%------------------------------------------------------------------------------
% File     : ITP215^3 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem Automation 00192_006252
% 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    : 0027_Automation_00192_006252 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 6335 (3386 unt; 887 typ;   0 def)
%            Number of atoms       : 12630 (6409 equ;   2 cnn)
%            Maximal formula atoms :   11 (   2 avg)
%            Number of connectives : 47430 (1200   ~; 192   |; 645   &;41507   @)
%                                         (   0 <=>;3886  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   5 avg)
%            Number of types       :   90 (  89 usr)
%            Number of type conns  : 2792 (2792   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  802 ( 798 usr;  52 con; 0-6 aty)
%            Number of variables   : 12463 ( 619   ^;11736   !; 108   ?;12463   :)
% 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 15:59:58.963
%------------------------------------------------------------------------------
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% Explicit typings (798)
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
    archim3151403230148437115or_rat: rat > int ).

thf(sy_c_Assertions_Oassn_OAbs__assn,type,
    abs_assn: ( produc3658429121746597890et_nat > $o ) > assn ).

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

thf(sy_c_Assertions_Oentails,type,
    entails: assn > assn > $o ).

thf(sy_c_Assertions_Oentailst,type,
    entailst: assn > assn > $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_Ois__pure__assn,type,
    is_pure_assn: assn > $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_Opure__assn__raw_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Nat__Onat,type,
    pure_a825153325127701367it_nat: $o > produc3658429121746597890et_nat > $o ).

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_Automation_OFI,type,
    fi: list_P8527749157015355191n_assn > assn > assn > assn > assn > assn > $o ).

thf(sy_c_Automation_OFI__QUERY,type,
    fI_QUERY: assn > assn > assn > $o ).

thf(sy_c_Automation_OFI__RESULT,type,
    fI_RESULT: list_P8527749157015355191n_assn > assn > assn > assn > $o ).

thf(sy_c_Automation_OSLN,type,
    sln: assn ).

thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

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_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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
    bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $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_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_Basic__BNFs_Opred__prod_001t__Int__Oint_001t__Int__Oint,type,
    basic_1567116559311922317nt_int: ( int > $o ) > ( int > $o ) > product_prod_int_int > $o ).

thf(sy_c_Basic__BNFs_Orel__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    basic_4387203522000727145nt_int: ( int > int > $o ) > ( int > int > $o ) > product_prod_int_int > product_prod_int_int > $o ).

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

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_Oor__not__num__neg,type,
    bit_or_not_num_neg: num > num > num ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Code____Numeral__Ointeger,type,
    bit_ri7632146776885996613nteger: code_integer > code_integer ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: int > int ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
    bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
    bit_ri631733984087533419it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
    bit_se3949692690581998587nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Onatural,type,
    bit_se2773287842338716102atural: code_natural > code_natural > code_natural ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
    bit_se725231765392027082nd_int: int > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Code____Numeral__Ointeger,type,
    bit_se3928097537394005634nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Code____Numeral__Onatural,type,
    bit_se2751692689150723149atural: nat > code_natural > code_natural ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
    bit_se8568078237143864401it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
    bit_se8570568707652914677it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
    bit_se1345352211410354436nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Onatural,type,
    bit_se168947363167071951atural: nat > code_natural > code_natural ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
    bit_se2159334234014336723it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
    bit_se2161824704523386999it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Code____Numeral__Ointeger,type,
    bit_se2119862282449309892nteger: nat > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Code____Numeral__Onatural,type,
    bit_se943457434206027407atural: nat > code_natural ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
    bit_se2000444600071755411sk_int: nat > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
    bit_se2002935070580805687sk_nat: nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Code____Numeral__Ointeger,type,
    bit_se1080825931792720795nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Code____Numeral__Onatural,type,
    bit_se9127793120404214118atural: code_natural > code_natural > code_natural ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
    bit_se1409905431419307370or_int: int > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
    bit_se1412395901928357646or_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Code____Numeral__Ointeger,type,
    bit_se7788150548672797655nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Code____Numeral__Onatural,type,
    bit_se6611745700429515170atural: nat > code_natural > code_natural ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
    bit_se545348938243370406it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
    bit_se547839408752420682it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
    bit_se2793503036327961859nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Onatural,type,
    bit_se1617098188084679374atural: nat > code_natural > code_natural ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
    bit_se7879613467334960850it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
    bit_se7882103937844011126it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Code____Numeral__Ointeger,type,
    bit_se1745604003318907178nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Code____Numeral__Onatural,type,
    bit_se569199155075624693atural: nat > code_natural > code_natural ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
    bit_se2923211474154528505it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
    bit_se2925701944663578781it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
    bit_se8260200283734997820nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Onatural,type,
    bit_se7083795435491715335atural: nat > code_natural > code_natural ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
    bit_se4203085406695923979it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
    bit_se4205575877204974255it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Code____Numeral__Ointeger,type,
    bit_se3222712562003087583nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Code____Numeral__Onatural,type,
    bit_se2046307713759805098atural: code_natural > code_natural > code_natural ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
    bit_se6526347334894502574or_int: int > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
    bit_se6528837805403552850or_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Code____Numeral__Ointeger,type,
    bit_se9216721137139052372nteger: code_integer > nat > $o ).

thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Code____Numeral__Onatural,type,
    bit_se8040316288895769887atural: code_natural > nat > $o ).

thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
    bit_se1146084159140164899it_int: int > nat > $o ).

thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
    bit_se1148574629649215175it_nat: nat > nat > $o ).

thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    boolea1849928419393854478_int_o: ( ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > product_prod_int_int > $o ) > ( ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > product_prod_int_int > $o ) > ( ( product_prod_int_int > $o ) > product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).

thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Assertions__Oassn,type,
    boolea5875918072483028530a_assn: ( assn > assn > assn ) > ( assn > assn > assn ) > ( assn > assn ) > assn > assn > $o ).

thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Code____Numeral__Ointeger,type,
    boolea5643492148547655015nteger: ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer ) > code_integer > code_integer > $o ).

thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Int__Oint,type,
    boolea5469654029284665142ra_int: ( int > int > int ) > ( int > int > int ) > ( int > int ) > int > int > $o ).

thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Set__Oset_It__Int__Oint_J,type,
    boolea5824373010784320748et_int: ( set_int > set_int > set_int ) > ( set_int > set_int > set_int ) > ( set_int > set_int ) > set_int > set_int > $o ).

thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Set__Oset_It__Nat__Onat_J,type,
    boolea778851993438741648et_nat: ( set_nat > set_nat > set_nat ) > ( set_nat > set_nat > set_nat ) > ( set_nat > set_nat ) > set_nat > set_nat > $o ).

thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    boolea949959532490082519at_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    boolea3191171876098928227t_unit: ( set_Product_unit > set_Product_unit > set_Product_unit ) > ( set_Product_unit > set_Product_unit > set_Product_unit ) > ( set_Product_unit > set_Product_unit ) > set_Product_unit > set_Product_unit > $o ).

thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff_001t__Code____Numeral__Ointeger,type,
    boolea2445317508997433345nteger: ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer > code_integer ) > ( code_integer > code_integer ) > code_integer > code_integer > ( code_integer > code_integer > code_integer ) > $o ).

thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff_001t__Int__Oint,type,
    boolea8527374999097803216ff_int: ( int > int > int ) > ( int > int > int ) > ( int > int ) > int > int > ( int > int > int ) > $o ).

thf(sy_c_Code__Numeral_OPos,type,
    code_Pos: num > code_integer ).

thf(sy_c_Code__Numeral_OSuc,type,
    code_Suc: code_natural > code_natural ).

thf(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).

thf(sy_c_Code__Numeral_Odivmod__abs,type,
    code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).

thf(sy_c_Code__Numeral_Odivmod__integer,type,
    code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).

thf(sy_c_Code__Numeral_Odup,type,
    code_dup: code_integer > code_integer ).

thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
    code_int_of_integer: code_integer > int ).

thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: int > code_integer ).

thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
    code_integer_of_nat: nat > code_integer ).

thf(sy_c_Code__Numeral_Ointeger__of__num,type,
    code_integer_of_num: num > code_integer ).

thf(sy_c_Code__Numeral_Onat__of__integer,type,
    code_nat_of_integer: code_integer > nat ).

thf(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
    code_nat_of_natural: code_natural > nat ).

thf(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
    code_natural_of_nat: nat > code_natural ).

thf(sy_c_Code__Numeral_Onum__of__integer,type,
    code_num_of_integer: code_integer > num ).

thf(sy_c_Code__Numeral_Opcr__integer,type,
    code_pcr_integer: int > code_integer > $o ).

thf(sy_c_Code__Numeral_Opcr__natural,type,
    code_pcr_natural: nat > code_natural > $o ).

thf(sy_c_Code__Numeral_Osub,type,
    code_sub: num > num > code_integer ).

thf(sy_c_Divides_Oadjust__div,type,
    adjust_div: product_prod_int_int > int ).

thf(sy_c_Divides_Oadjust__mod,type,
    adjust_mod: int > int > int ).

thf(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: int > int > product_prod_int_int > $o ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
    unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).

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    groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).

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    groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).

thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
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thf(sy_c_HOL_ONO__MATCH_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_HOL_ONO__MATCH_001t__Code____Numeral__Onatural_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_HOL_ONO__MATCH_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_HOL_ONO__MATCH_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_HOL_ONO__MATCH_001t__Int__Oint_001t__Rat__Orat,type,
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thf(sy_c_HOL_ONO__MATCH_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_HOL_ONO__MATCH_001t__Nat__Onat_001t__Code____Numeral__Onatural,type,
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thf(sy_c_HOL_ONO__MATCH_001t__Nat__Onat_001t__Int__Oint,type,
    nO_MATCH_nat_int: nat > int > $o ).

thf(sy_c_HOL_ONO__MATCH_001t__Nat__Onat_001t__Nat__Onat,type,
    nO_MATCH_nat_nat: nat > nat > $o ).

thf(sy_c_HOL_ONO__MATCH_001t__Nat__Onat_001t__Rat__Orat,type,
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thf(sy_c_HOL_ONO__MATCH_001t__Rat__Orat_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_HOL_ONO__MATCH_001t__Rat__Orat_001t__Int__Oint,type,
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thf(sy_c_HOL_ONO__MATCH_001t__Rat__Orat_001t__Rat__Orat,type,
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thf(sy_c_HOL_OThe_001t__Int__Oint,type,
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thf(sy_c_Heap_Oheap_Oarrays_001t__Product____Type__Ounit,type,
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thf(sy_c_Heap_Oheap_Olim_001t__Product____Type__Ounit,type,
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thf(sy_c_Heap_Oheap_Orefs_001t__Product____Type__Ounit,type,
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thf(sy_c_Heap__Time__Monad_Owait,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Product____Type__Ounit,type,
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thf(sy_c_If_001t__Code____Numeral__Onatural,type,
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thf(sy_c_If_001t__Int__Oint,type,
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thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Num__Onum,type,
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thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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thf(sy_c_If_001t__Rat__Orat,type,
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thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Int_OAbs__Integ,type,
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thf(sy_c_Int_OPos,type,
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thf(sy_c_Int_Odup,type,
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thf(sy_c_Int_Ointrel,type,
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thf(sy_c_Int_Onat,type,
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thf(sy_c_Int_Opcr__int,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
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thf(sy_c_Int_Osub,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_Eo,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Assertions__Oassn,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Product____Type__Ounit,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Rat__Orat,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Osemilattice_001t__Assertions__Oassn,type,
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thf(sy_c_Lattices_Osemilattice_001t__Int__Oint,type,
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thf(sy_c_Lattices_Osemilattice_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Osemilattice_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Osemilattice__neutr_001t__Assertions__Oassn,type,
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thf(sy_c_Lattices_Osemilattice__neutr_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Assertions__Oassn,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__Int__Oint,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Rat__Orat,type,
    sup_sup_rat: rat > rat > rat ).

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__Nat__Onat_J,type,
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thf(sy_c_Lattices__Big_Osemilattice__neutr__set_OF_001t__Nat__Onat,type,
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thf(sy_c_List_Oall__interval__int,type,
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thf(sy_c_List_Oall__interval__nat,type,
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thf(sy_c_List_Oappend_001t__Int__Oint,type,
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thf(sy_c_List_Oappend_001t__Nat__Onat,type,
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thf(sy_c_List_Obind_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_List_Obind_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_List_Obind_001t__Int__Oint_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
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thf(sy_c_List_Obind_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
    bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).

thf(sy_c_List_Obind_001t__Nat__Onat_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
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thf(sy_c_List_Obind_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001t__Int__Oint,type,
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thf(sy_c_List_Obind_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001t__Nat__Onat,type,
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thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
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thf(sy_c_List_Ocoset_001t__Int__Oint,type,
    coset_int: list_int > set_int ).

thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
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thf(sy_c_List_Ocoset_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
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thf(sy_c_List_Ocoset_001t__Product____Type__Ounit,type,
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thf(sy_c_List_Ocount__list_001t__Int__Oint,type,
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thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
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thf(sy_c_List_Ocount__list_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
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thf(sy_c_List_Odistinct_001t__Int__Oint,type,
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thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
    distinct_nat: list_nat > $o ).

thf(sy_c_List_Odrop_001t__Nat__Onat,type,
    drop_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Oenumerate_001t__Int__Oint,type,
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thf(sy_c_List_Oenumerate_001t__Nat__Onat,type,
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thf(sy_c_List_Oenumerate_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
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thf(sy_c_List_Ofold_001t__Int__Oint_001t__Int__Oint,type,
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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_Ofoldr_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
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thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Omeasures_001t__Int__Oint,type,
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thf(sy_c_List_Omeasures_001t__Nat__Onat,type,
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thf(sy_c_List_Omember_001t__Int__Oint,type,
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thf(sy_c_List_On__lists_001t__Int__Oint,type,
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thf(sy_c_List_On__lists_001t__Nat__Onat,type,
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thf(sy_c_List_Onths_001t__Int__Oint,type,
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thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_List_Oproduct__lists_001t__Int__Oint,type,
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thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
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thf(sy_c_List_Oremdups_001t__Nat__Onat,type,
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thf(sy_c_List_Orotate1_001t__Assertions__Oassn,type,
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thf(sy_c_List_Orotate1_001t__Int__Oint,type,
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thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
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thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
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thf(sy_c_List_Osplice_001t__Int__Oint,type,
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thf(sy_c_List_Osplice_001t__Nat__Onat,type,
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thf(sy_c_List_Osubseqs_001t__Int__Oint,type,
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thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
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thf(sy_c_List_Otake_001t__Nat__Onat,type,
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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__aux,type,
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thf(sy_c_List_Oupto__rel,type,
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thf(sy_c_List_Ozip_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,
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thf(sy_c_List_Ozip_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
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thf(sy_c_List_Ozip_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Misc_Olist__collect__set_001t__Assertions__Oassn_001t__Int__Oint,type,
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thf(sy_c_Misc_Olist__collect__set_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_Rat_Opcr__rat,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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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
    image_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(sy_c_Set_Oinsert_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,
    insert5175938949040314269_nat_o: ( produc3658429121746597890et_nat > $o ) > set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o ).

thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
    insert_int2: int > set_int > set_int ).

thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat2: nat > set_nat > set_nat ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
    set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Code____Numeral__Ointeger,type,
    set_or189985376899183464nteger: code_integer > code_integer > set_Code_integer ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
    set_or1266510415728281911st_int: int > int > set_int ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
    set_or1269000886237332187st_nat: nat > nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Code____Numeral__Ointeger,type,
    set_or8404916559141939852nteger: code_integer > code_integer > set_Code_integer ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
    set_or4662586982721622107an_int: int > int > set_int ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
    set_or4665077453230672383an_nat: nat > nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Code____Numeral__Ointeger,type,
    set_or2715278749043346189nteger: code_integer > code_integer > set_Code_integer ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
    set_or6656581121297822940st_int: int > int > set_int ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
    set_or6659071591806873216st_nat: nat > nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Code____Numeral__Ointeger,type,
    set_or4266950643985792945nteger: code_integer > code_integer > set_Code_integer ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
    set_or5832277885323065728an_int: int > int > set_int ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
    set_or5834768355832116004an_nat: nat > nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: nat > set_nat ).

thf(sy_c_Syntax__Match_Oac__operator_001t__Assertions__Oassn,type,
    syntax3673046151410588037r_assn: ( assn > assn > assn ) > $o ).

thf(sy_c_Syntax__Match_Oac__operator_001t__Code____Numeral__Ointeger,type,
    syntax414191934074104506nteger: ( code_integer > code_integer > code_integer ) > $o ).

thf(sy_c_Syntax__Match_Oac__operator_001t__Code____Numeral__Onatural,type,
    syntax8461159122685597829atural: ( code_natural > code_natural > code_natural ) > $o ).

thf(sy_c_Syntax__Match_Oac__operator_001t__Int__Oint,type,
    syntax8798048597141670025or_int: ( int > int > int ) > $o ).

thf(sy_c_Syntax__Match_Oac__operator_001t__Nat__Onat,type,
    syntax8800539067650720301or_nat: ( nat > nat > nat ) > $o ).

thf(sy_c_Syntax__Match_Oac__operator_001t__Rat__Orat,type,
    syntax8165409007564224565or_rat: ( rat > rat > rat ) > $o ).

thf(sy_c_Syntax__Match_Osyntax__fo__nomatch_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
    syntax7398250324933576852n_assn: assn > assn > $o ).

thf(sy_c_Syntax__Match_Osyntax__fo__nomatch_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    syntax380677951125133566nteger: code_integer > code_integer > $o ).

thf(sy_c_Syntax__Match_Osyntax__fo__nomatch_001t__Code____Numeral__Onatural_001t__Code____Numeral__Onatural,type,
    syntax2868745204708187284atural: code_natural > code_natural > $o ).

thf(sy_c_Syntax__Match_Osyntax__fo__nomatch_001t__Int__Oint_001t__Int__Oint,type,
    syntax5678989248478167196nt_int: int > int > $o ).

thf(sy_c_Syntax__Match_Osyntax__fo__nomatch_001t__Nat__Onat_001t__Nat__Onat,type,
    syntax4682126007086162916at_nat: nat > nat > $o ).

thf(sy_c_Syntax__Match_Osyntax__fo__nomatch_001t__Rat__Orat_001t__Rat__Orat,type,
    syntax3730441303064801268at_rat: rat > rat > $o ).

thf(sy_c_Typedef_Otype__definition_001t__Assertions__Oassn_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,
    type_d3909072315231072503_nat_o: ( assn > produc3658429121746597890et_nat > $o ) > ( ( produc3658429121746597890et_nat > $o ) > assn ) > set_Pr4532377907799695533_nat_o > $o ).

thf(sy_c_Wellfounded_Oaccp_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,
    accp_P1862375125659990705et_nat: ( produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o ) > produc2732055786443039994et_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    accp_P8126237942716283194atural: ( produc7822875418678951345atural > produc7822875418678951345atural > $o ) > produc7822875418678951345atural > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    accp_P5801069581201407417et_nat: ( produc3658429121746597890et_nat > produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_member_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,
    member6576561426505652726_nat_o: ( produc3658429121746597890et_nat > $o ) > set_Pr4532377907799695533_nat_o > $o ).

thf(sy_c_member_001t__Int__Oint,type,
    member_int2: int > set_int > $o ).

thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
    member_list_int: list_int > set_list_int > $o ).

thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
    member_list_nat: list_nat > set_list_nat > $o ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat2: nat > set_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    member7957490590177025114n_assn: produc6575502325842934193n_assn > set_Pr5949110396991348497n_assn > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_v_F,type,
    f: assn ).

thf(sy_v_P,type,
    p: assn ).

thf(sy_v_Q,type,
    q: assn ).

% Relevant facts (5418)
thf(fact_0_assms,axiom,
    fi @ nil_Pr5671120429643327159n_assn @ ( times_times_assn @ sln @ p ) @ ( times_times_assn @ sln @ q ) @ sln @ sln @ f ).

% assms
thf(fact_1_FI__QUERY__def,axiom,
    ( fI_QUERY
    = ( ^ [P: assn,Q: assn,F: assn] : ( entails @ P @ ( times_times_assn @ Q @ F ) ) ) ) ).

% FI_QUERY_def
thf(fact_2_assn__aci_I10_J,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C )
      = ( times_times_assn @ ( times_times_assn @ A @ C ) @ B ) ) ).

% assn_aci(10)
thf(fact_3_star__aci_I3_J,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( times_times_assn @ A @ ( times_times_assn @ B @ C ) )
      = ( times_times_assn @ B @ ( times_times_assn @ A @ C ) ) ) ).

% star_aci(3)
thf(fact_4_star__aci_I2_J,axiom,
    ( times_times_assn
    = ( ^ [A2: assn,B2: assn] : ( times_times_assn @ B2 @ A2 ) ) ) ).

% star_aci(2)
thf(fact_5_star__assoc,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 ) ) ) ).

% star_assoc
thf(fact_6_SLN__left,axiom,
    ! [P2: assn] :
      ( ( times_times_assn @ sln @ P2 )
      = P2 ) ).

% SLN_left
thf(fact_7_SLN__right,axiom,
    ! [P2: assn] :
      ( ( times_times_assn @ P2 @ sln )
      = P2 ) ).

% SLN_right
thf(fact_8_is__entails,axiom,
    ! [P2: assn,Q2: assn] :
      ( ( entails @ P2 @ Q2 )
     => ( entails @ P2 @ Q2 ) ) ).

% is_entails
thf(fact_9_ent__star__mono,axiom,
    ! [P2: assn,P3: assn,Q2: assn,Q3: assn] :
      ( ( entails @ P2 @ P3 )
     => ( ( entails @ Q2 @ Q3 )
       => ( entails @ ( times_times_assn @ P2 @ Q2 ) @ ( times_times_assn @ P3 @ Q3 ) ) ) ) ).

% ent_star_mono
thf(fact_10_ent__iffI,axiom,
    ! [A3: assn,B3: assn] :
      ( ( entails @ A3 @ B3 )
     => ( ( entails @ B3 @ A3 )
       => ( A3 = B3 ) ) ) ).

% ent_iffI
thf(fact_11_ent__refl,axiom,
    ! [P2: assn] : ( entails @ P2 @ P2 ) ).

% ent_refl
thf(fact_12_ent__trans,axiom,
    ! [P2: assn,Q2: assn,R: assn] :
      ( ( entails @ P2 @ Q2 )
     => ( ( entails @ Q2 @ R )
       => ( entails @ P2 @ R ) ) ) ).

% ent_trans
thf(fact_13_assn__times__comm,axiom,
    ( times_times_assn
    = ( ^ [P: assn,Q: assn] : ( times_times_assn @ Q @ P ) ) ) ).

% assn_times_comm
thf(fact_14_assn__times__assoc,axiom,
    ! [P2: assn,Q2: assn,R: assn] :
      ( ( times_times_assn @ ( times_times_assn @ P2 @ Q2 ) @ R )
      = ( times_times_assn @ P2 @ ( times_times_assn @ Q2 @ R ) ) ) ).

% assn_times_assoc
thf(fact_15_mult_Oright__assoc,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.right_assoc
thf(fact_16_mult_Oright__assoc,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.right_assoc
thf(fact_17_mult_Oright__assoc,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.right_assoc
thf(fact_18_mult_Oright__assoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
      = ( times_3573771949741848930nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% mult.right_assoc
thf(fact_19_mult_Oright__assoc,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( times_2397367101498566445atural @ ( times_2397367101498566445atural @ A @ B ) @ C )
      = ( times_2397367101498566445atural @ A @ ( times_2397367101498566445atural @ B @ C ) ) ) ).

% mult.right_assoc
thf(fact_20_mult_Oright__commute,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C )
      = ( times_times_assn @ ( times_times_assn @ A @ C ) @ B ) ) ).

% mult.right_commute
thf(fact_21_mult_Oright__commute,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ ( times_times_nat @ A @ C ) @ B ) ) ).

% mult.right_commute
thf(fact_22_mult_Oright__commute,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ ( times_times_int @ A @ C ) @ B ) ) ).

% mult.right_commute
thf(fact_23_mult_Oright__commute,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ).

% mult.right_commute
thf(fact_24_mult_Oright__commute,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( times_2397367101498566445atural @ ( times_2397367101498566445atural @ A @ B ) @ C )
      = ( times_2397367101498566445atural @ ( times_2397367101498566445atural @ A @ C ) @ B ) ) ).

% mult.right_commute
thf(fact_25_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_26_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_27_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_28_mult_Oassoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
      = ( times_3573771949741848930nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% mult.assoc
thf(fact_29_mult_Oassoc,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( times_2397367101498566445atural @ ( times_2397367101498566445atural @ A @ B ) @ C )
      = ( times_2397367101498566445atural @ A @ ( times_2397367101498566445atural @ B @ C ) ) ) ).

% mult.assoc
thf(fact_30_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_assn
    = ( ^ [A2: assn,B2: assn] : ( times_times_assn @ B2 @ A2 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_31_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_32_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A2: int,B2: int] : ( times_times_int @ B2 @ A2 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_33_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_3573771949741848930nteger
    = ( ^ [A2: code_integer,B2: code_integer] : ( times_3573771949741848930nteger @ B2 @ A2 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_34_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_2397367101498566445atural
    = ( ^ [A2: code_natural,B2: code_natural] : ( times_2397367101498566445atural @ B2 @ A2 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_35_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_36_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_37_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_38_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ B @ ( times_3573771949741848930nteger @ A @ C ) )
      = ( times_3573771949741848930nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_39_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( times_2397367101498566445atural @ B @ ( times_2397367101498566445atural @ A @ C ) )
      = ( times_2397367101498566445atural @ A @ ( times_2397367101498566445atural @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_40_ent__wandI,axiom,
    ! [Q2: assn,P2: assn,R: assn] :
      ( ( entails @ ( times_times_assn @ Q2 @ P2 ) @ R )
     => ( entails @ P2 @ ( wand_assn @ Q2 @ R ) ) ) ).

% ent_wandI
thf(fact_41_ent__mp,axiom,
    ! [P2: assn,Q2: assn] : ( entails @ ( times_times_assn @ P2 @ ( wand_assn @ P2 @ Q2 ) ) @ Q2 ) ).

% ent_mp
thf(fact_42_list__ex1__simps_I1_J,axiom,
    ! [P2: produc6575502325842934193n_assn > $o] :
      ~ ( list_e7761433933450087034n_assn @ P2 @ nil_Pr5671120429643327159n_assn ) ).

% list_ex1_simps(1)
thf(fact_43_list__ex1__simps_I1_J,axiom,
    ! [P2: int > $o] :
      ~ ( list_ex1_int @ P2 @ nil_int ) ).

% list_ex1_simps(1)
thf(fact_44_list__ex1__simps_I1_J,axiom,
    ! [P2: nat > $o] :
      ~ ( list_ex1_nat @ P2 @ nil_nat ) ).

% list_ex1_simps(1)
thf(fact_45_product_Osimps_I1_J,axiom,
    ! [Uu: list_assn] :
      ( ( product_assn_assn @ nil_assn @ Uu )
      = nil_Pr5671120429643327159n_assn ) ).

% product.simps(1)
thf(fact_46_is__pure__assn__starI,axiom,
    ! [A: assn,B: assn] :
      ( ( is_pure_assn @ A )
     => ( ( is_pure_assn @ B )
       => ( is_pure_assn @ ( times_times_assn @ A @ B ) ) ) ) ).

% is_pure_assn_starI
thf(fact_47_bind__simps_I1_J,axiom,
    ! [F2: produc6575502325842934193n_assn > list_P8527749157015355191n_assn] :
      ( ( bind_P5664457578022171847n_assn @ nil_Pr5671120429643327159n_assn @ F2 )
      = nil_Pr5671120429643327159n_assn ) ).

% bind_simps(1)
thf(fact_48_bind__simps_I1_J,axiom,
    ! [F2: produc6575502325842934193n_assn > list_int] :
      ( ( bind_P8081679045764635286sn_int @ nil_Pr5671120429643327159n_assn @ F2 )
      = nil_int ) ).

% bind_simps(1)
thf(fact_49_bind__simps_I1_J,axiom,
    ! [F2: produc6575502325842934193n_assn > list_nat] :
      ( ( bind_P8084169516273685562sn_nat @ nil_Pr5671120429643327159n_assn @ F2 )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_50_bind__simps_I1_J,axiom,
    ! [F2: int > list_P8527749157015355191n_assn] :
      ( ( bind_i4941608653131063484n_assn @ nil_int @ F2 )
      = nil_Pr5671120429643327159n_assn ) ).

% bind_simps(1)
thf(fact_51_bind__simps_I1_J,axiom,
    ! [F2: int > list_int] :
      ( ( bind_int_int @ nil_int @ F2 )
      = nil_int ) ).

% bind_simps(1)
thf(fact_52_bind__simps_I1_J,axiom,
    ! [F2: int > list_nat] :
      ( ( bind_int_nat @ nil_int @ F2 )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_53_bind__simps_I1_J,axiom,
    ! [F2: nat > list_P8527749157015355191n_assn] :
      ( ( bind_n3181571989377952280n_assn @ nil_nat @ F2 )
      = nil_Pr5671120429643327159n_assn ) ).

% bind_simps(1)
thf(fact_54_bind__simps_I1_J,axiom,
    ! [F2: nat > list_int] :
      ( ( bind_nat_int @ nil_nat @ F2 )
      = nil_int ) ).

% bind_simps(1)
thf(fact_55_bind__simps_I1_J,axiom,
    ! [F2: nat > list_nat] :
      ( ( bind_nat_nat @ nil_nat @ F2 )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_56_member__rec_I2_J,axiom,
    ! [Y: produc6575502325842934193n_assn] :
      ~ ( member743271071679453132n_assn @ nil_Pr5671120429643327159n_assn @ Y ) ).

% member_rec(2)
thf(fact_57_member__rec_I2_J,axiom,
    ! [Y: int] :
      ~ ( member_int @ nil_int @ Y ) ).

% member_rec(2)
thf(fact_58_member__rec_I2_J,axiom,
    ! [Y: nat] :
      ~ ( member_nat @ nil_nat @ Y ) ).

% member_rec(2)
thf(fact_59_mult_Osafe__commute,axiom,
    ! [X: nat,Y: nat,A: nat,B: nat] :
      ( ( syntax4682126007086162916at_nat @ ( times_times_nat @ X @ Y ) @ A )
     => ( ( times_times_nat @ A @ B )
        = ( times_times_nat @ B @ A ) ) ) ).

% mult.safe_commute
thf(fact_60_mult_Osafe__commute,axiom,
    ! [X: int,Y: int,A: int,B: int] :
      ( ( syntax5678989248478167196nt_int @ ( times_times_int @ X @ Y ) @ A )
     => ( ( times_times_int @ A @ B )
        = ( times_times_int @ B @ A ) ) ) ).

% mult.safe_commute
thf(fact_61_mult_Osafe__commute,axiom,
    ! [X: code_integer,Y: code_integer,A: code_integer,B: code_integer] :
      ( ( syntax380677951125133566nteger @ ( times_3573771949741848930nteger @ X @ Y ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ B )
        = ( times_3573771949741848930nteger @ B @ A ) ) ) ).

% mult.safe_commute
thf(fact_62_mult_Osafe__commute,axiom,
    ! [X: code_natural,Y: code_natural,A: code_natural,B: code_natural] :
      ( ( syntax2868745204708187284atural @ ( times_2397367101498566445atural @ X @ Y ) @ A )
     => ( ( times_2397367101498566445atural @ A @ B )
        = ( times_2397367101498566445atural @ B @ A ) ) ) ).

% mult.safe_commute
thf(fact_63_mult_Osafe__commute,axiom,
    ! [X: assn,Y: assn,A: assn,B: assn] :
      ( ( syntax7398250324933576852n_assn @ ( times_times_assn @ X @ Y ) @ A )
     => ( ( times_times_assn @ A @ B )
        = ( times_times_assn @ B @ A ) ) ) ).

% mult.safe_commute
thf(fact_64_assn__aci_I11_J,axiom,
    ! [X: assn,Y: assn,A: assn,B: assn] :
      ( ( syntax7398250324933576852n_assn @ ( times_times_assn @ X @ Y ) @ A )
     => ( ( times_times_assn @ A @ B )
        = ( times_times_assn @ B @ A ) ) ) ).

% assn_aci(11)
thf(fact_65_ent__pure__pre__iff,axiom,
    ! [P2: assn,B: $o,Q2: assn] :
      ( ( entails @ ( times_times_assn @ P2 @ ( pure_assn @ B ) ) @ Q2 )
      = ( B
       => ( entails @ P2 @ Q2 ) ) ) ).

% ent_pure_pre_iff
thf(fact_66_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_le1182329135810803268n_assn @ N @ nil_Pr5671120429643327159n_assn )
      = N ) ).

% gen_length_code(1)
thf(fact_67_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_int @ N @ nil_int )
      = N ) ).

% gen_length_code(1)
thf(fact_68_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_nat @ N @ nil_nat )
      = N ) ).

% gen_length_code(1)
thf(fact_69_maps__simps_I2_J,axiom,
    ! [F2: produc6575502325842934193n_assn > list_P8527749157015355191n_assn] :
      ( ( maps_P4274076103917212289n_assn @ F2 @ nil_Pr5671120429643327159n_assn )
      = nil_Pr5671120429643327159n_assn ) ).

% maps_simps(2)
thf(fact_70_maps__simps_I2_J,axiom,
    ! [F2: produc6575502325842934193n_assn > list_int] :
      ( ( maps_P5983552678929597788sn_int @ F2 @ nil_Pr5671120429643327159n_assn )
      = nil_int ) ).

% maps_simps(2)
thf(fact_71_maps__simps_I2_J,axiom,
    ! [F2: produc6575502325842934193n_assn > list_nat] :
      ( ( maps_P5986043149438648064sn_nat @ F2 @ nil_Pr5671120429643327159n_assn )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_72_maps__simps_I2_J,axiom,
    ! [F2: int > list_P8527749157015355191n_assn] :
      ( ( maps_i2843482286296025986n_assn @ F2 @ nil_int )
      = nil_Pr5671120429643327159n_assn ) ).

% maps_simps(2)
thf(fact_73_maps__simps_I2_J,axiom,
    ! [F2: int > list_int] :
      ( ( maps_int_int @ F2 @ nil_int )
      = nil_int ) ).

% maps_simps(2)
thf(fact_74_maps__simps_I2_J,axiom,
    ! [F2: int > list_nat] :
      ( ( maps_int_nat @ F2 @ nil_int )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_75_maps__simps_I2_J,axiom,
    ! [F2: nat > list_P8527749157015355191n_assn] :
      ( ( maps_n1083445622542914782n_assn @ F2 @ nil_nat )
      = nil_Pr5671120429643327159n_assn ) ).

% maps_simps(2)
thf(fact_76_maps__simps_I2_J,axiom,
    ! [F2: nat > list_int] :
      ( ( maps_nat_int @ F2 @ nil_nat )
      = nil_int ) ).

% maps_simps(2)
thf(fact_77_maps__simps_I2_J,axiom,
    ! [F2: nat > list_nat] :
      ( ( maps_nat_nat @ F2 @ nil_nat )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_78_pure__assn__eq__conv,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( ( pure_assn @ P2 )
        = ( pure_assn @ Q2 ) )
      = ( P2 = Q2 ) ) ).

% pure_assn_eq_conv
thf(fact_79_merge__pure__star,axiom,
    ! [A: $o,B: $o] :
      ( ( times_times_assn @ ( pure_assn @ A ) @ ( pure_assn @ B ) )
      = ( pure_assn
        @ ( A
          & B ) ) ) ).

% merge_pure_star
thf(fact_80_is__pure__assn__pure,axiom,
    ! [P2: $o] : ( is_pure_assn @ ( pure_assn @ P2 ) ) ).

% is_pure_assn_pure
thf(fact_81_is__pure__assn__def,axiom,
    ( is_pure_assn
    = ( ^ [A2: assn] :
        ? [P: $o] :
          ( A2
          = ( pure_assn @ P ) ) ) ) ).

% is_pure_assn_def
thf(fact_82_is__pure__assnE,axiom,
    ! [A: assn] :
      ( ( is_pure_assn @ A )
     => ~ ! [P4: $o] :
            ( A
           != ( pure_assn @ P4 ) ) ) ).

% is_pure_assnE
thf(fact_83_syntax__fo__nomatch__def,axiom,
    ( syntax7398250324933576852n_assn
    = ( ^ [Pat: assn,Obj: assn] : $true ) ) ).

% syntax_fo_nomatch_def
thf(fact_84_ent__pure__post__iff,axiom,
    ! [P2: assn,Q2: assn,B: $o] :
      ( ( entails @ P2 @ ( times_times_assn @ Q2 @ ( pure_assn @ B ) ) )
      = ( ! [H: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P2 @ H )
           => B )
        & ( entails @ P2 @ Q2 ) ) ) ).

% ent_pure_post_iff
thf(fact_85_ent__pure__pre__iff__sng,axiom,
    ! [B: $o,Q2: assn] :
      ( ( entails @ ( pure_assn @ B ) @ Q2 )
      = ( B
       => ( entails @ one_one_assn @ Q2 ) ) ) ).

% ent_pure_pre_iff_sng
thf(fact_86_ac__operator_Osafe__commute,axiom,
    ! [F2: assn > assn > assn,X: assn,Y: assn,A: assn,B: assn] :
      ( ( syntax3673046151410588037r_assn @ F2 )
     => ( ( syntax7398250324933576852n_assn @ ( F2 @ X @ Y ) @ A )
       => ( ( F2 @ A @ B )
          = ( F2 @ B @ A ) ) ) ) ).

% ac_operator.safe_commute
thf(fact_87_and__extract__pure__left__ctx__iff,axiom,
    ! [P2: assn,B: $o,Q2: assn] :
      ( ( inf_inf_assn @ ( times_times_assn @ P2 @ ( pure_assn @ B ) ) @ Q2 )
      = ( times_times_assn @ ( inf_inf_assn @ P2 @ Q2 ) @ ( pure_assn @ B ) ) ) ).

% and_extract_pure_left_ctx_iff
thf(fact_88_and__extract__pure__right__ctx__iff,axiom,
    ! [P2: assn,Q2: assn,B: $o] :
      ( ( inf_inf_assn @ P2 @ ( times_times_assn @ Q2 @ ( pure_assn @ B ) ) )
      = ( times_times_assn @ ( inf_inf_assn @ P2 @ Q2 ) @ ( pure_assn @ B ) ) ) ).

% and_extract_pure_right_ctx_iff
thf(fact_89_null__rec_I2_J,axiom,
    null_P7596217980778516671n_assn @ nil_Pr5671120429643327159n_assn ).

% null_rec(2)
thf(fact_90_null__rec_I2_J,axiom,
    null_int @ nil_int ).

% null_rec(2)
thf(fact_91_null__rec_I2_J,axiom,
    null_nat @ nil_nat ).

% null_rec(2)
thf(fact_92_mem__Collect__eq,axiom,
    ! [A: int,P2: int > $o] :
      ( ( member_int2 @ A @ ( collect_int @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_93_mem__Collect__eq,axiom,
    ! [A: produc3658429121746597890et_nat > $o,P2: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( member6576561426505652726_nat_o @ A @ ( collec939566748876313656_nat_o @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_94_mem__Collect__eq,axiom,
    ! [A: list_nat,P2: list_nat > $o] :
      ( ( member_list_nat @ A @ ( collect_list_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_95_mem__Collect__eq,axiom,
    ! [A: nat,P2: nat > $o] :
      ( ( member_nat2 @ A @ ( collect_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_96_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_97_Collect__mem__eq,axiom,
    ! [A3: set_int] :
      ( ( collect_int
        @ ^ [X2: int] : ( member_int2 @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_98_Collect__mem__eq,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o] :
      ( ( collec939566748876313656_nat_o
        @ ^ [X2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_99_Collect__mem__eq,axiom,
    ! [A3: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_100_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat2 @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_101_Collect__mem__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_102_Collect__cong,axiom,
    ! [P2: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ! [X3: produc3658429121746597890et_nat > $o] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collec939566748876313656_nat_o @ P2 )
        = ( collec939566748876313656_nat_o @ Q2 ) ) ) ).

% Collect_cong
thf(fact_103_Collect__cong,axiom,
    ! [P2: list_nat > $o,Q2: list_nat > $o] :
      ( ! [X3: list_nat] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collect_list_nat @ P2 )
        = ( collect_list_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_104_Collect__cong,axiom,
    ! [P2: nat > $o,Q2: nat > $o] :
      ( ! [X3: nat] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collect_nat @ P2 )
        = ( collect_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_105_Collect__cong,axiom,
    ! [P2: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P2 @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( collec3392354462482085612at_nat @ P2 )
        = ( collec3392354462482085612at_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_106_eq__Nil__null,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( Xs = nil_Pr5671120429643327159n_assn )
      = ( null_P7596217980778516671n_assn @ Xs ) ) ).

% eq_Nil_null
thf(fact_107_eq__Nil__null,axiom,
    ! [Xs: list_int] :
      ( ( Xs = nil_int )
      = ( null_int @ Xs ) ) ).

% eq_Nil_null
thf(fact_108_eq__Nil__null,axiom,
    ! [Xs: list_nat] :
      ( ( Xs = nil_nat )
      = ( null_nat @ Xs ) ) ).

% eq_Nil_null
thf(fact_109_zip__Nil,axiom,
    ! [Ys: list_assn] :
      ( ( zip_assn_assn @ nil_assn @ Ys )
      = nil_Pr5671120429643327159n_assn ) ).

% zip_Nil
thf(fact_110_Nil__eq__zip__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( nil_Pr4327835960982145133n_assn
        = ( zip_Pr8806919808394484215n_assn @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% Nil_eq_zip_iff
thf(fact_111_Nil__eq__zip__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( ( nil_Pr7991254345159696774sn_int
        = ( zip_Pr1015554993053626726sn_int @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_int ) ) ) ).

% Nil_eq_zip_iff
thf(fact_112_Nil__eq__zip__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( ( nil_Pr2945733327814117674sn_nat
        = ( zip_Pr1018045463562677002sn_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_113_Nil__eq__zip__iff,axiom,
    ! [Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( ( nil_Pr2440414457004260852n_assn
        = ( zip_in7098856637274830732n_assn @ Xs @ Ys ) )
      = ( ( Xs = nil_int )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% Nil_eq_zip_iff
thf(fact_114_Nil__eq__zip__iff,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( nil_Pr2300489316682597567nt_int
        = ( zip_int_int @ Xs @ Ys ) )
      = ( ( Xs = nil_int )
        | ( Ys = nil_int ) ) ) ).

% Nil_eq_zip_iff
thf(fact_115_Nil__eq__zip__iff,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( ( nil_Pr6478340336191794275nt_nat
        = ( zip_int_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_int )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_116_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( ( nil_Pr680377793251149648n_assn
        = ( zip_na5338819973521719528n_assn @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% Nil_eq_zip_iff
thf(fact_117_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( ( nil_Pr1301135604781543011at_int
        = ( zip_nat_int @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_int ) ) ) ).

% Nil_eq_zip_iff
thf(fact_118_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_Pr5478986624290739719at_nat
        = ( zip_nat_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_119_Nil__eq__zip__iff,axiom,
    ! [Xs: list_assn,Ys: list_assn] :
      ( ( nil_Pr5671120429643327159n_assn
        = ( zip_assn_assn @ Xs @ Ys ) )
      = ( ( Xs = nil_assn )
        | ( Ys = nil_assn ) ) ) ).

% Nil_eq_zip_iff
thf(fact_120_zip__eq__Nil__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( zip_Pr8806919808394484215n_assn @ Xs @ Ys )
        = nil_Pr4327835960982145133n_assn )
      = ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% zip_eq_Nil_iff
thf(fact_121_zip__eq__Nil__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( ( ( zip_Pr1015554993053626726sn_int @ Xs @ Ys )
        = nil_Pr7991254345159696774sn_int )
      = ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_int ) ) ) ).

% zip_eq_Nil_iff
thf(fact_122_zip__eq__Nil__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( ( ( zip_Pr1018045463562677002sn_nat @ Xs @ Ys )
        = nil_Pr2945733327814117674sn_nat )
      = ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_123_zip__eq__Nil__iff,axiom,
    ! [Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( ( ( zip_in7098856637274830732n_assn @ Xs @ Ys )
        = nil_Pr2440414457004260852n_assn )
      = ( ( Xs = nil_int )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% zip_eq_Nil_iff
thf(fact_124_zip__eq__Nil__iff,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( zip_int_int @ Xs @ Ys )
        = nil_Pr2300489316682597567nt_int )
      = ( ( Xs = nil_int )
        | ( Ys = nil_int ) ) ) ).

% zip_eq_Nil_iff
thf(fact_125_zip__eq__Nil__iff,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( ( ( zip_int_nat @ Xs @ Ys )
        = nil_Pr6478340336191794275nt_nat )
      = ( ( Xs = nil_int )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_126_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( ( ( zip_na5338819973521719528n_assn @ Xs @ Ys )
        = nil_Pr680377793251149648n_assn )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% zip_eq_Nil_iff
thf(fact_127_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( ( ( zip_nat_int @ Xs @ Ys )
        = nil_Pr1301135604781543011at_int )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_int ) ) ) ).

% zip_eq_Nil_iff
thf(fact_128_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( zip_nat_nat @ Xs @ Ys )
        = nil_Pr5478986624290739719at_nat )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_129_zip__eq__Nil__iff,axiom,
    ! [Xs: list_assn,Ys: list_assn] :
      ( ( ( zip_assn_assn @ Xs @ Ys )
        = nil_Pr5671120429643327159n_assn )
      = ( ( Xs = nil_assn )
        | ( Ys = nil_assn ) ) ) ).

% zip_eq_Nil_iff
thf(fact_130_mod__pure__star__dist,axiom,
    ! [P2: assn,B: $o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P2 @ ( pure_assn @ B ) ) @ H2 )
      = ( ( rep_assn @ P2 @ H2 )
        & B ) ) ).

% mod_pure_star_dist
thf(fact_131_list__ex__simps_I2_J,axiom,
    ! [P2: produc6575502325842934193n_assn > $o] :
      ~ ( list_e1087220904399170805n_assn @ P2 @ nil_Pr5671120429643327159n_assn ) ).

% list_ex_simps(2)
thf(fact_132_list__ex__simps_I2_J,axiom,
    ! [P2: int > $o] :
      ~ ( list_ex_int @ P2 @ nil_int ) ).

% list_ex_simps(2)
thf(fact_133_list__ex__simps_I2_J,axiom,
    ! [P2: nat > $o] :
      ~ ( list_ex_nat @ P2 @ nil_nat ) ).

% list_ex_simps(2)
thf(fact_134_Rep__assn__inject,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( rep_assn @ X )
        = ( rep_assn @ Y ) )
      = ( X = Y ) ) ).

% Rep_assn_inject
thf(fact_135_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_136_mult__1,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ one_one_assn @ A )
      = A ) ).

% mult_1
thf(fact_137_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_138_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_139_mult__1,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ one_one_Code_integer @ A )
      = A ) ).

% mult_1
thf(fact_140_mult__1,axiom,
    ! [A: code_natural] :
      ( ( times_2397367101498566445atural @ one_one_Code_natural @ A )
      = A ) ).

% mult_1
thf(fact_141_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_142_mult_Oright__neutral,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ A @ one_one_assn )
      = A ) ).

% mult.right_neutral
thf(fact_143_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_144_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_145_mult_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ one_one_Code_integer )
      = A ) ).

% mult.right_neutral
thf(fact_146_mult_Oright__neutral,axiom,
    ! [A: code_natural] :
      ( ( times_2397367101498566445atural @ A @ one_one_Code_natural )
      = A ) ).

% mult.right_neutral
thf(fact_147_merge__pure__and,axiom,
    ! [A: $o,B: $o] :
      ( ( inf_inf_assn @ ( pure_assn @ A ) @ ( pure_assn @ B ) )
      = ( pure_assn
        @ ( A
          & B ) ) ) ).

% merge_pure_and
thf(fact_148_pure__true,axiom,
    ( ( pure_assn @ $true )
    = one_one_assn ) ).

% pure_true
thf(fact_149_pure__assn__eq__emp__iff,axiom,
    ! [P2: $o] :
      ( ( ( pure_assn @ P2 )
        = one_one_assn )
      = P2 ) ).

% pure_assn_eq_emp_iff
thf(fact_150_ent__pure__post__iff__sng,axiom,
    ! [P2: assn,B: $o] :
      ( ( entails @ P2 @ ( pure_assn @ B ) )
      = ( ! [H: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P2 @ H )
           => B )
        & ( entails @ P2 @ one_one_assn ) ) ) ).

% ent_pure_post_iff_sng
thf(fact_151_and__extract__pure__right__iff,axiom,
    ! [P2: assn,B: $o] :
      ( ( inf_inf_assn @ P2 @ ( pure_assn @ B ) )
      = ( times_times_assn @ ( inf_inf_assn @ one_one_assn @ P2 ) @ ( pure_assn @ B ) ) ) ).

% and_extract_pure_right_iff
thf(fact_152_and__extract__pure__left__iff,axiom,
    ! [B: $o,Q2: assn] :
      ( ( inf_inf_assn @ ( pure_assn @ B ) @ Q2 )
      = ( times_times_assn @ ( inf_inf_assn @ one_one_assn @ Q2 ) @ ( pure_assn @ B ) ) ) ).

% and_extract_pure_left_iff
thf(fact_153_norm__assertion__simps_I14_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ X @ Y ) @ Z )
      = ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) ) ) ).

% norm_assertion_simps(14)
thf(fact_154_norm__assertion__simps_I31_J,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ X )
      = X ) ).

% norm_assertion_simps(31)
thf(fact_155_assn__aci_I1_J,axiom,
    ( inf_inf_assn
    = ( ^ [X2: assn,Y2: assn] : ( inf_inf_assn @ Y2 @ X2 ) ) ) ).

% assn_aci(1)
thf(fact_156_assn__aci_I3_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( inf_inf_assn @ Y @ ( inf_inf_assn @ X @ Z ) ) ) ).

% assn_aci(3)
thf(fact_157_assn__aci_I4_J,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ X @ Y ) )
      = ( inf_inf_assn @ X @ Y ) ) ).

% assn_aci(4)
thf(fact_158_one__reorient,axiom,
    ! [X: code_integer] :
      ( ( one_one_Code_integer = X )
      = ( X = one_one_Code_integer ) ) ).

% one_reorient
thf(fact_159_one__reorient,axiom,
    ! [X: assn] :
      ( ( one_one_assn = X )
      = ( X = one_one_assn ) ) ).

% one_reorient
thf(fact_160_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
      = ( X = one_one_rat ) ) ).

% one_reorient
thf(fact_161_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_162_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_163_mod__and__dist,axiom,
    ! [P2: assn,Q2: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( inf_inf_assn @ P2 @ Q2 ) @ H2 )
      = ( ( rep_assn @ P2 @ H2 )
        & ( rep_assn @ Q2 @ H2 ) ) ) ).

% mod_and_dist
thf(fact_164_mod__starE,axiom,
    ! [A: assn,B: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A @ B ) @ H2 )
     => ~ ( ? [X_1: produc3658429121746597890et_nat] : ( rep_assn @ A @ X_1 )
         => ! [H_2: produc3658429121746597890et_nat] :
              ~ ( rep_assn @ B @ H_2 ) ) ) ).

% mod_starE
thf(fact_165_mod__starD,axiom,
    ! [A3: assn,B3: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A3 @ B3 ) @ H2 )
     => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
          ( ( rep_assn @ A3 @ H1 )
          & ( rep_assn @ B3 @ H22 ) ) ) ).

% mod_starD
thf(fact_166_entails__def,axiom,
    ( entails
    = ( ^ [P: assn,Q: assn] :
        ! [H: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H )
         => ( rep_assn @ Q @ H ) ) ) ) ).

% entails_def
thf(fact_167_entailsI,axiom,
    ! [P2: assn,Q2: assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P2 @ H3 )
         => ( rep_assn @ Q2 @ H3 ) )
     => ( entails @ P2 @ Q2 ) ) ).

% entailsI
thf(fact_168_entailsD,axiom,
    ! [P2: assn,Q2: assn,H2: produc3658429121746597890et_nat] :
      ( ( entails @ P2 @ Q2 )
     => ( ( rep_assn @ P2 @ H2 )
       => ( rep_assn @ Q2 @ H2 ) ) ) ).

% entailsD
thf(fact_169_ent__fwd,axiom,
    ! [P2: assn,H2: produc3658429121746597890et_nat,Q2: assn] :
      ( ( rep_assn @ P2 @ H2 )
     => ( ( entails @ P2 @ Q2 )
       => ( rep_assn @ Q2 @ H2 ) ) ) ).

% ent_fwd
thf(fact_170_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_171_mult_Ocomm__neutral,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ A @ one_one_assn )
      = A ) ).

% mult.comm_neutral
thf(fact_172_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_173_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_174_mult_Ocomm__neutral,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ one_one_Code_integer )
      = A ) ).

% mult.comm_neutral
thf(fact_175_mult_Ocomm__neutral,axiom,
    ! [A: code_natural] :
      ( ( times_2397367101498566445atural @ A @ one_one_Code_natural )
      = A ) ).

% mult.comm_neutral
thf(fact_176_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_177_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_178_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_179_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_180_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_181_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: code_natural] :
      ( ( times_2397367101498566445atural @ one_one_Code_natural @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_182_ent__conjE2,axiom,
    ! [B3: assn,C2: assn,A3: assn] :
      ( ( entails @ B3 @ C2 )
     => ( entails @ ( inf_inf_assn @ A3 @ B3 ) @ C2 ) ) ).

