7864_nat__of__num__code_I3_J,axiom,
    ! [N2: num] :
      ( ( nat_of_num @ ( bit1 @ N2 ) )
      = ( suc @ ( plus_plus_nat @ ( nat_of_num @ N2 ) @ ( nat_of_num @ N2 ) ) ) ) ).

% nat_of_num_code(3)
thf(fact_7865_Gcd__nat__eq__one,axiom,
    ! [N6: set_nat] :
      ( ( member_nat @ one_one_nat @ N6 )
     => ( ( gcd_Gcd_nat @ N6 )
        = one_one_nat ) ) ).

% Gcd_nat_eq_one
thf(fact_7866_card__of__Field__natLeq,axiom,
    member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) ) @ bNF_Ca8665028551170535155natLeq ) @ bNF_We5258908940166488438at_nat ).

% card_of_Field_natLeq
thf(fact_7867_card__of__nat,axiom,
    member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ top_top_set_nat ) @ bNF_Ca8665028551170535155natLeq ) @ bNF_We5258908940166488438at_nat ).

% card_of_nat
thf(fact_7868_ctwo__Cnotzero,axiom,
    ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ bNF_Cardinal_ctwo @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
    & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ bNF_Cardinal_ctwo ) @ bNF_Cardinal_ctwo ) ) ).

% ctwo_Cnotzero
thf(fact_7869_ctwo__ordLess__natLeq,axiom,
    member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ bNF_Cardinal_ctwo @ bNF_Ca8665028551170535155natLeq ) @ bNF_We8182288985678559134_o_nat ).

% ctwo_ordLess_natLeq
thf(fact_7870_or__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: num] :
      ( ( ( bit_un6697907153464112080or_num @ X @ Xa )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ X @ Xa ) )
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y = one )
               => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X = one )
             => ! [N: num] :
                  ( ( Xa
                    = ( bit0 @ N ) )
                 => ( ( Y
                      = ( bit1 @ N ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N ) ) ) ) ) )
           => ( ( ( X = one )
               => ! [N: num] :
                    ( ( Xa
                      = ( bit1 @ N ) )
                   => ( ( Y
                        = ( bit1 @ N ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N ) ) ) ) ) )
             => ( ! [M5: num] :
                    ( ( X
                      = ( bit0 @ M5 ) )
                   => ( ( Xa = one )
                     => ( ( Y
                          = ( bit1 @ M5 ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X
                        = ( bit0 @ M5 ) )
                     => ! [N: num] :
                          ( ( Xa
                            = ( bit0 @ N ) )
                         => ( ( Y
                              = ( bit0 @ ( bit_un6697907153464112080or_num @ M5 @ N ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N ) ) ) ) ) )
                 => ( ! [M5: num] :
                        ( ( X
                          = ( bit0 @ M5 ) )
                       => ! [N: num] :
                            ( ( Xa
                              = ( bit1 @ N ) )
                           => ( ( Y
                                = ( bit1 @ ( bit_un6697907153464112080or_num @ M5 @ N ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N ) ) ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X
                            = ( bit1 @ M5 ) )
                         => ( ( Xa = one )
                           => ( ( Y
                                = ( bit1 @ M5 ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
                     => ( ! [M5: num] :
                            ( ( X
                              = ( bit1 @ M5 ) )
                           => ! [N: num] :
                                ( ( Xa
                                  = ( bit0 @ N ) )
                               => ( ( Y
                                    = ( bit1 @ ( bit_un6697907153464112080or_num @ M5 @ N ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N ) ) ) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X
                                = ( bit1 @ M5 ) )
                             => ! [N: num] :
                                  ( ( Xa
                                    = ( bit1 @ N ) )
                                 => ( ( Y
                                      = ( bit1 @ ( bit_un6697907153464112080or_num @ M5 @ N ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_un4773296044027857193um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_num.pelims
thf(fact_7871_Quotient__natural,axiom,
    ( quotie743640454666494480atural
    @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
    @ code_natural_of_nat
    @ code_nat_of_natural
    @ code_cr_natural ) ).

% Quotient_natural
thf(fact_7872_Quotient__integer,axiom,
    ( quotie2291698863146621601nteger
    @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 )
    @ code_integer_of_int
    @ code_int_of_integer
    @ code_cr_integer ) ).

% Quotient_integer
thf(fact_7873_Lcm__nat__def,axiom,
    ( gcd_Lcm_nat
    = ( ^ [M7: set_nat] : ( if_nat @ ( finite_finite_nat @ M7 ) @ ( lattic7826324295020591184_F_nat @ gcd_lcm_nat @ one_one_nat @ M7 ) @ zero_zero_nat ) ) ) ).

% Lcm_nat_def
thf(fact_7874_Lcm__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] :
      ( ( gcd_Lcm_nat @ ( set_nat2 @ Xs ) )
      = ( fold_nat_nat @ gcd_lcm_nat @ Xs @ one_one_nat ) ) ).