% ent_conjE2
thf(fact_183_ent__conjE1,axiom,
    ! [A3: assn,C2: assn,B3: assn] :
      ( ( entails @ A3 @ C2 )
     => ( entails @ ( inf_inf_assn @ A3 @ B3 ) @ C2 ) ) ).

% ent_conjE1
thf(fact_184_ent__conjI,axiom,
    ! [A3: assn,B3: assn,C2: assn] :
      ( ( entails @ A3 @ B3 )
     => ( ( entails @ A3 @ C2 )
       => ( entails @ A3 @ ( inf_inf_assn @ B3 @ C2 ) ) ) ) ).

% ent_conjI
thf(fact_185_assn__one__left,axiom,
    ! [P2: assn] :
      ( ( times_times_assn @ one_one_assn @ P2 )
      = P2 ) ).

% assn_one_left
thf(fact_186_norm__assertion__simps_I1_J,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ one_one_assn @ A )
      = A ) ).

% norm_assertion_simps(1)
thf(fact_187_norm__assertion__simps_I2_J,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ A @ one_one_assn )
      = A ) ).

% norm_assertion_simps(2)
thf(fact_188_is__pure__assn__basic__simps_I2_J,axiom,
    is_pure_assn @ one_one_assn ).

% is_pure_assn_basic_simps(2)
thf(fact_189_SLN__def,axiom,
    sln = one_one_assn ).

% SLN_def
thf(fact_190_mult_Oac__operator__axioms,axiom,
    syntax3673046151410588037r_assn @ times_times_assn ).

% mult.ac_operator_axioms
thf(fact_191_mult_Oac__operator__axioms,axiom,
    syntax8800539067650720301or_nat @ times_times_nat ).

% mult.ac_operator_axioms
thf(fact_192_mult_Oac__operator__axioms,axiom,
    syntax8798048597141670025or_int @ times_times_int ).

% mult.ac_operator_axioms
thf(fact_193_mult_Oac__operator__axioms,axiom,
    syntax414191934074104506nteger @ times_3573771949741848930nteger ).

% mult.ac_operator_axioms
thf(fact_194_mult_Oac__operator__axioms,axiom,
    syntax8461159122685597829atural @ times_2397367101498566445atural ).

% mult.ac_operator_axioms
thf(fact_195_zip_Osimps_I1_J,axiom,
    ! [Xs: list_assn] :
      ( ( zip_assn_assn @ Xs @ nil_assn )
      = nil_Pr5671120429643327159n_assn ) ).

% zip.simps(1)
thf(fact_196_inf__apply,axiom,
    ( inf_in3604695632404883862_int_o
    = ( ^ [F3: product_prod_int_int > $o,G: product_prod_int_int > $o,X2: product_prod_int_int] : ( inf_inf_o @ ( F3 @ X2 ) @ ( G @ X2 ) ) ) ) ).

% inf_apply
thf(fact_197_inf__right__idem,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ X @ Y ) @ Y )
      = ( inf_inf_assn @ X @ Y ) ) ).

% inf_right_idem
thf(fact_198_inf__right__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Y )
      = ( inf_in2572325071724192079at_nat @ X @ Y ) ) ).

% inf_right_idem
thf(fact_199_inf__right__idem,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y )
      = ( inf_inf_set_nat @ X @ Y ) ) ).

% inf_right_idem
thf(fact_200_inf__right__idem,axiom,
    ! [X: nat,Y: nat] :
      ( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Y )
      = ( inf_inf_nat @ X @ Y ) ) ).

% inf_right_idem
thf(fact_201_inf__right__idem,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ Y )
      = ( inf_in3604695632404883862_int_o @ X @ Y ) ) ).

% inf_right_idem
thf(fact_202_inf_Oright__idem,axiom,
    ! [A: assn,B: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ A @ B ) @ B )
      = ( inf_inf_assn @ A @ B ) ) ).

% inf.right_idem
thf(fact_203_inf_Oright__idem,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ B )
      = ( inf_in2572325071724192079at_nat @ A @ B ) ) ).

% inf.right_idem
thf(fact_204_inf_Oright__idem,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B )
      = ( inf_inf_set_nat @ A @ B ) ) ).

% inf.right_idem
thf(fact_205_inf_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ B )
      = ( inf_inf_nat @ A @ B ) ) ).

% inf.right_idem
thf(fact_206_inf_Oright__idem,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ B )
      = ( inf_in3604695632404883862_int_o @ A @ B ) ) ).

% inf.right_idem
thf(fact_207_inf__left__idem,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ X @ Y ) )
      = ( inf_inf_assn @ X @ Y ) ) ).

% inf_left_idem
thf(fact_208_inf__left__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ X @ Y ) )
      = ( inf_in2572325071724192079at_nat @ X @ Y ) ) ).

% inf_left_idem
thf(fact_209_inf__left__idem,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
      = ( inf_inf_set_nat @ X @ Y ) ) ).

% inf_left_idem
thf(fact_210_inf__left__idem,axiom,
    ! [X: nat,Y: nat] :
      ( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
      = ( inf_inf_nat @ X @ Y ) ) ).

% inf_left_idem
thf(fact_211_inf__left__idem,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ X @ ( inf_in3604695632404883862_int_o @ X @ Y ) )
      = ( inf_in3604695632404883862_int_o @ X @ Y ) ) ).

% inf_left_idem
thf(fact_212_inf_Oleft__idem,axiom,
    ! [A: assn,B: assn] :
      ( ( inf_inf_assn @ A @ ( inf_inf_assn @ A @ B ) )
      = ( inf_inf_assn @ A @ B ) ) ).

% inf.left_idem
thf(fact_213_inf_Oleft__idem,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A @ ( inf_in2572325071724192079at_nat @ A @ B ) )
      = ( inf_in2572325071724192079at_nat @ A @ B ) ) ).

% inf.left_idem
thf(fact_214_inf_Oleft__idem,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B ) )
      = ( inf_inf_set_nat @ A @ B ) ) ).

% inf.left_idem
thf(fact_215_inf_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( inf_inf_nat @ A @ ( inf_inf_nat @ A @ B ) )
      = ( inf_inf_nat @ A @ B ) ) ).

% inf.left_idem
thf(fact_216_inf_Oleft__idem,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ A @ ( inf_in3604695632404883862_int_o @ A @ B ) )
      = ( inf_in3604695632404883862_int_o @ A @ B ) ) ).

% inf.left_idem
thf(fact_217_inf__idem,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ X )
      = X ) ).

% inf_idem
thf(fact_218_inf__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ X )
      = X ) ).

% inf_idem
thf(fact_219_inf__idem,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ X )
      = X ) ).

% inf_idem
thf(fact_220_inf__idem,axiom,
    ! [X: nat] :
      ( ( inf_inf_nat @ X @ X )
      = X ) ).

% inf_idem
thf(fact_221_inf__idem,axiom,
    ! [X: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ X @ X )
      = X ) ).

% inf_idem
thf(fact_222_inf_Oidem,axiom,
    ! [A: assn] :
      ( ( inf_inf_assn @ A @ A )
      = A ) ).

% inf.idem
thf(fact_223_inf_Oidem,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A @ A )
      = A ) ).

% inf.idem
thf(fact_224_inf_Oidem,axiom,
    ! [A: set_nat] :
      ( ( inf_inf_set_nat @ A @ A )
      = A ) ).

% inf.idem
thf(fact_225_inf_Oidem,axiom,
    ! [A: nat] :
      ( ( inf_inf_nat @ A @ A )
      = A ) ).

% inf.idem
thf(fact_226_inf_Oidem,axiom,
    ! [A: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ A @ A )
      = A ) ).

% inf.idem
thf(fact_227_inf__fun__def,axiom,
    ( inf_in3604695632404883862_int_o
    = ( ^ [F3: product_prod_int_int > $o,G: product_prod_int_int > $o,X2: product_prod_int_int] : ( inf_inf_o @ ( F3 @ X2 ) @ ( G @ X2 ) ) ) ) ).

% inf_fun_def
thf(fact_228_inf__sup__aci_I4_J,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ X @ Y ) )
      = ( inf_inf_assn @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_229_inf__sup__aci_I4_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ X @ Y ) )
      = ( inf_in2572325071724192079at_nat @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_230_inf__sup__aci_I4_J,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
      = ( inf_inf_set_nat @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_231_inf__sup__aci_I4_J,axiom,
    ! [X: nat,Y: nat] :
      ( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
      = ( inf_inf_nat @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_232_inf__sup__aci_I4_J,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ X @ ( inf_in3604695632404883862_int_o @ X @ Y ) )
      = ( inf_in3604695632404883862_int_o @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_233_inf__sup__aci_I3_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( inf_inf_assn @ Y @ ( inf_inf_assn @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_234_inf__sup__aci_I3_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
      = ( inf_in2572325071724192079at_nat @ Y @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_235_inf__sup__aci_I3_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
      = ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_236_inf__sup__aci_I3_J,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
      = ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_237_inf__sup__aci_I3_J,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) )
      = ( inf_in3604695632404883862_int_o @ Y @ ( inf_in3604695632404883862_int_o @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_238_inf__sup__aci_I2_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ X @ Y ) @ Z )
      = ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_239_inf__sup__aci_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Z )
      = ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_240_inf__sup__aci_I2_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
      = ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_241_inf__sup__aci_I2_J,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
      = ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_242_inf__sup__aci_I2_J,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ Z )
      = ( inf_in3604695632404883862_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_243_inf__sup__aci_I1_J,axiom,
    ( inf_inf_assn
    = ( ^ [X2: assn,Y2: assn] : ( inf_inf_assn @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(1)
thf(fact_244_inf__sup__aci_I1_J,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(1)
thf(fact_245_inf__sup__aci_I1_J,axiom,
    ( inf_inf_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] : ( inf_inf_set_nat @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(1)
thf(fact_246_inf__sup__aci_I1_J,axiom,
    ( inf_inf_nat
    = ( ^ [X2: nat,Y2: nat] : ( inf_inf_nat @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(1)
thf(fact_247_inf__sup__aci_I1_J,axiom,
    ( inf_in3604695632404883862_int_o
    = ( ^ [X2: product_prod_int_int > $o,Y2: product_prod_int_int > $o] : ( inf_in3604695632404883862_int_o @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(1)
thf(fact_248_inf_Oassoc,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ A @ B ) @ C )
      = ( inf_inf_assn @ A @ ( inf_inf_assn @ B @ C ) ) ) ).

% inf.assoc
thf(fact_249_inf_Oassoc,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C )
      = ( inf_in2572325071724192079at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) ) ) ).

% inf.assoc
thf(fact_250_inf_Oassoc,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C )
      = ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ).

% inf.assoc
thf(fact_251_inf_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ C )
      = ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).

% inf.assoc
thf(fact_252_inf_Oassoc,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o,C: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ C )
      = ( inf_in3604695632404883862_int_o @ A @ ( inf_in3604695632404883862_int_o @ B @ C ) ) ) ).

% inf.assoc
thf(fact_253_inf__assoc,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ X @ Y ) @ Z )
      = ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_254_inf__assoc,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Z )
      = ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_255_inf__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
      = ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_256_inf__assoc,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
      = ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_257_inf__assoc,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ Z )
      = ( inf_in3604695632404883862_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_258_inf_Ocommute,axiom,
    ( inf_inf_assn
    = ( ^ [A2: assn,B2: assn] : ( inf_inf_assn @ B2 @ A2 ) ) ) ).

% inf.commute
thf(fact_259_inf_Ocommute,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ B2 @ A2 ) ) ) ).

% inf.commute
thf(fact_260_inf_Ocommute,axiom,
    ( inf_inf_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] : ( inf_inf_set_nat @ B2 @ A2 ) ) ) ).

% inf.commute
thf(fact_261_inf_Ocommute,axiom,
    ( inf_inf_nat
    = ( ^ [A2: nat,B2: nat] : ( inf_inf_nat @ B2 @ A2 ) ) ) ).

% inf.commute
thf(fact_262_inf_Ocommute,axiom,
    ( inf_in3604695632404883862_int_o
    = ( ^ [A2: product_prod_int_int > $o,B2: product_prod_int_int > $o] : ( inf_in3604695632404883862_int_o @ B2 @ A2 ) ) ) ).

% inf.commute
thf(fact_263_inf__commute,axiom,
    ( inf_inf_assn
    = ( ^ [X2: assn,Y2: assn] : ( inf_inf_assn @ Y2 @ X2 ) ) ) ).

% inf_commute
thf(fact_264_inf__commute,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ Y2 @ X2 ) ) ) ).

% inf_commute
thf(fact_265_inf__commute,axiom,
    ( inf_inf_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] : ( inf_inf_set_nat @ Y2 @ X2 ) ) ) ).

% inf_commute
thf(fact_266_inf__commute,axiom,
    ( inf_inf_nat
    = ( ^ [X2: nat,Y2: nat] : ( inf_inf_nat @ Y2 @ X2 ) ) ) ).

% inf_commute
thf(fact_267_inf__commute,axiom,
    ( inf_in3604695632404883862_int_o
    = ( ^ [X2: product_prod_int_int > $o,Y2: product_prod_int_int > $o] : ( inf_in3604695632404883862_int_o @ Y2 @ X2 ) ) ) ).

% inf_commute
thf(fact_268_inf_Oleft__commute,axiom,
    ! [B: assn,A: assn,C: assn] :
      ( ( inf_inf_assn @ B @ ( inf_inf_assn @ A @ C ) )
      = ( inf_inf_assn @ A @ ( inf_inf_assn @ B @ C ) ) ) ).

% inf.left_commute
thf(fact_269_inf_Oleft__commute,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ B @ ( inf_in2572325071724192079at_nat @ A @ C ) )
      = ( inf_in2572325071724192079at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) ) ) ).

% inf.left_commute
thf(fact_270_inf_Oleft__commute,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A @ C ) )
      = ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ).

% inf.left_commute
thf(fact_271_inf_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( inf_inf_nat @ B @ ( inf_inf_nat @ A @ C ) )
      = ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).

% inf.left_commute
thf(fact_272_inf_Oleft__commute,axiom,
    ! [B: product_prod_int_int > $o,A: product_prod_int_int > $o,C: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ B @ ( inf_in3604695632404883862_int_o @ A @ C ) )
      = ( inf_in3604695632404883862_int_o @ A @ ( inf_in3604695632404883862_int_o @ B @ C ) ) ) ).

% inf.left_commute
thf(fact_273_inf__left__commute,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( inf_inf_assn @ Y @ ( inf_inf_assn @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_274_inf__left__commute,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
      = ( inf_in2572325071724192079at_nat @ Y @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_275_inf__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
      = ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_276_inf__left__commute,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
      = ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_277_inf__left__commute,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) )
      = ( inf_in3604695632404883862_int_o @ Y @ ( inf_in3604695632404883862_int_o @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_278_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: assn,K: assn,A: assn,B: assn] :
      ( ( A3
        = ( inf_inf_assn @ K @ A ) )
     => ( ( inf_inf_assn @ A3 @ B )
        = ( inf_inf_assn @ K @ ( inf_inf_assn @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_279_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( A3
        = ( inf_in2572325071724192079at_nat @ K @ A ) )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ B )
        = ( inf_in2572325071724192079at_nat @ K @ ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_280_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: set_nat,K: set_nat,A: set_nat,B: set_nat] :
      ( ( A3
        = ( inf_inf_set_nat @ K @ A ) )
     => ( ( inf_inf_set_nat @ A3 @ B )
        = ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_281_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: nat,K: nat,A: nat,B: nat] :
      ( ( A3
        = ( inf_inf_nat @ K @ A ) )
     => ( ( inf_inf_nat @ A3 @ B )
        = ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_282_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: product_prod_int_int > $o,K: product_prod_int_int > $o,A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( A3
        = ( inf_in3604695632404883862_int_o @ K @ A ) )
     => ( ( inf_in3604695632404883862_int_o @ A3 @ B )
        = ( inf_in3604695632404883862_int_o @ K @ ( inf_in3604695632404883862_int_o @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_283_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: assn,K: assn,B: assn,A: assn] :
      ( ( B3
        = ( inf_inf_assn @ K @ B ) )
     => ( ( inf_inf_assn @ A @ B3 )
        = ( inf_inf_assn @ K @ ( inf_inf_assn @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_284_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( B3
        = ( inf_in2572325071724192079at_nat @ K @ B ) )
     => ( ( inf_in2572325071724192079at_nat @ A @ B3 )
        = ( inf_in2572325071724192079at_nat @ K @ ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_285_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: set_nat,K: set_nat,B: set_nat,A: set_nat] :
      ( ( B3
        = ( inf_inf_set_nat @ K @ B ) )
     => ( ( inf_inf_set_nat @ A @ B3 )
        = ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_286_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: nat,K: nat,B: nat,A: nat] :
      ( ( B3
        = ( inf_inf_nat @ K @ B ) )
     => ( ( inf_inf_nat @ A @ B3 )
        = ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_287_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: product_prod_int_int > $o,K: product_prod_int_int > $o,B: product_prod_int_int > $o,A: product_prod_int_int > $o] :
      ( ( B3
        = ( inf_in3604695632404883862_int_o @ K @ B ) )
     => ( ( inf_in3604695632404883862_int_o @ A @ B3 )
        = ( inf_in3604695632404883862_int_o @ K @ ( inf_in3604695632404883862_int_o @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_288_mult_Omonoid__axioms,axiom,
    monoid_rat @ times_times_rat @ one_one_rat ).

% mult.monoid_axioms
thf(fact_289_mult_Omonoid__axioms,axiom,
    monoid_assn @ times_times_assn @ one_one_assn ).

% mult.monoid_axioms
thf(fact_290_mult_Omonoid__axioms,axiom,
    monoid_nat @ times_times_nat @ one_one_nat ).

% mult.monoid_axioms
thf(fact_291_mult_Omonoid__axioms,axiom,
    monoid_int @ times_times_int @ one_one_int ).

% mult.monoid_axioms
thf(fact_292_mult_Omonoid__axioms,axiom,
    monoid_Code_integer @ times_3573771949741848930nteger @ one_one_Code_integer ).

% mult.monoid_axioms
thf(fact_293_mult_Omonoid__axioms,axiom,
    monoid_Code_natural @ times_2397367101498566445atural @ one_one_Code_natural ).

% mult.monoid_axioms
thf(fact_294_ent__false__iff,axiom,
    ! [P2: assn] :
      ( ( entails @ P2 @ bot_bot_assn )
      = ( ! [H: produc3658429121746597890et_nat] :
            ~ ( rep_assn @ P2 @ H ) ) ) ).

% ent_false_iff
thf(fact_295_mult_Ocomm__monoid__axioms,axiom,
    comm_monoid_rat @ times_times_rat @ one_one_rat ).

% mult.comm_monoid_axioms
thf(fact_296_mult_Ocomm__monoid__axioms,axiom,
    comm_monoid_assn @ times_times_assn @ one_one_assn ).

% mult.comm_monoid_axioms
thf(fact_297_mult_Ocomm__monoid__axioms,axiom,
    comm_monoid_nat @ times_times_nat @ one_one_nat ).

% mult.comm_monoid_axioms
thf(fact_298_mult_Ocomm__monoid__axioms,axiom,
    comm_monoid_int @ times_times_int @ one_one_int ).

% mult.comm_monoid_axioms
thf(fact_299_mult_Ocomm__monoid__axioms,axiom,
    comm_m1654649897431166824nteger @ times_3573771949741848930nteger @ one_one_Code_integer ).

% mult.comm_monoid_axioms
thf(fact_300_mult_Ocomm__monoid__axioms,axiom,
    comm_m478245049187884339atural @ times_2397367101498566445atural @ one_one_Code_natural ).

% mult.comm_monoid_axioms
thf(fact_301_map__filter__simps_I2_J,axiom,
    ! [F2: produc6575502325842934193n_assn > option7057359848497031415n_assn] :
      ( ( map_fi2680429960852639882n_assn @ F2 @ nil_Pr5671120429643327159n_assn )
      = nil_Pr5671120429643327159n_assn ) ).

% map_filter_simps(2)
thf(fact_302_map__filter__simps_I2_J,axiom,
    ! [F2: produc6575502325842934193n_assn > option_int] :
      ( ( map_fi8666144670822621843sn_int @ F2 @ nil_Pr5671120429643327159n_assn )
      = nil_int ) ).

% map_filter_simps(2)
thf(fact_303_map__filter__simps_I2_J,axiom,
    ! [F2: produc6575502325842934193n_assn > option_nat] :
      ( ( map_fi8668635141331672119sn_nat @ F2 @ nil_Pr5671120429643327159n_assn )
      = nil_nat ) ).

% map_filter_simps(2)
thf(fact_304_map__filter__simps_I2_J,axiom,
    ! [F2: int > option7057359848497031415n_assn] :
      ( ( map_fi5526074278189050041n_assn @ F2 @ nil_int )
      = nil_Pr5671120429643327159n_assn ) ).

% map_filter_simps(2)
thf(fact_305_map__filter__simps_I2_J,axiom,
    ! [F2: int > option_int] :
      ( ( map_filter_int_int @ F2 @ nil_int )
      = nil_int ) ).

% map_filter_simps(2)
thf(fact_306_map__filter__simps_I2_J,axiom,
    ! [F2: int > option_nat] :
      ( ( map_filter_int_nat @ F2 @ nil_int )
      = nil_nat ) ).

% map_filter_simps(2)
thf(fact_307_map__filter__simps_I2_J,axiom,
    ! [F2: nat > option7057359848497031415n_assn] :
      ( ( map_fi3766037614435938837n_assn @ F2 @ nil_nat )
      = nil_Pr5671120429643327159n_assn ) ).

% map_filter_simps(2)
thf(fact_308_map__filter__simps_I2_J,axiom,
    ! [F2: nat > option_int] :
      ( ( map_filter_nat_int @ F2 @ nil_nat )
      = nil_int ) ).

% map_filter_simps(2)
thf(fact_309_map__filter__simps_I2_J,axiom,
    ! [F2: nat > option_nat] :
      ( ( map_filter_nat_nat @ F2 @ nil_nat )
      = nil_nat ) ).

% map_filter_simps(2)
thf(fact_310_splice__Nil2,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( splice4588524007259616650n_assn @ Xs @ nil_Pr5671120429643327159n_assn )
      = Xs ) ).

% splice_Nil2
thf(fact_311_splice__Nil2,axiom,
    ! [Xs: list_int] :
      ( ( splice_int @ Xs @ nil_int )
      = Xs ) ).

% splice_Nil2
thf(fact_312_splice__Nil2,axiom,
    ! [Xs: list_nat] :
      ( ( splice_nat @ Xs @ nil_nat )
      = Xs ) ).

% splice_Nil2
thf(fact_313_split__Nil__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( splice4588524007259616650n_assn @ Xs @ Ys )
        = nil_Pr5671120429643327159n_assn )
      = ( ( Xs = nil_Pr5671120429643327159n_assn )
        & ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% split_Nil_iff
thf(fact_314_split__Nil__iff,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( splice_int @ Xs @ Ys )
        = nil_int )
      = ( ( Xs = nil_int )
        & ( Ys = nil_int ) ) ) ).

% split_Nil_iff
thf(fact_315_split__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( splice_nat @ Xs @ Ys )
        = nil_nat )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% split_Nil_iff
thf(fact_316_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( ( rotate328796349445179396n_assn @ Xs )
        = nil_Pr5671120429643327159n_assn )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% rotate1_is_Nil_conv
thf(fact_317_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( ( rotate1_int @ Xs )
        = nil_int )
      = ( Xs = nil_int ) ) ).

% rotate1_is_Nil_conv
thf(fact_318_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( rotate1_nat @ Xs )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% rotate1_is_Nil_conv
thf(fact_319_enumerate__simps_I1_J,axiom,
    ! [N: nat] :
      ( ( enumer9200277086812776726n_assn @ N @ nil_Pr5671120429643327159n_assn )
      = nil_Pr680377793251149648n_assn ) ).

% enumerate_simps(1)
thf(fact_320_enumerate__simps_I1_J,axiom,
    ! [N: nat] :
      ( ( enumerate_int @ N @ nil_int )
      = nil_Pr1301135604781543011at_int ) ).

% enumerate_simps(1)
thf(fact_321_enumerate__simps_I1_J,axiom,
    ! [N: nat] :
      ( ( enumerate_nat @ N @ nil_nat )
      = nil_Pr5478986624290739719at_nat ) ).

% enumerate_simps(1)
thf(fact_322_mod__star__trueI,axiom,
    ! [P2: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ P2 @ H2 )
     => ( rep_assn @ ( times_times_assn @ P2 @ top_top_assn ) @ H2 ) ) ).

% mod_star_trueI
thf(fact_323_mod__star__trueE,axiom,
    ! [P2: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P2 @ top_top_assn ) @ H2 )
     => ~ ! [H4: produc3658429121746597890et_nat] :
            ~ ( rep_assn @ P2 @ H4 ) ) ).

% mod_star_trueE
thf(fact_324_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_zero_right
thf(fact_325_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ X @ bot_bo8147686125503663512_int_o )
      = bot_bo8147686125503663512_int_o ) ).

% boolean_algebra.conj_zero_right
thf(fact_326_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ bot_bot_assn )
      = bot_bot_assn ) ).

% boolean_algebra.conj_zero_right
thf(fact_327_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_right
thf(fact_328_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ X @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_zero_right
thf(fact_329_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_zero_left
thf(fact_330_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ bot_bo8147686125503663512_int_o @ X )
      = bot_bo8147686125503663512_int_o ) ).

% boolean_algebra.conj_zero_left
thf(fact_331_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ bot_bot_assn @ X )
      = bot_bot_assn ) ).

% boolean_algebra.conj_zero_left
thf(fact_332_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_left
thf(fact_333_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ X )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_zero_left
thf(fact_334_inf__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% inf_bot_right
thf(fact_335_inf__bot__right,axiom,
    ! [X: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ X @ bot_bo8147686125503663512_int_o )
      = bot_bo8147686125503663512_int_o ) ).

% inf_bot_right
thf(fact_336_inf__bot__right,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ bot_bot_assn )
      = bot_bot_assn ) ).

% inf_bot_right
thf(fact_337_inf__bot__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% inf_bot_right
thf(fact_338_inf__bot__right,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ X @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% inf_bot_right
thf(fact_339_inf__bot__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X )
      = bot_bo2099793752762293965at_nat ) ).

% inf_bot_left
thf(fact_340_inf__bot__left,axiom,
    ! [X: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ bot_bo8147686125503663512_int_o @ X )
      = bot_bo8147686125503663512_int_o ) ).

% inf_bot_left
thf(fact_341_inf__bot__left,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ bot_bot_assn @ X )
      = bot_bot_assn ) ).

% inf_bot_left
thf(fact_342_inf__bot__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
      = bot_bot_set_nat ) ).

% inf_bot_left
thf(fact_343_inf__bot__left,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ X )
      = bot_bot_set_int ) ).

% inf_bot_left
thf(fact_344_inf__top_Oright__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A @ top_to4669805908274784177at_nat )
      = A ) ).

% inf_top.right_neutral
thf(fact_345_inf__top_Oright__neutral,axiom,
    ! [A: set_nat] :
      ( ( inf_inf_set_nat @ A @ top_top_set_nat )
      = A ) ).

% inf_top.right_neutral
thf(fact_346_inf__top_Oright__neutral,axiom,
    ! [A: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ A @ top_to1578927101902068148_int_o )
      = A ) ).

% inf_top.right_neutral
thf(fact_347_inf__top_Oright__neutral,axiom,
    ! [A: assn] :
      ( ( inf_inf_assn @ A @ top_top_assn )
      = A ) ).

% inf_top.right_neutral
thf(fact_348_inf__top_Oright__neutral,axiom,
    ! [A: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit @ A @ top_to1996260823553986621t_unit )
      = A ) ).

% inf_top.right_neutral
thf(fact_349_inf__top_Oneutr__eq__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( top_to4669805908274784177at_nat
        = ( inf_in2572325071724192079at_nat @ A @ B ) )
      = ( ( A = top_to4669805908274784177at_nat )
        & ( B = top_to4669805908274784177at_nat ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_350_inf__top_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( top_top_set_nat
        = ( inf_inf_set_nat @ A @ B ) )
      = ( ( A = top_top_set_nat )
        & ( B = top_top_set_nat ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_351_inf__top_Oneutr__eq__iff,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( top_to1578927101902068148_int_o
        = ( inf_in3604695632404883862_int_o @ A @ B ) )
      = ( ( A = top_to1578927101902068148_int_o )
        & ( B = top_to1578927101902068148_int_o ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_352_inf__top_Oneutr__eq__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( top_top_assn
        = ( inf_inf_assn @ A @ B ) )
      = ( ( A = top_top_assn )
        & ( B = top_top_assn ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_353_inf__top_Oneutr__eq__iff,axiom,
    ! [A: set_Product_unit,B: set_Product_unit] :
      ( ( top_to1996260823553986621t_unit
        = ( inf_in4660618365625256667t_unit @ A @ B ) )
      = ( ( A = top_to1996260823553986621t_unit )
        & ( B = top_to1996260823553986621t_unit ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_354_inf__top_Oleft__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ top_to4669805908274784177at_nat @ A )
      = A ) ).

% inf_top.left_neutral
thf(fact_355_inf__top_Oleft__neutral,axiom,
    ! [A: set_nat] :
      ( ( inf_inf_set_nat @ top_top_set_nat @ A )
      = A ) ).

% inf_top.left_neutral
thf(fact_356_inf__top_Oleft__neutral,axiom,
    ! [A: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ top_to1578927101902068148_int_o @ A )
      = A ) ).

% inf_top.left_neutral
thf(fact_357_inf__top_Oleft__neutral,axiom,
    ! [A: assn] :
      ( ( inf_inf_assn @ top_top_assn @ A )
      = A ) ).

% inf_top.left_neutral
thf(fact_358_inf__top_Oleft__neutral,axiom,
    ! [A: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ A )
      = A ) ).

% inf_top.left_neutral
thf(fact_359_inf__top_Oeq__neutr__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A @ B )
        = top_to4669805908274784177at_nat )
      = ( ( A = top_to4669805908274784177at_nat )
        & ( B = top_to4669805908274784177at_nat ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_360_inf__top_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( inf_inf_set_nat @ A @ B )
        = top_top_set_nat )
      = ( ( A = top_top_set_nat )
        & ( B = top_top_set_nat ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_361_inf__top_Oeq__neutr__iff,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( ( inf_in3604695632404883862_int_o @ A @ B )
        = top_to1578927101902068148_int_o )
      = ( ( A = top_to1578927101902068148_int_o )
        & ( B = top_to1578927101902068148_int_o ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_362_inf__top_Oeq__neutr__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( ( inf_inf_assn @ A @ B )
        = top_top_assn )
      = ( ( A = top_top_assn )
        & ( B = top_top_assn ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_363_inf__top_Oeq__neutr__iff,axiom,
    ! [A: set_Product_unit,B: set_Product_unit] :
      ( ( ( inf_in4660618365625256667t_unit @ A @ B )
        = top_to1996260823553986621t_unit )
      = ( ( A = top_to1996260823553986621t_unit )
        & ( B = top_to1996260823553986621t_unit ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_364_top__eq__inf__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( top_to4669805908274784177at_nat
        = ( inf_in2572325071724192079at_nat @ X @ Y ) )
      = ( ( X = top_to4669805908274784177at_nat )
        & ( Y = top_to4669805908274784177at_nat ) ) ) ).

% top_eq_inf_iff
thf(fact_365_top__eq__inf__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( top_top_set_nat
        = ( inf_inf_set_nat @ X @ Y ) )
      = ( ( X = top_top_set_nat )
        & ( Y = top_top_set_nat ) ) ) ).

% top_eq_inf_iff
thf(fact_366_top__eq__inf__iff,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( top_to1578927101902068148_int_o
        = ( inf_in3604695632404883862_int_o @ X @ Y ) )
      = ( ( X = top_to1578927101902068148_int_o )
        & ( Y = top_to1578927101902068148_int_o ) ) ) ).

% top_eq_inf_iff
thf(fact_367_top__eq__inf__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( top_top_assn
        = ( inf_inf_assn @ X @ Y ) )
      = ( ( X = top_top_assn )
        & ( Y = top_top_assn ) ) ) ).

% top_eq_inf_iff
thf(fact_368_top__eq__inf__iff,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] :
      ( ( top_to1996260823553986621t_unit
        = ( inf_in4660618365625256667t_unit @ X @ Y ) )
      = ( ( X = top_to1996260823553986621t_unit )
        & ( Y = top_to1996260823553986621t_unit ) ) ) ).

% top_eq_inf_iff
thf(fact_369_inf__eq__top__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ X @ Y )
        = top_to4669805908274784177at_nat )
      = ( ( X = top_to4669805908274784177at_nat )
        & ( Y = top_to4669805908274784177at_nat ) ) ) ).

% inf_eq_top_iff
thf(fact_370_inf__eq__top__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( inf_inf_set_nat @ X @ Y )
        = top_top_set_nat )
      = ( ( X = top_top_set_nat )
        & ( Y = top_top_set_nat ) ) ) ).

% inf_eq_top_iff
thf(fact_371_inf__eq__top__iff,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( ( inf_in3604695632404883862_int_o @ X @ Y )
        = top_to1578927101902068148_int_o )
      = ( ( X = top_to1578927101902068148_int_o )
        & ( Y = top_to1578927101902068148_int_o ) ) ) ).

% inf_eq_top_iff
thf(fact_372_inf__eq__top__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( inf_inf_assn @ X @ Y )
        = top_top_assn )
      = ( ( X = top_top_assn )
        & ( Y = top_top_assn ) ) ) ).

% inf_eq_top_iff
thf(fact_373_inf__eq__top__iff,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] :
      ( ( ( inf_in4660618365625256667t_unit @ X @ Y )
        = top_to1996260823553986621t_unit )
      = ( ( X = top_to1996260823553986621t_unit )
        & ( Y = top_to1996260823553986621t_unit ) ) ) ).

% inf_eq_top_iff
thf(fact_374_inf__top__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ top_to4669805908274784177at_nat )
      = X ) ).

% inf_top_right
thf(fact_375_inf__top__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ top_top_set_nat )
      = X ) ).

% inf_top_right
thf(fact_376_inf__top__right,axiom,
    ! [X: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ X @ top_to1578927101902068148_int_o )
      = X ) ).

% inf_top_right
thf(fact_377_inf__top__right,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ top_top_assn )
      = X ) ).

% inf_top_right
thf(fact_378_inf__top__right,axiom,
    ! [X: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit @ X @ top_to1996260823553986621t_unit )
      = X ) ).

% inf_top_right
thf(fact_379_inf__top__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ top_to4669805908274784177at_nat @ X )
      = X ) ).

% inf_top_left
thf(fact_380_inf__top__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ top_top_set_nat @ X )
      = X ) ).

% inf_top_left
thf(fact_381_inf__top__left,axiom,
    ! [X: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ top_to1578927101902068148_int_o @ X )
      = X ) ).

% inf_top_left
thf(fact_382_inf__top__left,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ top_top_assn @ X )
      = X ) ).

% inf_top_left
thf(fact_383_inf__top__left,axiom,
    ! [X: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ X )
      = X ) ).

% inf_top_left
thf(fact_384_merge__true__star,axiom,
    ( ( times_times_assn @ top_top_assn @ top_top_assn )
    = top_top_assn ) ).

% merge_true_star
thf(fact_385_star__false__right,axiom,
    ! [P2: assn] :
      ( ( times_times_assn @ P2 @ bot_bot_assn )
      = bot_bot_assn ) ).

% star_false_right
thf(fact_386_star__false__left,axiom,
    ! [P2: assn] :
      ( ( times_times_assn @ bot_bot_assn @ P2 )
      = bot_bot_assn ) ).

% star_false_left
thf(fact_387_pure__false,axiom,
    ( ( pure_assn @ $false )
    = bot_bot_assn ) ).

% pure_false
thf(fact_388_pure__assn__eq__false__iff,axiom,
    ! [P2: $o] :
      ( ( ( pure_assn @ P2 )
        = bot_bot_assn )
      = ~ P2 ) ).

% pure_assn_eq_false_iff
thf(fact_389_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
thf(fact_390_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
thf(fact_391_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
thf(fact_392_inf__top_Ocomm__monoid__axioms,axiom,
    comm_m5709050211238513046at_nat @ inf_in2572325071724192079at_nat @ top_to4669805908274784177at_nat ).

% inf_top.comm_monoid_axioms
thf(fact_393_inf__top_Ocomm__monoid__axioms,axiom,
    comm_monoid_set_nat @ inf_inf_set_nat @ top_top_set_nat ).

% inf_top.comm_monoid_axioms
thf(fact_394_inf__top_Ocomm__monoid__axioms,axiom,
    comm_m5839176995973721103_int_o @ inf_in3604695632404883862_int_o @ top_to1578927101902068148_int_o ).

% inf_top.comm_monoid_axioms
thf(fact_395_inf__top_Ocomm__monoid__axioms,axiom,
    comm_monoid_assn @ inf_inf_assn @ top_top_assn ).

% inf_top.comm_monoid_axioms
thf(fact_396_inf__top_Ocomm__monoid__axioms,axiom,
    comm_m7591727228438362018t_unit @ inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit ).

% inf_top.comm_monoid_axioms
thf(fact_397_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ top_to4669805908274784177at_nat )
      = X ) ).

% boolean_algebra.conj_one_right
thf(fact_398_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ top_top_set_nat )
      = X ) ).

% boolean_algebra.conj_one_right
thf(fact_399_boolean__algebra_Oconj__one__right,axiom,
    ! [X: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ X @ top_to1578927101902068148_int_o )
      = X ) ).

% boolean_algebra.conj_one_right
thf(fact_400_boolean__algebra_Oconj__one__right,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ top_top_assn )
      = X ) ).

% boolean_algebra.conj_one_right
thf(fact_401_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit @ X @ top_to1996260823553986621t_unit )
      = X ) ).

% boolean_algebra.conj_one_right
thf(fact_402_inf__top_Omonoid__axioms,axiom,
    monoid1519094961268053602at_nat @ inf_in2572325071724192079at_nat @ top_to4669805908274784177at_nat ).

% inf_top.monoid_axioms
thf(fact_403_inf__top_Omonoid__axioms,axiom,
    monoid_set_nat @ inf_inf_set_nat @ top_top_set_nat ).

% inf_top.monoid_axioms
thf(fact_404_inf__top_Omonoid__axioms,axiom,
    monoid6910981727789844035_int_o @ inf_in3604695632404883862_int_o @ top_to1578927101902068148_int_o ).

% inf_top.monoid_axioms
thf(fact_405_inf__top_Omonoid__axioms,axiom,
    monoid_assn @ inf_inf_assn @ top_top_assn ).

% inf_top.monoid_axioms
thf(fact_406_inf__top_Omonoid__axioms,axiom,
    monoid202360481972076142t_unit @ inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit ).

% inf_top.monoid_axioms
thf(fact_407_merge__true__star__ctx,axiom,
    ! [P2: assn] :
      ( ( times_times_assn @ top_top_assn @ ( times_times_assn @ top_top_assn @ P2 ) )
      = ( times_times_assn @ top_top_assn @ P2 ) ) ).

% merge_true_star_ctx
thf(fact_408_mod__false,axiom,
    ! [H2: produc3658429121746597890et_nat] :
      ~ ( rep_assn @ bot_bot_assn @ H2 ) ).

% mod_false
thf(fact_409_ent__false,axiom,
    ! [P2: assn] : ( entails @ bot_bot_assn @ P2 ) ).

% ent_false
thf(fact_410_ent__true,axiom,
    ! [P2: assn] : ( entails @ P2 @ top_top_assn ) ).

% ent_true
thf(fact_411_norm__assertion__simps_I9_J,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ bot_bot_assn @ X )
      = bot_bot_assn ) ).

% norm_assertion_simps(9)
thf(fact_412_norm__assertion__simps_I10_J,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ bot_bot_assn )
      = bot_bot_assn ) ).

% norm_assertion_simps(10)
thf(fact_413_norm__assertion__simps_I3_J,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ top_top_assn @ X )
      = X ) ).

% norm_assertion_simps(3)
thf(fact_414_norm__assertion__simps_I4_J,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ top_top_assn )
      = X ) ).

% norm_assertion_simps(4)
thf(fact_415_is__pure__assn__basic__simps_I1_J,axiom,
    is_pure_assn @ bot_bot_assn ).

% is_pure_assn_basic_simps(1)
thf(fact_416_rotate1_Osimps_I1_J,axiom,
    ( ( rotate328796349445179396n_assn @ nil_Pr5671120429643327159n_assn )
    = nil_Pr5671120429643327159n_assn ) ).

% rotate1.simps(1)
thf(fact_417_rotate1_Osimps_I1_J,axiom,
    ( ( rotate1_int @ nil_int )
    = nil_int ) ).

% rotate1.simps(1)
thf(fact_418_rotate1_Osimps_I1_J,axiom,
    ( ( rotate1_nat @ nil_nat )
    = nil_nat ) ).

% rotate1.simps(1)
thf(fact_419_splice_Osimps_I1_J,axiom,
    ! [Ys: list_P8527749157015355191n_assn] :
      ( ( splice4588524007259616650n_assn @ nil_Pr5671120429643327159n_assn @ Ys )
      = Ys ) ).

% splice.simps(1)
thf(fact_420_splice_Osimps_I1_J,axiom,
    ! [Ys: list_int] :
      ( ( splice_int @ nil_int @ Ys )
      = Ys ) ).

% splice.simps(1)
thf(fact_421_splice_Osimps_I1_J,axiom,
    ! [Ys: list_nat] :
      ( ( splice_nat @ nil_nat @ Ys )
      = Ys ) ).

% splice.simps(1)
thf(fact_422_entt__def__true,axiom,
    ( entailst
    = ( ^ [P: assn,Q: assn] : ( entails @ ( times_times_assn @ P @ top_top_assn ) @ ( times_times_assn @ Q @ top_top_assn ) ) ) ) ).

% entt_def_true
thf(fact_423_entailst__def,axiom,
    ( entailst
    = ( ^ [A4: assn,B4: assn] : ( entails @ A4 @ ( times_times_assn @ B4 @ top_top_assn ) ) ) ) ).

% entailst_def
thf(fact_424_enttI__true,axiom,
    ! [P2: assn,Q2: assn] :
      ( ( entails @ ( times_times_assn @ P2 @ top_top_assn ) @ ( times_times_assn @ Q2 @ top_top_assn ) )
     => ( entailst @ P2 @ Q2 ) ) ).

% enttI_true
thf(fact_425_enttI,axiom,
    ! [A3: assn,B3: assn] :
      ( ( entails @ A3 @ ( times_times_assn @ B3 @ top_top_assn ) )
     => ( entailst @ A3 @ B3 ) ) ).

% enttI
thf(fact_426_enttD,axiom,
    ! [A3: assn,B3: assn] :
      ( ( entailst @ A3 @ B3 )
     => ( entails @ A3 @ ( times_times_assn @ B3 @ top_top_assn ) ) ) ).

% enttD
thf(fact_427_entt__star__true__simp_I2_J,axiom,
    ! [A3: assn,B3: assn] :
      ( ( entailst @ ( times_times_assn @ A3 @ top_top_assn ) @ B3 )
      = ( entailst @ A3 @ B3 ) ) ).

% entt_star_true_simp(2)
thf(fact_428_entt__star__true__simp_I1_J,axiom,
    ! [A3: assn,B3: assn] :
      ( ( entailst @ A3 @ ( times_times_assn @ B3 @ top_top_assn ) )
      = ( entailst @ A3 @ B3 ) ) ).

% entt_star_true_simp(1)
thf(fact_429_inf__top_Osemilattice__neutr__axioms,axiom,
    semila858173770281612099at_nat @ inf_in2572325071724192079at_nat @ top_to4669805908274784177at_nat ).

% inf_top.semilattice_neutr_axioms
thf(fact_430_inf__top_Osemilattice__neutr__axioms,axiom,
    semila1241773964035338532et_nat @ inf_inf_set_nat @ top_top_set_nat ).

% inf_top.semilattice_neutr_axioms
thf(fact_431_inf__top_Osemilattice__neutr__axioms,axiom,
    semila6538661356715008162_int_o @ inf_in3604695632404883862_int_o @ top_to1578927101902068148_int_o ).

% inf_top.semilattice_neutr_axioms
thf(fact_432_inf__top_Osemilattice__neutr__axioms,axiom,
    semila5388846958306562758r_assn @ inf_inf_assn @ top_top_assn ).

% inf_top.semilattice_neutr_axioms
thf(fact_433_inf__top_Osemilattice__neutr__axioms,axiom,
    semila7638931295954862799t_unit @ inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit ).

% inf_top.semilattice_neutr_axioms
thf(fact_434_entt__refl,axiom,
    ! [A3: assn] : ( entailst @ A3 @ A3 ) ).

% entt_refl
thf(fact_435_entt__true,axiom,
    ! [A3: assn] : ( entailst @ A3 @ top_top_assn ) ).

% entt_true
thf(fact_436_entt__emp,axiom,
    ! [A3: assn] : ( entailst @ A3 @ one_one_assn ) ).

% entt_emp
thf(fact_437_entt__trans,axiom,
    ! [A3: assn,B3: assn,C2: assn] :
      ( ( entailst @ A3 @ B3 )
     => ( ( entailst @ B3 @ C2 )
       => ( entailst @ A3 @ C2 ) ) ) ).

% entt_trans
thf(fact_438_entt__frame__fwd,axiom,
    ! [P2: assn,Q2: assn,A3: assn,F4: assn,B3: assn] :
      ( ( entailst @ P2 @ Q2 )
     => ( ( entailst @ A3 @ ( times_times_assn @ P2 @ F4 ) )
       => ( ( entailst @ ( times_times_assn @ Q2 @ F4 ) @ B3 )
         => ( entailst @ A3 @ B3 ) ) ) ) ).

% entt_frame_fwd
thf(fact_439_entt__star__mono,axiom,
    ! [A3: assn,B3: assn,C2: assn,D: assn] :
      ( ( entailst @ A3 @ B3 )
     => ( ( entailst @ C2 @ D )
       => ( entailst @ ( times_times_assn @ A3 @ C2 ) @ ( times_times_assn @ B3 @ D ) ) ) ) ).

% entt_star_mono
thf(fact_440_ent__imp__entt,axiom,
    ! [P2: assn,Q2: assn] :
      ( ( entails @ P2 @ Q2 )
     => ( entailst @ P2 @ Q2 ) ) ).

% ent_imp_entt
thf(fact_441_mod__true,axiom,
    ! [H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ top_top_assn @ H2 )
      = ( in_range @ H2 ) ) ).

% mod_true
thf(fact_442_mod__h__bot__iff_I6_J,axiom,
    ! [P2: assn,Q2: assn,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( inf_inf_assn @ P2 @ Q2 ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        & ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(6)
thf(fact_443_mod__h__bot__iff_I1_J,axiom,
    ! [B: $o,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( pure_assn @ B ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = B ) ).

% mod_h_bot_iff(1)
thf(fact_444_in__range__empty,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : ( in_range @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).

% in_range_empty
thf(fact_445_mod__h__bot__iff_I5_J,axiom,
    ! [P2: assn,Q2: assn,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( times_times_assn @ P2 @ Q2 ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        & ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(5)
thf(fact_446_one__assn__raw_Ocases,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
          ( X
         != ( produc7507926704131184380et_nat @ H3 @ As ) ) ).

% one_assn_raw.cases
thf(fact_447_mod__h__bot__indep,axiom,
    ! [P2: assn,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H5 @ bot_bot_set_nat ) ) ) ).

% mod_h_bot_indep
thf(fact_448_mult_Oabel__semigroup__axioms,axiom,
    abel_semigroup_assn @ times_times_assn ).

% mult.abel_semigroup_axioms
thf(fact_449_mult_Oabel__semigroup__axioms,axiom,
    abel_semigroup_nat @ times_times_nat ).

% mult.abel_semigroup_axioms
thf(fact_450_mult_Oabel__semigroup__axioms,axiom,
    abel_semigroup_int @ times_times_int ).

% mult.abel_semigroup_axioms
thf(fact_451_mult_Oabel__semigroup__axioms,axiom,
    abel_s8772020223019846471nteger @ times_3573771949741848930nteger ).

% mult.abel_semigroup_axioms
thf(fact_452_mult_Oabel__semigroup__axioms,axiom,
    abel_s7595615374776563986atural @ times_2397367101498566445atural ).

% mult.abel_semigroup_axioms
thf(fact_453_models__in__range,axiom,
    ! [P2: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ P2 @ H2 )
     => ( in_range @ H2 ) ) ).

% models_in_range
thf(fact_454_inf_Oabel__semigroup__axioms,axiom,
    abel_semigroup_assn @ inf_inf_assn ).

% inf.abel_semigroup_axioms
thf(fact_455_inf_Oabel__semigroup__axioms,axiom,
    abel_s1357164767337380599at_nat @ inf_in2572325071724192079at_nat ).

% inf.abel_semigroup_axioms
thf(fact_456_inf_Oabel__semigroup__axioms,axiom,
    abel_s3907380067910933104et_nat @ inf_inf_set_nat ).

% inf.abel_semigroup_axioms
thf(fact_457_inf_Oabel__semigroup__axioms,axiom,
    abel_semigroup_nat @ inf_inf_nat ).

% inf.abel_semigroup_axioms
thf(fact_458_inf_Oabel__semigroup__axioms,axiom,
    abel_s3861158783995384302_int_o @ inf_in3604695632404883862_int_o ).

% inf.abel_semigroup_axioms
thf(fact_459_mult_Osemigroup__axioms,axiom,
    semigroup_assn @ times_times_assn ).

% mult.semigroup_axioms
thf(fact_460_mult_Osemigroup__axioms,axiom,
    semigroup_nat @ times_times_nat ).

% mult.semigroup_axioms
thf(fact_461_mult_Osemigroup__axioms,axiom,
    semigroup_int @ times_times_int ).

% mult.semigroup_axioms
thf(fact_462_mult_Osemigroup__axioms,axiom,
    semigr7036097719740861967nteger @ times_3573771949741848930nteger ).

% mult.semigroup_axioms
thf(fact_463_mult_Osemigroup__axioms,axiom,
    semigr5859692871497579482atural @ times_2397367101498566445atural ).

% mult.semigroup_axioms
thf(fact_464_inf_Osemilattice__axioms,axiom,
    semilattice_assn @ inf_inf_assn ).

% inf.semilattice_axioms
thf(fact_465_inf_Osemilattice__axioms,axiom,
    semila6306352305301525377at_nat @ inf_in2572325071724192079at_nat ).

% inf.semilattice_axioms
thf(fact_466_inf_Osemilattice__axioms,axiom,
    semilattice_set_nat @ inf_inf_set_nat ).

% inf.semilattice_axioms
thf(fact_467_inf_Osemilattice__axioms,axiom,
    semilattice_nat @ inf_inf_nat ).

% inf.semilattice_axioms
thf(fact_468_inf_Osemilattice__axioms,axiom,
    semila1175464891343163876_int_o @ inf_in3604695632404883862_int_o ).

% inf.semilattice_axioms
thf(fact_469_inf_Osemigroup__axioms,axiom,
    semigroup_assn @ inf_inf_assn ).

% inf.semigroup_axioms
thf(fact_470_inf_Osemigroup__axioms,axiom,
    semigr3814207799437385263at_nat @ inf_in2572325071724192079at_nat ).

% inf.semigroup_axioms
thf(fact_471_inf_Osemigroup__axioms,axiom,
    semigroup_set_nat @ inf_inf_set_nat ).

% inf.semigroup_axioms
thf(fact_472_inf_Osemigroup__axioms,axiom,
    semigroup_nat @ inf_inf_nat ).

% inf.semigroup_axioms
thf(fact_473_inf_Osemigroup__axioms,axiom,
    semigr8095932293087610038_int_o @ inf_in3604695632404883862_int_o ).

% inf.semigroup_axioms
thf(fact_474_mod__h__bot__iff_I2_J,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : ( rep_assn @ top_top_assn @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).

% mod_h_bot_iff(2)
thf(fact_475_mod__emp__simp,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : ( rep_assn @ one_one_assn @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).

% mod_emp_simp
thf(fact_476_pure__assn__raw_Oelims_I3_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat] :
      ( ~ ( pure_a825153325127701367it_nat @ X @ Xa )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xa
              = ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ( ( As = bot_bot_set_nat )
              & X ) ) ) ).

% pure_assn_raw.elims(3)
thf(fact_477_pure__assn__raw_Oelims_I2_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat] :
      ( ( pure_a825153325127701367it_nat @ X @ Xa )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xa
              = ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ~ ( ( As = bot_bot_set_nat )
                & X ) ) ) ).

% pure_assn_raw.elims(2)
thf(fact_478_pure__assn__raw_Oelims_I1_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( pure_a825153325127701367it_nat @ X @ Xa )
        = Y )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xa
              = ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ( Y
              = ( ~ ( ( As = bot_bot_set_nat )
                    & X ) ) ) ) ) ).

% pure_assn_raw.elims(1)
thf(fact_479_pure__assn__raw_Osimps,axiom,
    ! [B: $o,H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( pure_a825153325127701367it_nat @ B @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
      = ( ( As2 = bot_bot_set_nat )
        & B ) ) ).

% pure_assn_raw.simps
thf(fact_480_mod__star__conv,axiom,
    ! [A3: assn,B3: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A3 @ B3 ) @ H2 )
      = ( ? [Hr: heap_e7401611519738050253t_unit,As1: set_nat,As22: set_nat] :
            ( ( H2
              = ( produc7507926704131184380et_nat @ Hr @ ( sup_sup_set_nat @ As1 @ As22 ) ) )
            & ( ( inf_inf_set_nat @ As1 @ As22 )
              = bot_bot_set_nat )
            & ( rep_assn @ A3 @ ( produc7507926704131184380et_nat @ Hr @ As1 ) )
            & ( rep_assn @ B3 @ ( produc7507926704131184380et_nat @ Hr @ As22 ) ) ) ) ) ).

% mod_star_conv
thf(fact_481_star__assnI,axiom,
    ! [P2: assn,H2: heap_e7401611519738050253t_unit,As2: set_nat,Q2: assn,As3: set_nat] :
      ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
     => ( ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ As3 ) )
       => ( ( ( inf_inf_set_nat @ As2 @ As3 )
            = bot_bot_set_nat )
         => ( rep_assn @ ( times_times_assn @ P2 @ Q2 ) @ ( produc7507926704131184380et_nat @ H2 @ ( sup_sup_set_nat @ As2 @ As3 ) ) ) ) ) ) ).

% star_assnI
thf(fact_482_one__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( one_assn_raw @ X )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ( As = bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(3)
thf(fact_483_sup_Oright__idem,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ B )
      = ( sup_sup_set_nat @ A @ B ) ) ).

% sup.right_idem
thf(fact_484_sup_Oright__idem,axiom,
    ! [A: assn,B: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ A @ B ) @ B )
      = ( sup_sup_assn @ A @ B ) ) ).

% sup.right_idem
thf(fact_485_sup__left__idem,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
      = ( sup_sup_set_nat @ X @ Y ) ) ).

% sup_left_idem
thf(fact_486_sup__left__idem,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ X @ Y ) )
      = ( sup_sup_assn @ X @ Y ) ) ).

% sup_left_idem
thf(fact_487_sup_Oleft__idem,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
      = ( sup_sup_set_nat @ A @ B ) ) ).

% sup.left_idem
thf(fact_488_sup_Oleft__idem,axiom,
    ! [A: assn,B: assn] :
      ( ( sup_sup_assn @ A @ ( sup_sup_assn @ A @ B ) )
      = ( sup_sup_assn @ A @ B ) ) ).

% sup.left_idem
thf(fact_489_sup__idem,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_490_sup__idem,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ X )
      = X ) ).

% sup_idem
thf(fact_491_sup_Oidem,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ A )
      = A ) ).

% sup.idem
thf(fact_492_sup_Oidem,axiom,
    ! [A: assn] :
      ( ( sup_sup_assn @ A @ A )
      = A ) ).

% sup.idem
thf(fact_493_sup__bot_Oright__neutral,axiom,
    ! [A: assn] :
      ( ( sup_sup_assn @ A @ bot_bot_assn )
      = A ) ).

% sup_bot.right_neutral
thf(fact_494_sup__bot_Oright__neutral,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_495_sup__bot_Oright__neutral,axiom,
    ! [A: set_int] :
      ( ( sup_sup_set_int @ A @ bot_bot_set_int )
      = A ) ).

% sup_bot.right_neutral
thf(fact_496_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( bot_bot_assn
        = ( sup_sup_assn @ A @ B ) )
      = ( ( A = bot_bot_assn )
        & ( B = bot_bot_assn ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_497_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ A @ B ) )
      = ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_498_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_int
        = ( sup_sup_set_int @ A @ B ) )
      = ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_499_sup__bot_Oleft__neutral,axiom,
    ! [A: assn] :
      ( ( sup_sup_assn @ bot_bot_assn @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_500_sup__bot_Oleft__neutral,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_501_sup__bot_Oleft__neutral,axiom,
    ! [A: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_502_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( ( sup_sup_assn @ A @ B )
        = bot_bot_assn )
      = ( ( A = bot_bot_assn )
        & ( B = bot_bot_assn ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_503_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( sup_sup_set_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_504_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( sup_sup_set_int @ A @ B )
        = bot_bot_set_int )
      = ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_505_sup__eq__bot__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( sup_sup_assn @ X @ Y )
        = bot_bot_assn )
      = ( ( X = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% sup_eq_bot_iff
thf(fact_506_sup__eq__bot__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( sup_sup_set_nat @ X @ Y )
        = bot_bot_set_nat )
      = ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_507_sup__eq__bot__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( sup_sup_set_int @ X @ Y )
        = bot_bot_set_int )
      = ( ( X = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% sup_eq_bot_iff
thf(fact_508_bot__eq__sup__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( bot_bot_assn
        = ( sup_sup_assn @ X @ Y ) )
      = ( ( X = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% bot_eq_sup_iff
thf(fact_509_bot__eq__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ X @ Y ) )
      = ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_510_bot__eq__sup__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( bot_bot_set_int
        = ( sup_sup_set_int @ X @ Y ) )
      = ( ( X = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% bot_eq_sup_iff
thf(fact_511_sup__bot__right,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ bot_bot_assn )
      = X ) ).

% sup_bot_right
thf(fact_512_sup__bot__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% sup_bot_right
thf(fact_513_sup__bot__right,axiom,
    ! [X: set_int] :
      ( ( sup_sup_set_int @ X @ bot_bot_set_int )
      = X ) ).

% sup_bot_right
thf(fact_514_sup__bot__left,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ bot_bot_assn @ X )
      = X ) ).

% sup_bot_left
thf(fact_515_sup__bot__left,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_516_sup__bot__left,axiom,
    ! [X: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ X )
      = X ) ).

% sup_bot_left
thf(fact_517_sup__top__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ top_top_set_nat )
      = top_top_set_nat ) ).

% sup_top_right
thf(fact_518_sup__top__right,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ top_top_assn )
      = top_top_assn ) ).

% sup_top_right
thf(fact_519_sup__top__right,axiom,
    ! [X: set_Product_unit] :
      ( ( sup_su793286257634532545t_unit @ X @ top_to1996260823553986621t_unit )
      = top_to1996260823553986621t_unit ) ).

% sup_top_right
thf(fact_520_sup__top__left,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ top_top_set_nat @ X )
      = top_top_set_nat ) ).

% sup_top_left
thf(fact_521_sup__top__left,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ top_top_assn @ X )
      = top_top_assn ) ).

% sup_top_left
thf(fact_522_sup__top__left,axiom,
    ! [X: set_Product_unit] :
      ( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ X )
      = top_to1996260823553986621t_unit ) ).

% sup_top_left
thf(fact_523_inf__sup__absorb,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_524_inf__sup__absorb,axiom,
    ! [X: nat,Y: nat] :
      ( ( inf_inf_nat @ X @ ( sup_sup_nat @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_525_inf__sup__absorb,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ X @ ( sup_su8463660629351352368_int_o @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_526_inf__sup__absorb,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_527_inf__sup__absorb,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( sup_sup_assn @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_528_sup__inf__absorb,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_529_sup__inf__absorb,axiom,
    ! [X: nat,Y: nat] :
      ( ( sup_sup_nat @ X @ ( inf_inf_nat @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_530_sup__inf__absorb,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( sup_su8463660629351352368_int_o @ X @ ( inf_in3604695632404883862_int_o @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_531_sup__inf__absorb,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_532_sup__inf__absorb,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( inf_inf_assn @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_533_in__range__dist__union,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat,As3: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ ( sup_sup_set_nat @ As2 @ As3 ) ) )
      = ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
        & ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As3 ) ) ) ) ).

% in_range_dist_union
thf(fact_534_sup__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
      = ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_535_sup__left__commute,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ Y @ ( sup_sup_assn @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_536_sup_Oleft__commute,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C ) )
      = ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).

% sup.left_commute
thf(fact_537_sup_Oleft__commute,axiom,
    ! [B: assn,A: assn,C: assn] :
      ( ( sup_sup_assn @ B @ ( sup_sup_assn @ A @ C ) )
      = ( sup_sup_assn @ A @ ( sup_sup_assn @ B @ C ) ) ) ).

% sup.left_commute
thf(fact_538_sup__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] : ( sup_sup_set_nat @ Y2 @ X2 ) ) ) ).

% sup_commute
thf(fact_539_sup__commute,axiom,
    ( sup_sup_assn
    = ( ^ [X2: assn,Y2: assn] : ( sup_sup_assn @ Y2 @ X2 ) ) ) ).

% sup_commute
thf(fact_540_sup_Ocommute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] : ( sup_sup_set_nat @ B2 @ A2 ) ) ) ).

% sup.commute
thf(fact_541_sup_Ocommute,axiom,
    ( sup_sup_assn
    = ( ^ [A2: assn,B2: assn] : ( sup_sup_assn @ B2 @ A2 ) ) ) ).

% sup.commute
thf(fact_542_sup__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
      = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_543_sup__assoc,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ X @ Y ) @ Z )
      = ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_544_sup_Oassoc,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C )
      = ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).

% sup.assoc
thf(fact_545_sup_Oassoc,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ A @ B ) @ C )
      = ( sup_sup_assn @ A @ ( sup_sup_assn @ B @ C ) ) ) ).

% sup.assoc
thf(fact_546_inf__sup__aci_I5_J,axiom,
    ( sup_sup_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] : ( sup_sup_set_nat @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_547_inf__sup__aci_I5_J,axiom,
    ( sup_sup_assn
    = ( ^ [X2: assn,Y2: assn] : ( sup_sup_assn @ Y2 @ X2 ) ) ) ).

% inf_sup_aci(5)
thf(fact_548_inf__sup__aci_I6_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
      = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_549_inf__sup__aci_I6_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ X @ Y ) @ Z )
      = ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_550_inf__sup__aci_I7_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
      = ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_551_inf__sup__aci_I7_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ Y @ ( sup_sup_assn @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_552_inf__sup__aci_I8_J,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
      = ( sup_sup_set_nat @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_553_inf__sup__aci_I8_J,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ X @ Y ) )
      = ( sup_sup_assn @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_554_distrib__imp1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ X3 @ ( sup_su6327502436637775413at_nat @ Y3 @ Z2 ) )
          = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X3 @ Y3 ) @ ( inf_in2572325071724192079at_nat @ X3 @ Z2 ) ) )
     => ( ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
        = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_555_distrib__imp1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ! [X3: nat,Y3: nat,Z2: nat] :
          ( ( inf_inf_nat @ X3 @ ( sup_sup_nat @ Y3 @ Z2 ) )
          = ( sup_sup_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ ( inf_inf_nat @ X3 @ Z2 ) ) )
     => ( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
        = ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_556_distrib__imp1,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
      ( ! [X3: product_prod_int_int > $o,Y3: product_prod_int_int > $o,Z2: product_prod_int_int > $o] :
          ( ( inf_in3604695632404883862_int_o @ X3 @ ( sup_su8463660629351352368_int_o @ Y3 @ Z2 ) )
          = ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ X3 @ Y3 ) @ ( inf_in3604695632404883862_int_o @ X3 @ Z2 ) ) )
     => ( ( sup_su8463660629351352368_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) )
        = ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ X @ Y ) @ ( sup_su8463660629351352368_int_o @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_557_distrib__imp1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
          ( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
          = ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) )
     => ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
        = ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_558_distrib__imp1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ! [X3: assn,Y3: assn,Z2: assn] :
          ( ( inf_inf_assn @ X3 @ ( sup_sup_assn @ Y3 @ Z2 ) )
          = ( sup_sup_assn @ ( inf_inf_assn @ X3 @ Y3 ) @ ( inf_inf_assn @ X3 @ Z2 ) ) )
     => ( ( sup_sup_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
        = ( inf_inf_assn @ ( sup_sup_assn @ X @ Y ) @ ( sup_sup_assn @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_559_distrib__imp2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
          ( ( sup_su6327502436637775413at_nat @ X3 @ ( inf_in2572325071724192079at_nat @ Y3 @ Z2 ) )
          = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X3 @ Y3 ) @ ( sup_su6327502436637775413at_nat @ X3 @ Z2 ) ) )
     => ( ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z ) )
        = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_560_distrib__imp2,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ! [X3: nat,Y3: nat,Z2: nat] :
          ( ( sup_sup_nat @ X3 @ ( inf_inf_nat @ Y3 @ Z2 ) )
          = ( inf_inf_nat @ ( sup_sup_nat @ X3 @ Y3 ) @ ( sup_sup_nat @ X3 @ Z2 ) ) )
     => ( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
        = ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_561_distrib__imp2,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
      ( ! [X3: product_prod_int_int > $o,Y3: product_prod_int_int > $o,Z2: product_prod_int_int > $o] :
          ( ( sup_su8463660629351352368_int_o @ X3 @ ( inf_in3604695632404883862_int_o @ Y3 @ Z2 ) )
          = ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ X3 @ Y3 ) @ ( sup_su8463660629351352368_int_o @ X3 @ Z2 ) ) )
     => ( ( inf_in3604695632404883862_int_o @ X @ ( sup_su8463660629351352368_int_o @ Y @ Z ) )
        = ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ ( inf_in3604695632404883862_int_o @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_562_distrib__imp2,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
          ( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
          = ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) )
     => ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
        = ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_563_distrib__imp2,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ! [X3: assn,Y3: assn,Z2: assn] :
          ( ( sup_sup_assn @ X3 @ ( inf_inf_assn @ Y3 @ Z2 ) )
          = ( inf_inf_assn @ ( sup_sup_assn @ X3 @ Y3 ) @ ( sup_sup_assn @ X3 @ Z2 ) ) )
     => ( ( inf_inf_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
        = ( sup_sup_assn @ ( inf_inf_assn @ X @ Y ) @ ( inf_inf_assn @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_564_inf__sup__distrib1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z ) )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_565_inf__sup__distrib1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
      = ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_566_inf__sup__distrib1,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ X @ ( sup_su8463660629351352368_int_o @ Y @ Z ) )
      = ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ ( inf_in3604695632404883862_int_o @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_567_inf__sup__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_568_inf__sup__distrib1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ ( inf_inf_assn @ X @ Y ) @ ( inf_inf_assn @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_569_inf__sup__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ Y @ Z ) @ X )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ Y @ X ) @ ( inf_in2572325071724192079at_nat @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_570_inf__sup__distrib2,axiom,
    ! [Y: nat,Z: nat,X: nat] :
      ( ( inf_inf_nat @ ( sup_sup_nat @ Y @ Z ) @ X )
      = ( sup_sup_nat @ ( inf_inf_nat @ Y @ X ) @ ( inf_inf_nat @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_571_inf__sup__distrib2,axiom,
    ! [Y: product_prod_int_int > $o,Z: product_prod_int_int > $o,X: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ Y @ Z ) @ X )
      = ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ Y @ X ) @ ( inf_in3604695632404883862_int_o @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_572_inf__sup__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X ) @ ( inf_inf_set_nat @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_573_inf__sup__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] :
      ( ( inf_inf_assn @ ( sup_sup_assn @ Y @ Z ) @ X )
      = ( sup_sup_assn @ ( inf_inf_assn @ Y @ X ) @ ( inf_inf_assn @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_574_sup__inf__distrib1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_575_sup__inf__distrib1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
      = ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_576_sup__inf__distrib1,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
      ( ( sup_su8463660629351352368_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) )
      = ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ X @ Y ) @ ( sup_su8463660629351352368_int_o @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_577_sup__inf__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_578_sup__inf__distrib1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( inf_inf_assn @ ( sup_sup_assn @ X @ Y ) @ ( sup_sup_assn @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_579_sup__inf__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ Y @ Z ) @ X )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ Y @ X ) @ ( sup_su6327502436637775413at_nat @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_580_sup__inf__distrib2,axiom,
    ! [Y: nat,Z: nat,X: nat] :
      ( ( sup_sup_nat @ ( inf_inf_nat @ Y @ Z ) @ X )
      = ( inf_inf_nat @ ( sup_sup_nat @ Y @ X ) @ ( sup_sup_nat @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_581_sup__inf__distrib2,axiom,
    ! [Y: product_prod_int_int > $o,Z: product_prod_int_int > $o,X: product_prod_int_int > $o] :
      ( ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ Y @ Z ) @ X )
      = ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ Y @ X ) @ ( sup_su8463660629351352368_int_o @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_582_sup__inf__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z ) @ X )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X ) @ ( sup_sup_set_nat @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_583_sup__inf__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] :
      ( ( sup_sup_assn @ ( inf_inf_assn @ Y @ Z ) @ X )
      = ( inf_inf_assn @ ( sup_sup_assn @ Y @ X ) @ ( sup_sup_assn @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_584_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z ) )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_585_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ X @ ( sup_su8463660629351352368_int_o @ Y @ Z ) )
      = ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ ( inf_in3604695632404883862_int_o @ X @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_586_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_587_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ ( inf_inf_assn @ X @ Y ) @ ( inf_inf_assn @ X @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_588_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_589_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
      ( ( sup_su8463660629351352368_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) )
      = ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ X @ Y ) @ ( sup_su8463660629351352368_int_o @ X @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_590_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_591_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( inf_inf_assn @ ( sup_sup_assn @ X @ Y ) @ ( sup_sup_assn @ X @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_592_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ Y @ Z ) @ X )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ Y @ X ) @ ( inf_in2572325071724192079at_nat @ Z @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_593_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: product_prod_int_int > $o,Z: product_prod_int_int > $o,X: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ Y @ Z ) @ X )
      = ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ Y @ X ) @ ( inf_in3604695632404883862_int_o @ Z @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_594_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X ) @ ( inf_inf_set_nat @ Z @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_595_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] :
      ( ( inf_inf_assn @ ( sup_sup_assn @ Y @ Z ) @ X )
      = ( sup_sup_assn @ ( inf_inf_assn @ Y @ X ) @ ( inf_inf_assn @ Z @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_596_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ Y @ Z ) @ X )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ Y @ X ) @ ( sup_su6327502436637775413at_nat @ Z @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_597_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: product_prod_int_int > $o,Z: product_prod_int_int > $o,X: product_prod_int_int > $o] :
      ( ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ Y @ Z ) @ X )
      = ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ Y @ X ) @ ( sup_su8463660629351352368_int_o @ Z @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_598_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z ) @ X )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X ) @ ( sup_sup_set_nat @ Z @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_599_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] :
      ( ( sup_sup_assn @ ( inf_inf_assn @ Y @ Z ) @ X )
      = ( inf_inf_assn @ ( sup_sup_assn @ Y @ X ) @ ( sup_sup_assn @ Z @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_600_sup_Oabel__semigroup__axioms,axiom,
    abel_s3907380067910933104et_nat @ sup_sup_set_nat ).

% sup.abel_semigroup_axioms
thf(fact_601_sup_Oabel__semigroup__axioms,axiom,
    abel_semigroup_assn @ sup_sup_assn ).

% sup.abel_semigroup_axioms
thf(fact_602_sup_Osemilattice__axioms,axiom,
    semilattice_set_nat @ sup_sup_set_nat ).

% sup.semilattice_axioms
thf(fact_603_sup_Osemilattice__axioms,axiom,
    semilattice_assn @ sup_sup_assn ).

% sup.semilattice_axioms
thf(fact_604_sup_Osemigroup__axioms,axiom,
    semigroup_set_nat @ sup_sup_set_nat ).

% sup.semigroup_axioms
thf(fact_605_sup_Osemigroup__axioms,axiom,
    semigroup_assn @ sup_sup_assn ).

% sup.semigroup_axioms
thf(fact_606_sup__bot_Ocomm__monoid__axioms,axiom,
    comm_monoid_assn @ sup_sup_assn @ bot_bot_assn ).

% sup_bot.comm_monoid_axioms
thf(fact_607_sup__bot_Ocomm__monoid__axioms,axiom,
    comm_monoid_set_nat @ sup_sup_set_nat @ bot_bot_set_nat ).

% sup_bot.comm_monoid_axioms
thf(fact_608_sup__bot_Ocomm__monoid__axioms,axiom,
    comm_monoid_set_int @ sup_sup_set_int @ bot_bot_set_int ).

% sup_bot.comm_monoid_axioms
thf(fact_609_sup__bot_Omonoid__axioms,axiom,
    monoid_assn @ sup_sup_assn @ bot_bot_assn ).

% sup_bot.monoid_axioms
thf(fact_610_sup__bot_Omonoid__axioms,axiom,
    monoid_set_nat @ sup_sup_set_nat @ bot_bot_set_nat ).

% sup_bot.monoid_axioms
thf(fact_611_sup__bot_Omonoid__axioms,axiom,
    monoid_set_int @ sup_sup_set_int @ bot_bot_set_int ).

% sup_bot.monoid_axioms
thf(fact_612_sup__bot_Osemilattice__neutr__axioms,axiom,
    semila5388846958306562758r_assn @ sup_sup_assn @ bot_bot_assn ).

% sup_bot.semilattice_neutr_axioms
thf(fact_613_sup__bot_Osemilattice__neutr__axioms,axiom,
    semila1241773964035338532et_nat @ sup_sup_set_nat @ bot_bot_set_nat ).

% sup_bot.semilattice_neutr_axioms
thf(fact_614_sup__bot_Osemilattice__neutr__axioms,axiom,
    semila6287294981380917632et_int @ sup_sup_set_int @ bot_bot_set_int ).

% sup_bot.semilattice_neutr_axioms
thf(fact_615_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A @ X )
        = bot_bo2099793752762293965at_nat )
     => ( ( ( sup_su6327502436637775413at_nat @ A @ X )
          = top_to4669805908274784177at_nat )
       => ( ( ( inf_in2572325071724192079at_nat @ A @ Y )
            = bot_bo2099793752762293965at_nat )
         => ( ( ( sup_su6327502436637775413at_nat @ A @ Y )
              = top_to4669805908274784177at_nat )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
thf(fact_616_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: product_prod_int_int > $o,X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( ( inf_in3604695632404883862_int_o @ A @ X )
        = bot_bo8147686125503663512_int_o )
     => ( ( ( sup_su8463660629351352368_int_o @ A @ X )
          = top_to1578927101902068148_int_o )
       => ( ( ( inf_in3604695632404883862_int_o @ A @ Y )
            = bot_bo8147686125503663512_int_o )
         => ( ( ( sup_su8463660629351352368_int_o @ A @ Y )
              = top_to1578927101902068148_int_o )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
thf(fact_617_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: set_nat,X: set_nat,Y: set_nat] :
      ( ( ( inf_inf_set_nat @ A @ X )
        = bot_bot_set_nat )
     => ( ( ( sup_sup_set_nat @ A @ X )
          = top_top_set_nat )
       => ( ( ( inf_inf_set_nat @ A @ Y )
            = bot_bot_set_nat )
         => ( ( ( sup_sup_set_nat @ A @ Y )
              = top_top_set_nat )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
thf(fact_618_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: set_int,X: set_int,Y: set_int] :
      ( ( ( inf_inf_set_int @ A @ X )
        = bot_bot_set_int )
     => ( ( ( sup_sup_set_int @ A @ X )
          = top_top_set_int )
       => ( ( ( inf_inf_set_int @ A @ Y )
            = bot_bot_set_int )
         => ( ( ( sup_sup_set_int @ A @ Y )
              = top_top_set_int )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
thf(fact_619_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: assn,X: assn,Y: assn] :
      ( ( ( inf_inf_assn @ A @ X )
        = bot_bot_assn )
     => ( ( ( sup_sup_assn @ A @ X )
          = top_top_assn )
       => ( ( ( inf_inf_assn @ A @ Y )
            = bot_bot_assn )
         => ( ( ( sup_sup_assn @ A @ Y )
              = top_top_assn )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
thf(fact_620_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: set_Product_unit,X: set_Product_unit,Y: set_Product_unit] :
      ( ( ( inf_in4660618365625256667t_unit @ A @ X )
        = bot_bo3957492148770167129t_unit )
     => ( ( ( sup_su793286257634532545t_unit @ A @ X )
          = top_to1996260823553986621t_unit )
       => ( ( ( inf_in4660618365625256667t_unit @ A @ Y )
            = bot_bo3957492148770167129t_unit )
         => ( ( ( sup_su793286257634532545t_unit @ A @ Y )
              = top_to1996260823553986621t_unit )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
thf(fact_621_one__assn__raw_Osimps,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( one_assn_raw @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
      = ( As2 = bot_bot_set_nat ) ) ).

% one_assn_raw.simps
thf(fact_622_one__assn__raw_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( one_assn_raw @ X )
        = Y )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ( Y
              = ( As != bot_bot_set_nat ) ) ) ) ).

% one_assn_raw.elims(1)
thf(fact_623_one__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( one_assn_raw @ X )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ( As != bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(2)
thf(fact_624_Int__Un__eq_I4_J,axiom,
    ! [T: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ T @ ( inf_in2572325071724192079at_nat @ S @ T ) )
      = T ) ).

% Int_Un_eq(4)
thf(fact_625_Int__Un__eq_I4_J,axiom,
    ! [T: set_nat,S: set_nat] :
      ( ( sup_sup_set_nat @ T @ ( inf_inf_set_nat @ S @ T ) )
      = T ) ).

% Int_Un_eq(4)
thf(fact_626_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ S @ ( inf_in2572325071724192079at_nat @ S @ T ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_627_Int__Un__eq_I3_J,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_628_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S @ T ) @ T )
      = T ) ).

% Int_Un_eq(2)
thf(fact_629_Int__Un__eq_I2_J,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T ) @ T )
      = T ) ).

% Int_Un_eq(2)
thf(fact_630_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S @ T ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_631_Int__Un__eq_I1_J,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_632_Un__Int__eq_I4_J,axiom,
    ! [T: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ T @ ( sup_su6327502436637775413at_nat @ S @ T ) )
      = T ) ).

% Un_Int_eq(4)
thf(fact_633_Un__Int__eq_I4_J,axiom,
    ! [T: set_nat,S: set_nat] :
      ( ( inf_inf_set_nat @ T @ ( sup_sup_set_nat @ S @ T ) )
      = T ) ).

% Un_Int_eq(4)
thf(fact_634_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ S @ ( sup_su6327502436637775413at_nat @ S @ T ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_635_Un__Int__eq_I3_J,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_636_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S @ T ) @ T )
      = T ) ).

% Un_Int_eq(2)
thf(fact_637_Un__Int__eq_I2_J,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T ) @ T )
      = T ) ).

% Un_Int_eq(2)
thf(fact_638_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S @ T ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_639_Un__Int__eq_I1_J,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_640_wand__assnI,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat,Q2: assn,R: assn] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
     => ( ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( ( inf_inf_set_nat @ As2 @ As4 )
              = bot_bot_set_nat )
           => ( ( relH @ As2 @ H2 @ H4 )
             => ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As2 ) )
               => ( ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H4 @ As4 ) )
                 => ( rep_assn @ R @ ( produc7507926704131184380et_nat @ H4 @ ( sup_sup_set_nat @ As2 @ As4 ) ) ) ) ) ) )
       => ( rep_assn @ ( wand_assn @ Q2 @ R ) @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) ) ).

% wand_assnI
thf(fact_641_IntI,axiom,
    ! [C: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C @ A3 )
     => ( ( member6576561426505652726_nat_o @ C @ B3 )
       => ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) ) ) ) ).

% IntI
thf(fact_642_IntI,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( ( member_int2 @ C @ A3 )
     => ( ( member_int2 @ C @ B3 )
       => ( member_int2 @ C @ ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ).

% IntI
thf(fact_643_IntI,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ A3 )
     => ( ( member8440522571783428010at_nat @ C @ B3 )
       => ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ).

% IntI
thf(fact_644_IntI,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat2 @ C @ A3 )
     => ( ( member_nat2 @ C @ B3 )
       => ( member_nat2 @ C @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).

% IntI
thf(fact_645_Int__iff,axiom,
    ! [C: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) )
      = ( ( member6576561426505652726_nat_o @ C @ A3 )
        & ( member6576561426505652726_nat_o @ C @ B3 ) ) ) ).

% Int_iff
thf(fact_646_Int__iff,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( ( member_int2 @ C @ ( inf_inf_set_int @ A3 @ B3 ) )
      = ( ( member_int2 @ C @ A3 )
        & ( member_int2 @ C @ B3 ) ) ) ).

% Int_iff
thf(fact_647_Int__iff,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
      = ( ( member8440522571783428010at_nat @ C @ A3 )
        & ( member8440522571783428010at_nat @ C @ B3 ) ) ) ).

% Int_iff
thf(fact_648_Int__iff,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat2 @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
      = ( ( member_nat2 @ C @ A3 )
        & ( member_nat2 @ C @ B3 ) ) ) ).

% Int_iff
thf(fact_649_Int__UNIV,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = top_to4669805908274784177at_nat )
      = ( ( A3 = top_to4669805908274784177at_nat )
        & ( B3 = top_to4669805908274784177at_nat ) ) ) ).

% Int_UNIV
thf(fact_650_Int__UNIV,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B3 )
        = top_top_set_nat )
      = ( ( A3 = top_top_set_nat )
        & ( B3 = top_top_set_nat ) ) ) ).

% Int_UNIV
thf(fact_651_Int__UNIV,axiom,
    ! [A3: set_Product_unit,B3: set_Product_unit] :
      ( ( ( inf_in4660618365625256667t_unit @ A3 @ B3 )
        = top_to1996260823553986621t_unit )
      = ( ( A3 = top_to1996260823553986621t_unit )
        & ( B3 = top_to1996260823553986621t_unit ) ) ) ).

% Int_UNIV
thf(fact_652_mod__or__dist,axiom,
    ! [P2: assn,Q2: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( sup_sup_assn @ P2 @ Q2 ) @ H2 )
      = ( ( rep_assn @ P2 @ H2 )
        | ( rep_assn @ Q2 @ H2 ) ) ) ).

% mod_or_dist
thf(fact_653_merge__pure__or,axiom,
    ! [A: $o,B: $o] :
      ( ( sup_sup_assn @ ( pure_assn @ A ) @ ( pure_assn @ B ) )
      = ( pure_assn
        @ ( A
          | B ) ) ) ).

% merge_pure_or
thf(fact_654_relH__dist__union,axiom,
    ! [As2: set_nat,As3: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( relH @ ( sup_sup_set_nat @ As2 @ As3 ) @ H2 @ H5 )
      = ( ( relH @ As2 @ H2 @ H5 )
        & ( relH @ As3 @ H2 @ H5 ) ) ) ).

% relH_dist_union
thf(fact_655_mod__h__bot__iff_I7_J,axiom,
    ! [P2: assn,Q2: assn,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( sup_sup_assn @ P2 @ Q2 ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        | ( rep_assn @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(7)
thf(fact_656_relH__sym,axiom,
    ! [As2: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( relH @ As2 @ H2 @ H5 )
     => ( relH @ As2 @ H5 @ H2 ) ) ).

% relH_sym
thf(fact_657_relH__trans,axiom,
    ! [As2: set_nat,H12: heap_e7401611519738050253t_unit,H23: heap_e7401611519738050253t_unit,H32: heap_e7401611519738050253t_unit] :
      ( ( relH @ As2 @ H12 @ H23 )
     => ( ( relH @ As2 @ H23 @ H32 )
       => ( relH @ As2 @ H12 @ H32 ) ) ) ).

% relH_trans
thf(fact_658_star__or__dist2,axiom,
    ! [C2: assn,A3: assn,B3: assn] :
      ( ( times_times_assn @ C2 @ ( sup_sup_assn @ A3 @ B3 ) )
      = ( sup_sup_assn @ ( times_times_assn @ C2 @ A3 ) @ ( times_times_assn @ C2 @ B3 ) ) ) ).

% star_or_dist2
thf(fact_659_star__or__dist1,axiom,
    ! [A3: assn,B3: assn,C2: assn] :
      ( ( times_times_assn @ ( sup_sup_assn @ A3 @ B3 ) @ C2 )
      = ( sup_sup_assn @ ( times_times_assn @ A3 @ C2 ) @ ( times_times_assn @ B3 @ C2 ) ) ) ).

% star_or_dist1
thf(fact_660_ent__disjI2__direct,axiom,
    ! [B3: assn,A3: assn] : ( entails @ B3 @ ( sup_sup_assn @ A3 @ B3 ) ) ).

% ent_disjI2_direct
thf(fact_661_ent__disjI1__direct,axiom,
    ! [A3: assn,B3: assn] : ( entails @ A3 @ ( sup_sup_assn @ A3 @ B3 ) ) ).

% ent_disjI1_direct
thf(fact_662_ent__disjI2_H,axiom,
    ! [A3: assn,C2: assn,B3: assn] :
      ( ( entails @ A3 @ C2 )
     => ( entails @ A3 @ ( sup_sup_assn @ B3 @ C2 ) ) ) ).

% ent_disjI2'
thf(fact_663_ent__disjI1_H,axiom,
    ! [A3: assn,B3: assn,C2: assn] :
      ( ( entails @ A3 @ B3 )
     => ( entails @ A3 @ ( sup_sup_assn @ B3 @ C2 ) ) ) ).

% ent_disjI1'
thf(fact_664_ent__disjI2,axiom,
    ! [P2: assn,Q2: assn,R: assn] :
      ( ( entails @ ( sup_sup_assn @ P2 @ Q2 ) @ R )
     => ( entails @ Q2 @ R ) ) ).

% ent_disjI2
thf(fact_665_ent__disjI1,axiom,
    ! [P2: assn,Q2: assn,R: assn] :
      ( ( entails @ ( sup_sup_assn @ P2 @ Q2 ) @ R )
     => ( entails @ P2 @ R ) ) ).

% ent_disjI1
thf(fact_666_ent__disjE,axiom,
    ! [A3: assn,C2: assn,B3: assn] :
      ( ( entails @ A3 @ C2 )
     => ( ( entails @ B3 @ C2 )
       => ( entails @ ( sup_sup_assn @ A3 @ B3 ) @ C2 ) ) ) ).

% ent_disjE
thf(fact_667_entt__disjE,axiom,
    ! [A3: assn,M: assn,B3: assn] :
      ( ( entailst @ A3 @ M )
     => ( ( entailst @ B3 @ M )
       => ( entailst @ ( sup_sup_assn @ A3 @ B3 ) @ M ) ) ) ).

% entt_disjE
thf(fact_668_entt__disjD1,axiom,
    ! [A3: assn,B3: assn,C2: assn] :
      ( ( entailst @ ( sup_sup_assn @ A3 @ B3 ) @ C2 )
     => ( entailst @ A3 @ C2 ) ) ).

% entt_disjD1
thf(fact_669_entt__disjD2,axiom,
    ! [A3: assn,B3: assn,C2: assn] :
      ( ( entailst @ ( sup_sup_assn @ A3 @ B3 ) @ C2 )
     => ( entailst @ B3 @ C2 ) ) ).

% entt_disjD2
thf(fact_670_entt__disjI1_H,axiom,
    ! [A3: assn,B3: assn,C2: assn] :
      ( ( entailst @ A3 @ B3 )
     => ( entailst @ A3 @ ( sup_sup_assn @ B3 @ C2 ) ) ) ).

% entt_disjI1'
thf(fact_671_entt__disjI2_H,axiom,
    ! [A3: assn,C2: assn,B3: assn] :
      ( ( entailst @ A3 @ C2 )
     => ( entailst @ A3 @ ( sup_sup_assn @ B3 @ C2 ) ) ) ).

% entt_disjI2'
thf(fact_672_entt__disjI1__direct,axiom,
    ! [A3: assn,B3: assn] : ( entailst @ A3 @ ( sup_sup_assn @ A3 @ B3 ) ) ).

% entt_disjI1_direct
thf(fact_673_entt__disjI2__direct,axiom,
    ! [B3: assn,A3: assn] : ( entailst @ B3 @ ( sup_sup_assn @ A3 @ B3 ) ) ).

% entt_disjI2_direct
thf(fact_674_norm__assertion__simps_I11_J,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ top_top_assn @ X )
      = top_top_assn ) ).

% norm_assertion_simps(11)
thf(fact_675_norm__assertion__simps_I12_J,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ top_top_assn )
      = top_top_assn ) ).

% norm_assertion_simps(12)
thf(fact_676_norm__assertion__simps_I5_J,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ bot_bot_assn @ X )
      = X ) ).

% norm_assertion_simps(5)
thf(fact_677_norm__assertion__simps_I6_J,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ bot_bot_assn )
      = X ) ).

% norm_assertion_simps(6)
thf(fact_678_mod__relH,axiom,
    ! [As2: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,P2: assn] :
      ( ( relH @ As2 @ H2 @ H5 )
     => ( ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
        = ( rep_assn @ P2 @ ( produc7507926704131184380et_nat @ H5 @ As2 ) ) ) ) ).

% mod_relH
thf(fact_679_relH__in__rangeI_I2_J,axiom,
    ! [As2: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( relH @ As2 @ H2 @ H5 )
     => ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As2 ) ) ) ).

% relH_in_rangeI(2)
thf(fact_680_relH__in__rangeI_I1_J,axiom,
    ! [As2: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( relH @ As2 @ H2 @ H5 )
     => ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) ).

% relH_in_rangeI(1)
thf(fact_681_relH__refl,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
     => ( relH @ As2 @ H2 @ H2 ) ) ).

% relH_refl
thf(fact_682_Int__UNIV__right,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ top_to4669805908274784177at_nat )
      = A3 ) ).

% Int_UNIV_right
thf(fact_683_Int__UNIV__right,axiom,
    ! [A3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ top_top_set_nat )
      = A3 ) ).

% Int_UNIV_right
thf(fact_684_Int__UNIV__right,axiom,
    ! [A3: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit @ A3 @ top_to1996260823553986621t_unit )
      = A3 ) ).

% Int_UNIV_right
thf(fact_685_Int__UNIV__left,axiom,
    ! [B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ top_to4669805908274784177at_nat @ B3 )
      = B3 ) ).

% Int_UNIV_left
thf(fact_686_Int__UNIV__left,axiom,
    ! [B3: set_nat] :
      ( ( inf_inf_set_nat @ top_top_set_nat @ B3 )
      = B3 ) ).

% Int_UNIV_left
thf(fact_687_Int__UNIV__left,axiom,
    ! [B3: set_Product_unit] :
      ( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ B3 )
      = B3 ) ).

% Int_UNIV_left
thf(fact_688_Int__left__commute,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B3 @ C2 ) )
      = ( inf_in2572325071724192079at_nat @ B3 @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) ) ) ).

% Int_left_commute
thf(fact_689_Int__left__commute,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C2 ) )
      = ( inf_inf_set_nat @ B3 @ ( inf_inf_set_nat @ A3 @ C2 ) ) ) ).

% Int_left_commute
thf(fact_690_Int__left__absorb,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
      = ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ).

% Int_left_absorb
thf(fact_691_Int__left__absorb,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ A3 @ B3 ) )
      = ( inf_inf_set_nat @ A3 @ B3 ) ) ).

% Int_left_absorb
thf(fact_692_Int__commute,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ B4 @ A4 ) ) ) ).

% Int_commute
thf(fact_693_Int__commute,axiom,
    ( inf_inf_set_nat
    = ( ^ [A4: set_nat,B4: set_nat] : ( inf_inf_set_nat @ B4 @ A4 ) ) ) ).

% Int_commute
thf(fact_694_Int__absorb,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ A3 )
      = A3 ) ).

% Int_absorb
thf(fact_695_Int__absorb,axiom,
    ! [A3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ A3 )
      = A3 ) ).

% Int_absorb
thf(fact_696_Int__assoc,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ C2 )
      = ( inf_in2572325071724192079at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B3 @ C2 ) ) ) ).

% Int_assoc
thf(fact_697_Int__assoc,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ C2 )
      = ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C2 ) ) ) ).

% Int_assoc
thf(fact_698_IntD2,axiom,
    ! [C: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) )
     => ( member6576561426505652726_nat_o @ C @ B3 ) ) ).

% IntD2
thf(fact_699_IntD2,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( ( member_int2 @ C @ ( inf_inf_set_int @ A3 @ B3 ) )
     => ( member_int2 @ C @ B3 ) ) ).

% IntD2
thf(fact_700_IntD2,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
     => ( member8440522571783428010at_nat @ C @ B3 ) ) ).

% IntD2
thf(fact_701_IntD2,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat2 @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
     => ( member_nat2 @ C @ B3 ) ) ).

% IntD2
thf(fact_702_IntD1,axiom,
    ! [C: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) )
     => ( member6576561426505652726_nat_o @ C @ A3 ) ) ).

% IntD1
thf(fact_703_IntD1,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( ( member_int2 @ C @ ( inf_inf_set_int @ A3 @ B3 ) )
     => ( member_int2 @ C @ A3 ) ) ).

% IntD1
thf(fact_704_IntD1,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
     => ( member8440522571783428010at_nat @ C @ A3 ) ) ).

% IntD1
thf(fact_705_IntD1,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat2 @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
     => ( member_nat2 @ C @ A3 ) ) ).

% IntD1
thf(fact_706_IntE,axiom,
    ! [C: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C @ ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) )
     => ~ ( ( member6576561426505652726_nat_o @ C @ A3 )
         => ~ ( member6576561426505652726_nat_o @ C @ B3 ) ) ) ).

% IntE
thf(fact_707_IntE,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( ( member_int2 @ C @ ( inf_inf_set_int @ A3 @ B3 ) )
     => ~ ( ( member_int2 @ C @ A3 )
         => ~ ( member_int2 @ C @ B3 ) ) ) ).

% IntE
thf(fact_708_IntE,axiom,
    ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
     => ~ ( ( member8440522571783428010at_nat @ C @ A3 )
         => ~ ( member8440522571783428010at_nat @ C @ B3 ) ) ) ).

% IntE
thf(fact_709_IntE,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat2 @ C @ ( inf_inf_set_nat @ A3 @ B3 ) )
     => ~ ( ( member_nat2 @ C @ A3 )
         => ~ ( member_nat2 @ C @ B3 ) ) ) ).

% IntE
thf(fact_710_disjoint__iff__not__equal,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A3 )
           => ! [Y2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y2 @ B3 )
               => ( X2 != Y2 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_711_disjoint__iff__not__equal,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B3 )
        = bot_bot_set_nat )
      = ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ A3 )
           => ! [Y2: nat] :
                ( ( member_nat2 @ Y2 @ B3 )
               => ( X2 != Y2 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_712_disjoint__iff__not__equal,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ( inf_inf_set_int @ A3 @ B3 )
        = bot_bot_set_int )
      = ( ! [X2: int] :
            ( ( member_int2 @ X2 @ A3 )
           => ! [Y2: int] :
                ( ( member_int2 @ Y2 @ B3 )
               => ( X2 != Y2 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_713_Int__empty__right,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% Int_empty_right
thf(fact_714_Int__empty__right,axiom,
    ! [A3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% Int_empty_right
thf(fact_715_Int__empty__right,axiom,
    ! [A3: set_int] :
      ( ( inf_inf_set_int @ A3 @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% Int_empty_right
thf(fact_716_Int__empty__left,axiom,
    ! [B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ B3 )
      = bot_bo2099793752762293965at_nat ) ).

% Int_empty_left
thf(fact_717_Int__empty__left,axiom,
    ! [B3: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ B3 )
      = bot_bot_set_nat ) ).

% Int_empty_left
thf(fact_718_Int__empty__left,axiom,
    ! [B3: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ B3 )
      = bot_bot_set_int ) ).

% Int_empty_left
thf(fact_719_disjoint__iff,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( ( inf_in1906310914598751387_nat_o @ A3 @ B3 )
        = bot_bo7824918357723582233_nat_o )
      = ( ! [X2: produc3658429121746597890et_nat > $o] :
            ( ( member6576561426505652726_nat_o @ X2 @ A3 )
           => ~ ( member6576561426505652726_nat_o @ X2 @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_720_disjoint__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A3 )
           => ~ ( member8440522571783428010at_nat @ X2 @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_721_disjoint__iff,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B3 )
        = bot_bot_set_nat )
      = ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ A3 )
           => ~ ( member_nat2 @ X2 @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_722_disjoint__iff,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ( inf_inf_set_int @ A3 @ B3 )
        = bot_bot_set_int )
      = ( ! [X2: int] :
            ( ( member_int2 @ X2 @ A3 )
           => ~ ( member_int2 @ X2 @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_723_Int__emptyI,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ! [X3: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ X3 @ A3 )
         => ~ ( member6576561426505652726_nat_o @ X3 @ B3 ) )
     => ( ( inf_in1906310914598751387_nat_o @ A3 @ B3 )
        = bot_bo7824918357723582233_nat_o ) ) ).

% Int_emptyI
thf(fact_724_Int__emptyI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A3 )
         => ~ ( member8440522571783428010at_nat @ X3 @ B3 ) )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat ) ) ).

% Int_emptyI
thf(fact_725_Int__emptyI,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat2 @ X3 @ A3 )
         => ~ ( member_nat2 @ X3 @ B3 ) )
     => ( ( inf_inf_set_nat @ A3 @ B3 )
        = bot_bot_set_nat ) ) ).

% Int_emptyI
thf(fact_726_Int__emptyI,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ! [X3: int] :
          ( ( member_int2 @ X3 @ A3 )
         => ~ ( member_int2 @ X3 @ B3 ) )
     => ( ( inf_inf_set_int @ A3 @ B3 )
        = bot_bot_set_int ) ) ).

% Int_emptyI
thf(fact_727_Un__Int__distrib2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B3 @ C2 ) @ A3 )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B3 @ A3 ) @ ( sup_su6327502436637775413at_nat @ C2 @ A3 ) ) ) ).

% Un_Int_distrib2
thf(fact_728_Un__Int__distrib2,axiom,
    ! [B3: set_nat,C2: set_nat,A3: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B3 @ C2 ) @ A3 )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ B3 @ A3 ) @ ( sup_sup_set_nat @ C2 @ A3 ) ) ) ).

% Un_Int_distrib2
thf(fact_729_Int__Un__distrib2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B3 @ C2 ) @ A3 )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B3 @ A3 ) @ ( inf_in2572325071724192079at_nat @ C2 @ A3 ) ) ) ).

% Int_Un_distrib2
thf(fact_730_Int__Un__distrib2,axiom,
    ! [B3: set_nat,C2: set_nat,A3: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B3 @ C2 ) @ A3 )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ B3 @ A3 ) @ ( inf_inf_set_nat @ C2 @ A3 ) ) ) ).

% Int_Un_distrib2
thf(fact_731_Un__Int__distrib,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B3 @ C2 ) )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) @ ( sup_su6327502436637775413at_nat @ A3 @ C2 ) ) ) ).

% Un_Int_distrib
thf(fact_732_Un__Int__distrib,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C2 ) )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ ( sup_sup_set_nat @ A3 @ C2 ) ) ) ).

% Un_Int_distrib
thf(fact_733_Int__Un__distrib,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( sup_su6327502436637775413at_nat @ B3 @ C2 ) )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) ) ) ).

% Int_Un_distrib
thf(fact_734_Int__Un__distrib,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C2 ) )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( inf_inf_set_nat @ A3 @ C2 ) ) ) ).

% Int_Un_distrib
thf(fact_735_Un__Int__crazy,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ ( inf_in2572325071724192079at_nat @ B3 @ C2 ) ) @ ( inf_in2572325071724192079at_nat @ C2 @ A3 ) )
      = ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) @ ( sup_su6327502436637775413at_nat @ B3 @ C2 ) ) @ ( sup_su6327502436637775413at_nat @ C2 @ A3 ) ) ) ).

% Un_Int_crazy
thf(fact_736_Un__Int__crazy,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( inf_inf_set_nat @ B3 @ C2 ) ) @ ( inf_inf_set_nat @ C2 @ A3 ) )
      = ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ ( sup_sup_set_nat @ B3 @ C2 ) ) @ ( sup_sup_set_nat @ C2 @ A3 ) ) ) ).

% Un_Int_crazy
thf(fact_737_wand__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( wand_raw @ X @ Xa @ Xb )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) )
              & ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
                  ( ( ( ( inf_inf_set_nat @ As @ As4 )
                      = bot_bot_set_nat )
                    & ( relH @ As @ H3 @ H4 )
                    & ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As ) )
                    & ( X @ ( produc7507926704131184380et_nat @ H4 @ As4 ) ) )
                 => ( Xa @ ( produc7507926704131184380et_nat @ H4 @ ( sup_sup_set_nat @ As @ As4 ) ) ) ) ) ) ) ).

% wand_raw.elims(3)
thf(fact_738_wand__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( wand_raw @ X @ Xa @ Xb )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) )
                & ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
                    ( ( ( ( inf_inf_set_nat @ As @ As5 )
                        = bot_bot_set_nat )
                      & ( relH @ As @ H3 @ H6 )
                      & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As ) )
                      & ( X @ ( produc7507926704131184380et_nat @ H6 @ As5 ) ) )
                   => ( Xa @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As @ As5 ) ) ) ) ) ) ) ).

% wand_raw.elims(2)
thf(fact_739_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 )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ( Y
              = ( ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) )
                    & ! [H7: heap_e7401611519738050253t_unit,As6: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As @ As6 )
                            = bot_bot_set_nat )
                          & ( relH @ As @ H3 @ H7 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H7 @ As6 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As @ As6 ) ) ) ) ) ) ) ) ) ).

% wand_raw.elims(1)
thf(fact_740_wand__raw_Osimps,axiom,
    ! [P2: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( wand_raw @ P2 @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
      = ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
        & ! [H7: heap_e7401611519738050253t_unit,As6: set_nat] :
            ( ( ( ( inf_inf_set_nat @ As2 @ As6 )
                = bot_bot_set_nat )
              & ( relH @ As2 @ H2 @ H7 )
              & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As2 ) )
              & ( P2 @ ( produc7507926704131184380et_nat @ H7 @ As6 ) ) )
           => ( Q2 @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As2 @ As6 ) ) ) ) ) ) ).

% wand_raw.simps
thf(fact_741_times__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( times_assn_raw @ X @ Xa @ Xb )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ? [As12: set_nat,As23: set_nat] :
                ( ( As
                  = ( sup_sup_set_nat @ As12 @ As23 ) )
                & ( ( inf_inf_set_nat @ As12 @ As23 )
                  = bot_bot_set_nat )
                & ( X @ ( produc7507926704131184380et_nat @ H3 @ As12 ) )
                & ( Xa @ ( produc7507926704131184380et_nat @ H3 @ As23 ) ) ) ) ) ).

% times_assn_raw.elims(3)
thf(fact_742_times__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( times_assn_raw @ X @ Xa @ Xb )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ~ ? [As13: set_nat,As24: set_nat] :
                  ( ( As
                    = ( sup_sup_set_nat @ As13 @ As24 ) )
                  & ( ( inf_inf_set_nat @ As13 @ As24 )
                    = bot_bot_set_nat )
                  & ( X @ ( produc7507926704131184380et_nat @ H3 @ As13 ) )
                  & ( Xa @ ( produc7507926704131184380et_nat @ H3 @ As24 ) ) ) ) ) ).

% times_assn_raw.elims(2)
thf(fact_743_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 )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ( Y
              = ( ~ ? [As1: set_nat,As22: set_nat] :
                      ( ( As
                        = ( sup_sup_set_nat @ As1 @ As22 ) )
                      & ( ( inf_inf_set_nat @ As1 @ As22 )
                        = bot_bot_set_nat )
                      & ( X @ ( produc7507926704131184380et_nat @ H3 @ As1 ) )
                      & ( Xa @ ( produc7507926704131184380et_nat @ H3 @ As22 ) ) ) ) ) ) ) ).

% times_assn_raw.elims(1)
thf(fact_744_times__assn__raw_Osimps,axiom,
    ! [P2: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( times_assn_raw @ P2 @ Q2 @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
      = ( ? [As1: set_nat,As22: set_nat] :
            ( ( As2
              = ( sup_sup_set_nat @ As1 @ As22 ) )
            & ( ( inf_inf_set_nat @ As1 @ As22 )
              = bot_bot_set_nat )
            & ( P2 @ ( produc7507926704131184380et_nat @ H2 @ As1 ) )
            & ( Q2 @ ( produc7507926704131184380et_nat @ H2 @ As22 ) ) ) ) ) ).

% times_assn_raw.simps
thf(fact_745_disjointI,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ! [X3: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ X3 @ A )
         => ~ ( member6576561426505652726_nat_o @ X3 @ B ) )
     => ( ( inf_in1906310914598751387_nat_o @ A @ B )
        = bot_bo7824918357723582233_nat_o ) ) ).

% disjointI
thf(fact_746_disjointI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X3 @ A )
         => ~ ( member8440522571783428010at_nat @ X3 @ B ) )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat ) ) ).

% disjointI
thf(fact_747_disjointI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat2 @ X3 @ A )
         => ~ ( member_nat2 @ X3 @ B ) )
     => ( ( inf_inf_set_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% disjointI
thf(fact_748_disjointI,axiom,
    ! [A: set_int,B: set_int] :
      ( ! [X3: int] :
          ( ( member_int2 @ X3 @ A )
         => ~ ( member_int2 @ X3 @ B ) )
     => ( ( inf_inf_set_int @ A @ B )
        = bot_bot_set_int ) ) ).