% Lcm_nat_set_eq_fold
thf(fact_7875_mod__h__bot__iff_I7_J,axiom,
    ! [P: assn,Q: assn,H3: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( sup_sup_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) )
        | ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(7)
thf(fact_7876_mod__h__bot__iff_I5_J,axiom,
    ! [P: assn,Q: assn,H3: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( times_times_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) )
        & ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(5)
thf(fact_7877_mod__h__bot__iff_I6_J,axiom,
    ! [P: assn,Q: assn,H3: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( inf_inf_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) )
        & ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(6)
thf(fact_7878_mod__h__bot__iff_I2_J,axiom,
    ! [H3: heap_e7401611519738050253t_unit] : ( rep_assn @ top_top_assn @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) ) ).

% mod_h_bot_iff(2)
thf(fact_7879_times__assn__raw_Ocases,axiom,
    ! [X: produc2732055786443039994et_nat] :
      ~ ! [P7: produc3658429121746597890et_nat > $o,Q6: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As: set_nat] :
          ( X
         != ( produc2245416461498447860et_nat @ P7 @ ( produc5001842942810119800et_nat @ Q6 @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ) ).

% times_assn_raw.cases
thf(fact_7880_one__assn__raw_Ocases,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
          ( X
         != ( produc7507926704131184380et_nat @ H2 @ As ) ) ).

% one_assn_raw.cases
thf(fact_7881_mod__star__conv,axiom,
    ! [A2: assn,B5: assn,H3: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A2 @ B5 ) @ H3 )
      = ( ? [Hr: heap_e7401611519738050253t_unit,As1: set_nat,As2: set_nat] :
            ( ( H3
              = ( produc7507926704131184380et_nat @ Hr @ ( sup_sup_set_nat @ As1 @ As2 ) ) )
            & ( ( inf_inf_set_nat @ As1 @ As2 )
              = bot_bot_set_nat )
            & ( rep_assn @ A2 @ ( produc7507926704131184380et_nat @ Hr @ As1 ) )
            & ( rep_assn @ B5 @ ( produc7507926704131184380et_nat @ Hr @ As2 ) ) ) ) ) ).

% mod_star_conv
thf(fact_7882_star__assnI,axiom,
    ! [P: assn,H3: heap_e7401611519738050253t_unit,As3: set_nat,Q: assn,As4: set_nat] :
      ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
     => ( ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H3 @ As4 ) )
       => ( ( ( inf_inf_set_nat @ As3 @ As4 )
            = bot_bot_set_nat )
         => ( rep_assn @ ( times_times_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H3 @ ( sup_sup_set_nat @ As3 @ As4 ) ) ) ) ) ) ).

% star_assnI
thf(fact_7883_mod__emp__simp,axiom,
    ! [H3: heap_e7401611519738050253t_unit] : ( rep_assn @ one_one_assn @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) ) ).

% mod_emp_simp
thf(fact_7884_mod__h__bot__indep,axiom,
    ! [P: assn,H3: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) )
      = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H5 @ bot_bot_set_nat ) ) ) ).

% mod_h_bot_indep
thf(fact_7885_times__assn__raw_Osimps,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o,H3: heap_e7401611519738050253t_unit,As3: set_nat] :
      ( ( times_assn_raw @ P @ Q @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
      = ( ? [As1: set_nat,As2: set_nat] :
            ( ( As3
              = ( sup_sup_set_nat @ As1 @ As2 ) )
            & ( ( inf_inf_set_nat @ As1 @ As2 )
              = bot_bot_set_nat )
            & ( P @ ( produc7507926704131184380et_nat @ H3 @ As1 ) )
            & ( Q @ ( produc7507926704131184380et_nat @ H3 @ As2 ) ) ) ) ) ).

% times_assn_raw.simps
thf(fact_7886_times__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( times_assn_raw @ X @ Xa @ Xb )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H2 @ As ) )
           => ? [As12: set_nat,As22: set_nat] :
                ( ( As
                  = ( sup_sup_set_nat @ As12 @ As22 ) )
                & ( ( inf_inf_set_nat @ As12 @ As22 )
                  = bot_bot_set_nat )
                & ( X @ ( produc7507926704131184380et_nat @ H2 @ As12 ) )
                & ( Xa @ ( produc7507926704131184380et_nat @ H2 @ As22 ) ) ) ) ) ).

% times_assn_raw.elims(3)
thf(fact_7887_times__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( times_assn_raw @ X @ Xa @ Xb )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H2 @ As ) )
           => ~ ? [As13: set_nat,As23: set_nat] :
                  ( ( As
                    = ( sup_sup_set_nat @ As13 @ As23 ) )
                  & ( ( inf_inf_set_nat @ As13 @ As23 )
                    = bot_bot_set_nat )
                  & ( X @ ( produc7507926704131184380et_nat @ H2 @ As13 ) )
                  & ( Xa @ ( produc7507926704131184380et_nat @ H2 @ As23 ) ) ) ) ) ).

% times_assn_raw.elims(2)
thf(fact_7888_times__assn__raw_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( times_assn_raw @ X @ Xa @ Xb )
        = Y )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H2 @ As ) )
           => ( Y
              = ( ~ ? [As1: set_nat,As2: set_nat] :
                      ( ( As
                        = ( sup_sup_set_nat @ As1 @ As2 ) )
                      & ( ( inf_inf_set_nat @ As1 @ As2 )
                        = bot_bot_set_nat )
                      & ( X @ ( produc7507926704131184380et_nat @ H2 @ As1 ) )
                      & ( Xa @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) ) ) ) ) ).

% times_assn_raw.elims(1)
thf(fact_7889_times__assn__raw_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( times_assn_raw @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H2 @ As ) )
             => ( ( Y
                  = ( ? [As1: set_nat,As2: set_nat] :
                        ( ( As
                          = ( sup_sup_set_nat @ As1 @ As2 ) )
                        & ( ( inf_inf_set_nat @ As1 @ As2 )
                          = bot_bot_set_nat )
                        & ( X @ ( produc7507926704131184380et_nat @ H2 @ As1 ) )
                        & ( Xa @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) ) )
               => ~ ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(1)
thf(fact_7890_times__assn__raw_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( times_assn_raw @ X @ Xa @ Xb )
     => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H2 @ As ) )
             => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) )
               => ~ ? [As13: set_nat,As23: set_nat] :
                      ( ( As
                        = ( sup_sup_set_nat @ As13 @ As23 ) )
                      & ( ( inf_inf_set_nat @ As13 @ As23 )
                        = bot_bot_set_nat )
                      & ( X @ ( produc7507926704131184380et_nat @ H2 @ As13 ) )
                      & ( Xa @ ( produc7507926704131184380et_nat @ H2 @ As23 ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
thf(fact_7891_times__assn__raw_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( times_assn_raw @ X @ Xa @ Xb )
     => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H2 @ As ) )
             => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) )
               => ? [As12: set_nat,As22: set_nat] :
                    ( ( As
                      = ( sup_sup_set_nat @ As12 @ As22 ) )
                    & ( ( inf_inf_set_nat @ As12 @ As22 )
                      = bot_bot_set_nat )
                    & ( X @ ( produc7507926704131184380et_nat @ H2 @ As12 ) )
                    & ( Xa @ ( produc7507926704131184380et_nat @ H2 @ As22 ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
thf(fact_7892_mod__h__bot__iff_I1_J,axiom,
    ! [B: $o,H3: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( pure_assn @ B ) @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) )
      = B ) ).

% mod_h_bot_iff(1)
thf(fact_7893_mod__relH,axiom,
    ! [As3: set_nat,H3: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,P: assn] :
      ( ( relH @ As3 @ H3 @ H5 )
     => ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
        = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H5 @ As3 ) ) ) ) ).

% mod_relH
thf(fact_7894_in__range__empty,axiom,
    ! [H3: heap_e7401611519738050253t_unit] : ( in_range @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) ) ).

% in_range_empty
thf(fact_7895_in__range__dist__union,axiom,
    ! [H3: heap_e7401611519738050253t_unit,As3: set_nat,As4: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ ( sup_sup_set_nat @ As3 @ As4 ) ) )
      = ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
        & ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As4 ) ) ) ) ).

% in_range_dist_union
thf(fact_7896_relH__refl,axiom,
    ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
     => ( relH @ As3 @ H3 @ H3 ) ) ).

% relH_refl
thf(fact_7897_relH__in__rangeI_I1_J,axiom,
    ! [As3: set_nat,H3: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( relH @ As3 @ H3 @ H5 )
     => ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As3 ) ) ) ).

% relH_in_rangeI(1)
thf(fact_7898_relH__in__rangeI_I2_J,axiom,
    ! [As3: set_nat,H3: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( relH @ As3 @ H3 @ H5 )
     => ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As3 ) ) ) ).

% relH_in_rangeI(2)
thf(fact_7899_in__range__subset,axiom,
    ! [As3: set_nat,As4: set_nat,H3: heap_e7401611519738050253t_unit] :
      ( ( ord_less_eq_set_nat @ As3 @ As4 )
     => ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As4 ) )
       => ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As3 ) ) ) ) ).

% in_range_subset
thf(fact_7900_in__range_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( in_range @ X )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H2 @ As ) )
           => ! [X4: nat] :
                ( ( member_nat @ X4 @ As )
               => ( ord_less_nat @ X4 @ ( lim_Product_unit @ H2 ) ) ) ) ) ).