% disjointI
thf(fact_749_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ X @ Y )
        = bot_bo2099793752762293965at_nat )
     => ( ( ( sup_su6327502436637775413at_nat @ X @ Y )
          = top_to4669805908274784177at_nat )
       => ( ( uminus6524753893492686040at_nat @ X )
          = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_750_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( ( inf_in3604695632404883862_int_o @ X @ Y )
        = bot_bo8147686125503663512_int_o )
     => ( ( ( sup_su8463660629351352368_int_o @ X @ Y )
          = top_to1578927101902068148_int_o )
       => ( ( uminus7117520113953359693_int_o @ X )
          = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_751_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( inf_inf_set_nat @ X @ Y )
        = bot_bot_set_nat )
     => ( ( ( sup_sup_set_nat @ X @ Y )
          = top_top_set_nat )
       => ( ( uminus5710092332889474511et_nat @ X )
          = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_752_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( inf_inf_set_int @ X @ Y )
        = bot_bot_set_int )
     => ( ( ( sup_sup_set_int @ X @ Y )
          = top_top_set_int )
       => ( ( uminus1532241313380277803et_int @ X )
          = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_753_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] :
      ( ( ( inf_in4660618365625256667t_unit @ X @ Y )
        = bot_bo3957492148770167129t_unit )
     => ( ( ( sup_su793286257634532545t_unit @ X @ Y )
          = top_to1996260823553986621t_unit )
       => ( ( uminus5944136376168626660t_unit @ X )
          = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_754_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( inf_inf_assn @ X @ Y )
        = bot_bot_assn )
     => ( ( ( sup_sup_assn @ X @ Y )
          = top_top_assn )
       => ( ( uminus_uminus_assn @ X )
          = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_755_properI,axiom,
    ! [P2: produc3658429121746597890et_nat > $o] :
      ( ! [As: set_nat,H3: heap_e7401611519738050253t_unit] :
          ( ( P2 @ ( produc7507926704131184380et_nat @ H3 @ As ) )
         => ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) ) )
     => ( ! [As: set_nat,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
            ( ( P2 @ ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ( ( relH @ As @ H3 @ H4 )
             => ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As ) )
               => ( P2 @ ( produc7507926704131184380et_nat @ H4 @ As ) ) ) ) )
       => ( proper @ P2 ) ) ) ).

% properI
thf(fact_756_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_757_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_758_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_759_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_760_add_Oinverse__inverse,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_761_add_Oinverse__inverse,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_762_pure__assn__proper,axiom,
    ! [B: $o] : ( proper @ ( pure_a825153325127701367it_nat @ B ) ) ).

% pure_assn_proper
thf(fact_763_Compl__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( uminus6524753893492686040at_nat @ A3 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Compl_disjoint
thf(fact_764_Compl__disjoint,axiom,
    ! [A3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ A3 ) )
      = bot_bot_set_nat ) ).

% Compl_disjoint
thf(fact_765_Compl__disjoint,axiom,
    ! [A3: set_int] :
      ( ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ A3 ) )
      = bot_bot_set_int ) ).

% Compl_disjoint
thf(fact_766_Compl__disjoint2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ A3 ) @ A3 )
      = bot_bo2099793752762293965at_nat ) ).

% Compl_disjoint2
thf(fact_767_Compl__disjoint2,axiom,
    ! [A3: set_nat] :
      ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ A3 )
      = bot_bot_set_nat ) ).

% Compl_disjoint2
thf(fact_768_Compl__disjoint2,axiom,
    ! [A3: set_int] :
      ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ A3 ) @ A3 )
      = bot_bot_set_int ) ).

% Compl_disjoint2
thf(fact_769_mod__not__dist,axiom,
    ! [P2: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( uminus_uminus_assn @ P2 ) @ H2 )
      = ( ( in_range @ H2 )
        & ~ ( rep_assn @ P2 @ H2 ) ) ) ).

% mod_not_dist
thf(fact_770_bool__assn__proper_I1_J,axiom,
    proper @ in_range ).

% bool_assn_proper(1)
thf(fact_771_times__assn__proper,axiom,
    ! [P2: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] :
      ( ( proper @ P2 )
     => ( ( proper @ Q2 )
       => ( proper @ ( times_assn_raw @ P2 @ Q2 ) ) ) ) ).

% times_assn_proper
thf(fact_772_wand__proper,axiom,
    ! [P2: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] : ( proper @ ( wand_raw @ P2 @ Q2 ) ) ).

% wand_proper
thf(fact_773_one__assn__proper,axiom,
    proper @ one_assn_raw ).

% one_assn_proper
thf(fact_774_inf__compl__bot__left1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ ( inf_in2572325071724192079at_nat @ X @ Y ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_compl_bot_left1
thf(fact_775_inf__compl__bot__left1,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ ( uminus7117520113953359693_int_o @ X ) @ ( inf_in3604695632404883862_int_o @ X @ Y ) )
      = bot_bo8147686125503663512_int_o ) ).

% inf_compl_bot_left1
thf(fact_776_inf__compl__bot__left1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( inf_inf_set_nat @ X @ Y ) )
      = bot_bot_set_nat ) ).

% inf_compl_bot_left1
thf(fact_777_inf__compl__bot__left1,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( inf_inf_set_int @ X @ Y ) )
      = bot_bot_set_int ) ).

% inf_compl_bot_left1
thf(fact_778_inf__compl__bot__left1,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ ( inf_inf_assn @ X @ Y ) )
      = bot_bot_assn ) ).

% inf_compl_bot_left1
thf(fact_779_inf__compl__bot__left2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ Y ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_compl_bot_left2
thf(fact_780_inf__compl__bot__left2,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ X @ ( inf_in3604695632404883862_int_o @ ( uminus7117520113953359693_int_o @ X ) @ Y ) )
      = bot_bo8147686125503663512_int_o ) ).

% inf_compl_bot_left2
thf(fact_781_inf__compl__bot__left2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) )
      = bot_bot_set_nat ) ).

% inf_compl_bot_left2
thf(fact_782_inf__compl__bot__left2,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( inf_inf_set_int @ X @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) )
      = bot_bot_set_int ) ).

% inf_compl_bot_left2
thf(fact_783_inf__compl__bot__left2,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ Y ) )
      = bot_bot_assn ) ).

% inf_compl_bot_left2
thf(fact_784_inf__compl__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ ( uminus6524753893492686040at_nat @ X ) ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_compl_bot_right
thf(fact_785_inf__compl__bot__right,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ ( uminus7117520113953359693_int_o @ X ) ) )
      = bot_bo8147686125503663512_int_o ) ).

% inf_compl_bot_right
thf(fact_786_inf__compl__bot__right,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) ) )
      = bot_bot_set_nat ) ).

% inf_compl_bot_right
thf(fact_787_inf__compl__bot__right,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( inf_inf_set_int @ X @ ( inf_inf_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) ) )
      = bot_bot_set_int ) ).

% inf_compl_bot_right
thf(fact_788_inf__compl__bot__right,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ ( uminus_uminus_assn @ X ) ) )
      = bot_bot_assn ) ).

% inf_compl_bot_right
thf(fact_789_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ X )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_cancel_left
thf(fact_790_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ ( uminus7117520113953359693_int_o @ X ) @ X )
      = bot_bo8147686125503663512_int_o ) ).

% boolean_algebra.conj_cancel_left
thf(fact_791_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ X )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_cancel_left
thf(fact_792_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ X )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_cancel_left
thf(fact_793_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ X )
      = bot_bot_assn ) ).

% boolean_algebra.conj_cancel_left
thf(fact_794_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( uminus6524753893492686040at_nat @ X ) )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_cancel_right
thf(fact_795_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ X @ ( uminus7117520113953359693_int_o @ X ) )
      = bot_bo8147686125503663512_int_o ) ).

% boolean_algebra.conj_cancel_right
thf(fact_796_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( uminus5710092332889474511et_nat @ X ) )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_cancel_right
thf(fact_797_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ X @ ( uminus1532241313380277803et_int @ X ) )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_cancel_right
thf(fact_798_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ ( uminus_uminus_assn @ X ) )
      = bot_bot_assn ) ).

% boolean_algebra.conj_cancel_right
thf(fact_799_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) )
      = ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ X ) @ ( uminus6524753893492686040at_nat @ Y ) ) ) ).

% boolean_algebra.de_Morgan_conj
thf(fact_800_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( uminus7117520113953359693_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) )
      = ( sup_su8463660629351352368_int_o @ ( uminus7117520113953359693_int_o @ X ) @ ( uminus7117520113953359693_int_o @ Y ) ) ) ).

% boolean_algebra.de_Morgan_conj
thf(fact_801_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( inf_inf_set_nat @ X @ Y ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).

% boolean_algebra.de_Morgan_conj
thf(fact_802_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: assn,Y: assn] :
      ( ( uminus_uminus_assn @ ( inf_inf_assn @ X @ Y ) )
      = ( sup_sup_assn @ ( uminus_uminus_assn @ X ) @ ( uminus_uminus_assn @ Y ) ) ) ).

% boolean_algebra.de_Morgan_conj
thf(fact_803_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) )
      = ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ ( uminus6524753893492686040at_nat @ Y ) ) ) ).

% boolean_algebra.de_Morgan_disj
thf(fact_804_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( uminus7117520113953359693_int_o @ ( sup_su8463660629351352368_int_o @ X @ Y ) )
      = ( inf_in3604695632404883862_int_o @ ( uminus7117520113953359693_int_o @ X ) @ ( uminus7117520113953359693_int_o @ Y ) ) ) ).

% boolean_algebra.de_Morgan_disj
thf(fact_805_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( sup_sup_set_nat @ X @ Y ) )
      = ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).

% boolean_algebra.de_Morgan_disj
thf(fact_806_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: assn,Y: assn] :
      ( ( uminus_uminus_assn @ ( sup_sup_assn @ X @ Y ) )
      = ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ ( uminus_uminus_assn @ Y ) ) ) ).

% boolean_algebra.de_Morgan_disj
thf(fact_807_assn__aci_I8_J,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ X @ Y ) )
      = ( sup_sup_assn @ X @ Y ) ) ).

% assn_aci(8)
thf(fact_808_assn__aci_I7_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ Y @ ( sup_sup_assn @ X @ Z ) ) ) ).

% assn_aci(7)
thf(fact_809_assn__aci_I5_J,axiom,
    ( sup_sup_assn
    = ( ^ [X2: assn,Y2: assn] : ( sup_sup_assn @ Y2 @ X2 ) ) ) ).

% assn_aci(5)
thf(fact_810_norm__assertion__simps_I32_J,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ X )
      = X ) ).

% norm_assertion_simps(32)
thf(fact_811_norm__assertion__simps_I15_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ X @ Y ) @ Z )
      = ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% norm_assertion_simps(15)
thf(fact_812_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_813_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( uminus_uminus_rat @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_814_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( uminus1351360451143612070nteger @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_815_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_816_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% equation_minus_iff
thf(fact_817_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% equation_minus_iff
thf(fact_818_Rep__assn__induct,axiom,
    ! [Y: produc3658429121746597890et_nat > $o,P2: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
     => ( ! [X3: assn] : ( P2 @ ( rep_assn @ X3 ) )
       => ( P2 @ Y ) ) ) ).

% Rep_assn_induct
thf(fact_819_Rep__assn__cases,axiom,
    ! [Y: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
     => ~ ! [X3: assn] :
            ( Y
           != ( rep_assn @ X3 ) ) ) ).

% Rep_assn_cases
thf(fact_820_Rep__assn,axiom,
    ! [X: assn] : ( member6576561426505652726_nat_o @ ( rep_assn @ X ) @ ( collec939566748876313656_nat_o @ proper ) ) ).

% Rep_assn
thf(fact_821_inf__cancel__left1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X @ A ) @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ B ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_cancel_left1
thf(fact_822_inf__cancel__left1,axiom,
    ! [X: product_prod_int_int > $o,A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ ( inf_in3604695632404883862_int_o @ X @ A ) @ ( inf_in3604695632404883862_int_o @ ( uminus7117520113953359693_int_o @ X ) @ B ) )
      = bot_bo8147686125503663512_int_o ) ).

% inf_cancel_left1
thf(fact_823_inf__cancel__left1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ A ) @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ B ) )
      = bot_bot_set_nat ) ).

% inf_cancel_left1
thf(fact_824_inf__cancel__left1,axiom,
    ! [X: set_int,A: set_int,B: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ X @ A ) @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ B ) )
      = bot_bot_set_int ) ).

% inf_cancel_left1
thf(fact_825_inf__cancel__left1,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ X @ A ) @ ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ B ) )
      = bot_bot_assn ) ).

% inf_cancel_left1
thf(fact_826_inf__cancel__left2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X ) @ A ) @ ( inf_in2572325071724192079at_nat @ X @ B ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_cancel_left2
thf(fact_827_inf__cancel__left2,axiom,
    ! [X: product_prod_int_int > $o,A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( inf_in3604695632404883862_int_o @ ( inf_in3604695632404883862_int_o @ ( uminus7117520113953359693_int_o @ X ) @ A ) @ ( inf_in3604695632404883862_int_o @ X @ B ) )
      = bot_bo8147686125503663512_int_o ) ).

% inf_cancel_left2
thf(fact_828_inf__cancel__left2,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ A ) @ ( inf_inf_set_nat @ X @ B ) )
      = bot_bot_set_nat ) ).

% inf_cancel_left2
thf(fact_829_inf__cancel__left2,axiom,
    ! [X: set_int,A: set_int,B: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ A ) @ ( inf_inf_set_int @ X @ B ) )
      = bot_bot_set_int ) ).

% inf_cancel_left2
thf(fact_830_inf__cancel__left2,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ ( uminus_uminus_assn @ X ) @ A ) @ ( inf_inf_assn @ X @ B ) )
      = bot_bot_assn ) ).

% inf_cancel_left2
thf(fact_831_Compl__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) )
      = ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ A3 ) @ ( uminus6524753893492686040at_nat @ B3 ) ) ) ).

% Compl_Un
thf(fact_832_Compl__Un,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( sup_sup_set_nat @ A3 @ B3 ) )
      = ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ ( uminus5710092332889474511et_nat @ B3 ) ) ) ).

% Compl_Un
thf(fact_833_Compl__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
      = ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ A3 ) @ ( uminus6524753893492686040at_nat @ B3 ) ) ) ).

% Compl_Int
thf(fact_834_Compl__Int,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( inf_inf_set_nat @ A3 @ B3 ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ ( uminus5710092332889474511et_nat @ B3 ) ) ) ).

% Compl_Int
thf(fact_835_properD1,axiom,
    ! [P2: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( proper @ P2 )
     => ( ( P2 @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
       => ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) ) ).

% properD1
thf(fact_836_proper__iff,axiom,
    ! [P2: produc3658429121746597890et_nat > $o,As2: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( ( proper @ P2 )
     => ( ( relH @ As2 @ H2 @ H5 )
       => ( ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As2 ) )
         => ( ( P2 @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
            = ( P2 @ ( produc7507926704131184380et_nat @ H5 @ As2 ) ) ) ) ) ) ).

% proper_iff
thf(fact_837_proper__def,axiom,
    ( proper
    = ( ^ [P: produc3658429121746597890et_nat > $o] :
        ! [H: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,As7: set_nat] :
          ( ( ( P @ ( produc7507926704131184380et_nat @ H @ As7 ) )
           => ( in_range @ ( produc7507926704131184380et_nat @ H @ As7 ) ) )
          & ( ( ( P @ ( produc7507926704131184380et_nat @ H @ As7 ) )
              & ( relH @ As7 @ H @ H7 )
              & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As7 ) ) )
           => ( P @ ( produc7507926704131184380et_nat @ H7 @ As7 ) ) ) ) ) ) ).

% proper_def
thf(fact_838_properD2,axiom,
    ! [P2: produc3658429121746597890et_nat > $o,H2: heap_e7401611519738050253t_unit,As2: set_nat,H5: heap_e7401611519738050253t_unit] :
      ( ( proper @ P2 )
     => ( ( P2 @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
       => ( ( relH @ As2 @ H2 @ H5 )
         => ( ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As2 ) )
           => ( P2 @ ( produc7507926704131184380et_nat @ H5 @ As2 ) ) ) ) ) ) ).

% properD2
thf(fact_839_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_840_mult__minus1,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1
thf(fact_841_mult__minus1,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1
thf(fact_842_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_843_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_844_mult__minus1__right,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1_right
thf(fact_845_mult__minus__left,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_846_mult__minus__left,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_847_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_848_minus__mult__minus,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( times_times_int @ A @ B ) ) ).

% minus_mult_minus
thf(fact_849_minus__mult__minus,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( times_times_rat @ A @ B ) ) ).

% minus_mult_minus
thf(fact_850_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( times_3573771949741848930nteger @ A @ B ) ) ).

% minus_mult_minus
thf(fact_851_mult__minus__right,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_852_mult__minus__right,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_853_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_854_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_855_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_856_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_857_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_858_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_859_one__neq__neg__one,axiom,
    ( one_one_Code_integer
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% one_neq_neg_one
thf(fact_860_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ A )
        = ( times_times_int @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_int @ B ) ) ) ) ).

% square_eq_iff
thf(fact_861_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ A )
        = ( times_times_rat @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_rat @ B ) ) ) ) ).

% square_eq_iff
thf(fact_862_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ A )
        = ( times_3573771949741848930nteger @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus1351360451143612070nteger @ B ) ) ) ) ).

% square_eq_iff
thf(fact_863_minus__mult__commute,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
      = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).

% minus_mult_commute
thf(fact_864_minus__mult__commute,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_mult_commute
thf(fact_865_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).

% minus_mult_commute
thf(fact_866_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea949959532490082519at_nat @ inf_in2572325071724192079at_nat @ sup_su6327502436637775413at_nat @ uminus6524753893492686040at_nat @ bot_bo2099793752762293965at_nat @ top_to4669805908274784177at_nat ).

% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_867_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea1849928419393854478_int_o @ inf_in3604695632404883862_int_o @ sup_su8463660629351352368_int_o @ uminus7117520113953359693_int_o @ bot_bo8147686125503663512_int_o @ top_to1578927101902068148_int_o ).

% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_868_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea778851993438741648et_nat @ inf_inf_set_nat @ sup_sup_set_nat @ uminus5710092332889474511et_nat @ bot_bot_set_nat @ top_top_set_nat ).

% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_869_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea5824373010784320748et_int @ inf_inf_set_int @ sup_sup_set_int @ uminus1532241313380277803et_int @ bot_bot_set_int @ top_top_set_int ).

% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_870_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea3191171876098928227t_unit @ inf_in4660618365625256667t_unit @ sup_su793286257634532545t_unit @ uminus5944136376168626660t_unit @ bot_bo3957492148770167129t_unit @ top_to1996260823553986621t_unit ).

% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_871_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea5875918072483028530a_assn @ inf_inf_assn @ sup_sup_assn @ uminus_uminus_assn @ bot_bot_assn @ top_top_assn ).

% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_872_wand__assn__def,axiom,
    ( wand_assn
    = ( ^ [P: assn,Q: assn] : ( abs_assn @ ( wand_raw @ ( rep_assn @ P ) @ ( rep_assn @ Q ) ) ) ) ) ).

% wand_assn_def
thf(fact_873_Rep__assn__inverse,axiom,
    ! [X: assn] :
      ( ( abs_assn @ ( rep_assn @ X ) )
      = X ) ).

% Rep_assn_inverse
thf(fact_874_Abs__assn__eqI_I2_J,axiom,
    ! [P2: produc3658429121746597890et_nat > $o,Pr: assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( ( P2 @ H3 )
          = ( rep_assn @ Pr @ H3 ) )
     => ( Pr
        = ( abs_assn @ P2 ) ) ) ).

% Abs_assn_eqI(2)
thf(fact_875_Abs__assn__eqI_I1_J,axiom,
    ! [P2: produc3658429121746597890et_nat > $o,Pr: assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( ( P2 @ H3 )
          = ( rep_assn @ Pr @ H3 ) )
     => ( ( abs_assn @ P2 )
        = Pr ) ) ).

% Abs_assn_eqI(1)
thf(fact_876_pure__assn__def,axiom,
    ( pure_assn
    = ( ^ [B2: $o] : ( abs_assn @ ( pure_a825153325127701367it_nat @ B2 ) ) ) ) ).

% pure_assn_def
thf(fact_877_Abs__assn__cases,axiom,
    ! [X: assn] :
      ~ ! [Y3: produc3658429121746597890et_nat > $o] :
          ( ( X
            = ( abs_assn @ Y3 ) )
         => ~ ( member6576561426505652726_nat_o @ Y3 @ ( collec939566748876313656_nat_o @ proper ) ) ) ).

% Abs_assn_cases
thf(fact_878_Abs__assn__induct,axiom,
    ! [P2: assn > $o,X: assn] :
      ( ! [Y3: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ Y3 @ ( collec939566748876313656_nat_o @ proper ) )
         => ( P2 @ ( abs_assn @ Y3 ) ) )
     => ( P2 @ X ) ) ).

% Abs_assn_induct
thf(fact_879_Abs__assn__inject,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ X @ ( collec939566748876313656_nat_o @ proper ) )
     => ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
       => ( ( ( abs_assn @ X )
            = ( abs_assn @ Y ) )
          = ( X = Y ) ) ) ) ).

% Abs_assn_inject
thf(fact_880_Abs__assn__inverse,axiom,
    ! [Y: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
     => ( ( rep_assn @ ( abs_assn @ Y ) )
        = Y ) ) ).

% Abs_assn_inverse
thf(fact_881_top__assn__def,axiom,
    ( top_top_assn
    = ( abs_assn @ in_range ) ) ).

% top_assn_def
thf(fact_882_one__assn__def,axiom,
    ( one_one_assn
    = ( abs_assn @ one_assn_raw ) ) ).

% one_assn_def
thf(fact_883_times__assn__def,axiom,
    ( times_times_assn
    = ( ^ [P: assn,Q: assn] : ( abs_assn @ ( times_assn_raw @ ( rep_assn @ P ) @ ( rep_assn @ Q ) ) ) ) ) ).

% times_assn_def
thf(fact_884_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_885_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_886_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_inc_simps(4)
thf(fact_887_list__collect__set__simps_I1_J,axiom,
    ! [F2: produc6575502325842934193n_assn > set_nat] :
      ( ( list_c6061723043370948915sn_nat @ F2 @ nil_Pr5671120429643327159n_assn )
      = bot_bot_set_nat ) ).

% list_collect_set_simps(1)
thf(fact_888_list__collect__set__simps_I1_J,axiom,
    ! [F2: int > set_nat] :
      ( ( list_c3451693981498911948nt_nat @ F2 @ nil_int )
      = bot_bot_set_nat ) ).

% list_collect_set_simps(1)
thf(fact_889_list__collect__set__simps_I1_J,axiom,
    ! [F2: nat > set_nat] :
      ( ( list_c2452340269597857392at_nat @ F2 @ nil_nat )
      = bot_bot_set_nat ) ).

% list_collect_set_simps(1)
thf(fact_890_list__collect__set__simps_I1_J,axiom,
    ! [F2: produc6575502325842934193n_assn > set_int] :
      ( ( list_c6059232572861898639sn_int @ F2 @ nil_Pr5671120429643327159n_assn )
      = bot_bot_set_int ) ).

% list_collect_set_simps(1)
thf(fact_891_list__collect__set__simps_I1_J,axiom,
    ! [F2: int > set_int] :
      ( ( list_c3449203510989861672nt_int @ F2 @ nil_int )
      = bot_bot_set_int ) ).

% list_collect_set_simps(1)
thf(fact_892_list__collect__set__simps_I1_J,axiom,
    ! [F2: nat > set_int] :
      ( ( list_c2449849799088807116at_int @ F2 @ nil_nat )
      = bot_bot_set_int ) ).

% list_collect_set_simps(1)
thf(fact_893_mod__pure,axiom,
    ! [B: $o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( pure_assn @ B ) @ H2 )
      = ( ( ( produc8586169260539613262et_nat @ H2 )
          = bot_bot_set_nat )
        & B ) ) ).

% mod_pure
thf(fact_894_type__definition__assn,axiom,
    type_d3909072315231072503_nat_o @ rep_assn @ abs_assn @ ( collec939566748876313656_nat_o @ proper ) ).

% type_definition_assn
thf(fact_895_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% dbl_dec_simps(3)
thf(fact_896_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
    = one_one_rat ) ).

% dbl_dec_simps(3)
thf(fact_897_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_898_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ( ( linord2612477271533052124et_int @ bot_bot_set_int )
    = nil_int ) ).

% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_899_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ( ( linord2614967742042102400et_nat @ bot_bot_set_nat )
    = nil_nat ) ).

% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_900_UNIV__coset,axiom,
    ( top_to5722533637347664993n_assn
    = ( coset_703488708621576356n_assn @ nil_Pr5671120429643327159n_assn ) ) ).

% UNIV_coset
thf(fact_901_UNIV__coset,axiom,
    ( top_top_set_int
    = ( coset_int @ nil_int ) ) ).

% UNIV_coset
thf(fact_902_UNIV__coset,axiom,
    ( top_top_set_nat
    = ( coset_nat @ nil_nat ) ) ).

% UNIV_coset
thf(fact_903_UNIV__coset,axiom,
    ( top_to1996260823553986621t_unit
    = ( coset_Product_unit @ nil_Product_unit ) ) ).

% UNIV_coset
thf(fact_904_mod__emp,axiom,
    ! [H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ one_one_assn @ H2 )
      = ( ( produc8586169260539613262et_nat @ H2 )
        = bot_bot_set_nat ) ) ).

% mod_emp
thf(fact_905_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_906_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_907_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_dec_simps(2)
thf(fact_908_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( ( linord2612477271533052124et_int @ A3 )
          = nil_int )
        = ( A3 = bot_bot_set_int ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_909_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( linord2614967742042102400et_nat @ A3 )
          = nil_nat )
        = ( A3 = bot_bot_set_nat ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_910_list__collect__set__map__simps_I1_J,axiom,
    ! [F2: nat > set_nat,X: nat > nat] :
      ( ( list_c2452340269597857392at_nat @ F2 @ ( map_nat_nat @ X @ nil_nat ) )
      = bot_bot_set_nat ) ).

% list_collect_set_map_simps(1)
thf(fact_911_list__collect__set__map__simps_I1_J,axiom,
    ! [F2: assn > set_nat,X: produc6575502325842934193n_assn > assn] :
      ( ( list_c1844713377658005960sn_nat @ F2 @ ( map_Pr8991440229025900053n_assn @ X @ nil_Pr5671120429643327159n_assn ) )
      = bot_bot_set_nat ) ).

% list_collect_set_map_simps(1)
thf(fact_912_list__collect__set__map__simps_I1_J,axiom,
    ! [F2: nat > set_int,X: nat > nat] :
      ( ( list_c2449849799088807116at_int @ F2 @ ( map_nat_nat @ X @ nil_nat ) )
      = bot_bot_set_int ) ).

% list_collect_set_map_simps(1)
thf(fact_913_list__collect__set__map__simps_I1_J,axiom,
    ! [F2: assn > set_int,X: produc6575502325842934193n_assn > assn] :
      ( ( list_c1842222907148955684sn_int @ F2 @ ( map_Pr8991440229025900053n_assn @ X @ nil_Pr5671120429643327159n_assn ) )
      = bot_bot_set_int ) ).

% list_collect_set_map_simps(1)
thf(fact_914_nths__empty,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( nths_P5537322223065504949n_assn @ Xs @ bot_bot_set_nat )
      = nil_Pr5671120429643327159n_assn ) ).

% nths_empty
thf(fact_915_nths__empty,axiom,
    ! [Xs: list_int] :
      ( ( nths_int @ Xs @ bot_bot_set_nat )
      = nil_int ) ).

% nths_empty
thf(fact_916_nths__empty,axiom,
    ! [Xs: list_nat] :
      ( ( nths_nat @ Xs @ bot_bot_set_nat )
      = nil_nat ) ).

% nths_empty
thf(fact_917_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_918_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_919_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ zero_z3403309356797280102nteger )
    = one_one_Code_integer ) ).

% dbl_inc_simps(2)
thf(fact_920_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ A3 @ ( uminus6524753893492686040at_nat @ B3 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_921_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ( inf_inf_set_int @ A3 @ B3 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A3 @ ( uminus1532241313380277803et_int @ B3 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_922_disjoint__eq__subset__Compl,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B3 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ B3 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_923_in__measures_I1_J,axiom,
    ! [X: nat,Y: nat] :
      ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ nil_nat_nat ) ) ).

% in_measures(1)
thf(fact_924_in__measures_I1_J,axiom,
    ! [X: int,Y: int] :
      ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ nil_int_nat ) ) ).

% in_measures(1)
thf(fact_925_sup__neg__inf,axiom,
    ! [P5: set_Pr1261947904930325089at_nat,Q4: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ P5 @ ( sup_su6327502436637775413at_nat @ Q4 @ R2 ) )
      = ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ P5 @ ( uminus6524753893492686040at_nat @ Q4 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_926_sup__neg__inf,axiom,
    ! [P5: product_prod_int_int > $o,Q4: product_prod_int_int > $o,R2: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ P5 @ ( sup_su8463660629351352368_int_o @ Q4 @ R2 ) )
      = ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ P5 @ ( uminus7117520113953359693_int_o @ Q4 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_927_sup__neg__inf,axiom,
    ! [P5: assn,Q4: assn,R2: assn] :
      ( ( ord_less_eq_assn @ P5 @ ( sup_sup_assn @ Q4 @ R2 ) )
      = ( ord_less_eq_assn @ ( inf_inf_assn @ P5 @ ( uminus_uminus_assn @ Q4 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_928_sup__neg__inf,axiom,
    ! [P5: set_nat,Q4: set_nat,R2: set_nat] :
      ( ( ord_less_eq_set_nat @ P5 @ ( sup_sup_set_nat @ Q4 @ R2 ) )
      = ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ P5 @ ( uminus5710092332889474511et_nat @ Q4 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_929_shunt2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ ( uminus6524753893492686040at_nat @ Y ) ) @ Z )
      = ( ord_le3146513528884898305at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z ) ) ) ).

% shunt2
thf(fact_930_shunt2,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ X @ ( uminus7117520113953359693_int_o @ Y ) ) @ Z )
      = ( ord_le8369615600986905444_int_o @ X @ ( sup_su8463660629351352368_int_o @ Y @ Z ) ) ) ).

% shunt2
thf(fact_931_shunt2,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ ( inf_inf_assn @ X @ ( uminus_uminus_assn @ Y ) ) @ Z )
      = ( ord_less_eq_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% shunt2
thf(fact_932_shunt2,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) @ Z )
      = ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).

% shunt2
thf(fact_933_shunt1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Z )
      = ( ord_le3146513528884898305at_nat @ X @ ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ Y ) @ Z ) ) ) ).

% shunt1
thf(fact_934_shunt1,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ Z )
      = ( ord_le8369615600986905444_int_o @ X @ ( sup_su8463660629351352368_int_o @ ( uminus7117520113953359693_int_o @ Y ) @ Z ) ) ) ).

% shunt1
thf(fact_935_shunt1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ Z )
      = ( ord_less_eq_assn @ X @ ( sup_sup_assn @ ( uminus_uminus_assn @ Y ) @ Z ) ) ) ).

% shunt1
thf(fact_936_shunt1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
      = ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ Z ) ) ) ).

% shunt1
thf(fact_937_relH__def,axiom,
    ( relH
    = ( ^ [As7: set_nat,H: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit] :
          ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As7 ) )
          & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As7 ) )
          & ! [T2: typerep,X2: nat] :
              ( ( member_nat2 @ X2 @ As7 )
             => ( ( ( refs_Product_unit @ H @ T2 @ X2 )
                  = ( refs_Product_unit @ H7 @ T2 @ X2 ) )
                & ( ( arrays_Product_unit @ H @ T2 @ X2 )
                  = ( arrays_Product_unit @ H7 @ T2 @ X2 ) ) ) ) ) ) ) ).

% relH_def
thf(fact_938_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_939_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( times_times_rat @ A @ C )
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_940_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( times_times_nat @ A @ C )
        = ( times_times_nat @ B @ C ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_941_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( times_times_int @ A @ C )
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_942_mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ C )
        = ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_943_mult__cancel__right,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( ( times_2397367101498566445atural @ A @ C )
        = ( times_2397367101498566445atural @ B @ C ) )
      = ( ( C = zero_z2226904508553997617atural )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_944_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( times_times_rat @ C @ A )
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_945_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( times_times_nat @ C @ A )
        = ( times_times_nat @ C @ B ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_946_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( times_times_int @ C @ A )
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_947_mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ C @ A )
        = ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_948_mult__cancel__left,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( ( times_2397367101498566445atural @ C @ A )
        = ( times_2397367101498566445atural @ C @ B ) )
      = ( ( C = zero_z2226904508553997617atural )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_949_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
thf(fact_950_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_951_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
thf(fact_952_mult__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( ( A = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% mult_eq_0_iff
thf(fact_953_mult__eq__0__iff,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( times_2397367101498566445atural @ A @ B )
        = zero_z2226904508553997617atural )
      = ( ( A = zero_z2226904508553997617atural )
        | ( B = zero_z2226904508553997617atural ) ) ) ).

% mult_eq_0_iff
thf(fact_954_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_955_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_956_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_957_mult__zero__right,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% mult_zero_right
thf(fact_958_mult__zero__right,axiom,
    ! [A: code_natural] :
      ( ( times_2397367101498566445atural @ A @ zero_z2226904508553997617atural )
      = zero_z2226904508553997617atural ) ).

% mult_zero_right
thf(fact_959_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_960_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_961_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_962_mult__zero__left,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% mult_zero_left
thf(fact_963_mult__zero__left,axiom,
    ! [A: code_natural] :
      ( ( times_2397367101498566445atural @ zero_z2226904508553997617atural @ A )
      = zero_z2226904508553997617atural ) ).

% mult_zero_left
thf(fact_964_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_965_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_966_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_967_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_968_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_969_add_Oinverse__neutral,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% add.inverse_neutral
thf(fact_970_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_971_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_972_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( uminus1351360451143612070nteger @ A ) )
      = ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_973_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_974_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_975_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
thf(fact_976_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_977_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A
        = ( uminus_uminus_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_978_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
thf(fact_979_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_980_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = A )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_981_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
thf(fact_982_inf_Obounded__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) )
      = ( ( ord_le3146513528884898305at_nat @ A @ B )
        & ( ord_le3146513528884898305at_nat @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_983_inf_Obounded__iff,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o,C: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ A @ ( inf_in3604695632404883862_int_o @ B @ C ) )
      = ( ( ord_le8369615600986905444_int_o @ A @ B )
        & ( ord_le8369615600986905444_int_o @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_984_inf_Obounded__iff,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( ord_less_eq_assn @ A @ ( inf_inf_assn @ B @ C ) )
      = ( ( ord_less_eq_assn @ A @ B )
        & ( ord_less_eq_assn @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_985_inf_Obounded__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
      = ( ( ord_less_eq_set_nat @ A @ B )
        & ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_986_inf_Obounded__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( inf_inf_rat @ B @ C ) )
      = ( ( ord_less_eq_rat @ A @ B )
        & ( ord_less_eq_rat @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_987_inf_Obounded__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
      = ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_eq_nat @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_988_inf_Obounded__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) )
      = ( ( ord_less_eq_int @ A @ B )
        & ( ord_less_eq_int @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_989_le__inf__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
      = ( ( ord_le3146513528884898305at_nat @ X @ Y )
        & ( ord_le3146513528884898305at_nat @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_990_le__inf__iff,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) )
      = ( ( ord_le8369615600986905444_int_o @ X @ Y )
        & ( ord_le8369615600986905444_int_o @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_991_le__inf__iff,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( ( ord_less_eq_assn @ X @ Y )
        & ( ord_less_eq_assn @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_992_le__inf__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
      = ( ( ord_less_eq_set_nat @ X @ Y )
        & ( ord_less_eq_set_nat @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_993_le__inf__iff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X @ ( inf_inf_rat @ Y @ Z ) )
      = ( ( ord_less_eq_rat @ X @ Y )
        & ( ord_less_eq_rat @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_994_le__inf__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
      = ( ( ord_less_eq_nat @ X @ Y )
        & ( ord_less_eq_nat @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_995_le__inf__iff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) )
      = ( ( ord_less_eq_int @ X @ Y )
        & ( ord_less_eq_int @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_996_le__sup__iff,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ X @ Y ) @ Z )
      = ( ( ord_less_eq_assn @ X @ Z )
        & ( ord_less_eq_assn @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_997_le__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_set_nat @ X @ Z )
        & ( ord_less_eq_set_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_998_le__sup__iff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_rat @ X @ Z )
        & ( ord_less_eq_rat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_999_le__sup__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_nat @ X @ Z )
        & ( ord_less_eq_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_1000_le__sup__iff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y ) @ Z )
      = ( ( ord_less_eq_int @ X @ Z )
        & ( ord_less_eq_int @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_1001_sup_Obounded__iff,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ B @ C ) @ A )
      = ( ( ord_less_eq_assn @ B @ A )
        & ( ord_less_eq_assn @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1002_sup_Obounded__iff,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_set_nat @ B @ A )
        & ( ord_less_eq_set_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1003_sup_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C ) @ A )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( ord_less_eq_rat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1004_sup_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1005_sup_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1006_map__is__Nil__conv,axiom,
    ! [F2: produc6575502325842934193n_assn > produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( map_Pr7925354932063753860n_assn @ F2 @ Xs )
        = nil_Pr5671120429643327159n_assn )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% map_is_Nil_conv
thf(fact_1007_map__is__Nil__conv,axiom,
    ! [F2: int > produc6575502325842934193n_assn,Xs: list_int] :
      ( ( ( map_in4427992030928829247n_assn @ F2 @ Xs )
        = nil_Pr5671120429643327159n_assn )
      = ( Xs = nil_int ) ) ).

% map_is_Nil_conv
thf(fact_1008_map__is__Nil__conv,axiom,
    ! [F2: nat > produc6575502325842934193n_assn,Xs: list_nat] :
      ( ( ( map_na2667955367175718043n_assn @ F2 @ Xs )
        = nil_Pr5671120429643327159n_assn )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_1009_map__is__Nil__conv,axiom,
    ! [F2: produc6575502325842934193n_assn > int,Xs: list_P8527749157015355191n_assn] :
      ( ( ( map_Pr7568062423562401049sn_int @ F2 @ Xs )
        = nil_int )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% map_is_Nil_conv
thf(fact_1010_map__is__Nil__conv,axiom,
    ! [F2: int > int,Xs: list_int] :
      ( ( ( map_int_int @ F2 @ Xs )
        = nil_int )
      = ( Xs = nil_int ) ) ).

% map_is_Nil_conv
thf(fact_1011_map__is__Nil__conv,axiom,
    ! [F2: nat > int,Xs: list_nat] :
      ( ( ( map_nat_int @ F2 @ Xs )
        = nil_int )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_1012_map__is__Nil__conv,axiom,
    ! [F2: produc6575502325842934193n_assn > nat,Xs: list_P8527749157015355191n_assn] :
      ( ( ( map_Pr7570552894071451325sn_nat @ F2 @ Xs )
        = nil_nat )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% map_is_Nil_conv
thf(fact_1013_map__is__Nil__conv,axiom,
    ! [F2: int > nat,Xs: list_int] :
      ( ( ( map_int_nat @ F2 @ Xs )
        = nil_nat )
      = ( Xs = nil_int ) ) ).

% map_is_Nil_conv
thf(fact_1014_map__is__Nil__conv,axiom,
    ! [F2: nat > nat,Xs: list_nat] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_1015_map__is__Nil__conv,axiom,
    ! [F2: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( map_Pr8991440229025900053n_assn @ F2 @ Xs )
        = nil_assn )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% map_is_Nil_conv
thf(fact_1016_Nil__is__map__conv,axiom,
    ! [F2: produc6575502325842934193n_assn > produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( nil_Pr5671120429643327159n_assn
        = ( map_Pr7925354932063753860n_assn @ F2 @ Xs ) )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% Nil_is_map_conv
thf(fact_1017_Nil__is__map__conv,axiom,
    ! [F2: int > produc6575502325842934193n_assn,Xs: list_int] :
      ( ( nil_Pr5671120429643327159n_assn
        = ( map_in4427992030928829247n_assn @ F2 @ Xs ) )
      = ( Xs = nil_int ) ) ).

% Nil_is_map_conv
thf(fact_1018_Nil__is__map__conv,axiom,
    ! [F2: nat > produc6575502325842934193n_assn,Xs: list_nat] :
      ( ( nil_Pr5671120429643327159n_assn
        = ( map_na2667955367175718043n_assn @ F2 @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_1019_Nil__is__map__conv,axiom,
    ! [F2: produc6575502325842934193n_assn > int,Xs: list_P8527749157015355191n_assn] :
      ( ( nil_int
        = ( map_Pr7568062423562401049sn_int @ F2 @ Xs ) )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% Nil_is_map_conv
thf(fact_1020_Nil__is__map__conv,axiom,
    ! [F2: int > int,Xs: list_int] :
      ( ( nil_int
        = ( map_int_int @ F2 @ Xs ) )
      = ( Xs = nil_int ) ) ).

% Nil_is_map_conv
thf(fact_1021_Nil__is__map__conv,axiom,
    ! [F2: nat > int,Xs: list_nat] :
      ( ( nil_int
        = ( map_nat_int @ F2 @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_1022_Nil__is__map__conv,axiom,
    ! [F2: produc6575502325842934193n_assn > nat,Xs: list_P8527749157015355191n_assn] :
      ( ( nil_nat
        = ( map_Pr7570552894071451325sn_nat @ F2 @ Xs ) )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% Nil_is_map_conv
thf(fact_1023_Nil__is__map__conv,axiom,
    ! [F2: int > nat,Xs: list_int] :
      ( ( nil_nat
        = ( map_int_nat @ F2 @ Xs ) )
      = ( Xs = nil_int ) ) ).

% Nil_is_map_conv
thf(fact_1024_Nil__is__map__conv,axiom,
    ! [F2: nat > nat,Xs: list_nat] :
      ( ( nil_nat
        = ( map_nat_nat @ F2 @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_1025_Nil__is__map__conv,axiom,
    ! [F2: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn] :
      ( ( nil_assn
        = ( map_Pr8991440229025900053n_assn @ F2 @ Xs ) )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% Nil_is_map_conv
thf(fact_1026_list_Omap__disc__iff,axiom,
    ! [F2: produc6575502325842934193n_assn > produc6575502325842934193n_assn,A: list_P8527749157015355191n_assn] :
      ( ( ( map_Pr7925354932063753860n_assn @ F2 @ A )
        = nil_Pr5671120429643327159n_assn )
      = ( A = nil_Pr5671120429643327159n_assn ) ) ).

% list.map_disc_iff
thf(fact_1027_list_Omap__disc__iff,axiom,
    ! [F2: int > produc6575502325842934193n_assn,A: list_int] :
      ( ( ( map_in4427992030928829247n_assn @ F2 @ A )
        = nil_Pr5671120429643327159n_assn )
      = ( A = nil_int ) ) ).

% list.map_disc_iff
thf(fact_1028_list_Omap__disc__iff,axiom,
    ! [F2: nat > produc6575502325842934193n_assn,A: list_nat] :
      ( ( ( map_na2667955367175718043n_assn @ F2 @ A )
        = nil_Pr5671120429643327159n_assn )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_1029_list_Omap__disc__iff,axiom,
    ! [F2: produc6575502325842934193n_assn > int,A: list_P8527749157015355191n_assn] :
      ( ( ( map_Pr7568062423562401049sn_int @ F2 @ A )
        = nil_int )
      = ( A = nil_Pr5671120429643327159n_assn ) ) ).

% list.map_disc_iff
thf(fact_1030_list_Omap__disc__iff,axiom,
    ! [F2: int > int,A: list_int] :
      ( ( ( map_int_int @ F2 @ A )
        = nil_int )
      = ( A = nil_int ) ) ).

% list.map_disc_iff
thf(fact_1031_list_Omap__disc__iff,axiom,
    ! [F2: nat > int,A: list_nat] :
      ( ( ( map_nat_int @ F2 @ A )
        = nil_int )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_1032_list_Omap__disc__iff,axiom,
    ! [F2: produc6575502325842934193n_assn > nat,A: list_P8527749157015355191n_assn] :
      ( ( ( map_Pr7570552894071451325sn_nat @ F2 @ A )
        = nil_nat )
      = ( A = nil_Pr5671120429643327159n_assn ) ) ).

% list.map_disc_iff
thf(fact_1033_list_Omap__disc__iff,axiom,
    ! [F2: int > nat,A: list_int] :
      ( ( ( map_int_nat @ F2 @ A )
        = nil_nat )
      = ( A = nil_int ) ) ).

% list.map_disc_iff
thf(fact_1034_list_Omap__disc__iff,axiom,
    ! [F2: nat > nat,A: list_nat] :
      ( ( ( map_nat_nat @ F2 @ A )
        = nil_nat )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_1035_list_Omap__disc__iff,axiom,
    ! [F2: produc6575502325842934193n_assn > assn,A: list_P8527749157015355191n_assn] :
      ( ( ( map_Pr8991440229025900053n_assn @ F2 @ A )
        = nil_assn )
      = ( A = nil_Pr5671120429643327159n_assn ) ) ).

% list.map_disc_iff
thf(fact_1036_Int__subset__iff,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
      = ( ( ord_le3146513528884898305at_nat @ C2 @ A3 )
        & ( ord_le3146513528884898305at_nat @ C2 @ B3 ) ) ) ).

% Int_subset_iff
thf(fact_1037_Int__subset__iff,axiom,
    ! [C2: set_nat,A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A3 @ B3 ) )
      = ( ( ord_less_eq_set_nat @ C2 @ A3 )
        & ( ord_less_eq_set_nat @ C2 @ B3 ) ) ) ).

% Int_subset_iff
thf(fact_1038_nths__nil,axiom,
    ! [A3: set_nat] :
      ( ( nths_P5537322223065504949n_assn @ nil_Pr5671120429643327159n_assn @ A3 )
      = nil_Pr5671120429643327159n_assn ) ).

% nths_nil
thf(fact_1039_nths__nil,axiom,
    ! [A3: set_nat] :
      ( ( nths_int @ nil_int @ A3 )
      = nil_int ) ).

% nths_nil
thf(fact_1040_nths__nil,axiom,
    ! [A3: set_nat] :
      ( ( nths_nat @ nil_nat @ A3 )
      = nil_nat ) ).

% nths_nil
thf(fact_1041_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_1042_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_1043_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_1044_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_1045_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_1046_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_1047_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_1048_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_1049_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_1050_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_1051_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_1052_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_1053_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_1054_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_1055_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_1056_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_1057_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_1058_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_1059_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_1060_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_1061_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_1062_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_1063_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_1064_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_1065_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A3: set_int] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( linord2612477271533052124et_int @ A3 )
        = nil_int ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
thf(fact_1066_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A3: set_nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( linord2614967742042102400et_nat @ A3 )
        = nil_nat ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
thf(fact_1067_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_1068_nths__map,axiom,
    ! [F2: nat > nat,Xs: list_nat,I: set_nat] :
      ( ( nths_nat @ ( map_nat_nat @ F2 @ Xs ) @ I )
      = ( map_nat_nat @ F2 @ ( nths_nat @ Xs @ I ) ) ) ).

% nths_map
thf(fact_1069_nths__map,axiom,
    ! [F2: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn,I: set_nat] :
      ( ( nths_assn @ ( map_Pr8991440229025900053n_assn @ F2 @ Xs ) @ I )
      = ( map_Pr8991440229025900053n_assn @ F2 @ ( nths_P5537322223065504949n_assn @ Xs @ I ) ) ) ).

% nths_map
thf(fact_1070_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_1071_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_1072_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
      = ( X = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_1073_zero__reorient,axiom,
    ! [X: code_integer] :
      ( ( zero_z3403309356797280102nteger = X )
      = ( X = zero_z3403309356797280102nteger ) ) ).

% zero_reorient
thf(fact_1074_zero__reorient,axiom,
    ! [X: code_natural] :
      ( ( zero_z2226904508553997617atural = X )
      = ( X = zero_z2226904508553997617atural ) ) ).

% zero_reorient
thf(fact_1075_times__assn__raw_Ocases,axiom,
    ! [X: produc2732055786443039994et_nat] :
      ~ ! [P4: produc3658429121746597890et_nat > $o,Q5: produc3658429121746597890et_nat > $o,H3: heap_e7401611519738050253t_unit,As: set_nat] :
          ( X
         != ( produc2245416461498447860et_nat @ P4 @ ( produc5001842942810119800et_nat @ Q5 @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ) ).

% times_assn_raw.cases
thf(fact_1076_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1077_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ B )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1078_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1079_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1080_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1081_zero__le__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) )
        | ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) ) ) ).

% zero_le_mult_iff
thf(fact_1082_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_le_mult_iff
thf(fact_1083_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).

% zero_le_mult_iff
thf(fact_1084_mult__nonneg__nonpos2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ B @ A ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_1085_mult__nonneg__nonpos2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le1926595141338095240atural @ B @ zero_z2226904508553997617atural )
       => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ B @ A ) @ zero_z2226904508553997617atural ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_1086_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_1087_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_1088_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_1089_mult__nonpos__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_nonpos_nonneg
thf(fact_1090_mult__nonpos__nonneg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ zero_z2226904508553997617atural )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
       => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A @ B ) @ zero_z2226904508553997617atural ) ) ) ).

% mult_nonpos_nonneg
thf(fact_1091_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_1092_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_1093_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonpos_nonneg
thf(fact_1094_mult__nonneg__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_nonneg_nonpos
thf(fact_1095_mult__nonneg__nonpos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le1926595141338095240atural @ B @ zero_z2226904508553997617atural )
       => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A @ B ) @ zero_z2226904508553997617atural ) ) ) ).

% mult_nonneg_nonpos
thf(fact_1096_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_1097_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_1098_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos
thf(fact_1099_mult__nonneg__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_1100_mult__nonneg__nonneg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
       => ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( times_2397367101498566445atural @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_1101_mult__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 @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_1102_mult__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 @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_1103_mult__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 @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_1104_split__mult__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) )
        | ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) )
     => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ).

% split_mult_neg_le
thf(fact_1105_split__mult__neg__le,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
          & ( ord_le1926595141338095240atural @ B @ zero_z2226904508553997617atural ) )
        | ( ( ord_le1926595141338095240atural @ A @ zero_z2226904508553997617atural )
          & ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B ) ) )
     => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A @ B ) @ zero_z2226904508553997617atural ) ) ).

% split_mult_neg_le
thf(fact_1106_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).

% split_mult_neg_le
thf(fact_1107_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
          & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_1108_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).

% split_mult_neg_le
thf(fact_1109_mult__le__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) )
        | ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_1110_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_1111_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_1112_mult__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_1113_mult__right__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ B )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_1114_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_1115_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_1116_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_1117_mult__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_1118_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_1119_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_1120_mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_1121_mult__left__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ B )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_1122_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_1123_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_1124_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_1125_mult__nonpos__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_1126_mult__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 @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_1127_mult__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 @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_1128_mult__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_1129_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_1130_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_1131_split__mult__pos__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B ) )
        | ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_1132_split__mult__pos__le,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 @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_1133_split__mult__pos__le,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 @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_1134_zero__le__square,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ A ) ) ).

% zero_le_square
thf(fact_1135_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_1136_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_1137_mult__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_1138_mult__mono_H,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ B )
     => ( ( ord_le1926595141338095240atural @ C @ D2 )
       => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
         => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
           => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_1139_mult__mono_H,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 @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_1140_mult__mono_H,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 @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_1141_mult__mono_H,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 @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_1142_mult__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_1143_mult__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ B )
     => ( ( ord_le1926595141338095240atural @ C @ D2 )
       => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
         => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
           => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_1144_mult__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 @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_1145_mult__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 @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_1146_mult__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 @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_1147_not__one__le__zero,axiom,
    ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).

% not_one_le_zero
thf(fact_1148_not__one__le__zero,axiom,
    ~ ( ord_le1926595141338095240atural @ one_one_Code_natural @ zero_z2226904508553997617atural ) ).

% not_one_le_zero
thf(fact_1149_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_1150_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_1151_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_1152_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_1153_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1154_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_1155_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_1156_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_1157_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_1158_zero__less__one__class_Ozero__le__one,axiom,
    ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).

% zero_less_one_class.zero_le_one
thf(fact_1159_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_1160_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_1161_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_1162_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_1163_mult__left__le,axiom,
    ! [C: code_natural,A: code_natural] :
      ( ( ord_le1926595141338095240atural @ C @ one_one_Code_natural )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
       => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_1164_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_1165_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_1166_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_1167_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_1168_mult__le__one,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ one_one_Code_natural )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
       => ( ( ord_le1926595141338095240atural @ B @ one_one_Code_natural )
         => ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A @ B ) @ one_one_Code_natural ) ) ) ) ).

% mult_le_one
thf(fact_1169_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_1170_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_1171_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_1172_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_1173_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_1174_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_1175_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_1176_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_1177_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_1178_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_1179_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_1180_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_1181_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_1182_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_1183_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_1184_list_Osimps_I8_J,axiom,
    ! [F2: produc6575502325842934193n_assn > produc6575502325842934193n_assn] :
      ( ( map_Pr7925354932063753860n_assn @ F2 @ nil_Pr5671120429643327159n_assn )
      = nil_Pr5671120429643327159n_assn ) ).

% list.simps(8)
thf(fact_1185_list_Osimps_I8_J,axiom,
    ! [F2: produc6575502325842934193n_assn > int] :
      ( ( map_Pr7568062423562401049sn_int @ F2 @ nil_Pr5671120429643327159n_assn )
      = nil_int ) ).

% list.simps(8)
thf(fact_1186_list_Osimps_I8_J,axiom,
    ! [F2: produc6575502325842934193n_assn > nat] :
      ( ( map_Pr7570552894071451325sn_nat @ F2 @ nil_Pr5671120429643327159n_assn )
      = nil_nat ) ).