% in_range.elims(3)
thf(fact_7901_in__range_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( in_range @ X )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H2 @ As ) )
           => ~ ! [X8: nat] :
                  ( ( member_nat @ X8 @ As )
                 => ( ord_less_nat @ X8 @ ( lim_Product_unit @ H2 ) ) ) ) ) ).

% in_range.elims(2)
thf(fact_7902_in__range_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( in_range @ X )
        = Y )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H2 @ As ) )
           => ( Y
              = ( ~ ! [X3: nat] :
                      ( ( member_nat @ X3 @ As )
                     => ( ord_less_nat @ X3 @ ( lim_Product_unit @ H2 ) ) ) ) ) ) ) ).

% in_range.elims(1)
thf(fact_7903_in__range_Osimps,axiom,
    ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ As3 )
           => ( ord_less_nat @ X3 @ ( lim_Product_unit @ H3 ) ) ) ) ) ).

% in_range.simps
thf(fact_7904_wand__assnI,axiom,
    ! [H3: heap_e7401611519738050253t_unit,As3: set_nat,Q: assn,R4: assn] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
     => ( ! [H4: heap_e7401611519738050253t_unit,As5: set_nat] :
            ( ( ( inf_inf_set_nat @ As3 @ As5 )
              = bot_bot_set_nat )
           => ( ( relH @ As3 @ H3 @ H4 )
             => ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As3 ) )
               => ( ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H4 @ As5 ) )
                 => ( rep_assn @ R4 @ ( produc7507926704131184380et_nat @ H4 @ ( sup_sup_set_nat @ As3 @ As5 ) ) ) ) ) ) )
       => ( rep_assn @ ( wand_assn @ Q @ R4 ) @ ( produc7507926704131184380et_nat @ H3 @ As3 ) ) ) ) ).

% wand_assnI
thf(fact_7905_wand__raw_Osimps,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o,H3: heap_e7401611519738050253t_unit,As3: set_nat] :
      ( ( wand_raw @ P @ Q @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
      = ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
        & ! [H6: heap_e7401611519738050253t_unit,As6: set_nat] :
            ( ( ( ( inf_inf_set_nat @ As3 @ As6 )
                = bot_bot_set_nat )
              & ( relH @ As3 @ H3 @ H6 )
              & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As3 ) )
              & ( P @ ( produc7507926704131184380et_nat @ H6 @ As6 ) ) )
           => ( Q @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As3 @ As6 ) ) ) ) ) ) ).

% wand_raw.simps
thf(fact_7906_wand__raw_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( wand_raw @ X @ Xa @ Xb )
        = Y )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H2 @ As ) )
           => ( Y
              = ( ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
                    & ! [H6: heap_e7401611519738050253t_unit,As6: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As @ As6 )
                            = bot_bot_set_nat )
                          & ( relH @ As @ H2 @ H6 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H6 @ As6 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As @ As6 ) ) ) ) ) ) ) ) ) ).

% wand_raw.elims(1)
thf(fact_7907_wand__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( wand_raw @ X @ Xa @ Xb )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H2 @ As ) )
           => ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
              & ! [H4: heap_e7401611519738050253t_unit,As5: set_nat] :
                  ( ( ( ( inf_inf_set_nat @ As @ As5 )
                      = bot_bot_set_nat )
                    & ( relH @ As @ H2 @ H4 )
                    & ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As ) )
                    & ( X @ ( produc7507926704131184380et_nat @ H4 @ As5 ) ) )
                 => ( Xa @ ( produc7507926704131184380et_nat @ H4 @ ( sup_sup_set_nat @ As @ As5 ) ) ) ) ) ) ) ).

% wand_raw.elims(3)
thf(fact_7908_wand__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( wand_raw @ X @ Xa @ Xb )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H2 @ As ) )
           => ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
                & ! [H8: heap_e7401611519738050253t_unit,As7: set_nat] :
                    ( ( ( ( inf_inf_set_nat @ As @ As7 )
                        = bot_bot_set_nat )
                      & ( relH @ As @ H2 @ H8 )
                      & ( in_range @ ( produc7507926704131184380et_nat @ H8 @ As ) )
                      & ( X @ ( produc7507926704131184380et_nat @ H8 @ As7 ) ) )
                   => ( Xa @ ( produc7507926704131184380et_nat @ H8 @ ( sup_sup_set_nat @ As @ As7 ) ) ) ) ) ) ) ).

% wand_raw.elims(2)
thf(fact_7909_wand__raw_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( wand_raw @ X @ Xa @ Xb )
     => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H2 @ As ) )
             => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) )
               => ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
                  & ! [H4: heap_e7401611519738050253t_unit,As5: set_nat] :
                      ( ( ( ( inf_inf_set_nat @ As @ As5 )
                          = bot_bot_set_nat )
                        & ( relH @ As @ H2 @ H4 )
                        & ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As ) )
                        & ( X @ ( produc7507926704131184380et_nat @ H4 @ As5 ) ) )
                     => ( Xa @ ( produc7507926704131184380et_nat @ H4 @ ( sup_sup_set_nat @ As @ As5 ) ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(3)
thf(fact_7910_wand__raw_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( wand_raw @ X @ Xa @ Xb )
     => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H2 @ As ) )
             => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) )
               => ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
                    & ! [H8: heap_e7401611519738050253t_unit,As7: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As @ As7 )
                            = bot_bot_set_nat )
                          & ( relH @ As @ H2 @ H8 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H8 @ As ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H8 @ As7 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H8 @ ( sup_sup_set_nat @ As @ As7 ) ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(2)
thf(fact_7911_wand__raw_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( wand_raw @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H2 @ As ) )
             => ( ( Y
                  = ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) )
                    & ! [H6: heap_e7401611519738050253t_unit,As6: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As @ As6 )
                            = bot_bot_set_nat )
                          & ( relH @ As @ H2 @ H6 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H6 @ As6 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As @ As6 ) ) ) ) ) )
               => ~ ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(1)
thf(fact_7912_in__range_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( in_range @ X )
     => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H2 @ As ) )
             => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H2 @ As ) )
               => ! [X4: nat] :
                    ( ( member_nat @ X4 @ As )
                   => ( ord_less_nat @ X4 @ ( lim_Product_unit @ H2 ) ) ) ) ) ) ) ).

% in_range.pelims(3)
thf(fact_7913_in__range_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( in_range @ X )
     => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H2 @ As ) )
             => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H2 @ As ) )
               => ~ ! [X8: nat] :
                      ( ( member_nat @ X8 @ As )
                     => ( ord_less_nat @ X8 @ ( lim_Product_unit @ H2 ) ) ) ) ) ) ) ).

% in_range.pelims(2)
thf(fact_7914_in__range_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( in_range @ X )
        = Y )
     => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H2 @ As ) )
             => ( ( Y
                  = ( ! [X3: nat] :
                        ( ( member_nat @ X3 @ As )
                       => ( ord_less_nat @ X3 @ ( lim_Product_unit @ H2 ) ) ) ) )
               => ~ ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ) ) ) ).

% in_range.pelims(1)
thf(fact_7915_one__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( one_assn_raw @ X )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H2 @ As ) )
           => ( As = bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(3)
thf(fact_7916_one__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( one_assn_raw @ X )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H2 @ As ) )
           => ( As != bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(2)
thf(fact_7917_one__assn__raw_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( one_assn_raw @ X )
        = Y )
     => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H2 @ As ) )
           => ( Y
              = ( As != bot_bot_set_nat ) ) ) ) ).

% one_assn_raw.elims(1)
thf(fact_7918_one__assn__raw_Osimps,axiom,
    ! [H3: heap_e7401611519738050253t_unit,As3: set_nat] :
      ( ( one_assn_raw @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
      = ( As3 = bot_bot_set_nat ) ) ).

% one_assn_raw.simps
thf(fact_7919_one__assn__raw_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( one_assn_raw @ X )
     => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H2 @ As ) )
             => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H2 @ As ) )
               => ( As = bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(3)
thf(fact_7920_one__assn__raw_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( one_assn_raw @ X )
     => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H2 @ As ) )
             => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H2 @ As ) )
               => ( As != bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(2)
thf(fact_7921_one__assn__raw_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( one_assn_raw @ X )
        = Y )
     => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
       => ~ ! [H2: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H2 @ As ) )
             => ( ( Y
                  = ( As = bot_bot_set_nat ) )
               => ~ ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H2 @ As ) ) ) ) ) ) ).

% one_assn_raw.pelims(1)
thf(fact_7922_proper__iff,axiom,
    ! [P: produc3658429121746597890et_nat > $o,As3: set_nat,H3: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( proper @ P )
     => ( ( relH @ As3 @ H3 @ H5 )
       => ( ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As3 ) )
         => ( ( P @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
            = ( P @ ( produc7507926704131184380et_nat @ H5 @ As3 ) ) ) ) ) ) ).

% proper_iff
thf(fact_7923_proper__def,axiom,
    ( proper
    = ( ^ [P3: produc3658429121746597890et_nat > $o] :
        ! [H: heap_e7401611519738050253t_unit,H6: heap_e7401611519738050253t_unit,As8: set_nat] :
          ( ( ( P3 @ ( produc7507926704131184380et_nat @ H @ As8 ) )
           => ( in_range @ ( produc7507926704131184380et_nat @ H @ As8 ) ) )
          & ( ( ( P3 @ ( produc7507926704131184380et_nat @ H @ As8 ) )
              & ( relH @ As8 @ H @ H6 )
              & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As8 ) ) )
           => ( P3 @ ( produc7507926704131184380et_nat @ H6 @ As8 ) ) ) ) ) ) ).