% list.simps(8)
thf(fact_1187_list_Osimps_I8_J,axiom,
    ! [F2: int > produc6575502325842934193n_assn] :
      ( ( map_in4427992030928829247n_assn @ F2 @ nil_int )
      = nil_Pr5671120429643327159n_assn ) ).

% list.simps(8)
thf(fact_1188_list_Osimps_I8_J,axiom,
    ! [F2: int > int] :
      ( ( map_int_int @ F2 @ nil_int )
      = nil_int ) ).

% list.simps(8)
thf(fact_1189_list_Osimps_I8_J,axiom,
    ! [F2: int > nat] :
      ( ( map_int_nat @ F2 @ nil_int )
      = nil_nat ) ).

% list.simps(8)
thf(fact_1190_list_Osimps_I8_J,axiom,
    ! [F2: nat > produc6575502325842934193n_assn] :
      ( ( map_na2667955367175718043n_assn @ F2 @ nil_nat )
      = nil_Pr5671120429643327159n_assn ) ).

% list.simps(8)
thf(fact_1191_list_Osimps_I8_J,axiom,
    ! [F2: nat > int] :
      ( ( map_nat_int @ F2 @ nil_nat )
      = nil_int ) ).

% list.simps(8)
thf(fact_1192_list_Osimps_I8_J,axiom,
    ! [F2: nat > nat] :
      ( ( map_nat_nat @ F2 @ nil_nat )
      = nil_nat ) ).

% list.simps(8)
thf(fact_1193_list_Osimps_I8_J,axiom,
    ! [F2: produc6575502325842934193n_assn > assn] :
      ( ( map_Pr8991440229025900053n_assn @ F2 @ nil_Pr5671120429643327159n_assn )
      = nil_assn ) ).

% list.simps(8)
thf(fact_1194_le__numeral__extra_I4_J,axiom,
    ord_le3102999989581377725nteger @ one_one_Code_integer @ one_one_Code_integer ).

% le_numeral_extra(4)
thf(fact_1195_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_1196_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_1197_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_1198_pure__assn__raw_Ocases,axiom,
    ! [X: produc6197004810343482825et_nat] :
      ~ ! [B5: $o,H3: heap_e7401611519738050253t_unit,As: set_nat] :
          ( X
         != ( produc3762353314782720579et_nat @ B5 @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ).

% pure_assn_raw.cases
thf(fact_1199_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_1200_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_1201_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_1202_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_1203_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_1204_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_1205_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_1206_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_1207_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_1208_inf_OcoboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B @ C )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_1209_inf_OcoboundedI2,axiom,
    ! [B: product_prod_int_int > $o,C: product_prod_int_int > $o,A: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ B @ C )
     => ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_1210_inf_OcoboundedI2,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ( ord_less_eq_assn @ B @ C )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_1211_inf_OcoboundedI2,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ C )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_1212_inf_OcoboundedI2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ C )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_1213_inf_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_1214_inf_OcoboundedI2,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ B @ C )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_1215_inf_OcoboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ C )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_1216_inf_OcoboundedI1,axiom,
    ! [A: product_prod_int_int > $o,C: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ A @ C )
     => ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_1217_inf_OcoboundedI1,axiom,
    ! [A: assn,C: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ C )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_1218_inf_OcoboundedI1,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_1219_inf_OcoboundedI1,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ C )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_1220_inf_OcoboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_1221_inf_OcoboundedI1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ C )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_1222_inf_Oabsorb__iff2,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% inf.absorb_iff2
thf(fact_1223_inf_Oabsorb__iff2,axiom,
    ( ord_le8369615600986905444_int_o
    = ( ^ [B2: product_prod_int_int > $o,A2: product_prod_int_int > $o] :
          ( ( inf_in3604695632404883862_int_o @ A2 @ B2 )
          = B2 ) ) ) ).

% inf.absorb_iff2
thf(fact_1224_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_assn
    = ( ^ [B2: assn,A2: assn] :
          ( ( inf_inf_assn @ A2 @ B2 )
          = B2 ) ) ) ).

% inf.absorb_iff2
thf(fact_1225_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( ( inf_inf_set_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% inf.absorb_iff2
thf(fact_1226_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( inf_inf_rat @ A2 @ B2 )
          = B2 ) ) ) ).

% inf.absorb_iff2
thf(fact_1227_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( inf_inf_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% inf.absorb_iff2
thf(fact_1228_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( ( inf_inf_int @ A2 @ B2 )
          = B2 ) ) ) ).

% inf.absorb_iff2
thf(fact_1229_inf_Oabsorb__iff1,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% inf.absorb_iff1
thf(fact_1230_inf_Oabsorb__iff1,axiom,
    ( ord_le8369615600986905444_int_o
    = ( ^ [A2: product_prod_int_int > $o,B2: product_prod_int_int > $o] :
          ( ( inf_in3604695632404883862_int_o @ A2 @ B2 )
          = A2 ) ) ) ).

% inf.absorb_iff1
thf(fact_1231_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_assn
    = ( ^ [A2: assn,B2: assn] :
          ( ( inf_inf_assn @ A2 @ B2 )
          = A2 ) ) ) ).

% inf.absorb_iff1
thf(fact_1232_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( inf_inf_set_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% inf.absorb_iff1
thf(fact_1233_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( inf_inf_rat @ A2 @ B2 )
          = A2 ) ) ) ).

% inf.absorb_iff1
thf(fact_1234_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( inf_inf_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% inf.absorb_iff1
thf(fact_1235_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( ( inf_inf_int @ A2 @ B2 )
          = A2 ) ) ) ).

% inf.absorb_iff1
thf(fact_1236_inf_Ocobounded2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_1237_inf_Ocobounded2,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_1238_inf_Ocobounded2,axiom,
    ! [A: assn,B: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_1239_inf_Ocobounded2,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_1240_inf_Ocobounded2,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_1241_inf_Ocobounded2,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_1242_inf_Ocobounded2,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_1243_inf_Ocobounded1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_1244_inf_Ocobounded1,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_1245_inf_Ocobounded1,axiom,
    ! [A: assn,B: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_1246_inf_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_1247_inf_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_1248_inf_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_1249_inf_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_1250_inf_Oorder__iff,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
          ( A2
          = ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ).

% inf.order_iff
thf(fact_1251_inf_Oorder__iff,axiom,
    ( ord_le8369615600986905444_int_o
    = ( ^ [A2: product_prod_int_int > $o,B2: product_prod_int_int > $o] :
          ( A2
          = ( inf_in3604695632404883862_int_o @ A2 @ B2 ) ) ) ) ).

% inf.order_iff
thf(fact_1252_inf_Oorder__iff,axiom,
    ( ord_less_eq_assn
    = ( ^ [A2: assn,B2: assn] :
          ( A2
          = ( inf_inf_assn @ A2 @ B2 ) ) ) ) ).

% inf.order_iff
thf(fact_1253_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( A2
          = ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).

% inf.order_iff
thf(fact_1254_inf_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [A2: rat,B2: rat] :
          ( A2
          = ( inf_inf_rat @ A2 @ B2 ) ) ) ) ).

% inf.order_iff
thf(fact_1255_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( A2
          = ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).

% inf.order_iff
thf(fact_1256_inf_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( A2
          = ( inf_inf_int @ A2 @ B2 ) ) ) ) ).

% inf.order_iff
thf(fact_1257_inf__greatest,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ Y )
     => ( ( ord_le3146513528884898305at_nat @ X @ Z )
       => ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1258_inf__greatest,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ X @ Y )
     => ( ( ord_le8369615600986905444_int_o @ X @ Z )
       => ( ord_le8369615600986905444_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1259_inf__greatest,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ X @ Z )
       => ( ord_less_eq_assn @ X @ ( inf_inf_assn @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1260_inf__greatest,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ X @ Z )
       => ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1261_inf__greatest,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ X @ Z )
       => ( ord_less_eq_rat @ X @ ( inf_inf_rat @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1262_inf__greatest,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Z )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1263_inf__greatest,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Z )
       => ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_1264_inf_OboundedI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ( ord_le3146513528884898305at_nat @ A @ C )
       => ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_1265_inf_OboundedI,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o,C: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ A @ B )
     => ( ( ord_le8369615600986905444_int_o @ A @ C )
       => ( ord_le8369615600986905444_int_o @ A @ ( inf_in3604695632404883862_int_o @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_1266_inf_OboundedI,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ord_less_eq_assn @ A @ C )
       => ( ord_less_eq_assn @ A @ ( inf_inf_assn @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_1267_inf_OboundedI,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ A @ C )
       => ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_1268_inf_OboundedI,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ A @ C )
       => ( ord_less_eq_rat @ A @ ( inf_inf_rat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_1269_inf_OboundedI,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ C )
       => ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_1270_inf_OboundedI,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ A @ C )
       => ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_1271_inf_OboundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) )
     => ~ ( ( ord_le3146513528884898305at_nat @ A @ B )
         => ~ ( ord_le3146513528884898305at_nat @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_1272_inf_OboundedE,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o,C: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ A @ ( inf_in3604695632404883862_int_o @ B @ C ) )
     => ~ ( ( ord_le8369615600986905444_int_o @ A @ B )
         => ~ ( ord_le8369615600986905444_int_o @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_1273_inf_OboundedE,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( ord_less_eq_assn @ A @ ( inf_inf_assn @ B @ C ) )
     => ~ ( ( ord_less_eq_assn @ A @ B )
         => ~ ( ord_less_eq_assn @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_1274_inf_OboundedE,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
     => ~ ( ( ord_less_eq_set_nat @ A @ B )
         => ~ ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_1275_inf_OboundedE,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( inf_inf_rat @ B @ C ) )
     => ~ ( ( ord_less_eq_rat @ A @ B )
         => ~ ( ord_less_eq_rat @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_1276_inf_OboundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
     => ~ ( ( ord_less_eq_nat @ A @ B )
         => ~ ( ord_less_eq_nat @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_1277_inf_OboundedE,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) )
     => ~ ( ( ord_less_eq_int @ A @ B )
         => ~ ( ord_less_eq_int @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_1278_inf__absorb2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ Y @ X )
     => ( ( inf_in2572325071724192079at_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1279_inf__absorb2,axiom,
    ! [Y: product_prod_int_int > $o,X: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ Y @ X )
     => ( ( inf_in3604695632404883862_int_o @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1280_inf__absorb2,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( inf_inf_assn @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1281_inf__absorb2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( inf_inf_set_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1282_inf__absorb2,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( inf_inf_rat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1283_inf__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( inf_inf_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1284_inf__absorb2,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( inf_inf_int @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_1285_inf__absorb1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ Y )
     => ( ( inf_in2572325071724192079at_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_1286_inf__absorb1,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ X @ Y )
     => ( ( inf_in3604695632404883862_int_o @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_1287_inf__absorb1,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( inf_inf_assn @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_1288_inf__absorb1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( inf_inf_set_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_1289_inf__absorb1,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( inf_inf_rat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_1290_inf__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( inf_inf_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_1291_inf__absorb1,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( inf_inf_int @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_1292_inf_Oabsorb2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B @ A )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_1293_inf_Oabsorb2,axiom,
    ! [B: product_prod_int_int > $o,A: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ B @ A )
     => ( ( inf_in3604695632404883862_int_o @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_1294_inf_Oabsorb2,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_eq_assn @ B @ A )
     => ( ( inf_inf_assn @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_1295_inf_Oabsorb2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( inf_inf_set_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_1296_inf_Oabsorb2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( inf_inf_rat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_1297_inf_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( inf_inf_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_1298_inf_Oabsorb2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( inf_inf_int @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_1299_inf_Oabsorb1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_1300_inf_Oabsorb1,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ A @ B )
     => ( ( inf_in3604695632404883862_int_o @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_1301_inf_Oabsorb1,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( inf_inf_assn @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_1302_inf_Oabsorb1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( inf_inf_set_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_1303_inf_Oabsorb1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( inf_inf_rat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_1304_inf_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( inf_inf_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_1305_inf_Oabsorb1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( inf_inf_int @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_1306_le__iff__inf,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ X2 @ Y2 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_1307_le__iff__inf,axiom,
    ( ord_le8369615600986905444_int_o
    = ( ^ [X2: product_prod_int_int > $o,Y2: product_prod_int_int > $o] :
          ( ( inf_in3604695632404883862_int_o @ X2 @ Y2 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_1308_le__iff__inf,axiom,
    ( ord_less_eq_assn
    = ( ^ [X2: assn,Y2: assn] :
          ( ( inf_inf_assn @ X2 @ Y2 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_1309_le__iff__inf,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] :
          ( ( inf_inf_set_nat @ X2 @ Y2 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_1310_le__iff__inf,axiom,
    ( ord_less_eq_rat
    = ( ^ [X2: rat,Y2: rat] :
          ( ( inf_inf_rat @ X2 @ Y2 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_1311_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y2: nat] :
          ( ( inf_inf_nat @ X2 @ Y2 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_1312_le__iff__inf,axiom,
    ( ord_less_eq_int
    = ( ^ [X2: int,Y2: int] :
          ( ( inf_inf_int @ X2 @ Y2 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_1313_inf__unique,axiom,
    ! [F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F2 @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F2 @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
              ( ( ord_le3146513528884898305at_nat @ X3 @ Y3 )
             => ( ( ord_le3146513528884898305at_nat @ X3 @ Z2 )
               => ( ord_le3146513528884898305at_nat @ X3 @ ( F2 @ Y3 @ Z2 ) ) ) )
         => ( ( inf_in2572325071724192079at_nat @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1314_inf__unique,axiom,
    ! [F2: ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > product_prod_int_int > $o,X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ! [X3: product_prod_int_int > $o,Y3: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( F2 @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: product_prod_int_int > $o,Y3: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( F2 @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: product_prod_int_int > $o,Y3: product_prod_int_int > $o,Z2: product_prod_int_int > $o] :
              ( ( ord_le8369615600986905444_int_o @ X3 @ Y3 )
             => ( ( ord_le8369615600986905444_int_o @ X3 @ Z2 )
               => ( ord_le8369615600986905444_int_o @ X3 @ ( F2 @ Y3 @ Z2 ) ) ) )
         => ( ( inf_in3604695632404883862_int_o @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1315_inf__unique,axiom,
    ! [F2: assn > assn > assn,X: assn,Y: assn] :
      ( ! [X3: assn,Y3: assn] : ( ord_less_eq_assn @ ( F2 @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: assn,Y3: assn] : ( ord_less_eq_assn @ ( F2 @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: assn,Y3: assn,Z2: assn] :
              ( ( ord_less_eq_assn @ X3 @ Y3 )
             => ( ( ord_less_eq_assn @ X3 @ Z2 )
               => ( ord_less_eq_assn @ X3 @ ( F2 @ Y3 @ Z2 ) ) ) )
         => ( ( inf_inf_assn @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1316_inf__unique,axiom,
    ! [F2: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( F2 @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( F2 @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y3 )
             => ( ( ord_less_eq_set_nat @ X3 @ Z2 )
               => ( ord_less_eq_set_nat @ X3 @ ( F2 @ Y3 @ Z2 ) ) ) )
         => ( ( inf_inf_set_nat @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1317_inf__unique,axiom,
    ! [F2: rat > rat > rat,X: rat,Y: rat] :
      ( ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ ( F2 @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ ( F2 @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: rat,Y3: rat,Z2: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ( ord_less_eq_rat @ X3 @ Z2 )
               => ( ord_less_eq_rat @ X3 @ ( F2 @ Y3 @ Z2 ) ) ) )
         => ( ( inf_inf_rat @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1318_inf__unique,axiom,
    ! [F2: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F2 @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F2 @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ( ord_less_eq_nat @ X3 @ Z2 )
               => ( ord_less_eq_nat @ X3 @ ( F2 @ Y3 @ Z2 ) ) ) )
         => ( ( inf_inf_nat @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1319_inf__unique,axiom,
    ! [F2: int > int > int,X: int,Y: int] :
      ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ ( F2 @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ ( F2 @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: int,Y3: int,Z2: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ( ord_less_eq_int @ X3 @ Z2 )
               => ( ord_less_eq_int @ X3 @ ( F2 @ Y3 @ Z2 ) ) ) )
         => ( ( inf_inf_int @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_1320_inf_OorderI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( A
        = ( inf_in2572325071724192079at_nat @ A @ B ) )
     => ( ord_le3146513528884898305at_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_1321_inf_OorderI,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( A
        = ( inf_in3604695632404883862_int_o @ A @ B ) )
     => ( ord_le8369615600986905444_int_o @ A @ B ) ) ).

% inf.orderI
thf(fact_1322_inf_OorderI,axiom,
    ! [A: assn,B: assn] :
      ( ( A
        = ( inf_inf_assn @ A @ B ) )
     => ( ord_less_eq_assn @ A @ B ) ) ).

% inf.orderI
thf(fact_1323_inf_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A
        = ( inf_inf_set_nat @ A @ B ) )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_1324_inf_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( inf_inf_rat @ A @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% inf.orderI
thf(fact_1325_inf_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( inf_inf_nat @ A @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_1326_inf_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( inf_inf_int @ A @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% inf.orderI
thf(fact_1327_inf_OorderE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( A
        = ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_1328_inf_OorderE,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ A @ B )
     => ( A
        = ( inf_in3604695632404883862_int_o @ A @ B ) ) ) ).

% inf.orderE
thf(fact_1329_inf_OorderE,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( A
        = ( inf_inf_assn @ A @ B ) ) ) ).

% inf.orderE
thf(fact_1330_inf_OorderE,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( A
        = ( inf_inf_set_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_1331_inf_OorderE,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( A
        = ( inf_inf_rat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_1332_inf_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( A
        = ( inf_inf_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_1333_inf_OorderE,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( A
        = ( inf_inf_int @ A @ B ) ) ) ).

% inf.orderE
thf(fact_1334_le__infI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B @ X )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_1335_le__infI2,axiom,
    ! [B: product_prod_int_int > $o,X: product_prod_int_int > $o,A: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ B @ X )
     => ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_1336_le__infI2,axiom,
    ! [B: assn,X: assn,A: assn] :
      ( ( ord_less_eq_assn @ B @ X )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_1337_le__infI2,axiom,
    ! [B: set_nat,X: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ X )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_1338_le__infI2,axiom,
    ! [B: rat,X: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ X )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_1339_le__infI2,axiom,
    ! [B: nat,X: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_1340_le__infI2,axiom,
    ! [B: int,X: int,A: int] :
      ( ( ord_less_eq_int @ B @ X )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_1341_le__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ X )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_1342_le__infI1,axiom,
    ! [A: product_prod_int_int > $o,X: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ A @ X )
     => ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_1343_le__infI1,axiom,
    ! [A: assn,X: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ X )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_1344_le__infI1,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ X )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_1345_le__infI1,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ X )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_1346_le__infI1,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_1347_le__infI1,axiom,
    ! [A: int,X: int,B: int] :
      ( ( ord_less_eq_int @ A @ X )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_1348_inf__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ C )
     => ( ( ord_le3146513528884898305at_nat @ B @ D2 )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ ( inf_in2572325071724192079at_nat @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1349_inf__mono,axiom,
    ! [A: product_prod_int_int > $o,C: product_prod_int_int > $o,B: product_prod_int_int > $o,D2: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ A @ C )
     => ( ( ord_le8369615600986905444_int_o @ B @ D2 )
       => ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ ( inf_in3604695632404883862_int_o @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1350_inf__mono,axiom,
    ! [A: assn,C: assn,B: assn,D2: assn] :
      ( ( ord_less_eq_assn @ A @ C )
     => ( ( ord_less_eq_assn @ B @ D2 )
       => ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ ( inf_inf_assn @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1351_inf__mono,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ( ord_less_eq_set_nat @ B @ D2 )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1352_inf__mono,axiom,
    ! [A: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ C )
     => ( ( ord_less_eq_rat @ B @ D2 )
       => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ ( inf_inf_rat @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1353_inf__mono,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ D2 )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1354_inf__mono,axiom,
    ! [A: int,C: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ A @ C )
     => ( ( ord_less_eq_int @ B @ D2 )
       => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ ( inf_inf_int @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_1355_le__infI,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ A )
     => ( ( ord_le3146513528884898305at_nat @ X @ B )
       => ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_1356_le__infI,axiom,
    ! [X: product_prod_int_int > $o,A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ X @ A )
     => ( ( ord_le8369615600986905444_int_o @ X @ B )
       => ( ord_le8369615600986905444_int_o @ X @ ( inf_in3604695632404883862_int_o @ A @ B ) ) ) ) ).

% le_infI
thf(fact_1357_le__infI,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( ( ord_less_eq_assn @ X @ A )
     => ( ( ord_less_eq_assn @ X @ B )
       => ( ord_less_eq_assn @ X @ ( inf_inf_assn @ A @ B ) ) ) ) ).

% le_infI
thf(fact_1358_le__infI,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ A )
     => ( ( ord_less_eq_set_nat @ X @ B )
       => ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_1359_le__infI,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ X @ A )
     => ( ( ord_less_eq_rat @ X @ B )
       => ( ord_less_eq_rat @ X @ ( inf_inf_rat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_1360_le__infI,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ A )
     => ( ( ord_less_eq_nat @ X @ B )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_1361_le__infI,axiom,
    ! [X: int,A: int,B: int] :
      ( ( ord_less_eq_int @ X @ A )
     => ( ( ord_less_eq_int @ X @ B )
       => ( ord_less_eq_int @ X @ ( inf_inf_int @ A @ B ) ) ) ) ).

% le_infI
thf(fact_1362_le__infE,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ A @ B ) )
     => ~ ( ( ord_le3146513528884898305at_nat @ X @ A )
         => ~ ( ord_le3146513528884898305at_nat @ X @ B ) ) ) ).

% le_infE
thf(fact_1363_le__infE,axiom,
    ! [X: product_prod_int_int > $o,A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( ord_le8369615600986905444_int_o @ X @ ( inf_in3604695632404883862_int_o @ A @ B ) )
     => ~ ( ( ord_le8369615600986905444_int_o @ X @ A )
         => ~ ( ord_le8369615600986905444_int_o @ X @ B ) ) ) ).

% le_infE
thf(fact_1364_le__infE,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( ( ord_less_eq_assn @ X @ ( inf_inf_assn @ A @ B ) )
     => ~ ( ( ord_less_eq_assn @ X @ A )
         => ~ ( ord_less_eq_assn @ X @ B ) ) ) ).

% le_infE
thf(fact_1365_le__infE,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B ) )
     => ~ ( ( ord_less_eq_set_nat @ X @ A )
         => ~ ( ord_less_eq_set_nat @ X @ B ) ) ) ).

% le_infE
thf(fact_1366_le__infE,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ X @ ( inf_inf_rat @ A @ B ) )
     => ~ ( ( ord_less_eq_rat @ X @ A )
         => ~ ( ord_less_eq_rat @ X @ B ) ) ) ).

% le_infE
thf(fact_1367_le__infE,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
     => ~ ( ( ord_less_eq_nat @ X @ A )
         => ~ ( ord_less_eq_nat @ X @ B ) ) ) ).

% le_infE
thf(fact_1368_le__infE,axiom,
    ! [X: int,A: int,B: int] :
      ( ( ord_less_eq_int @ X @ ( inf_inf_int @ A @ B ) )
     => ~ ( ( ord_less_eq_int @ X @ A )
         => ~ ( ord_less_eq_int @ X @ B ) ) ) ).

% le_infE
thf(fact_1369_inf__le2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_1370_inf__le2,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_1371_inf__le2,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_1372_inf__le2,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_1373_inf__le2,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_1374_inf__le2,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_1375_inf__le2,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_1376_inf__le1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_1377_inf__le1,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_1378_inf__le1,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_1379_inf__le1,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_1380_inf__le1,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_1381_inf__le1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_1382_inf__le1,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_1383_inf__sup__ord_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_1384_inf__sup__ord_I1_J,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_1385_inf__sup__ord_I1_J,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_1386_inf__sup__ord_I1_J,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_1387_inf__sup__ord_I1_J,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_1388_inf__sup__ord_I1_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_1389_inf__sup__ord_I1_J,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_1390_inf__sup__ord_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1391_inf__sup__ord_I2_J,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1392_inf__sup__ord_I2_J,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1393_inf__sup__ord_I2_J,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1394_inf__sup__ord_I2_J,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1395_inf__sup__ord_I2_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1396_inf__sup__ord_I2_J,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_1397_inf__sup__ord_I4_J,axiom,
    ! [Y: assn,X: assn] : ( ord_less_eq_assn @ Y @ ( sup_sup_assn @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_1398_inf__sup__ord_I4_J,axiom,
    ! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_1399_inf__sup__ord_I4_J,axiom,
    ! [Y: rat,X: rat] : ( ord_less_eq_rat @ Y @ ( sup_sup_rat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_1400_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_1401_inf__sup__ord_I4_J,axiom,
    ! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_1402_inf__sup__ord_I3_J,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ X @ ( sup_sup_assn @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_1403_inf__sup__ord_I3_J,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_1404_inf__sup__ord_I3_J,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ X @ ( sup_sup_rat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_1405_inf__sup__ord_I3_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_1406_inf__sup__ord_I3_J,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_1407_le__supE,axiom,
    ! [A: assn,B: assn,X: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_assn @ A @ X )
         => ~ ( ord_less_eq_assn @ B @ X ) ) ) ).

% le_supE
thf(fact_1408_le__supE,axiom,
    ! [A: set_nat,B: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_set_nat @ A @ X )
         => ~ ( ord_less_eq_set_nat @ B @ X ) ) ) ).

% le_supE
thf(fact_1409_le__supE,axiom,
    ! [A: rat,B: rat,X: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_rat @ A @ X )
         => ~ ( ord_less_eq_rat @ B @ X ) ) ) ).

% le_supE
thf(fact_1410_le__supE,axiom,
    ! [A: nat,B: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_nat @ A @ X )
         => ~ ( ord_less_eq_nat @ B @ X ) ) ) ).

% le_supE
thf(fact_1411_le__supE,axiom,
    ! [A: int,B: int,X: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_int @ A @ X )
         => ~ ( ord_less_eq_int @ B @ X ) ) ) ).

% le_supE
thf(fact_1412_le__supI,axiom,
    ! [A: assn,X: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ X )
     => ( ( ord_less_eq_assn @ B @ X )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_1413_le__supI,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ X )
     => ( ( ord_less_eq_set_nat @ B @ X )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_1414_le__supI,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ X )
     => ( ( ord_less_eq_rat @ B @ X )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_1415_le__supI,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X )
     => ( ( ord_less_eq_nat @ B @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_1416_le__supI,axiom,
    ! [A: int,X: int,B: int] :
      ( ( ord_less_eq_int @ A @ X )
     => ( ( ord_less_eq_int @ B @ X )
       => ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_1417_sup__ge1,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ X @ ( sup_sup_assn @ X @ Y ) ) ).

% sup_ge1
thf(fact_1418_sup__ge1,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_1419_sup__ge1,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ X @ ( sup_sup_rat @ X @ Y ) ) ).

% sup_ge1
thf(fact_1420_sup__ge1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_1421_sup__ge1,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).

% sup_ge1
thf(fact_1422_sup__ge2,axiom,
    ! [Y: assn,X: assn] : ( ord_less_eq_assn @ Y @ ( sup_sup_assn @ X @ Y ) ) ).

% sup_ge2
thf(fact_1423_sup__ge2,axiom,
    ! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_1424_sup__ge2,axiom,
    ! [Y: rat,X: rat] : ( ord_less_eq_rat @ Y @ ( sup_sup_rat @ X @ Y ) ) ).

% sup_ge2
thf(fact_1425_sup__ge2,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_1426_sup__ge2,axiom,
    ! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).

% sup_ge2
thf(fact_1427_le__supI1,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( ( ord_less_eq_assn @ X @ A )
     => ( ord_less_eq_assn @ X @ ( sup_sup_assn @ A @ B ) ) ) ).

% le_supI1
thf(fact_1428_le__supI1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ A )
     => ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_1429_le__supI1,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ X @ A )
     => ( ord_less_eq_rat @ X @ ( sup_sup_rat @ A @ B ) ) ) ).

% le_supI1
thf(fact_1430_le__supI1,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ A )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_1431_le__supI1,axiom,
    ! [X: int,A: int,B: int] :
      ( ( ord_less_eq_int @ X @ A )
     => ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).

% le_supI1
thf(fact_1432_le__supI2,axiom,
    ! [X: assn,B: assn,A: assn] :
      ( ( ord_less_eq_assn @ X @ B )
     => ( ord_less_eq_assn @ X @ ( sup_sup_assn @ A @ B ) ) ) ).

% le_supI2
thf(fact_1433_le__supI2,axiom,
    ! [X: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ B )
     => ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_1434_le__supI2,axiom,
    ! [X: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ X @ B )
     => ( ord_less_eq_rat @ X @ ( sup_sup_rat @ A @ B ) ) ) ).

% le_supI2
thf(fact_1435_le__supI2,axiom,
    ! [X: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ X @ B )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_1436_le__supI2,axiom,
    ! [X: int,B: int,A: int] :
      ( ( ord_less_eq_int @ X @ B )
     => ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).

% le_supI2
thf(fact_1437_sup_Omono,axiom,
    ! [C: assn,A: assn,D2: assn,B: assn] :
      ( ( ord_less_eq_assn @ C @ A )
     => ( ( ord_less_eq_assn @ D2 @ B )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ C @ D2 ) @ ( sup_sup_assn @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_1438_sup_Omono,axiom,
    ! [C: set_nat,A: set_nat,D2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ A )
     => ( ( ord_less_eq_set_nat @ D2 @ B )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D2 ) @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_1439_sup_Omono,axiom,
    ! [C: rat,A: rat,D2: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ( ord_less_eq_rat @ D2 @ B )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ C @ D2 ) @ ( sup_sup_rat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_1440_sup_Omono,axiom,
    ! [C: nat,A: nat,D2: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D2 @ B )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_1441_sup_Omono,axiom,
    ! [C: int,A: int,D2: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ( ord_less_eq_int @ D2 @ B )
       => ( ord_less_eq_int @ ( sup_sup_int @ C @ D2 ) @ ( sup_sup_int @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_1442_sup__mono,axiom,
    ! [A: assn,C: assn,B: assn,D2: assn] :
      ( ( ord_less_eq_assn @ A @ C )
     => ( ( ord_less_eq_assn @ B @ D2 )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ A @ B ) @ ( sup_sup_assn @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1443_sup__mono,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ( ord_less_eq_set_nat @ B @ D2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1444_sup__mono,axiom,
    ! [A: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ C )
     => ( ( ord_less_eq_rat @ B @ D2 )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ A @ B ) @ ( sup_sup_rat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1445_sup__mono,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ D2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1446_sup__mono,axiom,
    ! [A: int,C: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ A @ C )
     => ( ( ord_less_eq_int @ B @ D2 )
       => ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ ( sup_sup_int @ C @ D2 ) ) ) ) ).

% sup_mono
thf(fact_1447_sup__least,axiom,
    ! [Y: assn,X: assn,Z: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( ord_less_eq_assn @ Z @ X )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_1448_sup__least,axiom,
    ! [Y: set_nat,X: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_less_eq_set_nat @ Z @ X )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_1449_sup__least,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( ord_less_eq_rat @ Z @ X )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_1450_sup__least,axiom,
    ! [Y: nat,X: nat,Z: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ Z @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_1451_sup__least,axiom,
    ! [Y: int,X: int,Z: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_less_eq_int @ Z @ X )
       => ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_1452_le__iff__sup,axiom,
    ( ord_less_eq_assn
    = ( ^ [X2: assn,Y2: assn] :
          ( ( sup_sup_assn @ X2 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_1453_le__iff__sup,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] :
          ( ( sup_sup_set_nat @ X2 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_1454_le__iff__sup,axiom,
    ( ord_less_eq_rat
    = ( ^ [X2: rat,Y2: rat] :
          ( ( sup_sup_rat @ X2 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_1455_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y2: nat] :
          ( ( sup_sup_nat @ X2 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_1456_le__iff__sup,axiom,
    ( ord_less_eq_int
    = ( ^ [X2: int,Y2: int] :
          ( ( sup_sup_int @ X2 @ Y2 )
          = Y2 ) ) ) ).

% le_iff_sup
thf(fact_1457_sup_OorderE,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_eq_assn @ B @ A )
     => ( A
        = ( sup_sup_assn @ A @ B ) ) ) ).

% sup.orderE
thf(fact_1458_sup_OorderE,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( A
        = ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_1459_sup_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( sup_sup_rat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_1460_sup_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( sup_sup_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_1461_sup_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( sup_sup_int @ A @ B ) ) ) ).

% sup.orderE
thf(fact_1462_sup_OorderI,axiom,
    ! [A: assn,B: assn] :
      ( ( A
        = ( sup_sup_assn @ A @ B ) )
     => ( ord_less_eq_assn @ B @ A ) ) ).

% sup.orderI
thf(fact_1463_sup_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A
        = ( sup_sup_set_nat @ A @ B ) )
     => ( ord_less_eq_set_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_1464_sup_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( sup_sup_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% sup.orderI
thf(fact_1465_sup_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( sup_sup_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_1466_sup_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( sup_sup_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% sup.orderI
thf(fact_1467_sup__unique,axiom,
    ! [F2: assn > assn > assn,X: assn,Y: assn] :
      ( ! [X3: assn,Y3: assn] : ( ord_less_eq_assn @ X3 @ ( F2 @ X3 @ Y3 ) )
     => ( ! [X3: assn,Y3: assn] : ( ord_less_eq_assn @ Y3 @ ( F2 @ X3 @ Y3 ) )
       => ( ! [X3: assn,Y3: assn,Z2: assn] :
              ( ( ord_less_eq_assn @ Y3 @ X3 )
             => ( ( ord_less_eq_assn @ Z2 @ X3 )
               => ( ord_less_eq_assn @ ( F2 @ Y3 @ Z2 ) @ X3 ) ) )
         => ( ( sup_sup_assn @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_1468_sup__unique,axiom,
    ! [F2: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( F2 @ X3 @ Y3 ) )
     => ( ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( F2 @ X3 @ Y3 ) )
       => ( ! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
              ( ( ord_less_eq_set_nat @ Y3 @ X3 )
             => ( ( ord_less_eq_set_nat @ Z2 @ X3 )
               => ( ord_less_eq_set_nat @ ( F2 @ Y3 @ Z2 ) @ X3 ) ) )
         => ( ( sup_sup_set_nat @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_1469_sup__unique,axiom,
    ! [F2: rat > rat > rat,X: rat,Y: rat] :
      ( ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ X3 @ ( F2 @ X3 @ Y3 ) )
     => ( ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ Y3 @ ( F2 @ X3 @ Y3 ) )
       => ( ! [X3: rat,Y3: rat,Z2: rat] :
              ( ( ord_less_eq_rat @ Y3 @ X3 )
             => ( ( ord_less_eq_rat @ Z2 @ X3 )
               => ( ord_less_eq_rat @ ( F2 @ Y3 @ Z2 ) @ X3 ) ) )
         => ( ( sup_sup_rat @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_1470_sup__unique,axiom,
    ! [F2: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( F2 @ X3 @ Y3 ) )
     => ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F2 @ X3 @ Y3 ) )
       => ( ! [X3: nat,Y3: nat,Z2: nat] :
              ( ( ord_less_eq_nat @ Y3 @ X3 )
             => ( ( ord_less_eq_nat @ Z2 @ X3 )
               => ( ord_less_eq_nat @ ( F2 @ Y3 @ Z2 ) @ X3 ) ) )
         => ( ( sup_sup_nat @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_1471_sup__unique,axiom,
    ! [F2: int > int > int,X: int,Y: int] :
      ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ X3 @ ( F2 @ X3 @ Y3 ) )
     => ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ Y3 @ ( F2 @ X3 @ Y3 ) )
       => ( ! [X3: int,Y3: int,Z2: int] :
              ( ( ord_less_eq_int @ Y3 @ X3 )
             => ( ( ord_less_eq_int @ Z2 @ X3 )
               => ( ord_less_eq_int @ ( F2 @ Y3 @ Z2 ) @ X3 ) ) )
         => ( ( sup_sup_int @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_1472_sup_Oabsorb1,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_eq_assn @ B @ A )
     => ( ( sup_sup_assn @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_1473_sup_Oabsorb1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( sup_sup_set_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_1474_sup_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( sup_sup_rat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_1475_sup_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_1476_sup_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( sup_sup_int @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_1477_sup_Oabsorb2,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( sup_sup_assn @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_1478_sup_Oabsorb2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( sup_sup_set_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_1479_sup_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( sup_sup_rat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_1480_sup_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_1481_sup_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( sup_sup_int @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_1482_sup__absorb1,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( sup_sup_assn @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_1483_sup__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( sup_sup_set_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_1484_sup__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( sup_sup_rat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_1485_sup__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( sup_sup_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_1486_sup__absorb1,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( sup_sup_int @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_1487_sup__absorb2,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( sup_sup_assn @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_1488_sup__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( sup_sup_set_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_1489_sup__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( sup_sup_rat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_1490_sup__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( sup_sup_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_1491_sup__absorb2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( sup_sup_int @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_1492_sup_OboundedE,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_assn @ B @ A )
         => ~ ( ord_less_eq_assn @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_1493_sup_OboundedE,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_set_nat @ B @ A )
         => ~ ( ord_less_eq_set_nat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_1494_sup_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_rat @ B @ A )
         => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_1495_sup_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_1496_sup_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_1497_sup_OboundedI,axiom,
    ! [B: assn,A: assn,C: assn] :
      ( ( ord_less_eq_assn @ B @ A )
     => ( ( ord_less_eq_assn @ C @ A )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_1498_sup_OboundedI,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C @ A )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_1499_sup_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ A )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_1500_sup_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_1501_sup_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ A )
       => ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_1502_sup_Oorder__iff,axiom,
    ( ord_less_eq_assn
    = ( ^ [B2: assn,A2: assn] :
          ( A2
          = ( sup_sup_assn @ A2 @ B2 ) ) ) ) ).

% sup.order_iff
thf(fact_1503_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( A2
          = ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).

% sup.order_iff
thf(fact_1504_sup_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A2: rat] :
          ( A2
          = ( sup_sup_rat @ A2 @ B2 ) ) ) ) ).

% sup.order_iff
thf(fact_1505_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( A2
          = ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).

% sup.order_iff
thf(fact_1506_sup_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( A2
          = ( sup_sup_int @ A2 @ B2 ) ) ) ) ).

% sup.order_iff
thf(fact_1507_sup_Ocobounded1,axiom,
    ! [A: assn,B: assn] : ( ord_less_eq_assn @ A @ ( sup_sup_assn @ A @ B ) ) ).

% sup.cobounded1
thf(fact_1508_sup_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_1509_sup_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( sup_sup_rat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_1510_sup_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_1511_sup_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( sup_sup_int @ A @ B ) ) ).

% sup.cobounded1
thf(fact_1512_sup_Ocobounded2,axiom,
    ! [B: assn,A: assn] : ( ord_less_eq_assn @ B @ ( sup_sup_assn @ A @ B ) ) ).

% sup.cobounded2
thf(fact_1513_sup_Ocobounded2,axiom,
    ! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_1514_sup_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( sup_sup_rat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_1515_sup_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_1516_sup_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( sup_sup_int @ A @ B ) ) ).

% sup.cobounded2
thf(fact_1517_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_assn
    = ( ^ [B2: assn,A2: assn] :
          ( ( sup_sup_assn @ A2 @ B2 )
          = A2 ) ) ) ).

% sup.absorb_iff1
thf(fact_1518_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( ( sup_sup_set_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% sup.absorb_iff1
thf(fact_1519_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( sup_sup_rat @ A2 @ B2 )
          = A2 ) ) ) ).

% sup.absorb_iff1
thf(fact_1520_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( sup_sup_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% sup.absorb_iff1
thf(fact_1521_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( ( sup_sup_int @ A2 @ B2 )
          = A2 ) ) ) ).

% sup.absorb_iff1
thf(fact_1522_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_assn
    = ( ^ [A2: assn,B2: assn] :
          ( ( sup_sup_assn @ A2 @ B2 )
          = B2 ) ) ) ).

% sup.absorb_iff2
thf(fact_1523_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( sup_sup_set_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% sup.absorb_iff2
thf(fact_1524_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( sup_sup_rat @ A2 @ B2 )
          = B2 ) ) ) ).

% sup.absorb_iff2
thf(fact_1525_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( sup_sup_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% sup.absorb_iff2
thf(fact_1526_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( ( sup_sup_int @ A2 @ B2 )
          = B2 ) ) ) ).

% sup.absorb_iff2
thf(fact_1527_sup_OcoboundedI1,axiom,
    ! [C: assn,A: assn,B: assn] :
      ( ( ord_less_eq_assn @ C @ A )
     => ( ord_less_eq_assn @ C @ ( sup_sup_assn @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_1528_sup_OcoboundedI1,axiom,
    ! [C: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ A )
     => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_1529_sup_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_1530_sup_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_1531_sup_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_1532_sup_OcoboundedI2,axiom,
    ! [C: assn,B: assn,A: assn] :
      ( ( ord_less_eq_assn @ C @ B )
     => ( ord_less_eq_assn @ C @ ( sup_sup_assn @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_1533_sup_OcoboundedI2,axiom,
    ! [C: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ B )
     => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_1534_sup_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ B )
     => ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_1535_sup_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_1536_sup_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C @ B )
     => ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_1537_Int__Collect__mono,axiom,
    ! [A3: set_int,B3: set_int,P2: int > $o,Q2: int > $o] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ! [X3: int] :
            ( ( member_int2 @ X3 @ A3 )
           => ( ( P2 @ X3 )
             => ( Q2 @ X3 ) ) )
       => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A3 @ ( collect_int @ P2 ) ) @ ( inf_inf_set_int @ B3 @ ( collect_int @ Q2 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1538_Int__Collect__mono,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o,P2: ( produc3658429121746597890et_nat > $o ) > $o,Q2: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( ord_le2965882846123202637_nat_o @ A3 @ B3 )
     => ( ! [X3: produc3658429121746597890et_nat > $o] :
            ( ( member6576561426505652726_nat_o @ X3 @ A3 )
           => ( ( P2 @ X3 )
             => ( Q2 @ X3 ) ) )
       => ( ord_le2965882846123202637_nat_o @ ( inf_in1906310914598751387_nat_o @ A3 @ ( collec939566748876313656_nat_o @ P2 ) ) @ ( inf_in1906310914598751387_nat_o @ B3 @ ( collec939566748876313656_nat_o @ Q2 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1539_Int__Collect__mono,axiom,
    ! [A3: set_list_nat,B3: set_list_nat,P2: list_nat > $o,Q2: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ A3 @ B3 )
     => ( ! [X3: list_nat] :
            ( ( member_list_nat @ X3 @ A3 )
           => ( ( P2 @ X3 )
             => ( Q2 @ X3 ) ) )
       => ( ord_le6045566169113846134st_nat @ ( inf_inf_set_list_nat @ A3 @ ( collect_list_nat @ P2 ) ) @ ( inf_inf_set_list_nat @ B3 @ ( collect_list_nat @ Q2 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1540_Int__Collect__mono,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat > $o,Q2: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A3 )
           => ( ( P2 @ X3 )
             => ( Q2 @ X3 ) ) )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ ( collec3392354462482085612at_nat @ P2 ) ) @ ( inf_in2572325071724192079at_nat @ B3 @ ( collec3392354462482085612at_nat @ Q2 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1541_Int__Collect__mono,axiom,
    ! [A3: set_nat,B3: set_nat,P2: nat > $o,Q2: nat > $o] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ! [X3: nat] :
            ( ( member_nat2 @ X3 @ A3 )
           => ( ( P2 @ X3 )
             => ( Q2 @ X3 ) ) )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P2 ) ) @ ( inf_inf_set_nat @ B3 @ ( collect_nat @ Q2 ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1542_Int__greatest,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ C2 @ A3 )
     => ( ( ord_le3146513528884898305at_nat @ C2 @ B3 )
       => ( ord_le3146513528884898305at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ).

% Int_greatest
thf(fact_1543_Int__greatest,axiom,
    ! [C2: set_nat,A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ A3 )
     => ( ( ord_less_eq_set_nat @ C2 @ B3 )
       => ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).

% Int_greatest
thf(fact_1544_Int__absorb2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = A3 ) ) ).

% Int_absorb2
thf(fact_1545_Int__absorb2,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( inf_inf_set_nat @ A3 @ B3 )
        = A3 ) ) ).

% Int_absorb2
thf(fact_1546_Int__absorb1,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B3 @ A3 )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = B3 ) ) ).

% Int_absorb1
thf(fact_1547_Int__absorb1,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ( inf_inf_set_nat @ A3 @ B3 )
        = B3 ) ) ).

% Int_absorb1
thf(fact_1548_Int__lower2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ B3 ) ).

% Int_lower2
thf(fact_1549_Int__lower2,axiom,
    ! [A3: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ B3 ) ).

% Int_lower2
thf(fact_1550_Int__lower1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ A3 ) ).

% Int_lower1
thf(fact_1551_Int__lower1,axiom,
    ! [A3: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ A3 ) ).

% Int_lower1
thf(fact_1552_Int__mono,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ C2 )
     => ( ( ord_le3146513528884898305at_nat @ B3 @ D )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ ( inf_in2572325071724192079at_nat @ C2 @ D ) ) ) ) ).

% Int_mono
thf(fact_1553_Int__mono,axiom,
    ! [A3: set_nat,C2: set_nat,B3: set_nat,D: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ C2 )
     => ( ( ord_less_eq_set_nat @ B3 @ D )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( inf_inf_set_nat @ C2 @ D ) ) ) ) ).

% Int_mono
thf(fact_1554_inter__eq__subsetI,axiom,
    ! [S: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ S @ S2 )
     => ( ( ( inf_in2572325071724192079at_nat @ A3 @ S2 )
          = ( inf_in2572325071724192079at_nat @ B3 @ S2 ) )
       => ( ( inf_in2572325071724192079at_nat @ A3 @ S )
          = ( inf_in2572325071724192079at_nat @ B3 @ S ) ) ) ) ).

% inter_eq_subsetI
thf(fact_1555_inter__eq__subsetI,axiom,
    ! [S: set_nat,S2: set_nat,A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ S2 )
     => ( ( ( inf_inf_set_nat @ A3 @ S2 )
          = ( inf_inf_set_nat @ B3 @ S2 ) )
       => ( ( inf_inf_set_nat @ A3 @ S )
          = ( inf_inf_set_nat @ B3 @ S ) ) ) ) ).

% inter_eq_subsetI
thf(fact_1556_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = ( times_times_rat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_1557_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A @ C )
          = ( times_times_nat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_1558_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ A @ C )
          = ( times_times_int @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_1559_mult__right__cancel,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( ( times_3573771949741848930nteger @ A @ C )
          = ( times_3573771949741848930nteger @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_1560_mult__right__cancel,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( ( ( times_2397367101498566445atural @ A @ C )
          = ( times_2397367101498566445atural @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_1561_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ C @ A )
          = ( times_times_rat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_1562_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A )
          = ( times_times_nat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_1563_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ C @ A )
          = ( times_times_int @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_1564_mult__left__cancel,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( ( times_3573771949741848930nteger @ C @ A )
          = ( times_3573771949741848930nteger @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_1565_mult__left__cancel,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( ( ( times_2397367101498566445atural @ C @ A )
          = ( times_2397367101498566445atural @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_1566_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( times_times_rat @ A @ B )
         != zero_zero_rat ) ) ) ).

% no_zero_divisors
thf(fact_1567_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( ( times_times_nat @ A @ B )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_1568_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( ( times_times_int @ A @ B )
         != zero_zero_int ) ) ) ).

% no_zero_divisors
thf(fact_1569_no__zero__divisors,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( B != zero_z3403309356797280102nteger )
       => ( ( times_3573771949741848930nteger @ A @ B )
         != zero_z3403309356797280102nteger ) ) ) ).

% no_zero_divisors
thf(fact_1570_no__zero__divisors,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( B != zero_z2226904508553997617atural )
       => ( ( times_2397367101498566445atural @ A @ B )
         != zero_z2226904508553997617atural ) ) ) ).

% no_zero_divisors
thf(fact_1571_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
thf(fact_1572_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_1573_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
        = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
thf(fact_1574_divisors__zero,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( ( A = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% divisors_zero
thf(fact_1575_divisors__zero,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( times_2397367101498566445atural @ A @ B )
        = zero_z2226904508553997617atural )
     => ( ( A = zero_z2226904508553997617atural )
        | ( B = zero_z2226904508553997617atural ) ) ) ).

% divisors_zero
thf(fact_1576_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
       != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
thf(fact_1577_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
       != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_1578_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
       != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
thf(fact_1579_mult__not__zero,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ B )
       != zero_z3403309356797280102nteger )
     => ( ( A != zero_z3403309356797280102nteger )
        & ( B != zero_z3403309356797280102nteger ) ) ) ).

% mult_not_zero
thf(fact_1580_mult__not__zero,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( times_2397367101498566445atural @ A @ B )
       != zero_z2226904508553997617atural )
     => ( ( A != zero_z2226904508553997617atural )
        & ( B != zero_z2226904508553997617atural ) ) ) ).

% mult_not_zero
thf(fact_1581_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_1582_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_1583_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_1584_zero__neq__one,axiom,
    zero_z3403309356797280102nteger != one_one_Code_integer ).

% zero_neq_one
thf(fact_1585_zero__neq__one,axiom,
    zero_z2226904508553997617atural != one_one_Code_natural ).

% zero_neq_one
thf(fact_1586_rotate1__map,axiom,
    ! [F2: nat > nat,Xs: list_nat] :
      ( ( rotate1_nat @ ( map_nat_nat @ F2 @ Xs ) )
      = ( map_nat_nat @ F2 @ ( rotate1_nat @ Xs ) ) ) ).

% rotate1_map
thf(fact_1587_rotate1__map,axiom,
    ! [F2: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn] :
      ( ( rotate1_assn @ ( map_Pr8991440229025900053n_assn @ F2 @ Xs ) )
      = ( map_Pr8991440229025900053n_assn @ F2 @ ( rotate328796349445179396n_assn @ Xs ) ) ) ).

% rotate1_map
thf(fact_1588_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( linord2614967742042102400et_nat @ A3 )
        = ( linord2614967742042102400et_nat @ B3 ) )
     => ( ( finite_finite_nat @ A3 )
       => ( ( finite_finite_nat @ B3 )
         => ( A3 = B3 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_1589_less__eq__assn__def,axiom,
    ( ord_less_eq_assn
    = ( ^ [A2: assn,B2: assn] :
          ( A2
          = ( inf_inf_assn @ A2 @ B2 ) ) ) ) ).

% less_eq_assn_def
thf(fact_1590_relH__subset,axiom,
    ! [Bs: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( relH @ Bs @ H2 @ H5 )
     => ( ( ord_less_eq_set_nat @ As2 @ Bs )
       => ( relH @ As2 @ H2 @ H5 ) ) ) ).

% relH_subset
thf(fact_1591_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_1592_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_1593_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_1594_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_1595_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_1596_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_1597_distrib__inf__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) @ ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1598_distrib__inf__le,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( sup_su8463660629351352368_int_o @ ( inf_in3604695632404883862_int_o @ X @ Y ) @ ( inf_in3604695632404883862_int_o @ X @ Z ) ) @ ( inf_in3604695632404883862_int_o @ X @ ( sup_su8463660629351352368_int_o @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1599_distrib__inf__le,axiom,
    ! [X: assn,Y: assn,Z: assn] : ( ord_less_eq_assn @ ( sup_sup_assn @ ( inf_inf_assn @ X @ Y ) @ ( inf_inf_assn @ X @ Z ) ) @ ( inf_inf_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1600_distrib__inf__le,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) @ ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1601_distrib__inf__le,axiom,
    ! [X: rat,Y: rat,Z: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ ( inf_inf_rat @ X @ Y ) @ ( inf_inf_rat @ X @ Z ) ) @ ( inf_inf_rat @ X @ ( sup_sup_rat @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1602_distrib__inf__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1603_distrib__inf__le,axiom,
    ! [X: int,Y: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ ( inf_inf_int @ X @ Y ) @ ( inf_inf_int @ X @ Z ) ) @ ( inf_inf_int @ X @ ( sup_sup_int @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_1604_distrib__sup__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) ) @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1605_distrib__sup__le,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o,Z: product_prod_int_int > $o] : ( ord_le8369615600986905444_int_o @ ( sup_su8463660629351352368_int_o @ X @ ( inf_in3604695632404883862_int_o @ Y @ Z ) ) @ ( inf_in3604695632404883862_int_o @ ( sup_su8463660629351352368_int_o @ X @ Y ) @ ( sup_su8463660629351352368_int_o @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1606_distrib__sup__le,axiom,
    ! [X: assn,Y: assn,Z: assn] : ( ord_less_eq_assn @ ( sup_sup_assn @ X @ ( inf_inf_assn @ Y @ Z ) ) @ ( inf_inf_assn @ ( sup_sup_assn @ X @ Y ) @ ( sup_sup_assn @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1607_distrib__sup__le,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1608_distrib__sup__le,axiom,
    ! [X: rat,Y: rat,Z: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ X @ ( inf_inf_rat @ Y @ Z ) ) @ ( inf_inf_rat @ ( sup_sup_rat @ X @ Y ) @ ( sup_sup_rat @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1609_distrib__sup__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1610_distrib__sup__le,axiom,
    ! [X: int,Y: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ X @ ( inf_inf_int @ Y @ Z ) ) @ ( inf_inf_int @ ( sup_sup_int @ X @ Y ) @ ( sup_sup_int @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_1611_disjoint__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A5: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ A5 )
     => ( ( ord_le3146513528884898305at_nat @ B @ B6 )
       => ( ( ( inf_in2572325071724192079at_nat @ A5 @ B6 )
            = bot_bo2099793752762293965at_nat )
         => ( ( inf_in2572325071724192079at_nat @ A @ B )
            = bot_bo2099793752762293965at_nat ) ) ) ) ).