% proper_def
thf(fact_7924_properD2,axiom,
    ! [P: produc3658429121746597890et_nat > $o,H3: heap_e7401611519738050253t_unit,As3: set_nat,H5: heap_e7401611519738050253t_unit] :
      ( ( proper @ P )
     => ( ( P @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
       => ( ( relH @ As3 @ H3 @ H5 )
         => ( ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As3 ) )
           => ( P @ ( produc7507926704131184380et_nat @ H5 @ As3 ) ) ) ) ) ) ).

% properD2
thf(fact_7925_properD1,axiom,
    ! [P: produc3658429121746597890et_nat > $o,H3: heap_e7401611519738050253t_unit,As3: set_nat] :
      ( ( proper @ P )
     => ( ( P @ ( produc7507926704131184380et_nat @ H3 @ As3 ) )
       => ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As3 ) ) ) ) ).

% properD1
thf(fact_7926_properI,axiom,
    ! [P: produc3658429121746597890et_nat > $o] :
      ( ! [As: set_nat,H2: heap_e7401611519738050253t_unit] :
          ( ( P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
         => ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As ) ) )
     => ( ! [As: set_nat,H2: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
            ( ( P @ ( produc7507926704131184380et_nat @ H2 @ As ) )
           => ( ( relH @ As @ H2 @ H4 )
             => ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As ) )
               => ( P @ ( produc7507926704131184380et_nat @ H4 @ As ) ) ) ) )
       => ( proper @ P ) ) ) ).

% properI
thf(fact_7927_Int_Osub__code_I8_J,axiom,
    ! [M2: num,N2: num] :
      ( ( sub @ ( bit1 @ M2 ) @ ( bit0 @ N2 ) )
      = ( plus_plus_int @ ( dup @ ( sub @ M2 @ N2 ) ) @ one_one_int ) ) ).

% Int.sub_code(8)
thf(fact_7928_Int_Odup__def,axiom,
    ( dup
    = ( ^ [K4: int] : ( plus_plus_int @ K4 @ K4 ) ) ) ).

% Int.dup_def
thf(fact_7929_relH__def,axiom,
    ( relH
    = ( ^ [As8: set_nat,H: heap_e7401611519738050253t_unit,H6: heap_e7401611519738050253t_unit] :
          ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As8 ) )
          & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As8 ) )
          & ! [T2: typerep,X3: nat] :
              ( ( member_nat @ X3 @ As8 )
             => ( ( ( refs_Product_unit @ H @ T2 @ X3 )
                  = ( refs_Product_unit @ H6 @ T2 @ X3 ) )
                & ( ( arrays_Product_unit @ H @ T2 @ X3 )
                  = ( arrays_Product_unit @ H6 @ T2 @ X3 ) ) ) ) ) ) ) ).

% relH_def
thf(fact_7930_plus__int__code_I6_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_int @ ( neg @ M2 ) @ ( neg @ N2 ) )
      = ( neg @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% plus_int_code(6)
thf(fact_7931_plus__int__code_I5_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_int @ ( neg @ M2 ) @ ( pos @ N2 ) )
      = ( sub @ N2 @ M2 ) ) ).

% plus_int_code(5)
thf(fact_7932_plus__int__code_I4_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_int @ ( pos @ M2 ) @ ( neg @ N2 ) )
      = ( sub @ M2 @ N2 ) ) ).

% plus_int_code(4)
thf(fact_7933_minus__int__code_I4_J,axiom,
    ! [M2: num,N2: num] :
      ( ( minus_minus_int @ ( pos @ M2 ) @ ( neg @ N2 ) )
      = ( pos @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% minus_int_code(4)
thf(fact_7934_plus__int__code_I3_J,axiom,
    ! [M2: num,N2: num] :
      ( ( plus_plus_int @ ( pos @ M2 ) @ ( pos @ N2 ) )
      = ( pos @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% plus_int_code(3)
thf(fact_7935_minus__int__code_I5_J,axiom,
    ! [M2: num,N2: num] :
      ( ( minus_minus_int @ ( neg @ M2 ) @ ( pos @ N2 ) )
      = ( neg @ ( plus_plus_num @ M2 @ N2 ) ) ) ).

% minus_int_code(5)
thf(fact_7936_Unity__def,axiom,
    ( product_Unity
    = ( product_Abs_unit @ $true ) ) ).

% Unity_def
thf(fact_7937_Abs__unit__inject,axiom,
    ! [X: $o,Y: $o] :
      ( ( member_o @ X @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
       => ( ( ( product_Abs_unit @ X )
            = ( product_Abs_unit @ Y ) )
          = ( X = Y ) ) ) ) ).