% disjoint_mono
thf(fact_1612_disjoint__mono,axiom,
    ! [A: set_int,A5: set_int,B: set_int,B6: set_int] :
      ( ( ord_less_eq_set_int @ A @ A5 )
     => ( ( ord_less_eq_set_int @ B @ B6 )
       => ( ( ( inf_inf_set_int @ A5 @ B6 )
            = bot_bot_set_int )
         => ( ( inf_inf_set_int @ A @ B )
            = bot_bot_set_int ) ) ) ) ).

% disjoint_mono
thf(fact_1613_disjoint__mono,axiom,
    ! [A: set_nat,A5: set_nat,B: set_nat,B6: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ A5 )
     => ( ( ord_less_eq_set_nat @ B @ B6 )
       => ( ( ( inf_inf_set_nat @ A5 @ B6 )
            = bot_bot_set_nat )
         => ( ( inf_inf_set_nat @ A @ B )
            = bot_bot_set_nat ) ) ) ) ).

% disjoint_mono
thf(fact_1614_Un__Int__assoc__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ C2 )
        = ( inf_in2572325071724192079at_nat @ A3 @ ( sup_su6327502436637775413at_nat @ B3 @ C2 ) ) )
      = ( ord_le3146513528884898305at_nat @ C2 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_1615_Un__Int__assoc__eq,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] :
      ( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ C2 )
        = ( inf_inf_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C2 ) ) )
      = ( ord_less_eq_set_nat @ C2 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_1616_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_1617_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_1618_zero__neq__neg__one,axiom,
    ( zero_z3403309356797280102nteger
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% zero_neq_neg_one
thf(fact_1619_in__range__subset,axiom,
    ! [As2: set_nat,As3: set_nat,H2: heap_e7401611519738050253t_unit] :
      ( ( ord_less_eq_set_nat @ As2 @ As3 )
     => ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As3 ) )
       => ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) ) ) ) ).

% in_range_subset
thf(fact_1620_inf__shunt,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ X @ Y )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ X @ ( uminus6524753893492686040at_nat @ Y ) ) ) ).

% inf_shunt
thf(fact_1621_inf__shunt,axiom,
    ! [X: product_prod_int_int > $o,Y: product_prod_int_int > $o] :
      ( ( ( inf_in3604695632404883862_int_o @ X @ Y )
        = bot_bo8147686125503663512_int_o )
      = ( ord_le8369615600986905444_int_o @ X @ ( uminus7117520113953359693_int_o @ Y ) ) ) ).

% inf_shunt
thf(fact_1622_inf__shunt,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( inf_inf_set_int @ X @ Y )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).

% inf_shunt
thf(fact_1623_inf__shunt,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( inf_inf_assn @ X @ Y )
        = bot_bot_assn )
      = ( ord_less_eq_assn @ X @ ( uminus_uminus_assn @ Y ) ) ) ).

% inf_shunt
thf(fact_1624_inf__shunt,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( inf_inf_set_nat @ X @ Y )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).

% inf_shunt
thf(fact_1625_finite__Int,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
      ( ( ( finite6177210948735845034at_nat @ F4 )
        | ( finite6177210948735845034at_nat @ G2 ) )
     => ( finite6177210948735845034at_nat @ ( inf_in2572325071724192079at_nat @ F4 @ G2 ) ) ) ).

% finite_Int
thf(fact_1626_finite__Int,axiom,
    ! [F4: set_nat,G2: set_nat] :
      ( ( ( finite_finite_nat @ F4 )
        | ( finite_finite_nat @ G2 ) )
     => ( finite_finite_nat @ ( inf_inf_set_nat @ F4 @ G2 ) ) ) ).

% finite_Int
thf(fact_1627_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 ) ) )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H3 @ As ) )
             => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) )
               => ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) )
                  & ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
                      ( ( ( ( inf_inf_set_nat @ As @ As4 )
                          = bot_bot_set_nat )
                        & ( relH @ As @ H3 @ H4 )
                        & ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As ) )
                        & ( X @ ( produc7507926704131184380et_nat @ H4 @ As4 ) ) )
                     => ( Xa @ ( produc7507926704131184380et_nat @ H4 @ ( sup_sup_set_nat @ As @ As4 ) ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(3)
thf(fact_1628_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 ) ) )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H3 @ As ) )
             => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) )
               => ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) )
                    & ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As @ As5 )
                            = bot_bot_set_nat )
                          & ( relH @ As @ H3 @ H6 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H6 @ As5 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As @ As5 ) ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(2)
thf(fact_1629_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 ) ) )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H3 @ As ) )
             => ( ( Y
                  = ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As ) )
                    & ! [H7: heap_e7401611519738050253t_unit,As6: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As @ As6 )
                            = bot_bot_set_nat )
                          & ( relH @ As @ H3 @ H7 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H7 @ As6 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As @ As6 ) ) ) ) ) )
               => ~ ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(1)
thf(fact_1630_mod__star__trueE_H,axiom,
    ! [P2: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P2 @ top_top_assn ) @ H2 )
     => ~ ! [H4: produc3658429121746597890et_nat] :
            ( ( ( produc1824681642469235216et_nat @ H4 )
              = ( produc1824681642469235216et_nat @ H2 ) )
           => ( ( ord_less_eq_set_nat @ ( produc8586169260539613262et_nat @ H4 ) @ ( produc8586169260539613262et_nat @ H2 ) )
             => ~ ( rep_assn @ P2 @ H4 ) ) ) ) ).

% mod_star_trueE'
thf(fact_1631_map__eq__map__tailrec,axiom,
    map_nat_nat = map_tailrec_nat_nat ).

% map_eq_map_tailrec
thf(fact_1632_map__eq__map__tailrec,axiom,
    map_Pr8991440229025900053n_assn = map_ta2194206859901583947n_assn ).

% map_eq_map_tailrec
thf(fact_1633_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 ) ) )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H3 @ As ) )
             => ( ( Y
                  = ( ? [As1: set_nat,As22: set_nat] :
                        ( ( As
                          = ( sup_sup_set_nat @ As1 @ As22 ) )
                        & ( ( inf_inf_set_nat @ As1 @ As22 )
                          = bot_bot_set_nat )
                        & ( X @ ( produc7507926704131184380et_nat @ H3 @ As1 ) )
                        & ( Xa @ ( produc7507926704131184380et_nat @ H3 @ As22 ) ) ) ) )
               => ~ ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(1)
thf(fact_1634_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 ) ) )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H3 @ As ) )
             => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) )
               => ~ ? [As13: set_nat,As24: set_nat] :
                      ( ( As
                        = ( sup_sup_set_nat @ As13 @ As24 ) )
                      & ( ( inf_inf_set_nat @ As13 @ As24 )
                        = bot_bot_set_nat )
                      & ( X @ ( produc7507926704131184380et_nat @ H3 @ As13 ) )
                      & ( Xa @ ( produc7507926704131184380et_nat @ H3 @ As24 ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
thf(fact_1635_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 ) ) )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H3 @ As ) )
             => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) )
               => ? [As12: set_nat,As23: set_nat] :
                    ( ( As
                      = ( sup_sup_set_nat @ As12 @ As23 ) )
                    & ( ( inf_inf_set_nat @ As12 @ As23 )
                      = bot_bot_set_nat )
                    & ( X @ ( produc7507926704131184380et_nat @ H3 @ As12 ) )
                    & ( Xa @ ( produc7507926704131184380et_nat @ H3 @ As23 ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
thf(fact_1636_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_1637_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_1638_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_1639_list_Oinject,axiom,
    ! [X21: int,X22: list_int,Y21: int,Y22: list_int] :
      ( ( ( cons_int @ X21 @ X22 )
        = ( cons_int @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_1640_list_Oinject,axiom,
    ! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
      ( ( ( cons_nat @ X21 @ X22 )
        = ( cons_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_1641_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_1642_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_1643_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_1644_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_1645_list__ex__simps_I1_J,axiom,
    ! [P2: int > $o,X: int,Xs: list_int] :
      ( ( list_ex_int @ P2 @ ( cons_int @ X @ Xs ) )
      = ( ( P2 @ X )
        | ( list_ex_int @ P2 @ Xs ) ) ) ).

% list_ex_simps(1)
thf(fact_1646_list__ex__simps_I1_J,axiom,
    ! [P2: nat > $o,X: nat,Xs: list_nat] :
      ( ( list_ex_nat @ P2 @ ( cons_nat @ X @ Xs ) )
      = ( ( P2 @ X )
        | ( list_ex_nat @ P2 @ Xs ) ) ) ).

% list_ex_simps(1)
thf(fact_1647_in__measures_I2_J,axiom,
    ! [X: nat,Y: nat,F2: nat > nat,Fs: list_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F2 @ Fs ) ) )
      = ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
        | ( ( ( F2 @ X )
            = ( F2 @ Y ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_1648_in__measures_I2_J,axiom,
    ! [X: int,Y: int,F2: int > nat,Fs: list_int_nat] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F2 @ Fs ) ) )
      = ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
        | ( ( ( F2 @ X )
            = ( F2 @ Y ) )
          & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_1649_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_1650_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_1651_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_1652_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_1653_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_1654_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_1655_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_1656_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_1657_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_pos
thf(fact_1658_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_1659_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_1660_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_1661_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: nat,Ys: list_nat] :
      ( ( zip_int_nat @ ( cons_int @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( cons_P7512249878480867347nt_nat @ ( product_Pair_int_nat @ X @ Y ) @ ( zip_int_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1662_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: int,Ys: list_int] :
      ( ( zip_nat_int @ ( cons_nat @ X @ Xs ) @ ( cons_int @ Y @ Ys ) )
      = ( cons_P2335045147070616083at_int @ ( product_Pair_nat_int @ X @ Y ) @ ( zip_nat_int @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1663_zip__Cons__Cons,axiom,
    ! [X: heap_e7401611519738050253t_unit,Xs: list_h2745205591237434579t_unit,Y: set_nat,Ys: list_set_nat] :
      ( ( zip_he6711929980790699996et_nat @ ( cons_h4161061858534515843t_unit @ X @ Xs ) @ ( cons_set_nat @ Y @ Ys ) )
      = ( cons_P2869003950965171916et_nat @ ( produc7507926704131184380et_nat @ X @ Y ) @ ( zip_he6711929980790699996et_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1664_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1665_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int] :
      ( ( zip_int_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) )
      = ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( zip_int_int @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1666_zip__Cons__Cons,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o,Y: produc3658429121746597890et_nat,Ys: list_P9062070895058802706et_nat] :
      ( ( zip_Pr7134870689397686104et_nat @ ( cons_P6219271836124797827_nat_o @ X @ Xs ) @ ( cons_P2869003950965171916et_nat @ Y @ Ys ) )
      = ( cons_P3419706843779801160et_nat @ ( produc5001842942810119800et_nat @ X @ Y ) @ ( zip_Pr7134870689397686104et_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1667_zip__Cons__Cons,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xs: list_P7985473006766602707_nat_o,Y: produc3925858234332021118et_nat,Ys: list_P2321686559999237006et_nat] :
      ( ( zip_Pr8136144321567152340et_nat @ ( cons_P6219271836124797827_nat_o @ X @ Xs ) @ ( cons_P3419706843779801160et_nat @ Y @ Ys ) )
      = ( cons_P6329468272379687876et_nat @ ( produc2245416461498447860et_nat @ X @ Y ) @ ( zip_Pr8136144321567152340et_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1668_zip__map__fst__snd,axiom,
    ! [Zs: list_P9062070895058802706et_nat] :
      ( ( zip_he6711929980790699996et_nat @ ( map_Pr7042224834074979589t_unit @ produc1824681642469235216et_nat @ Zs ) @ ( map_Pr4871828589360411470et_nat @ produc8586169260539613262et_nat @ Zs ) )
      = Zs ) ).

% zip_map_fst_snd
thf(fact_1669_zip__map__fst__snd,axiom,
    ! [Zs: list_P5707943133018811711nt_int] :
      ( ( zip_int_int @ ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Zs ) @ ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ Zs ) )
      = Zs ) ).

% zip_map_fst_snd
thf(fact_1670_zip__map__fst__snd,axiom,
    ! [Zs: list_P8527749157015355191n_assn] :
      ( ( zip_assn_assn @ ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ Zs ) @ ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn @ Zs ) )
      = Zs ) ).

% zip_map_fst_snd
thf(fact_1671_nths__singleton,axiom,
    ! [A3: set_nat,X: produc6575502325842934193n_assn] :
      ( ( ( member_nat2 @ zero_zero_nat @ A3 )
       => ( ( nths_P5537322223065504949n_assn @ ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) @ A3 )
          = ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) ) )
      & ( ~ ( member_nat2 @ zero_zero_nat @ A3 )
       => ( ( nths_P5537322223065504949n_assn @ ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) @ A3 )
          = nil_Pr5671120429643327159n_assn ) ) ) ).

% nths_singleton
thf(fact_1672_nths__singleton,axiom,
    ! [A3: set_nat,X: int] :
      ( ( ( member_nat2 @ zero_zero_nat @ A3 )
       => ( ( nths_int @ ( cons_int @ X @ nil_int ) @ A3 )
          = ( cons_int @ X @ nil_int ) ) )
      & ( ~ ( member_nat2 @ zero_zero_nat @ A3 )
       => ( ( nths_int @ ( cons_int @ X @ nil_int ) @ A3 )
          = nil_int ) ) ) ).

% nths_singleton
thf(fact_1673_nths__singleton,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( ( member_nat2 @ zero_zero_nat @ A3 )
       => ( ( nths_nat @ ( cons_nat @ X @ nil_nat ) @ A3 )
          = ( cons_nat @ X @ nil_nat ) ) )
      & ( ~ ( member_nat2 @ zero_zero_nat @ A3 )
       => ( ( nths_nat @ ( cons_nat @ X @ nil_nat ) @ A3 )
          = nil_nat ) ) ) ).

% nths_singleton
thf(fact_1674_less__assn__def,axiom,
    ( ord_less_assn
    = ( ^ [A2: assn,B2: assn] :
          ( ( ord_less_eq_assn @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% less_assn_def
thf(fact_1675_not__Cons__self2,axiom,
    ! [X: int,Xs: list_int] :
      ( ( cons_int @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_1676_not__Cons__self2,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( cons_nat @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_1677_transpose_Ocases,axiom,
    ! [X: list_l6351802567095793725n_assn] :
      ( ( X != nil_li5476096274760905021n_assn )
     => ( ! [Xss: list_l6351802567095793725n_assn] :
            ( X
           != ( cons_l2423627976422276333n_assn @ nil_Pr5671120429643327159n_assn @ Xss ) )
       => ~ ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Xss: list_l6351802567095793725n_assn] :
              ( X
             != ( cons_l2423627976422276333n_assn @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_1678_transpose_Ocases,axiom,
    ! [X: list_list_int] :
      ( ( X != nil_list_int )
     => ( ! [Xss: list_list_int] :
            ( X
           != ( cons_list_int @ nil_int @ Xss ) )
       => ~ ! [X3: int,Xs2: list_int,Xss: list_list_int] :
              ( X
             != ( cons_list_int @ ( cons_int @ X3 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_1679_transpose_Ocases,axiom,
    ! [X: list_list_nat] :
      ( ( X != nil_list_nat )
     => ( ! [Xss: list_list_nat] :
            ( X
           != ( cons_list_nat @ nil_nat @ Xss ) )
       => ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
              ( X
             != ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_1680_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_nat,Xy: product_prod_int_nat,Xys: list_P8198026277950538467nt_nat] :
      ( ( ( zip_int_nat @ Xs @ Ys )
        = ( cons_P7512249878480867347nt_nat @ Xy @ Xys ) )
     => ~ ! [X3: int,Xs3: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Xs3 ) )
           => ! [Y3: nat,Ys2: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( product_Pair_int_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_int_nat @ Xs3 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1681_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_int,Xy: product_prod_nat_int,Xys: list_P3521021558325789923at_int] :
      ( ( ( zip_nat_int @ Xs @ Ys )
        = ( cons_P2335045147070616083at_int @ Xy @ Xys ) )
     => ~ ! [X3: nat,Xs3: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Xs3 ) )
           => ! [Y3: int,Ys2: list_int] :
                ( ( Ys
                  = ( cons_int @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( product_Pair_nat_int @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_nat_int @ Xs3 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1682_zip__eq__ConsE,axiom,
    ! [Xs: list_h2745205591237434579t_unit,Ys: list_set_nat,Xy: produc3658429121746597890et_nat,Xys: list_P9062070895058802706et_nat] :
      ( ( ( zip_he6711929980790699996et_nat @ Xs @ Ys )
        = ( cons_P2869003950965171916et_nat @ Xy @ Xys ) )
     => ~ ! [X3: heap_e7401611519738050253t_unit,Xs3: list_h2745205591237434579t_unit] :
            ( ( Xs
              = ( cons_h4161061858534515843t_unit @ X3 @ Xs3 ) )
           => ! [Y3: set_nat,Ys2: list_set_nat] :
                ( ( Ys
                  = ( cons_set_nat @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( produc7507926704131184380et_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_he6711929980790699996et_nat @ Xs3 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1683_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
      ( ( ( zip_nat_nat @ Xs @ Ys )
        = ( cons_P6512896166579812791at_nat @ Xy @ Xys ) )
     => ~ ! [X3: nat,Xs3: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Xs3 ) )
           => ! [Y3: nat,Ys2: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( product_Pair_nat_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_nat_nat @ Xs3 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1684_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_int,Xy: product_prod_int_int,Xys: list_P5707943133018811711nt_int] :
      ( ( ( zip_int_int @ Xs @ Ys )
        = ( cons_P3334398858971670639nt_int @ Xy @ Xys ) )
     => ~ ! [X3: int,Xs3: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Xs3 ) )
           => ! [Y3: int,Ys2: list_int] :
                ( ( Ys
                  = ( cons_int @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( product_Pair_int_int @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_int_int @ Xs3 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1685_zip__eq__ConsE,axiom,
    ! [Xs: list_P7985473006766602707_nat_o,Ys: list_P9062070895058802706et_nat,Xy: produc3925858234332021118et_nat,Xys: list_P2321686559999237006et_nat] :
      ( ( ( zip_Pr7134870689397686104et_nat @ Xs @ Ys )
        = ( cons_P3419706843779801160et_nat @ Xy @ Xys ) )
     => ~ ! [X3: produc3658429121746597890et_nat > $o,Xs3: list_P7985473006766602707_nat_o] :
            ( ( Xs
              = ( cons_P6219271836124797827_nat_o @ X3 @ Xs3 ) )
           => ! [Y3: produc3658429121746597890et_nat,Ys2: list_P9062070895058802706et_nat] :
                ( ( Ys
                  = ( cons_P2869003950965171916et_nat @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( produc5001842942810119800et_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_Pr7134870689397686104et_nat @ Xs3 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1686_zip__eq__ConsE,axiom,
    ! [Xs: list_P7985473006766602707_nat_o,Ys: list_P2321686559999237006et_nat,Xy: produc2732055786443039994et_nat,Xys: list_P362550909693114634et_nat] :
      ( ( ( zip_Pr8136144321567152340et_nat @ Xs @ Ys )
        = ( cons_P6329468272379687876et_nat @ Xy @ Xys ) )
     => ~ ! [X3: produc3658429121746597890et_nat > $o,Xs3: list_P7985473006766602707_nat_o] :
            ( ( Xs
              = ( cons_P6219271836124797827_nat_o @ X3 @ Xs3 ) )
           => ! [Y3: produc3925858234332021118et_nat,Ys2: list_P2321686559999237006et_nat] :
                ( ( Ys
                  = ( cons_P3419706843779801160et_nat @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( produc2245416461498447860et_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_Pr8136144321567152340et_nat @ Xs3 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1687_pair__list__eqI,axiom,
    ! [Xs: list_P9062070895058802706et_nat,Ys: list_P9062070895058802706et_nat] :
      ( ( ( map_Pr7042224834074979589t_unit @ produc1824681642469235216et_nat @ Xs )
        = ( map_Pr7042224834074979589t_unit @ produc1824681642469235216et_nat @ Ys ) )
     => ( ( ( map_Pr4871828589360411470et_nat @ produc8586169260539613262et_nat @ Xs )
          = ( map_Pr4871828589360411470et_nat @ produc8586169260539613262et_nat @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_1688_pair__list__eqI,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Xs )
        = ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Ys ) )
     => ( ( ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ Xs )
          = ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_1689_pair__list__eqI,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ Xs )
        = ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ Ys ) )
     => ( ( ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn @ Xs )
          = ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_1690_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_1691_gr__implies__not__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_1692_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_1693_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_1694_less__numeral__extra_I4_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ one_one_Code_integer ) ).

% less_numeral_extra(4)
thf(fact_1695_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_1696_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_1697_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_1698_neq__NilE,axiom,
    ! [L: list_P8527749157015355191n_assn] :
      ( ( L != nil_Pr5671120429643327159n_assn )
     => ~ ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
            ( L
           != ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) ) ) ).

% neq_NilE
thf(fact_1699_neq__NilE,axiom,
    ! [L: list_int] :
      ( ( L != nil_int )
     => ~ ! [X3: int,Xs2: list_int] :
            ( L
           != ( cons_int @ X3 @ Xs2 ) ) ) ).

% neq_NilE
thf(fact_1700_neq__NilE,axiom,
    ! [L: list_nat] :
      ( ( L != nil_nat )
     => ~ ! [X3: nat,Xs2: list_nat] :
            ( L
           != ( cons_nat @ X3 @ Xs2 ) ) ) ).

% neq_NilE
thf(fact_1701_list__2pre__induct,axiom,
    ! [P2: list_P8527749157015355191n_assn > list_P8527749157015355191n_assn > $o,W1: list_P8527749157015355191n_assn,W2: list_P8527749157015355191n_assn] :
      ( ( P2 @ nil_Pr5671120429643327159n_assn @ nil_Pr5671120429643327159n_assn )
     => ( ! [E: produc6575502325842934193n_assn,W12: list_P8527749157015355191n_assn,W22: list_P8527749157015355191n_assn] :
            ( ( P2 @ W12 @ W22 )
           => ( P2 @ ( cons_P2971678138204555879n_assn @ E @ W12 ) @ W22 ) )
       => ( ! [E: produc6575502325842934193n_assn,W13: list_P8527749157015355191n_assn,W23: list_P8527749157015355191n_assn] :
              ( ( P2 @ W13 @ W23 )
             => ( P2 @ W13 @ ( cons_P2971678138204555879n_assn @ E @ W23 ) ) )
         => ( P2 @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_1702_list__2pre__induct,axiom,
    ! [P2: list_P8527749157015355191n_assn > list_int > $o,W1: list_P8527749157015355191n_assn,W2: list_int] :
      ( ( P2 @ nil_Pr5671120429643327159n_assn @ nil_int )
     => ( ! [E: produc6575502325842934193n_assn,W12: list_P8527749157015355191n_assn,W22: list_int] :
            ( ( P2 @ W12 @ W22 )
           => ( P2 @ ( cons_P2971678138204555879n_assn @ E @ W12 ) @ W22 ) )
       => ( ! [E: int,W13: list_P8527749157015355191n_assn,W23: list_int] :
              ( ( P2 @ W13 @ W23 )
             => ( P2 @ W13 @ ( cons_int @ E @ W23 ) ) )
         => ( P2 @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_1703_list__2pre__induct,axiom,
    ! [P2: list_P8527749157015355191n_assn > list_nat > $o,W1: list_P8527749157015355191n_assn,W2: list_nat] :
      ( ( P2 @ nil_Pr5671120429643327159n_assn @ nil_nat )
     => ( ! [E: produc6575502325842934193n_assn,W12: list_P8527749157015355191n_assn,W22: list_nat] :
            ( ( P2 @ W12 @ W22 )
           => ( P2 @ ( cons_P2971678138204555879n_assn @ E @ W12 ) @ W22 ) )
       => ( ! [E: nat,W13: list_P8527749157015355191n_assn,W23: list_nat] :
              ( ( P2 @ W13 @ W23 )
             => ( P2 @ W13 @ ( cons_nat @ E @ W23 ) ) )
         => ( P2 @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_1704_list__2pre__induct,axiom,
    ! [P2: list_int > list_P8527749157015355191n_assn > $o,W1: list_int,W2: list_P8527749157015355191n_assn] :
      ( ( P2 @ nil_int @ nil_Pr5671120429643327159n_assn )
     => ( ! [E: int,W12: list_int,W22: list_P8527749157015355191n_assn] :
            ( ( P2 @ W12 @ W22 )
           => ( P2 @ ( cons_int @ E @ W12 ) @ W22 ) )
       => ( ! [E: produc6575502325842934193n_assn,W13: list_int,W23: list_P8527749157015355191n_assn] :
              ( ( P2 @ W13 @ W23 )
             => ( P2 @ W13 @ ( cons_P2971678138204555879n_assn @ E @ W23 ) ) )
         => ( P2 @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_1705_list__2pre__induct,axiom,
    ! [P2: list_int > list_int > $o,W1: list_int,W2: list_int] :
      ( ( P2 @ nil_int @ nil_int )
     => ( ! [E: int,W12: list_int,W22: list_int] :
            ( ( P2 @ W12 @ W22 )
           => ( P2 @ ( cons_int @ E @ W12 ) @ W22 ) )
       => ( ! [E: int,W13: list_int,W23: list_int] :
              ( ( P2 @ W13 @ W23 )
             => ( P2 @ W13 @ ( cons_int @ E @ W23 ) ) )
         => ( P2 @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_1706_list__2pre__induct,axiom,
    ! [P2: list_int > list_nat > $o,W1: list_int,W2: list_nat] :
      ( ( P2 @ nil_int @ nil_nat )
     => ( ! [E: int,W12: list_int,W22: list_nat] :
            ( ( P2 @ W12 @ W22 )
           => ( P2 @ ( cons_int @ E @ W12 ) @ W22 ) )
       => ( ! [E: nat,W13: list_int,W23: list_nat] :
              ( ( P2 @ W13 @ W23 )
             => ( P2 @ W13 @ ( cons_nat @ E @ W23 ) ) )
         => ( P2 @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_1707_list__2pre__induct,axiom,
    ! [P2: list_nat > list_P8527749157015355191n_assn > $o,W1: list_nat,W2: list_P8527749157015355191n_assn] :
      ( ( P2 @ nil_nat @ nil_Pr5671120429643327159n_assn )
     => ( ! [E: nat,W12: list_nat,W22: list_P8527749157015355191n_assn] :
            ( ( P2 @ W12 @ W22 )
           => ( P2 @ ( cons_nat @ E @ W12 ) @ W22 ) )
       => ( ! [E: produc6575502325842934193n_assn,W13: list_nat,W23: list_P8527749157015355191n_assn] :
              ( ( P2 @ W13 @ W23 )
             => ( P2 @ W13 @ ( cons_P2971678138204555879n_assn @ E @ W23 ) ) )
         => ( P2 @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_1708_list__2pre__induct,axiom,
    ! [P2: list_nat > list_int > $o,W1: list_nat,W2: list_int] :
      ( ( P2 @ nil_nat @ nil_int )
     => ( ! [E: nat,W12: list_nat,W22: list_int] :
            ( ( P2 @ W12 @ W22 )
           => ( P2 @ ( cons_nat @ E @ W12 ) @ W22 ) )
       => ( ! [E: int,W13: list_nat,W23: list_int] :
              ( ( P2 @ W13 @ W23 )
             => ( P2 @ W13 @ ( cons_int @ E @ W23 ) ) )
         => ( P2 @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_1709_list__2pre__induct,axiom,
    ! [P2: list_nat > list_nat > $o,W1: list_nat,W2: list_nat] :
      ( ( P2 @ nil_nat @ nil_nat )
     => ( ! [E: nat,W12: list_nat,W22: list_nat] :
            ( ( P2 @ W12 @ W22 )
           => ( P2 @ ( cons_nat @ E @ W12 ) @ W22 ) )
       => ( ! [E: nat,W13: list_nat,W23: list_nat] :
              ( ( P2 @ W13 @ W23 )
             => ( P2 @ W13 @ ( cons_nat @ E @ W23 ) ) )
         => ( P2 @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_1710_list__induct__first2,axiom,
    ! [P2: list_P8527749157015355191n_assn > $o,Xs: list_P8527749157015355191n_assn] :
      ( ( P2 @ nil_Pr5671120429643327159n_assn )
     => ( ! [X3: produc6575502325842934193n_assn] : ( P2 @ ( cons_P2971678138204555879n_assn @ X3 @ nil_Pr5671120429643327159n_assn ) )
       => ( ! [X1: produc6575502325842934193n_assn,X23: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( ( P2 @ Xs2 )
             => ( P2 @ ( cons_P2971678138204555879n_assn @ X1 @ ( cons_P2971678138204555879n_assn @ X23 @ Xs2 ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% list_induct_first2
thf(fact_1711_list__induct__first2,axiom,
    ! [P2: list_int > $o,Xs: list_int] :
      ( ( P2 @ nil_int )
     => ( ! [X3: int] : ( P2 @ ( cons_int @ X3 @ nil_int ) )
       => ( ! [X1: int,X23: int,Xs2: list_int] :
              ( ( P2 @ Xs2 )
             => ( P2 @ ( cons_int @ X1 @ ( cons_int @ X23 @ Xs2 ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% list_induct_first2
thf(fact_1712_list__induct__first2,axiom,
    ! [P2: list_nat > $o,Xs: list_nat] :
      ( ( P2 @ nil_nat )
     => ( ! [X3: nat] : ( P2 @ ( cons_nat @ X3 @ nil_nat ) )
       => ( ! [X1: nat,X23: nat,Xs2: list_nat] :
              ( ( P2 @ Xs2 )
             => ( P2 @ ( cons_nat @ X1 @ ( cons_nat @ X23 @ Xs2 ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% list_induct_first2
thf(fact_1713_mergesort__by__rel__merge__induct,axiom,
    ! [P2: list_P8527749157015355191n_assn > list_P8527749157015355191n_assn > $o,R: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ! [Xs2: list_P8527749157015355191n_assn] : ( P2 @ Xs2 @ nil_Pr5671120429643327159n_assn )
     => ( ! [X_1: list_P8527749157015355191n_assn] : ( P2 @ nil_Pr5671120429643327159n_assn @ X_1 )
       => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
              ( ( R @ X3 @ Y3 )
             => ( ( P2 @ Xs2 @ ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) )
               => ( P2 @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) ) ) )
         => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
                ( ~ ( R @ X3 @ Y3 )
               => ( ( P2 @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ Ys3 )
                 => ( P2 @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_1714_mergesort__by__rel__merge__induct,axiom,
    ! [P2: list_int > list_P8527749157015355191n_assn > $o,R: int > produc6575502325842934193n_assn > $o,Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( ! [Xs2: list_int] : ( P2 @ Xs2 @ nil_Pr5671120429643327159n_assn )
     => ( ! [X_1: list_P8527749157015355191n_assn] : ( P2 @ nil_int @ X_1 )
       => ( ! [X3: int,Xs2: list_int,Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
              ( ( R @ X3 @ Y3 )
             => ( ( P2 @ Xs2 @ ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) )
               => ( P2 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) ) ) )
         => ( ! [X3: int,Xs2: list_int,Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
                ( ~ ( R @ X3 @ Y3 )
               => ( ( P2 @ ( cons_int @ X3 @ Xs2 ) @ Ys3 )
                 => ( P2 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_1715_mergesort__by__rel__merge__induct,axiom,
    ! [P2: list_nat > list_P8527749157015355191n_assn > $o,R: nat > produc6575502325842934193n_assn > $o,Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( ! [Xs2: list_nat] : ( P2 @ Xs2 @ nil_Pr5671120429643327159n_assn )
     => ( ! [X_1: list_P8527749157015355191n_assn] : ( P2 @ nil_nat @ X_1 )
       => ( ! [X3: nat,Xs2: list_nat,Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
              ( ( R @ X3 @ Y3 )
             => ( ( P2 @ Xs2 @ ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) )
               => ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) ) ) )
         => ( ! [X3: nat,Xs2: list_nat,Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
                ( ~ ( R @ X3 @ Y3 )
               => ( ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ Ys3 )
                 => ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_1716_mergesort__by__rel__merge__induct,axiom,
    ! [P2: list_P8527749157015355191n_assn > list_int > $o,R: produc6575502325842934193n_assn > int > $o,Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( ! [Xs2: list_P8527749157015355191n_assn] : ( P2 @ Xs2 @ nil_int )
     => ( ! [X_1: list_int] : ( P2 @ nil_Pr5671120429643327159n_assn @ X_1 )
       => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y3: int,Ys3: list_int] :
              ( ( R @ X3 @ Y3 )
             => ( ( P2 @ Xs2 @ ( cons_int @ Y3 @ Ys3 ) )
               => ( P2 @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) ) ) )
         => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y3: int,Ys3: list_int] :
                ( ~ ( R @ X3 @ Y3 )
               => ( ( P2 @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ Ys3 )
                 => ( P2 @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_1717_mergesort__by__rel__merge__induct,axiom,
    ! [P2: list_int > list_int > $o,R: int > int > $o,Xs: list_int,Ys: list_int] :
      ( ! [Xs2: list_int] : ( P2 @ Xs2 @ nil_int )
     => ( ! [X_1: list_int] : ( P2 @ nil_int @ X_1 )
       => ( ! [X3: int,Xs2: list_int,Y3: int,Ys3: list_int] :
              ( ( R @ X3 @ Y3 )
             => ( ( P2 @ Xs2 @ ( cons_int @ Y3 @ Ys3 ) )
               => ( P2 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) ) ) )
         => ( ! [X3: int,Xs2: list_int,Y3: int,Ys3: list_int] :
                ( ~ ( R @ X3 @ Y3 )
               => ( ( P2 @ ( cons_int @ X3 @ Xs2 ) @ Ys3 )
                 => ( P2 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_1718_mergesort__by__rel__merge__induct,axiom,
    ! [P2: list_nat > list_int > $o,R: nat > int > $o,Xs: list_nat,Ys: list_int] :
      ( ! [Xs2: list_nat] : ( P2 @ Xs2 @ nil_int )
     => ( ! [X_1: list_int] : ( P2 @ nil_nat @ X_1 )
       => ( ! [X3: nat,Xs2: list_nat,Y3: int,Ys3: list_int] :
              ( ( R @ X3 @ Y3 )
             => ( ( P2 @ Xs2 @ ( cons_int @ Y3 @ Ys3 ) )
               => ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) ) ) )
         => ( ! [X3: nat,Xs2: list_nat,Y3: int,Ys3: list_int] :
                ( ~ ( R @ X3 @ Y3 )
               => ( ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ Ys3 )
                 => ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_1719_mergesort__by__rel__merge__induct,axiom,
    ! [P2: list_P8527749157015355191n_assn > list_nat > $o,R: produc6575502325842934193n_assn > nat > $o,Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( ! [Xs2: list_P8527749157015355191n_assn] : ( P2 @ Xs2 @ nil_nat )
     => ( ! [X_1: list_nat] : ( P2 @ nil_Pr5671120429643327159n_assn @ X_1 )
       => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y3: nat,Ys3: list_nat] :
              ( ( R @ X3 @ Y3 )
             => ( ( P2 @ Xs2 @ ( cons_nat @ Y3 @ Ys3 ) )
               => ( P2 @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) ) ) )
         => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y3: nat,Ys3: list_nat] :
                ( ~ ( R @ X3 @ Y3 )
               => ( ( P2 @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ Ys3 )
                 => ( P2 @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_1720_mergesort__by__rel__merge__induct,axiom,
    ! [P2: list_int > list_nat > $o,R: int > nat > $o,Xs: list_int,Ys: list_nat] :
      ( ! [Xs2: list_int] : ( P2 @ Xs2 @ nil_nat )
     => ( ! [X_1: list_nat] : ( P2 @ nil_int @ X_1 )
       => ( ! [X3: int,Xs2: list_int,Y3: nat,Ys3: list_nat] :
              ( ( R @ X3 @ Y3 )
             => ( ( P2 @ Xs2 @ ( cons_nat @ Y3 @ Ys3 ) )
               => ( P2 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) ) ) )
         => ( ! [X3: int,Xs2: list_int,Y3: nat,Ys3: list_nat] :
                ( ~ ( R @ X3 @ Y3 )
               => ( ( P2 @ ( cons_int @ X3 @ Xs2 ) @ Ys3 )
                 => ( P2 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_1721_mergesort__by__rel__merge__induct,axiom,
    ! [P2: list_nat > list_nat > $o,R: nat > nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ! [Xs2: list_nat] : ( P2 @ Xs2 @ nil_nat )
     => ( ! [X_1: list_nat] : ( P2 @ nil_nat @ X_1 )
       => ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys3: list_nat] :
              ( ( R @ X3 @ Y3 )
             => ( ( P2 @ Xs2 @ ( cons_nat @ Y3 @ Ys3 ) )
               => ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) ) ) )
         => ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys3: list_nat] :
                ( ~ ( R @ X3 @ Y3 )
               => ( ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ Ys3 )
                 => ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_1722_list_Odistinct_I1_J,axiom,
    ! [X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn] :
      ( nil_Pr5671120429643327159n_assn
     != ( cons_P2971678138204555879n_assn @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_1723_list_Odistinct_I1_J,axiom,
    ! [X21: int,X22: list_int] :
      ( nil_int
     != ( cons_int @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_1724_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( nil_nat
     != ( cons_nat @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_1725_list_OdiscI,axiom,
    ! [List: list_P8527749157015355191n_assn,X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn] :
      ( ( List
        = ( cons_P2971678138204555879n_assn @ X21 @ X22 ) )
     => ( List != nil_Pr5671120429643327159n_assn ) ) ).

% list.discI
thf(fact_1726_list_OdiscI,axiom,
    ! [List: list_int,X21: int,X22: list_int] :
      ( ( List
        = ( cons_int @ X21 @ X22 ) )
     => ( List != nil_int ) ) ).

% list.discI
thf(fact_1727_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X22: list_nat] :
      ( ( List
        = ( cons_nat @ X21 @ X22 ) )
     => ( List != nil_nat ) ) ).

% list.discI
thf(fact_1728_list_Oexhaust,axiom,
    ! [Y: list_P8527749157015355191n_assn] :
      ( ( Y != nil_Pr5671120429643327159n_assn )
     => ~ ! [X212: produc6575502325842934193n_assn,X222: list_P8527749157015355191n_assn] :
            ( Y
           != ( cons_P2971678138204555879n_assn @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_1729_list_Oexhaust,axiom,
    ! [Y: list_int] :
      ( ( Y != nil_int )
     => ~ ! [X212: int,X222: list_int] :
            ( Y
           != ( cons_int @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_1730_list_Oexhaust,axiom,
    ! [Y: list_nat] :
      ( ( Y != nil_nat )
     => ~ ! [X212: nat,X222: list_nat] :
            ( Y
           != ( cons_nat @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_1731_min__list_Ocases,axiom,
    ! [X: list_int] :
      ( ! [X3: int,Xs2: list_int] :
          ( X
         != ( cons_int @ X3 @ Xs2 ) )
     => ( X = nil_int ) ) ).

% min_list.cases
thf(fact_1732_min__list_Ocases,axiom,
    ! [X: list_nat] :
      ( ! [X3: nat,Xs2: list_nat] :
          ( X
         != ( cons_nat @ X3 @ Xs2 ) )
     => ( X = nil_nat ) ) ).

% min_list.cases
thf(fact_1733_remdups__adj_Ocases,axiom,
    ! [X: list_P8527749157015355191n_assn] :
      ( ( X != nil_Pr5671120429643327159n_assn )
     => ( ! [X3: produc6575502325842934193n_assn] :
            ( X
           != ( cons_P2971678138204555879n_assn @ X3 @ nil_Pr5671120429643327159n_assn ) )
       => ~ ! [X3: produc6575502325842934193n_assn,Y3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( X
             != ( cons_P2971678138204555879n_assn @ X3 @ ( cons_P2971678138204555879n_assn @ Y3 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_1734_remdups__adj_Ocases,axiom,
    ! [X: list_int] :
      ( ( X != nil_int )
     => ( ! [X3: int] :
            ( X
           != ( cons_int @ X3 @ nil_int ) )
       => ~ ! [X3: int,Y3: int,Xs2: list_int] :
              ( X
             != ( cons_int @ X3 @ ( cons_int @ Y3 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_1735_remdups__adj_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ( ! [X3: nat] :
            ( X
           != ( cons_nat @ X3 @ nil_nat ) )
       => ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
              ( X
             != ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_1736_neq__Nil__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
      = ( ? [Y2: produc6575502325842934193n_assn,Ys4: list_P8527749157015355191n_assn] :
            ( Xs
            = ( cons_P2971678138204555879n_assn @ Y2 @ Ys4 ) ) ) ) ).

% neq_Nil_conv
thf(fact_1737_neq__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( Xs != nil_int )
      = ( ? [Y2: int,Ys4: list_int] :
            ( Xs
            = ( cons_int @ Y2 @ Ys4 ) ) ) ) ).

% neq_Nil_conv
thf(fact_1738_neq__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
      = ( ? [Y2: nat,Ys4: list_nat] :
            ( Xs
            = ( cons_nat @ Y2 @ Ys4 ) ) ) ) ).

% neq_Nil_conv
thf(fact_1739_list__induct2_H,axiom,
    ! [P2: list_P8527749157015355191n_assn > list_P8527749157015355191n_assn > $o,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( P2 @ nil_Pr5671120429643327159n_assn @ nil_Pr5671120429643327159n_assn )
     => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : ( P2 @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ nil_Pr5671120429643327159n_assn )
       => ( ! [Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] : ( P2 @ nil_Pr5671120429643327159n_assn @ ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) )
         => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( P2 @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1740_list__induct2_H,axiom,
    ! [P2: list_P8527749157015355191n_assn > list_int > $o,Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( ( P2 @ nil_Pr5671120429643327159n_assn @ nil_int )
     => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : ( P2 @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ nil_int )
       => ( ! [Y3: int,Ys3: list_int] : ( P2 @ nil_Pr5671120429643327159n_assn @ ( cons_int @ Y3 @ Ys3 ) )
         => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y3: int,Ys3: list_int] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( P2 @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1741_list__induct2_H,axiom,
    ! [P2: list_P8527749157015355191n_assn > list_nat > $o,Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( ( P2 @ nil_Pr5671120429643327159n_assn @ nil_nat )
     => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : ( P2 @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ nil_nat )
       => ( ! [Y3: nat,Ys3: list_nat] : ( P2 @ nil_Pr5671120429643327159n_assn @ ( cons_nat @ Y3 @ Ys3 ) )
         => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y3: nat,Ys3: list_nat] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( P2 @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1742_list__induct2_H,axiom,
    ! [P2: list_int > list_P8527749157015355191n_assn > $o,Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( ( P2 @ nil_int @ nil_Pr5671120429643327159n_assn )
     => ( ! [X3: int,Xs2: list_int] : ( P2 @ ( cons_int @ X3 @ Xs2 ) @ nil_Pr5671120429643327159n_assn )
       => ( ! [Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] : ( P2 @ nil_int @ ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) )
         => ( ! [X3: int,Xs2: list_int,Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( P2 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1743_list__induct2_H,axiom,
    ! [P2: list_int > list_int > $o,Xs: list_int,Ys: list_int] :
      ( ( P2 @ nil_int @ nil_int )
     => ( ! [X3: int,Xs2: list_int] : ( P2 @ ( cons_int @ X3 @ Xs2 ) @ nil_int )
       => ( ! [Y3: int,Ys3: list_int] : ( P2 @ nil_int @ ( cons_int @ Y3 @ Ys3 ) )
         => ( ! [X3: int,Xs2: list_int,Y3: int,Ys3: list_int] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( P2 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1744_list__induct2_H,axiom,
    ! [P2: list_int > list_nat > $o,Xs: list_int,Ys: list_nat] :
      ( ( P2 @ nil_int @ nil_nat )
     => ( ! [X3: int,Xs2: list_int] : ( P2 @ ( cons_int @ X3 @ Xs2 ) @ nil_nat )
       => ( ! [Y3: nat,Ys3: list_nat] : ( P2 @ nil_int @ ( cons_nat @ Y3 @ Ys3 ) )
         => ( ! [X3: int,Xs2: list_int,Y3: nat,Ys3: list_nat] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( P2 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1745_list__induct2_H,axiom,
    ! [P2: list_nat > list_P8527749157015355191n_assn > $o,Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( ( P2 @ nil_nat @ nil_Pr5671120429643327159n_assn )
     => ( ! [X3: nat,Xs2: list_nat] : ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ nil_Pr5671120429643327159n_assn )
       => ( ! [Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] : ( P2 @ nil_nat @ ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) )
         => ( ! [X3: nat,Xs2: list_nat,Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1746_list__induct2_H,axiom,
    ! [P2: list_nat > list_int > $o,Xs: list_nat,Ys: list_int] :
      ( ( P2 @ nil_nat @ nil_int )
     => ( ! [X3: nat,Xs2: list_nat] : ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ nil_int )
       => ( ! [Y3: int,Ys3: list_int] : ( P2 @ nil_nat @ ( cons_int @ Y3 @ Ys3 ) )
         => ( ! [X3: nat,Xs2: list_nat,Y3: int,Ys3: list_int] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1747_list__induct2_H,axiom,
    ! [P2: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ( P2 @ nil_nat @ nil_nat )
     => ( ! [X3: nat,Xs2: list_nat] : ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ nil_nat )
       => ( ! [Y3: nat,Ys3: list_nat] : ( P2 @ nil_nat @ ( cons_nat @ Y3 @ Ys3 ) )
         => ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys3: list_nat] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1748_list__nonempty__induct,axiom,
    ! [Xs: list_P8527749157015355191n_assn,P2: list_P8527749157015355191n_assn > $o] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ( ! [X3: produc6575502325842934193n_assn] : ( P2 @ ( cons_P2971678138204555879n_assn @ X3 @ nil_Pr5671120429643327159n_assn ) )
       => ( ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( ( Xs2 != nil_Pr5671120429643327159n_assn )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_1749_list__nonempty__induct,axiom,
    ! [Xs: list_int,P2: list_int > $o] :
      ( ( Xs != nil_int )
     => ( ! [X3: int] : ( P2 @ ( cons_int @ X3 @ nil_int ) )
       => ( ! [X3: int,Xs2: list_int] :
              ( ( Xs2 != nil_int )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( cons_int @ X3 @ Xs2 ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_1750_list__nonempty__induct,axiom,
    ! [Xs: list_nat,P2: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X3: nat] : ( P2 @ ( cons_nat @ X3 @ nil_nat ) )
       => ( ! [X3: nat,Xs2: list_nat] :
              ( ( Xs2 != nil_nat )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( cons_nat @ X3 @ Xs2 ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_1751_merge_Ocases,axiom,
    ! [X: produc1186641810826059865st_int] :
      ( ! [L2: list_int] :
          ( X
         != ( produc364263696895485585st_int @ nil_int @ L2 ) )
     => ( ! [V: int,Va: list_int] :
            ( X
           != ( produc364263696895485585st_int @ ( cons_int @ V @ Va ) @ nil_int ) )
       => ~ ! [X1: int,L1: list_int,X23: int,L2: list_int] :
              ( X
             != ( produc364263696895485585st_int @ ( cons_int @ X1 @ L1 ) @ ( cons_int @ X23 @ L2 ) ) ) ) ) ).

% merge.cases
thf(fact_1752_merge_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [L2: list_nat] :
          ( X
         != ( produc2694037385005941721st_nat @ nil_nat @ L2 ) )
     => ( ! [V: nat,Va: list_nat] :
            ( X
           != ( produc2694037385005941721st_nat @ ( cons_nat @ V @ Va ) @ nil_nat ) )
       => ~ ! [X1: nat,L1: list_nat,X23: nat,L2: list_nat] :
              ( X
             != ( produc2694037385005941721st_nat @ ( cons_nat @ X1 @ L1 ) @ ( cons_nat @ X23 @ L2 ) ) ) ) ) ).

% merge.cases
thf(fact_1753_shuffles_Ocases,axiom,
    ! [X: produc337509097454979815n_assn] :
      ( ! [Ys3: list_P8527749157015355191n_assn] :
          ( X
         != ( produc3014731801419794263n_assn @ nil_Pr5671120429643327159n_assn @ Ys3 ) )
     => ( ! [Xs2: list_P8527749157015355191n_assn] :
            ( X
           != ( produc3014731801419794263n_assn @ Xs2 @ nil_Pr5671120429643327159n_assn ) )
       => ~ ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
              ( X
             != ( produc3014731801419794263n_assn @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) ) ) ) ) ).

% shuffles.cases
thf(fact_1754_shuffles_Ocases,axiom,
    ! [X: produc1186641810826059865st_int] :
      ( ! [Ys3: list_int] :
          ( X
         != ( produc364263696895485585st_int @ nil_int @ Ys3 ) )
     => ( ! [Xs2: list_int] :
            ( X
           != ( produc364263696895485585st_int @ Xs2 @ nil_int ) )
       => ~ ! [X3: int,Xs2: list_int,Y3: int,Ys3: list_int] :
              ( X
             != ( produc364263696895485585st_int @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) ) ) ) ) ).

% shuffles.cases
thf(fact_1755_shuffles_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [Ys3: list_nat] :
          ( X
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys3 ) )
     => ( ! [Xs2: list_nat] :
            ( X
           != ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) )
       => ~ ! [X3: nat,Xs2: list_nat,Y3: nat,Ys3: list_nat] :
              ( X
             != ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) ) ) ) ) ).

% shuffles.cases
thf(fact_1756_successively_Ocases,axiom,
    ! [X: produc4530688714081574576n_assn] :
      ( ! [P4: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o] :
          ( X
         != ( produc6115016301897879072n_assn @ P4 @ nil_Pr5671120429643327159n_assn ) )
     => ( ! [P4: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,X3: produc6575502325842934193n_assn] :
            ( X
           != ( produc6115016301897879072n_assn @ P4 @ ( cons_P2971678138204555879n_assn @ X3 @ nil_Pr5671120429643327159n_assn ) ) )
       => ~ ! [P4: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,X3: produc6575502325842934193n_assn,Y3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( X
             != ( produc6115016301897879072n_assn @ P4 @ ( cons_P2971678138204555879n_assn @ X3 @ ( cons_P2971678138204555879n_assn @ Y3 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_1757_successively_Ocases,axiom,
    ! [X: produc5834231552977413017st_int] :
      ( ! [P4: int > int > $o] :
          ( X
         != ( produc8618682346314911123st_int @ P4 @ nil_int ) )
     => ( ! [P4: int > int > $o,X3: int] :
            ( X
           != ( produc8618682346314911123st_int @ P4 @ ( cons_int @ X3 @ nil_int ) ) )
       => ~ ! [P4: int > int > $o,X3: int,Y3: int,Xs2: list_int] :
              ( X
             != ( produc8618682346314911123st_int @ P4 @ ( cons_int @ X3 @ ( cons_int @ Y3 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_1758_successively_Ocases,axiom,
    ! [X: produc254973753779126261st_nat] :
      ( ! [P4: nat > nat > $o] :
          ( X
         != ( produc4727192421694094319st_nat @ P4 @ nil_nat ) )
     => ( ! [P4: nat > nat > $o,X3: nat] :
            ( X
           != ( produc4727192421694094319st_nat @ P4 @ ( cons_nat @ X3 @ nil_nat ) ) )
       => ~ ! [P4: nat > nat > $o,X3: nat,Y3: nat,Xs2: list_nat] :
              ( X
             != ( produc4727192421694094319st_nat @ P4 @ ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_1759_sorted__wrt_Ocases,axiom,
    ! [X: produc4530688714081574576n_assn] :
      ( ! [P4: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o] :
          ( X
         != ( produc6115016301897879072n_assn @ P4 @ nil_Pr5671120429643327159n_assn ) )
     => ~ ! [P4: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,X3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
            ( X
           != ( produc6115016301897879072n_assn @ P4 @ ( cons_P2971678138204555879n_assn @ X3 @ Ys3 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_1760_sorted__wrt_Ocases,axiom,
    ! [X: produc5834231552977413017st_int] :
      ( ! [P4: int > int > $o] :
          ( X
         != ( produc8618682346314911123st_int @ P4 @ nil_int ) )
     => ~ ! [P4: int > int > $o,X3: int,Ys3: list_int] :
            ( X
           != ( produc8618682346314911123st_int @ P4 @ ( cons_int @ X3 @ Ys3 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_1761_sorted__wrt_Ocases,axiom,
    ! [X: produc254973753779126261st_nat] :
      ( ! [P4: nat > nat > $o] :
          ( X
         != ( produc4727192421694094319st_nat @ P4 @ nil_nat ) )
     => ~ ! [P4: nat > nat > $o,X3: nat,Ys3: list_nat] :
            ( X
           != ( produc4727192421694094319st_nat @ P4 @ ( cons_nat @ X3 @ Ys3 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_1762_mergesort__by__rel__split_Ocases,axiom,
    ! [X: produc858023921370090807n_assn] :
      ( ! [Xs1: list_P8527749157015355191n_assn,Xs22: list_P8527749157015355191n_assn] :
          ( X
         != ( produc7497315745449715623n_assn @ ( produc3014731801419794263n_assn @ Xs1 @ Xs22 ) @ nil_Pr5671120429643327159n_assn ) )
     => ( ! [Xs1: list_P8527749157015355191n_assn,Xs22: list_P8527749157015355191n_assn,X3: produc6575502325842934193n_assn] :
            ( X
           != ( produc7497315745449715623n_assn @ ( produc3014731801419794263n_assn @ Xs1 @ Xs22 ) @ ( cons_P2971678138204555879n_assn @ X3 @ nil_Pr5671120429643327159n_assn ) ) )
       => ~ ! [Xs1: list_P8527749157015355191n_assn,Xs22: list_P8527749157015355191n_assn,X1: produc6575502325842934193n_assn,X23: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( X
             != ( produc7497315745449715623n_assn @ ( produc3014731801419794263n_assn @ Xs1 @ Xs22 ) @ ( cons_P2971678138204555879n_assn @ X1 @ ( cons_P2971678138204555879n_assn @ X23 @ Xs2 ) ) ) ) ) ) ).

% mergesort_by_rel_split.cases
thf(fact_1763_mergesort__by__rel__split_Ocases,axiom,
    ! [X: produc5625914783551145028st_int] :
      ( ! [Xs1: list_int,Xs22: list_int] :
          ( X
         != ( produc6571967182415643454st_int @ ( produc364263696895485585st_int @ Xs1 @ Xs22 ) @ nil_int ) )
     => ( ! [Xs1: list_int,Xs22: list_int,X3: int] :
            ( X
           != ( produc6571967182415643454st_int @ ( produc364263696895485585st_int @ Xs1 @ Xs22 ) @ ( cons_int @ X3 @ nil_int ) ) )
       => ~ ! [Xs1: list_int,Xs22: list_int,X1: int,X23: int,Xs2: list_int] :
              ( X
             != ( produc6571967182415643454st_int @ ( produc364263696895485585st_int @ Xs1 @ Xs22 ) @ ( cons_int @ X1 @ ( cons_int @ X23 @ Xs2 ) ) ) ) ) ) ).

% mergesort_by_rel_split.cases
thf(fact_1764_mergesort__by__rel__split_Ocases,axiom,
    ! [X: produc8368264342088597808st_nat] :
      ( ! [Xs1: list_nat,Xs22: list_nat] :
          ( X
         != ( produc1778712578675790378st_nat @ ( produc2694037385005941721st_nat @ Xs1 @ Xs22 ) @ nil_nat ) )
     => ( ! [Xs1: list_nat,Xs22: list_nat,X3: nat] :
            ( X
           != ( produc1778712578675790378st_nat @ ( produc2694037385005941721st_nat @ Xs1 @ Xs22 ) @ ( cons_nat @ X3 @ nil_nat ) ) )
       => ~ ! [Xs1: list_nat,Xs22: list_nat,X1: nat,X23: nat,Xs2: list_nat] :
              ( X
             != ( produc1778712578675790378st_nat @ ( produc2694037385005941721st_nat @ Xs1 @ Xs22 ) @ ( cons_nat @ X1 @ ( cons_nat @ X23 @ Xs2 ) ) ) ) ) ) ).

% mergesort_by_rel_split.cases
thf(fact_1765_mergesort__by__rel__merge_Ocases,axiom,
    ! [X: produc6691681559177096800n_assn] :
      ( ! [R3: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
          ( X
         != ( produc218933061472242640n_assn @ R3 @ ( produc3014731801419794263n_assn @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) ) ) )
     => ( ! [R3: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,Xs2: list_P8527749157015355191n_assn] :
            ( X
           != ( produc218933061472242640n_assn @ R3 @ ( produc3014731801419794263n_assn @ Xs2 @ nil_Pr5671120429643327159n_assn ) ) )
       => ~ ! [R3: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,V: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] :
              ( X
             != ( produc218933061472242640n_assn @ R3 @ ( produc3014731801419794263n_assn @ nil_Pr5671120429643327159n_assn @ ( cons_P2971678138204555879n_assn @ V @ Va ) ) ) ) ) ) ).

% mergesort_by_rel_merge.cases
thf(fact_1766_mergesort__by__rel__merge_Ocases,axiom,
    ! [X: produc6134427711877206866st_int] :
      ( ! [R3: int > int > $o,X3: int,Xs2: list_int,Y3: int,Ys3: list_int] :
          ( X
         != ( produc6519019827157805250st_int @ R3 @ ( produc364263696895485585st_int @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) ) ) )
     => ( ! [R3: int > int > $o,Xs2: list_int] :
            ( X
           != ( produc6519019827157805250st_int @ R3 @ ( produc364263696895485585st_int @ Xs2 @ nil_int ) ) )
       => ~ ! [R3: int > int > $o,V: int,Va: list_int] :
              ( X
             != ( produc6519019827157805250st_int @ R3 @ ( produc364263696895485585st_int @ nil_int @ ( cons_int @ V @ Va ) ) ) ) ) ) ).

% mergesort_by_rel_merge.cases
thf(fact_1767_mergesort__by__rel__merge_Ocases,axiom,
    ! [X: produc4787317212837456354st_nat] :
      ( ! [R3: nat > nat > $o,X3: nat,Xs2: list_nat,Y3: nat,Ys3: list_nat] :
          ( X
         != ( produc3127733452865184594st_nat @ R3 @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) ) ) )
     => ( ! [R3: nat > nat > $o,Xs2: list_nat] :
            ( X
           != ( produc3127733452865184594st_nat @ R3 @ ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) ) )
       => ~ ! [R3: nat > nat > $o,V: nat,Va: list_nat] :
              ( X
             != ( produc3127733452865184594st_nat @ R3 @ ( produc2694037385005941721st_nat @ nil_nat @ ( cons_nat @ V @ Va ) ) ) ) ) ) ).

% mergesort_by_rel_merge.cases
thf(fact_1768_quicksort__by__rel_Ocases,axiom,
    ! [X: produc6691681559177096800n_assn] :
      ( ! [R3: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,Sl: list_P8527749157015355191n_assn] :
          ( X
         != ( produc218933061472242640n_assn @ R3 @ ( produc3014731801419794263n_assn @ Sl @ nil_Pr5671120429643327159n_assn ) ) )
     => ~ ! [R3: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,Sl: list_P8527749157015355191n_assn,X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
            ( X
           != ( produc218933061472242640n_assn @ R3 @ ( produc3014731801419794263n_assn @ Sl @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) ) ) ) ) ).

% quicksort_by_rel.cases
thf(fact_1769_quicksort__by__rel_Ocases,axiom,
    ! [X: produc6134427711877206866st_int] :
      ( ! [R3: int > int > $o,Sl: list_int] :
          ( X
         != ( produc6519019827157805250st_int @ R3 @ ( produc364263696895485585st_int @ Sl @ nil_int ) ) )
     => ~ ! [R3: int > int > $o,Sl: list_int,X3: int,Xs2: list_int] :
            ( X
           != ( produc6519019827157805250st_int @ R3 @ ( produc364263696895485585st_int @ Sl @ ( cons_int @ X3 @ Xs2 ) ) ) ) ) ).

% quicksort_by_rel.cases
thf(fact_1770_quicksort__by__rel_Ocases,axiom,
    ! [X: produc4787317212837456354st_nat] :
      ( ! [R3: nat > nat > $o,Sl: list_nat] :
          ( X
         != ( produc3127733452865184594st_nat @ R3 @ ( produc2694037385005941721st_nat @ Sl @ nil_nat ) ) )
     => ~ ! [R3: nat > nat > $o,Sl: list_nat,X3: nat,Xs2: list_nat] :
            ( X
           != ( produc3127733452865184594st_nat @ R3 @ ( produc2694037385005941721st_nat @ Sl @ ( cons_nat @ X3 @ Xs2 ) ) ) ) ) ).

% quicksort_by_rel.cases
thf(fact_1771_partition__rev_Ocases,axiom,
    ! [X: produc3251403909973641504n_assn] :
      ( ! [P4: produc6575502325842934193n_assn > $o,Yes: list_P8527749157015355191n_assn,No: list_P8527749157015355191n_assn] :
          ( X
         != ( produc7995034776595945426n_assn @ P4 @ ( produc7497315745449715623n_assn @ ( produc3014731801419794263n_assn @ Yes @ No ) @ nil_Pr5671120429643327159n_assn ) ) )
     => ~ ! [P4: produc6575502325842934193n_assn > $o,Yes: list_P8527749157015355191n_assn,No: list_P8527749157015355191n_assn,X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
            ( X
           != ( produc7995034776595945426n_assn @ P4 @ ( produc7497315745449715623n_assn @ ( produc3014731801419794263n_assn @ Yes @ No ) @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) ) ) ) ) ).

% partition_rev.cases
thf(fact_1772_partition__rev_Ocases,axiom,
    ! [X: produc141556263843406102st_int] :
      ( ! [P4: int > $o,Yes: list_int,No: list_int] :
          ( X
         != ( produc3116401902672145424st_int @ P4 @ ( produc6571967182415643454st_int @ ( produc364263696895485585st_int @ Yes @ No ) @ nil_int ) ) )
     => ~ ! [P4: int > $o,Yes: list_int,No: list_int,X3: int,Xs2: list_int] :
            ( X
           != ( produc3116401902672145424st_int @ P4 @ ( produc6571967182415643454st_int @ ( produc364263696895485585st_int @ Yes @ No ) @ ( cons_int @ X3 @ Xs2 ) ) ) ) ) ).

% partition_rev.cases
thf(fact_1773_partition__rev_Ocases,axiom,
    ! [X: produc6155704871766690142st_nat] :
      ( ! [P4: nat > $o,Yes: list_nat,No: list_nat] :
          ( X
         != ( produc2404789219858123352st_nat @ P4 @ ( produc1778712578675790378st_nat @ ( produc2694037385005941721st_nat @ Yes @ No ) @ nil_nat ) ) )
     => ~ ! [P4: nat > $o,Yes: list_nat,No: list_nat,X3: nat,Xs2: list_nat] :
            ( X
           != ( produc2404789219858123352st_nat @ P4 @ ( produc1778712578675790378st_nat @ ( produc2694037385005941721st_nat @ Yes @ No ) @ ( cons_nat @ X3 @ Xs2 ) ) ) ) ) ).

% partition_rev.cases
thf(fact_1774_list__all__zip_Ocases,axiom,
    ! [X: produc6691681559177096800n_assn] :
      ( ! [P4: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o] :
          ( X
         != ( produc218933061472242640n_assn @ P4 @ ( produc3014731801419794263n_assn @ nil_Pr5671120429643327159n_assn @ nil_Pr5671120429643327159n_assn ) ) )
     => ( ! [P4: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,A6: produc6575502325842934193n_assn,As: list_P8527749157015355191n_assn,B5: produc6575502325842934193n_assn,Bs2: list_P8527749157015355191n_assn] :
            ( X
           != ( produc218933061472242640n_assn @ P4 @ ( produc3014731801419794263n_assn @ ( cons_P2971678138204555879n_assn @ A6 @ As ) @ ( cons_P2971678138204555879n_assn @ B5 @ Bs2 ) ) ) )
       => ( ! [P4: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,V: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] :
              ( X
             != ( produc218933061472242640n_assn @ P4 @ ( produc3014731801419794263n_assn @ ( cons_P2971678138204555879n_assn @ V @ Va ) @ nil_Pr5671120429643327159n_assn ) ) )
         => ~ ! [P4: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,V: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] :
                ( X
               != ( produc218933061472242640n_assn @ P4 @ ( produc3014731801419794263n_assn @ nil_Pr5671120429643327159n_assn @ ( cons_P2971678138204555879n_assn @ V @ Va ) ) ) ) ) ) ) ).

% list_all_zip.cases
thf(fact_1775_list__all__zip_Ocases,axiom,
    ! [X: produc4920635200459206554st_int] :
      ( ! [P4: produc6575502325842934193n_assn > int > $o] :
          ( X
         != ( produc7143791561355829714st_int @ P4 @ ( produc1996262876970182428st_int @ nil_Pr5671120429643327159n_assn @ nil_int ) ) )
     => ( ! [P4: produc6575502325842934193n_assn > int > $o,A6: produc6575502325842934193n_assn,As: list_P8527749157015355191n_assn,B5: int,Bs2: list_int] :
            ( X
           != ( produc7143791561355829714st_int @ P4 @ ( produc1996262876970182428st_int @ ( cons_P2971678138204555879n_assn @ A6 @ As ) @ ( cons_int @ B5 @ Bs2 ) ) ) )
       => ( ! [P4: produc6575502325842934193n_assn > int > $o,V: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] :
              ( X
             != ( produc7143791561355829714st_int @ P4 @ ( produc1996262876970182428st_int @ ( cons_P2971678138204555879n_assn @ V @ Va ) @ nil_int ) ) )
         => ~ ! [P4: produc6575502325842934193n_assn > int > $o,V: int,Va: list_int] :
                ( X
               != ( produc7143791561355829714st_int @ P4 @ ( produc1996262876970182428st_int @ nil_Pr5671120429643327159n_assn @ ( cons_int @ V @ Va ) ) ) ) ) ) ) ).

% list_all_zip.cases
thf(fact_1776_list__all__zip_Ocases,axiom,
    ! [X: produc6280221377895650458st_nat] :
      ( ! [P4: produc6575502325842934193n_assn > nat > $o] :
          ( X
         != ( produc6179912464936081618st_nat @ P4 @ ( produc6174113896479379136st_nat @ nil_Pr5671120429643327159n_assn @ nil_nat ) ) )
     => ( ! [P4: produc6575502325842934193n_assn > nat > $o,A6: produc6575502325842934193n_assn,As: list_P8527749157015355191n_assn,B5: nat,Bs2: list_nat] :
            ( X
           != ( produc6179912464936081618st_nat @ P4 @ ( produc6174113896479379136st_nat @ ( cons_P2971678138204555879n_assn @ A6 @ As ) @ ( cons_nat @ B5 @ Bs2 ) ) ) )
       => ( ! [P4: produc6575502325842934193n_assn > nat > $o,V: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] :
              ( X
             != ( produc6179912464936081618st_nat @ P4 @ ( produc6174113896479379136st_nat @ ( cons_P2971678138204555879n_assn @ V @ Va ) @ nil_nat ) ) )
         => ~ ! [P4: produc6575502325842934193n_assn > nat > $o,V: nat,Va: list_nat] :
                ( X
               != ( produc6179912464936081618st_nat @ P4 @ ( produc6174113896479379136st_nat @ nil_Pr5671120429643327159n_assn @ ( cons_nat @ V @ Va ) ) ) ) ) ) ) ).

% list_all_zip.cases
thf(fact_1777_list__all__zip_Ocases,axiom,
    ! [X: produc4944321512348556200n_assn] :
      ( ! [P4: int > produc6575502325842934193n_assn > $o] :
          ( X
         != ( produc3092413063583913184n_assn @ P4 @ ( produc4510294837168919970n_assn @ nil_int @ nil_Pr5671120429643327159n_assn ) ) )
     => ( ! [P4: int > produc6575502325842934193n_assn > $o,A6: int,As: list_int,B5: produc6575502325842934193n_assn,Bs2: list_P8527749157015355191n_assn] :
            ( X
           != ( produc3092413063583913184n_assn @ P4 @ ( produc4510294837168919970n_assn @ ( cons_int @ A6 @ As ) @ ( cons_P2971678138204555879n_assn @ B5 @ Bs2 ) ) ) )
       => ( ! [P4: int > produc6575502325842934193n_assn > $o,V: int,Va: list_int] :
              ( X
             != ( produc3092413063583913184n_assn @ P4 @ ( produc4510294837168919970n_assn @ ( cons_int @ V @ Va ) @ nil_Pr5671120429643327159n_assn ) ) )
         => ~ ! [P4: int > produc6575502325842934193n_assn > $o,V: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] :
                ( X
               != ( produc3092413063583913184n_assn @ P4 @ ( produc4510294837168919970n_assn @ nil_int @ ( cons_P2971678138204555879n_assn @ V @ Va ) ) ) ) ) ) ) ).

% list_all_zip.cases
thf(fact_1778_list__all__zip_Ocases,axiom,
    ! [X: produc6134427711877206866st_int] :
      ( ! [P4: int > int > $o] :
          ( X
         != ( produc6519019827157805250st_int @ P4 @ ( produc364263696895485585st_int @ nil_int @ nil_int ) ) )
     => ( ! [P4: int > int > $o,A6: int,As: list_int,B5: int,Bs2: list_int] :
            ( X
           != ( produc6519019827157805250st_int @ P4 @ ( produc364263696895485585st_int @ ( cons_int @ A6 @ As ) @ ( cons_int @ B5 @ Bs2 ) ) ) )
       => ( ! [P4: int > int > $o,V: int,Va: list_int] :
              ( X
             != ( produc6519019827157805250st_int @ P4 @ ( produc364263696895485585st_int @ ( cons_int @ V @ Va ) @ nil_int ) ) )
         => ~ ! [P4: int > int > $o,V: int,Va: list_int] :
                ( X
               != ( produc6519019827157805250st_int @ P4 @ ( produc364263696895485585st_int @ nil_int @ ( cons_int @ V @ Va ) ) ) ) ) ) ) ).

% list_all_zip.cases
thf(fact_1779_list__all__zip_Ocases,axiom,
    ! [X: produc5074578776457504922st_nat] :
      ( ! [P4: int > nat > $o] :
          ( X
         != ( produc3135705617881911306st_nat @ P4 @ ( produc4542114716404682293st_nat @ nil_int @ nil_nat ) ) )
     => ( ! [P4: int > nat > $o,A6: int,As: list_int,B5: nat,Bs2: list_nat] :
            ( X
           != ( produc3135705617881911306st_nat @ P4 @ ( produc4542114716404682293st_nat @ ( cons_int @ A6 @ As ) @ ( cons_nat @ B5 @ Bs2 ) ) ) )
       => ( ! [P4: int > nat > $o,V: int,Va: list_int] :
              ( X
             != ( produc3135705617881911306st_nat @ P4 @ ( produc4542114716404682293st_nat @ ( cons_int @ V @ Va ) @ nil_nat ) ) )
         => ~ ! [P4: int > nat > $o,V: nat,Va: list_nat] :
                ( X
               != ( produc3135705617881911306st_nat @ P4 @ ( produc4542114716404682293st_nat @ nil_int @ ( cons_nat @ V @ Va ) ) ) ) ) ) ) ).

% list_all_zip.cases
thf(fact_1780_list__all__zip_Ocases,axiom,
    ! [X: produc3272579373640082088n_assn] :
      ( ! [P4: nat > produc6575502325842934193n_assn > $o] :
          ( X
         != ( produc1692222275734866400n_assn @ P4 @ ( produc7706768604797473534n_assn @ nil_nat @ nil_Pr5671120429643327159n_assn ) ) )
     => ( ! [P4: nat > produc6575502325842934193n_assn > $o,A6: nat,As: list_nat,B5: produc6575502325842934193n_assn,Bs2: list_P8527749157015355191n_assn] :
            ( X
           != ( produc1692222275734866400n_assn @ P4 @ ( produc7706768604797473534n_assn @ ( cons_nat @ A6 @ As ) @ ( cons_P2971678138204555879n_assn @ B5 @ Bs2 ) ) ) )
       => ( ! [P4: nat > produc6575502325842934193n_assn > $o,V: nat,Va: list_nat] :
              ( X
             != ( produc1692222275734866400n_assn @ P4 @ ( produc7706768604797473534n_assn @ ( cons_nat @ V @ Va ) @ nil_Pr5671120429643327159n_assn ) ) )
         => ~ ! [P4: nat > produc6575502325842934193n_assn > $o,V: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] :
                ( X
               != ( produc1692222275734866400n_assn @ P4 @ ( produc7706768604797473534n_assn @ nil_nat @ ( cons_P2971678138204555879n_assn @ V @ Va ) ) ) ) ) ) ) ).

% list_all_zip.cases
thf(fact_1781_list__all__zip_Ocases,axiom,
    ! [X: produc5847166148257158298st_int] :
      ( ! [P4: nat > int > $o] :
          ( X
         != ( produc6511047662141078538st_int @ P4 @ ( produc7739558402351520821st_int @ nil_nat @ nil_int ) ) )
     => ( ! [P4: nat > int > $o,A6: nat,As: list_nat,B5: int,Bs2: list_int] :
            ( X
           != ( produc6511047662141078538st_int @ P4 @ ( produc7739558402351520821st_int @ ( cons_nat @ A6 @ As ) @ ( cons_int @ B5 @ Bs2 ) ) ) )
       => ( ! [P4: nat > int > $o,V: nat,Va: list_nat] :
              ( X
             != ( produc6511047662141078538st_int @ P4 @ ( produc7739558402351520821st_int @ ( cons_nat @ V @ Va ) @ nil_int ) ) )
         => ~ ! [P4: nat > int > $o,V: int,Va: list_int] :
                ( X
               != ( produc6511047662141078538st_int @ P4 @ ( produc7739558402351520821st_int @ nil_nat @ ( cons_int @ V @ Va ) ) ) ) ) ) ) ).

% list_all_zip.cases
thf(fact_1782_list__all__zip_Ocases,axiom,
    ! [X: produc4787317212837456354st_nat] :
      ( ! [P4: nat > nat > $o] :
          ( X
         != ( produc3127733452865184594st_nat @ P4 @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) ) )
     => ( ! [P4: nat > nat > $o,A6: nat,As: list_nat,B5: nat,Bs2: list_nat] :
            ( X
           != ( produc3127733452865184594st_nat @ P4 @ ( produc2694037385005941721st_nat @ ( cons_nat @ A6 @ As ) @ ( cons_nat @ B5 @ Bs2 ) ) ) )
       => ( ! [P4: nat > nat > $o,V: nat,Va: list_nat] :
              ( X
             != ( produc3127733452865184594st_nat @ P4 @ ( produc2694037385005941721st_nat @ ( cons_nat @ V @ Va ) @ nil_nat ) ) )
         => ~ ! [P4: nat > nat > $o,V: nat,Va: list_nat] :
                ( X
               != ( produc3127733452865184594st_nat @ P4 @ ( produc2694037385005941721st_nat @ nil_nat @ ( cons_nat @ V @ Va ) ) ) ) ) ) ) ).

% list_all_zip.cases
thf(fact_1783_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_1784_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_1785_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_1786_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_1787_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_1788_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_1789_measures__less,axiom,
    ! [F2: nat > nat,X: nat,Y: nat,Fs: list_nat_nat] :
      ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F2 @ Fs ) ) ) ) ).

% measures_less
thf(fact_1790_measures__less,axiom,
    ! [F2: int > nat,X: int,Y: int,Fs: list_int_nat] :
      ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F2 @ Fs ) ) ) ) ).

% measures_less
thf(fact_1791_less__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A @ X )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_1792_less__infI1,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ X )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_1793_less__infI1,axiom,
    ! [A: product_prod_int_int > $o,X: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( ord_le8213806771718485336_int_o @ A @ X )
     => ( ord_le8213806771718485336_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_1794_less__infI1,axiom,
    ! [A: assn,X: assn,B: assn] :
      ( ( ord_less_assn @ A @ X )
     => ( ord_less_assn @ ( inf_inf_assn @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_1795_less__infI1,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( ( ord_less_rat @ A @ X )
     => ( ord_less_rat @ ( inf_inf_rat @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_1796_less__infI1,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( ( ord_less_nat @ A @ X )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_1797_less__infI1,axiom,
    ! [A: int,X: int,B: int] :
      ( ( ord_less_int @ A @ X )
     => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_1798_less__infI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ B @ X )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_1799_less__infI2,axiom,
    ! [B: set_nat,X: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ X )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_1800_less__infI2,axiom,
    ! [B: product_prod_int_int > $o,X: product_prod_int_int > $o,A: product_prod_int_int > $o] :
      ( ( ord_le8213806771718485336_int_o @ B @ X )
     => ( ord_le8213806771718485336_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_1801_less__infI2,axiom,
    ! [B: assn,X: assn,A: assn] :
      ( ( ord_less_assn @ B @ X )
     => ( ord_less_assn @ ( inf_inf_assn @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_1802_less__infI2,axiom,
    ! [B: rat,X: rat,A: rat] :
      ( ( ord_less_rat @ B @ X )
     => ( ord_less_rat @ ( inf_inf_rat @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_1803_less__infI2,axiom,
    ! [B: nat,X: nat,A: nat] :
      ( ( ord_less_nat @ B @ X )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_1804_less__infI2,axiom,
    ! [B: int,X: int,A: int] :
      ( ( ord_less_int @ B @ X )
     => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_1805_inf_Oabsorb3,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A @ B )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_1806_inf_Oabsorb3,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( inf_inf_set_nat @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_1807_inf_Oabsorb3,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( ord_le8213806771718485336_int_o @ A @ B )
     => ( ( inf_in3604695632404883862_int_o @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_1808_inf_Oabsorb3,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( inf_inf_assn @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_1809_inf_Oabsorb3,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( inf_inf_rat @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_1810_inf_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( inf_inf_nat @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_1811_inf_Oabsorb3,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( inf_inf_int @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_1812_inf_Oabsorb4,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ B @ A )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_1813_inf_Oabsorb4,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ( inf_inf_set_nat @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_1814_inf_Oabsorb4,axiom,
    ! [B: product_prod_int_int > $o,A: product_prod_int_int > $o] :
      ( ( ord_le8213806771718485336_int_o @ B @ A )
     => ( ( inf_in3604695632404883862_int_o @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_1815_inf_Oabsorb4,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( ( inf_inf_assn @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_1816_inf_Oabsorb4,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( inf_inf_rat @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_1817_inf_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( inf_inf_nat @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_1818_inf_Oabsorb4,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( inf_inf_int @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_1819_inf_Ostrict__boundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) )
     => ~ ( ( ord_le7866589430770878221at_nat @ A @ B )
         => ~ ( ord_le7866589430770878221at_nat @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_1820_inf_Ostrict__boundedE,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
     => ~ ( ( ord_less_set_nat @ A @ B )
         => ~ ( ord_less_set_nat @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_1821_inf_Ostrict__boundedE,axiom,
    ! [A: product_prod_int_int > $o,B: product_prod_int_int > $o,C: product_prod_int_int > $o] :
      ( ( ord_le8213806771718485336_int_o @ A @ ( inf_in3604695632404883862_int_o @ B @ C ) )
     => ~ ( ( ord_le8213806771718485336_int_o @ A @ B )
         => ~ ( ord_le8213806771718485336_int_o @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_1822_inf_Ostrict__boundedE,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( ord_less_assn @ A @ ( inf_inf_assn @ B @ C ) )
     => ~ ( ( ord_less_assn @ A @ B )
         => ~ ( ord_less_assn @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_1823_inf_Ostrict__boundedE,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( inf_inf_rat @ B @ C ) )
     => ~ ( ( ord_less_rat @ A @ B )
         => ~ ( ord_less_rat @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_1824_inf_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
     => ~ ( ( ord_less_nat @ A @ B )
         => ~ ( ord_less_nat @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_1825_inf_Ostrict__boundedE,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ ( inf_inf_int @ B @ C ) )
     => ~ ( ( ord_less_int @ A @ B )
         => ~ ( ord_less_int @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_1826_inf_Ostrict__order__iff,axiom,
    ( ord_le7866589430770878221at_nat
    = ( ^ [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
          ( ( A2
            = ( inf_in2572325071724192079at_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_1827_inf_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( A2
            = ( inf_inf_set_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_1828_inf_Ostrict__order__iff,axiom,
    ( ord_le8213806771718485336_int_o
    = ( ^ [A2: product_prod_int_int > $o,B2: product_prod_int_int > $o] :
          ( ( A2
            = ( inf_in3604695632404883862_int_o @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_1829_inf_Ostrict__order__iff,axiom,
    ( ord_less_assn
    = ( ^ [A2: assn,B2: assn] :
          ( ( A2
            = ( inf_inf_assn @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_1830_inf_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( A2
            = ( inf_inf_rat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_1831_inf_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( A2
            = ( inf_inf_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_1832_inf_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] :
          ( ( A2
            = ( inf_inf_int @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_1833_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A @ C )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_1834_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ C )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_1835_inf_Ostrict__coboundedI1,axiom,
    ! [A: product_prod_int_int > $o,C: product_prod_int_int > $o,B: product_prod_int_int > $o] :
      ( ( ord_le8213806771718485336_int_o @ A @ C )
     => ( ord_le8213806771718485336_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_1836_inf_Ostrict__coboundedI1,axiom,
    ! [A: assn,C: assn,B: assn] :
      ( ( ord_less_assn @ A @ C )
     => ( ord_less_assn @ ( inf_inf_assn @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_1837_inf_Ostrict__coboundedI1,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ A @ C )
     => ( ord_less_rat @ ( inf_inf_rat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_1838_inf_Ostrict__coboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ A @ C )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_1839_inf_Ostrict__coboundedI1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ A @ C )
     => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_1840_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ B @ C )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_1841_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ C )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_1842_inf_Ostrict__coboundedI2,axiom,
    ! [B: product_prod_int_int > $o,C: product_prod_int_int > $o,A: product_prod_int_int > $o] :
      ( ( ord_le8213806771718485336_int_o @ B @ C )
     => ( ord_le8213806771718485336_int_o @ ( inf_in3604695632404883862_int_o @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_1843_inf_Ostrict__coboundedI2,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ( ord_less_assn @ B @ C )
     => ( ord_less_assn @ ( inf_inf_assn @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_1844_inf_Ostrict__coboundedI2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ B @ C )
     => ( ord_less_rat @ ( inf_inf_rat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_1845_inf_Ostrict__coboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ B @ C )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_1846_inf_Ostrict__coboundedI2,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_int @ B @ C )
     => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_1847_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ C @ B )
     => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_1848_sup_Ostrict__coboundedI2,axiom,
    ! [C: assn,B: assn,A: assn] :
      ( ( ord_less_assn @ C @ B )
     => ( ord_less_assn @ C @ ( sup_sup_assn @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_1849_sup_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ B )
     => ( ord_less_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_1850_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C @ B )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_1851_sup_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_int @ C @ B )
     => ( ord_less_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_1852_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ C @ A )
     => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_1853_sup_Ostrict__coboundedI1,axiom,
    ! [C: assn,A: assn,B: assn] :
      ( ( ord_less_assn @ C @ A )
     => ( ord_less_assn @ C @ ( sup_sup_assn @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_1854_sup_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ A )
     => ( ord_less_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_1855_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_1856_sup_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ A )
     => ( ord_less_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_1857_sup_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( ( A2
            = ( sup_sup_set_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_1858_sup_Ostrict__order__iff,axiom,
    ( ord_less_assn
    = ( ^ [B2: assn,A2: assn] :
          ( ( A2
            = ( sup_sup_assn @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_1859_sup_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( A2
            = ( sup_sup_rat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_1860_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( A2
            = ( sup_sup_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_1861_sup_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A2: int] :
          ( ( A2
            = ( sup_sup_int @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_1862_sup_Ostrict__boundedE,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_set_nat @ B @ A )
         => ~ ( ord_less_set_nat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_1863_sup_Ostrict__boundedE,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( ( ord_less_assn @ ( sup_sup_assn @ B @ C ) @ A )
     => ~ ( ( ord_less_assn @ B @ A )
         => ~ ( ord_less_assn @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_1864_sup_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( sup_sup_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_rat @ B @ A )
         => ~ ( ord_less_rat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_1865_sup_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_1866_sup_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_int @ ( sup_sup_int @ B @ C ) @ A )
     => ~ ( ( ord_less_int @ B @ A )
         => ~ ( ord_less_int @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_1867_sup_Oabsorb4,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( sup_sup_set_nat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_1868_sup_Oabsorb4,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( sup_sup_assn @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_1869_sup_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( sup_sup_rat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_1870_sup_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_1871_sup_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( sup_sup_int @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_1872_sup_Oabsorb3,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ( sup_sup_set_nat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_1873_sup_Oabsorb3,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( ( sup_sup_assn @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_1874_sup_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( sup_sup_rat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_1875_sup_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_1876_sup_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( sup_sup_int @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_1877_less__supI2,axiom,
    ! [X: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ X @ B )
     => ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% less_supI2
thf(fact_1878_less__supI2,axiom,
    ! [X: assn,B: assn,A: assn] :
      ( ( ord_less_assn @ X @ B )
     => ( ord_less_assn @ X @ ( sup_sup_assn @ A @ B ) ) ) ).

% less_supI2
thf(fact_1879_less__supI2,axiom,
    ! [X: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ X @ B )
     => ( ord_less_rat @ X @ ( sup_sup_rat @ A @ B ) ) ) ).

% less_supI2
thf(fact_1880_less__supI2,axiom,
    ! [X: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ X @ B )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% less_supI2
thf(fact_1881_less__supI2,axiom,
    ! [X: int,B: int,A: int] :
      ( ( ord_less_int @ X @ B )
     => ( ord_less_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).

% less_supI2
thf(fact_1882_less__supI1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ X @ A )
     => ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% less_supI1
thf(fact_1883_less__supI1,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( ( ord_less_assn @ X @ A )
     => ( ord_less_assn @ X @ ( sup_sup_assn @ A @ B ) ) ) ).

% less_supI1
thf(fact_1884_less__supI1,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ X @ A )
     => ( ord_less_rat @ X @ ( sup_sup_rat @ A @ B ) ) ) ).

% less_supI1
thf(fact_1885_less__supI1,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ X @ A )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% less_supI1
thf(fact_1886_less__supI1,axiom,
    ! [X: int,A: int,B: int] :
      ( ( ord_less_int @ X @ A )
     => ( ord_less_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).

% less_supI1
thf(fact_1887_map__eq__Cons__conv,axiom,
    ! [F2: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn,Y: assn,Ys: list_assn] :
      ( ( ( map_Pr8991440229025900053n_assn @ F2 @ Xs )
        = ( cons_assn @ Y @ Ys ) )
      = ( ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ Z3 @ Zs2 ) )
            & ( ( F2 @ Z3 )
              = Y )
            & ( ( map_Pr8991440229025900053n_assn @ F2 @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_1888_map__eq__Cons__conv,axiom,
    ! [F2: int > int,Xs: list_int,Y: int,Ys: list_int] :
      ( ( ( map_int_int @ F2 @ Xs )
        = ( cons_int @ Y @ Ys ) )
      = ( ? [Z3: int,Zs2: list_int] :
            ( ( Xs
              = ( cons_int @ Z3 @ Zs2 ) )
            & ( ( F2 @ Z3 )
              = Y )
            & ( ( map_int_int @ F2 @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_1889_map__eq__Cons__conv,axiom,
    ! [F2: nat > int,Xs: list_nat,Y: int,Ys: list_int] :
      ( ( ( map_nat_int @ F2 @ Xs )
        = ( cons_int @ Y @ Ys ) )
      = ( ? [Z3: nat,Zs2: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z3 @ Zs2 ) )
            & ( ( F2 @ Z3 )
              = Y )
            & ( ( map_nat_int @ F2 @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_1890_map__eq__Cons__conv,axiom,
    ! [F2: int > nat,Xs: list_int,Y: nat,Ys: list_nat] :
      ( ( ( map_int_nat @ F2 @ Xs )
        = ( cons_nat @ Y @ Ys ) )
      = ( ? [Z3: int,Zs2: list_int] :
            ( ( Xs
              = ( cons_int @ Z3 @ Zs2 ) )
            & ( ( F2 @ Z3 )
              = Y )
            & ( ( map_int_nat @ F2 @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_1891_map__eq__Cons__conv,axiom,
    ! [F2: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = ( cons_nat @ Y @ Ys ) )
      = ( ? [Z3: nat,Zs2: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z3 @ Zs2 ) )
            & ( ( F2 @ Z3 )
              = Y )
            & ( ( map_nat_nat @ F2 @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_1892_Cons__eq__map__conv,axiom,
    ! [X: assn,Xs: list_assn,F2: produc6575502325842934193n_assn > assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( cons_assn @ X @ Xs )
        = ( map_Pr8991440229025900053n_assn @ F2 @ Ys ) )
      = ( ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
            ( ( Ys
              = ( cons_P2971678138204555879n_assn @ Z3 @ Zs2 ) )
            & ( X
              = ( F2 @ Z3 ) )
            & ( Xs
              = ( map_Pr8991440229025900053n_assn @ F2 @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_1893_Cons__eq__map__conv,axiom,
    ! [X: int,Xs: list_int,F2: int > int,Ys: list_int] :
      ( ( ( cons_int @ X @ Xs )
        = ( map_int_int @ F2 @ Ys ) )
      = ( ? [Z3: int,Zs2: list_int] :
            ( ( Ys
              = ( cons_int @ Z3 @ Zs2 ) )
            & ( X
              = ( F2 @ Z3 ) )
            & ( Xs
              = ( map_int_int @ F2 @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_1894_Cons__eq__map__conv,axiom,
    ! [X: int,Xs: list_int,F2: nat > int,Ys: list_nat] :
      ( ( ( cons_int @ X @ Xs )
        = ( map_nat_int @ F2 @ Ys ) )
      = ( ? [Z3: nat,Zs2: list_nat] :
            ( ( Ys
              = ( cons_nat @ Z3 @ Zs2 ) )
            & ( X
              = ( F2 @ Z3 ) )
            & ( Xs
              = ( map_nat_int @ F2 @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_1895_Cons__eq__map__conv,axiom,
    ! [X: nat,Xs: list_nat,F2: int > nat,Ys: list_int] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_int_nat @ F2 @ Ys ) )
      = ( ? [Z3: int,Zs2: list_int] :
            ( ( Ys
              = ( cons_int @ Z3 @ Zs2 ) )
            & ( X
              = ( F2 @ Z3 ) )
            & ( Xs
              = ( map_int_nat @ F2 @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_1896_Cons__eq__map__conv,axiom,
    ! [X: nat,Xs: list_nat,F2: nat > nat,Ys: list_nat] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_nat_nat @ F2 @ Ys ) )
      = ( ? [Z3: nat,Zs2: list_nat] :
            ( ( Ys
              = ( cons_nat @ Z3 @ Zs2 ) )
            & ( X
              = ( F2 @ Z3 ) )
            & ( Xs
              = ( map_nat_nat @ F2 @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_1897_map__eq__Cons__D,axiom,
    ! [F2: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn,Y: assn,Ys: list_assn] :
      ( ( ( map_Pr8991440229025900053n_assn @ F2 @ Xs )
        = ( cons_assn @ Y @ Ys ) )
     => ? [Z2: produc6575502325842934193n_assn,Zs3: list_P8527749157015355191n_assn] :
          ( ( Xs
            = ( cons_P2971678138204555879n_assn @ Z2 @ Zs3 ) )
          & ( ( F2 @ Z2 )
            = Y )
          & ( ( map_Pr8991440229025900053n_assn @ F2 @ Zs3 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_1898_map__eq__Cons__D,axiom,
    ! [F2: int > int,Xs: list_int,Y: int,Ys: list_int] :
      ( ( ( map_int_int @ F2 @ Xs )
        = ( cons_int @ Y @ Ys ) )
     => ? [Z2: int,Zs3: list_int] :
          ( ( Xs
            = ( cons_int @ Z2 @ Zs3 ) )
          & ( ( F2 @ Z2 )
            = Y )
          & ( ( map_int_int @ F2 @ Zs3 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_1899_map__eq__Cons__D,axiom,
    ! [F2: nat > int,Xs: list_nat,Y: int,Ys: list_int] :
      ( ( ( map_nat_int @ F2 @ Xs )
        = ( cons_int @ Y @ Ys ) )
     => ? [Z2: nat,Zs3: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z2 @ Zs3 ) )
          & ( ( F2 @ Z2 )
            = Y )
          & ( ( map_nat_int @ F2 @ Zs3 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_1900_map__eq__Cons__D,axiom,
    ! [F2: int > nat,Xs: list_int,Y: nat,Ys: list_nat] :
      ( ( ( map_int_nat @ F2 @ Xs )
        = ( cons_nat @ Y @ Ys ) )
     => ? [Z2: int,Zs3: list_int] :
          ( ( Xs
            = ( cons_int @ Z2 @ Zs3 ) )
          & ( ( F2 @ Z2 )
            = Y )
          & ( ( map_int_nat @ F2 @ Zs3 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_1901_map__eq__Cons__D,axiom,
    ! [F2: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = ( cons_nat @ Y @ Ys ) )
     => ? [Z2: nat,Zs3: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z2 @ Zs3 ) )
          & ( ( F2 @ Z2 )
            = Y )
          & ( ( map_nat_nat @ F2 @ Zs3 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_1902_Cons__eq__map__D,axiom,
    ! [X: assn,Xs: list_assn,F2: produc6575502325842934193n_assn > assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( cons_assn @ X @ Xs )
        = ( map_Pr8991440229025900053n_assn @ F2 @ Ys ) )
     => ? [Z2: produc6575502325842934193n_assn,Zs3: list_P8527749157015355191n_assn] :
          ( ( Ys
            = ( cons_P2971678138204555879n_assn @ Z2 @ Zs3 ) )
          & ( X
            = ( F2 @ Z2 ) )
          & ( Xs
            = ( map_Pr8991440229025900053n_assn @ F2 @ Zs3 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_1903_Cons__eq__map__D,axiom,
    ! [X: int,Xs: list_int,F2: int > int,Ys: list_int] :
      ( ( ( cons_int @ X @ Xs )
        = ( map_int_int @ F2 @ Ys ) )
     => ? [Z2: int,Zs3: list_int] :
          ( ( Ys
            = ( cons_int @ Z2 @ Zs3 ) )
          & ( X
            = ( F2 @ Z2 ) )
          & ( Xs
            = ( map_int_int @ F2 @ Zs3 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_1904_Cons__eq__map__D,axiom,
    ! [X: int,Xs: list_int,F2: nat > int,Ys: list_nat] :
      ( ( ( cons_int @ X @ Xs )
        = ( map_nat_int @ F2 @ Ys ) )
     => ? [Z2: nat,Zs3: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z2 @ Zs3 ) )
          & ( X
            = ( F2 @ Z2 ) )
          & ( Xs
            = ( map_nat_int @ F2 @ Zs3 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_1905_Cons__eq__map__D,axiom,
    ! [X: nat,Xs: list_nat,F2: int > nat,Ys: list_int] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_int_nat @ F2 @ Ys ) )
     => ? [Z2: int,Zs3: list_int] :
          ( ( Ys
            = ( cons_int @ Z2 @ Zs3 ) )
          & ( X
            = ( F2 @ Z2 ) )
          & ( Xs
            = ( map_int_nat @ F2 @ Zs3 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_1906_Cons__eq__map__D,axiom,
    ! [X: nat,Xs: list_nat,F2: nat > nat,Ys: list_nat] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_nat_nat @ F2 @ Ys ) )
     => ? [Z2: nat,Zs3: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z2 @ Zs3 ) )
          & ( X
            = ( F2 @ Z2 ) )
          & ( Xs
            = ( map_nat_nat @ F2 @ Zs3 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_1907_list_Osimps_I9_J,axiom,
    ! [F2: produc6575502325842934193n_assn > assn,X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn] :
      ( ( map_Pr8991440229025900053n_assn @ F2 @ ( cons_P2971678138204555879n_assn @ X21 @ X22 ) )
      = ( cons_assn @ ( F2 @ X21 ) @ ( map_Pr8991440229025900053n_assn @ F2 @ X22 ) ) ) ).

% list.simps(9)
thf(fact_1908_list_Osimps_I9_J,axiom,
    ! [F2: int > int,X21: int,X22: list_int] :
      ( ( map_int_int @ F2 @ ( cons_int @ X21 @ X22 ) )
      = ( cons_int @ ( F2 @ X21 ) @ ( map_int_int @ F2 @ X22 ) ) ) ).

% list.simps(9)
thf(fact_1909_list_Osimps_I9_J,axiom,
    ! [F2: int > nat,X21: int,X22: list_int] :
      ( ( map_int_nat @ F2 @ ( cons_int @ X21 @ X22 ) )
      = ( cons_nat @ ( F2 @ X21 ) @ ( map_int_nat @ F2 @ X22 ) ) ) ).

% list.simps(9)
thf(fact_1910_list_Osimps_I9_J,axiom,
    ! [F2: nat > int,X21: nat,X22: list_nat] :
      ( ( map_nat_int @ F2 @ ( cons_nat @ X21 @ X22 ) )
      = ( cons_int @ ( F2 @ X21 ) @ ( map_nat_int @ F2 @ X22 ) ) ) ).

% list.simps(9)
thf(fact_1911_list_Osimps_I9_J,axiom,
    ! [F2: nat > nat,X21: nat,X22: list_nat] :
      ( ( map_nat_nat @ F2 @ ( cons_nat @ X21 @ X22 ) )
      = ( cons_nat @ ( F2 @ X21 ) @ ( map_nat_nat @ F2 @ X22 ) ) ) ).

% list.simps(9)
thf(fact_1912_splice_Osimps_I2_J,axiom,
    ! [X: int,Xs: list_int,Ys: list_int] :
      ( ( splice_int @ ( cons_int @ X @ Xs ) @ Ys )
      = ( cons_int @ X @ ( splice_int @ Ys @ Xs ) ) ) ).

% splice.simps(2)
thf(fact_1913_splice_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat] :
      ( ( splice_nat @ ( cons_nat @ X @ Xs ) @ Ys )
      = ( cons_nat @ X @ ( splice_nat @ Ys @ Xs ) ) ) ).

% splice.simps(2)
thf(fact_1914_null__rec_I1_J,axiom,
    ! [X: int,Xs: list_int] :
      ~ ( null_int @ ( cons_int @ X @ Xs ) ) ).

% null_rec(1)
thf(fact_1915_null__rec_I1_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ~ ( null_nat @ ( cons_nat @ X @ Xs ) ) ).

% null_rec(1)
thf(fact_1916_member__rec_I1_J,axiom,
    ! [X: int,Xs: list_int,Y: int] :
      ( ( member_int @ ( cons_int @ X @ Xs ) @ Y )
      = ( ( X = Y )
        | ( member_int @ Xs @ Y ) ) ) ).

% member_rec(1)
thf(fact_1917_member__rec_I1_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat] :
      ( ( member_nat @ ( cons_nat @ X @ Xs ) @ Y )
      = ( ( X = Y )
        | ( member_nat @ Xs @ Y ) ) ) ).

% member_rec(1)
thf(fact_1918_mult__neg__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_1919_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_1920_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_1921_not__square__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ A ) @ zero_z3403309356797280102nteger ) ).

% not_square_less_zero
thf(fact_1922_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_1923_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_1924_mult__less__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger ) )
        | ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_1925_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_1926_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ B @ zero_zero_int ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_1927_mult__neg__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ zero_z2226904508553997617atural )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A @ B ) @ zero_z2226904508553997617atural ) ) ) ).

% mult_neg_pos
thf(fact_1928_mult__neg__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_neg_pos
thf(fact_1929_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_neg_pos
thf(fact_1930_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_1931_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_neg_pos
thf(fact_1932_mult__pos__neg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le5570908160329646204atural @ B @ zero_z2226904508553997617atural )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A @ B ) @ zero_z2226904508553997617atural ) ) ) ).

% mult_pos_neg
thf(fact_1933_mult__pos__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ B ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_pos_neg
thf(fact_1934_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg
thf(fact_1935_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_1936_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_pos_neg
thf(fact_1937_mult__pos__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
       => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( times_2397367101498566445atural @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_1938_mult__pos__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_1939_mult__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 @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_1940_mult__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 @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_1941_mult__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 @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_1942_mult__pos__neg2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le5570908160329646204atural @ B @ zero_z2226904508553997617atural )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ B @ A ) @ zero_z2226904508553997617atural ) ) ) ).

% mult_pos_neg2
thf(fact_1943_mult__pos__neg2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ B @ A ) @ zero_z3403309356797280102nteger ) ) ) ).

% mult_pos_neg2
thf(fact_1944_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg2
thf(fact_1945_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_1946_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_pos_neg2
thf(fact_1947_zero__less__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) )
        | ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B @ zero_z3403309356797280102nteger ) ) ) ) ).

% zero_less_mult_iff
thf(fact_1948_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_less_mult_iff
thf(fact_1949_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ zero_zero_int @ B ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).

% zero_less_mult_iff
thf(fact_1950_zero__less__mult__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( times_2397367101498566445atural @ A @ B ) )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
       => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_1951_zero__less__mult__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ A @ B ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_1952_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_1953_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_1954_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_1955_zero__less__mult__pos2,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( times_2397367101498566445atural @ B @ A ) )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
       => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_1956_zero__less__mult__pos2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( times_3573771949741848930nteger @ B @ A ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_1957_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_1958_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_1959_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_1960_mult__less__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( ord_le6747313008572928689nteger @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_1961_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_1962_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_1963_mult__less__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_1964_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_1965_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_1966_mult__strict__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_1967_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_1968_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_1969_mult__strict__left__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ B )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_1970_mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_1971_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_1972_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_1973_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_1974_mult__less__cancel__left__disj,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
          & ( ord_le6747313008572928689nteger @ A @ B ) )
        | ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_1975_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_1976_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_1977_mult__strict__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_1978_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_1979_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_1980_mult__strict__right__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ B )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_1981_mult__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_1982_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_1983_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_1984_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_1985_mult__less__cancel__right__disj,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
          & ( ord_le6747313008572928689nteger @ A @ B ) )
        | ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
          & ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_1986_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_1987_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_1988_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ B )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1989_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1990_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1991_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1992_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1993_less__numeral__extra_I1_J,axiom,
    ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).

% less_numeral_extra(1)
thf(fact_1994_less__numeral__extra_I1_J,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% less_numeral_extra(1)
thf(fact_1995_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_1996_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_1997_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_1998_zero__less__one,axiom,
    ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ one_one_Code_natural ).

% zero_less_one
thf(fact_1999_zero__less__one,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% zero_less_one
thf(fact_2000_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_2001_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_2002_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_2003_not__one__less__zero,axiom,
    ~ ( ord_le5570908160329646204atural @ one_one_Code_natural @ zero_z2226904508553997617atural ) ).

% not_one_less_zero
thf(fact_2004_not__one__less__zero,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).

% not_one_less_zero
thf(fact_2005_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_2006_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_2007_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_2008_less__1__mult,axiom,
    ! [M2: code_natural,N: code_natural] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ M2 )
     => ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ N )
       => ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_2009_less__1__mult,axiom,
    ! [M2: code_integer,N: code_integer] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ M2 )
     => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ N )
       => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_2010_less__1__mult,axiom,
    ! [M2: rat,N: rat] :
      ( ( ord_less_rat @ one_one_rat @ M2 )
     => ( ( ord_less_rat @ one_one_rat @ N )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_2011_less__1__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ M2 )
     => ( ( ord_less_nat @ one_one_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_2012_less__1__mult,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_int @ one_one_int @ M2 )
     => ( ( ord_less_int @ one_one_int @ N )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_2013_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_2014_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_2015_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_2016_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_2017_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_2018_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_2019_splice_Oelims,axiom,
    ! [X: list_P8527749157015355191n_assn,Xa: list_P8527749157015355191n_assn,Y: list_P8527749157015355191n_assn] :
      ( ( ( splice4588524007259616650n_assn @ X @ Xa )
        = Y )
     => ( ( ( X = nil_Pr5671120429643327159n_assn )
         => ( Y != Xa ) )
       => ~ ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( ( X
                = ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) )
             => ( Y
               != ( cons_P2971678138204555879n_assn @ X3 @ ( splice4588524007259616650n_assn @ Xa @ Xs2 ) ) ) ) ) ) ).

% splice.elims
thf(fact_2020_splice_Oelims,axiom,
    ! [X: list_int,Xa: list_int,Y: list_int] :
      ( ( ( splice_int @ X @ Xa )
        = Y )
     => ( ( ( X = nil_int )
         => ( Y != Xa ) )
       => ~ ! [X3: int,Xs2: list_int] :
              ( ( X
                = ( cons_int @ X3 @ Xs2 ) )
             => ( Y
               != ( cons_int @ X3 @ ( splice_int @ Xa @ Xs2 ) ) ) ) ) ) ).

% splice.elims
thf(fact_2021_splice_Oelims,axiom,
    ! [X: list_nat,Xa: list_nat,Y: list_nat] :
      ( ( ( splice_nat @ X @ Xa )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != Xa ) )
       => ~ ! [X3: nat,Xs2: list_nat] :
              ( ( X
                = ( cons_nat @ X3 @ Xs2 ) )
             => ( Y
               != ( cons_nat @ X3 @ ( splice_nat @ Xa @ Xs2 ) ) ) ) ) ) ).