% Abs_unit_inject
thf(fact_7938_Abs__unit__induct,axiom,
    ! [P: product_unit > $o,X: product_unit] :
      ( ! [Y5: $o] :
          ( ( member_o @ Y5 @ ( insert_o @ $true @ bot_bot_set_o ) )
         => ( P @ ( product_Abs_unit @ Y5 ) ) )
     => ( P @ X ) ) ).

% Abs_unit_induct
thf(fact_7939_Abs__unit__cases,axiom,
    ! [X: product_unit] :
      ~ ! [Y5: $o] :
          ( ( X
            = ( product_Abs_unit @ Y5 ) )
         => ~ ( member_o @ Y5 @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% Abs_unit_cases
thf(fact_7940_integer_Odomain,axiom,
    ! [P: int > $o] :
      ( ( ( domainp_int_int
          @ ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
        = P )
     => ( ( domain452967906960104687nteger @ code_pcr_integer )
        = P ) ) ).

% integer.domain
thf(fact_7941_natural_Odomain,axiom,
    ! [P: nat > $o] :
      ( ( ( domainp_nat_nat
          @ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
        = P )
     => ( ( domain8128281535334753374atural @ code_pcr_natural )
        = P ) ) ).

% natural.domain
thf(fact_7942_Abs__unit__inverse,axiom,
    ! [Y: $o] :
      ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ( product_Rep_unit @ ( product_Abs_unit @ Y ) )
        = Y ) ) ).

% Abs_unit_inverse
thf(fact_7943_type__definition__unit,axiom,
    type_d6188575255521822967unit_o @ product_Rep_unit @ product_Abs_unit @ ( insert_o @ $true @ bot_bot_set_o ) ).

% type_definition_unit
thf(fact_7944_Rep__unit__inverse,axiom,
    ! [X: product_unit] :
      ( ( product_Abs_unit @ ( product_Rep_unit @ X ) )
      = X ) ).

% Rep_unit_inverse
thf(fact_7945_Rep__unit__inject,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( ( product_Rep_unit @ X )
        = ( product_Rep_unit @ Y ) )
      = ( X = Y ) ) ).

% Rep_unit_inject
thf(fact_7946_Rep__unit,axiom,
    ! [X: product_unit] : ( member_o @ ( product_Rep_unit @ X ) @ ( insert_o @ $true @ bot_bot_set_o ) ) ).

% Rep_unit
thf(fact_7947_Rep__unit__cases,axiom,
    ! [Y: $o] :
      ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ~ ! [X4: product_unit] :
            ( Y
            = ( ~ ( product_Rep_unit @ X4 ) ) ) ) ).

% Rep_unit_cases
thf(fact_7948_Rep__unit__induct,axiom,
    ! [Y: $o,P: $o > $o] :
      ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ! [X4: product_unit] : ( P @ ( product_Rep_unit @ X4 ) )
       => ( P @ Y ) ) ) ).

% Rep_unit_induct
thf(fact_7949_positive__add,axiom,
    ! [X: rat,Y: rat] :
      ( ( positive @ X )
     => ( ( positive @ Y )
       => ( positive @ ( plus_plus_rat @ X @ Y ) ) ) ) ).

% positive_add
thf(fact_7950_nth__item_Opinduct,axiom,
    ! [A0: nat,P: nat > $o] :
      ( ( accp_nat @ nth_item_rel @ A0 )
     => ( ( ( accp_nat @ nth_item_rel @ zero_zero_nat )
         => ( P @ zero_zero_nat ) )
       => ( ! [N: nat] :
              ( ( accp_nat @ nth_item_rel @ ( suc @ N ) )
             => ( ! [A9: nat,Aa2: nat] :
                    ( ( ( nat_sum_decode @ N )
                      = ( sum_Inl_nat_nat @ A9 ) )
                   => ( ( ( nat_sum_decode @ A9 )
                        = ( sum_Inl_nat_nat @ Aa2 ) )
                     => ( P @ Aa2 ) ) )
               => ( ! [A9: nat,B7: nat] :
                      ( ( ( nat_sum_decode @ N )
                        = ( sum_Inl_nat_nat @ A9 ) )
                     => ( ( ( nat_sum_decode @ A9 )
                          = ( sum_Inr_nat_nat @ B7 ) )
                       => ( P @ B7 ) ) )
                 => ( ! [B7: nat,Ba: nat,X8: nat,Y6: nat] :
                        ( ( ( nat_sum_decode @ N )
                          = ( sum_Inr_nat_nat @ B7 ) )
                       => ( ( ( nat_sum_decode @ B7 )
                            = ( sum_Inr_nat_nat @ Ba ) )
                         => ( ( ( product_Pair_nat_nat @ X8 @ Y6 )
                              = ( nat_prod_decode @ Ba ) )
                           => ( P @ X8 ) ) ) )
                   => ( ! [B7: nat,Ba: nat,X8: nat,Y6: nat] :
                          ( ( ( nat_sum_decode @ N )
                            = ( sum_Inr_nat_nat @ B7 ) )
                         => ( ( ( nat_sum_decode @ B7 )
                              = ( sum_Inr_nat_nat @ Ba ) )
                           => ( ( ( product_Pair_nat_nat @ X8 @ Y6 )
                                = ( nat_prod_decode @ Ba ) )
                             => ( P @ Y6 ) ) ) )
                     => ( P @ ( suc @ N ) ) ) ) ) ) )
         => ( P @ A0 ) ) ) ) ).

% nth_item.pinduct

% Helper facts (51)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Assertions__Oassn_T,axiom,
    ! [X: assn,Y: assn] :
      ( ( if_assn @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Assertions__Oassn_T,axiom,
    ! [X: assn,Y: assn] :
      ( ( if_assn @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Onatural_T,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( if_Code_natural @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Onatural_T,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( if_Code_natural @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( if_set_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( if_set_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ( if_option_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ( if_option_num @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Set__Oset_I_Eo_J_J_T,axiom,
    ! [X: option_set_o,Y: option_set_o] :
      ( ( if_option_set_o @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Set__Oset_I_Eo_J_J_T,axiom,
    ! [X: option_set_o,Y: option_set_o] :
      ( ( if_option_set_o @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Product____Type__Ounit_J_T,axiom,
    ! [X: option_Product_unit,Y: option_Product_unit] :
      ( ( if_opt6227717841141446553t_unit @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Product____Type__Ounit_J_T,axiom,
    ! [X: option_Product_unit,Y: option_Product_unit] :
      ( ( if_opt6227717841141446553t_unit @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Set__Oset_It__Int__Oint_J_J_T,axiom,
    ! [X: option_set_int,Y: option_set_int] :
      ( ( if_option_set_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Set__Oset_It__Int__Oint_J_J_T,axiom,
    ! [X: option_set_int,Y: option_set_int] :
      ( ( if_option_set_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
    ! [X: option_set_nat,Y: option_set_nat] :
      ( ( if_option_set_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J_T,axiom,
    ! [X: option_set_nat,Y: option_set_nat] :
      ( ( if_option_set_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__Heap__Oarray_Itf__a_J_J_T,axiom,
    ! [X: heap_T6362127782908069795rray_a,Y: heap_T6362127782908069795rray_a] :
      ( ( if_Hea5896630067203373929rray_a @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__Heap__Oarray_Itf__a_J_J_T,axiom,
    ! [X: heap_T6362127782908069795rray_a,Y: heap_T6362127782908069795rray_a] :
      ( ( if_Hea5896630067203373929rray_a @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J_T,axiom,
    ! [X: heap_T5738788834812785303t_unit,Y: heap_T5738788834812785303t_unit] :
      ( ( if_Hea8138950348631371857t_unit @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J_T,axiom,
    ! [X: heap_T5738788834812785303t_unit,Y: heap_T5738788834812785303t_unit] :
      ( ( if_Hea8138950348631371857t_unit @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: multis2468970476368604999at_nat,Y: multis2468970476368604999at_nat] :
      ( ( if_mul8430962117462786573at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: multis2468970476368604999at_nat,Y: multis2468970476368604999at_nat] :
      ( ( if_mul8430962117462786573at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: option8963830502488799655at_nat,Y: option8963830502488799655at_nat] :
      ( ( if_opt7704869406773131885at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: option8963830502488799655at_nat,Y: option8963830502488799655at_nat] :
      ( ( if_opt7704869406773131885at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Product____Type__Oprod_It__Heap__Oarray_Itf__a_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: option3216581401566546698it_nat,Y: option3216581401566546698it_nat] :
      ( ( if_opt7017092727090204112it_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Product____Type__Oprod_It__Heap__Oarray_Itf__a_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: option3216581401566546698it_nat,Y: option3216581401566546698it_nat] :
      ( ( if_opt7017092727090204112it_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: option8956607266484857688it_nat,Y: option8956607266484857688it_nat] :
      ( ( if_opt1729522071442692626it_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X: option8956607266484857688it_nat,Y: option8956607266484857688it_nat] :
      ( ( if_opt1729522071442692626it_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_T,axiom,
    ! [X: option1583680563626158861at_nat,Y: option1583680563626158861at_nat] :
      ( ( if_opt5121585740920863443at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_T,axiom,
    ! [X: option1583680563626158861at_nat,Y: option1583680563626158861at_nat] :
      ( ( if_opt5121585740920863443at_nat @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( heap_T5591663578999710211rray_a @ ( array_of_list_a @ xs ) @ h )
   != none_P2587165478634041381it_nat ) ).

%------------------------------------------------------------------------------