% splice.elims
thf(fact_2022_measures__lesseq,axiom,
    ! [F2: nat > nat,X: nat,Y: nat,Fs: list_nat_nat] :
      ( ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ Fs ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F2 @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_2023_measures__lesseq,axiom,
    ! [F2: int > nat,X: int,Y: int,Fs: list_int_nat] :
      ( ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ Fs ) )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F2 @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_2024_mult__less__le__imp__less,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ B )
     => ( ( ord_le1926595141338095240atural @ C @ D2 )
       => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
         => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ C )
           => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2025_mult__less__le__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2026_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2027_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2028_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2029_mult__le__less__imp__less,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ B )
     => ( ( ord_le5570908160329646204atural @ C @ D2 )
       => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
         => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
           => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2030_mult__le__less__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2031_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ( ord_less_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2032_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( ord_less_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2033_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( ord_less_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2034_mult__right__le__imp__le,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le1926595141338095240atural @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2035_mult__right__le__imp__le,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2036_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2037_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2038_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_2039_mult__left__le__imp__le,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le1926595141338095240atural @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2040_mult__left__le__imp__le,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2041_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2042_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2043_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_2044_mult__le__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( ord_le3102999989581377725nteger @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2045_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2046_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2047_mult__le__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( ord_le3102999989581377725nteger @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2048_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2049_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2050_mult__less__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2051_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2052_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2053_mult__strict__mono_H,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ B )
     => ( ( ord_le5570908160329646204atural @ C @ D2 )
       => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
         => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
           => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2054_mult__strict__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_le6747313008572928689nteger @ C @ D2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2055_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2056_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2057_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2058_mult__right__less__imp__less,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le5570908160329646204atural @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2059_mult__right__less__imp__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2060_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2061_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2062_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_2063_mult__less__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le6747313008572928689nteger @ A @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2064_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2065_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2066_mult__strict__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ B )
     => ( ( ord_le5570908160329646204atural @ C @ D2 )
       => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
         => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
           => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2067_mult__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 @ zero_z3403309356797280102nteger @ B )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
           => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2068_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D2 )
       => ( ( ord_less_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2069_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( ord_less_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2070_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( ord_less_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2071_mult__left__less__imp__less,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
       => ( ord_le5570908160329646204atural @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2072_mult__left__less__imp__less,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
       => ( ord_le6747313008572928689nteger @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2073_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2074_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2075_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_2076_mult__le__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2077_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2078_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2079_mult__le__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C )
         => ( ord_le3102999989581377725nteger @ A @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2080_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2081_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2082_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_2083_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_2084_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_2085_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_2086_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_2087_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_2088_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_2089_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_2090_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_2091_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_2092_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_2093_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_2094_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_2095_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_2096_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_2097_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_2098_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_2099_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_2100_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_2101_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_2102_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_2103_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_2104_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_2105_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_2106_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_2107_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_2108_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_2109_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z2 )
         => ( ( ord_less_rat @ Z2 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X ) @ Y ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_2110_mult__le__cancel__iff1,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ X @ Z ) @ ( times_3573771949741848930nteger @ Y @ Z ) )
        = ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2111_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2112_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2113_mult__le__cancel__iff2,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
     => ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ Z @ X ) @ ( times_3573771949741848930nteger @ Z @ Y ) )
        = ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2114_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2115_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2116_mult__less__iff1,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ Z )
     => ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ X @ Z ) @ ( times_3573771949741848930nteger @ Y @ Z ) )
        = ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_2117_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_rat @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_2118_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_int @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_2119_quicksort_Ocases,axiom,
    ! [X: list_int] :
      ( ( X != nil_int )
     => ~ ! [X3: int,Xs2: list_int] :
            ( X
           != ( cons_int @ X3 @ Xs2 ) ) ) ).

% quicksort.cases
thf(fact_2120_quicksort_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ~ ! [X3: nat,Xs2: list_nat] :
            ( X
           != ( cons_nat @ X3 @ Xs2 ) ) ) ).

% quicksort.cases
thf(fact_2121_subset__eq__mset__impl_Ocases,axiom,
    ! [X: produc337509097454979815n_assn] :
      ( ! [Ys3: list_P8527749157015355191n_assn] :
          ( X
         != ( produc3014731801419794263n_assn @ nil_Pr5671120429643327159n_assn @ Ys3 ) )
     => ~ ! [X3: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Ys3: list_P8527749157015355191n_assn] :
            ( X
           != ( produc3014731801419794263n_assn @ ( cons_P2971678138204555879n_assn @ X3 @ Xs2 ) @ Ys3 ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_2122_subset__eq__mset__impl_Ocases,axiom,
    ! [X: produc1186641810826059865st_int] :
      ( ! [Ys3: list_int] :
          ( X
         != ( produc364263696895485585st_int @ nil_int @ Ys3 ) )
     => ~ ! [X3: int,Xs2: list_int,Ys3: list_int] :
            ( X
           != ( produc364263696895485585st_int @ ( cons_int @ X3 @ Xs2 ) @ Ys3 ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_2123_subset__eq__mset__impl_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [Ys3: list_nat] :
          ( X
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys3 ) )
     => ~ ! [X3: nat,Xs2: list_nat,Ys3: list_nat] :
            ( X
           != ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs2 ) @ Ys3 ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_2124_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila4918476307565957903at_nat @ inf_in2572325071724192079at_nat @ top_to4669805908274784177at_nat @ ord_le3146513528884898305at_nat @ ord_le7866589430770878221at_nat ).

% inf_top.semilattice_neutr_order_axioms
thf(fact_2125_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila8052717425514686422_int_o @ inf_in3604695632404883862_int_o @ top_to1578927101902068148_int_o @ ord_le8369615600986905444_int_o @ ord_le8213806771718485336_int_o ).

% inf_top.semilattice_neutr_order_axioms
thf(fact_2126_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila4423693302617054363t_unit @ inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ ord_le3507040750410214029t_unit @ ord_le8056459307392131481t_unit ).

% inf_top.semilattice_neutr_order_axioms
thf(fact_2127_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila8603258263270017530r_assn @ inf_inf_assn @ top_top_assn @ ord_less_eq_assn @ ord_less_assn ).

% inf_top.semilattice_neutr_order_axioms
thf(fact_2128_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila1667268886620078168et_nat @ inf_inf_set_nat @ top_top_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).

% inf_top.semilattice_neutr_order_axioms
thf(fact_2129_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_2130_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_2131_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_2132_insert__Nil,axiom,
    ! [X: produc6575502325842934193n_assn] :
      ( ( insert3246601298802261197n_assn @ X @ nil_Pr5671120429643327159n_assn )
      = ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) ) ).

% insert_Nil
thf(fact_2133_insert__Nil,axiom,
    ! [X: int] :
      ( ( insert_int @ X @ nil_int )
      = ( cons_int @ X @ nil_int ) ) ).

% insert_Nil
thf(fact_2134_insert__Nil,axiom,
    ! [X: nat] :
      ( ( insert_nat @ X @ nil_nat )
      = ( cons_nat @ X @ nil_nat ) ) ).

% insert_Nil
thf(fact_2135_convex__bound__lt,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ A )
     => ( ( ord_le6747313008572928689nteger @ Y @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V2 )
           => ( ( ( plus_p5714425477246183910nteger @ U @ V2 )
                = one_one_Code_integer )
             => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2136_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V2: 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 @ V2 )
           => ( ( ( plus_plus_rat @ U @ V2 )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2137_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V2: 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 @ V2 )
           => ( ( ( plus_plus_int @ U @ V2 )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2138_add__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = ( plus_p5714425477246183910nteger @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_2139_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_2140_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_2141_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_2142_add__right__cancel,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ B @ A )
        = ( plus_p5714425477246183910nteger @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_2143_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_2144_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_2145_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_2146_add__le__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ C @ A ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
      = ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_2147_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_2148_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_2149_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_2150_add__le__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
      = ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_2151_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_2152_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_2153_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_2154_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_2155_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_2156_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_2157_add_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% add.right_neutral
thf(fact_2158_add_Oright__neutral,axiom,
    ! [A: code_natural] :
      ( ( plus_p4538020629002901425atural @ A @ zero_z2226904508553997617atural )
      = A ) ).

% add.right_neutral
thf(fact_2159_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_2160_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_2161_double__zero__sym,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( plus_p5714425477246183910nteger @ A @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% double_zero_sym
thf(fact_2162_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_2163_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_2164_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_2165_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_2166_add__cancel__left__left,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( ( plus_p4538020629002901425atural @ B @ A )
        = A )
      = ( B = zero_z2226904508553997617atural ) ) ).

% add_cancel_left_left
thf(fact_2167_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_2168_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_2169_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_2170_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_2171_add__cancel__left__right,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( plus_p4538020629002901425atural @ A @ B )
        = A )
      = ( B = zero_z2226904508553997617atural ) ) ).

% add_cancel_left_right
thf(fact_2172_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_2173_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_2174_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_2175_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_2176_add__cancel__right__left,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A
        = ( plus_p4538020629002901425atural @ B @ A ) )
      = ( B = zero_z2226904508553997617atural ) ) ).

% add_cancel_right_left
thf(fact_2177_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_2178_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_2179_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_2180_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_2181_add__cancel__right__right,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A
        = ( plus_p4538020629002901425atural @ A @ B ) )
      = ( B = zero_z2226904508553997617atural ) ) ).

% add_cancel_right_right
thf(fact_2182_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_2183_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_2184_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_2185_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_2186_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_2187_add__0,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A )
      = A ) ).

% add_0
thf(fact_2188_add__0,axiom,
    ! [A: code_natural] :
      ( ( plus_p4538020629002901425atural @ zero_z2226904508553997617atural @ A )
      = A ) ).

% add_0
thf(fact_2189_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_2190_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_2191_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_2192_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_2193_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_2194_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_2195_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_2196_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_2197_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_2198_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_2199_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_2200_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_2201_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_2202_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_2203_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_2204_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_2205_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_2206_sgn__1,axiom,
    ( ( sgn_sgn_int @ one_one_int )
    = one_one_int ) ).

% sgn_1
thf(fact_2207_sgn__1,axiom,
    ( ( sgn_sgn_rat @ one_one_rat )
    = one_one_rat ) ).

% sgn_1
thf(fact_2208_sgn__1,axiom,
    ( ( sgn_sgn_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% sgn_1
thf(fact_2209_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_2210_add__le__same__cancel1,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( ord_le1926595141338095240atural @ ( plus_p4538020629002901425atural @ B @ A ) @ B )
      = ( ord_le1926595141338095240atural @ A @ zero_z2226904508553997617atural ) ) ).

% add_le_same_cancel1
thf(fact_2211_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_2212_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_2213_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_2214_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_2215_add__le__same__cancel2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ ( plus_p4538020629002901425atural @ A @ B ) @ B )
      = ( ord_le1926595141338095240atural @ A @ zero_z2226904508553997617atural ) ) ).

% add_le_same_cancel2
thf(fact_2216_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_2217_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_2218_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_2219_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_2220_le__add__same__cancel1,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ ( plus_p4538020629002901425atural @ A @ B ) )
      = ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B ) ) ).

% le_add_same_cancel1
thf(fact_2221_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_2222_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_2223_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_2224_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_2225_le__add__same__cancel2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ ( plus_p4538020629002901425atural @ B @ A ) )
      = ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B ) ) ).

% le_add_same_cancel2
thf(fact_2226_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_2227_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_2228_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_2229_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_2230_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_2231_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_2232_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_2233_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_2234_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_2235_add__less__same__cancel1,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ B @ A ) @ B )
      = ( ord_le5570908160329646204atural @ A @ zero_z2226904508553997617atural ) ) ).

% add_less_same_cancel1
thf(fact_2236_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_2237_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_2238_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_2239_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_2240_add__less__same__cancel2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A @ B ) @ B )
      = ( ord_le5570908160329646204atural @ A @ zero_z2226904508553997617atural ) ) ).

% add_less_same_cancel2
thf(fact_2241_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_2242_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_2243_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_2244_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_2245_less__add__same__cancel1,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ ( plus_p4538020629002901425atural @ A @ B ) )
      = ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B ) ) ).

% less_add_same_cancel1
thf(fact_2246_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_2247_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_2248_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_2249_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_2250_less__add__same__cancel2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ ( plus_p4538020629002901425atural @ B @ A ) )
      = ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B ) ) ).

% less_add_same_cancel2
thf(fact_2251_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_2252_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_2253_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_2254_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_2255_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_2256_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_2257_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_2258_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_2259_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_2260_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_2261_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_2262_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_2263_add_Oright__inverse,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_2264_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_2265_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_2266_ab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_2267_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_2268_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_2269_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_2270_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_2271_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_2272_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_2273_sgn__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( sgn_sgn_Code_integer @ A )
        = one_one_Code_integer ) ) ).

% sgn_pos
thf(fact_2274_sgn__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( sgn_sgn_rat @ A )
        = one_one_rat ) ) ).

% sgn_pos
thf(fact_2275_sgn__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( sgn_sgn_int @ A )
        = one_one_int ) ) ).

% sgn_pos
thf(fact_2276_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_2277_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( ( plus_p5714425477246183910nteger @ I2 @ K )
        = ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_2278_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( ( plus_plus_rat @ I2 @ K )
        = ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_2279_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I2 @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_2280_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( ( plus_plus_int @ I2 @ K )
        = ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_2281_group__cancel_Oadd1,axiom,
    ! [A3: code_integer,K: code_integer,A: code_integer,B: code_integer] :
      ( ( A3
        = ( plus_p5714425477246183910nteger @ K @ A ) )
     => ( ( plus_p5714425477246183910nteger @ A3 @ B )
        = ( plus_p5714425477246183910nteger @ K @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_2282_group__cancel_Oadd1,axiom,
    ! [A3: rat,K: rat,A: rat,B: rat] :
      ( ( A3
        = ( plus_plus_rat @ K @ A ) )
     => ( ( plus_plus_rat @ A3 @ B )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_2283_group__cancel_Oadd1,axiom,
    ! [A3: nat,K: nat,A: nat,B: nat] :
      ( ( A3
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A3 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_2284_group__cancel_Oadd1,axiom,
    ! [A3: int,K: int,A: int,B: int] :
      ( ( A3
        = ( plus_plus_int @ K @ A ) )
     => ( ( plus_plus_int @ A3 @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_2285_group__cancel_Oadd2,axiom,
    ! [B3: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B3
        = ( plus_p5714425477246183910nteger @ K @ B ) )
     => ( ( plus_p5714425477246183910nteger @ A @ B3 )
        = ( plus_p5714425477246183910nteger @ K @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_2286_group__cancel_Oadd2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3
        = ( plus_plus_rat @ K @ B ) )
     => ( ( plus_plus_rat @ A @ B3 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_2287_group__cancel_Oadd2,axiom,
    ! [B3: nat,K: nat,B: nat,A: nat] :
      ( ( B3
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B3 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_2288_group__cancel_Oadd2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A @ B3 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_2289_add_Oright__assoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).

% add.right_assoc
thf(fact_2290_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_2291_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_2292_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_2293_add_Oright__commute,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ B ) ) ).

% add.right_commute
thf(fact_2294_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_2295_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_2296_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_2297_add_Oassoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).

% add.assoc
thf(fact_2298_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_2299_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_2300_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_2301_add_Oleft__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = ( plus_p5714425477246183910nteger @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_2302_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_2303_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_2304_add_Oright__cancel,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ B @ A )
        = ( plus_p5714425477246183910nteger @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_2305_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_2306_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_2307_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_p5714425477246183910nteger
    = ( ^ [A2: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ B2 @ A2 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2308_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A2: rat,B2: rat] : ( plus_plus_rat @ B2 @ A2 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2309_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2310_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A2: int,B2: int] : ( plus_plus_int @ B2 @ A2 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2311_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ B @ ( plus_p5714425477246183910nteger @ A @ C ) )
      = ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_2312_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_2313_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_2314_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_2315_add__left__imp__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = ( plus_p5714425477246183910nteger @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_2316_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_2317_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_2318_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_2319_add__right__imp__eq,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ B @ A )
        = ( plus_p5714425477246183910nteger @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_2320_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_2321_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_2322_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_2323_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ I2 @ J )
        & ( K = L ) )
     => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2324_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2325_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2326_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2327_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( I2 = J )
        & ( ord_le3102999989581377725nteger @ K @ L ) )
     => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2328_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( I2 = J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2329_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2330_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( I2 = J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2331_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ I2 @ J )
        & ( ord_le3102999989581377725nteger @ K @ L ) )
     => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2332_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I2 @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2333_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2334_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I2 @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2335_add__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ C @ D2 )
       => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ D2 ) ) ) ) ).

% add_mono
thf(fact_2336_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_2337_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_2338_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_2339_add__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ C @ A ) @ ( plus_p5714425477246183910nteger @ C @ B ) ) ) ).

% add_left_mono
thf(fact_2340_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_2341_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_2342_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_2343_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_2344_add__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ).

% add_right_mono
thf(fact_2345_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_2346_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_2347_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_2348_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
        ? [C4: nat] :
          ( B2
          = ( plus_plus_nat @ A2 @ C4 ) ) ) ) ).

% le_iff_add
thf(fact_2349_add__le__imp__le__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ C @ A ) @ ( plus_p5714425477246183910nteger @ C @ B ) )
     => ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_2350_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_2351_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_2352_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_2353_add__le__imp__le__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A @ C ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
     => ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_2354_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_2355_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_2356_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_2357_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_2358_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_2359_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_2360_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2361_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: code_natural] :
      ( ( plus_p4538020629002901425atural @ zero_z2226904508553997617atural @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_2362_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_2363_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_2364_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_2365_add_Ocomm__neutral,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% add.comm_neutral
thf(fact_2366_add_Ocomm__neutral,axiom,
    ! [A: code_natural] :
      ( ( plus_p4538020629002901425atural @ A @ zero_z2226904508553997617atural )
      = A ) ).

% add.comm_neutral
thf(fact_2367_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2368_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2369_add_Ogroup__left__neutral,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_2370_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le6747313008572928689nteger @ I2 @ J )
        & ( ord_le6747313008572928689nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_2371_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I2 @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_2372_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_2373_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I2 @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_2374_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( I2 = J )
        & ( ord_le6747313008572928689nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_2375_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( I2 = J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_2376_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_2377_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( I2 = J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_2378_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le6747313008572928689nteger @ I2 @ J )
        & ( K = L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_2379_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_2380_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_2381_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_2382_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_2383_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_2384_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_2385_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_2386_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_2387_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_2388_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_2389_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_2390_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_2391_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_2392_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_2393_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_2394_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_2395_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_2396_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_2397_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_2398_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_2399_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_2400_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_2401_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_2402_combine__common__factor,axiom,
    ! [A: rat,E2: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2403_combine__common__factor,axiom,
    ! [A: nat,E2: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2404_combine__common__factor,axiom,
    ! [A: int,E2: int,B: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2405_combine__common__factor,axiom,
    ! [A: code_integer,E2: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E2 ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ C ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2406_combine__common__factor,axiom,
    ! [A: code_natural,E2: code_natural,B: code_natural,C: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ E2 ) @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ E2 ) @ C ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2407_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_2408_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_2409_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_2410_distrib__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% distrib_right
thf(fact_2411_distrib__right,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) ) ) ).

% distrib_right
thf(fact_2412_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_2413_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_2414_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_2415_distrib__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ).

% distrib_left
thf(fact_2416_distrib__left,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( times_2397367101498566445atural @ A @ ( plus_p4538020629002901425atural @ B @ C ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ B ) @ ( times_2397367101498566445atural @ A @ C ) ) ) ).

% distrib_left
thf(fact_2417_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_2418_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_2419_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_2420_comm__semiring__class_Odistrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2421_comm__semiring__class_Odistrib,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2422_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_2423_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_2424_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2425_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_2426_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_2427_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2428_group__cancel_Oneg1,axiom,
    ! [A3: int,K: int,A: int] :
      ( ( A3
        = ( plus_plus_int @ K @ A ) )
     => ( ( uminus_uminus_int @ A3 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_2429_group__cancel_Oneg1,axiom,
    ! [A3: rat,K: rat,A: rat] :
      ( ( A3
        = ( plus_plus_rat @ K @ A ) )
     => ( ( uminus_uminus_rat @ A3 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_2430_group__cancel_Oneg1,axiom,
    ! [A3: code_integer,K: code_integer,A: code_integer] :
      ( ( A3
        = ( plus_p5714425477246183910nteger @ K @ A ) )
     => ( ( uminus1351360451143612070nteger @ A3 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_2431_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_2432_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_2433_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_2434_sgn__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( sgn_sgn_rat @ ( times_times_rat @ A @ B ) )
      = ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).

% sgn_mult
thf(fact_2435_sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( sgn_sgn_int @ ( times_times_int @ A @ B ) )
      = ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ).

% sgn_mult
thf(fact_2436_sgn__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( sgn_sgn_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ B ) ) ) ).

% sgn_mult
thf(fact_2437_add_Oabel__semigroup__axioms,axiom,
    abel_s8772020223019846471nteger @ plus_p5714425477246183910nteger ).

% add.abel_semigroup_axioms
thf(fact_2438_add_Oabel__semigroup__axioms,axiom,
    abel_semigroup_rat @ plus_plus_rat ).

% add.abel_semigroup_axioms
thf(fact_2439_add_Oabel__semigroup__axioms,axiom,
    abel_semigroup_nat @ plus_plus_nat ).

% add.abel_semigroup_axioms
thf(fact_2440_add_Oabel__semigroup__axioms,axiom,
    abel_semigroup_int @ plus_plus_int ).

% add.abel_semigroup_axioms
thf(fact_2441_add_Osemigroup__axioms,axiom,
    semigr7036097719740861967nteger @ plus_p5714425477246183910nteger ).

% add.semigroup_axioms
thf(fact_2442_add_Osemigroup__axioms,axiom,
    semigroup_rat @ plus_plus_rat ).

% add.semigroup_axioms
thf(fact_2443_add_Osemigroup__axioms,axiom,
    semigroup_nat @ plus_plus_nat ).

% add.semigroup_axioms
thf(fact_2444_add_Osemigroup__axioms,axiom,
    semigroup_int @ plus_plus_int ).

% add.semigroup_axioms
thf(fact_2445_add_Oac__operator__axioms,axiom,
    syntax414191934074104506nteger @ plus_p5714425477246183910nteger ).

% add.ac_operator_axioms
thf(fact_2446_add_Oac__operator__axioms,axiom,
    syntax8165409007564224565or_rat @ plus_plus_rat ).

% add.ac_operator_axioms
thf(fact_2447_add_Oac__operator__axioms,axiom,
    syntax8800539067650720301or_nat @ plus_plus_nat ).

% add.ac_operator_axioms
thf(fact_2448_add_Oac__operator__axioms,axiom,
    syntax8798048597141670025or_int @ plus_plus_int ).

% add.ac_operator_axioms
thf(fact_2449_add_Osafe__commute,axiom,
    ! [X: code_integer,Y: code_integer,A: code_integer,B: code_integer] :
      ( ( syntax380677951125133566nteger @ ( plus_p5714425477246183910nteger @ X @ Y ) @ A )
     => ( ( plus_p5714425477246183910nteger @ A @ B )
        = ( plus_p5714425477246183910nteger @ B @ A ) ) ) ).

% add.safe_commute
thf(fact_2450_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_2451_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_2452_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_2453_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_2454_add__decreasing,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ zero_z2226904508553997617atural )
     => ( ( ord_le1926595141338095240atural @ C @ B )
       => ( ord_le1926595141338095240atural @ ( plus_p4538020629002901425atural @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_2455_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_2456_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_2457_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_2458_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_2459_add__increasing,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le1926595141338095240atural @ B @ C )
       => ( ord_le1926595141338095240atural @ B @ ( plus_p4538020629002901425atural @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_2460_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_2461_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_2462_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_2463_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_2464_add__decreasing2,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ C @ zero_z2226904508553997617atural )
     => ( ( ord_le1926595141338095240atural @ A @ B )
       => ( ord_le1926595141338095240atural @ ( plus_p4538020629002901425atural @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_2465_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_2466_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_2467_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_2468_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_2469_add__increasing2,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ C )
     => ( ( ord_le1926595141338095240atural @ B @ A )
       => ( ord_le1926595141338095240atural @ B @ ( plus_p4538020629002901425atural @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_2470_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_2471_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_2472_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_2473_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_2474_add__nonneg__nonneg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
       => ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2475_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_2476_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_2477_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_2478_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_2479_add__nonpos__nonpos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ zero_z2226904508553997617atural )
     => ( ( ord_le1926595141338095240atural @ B @ zero_z2226904508553997617atural )
       => ( ord_le1926595141338095240atural @ ( plus_p4538020629002901425atural @ A @ B ) @ zero_z2226904508553997617atural ) ) ) ).

% add_nonpos_nonpos
thf(fact_2480_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_2481_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_2482_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_2483_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_2484_add__nonneg__eq__0__iff,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ X )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ Y )
       => ( ( ( plus_p4538020629002901425atural @ X @ Y )
            = zero_z2226904508553997617atural )
          = ( ( X = zero_z2226904508553997617atural )
            & ( Y = zero_z2226904508553997617atural ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2485_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_2486_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_2487_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_2488_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_2489_add__nonpos__eq__0__iff,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( ( ord_le1926595141338095240atural @ X @ zero_z2226904508553997617atural )
     => ( ( ord_le1926595141338095240atural @ Y @ zero_z2226904508553997617atural )
       => ( ( ( plus_p4538020629002901425atural @ X @ Y )
            = zero_z2226904508553997617atural )
          = ( ( X = zero_z2226904508553997617atural )
            & ( Y = zero_z2226904508553997617atural ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2490_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_2491_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_2492_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_2493_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ I2 @ J )
        & ( ord_le6747313008572928689nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2494_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I2 @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2495_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2496_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I2 @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2497_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( ord_le6747313008572928689nteger @ I2 @ J )
        & ( ord_le3102999989581377725nteger @ K @ L ) )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ I2 @ K ) @ ( plus_p5714425477246183910nteger @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2498_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I2 @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2499_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2500_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I2 @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2501_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_2502_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_2503_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_2504_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_2505_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_2506_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_2507_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_2508_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_2509_add__neg__neg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ zero_z2226904508553997617atural )
     => ( ( ord_le5570908160329646204atural @ B @ zero_z2226904508553997617atural )
       => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A @ B ) @ zero_z2226904508553997617atural ) ) ) ).

% add_neg_neg
thf(fact_2510_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_2511_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_2512_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_2513_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_2514_add__pos__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
       => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_2515_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_2516_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_2517_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_2518_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_2519_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_2520_pos__add__strict,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le5570908160329646204atural @ B @ C )
       => ( ord_le5570908160329646204atural @ B @ ( plus_p4538020629002901425atural @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_2521_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_2522_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_2523_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_2524_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_2525_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_2526_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_2527_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_2528_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_2529_less__add__one,axiom,
    ! [A: code_integer] : ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) ) ).

% less_add_one
thf(fact_2530_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_2531_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_2532_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_2533_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_2534_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_2535_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_2536_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_2537_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_2538_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_2539_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_2540_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_2541_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_2542_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_2543_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_2544_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_2545_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_2546_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_2547_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_2548_sgn__minus__1,axiom,
    ( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% sgn_minus_1
thf(fact_2549_sgn__minus__1,axiom,
    ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% sgn_minus_1
thf(fact_2550_sgn__minus__1,axiom,
    ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% sgn_minus_1
thf(fact_2551_add_Ocomm__monoid__axioms,axiom,
    comm_monoid_nat @ plus_plus_nat @ zero_zero_nat ).

% add.comm_monoid_axioms
thf(fact_2552_add_Ocomm__monoid__axioms,axiom,
    comm_monoid_int @ plus_plus_int @ zero_zero_int ).

% add.comm_monoid_axioms
thf(fact_2553_add_Ocomm__monoid__axioms,axiom,
    comm_monoid_rat @ plus_plus_rat @ zero_zero_rat ).

% add.comm_monoid_axioms
thf(fact_2554_add_Ocomm__monoid__axioms,axiom,
    comm_m1654649897431166824nteger @ plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger ).

% add.comm_monoid_axioms
thf(fact_2555_add_Ocomm__monoid__axioms,axiom,
    comm_m478245049187884339atural @ plus_p4538020629002901425atural @ zero_z2226904508553997617atural ).

% add.comm_monoid_axioms
thf(fact_2556_add_Omonoid__axioms,axiom,
    monoid_nat @ plus_plus_nat @ zero_zero_nat ).

% add.monoid_axioms
thf(fact_2557_add_Omonoid__axioms,axiom,
    monoid_int @ plus_plus_int @ zero_zero_int ).

% add.monoid_axioms
thf(fact_2558_add_Omonoid__axioms,axiom,
    monoid_rat @ plus_plus_rat @ zero_zero_rat ).

% add.monoid_axioms
thf(fact_2559_add_Omonoid__axioms,axiom,
    monoid_Code_integer @ plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger ).

% add.monoid_axioms
thf(fact_2560_add_Omonoid__axioms,axiom,
    monoid_Code_natural @ plus_p4538020629002901425atural @ zero_z2226904508553997617atural ).

% add.monoid_axioms
thf(fact_2561_dbl__inc__def,axiom,
    ( neg_nu5831290666863070958nteger
    = ( ^ [X2: code_integer] : ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ X2 @ X2 ) @ one_one_Code_integer ) ) ) ).

% dbl_inc_def
thf(fact_2562_dbl__inc__def,axiom,
    ( neg_nu5219082963157363817nc_rat
    = ( ^ [X2: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).

% dbl_inc_def
thf(fact_2563_dbl__inc__def,axiom,
    ( neg_nu5851722552734809277nc_int
    = ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).

% dbl_inc_def
thf(fact_2564_add__neg__nonpos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ A @ zero_z2226904508553997617atural )
     => ( ( ord_le1926595141338095240atural @ B @ zero_z2226904508553997617atural )
       => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A @ B ) @ zero_z2226904508553997617atural ) ) ) ).

% add_neg_nonpos
thf(fact_2565_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_2566_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_2567_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_2568_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_2569_add__nonneg__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
       => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_2570_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_2571_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_2572_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_2573_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_2574_add__nonpos__neg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le1926595141338095240atural @ A @ zero_z2226904508553997617atural )
     => ( ( ord_le5570908160329646204atural @ B @ zero_z2226904508553997617atural )
       => ( ord_le5570908160329646204atural @ ( plus_p4538020629002901425atural @ A @ B ) @ zero_z2226904508553997617atural ) ) ) ).

% add_nonpos_neg
thf(fact_2575_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_2576_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_2577_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_2578_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_2579_add__pos__nonneg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ B )
       => ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_2580_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_2581_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_2582_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_2583_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_2584_add__strict__increasing,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le1926595141338095240atural @ B @ C )
       => ( ord_le5570908160329646204atural @ B @ ( plus_p4538020629002901425atural @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2585_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_2586_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_2587_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_2588_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_2589_add__strict__increasing2,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le5570908160329646204atural @ B @ C )
       => ( ord_le5570908160329646204atural @ B @ ( plus_p4538020629002901425atural @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2590_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_2591_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_2592_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_2593_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_2594_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_2595_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_2596_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_2597_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_2598_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_2599_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_2600_zero__less__two,axiom,
    ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ one_one_Code_natural ) ).

% zero_less_two
thf(fact_2601_zero__less__two,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer ) ).

% zero_less_two
thf(fact_2602_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_2603_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_2604_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_2605_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_2606_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_2607_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_2608_convex__bound__le,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ A )
     => ( ( ord_le3102999989581377725nteger @ Y @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V2 )
           => ( ( ( plus_p5714425477246183910nteger @ U @ V2 )
                = one_one_Code_integer )
             => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2609_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V2: 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 @ V2 )
           => ( ( ( plus_plus_rat @ U @ V2 )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2610_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V2: 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 @ V2 )
           => ( ( ( plus_plus_int @ U @ V2 )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2611_sgn__if,axiom,
    ( sgn_sgn_int
    = ( ^ [X2: int] : ( if_int @ ( X2 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X2 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% sgn_if
thf(fact_2612_sgn__if,axiom,
    ( sgn_sgn_rat
    = ( ^ [X2: rat] : ( if_rat @ ( X2 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X2 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_if
thf(fact_2613_sgn__if,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [X2: code_integer] : ( if_Code_integer @ ( X2 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X2 ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).

% sgn_if
thf(fact_2614_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_2615_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_2616_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_2617_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_2618_discrete,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A2: code_integer] : ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A2 @ one_one_Code_integer ) ) ) ) ).

% discrete
thf(fact_2619_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_2620_discrete,axiom,
    ( ord_less_int
    = ( ^ [A2: int] : ( ord_less_eq_int @ ( plus_plus_int @ A2 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_2621_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_2622_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_2623_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_2624_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_2625_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_2626_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_2627_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_list679713369942834879n_assn @ N @ nil_Pr5671120429643327159n_assn )
          = ( cons_l2423627976422276333n_assn @ nil_Pr5671120429643327159n_assn @ nil_li5476096274760905021n_assn ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_list679713369942834879n_assn @ N @ nil_Pr5671120429643327159n_assn )
          = nil_li5476096274760905021n_assn ) ) ) ).

% n_lists_Nil
thf(fact_2628_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_lists_int @ N @ nil_int )
          = ( cons_list_int @ nil_int @ nil_list_int ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_lists_int @ N @ nil_int )
          = nil_list_int ) ) ) ).

% n_lists_Nil
thf(fact_2629_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_lists_nat @ N @ nil_nat )
          = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_lists_nat @ N @ nil_nat )
          = nil_list_nat ) ) ) ).

% n_lists_Nil
thf(fact_2630_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_2631_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_2632_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_2633_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_2634_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_2635_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_2636_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_2637_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( n_list679713369942834879n_assn @ zero_zero_nat @ Xs )
      = ( cons_l2423627976422276333n_assn @ nil_Pr5671120429643327159n_assn @ nil_li5476096274760905021n_assn ) ) ).

% n_lists.simps(1)
thf(fact_2638_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_int] :
      ( ( n_lists_int @ zero_zero_nat @ Xs )
      = ( cons_list_int @ nil_int @ nil_list_int ) ) ).

% n_lists.simps(1)
thf(fact_2639_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_nat] :
      ( ( n_lists_nat @ zero_zero_nat @ Xs )
      = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% n_lists.simps(1)
thf(fact_2640_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_2641_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_2642_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_2643_crossproduct__noteq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ( A != B )
        & ( C != D2 ) )
      = ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) )
       != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ D2 ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2644_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_2645_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_2646_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_2647_crossproduct__eq,axiom,
    ! [W: code_integer,Y: code_integer,X: code_integer,Z: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ W @ Y ) @ ( times_3573771949741848930nteger @ X @ Z ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ W @ Z ) @ ( times_3573771949741848930nteger @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_2648_nat__1__eq__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M2 @ N ) )
      = ( ( M2 = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_2649_nat__mult__eq__1__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( times_times_nat @ M2 @ N )
        = one_one_nat )
      = ( ( M2 = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_2650_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_2651_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_2652_mult__eq__self__implies__10,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2
        = ( times_times_nat @ M2 @ N ) )
     => ( ( N = one_one_nat )
        | ( M2 = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_2653_product__lists_Osimps_I1_J,axiom,
    ( ( produc1566369527784499744n_assn @ nil_li5476096274760905021n_assn )
    = ( cons_l2423627976422276333n_assn @ nil_Pr5671120429643327159n_assn @ nil_li5476096274760905021n_assn ) ) ).

% product_lists.simps(1)
thf(fact_2654_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_int @ nil_list_int )
    = ( cons_list_int @ nil_int @ nil_list_int ) ) ).

% product_lists.simps(1)
thf(fact_2655_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_nat @ nil_list_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% product_lists.simps(1)
thf(fact_2656_power__decreasing__iff,axiom,
    ! [B: code_natural,M2: nat,N: nat] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
     => ( ( ord_le5570908160329646204atural @ B @ one_one_Code_natural )
       => ( ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ B @ M2 ) @ ( power_7079662738309270450atural @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_2657_power__decreasing__iff,axiom,
    ! [B: code_integer,M2: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
       => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ M2 ) @ ( power_8256067586552552935nteger @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_2658_power__decreasing__iff,axiom,
    ! [B: rat,M2: nat,N: 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 @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_2659_power__decreasing__iff,axiom,
    ! [B: nat,M2: nat,N: 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 @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_2660_power__decreasing__iff,axiom,
    ! [B: int,M2: nat,N: 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 @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_2661_subseqs_Osimps_I1_J,axiom,
    ( ( subseq184808802919281286n_assn @ nil_Pr5671120429643327159n_assn )
    = ( cons_l2423627976422276333n_assn @ nil_Pr5671120429643327159n_assn @ nil_li5476096274760905021n_assn ) ) ).

% subseqs.simps(1)
thf(fact_2662_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_int @ nil_int )
    = ( cons_list_int @ nil_int @ nil_list_int ) ) ).

% subseqs.simps(1)
thf(fact_2663_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_nat @ nil_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% subseqs.simps(1)
thf(fact_2664_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_2665_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_2666_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_2667_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_2668_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_2669_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_2670_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_2671_div__by__1,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ one_one_Code_integer )
      = A ) ).

% div_by_1
thf(fact_2672_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_2673_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_2674_div__by__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ one_one_rat )
      = A ) ).

% div_by_1
thf(fact_2675_div__by__1,axiom,
    ! [A: code_natural] :
      ( ( divide5121882707175180666atural @ A @ one_one_Code_natural )
      = A ) ).

% div_by_1
thf(fact_2676_power__one,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ one_one_Code_integer @ N )
      = one_one_Code_integer ) ).

% power_one
thf(fact_2677_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_assn @ one_one_assn @ N )
      = one_one_assn ) ).

% power_one
thf(fact_2678_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ one_one_rat @ N )
      = one_one_rat ) ).

% power_one
thf(fact_2679_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_int @ one_one_int @ N )
      = one_one_int ) ).

% power_one
thf(fact_2680_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ one_one_nat @ N )
      = one_one_nat ) ).

% power_one
thf(fact_2681_power__one__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_2682_power__one__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_2683_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_2684_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_2685_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_2686_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_2687_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( C = zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
          = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
          = ( divide_divide_rat @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_2688_nonzero__mult__div__cancel__right,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2689_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2690_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2691_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2692_nonzero__mult__div__cancel__right,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2693_nonzero__mult__div__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2694_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2695_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2696_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2697_nonzero__mult__div__cancel__left,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2698_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_2699_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_2700_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_2701_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_2702_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_2703_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% divide_self
thf(fact_2704_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_2705_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_2706_div__self,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ A @ A )
        = one_one_Code_integer ) ) ).

% div_self
thf(fact_2707_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_2708_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_2709_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% div_self
thf(fact_2710_div__self,axiom,
    ! [A: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ A @ A )
        = one_one_Code_natural ) ) ).

% div_self
thf(fact_2711_power__inject__exp,axiom,
    ! [A: code_integer,M2: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ( power_8256067586552552935nteger @ A @ M2 )
          = ( power_8256067586552552935nteger @ A @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_2712_power__inject__exp,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ( power_power_rat @ A @ M2 )
          = ( power_power_rat @ A @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_2713_power__inject__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M2 )
          = ( power_power_nat @ A @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_2714_power__inject__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M2 )
          = ( power_power_int @ A @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_2715_divide__minus1,axiom,
    ! [X: rat] :
      ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X ) ) ).

% divide_minus1
thf(fact_2716_divide__sgn,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat @ A @ ( sgn_sgn_rat @ B ) )
      = ( times_times_rat @ A @ ( sgn_sgn_rat @ B ) ) ) ).

% divide_sgn
thf(fact_2717_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_2718_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_2719_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_2720_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_2721_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_2722_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_2723_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_2724_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_2725_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_2726_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_2727_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_2728_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_2729_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_2730_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_2731_left__minus__one__mult__self,axiom,
    ! [N: nat,A: int] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_2732_left__minus__one__mult__self,axiom,
    ! [N: nat,A: rat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_2733_left__minus__one__mult__self,axiom,
    ! [N: nat,A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_2734_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
      = one_one_int ) ).

% minus_one_mult_self
thf(fact_2735_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
      = one_one_rat ) ).

% minus_one_mult_self
thf(fact_2736_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
      = one_one_Code_integer ) ).

% minus_one_mult_self
thf(fact_2737_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_2738_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_2739_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_2740_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_2741_power__strict__decreasing__iff,axiom,
    ! [B: code_natural,M2: nat,N: nat] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ B )
     => ( ( ord_le5570908160329646204atural @ B @ one_one_Code_natural )
       => ( ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ B @ M2 ) @ ( power_7079662738309270450atural @ B @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2742_power__strict__decreasing__iff,axiom,
    ! [B: code_integer,M2: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
       => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ M2 ) @ ( power_8256067586552552935nteger @ B @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2743_power__strict__decreasing__iff,axiom,
    ! [B: rat,M2: nat,N: 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 @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2744_power__strict__decreasing__iff,axiom,
    ! [B: nat,M2: nat,N: 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 @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2745_power__strict__decreasing__iff,axiom,
    ! [B: int,M2: nat,N: 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 @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2746_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_2747_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_2748_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_2749_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_2750_power__one__over,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
      = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).

% power_one_over
thf(fact_2751_power__add,axiom,
    ! [A: assn,M2: nat,N: nat] :
      ( ( power_power_assn @ A @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_assn @ ( power_power_assn @ A @ M2 ) @ ( power_power_assn @ A @ N ) ) ) ).

% power_add
thf(fact_2752_power__add,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) ) ) ).

% power_add
thf(fact_2753_power__add,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( power_power_int @ A @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) ) ) ).

% power_add
thf(fact_2754_power__add,axiom,
    ! [A: code_integer,M2: nat,N: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ M2 ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_add
thf(fact_2755_power__add,axiom,
    ! [A: code_natural,M2: nat,N: nat] :
      ( ( power_7079662738309270450atural @ A @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_2397367101498566445atural @ ( power_7079662738309270450atural @ A @ M2 ) @ ( power_7079662738309270450atural @ A @ N ) ) ) ).

% power_add
thf(fact_2756_power__commuting__commutes,axiom,
    ! [X: assn,Y: assn,N: nat] :
      ( ( ( times_times_assn @ X @ Y )
        = ( times_times_assn @ Y @ X ) )
     => ( ( times_times_assn @ ( power_power_assn @ X @ N ) @ Y )
        = ( times_times_assn @ Y @ ( power_power_assn @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_2757_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = ( times_times_nat @ Y @ X ) )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
        = ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_2758_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X @ Y )
        = ( times_times_int @ Y @ X ) )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
        = ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_2759_power__commuting__commutes,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y )
        = ( times_3573771949741848930nteger @ Y @ X ) )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N ) @ Y )
        = ( times_3573771949741848930nteger @ Y @ ( power_8256067586552552935nteger @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_2760_power__commuting__commutes,axiom,
    ! [X: code_natural,Y: code_natural,N: nat] :
      ( ( ( times_2397367101498566445atural @ X @ Y )
        = ( times_2397367101498566445atural @ Y @ X ) )
     => ( ( times_2397367101498566445atural @ ( power_7079662738309270450atural @ X @ N ) @ Y )
        = ( times_2397367101498566445atural @ Y @ ( power_7079662738309270450atural @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_2761_power__mult__distrib,axiom,
    ! [A: assn,B: assn,N: nat] :
      ( ( power_power_assn @ ( times_times_assn @ A @ B ) @ N )
      = ( times_times_assn @ ( power_power_assn @ A @ N ) @ ( power_power_assn @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_2762_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_2763_power__mult__distrib,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
      = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_2764_power__mult__distrib,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( times_3573771949741848930nteger @ A @ B ) @ N )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_2765_power__mult__distrib,axiom,
    ! [A: code_natural,B: code_natural,N: nat] :
      ( ( power_7079662738309270450atural @ ( times_2397367101498566445atural @ A @ B ) @ N )
      = ( times_2397367101498566445atural @ ( power_7079662738309270450atural @ A @ N ) @ ( power_7079662738309270450atural @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_2766_power__commutes,axiom,
    ! [A: assn,N: nat] :
      ( ( times_times_assn @ ( power_power_assn @ A @ N ) @ A )
      = ( times_times_assn @ A @ ( power_power_assn @ A @ N ) ) ) ).

% power_commutes
thf(fact_2767_power__commutes,axiom,
    ! [A: nat,N: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).

% power_commutes
thf(fact_2768_power__commutes,axiom,
    ! [A: int,N: nat] :
      ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
      = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).

% power_commutes
thf(fact_2769_power__commutes,axiom,
    ! [A: code_integer,N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ N ) @ A )
      = ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_commutes
thf(fact_2770_power__commutes,axiom,
    ! [A: code_natural,N: nat] :
      ( ( times_2397367101498566445atural @ ( power_7079662738309270450atural @ A @ N ) @ A )
      = ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ A @ N ) ) ) ).

% power_commutes
thf(fact_2771_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_2772_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_2773_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_2774_one__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% one_le_power
thf(fact_2775_one__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).

% one_le_power
thf(fact_2776_one__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).

% one_le_power
thf(fact_2777_one__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).

% one_le_power
thf(fact_2778_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_2779_left__right__inverse__power,axiom,
    ! [X: assn,Y: assn,N: nat] :
      ( ( ( times_times_assn @ X @ Y )
        = one_one_assn )
     => ( ( times_times_assn @ ( power_power_assn @ X @ N ) @ ( power_power_assn @ Y @ N ) )
        = one_one_assn ) ) ).

% left_right_inverse_power
thf(fact_2780_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_2781_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X @ Y )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_2782_left__right__inverse__power,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y )
        = one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ N ) )
        = one_one_Code_integer ) ) ).

% left_right_inverse_power
thf(fact_2783_left__right__inverse__power,axiom,
    ! [X: code_natural,Y: code_natural,N: nat] :
      ( ( ( times_2397367101498566445atural @ X @ Y )
        = one_one_Code_natural )
     => ( ( times_2397367101498566445atural @ ( power_7079662738309270450atural @ X @ N ) @ ( power_7079662738309270450atural @ Y @ N ) )
        = one_one_Code_natural ) ) ).

% left_right_inverse_power
thf(fact_2784_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A
          = ( divide_divide_rat @ B @ C ) )
        = ( ( times_times_rat @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2785_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( divide_divide_rat @ B @ C )
          = A )
        = ( B
          = ( times_times_rat @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2786_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = B )
       => ( A
          = ( divide_divide_rat @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2787_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B
          = ( times_times_rat @ A @ C ) )
       => ( ( divide_divide_rat @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_2788_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
thf(fact_2789_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ( divide_divide_rat @ B @ C )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
thf(fact_2790_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X @ Y )
            = ( divide_divide_rat @ W @ Z ) )
          = ( ( times_times_rat @ X @ Z )
            = ( times_times_rat @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2791_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_2792_power__0,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ zero_zero_nat )
      = one_one_Code_integer ) ).

% power_0
thf(fact_2793_power__0,axiom,
    ! [A: assn] :
      ( ( power_power_assn @ A @ zero_zero_nat )
      = one_one_assn ) ).

% power_0
thf(fact_2794_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_2795_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_2796_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_2797_power__le__one,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer ) ) ) ).

% power_le_one
thf(fact_2798_power__le__one,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le1926595141338095240atural @ A @ one_one_Code_natural )
       => ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A @ N ) @ one_one_Code_natural ) ) ) ).

% power_le_one
thf(fact_2799_power__le__one,axiom,
    ! [A: rat,N: 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 @ N ) @ one_one_rat ) ) ) ).

% power_le_one
thf(fact_2800_power__le__one,axiom,
    ! [A: nat,N: 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 @ N ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_2801_power__le__one,axiom,
    ! [A: int,N: 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 @ N ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_2802_power__less__power__Suc,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A )
     => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A @ N ) @ ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_2803_power__less__power__Suc,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_2804_power__less__power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_2805_power__less__power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_2806_power__less__power__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_2807_power__gt1__lemma,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ A )
     => ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_2808_power__gt1__lemma,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_2809_power__gt1__lemma,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_2810_power__gt1__lemma,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_2811_power__gt1__lemma,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_2812_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2813_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2814_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2815_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_2816_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_2817_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y ) @ X )
       => ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_2818_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_2819_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% pos_less_divide_eq
thf(fact_2820_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_2821_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_2822_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% neg_divide_less_eq
thf(fact_2823_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_2824_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_2825_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N )
          = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N )
          = zero_zero_rat ) ) ) ).

% power_0_left
thf(fact_2826_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N )
          = one_one_Code_integer ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N )
          = zero_z3403309356797280102nteger ) ) ) ).

% power_0_left
thf(fact_2827_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_7079662738309270450atural @ zero_z2226904508553997617atural @ N )
          = one_one_Code_natural ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_7079662738309270450atural @ zero_z2226904508553997617atural @ N )
          = zero_z2226904508553997617atural ) ) ) ).

% power_0_left
thf(fact_2828_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = zero_zero_int ) ) ) ).

% power_0_left
thf(fact_2829_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = zero_zero_nat ) ) ) ).

% power_0_left
thf(fact_2830_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_2831_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_2832_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_2833_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_2834_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_2835_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_2836_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_2837_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_2838_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_2839_power__strict__increasing,axiom,
    ! [N: nat,N2: nat,A: code_integer] :
      ( ( ord_less_nat @ N @ N2 )
     => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ).

% power_strict_increasing
thf(fact_2840_power__strict__increasing,axiom,
    ! [N: nat,N2: nat,A: rat] :
      ( ( ord_less_nat @ N @ N2 )
     => ( ( ord_less_rat @ one_one_rat @ A )
       => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_strict_increasing
thf(fact_2841_power__strict__increasing,axiom,
    ! [N: nat,N2: nat,A: nat] :
      ( ( ord_less_nat @ N @ N2 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_strict_increasing
thf(fact_2842_power__strict__increasing,axiom,
    ! [N: nat,N2: nat,A: int] :
      ( ( ord_less_nat @ N @ N2 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_strict_increasing
thf(fact_2843_power__less__imp__less__exp,axiom,
    ! [A: code_integer,M2: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ M2 ) @ ( power_8256067586552552935nteger @ A @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_2844_power__less__imp__less__exp,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_2845_power__less__imp__less__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_2846_power__less__imp__less__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_2847_power__minus,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).

% power_minus
thf(fact_2848_power__minus,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).

% power_minus
thf(fact_2849_power__minus,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_minus
thf(fact_2850_power__increasing,axiom,
    ! [N: nat,N2: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ N @ N2 )
     => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ).

% power_increasing
thf(fact_2851_power__increasing,axiom,
    ! [N: nat,N2: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N2 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_increasing
thf(fact_2852_power__increasing,axiom,
    ! [N: nat,N2: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N2 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).

% power_increasing
thf(fact_2853_power__increasing,axiom,
    ! [N: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N2 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_increasing
thf(fact_2854_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_2855_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_2856_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C
          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
        = ( ( times_times_rat @ C @ B )
          = ( uminus_uminus_rat @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_2857_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
          = C )
        = ( ( uminus_uminus_rat @ A )
          = ( times_times_rat @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_2858_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( ( uminus_uminus_rat @ B )
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_2859_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A
        = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = ( uminus_uminus_rat @ B ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_2860_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_2861_power__Suc__less,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le5570908160329646204atural @ A @ one_one_Code_natural )
       => ( ord_le5570908160329646204atural @ ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ A @ N ) ) @ ( power_7079662738309270450atural @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2862_power__Suc__less,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2863_power__Suc__less,axiom,
    ! [A: rat,N: 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 @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2864_power__Suc__less,axiom,
    ! [A: nat,N: 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 @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2865_power__Suc__less,axiom,
    ! [A: int,N: 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 @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2866_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_2867_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y ) @ X )
       => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_2868_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_2869_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_2870_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_2871_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_2872_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_2873_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_2874_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_2875_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_2876_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_2877_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_2878_power__strict__decreasing,axiom,
    ! [N: nat,N2: nat,A: code_natural] :
      ( ( ord_less_nat @ N @ N2 )
     => ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
       => ( ( ord_le5570908160329646204atural @ A @ one_one_Code_natural )
         => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A @ N2 ) @ ( power_7079662738309270450atural @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2879_power__strict__decreasing,axiom,
    ! [N: nat,N2: nat,A: code_integer] :
      ( ( ord_less_nat @ N @ N2 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
         => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2880_power__strict__decreasing,axiom,
    ! [N: nat,N2: nat,A: rat] :
      ( ( ord_less_nat @ N @ N2 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ( ord_less_rat @ A @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2881_power__strict__decreasing,axiom,
    ! [N: nat,N2: nat,A: nat] :
      ( ( ord_less_nat @ N @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2882_power__strict__decreasing,axiom,
    ! [N: nat,N2: nat,A: int] :
      ( ( ord_less_nat @ N @ N2 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2883_power__decreasing,axiom,
    ! [N: nat,N2: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ N @ N2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2884_power__decreasing,axiom,
    ! [N: nat,N2: nat,A: code_natural] :
      ( ( ord_less_eq_nat @ N @ N2 )
     => ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
       => ( ( ord_le1926595141338095240atural @ A @ one_one_Code_natural )
         => ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A @ N2 ) @ ( power_7079662738309270450atural @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2885_power__decreasing,axiom,
    ! [N: nat,N2: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N2 )
     => ( ( 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 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2886_power__decreasing,axiom,
    ! [N: nat,N2: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N2 )
     => ( ( 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 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2887_power__decreasing,axiom,
    ! [N: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N2 )
     => ( ( 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 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2888_power__le__imp__le__exp,axiom,
    ! [A: code_integer,M2: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ M2 ) @ ( power_8256067586552552935nteger @ A @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_2889_power__le__imp__le__exp,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_2890_power__le__imp__le__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_2891_power__le__imp__le__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_2892_self__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_le3102999989581377725nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_2893_self__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_2894_self__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_2895_self__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_2896_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_2897_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_2898_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_2899_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_2900_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_2901_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_2902_one__less__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_2903_one__less__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_2904_one__less__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_2905_one__less__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_2906_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_2907_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_2908_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_2909_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_2910_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_2911_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_2912_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_2913_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_2914_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_2915_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_2916_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_2917_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_2918_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_2919_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_2920_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_2921_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_2922_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_2923_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_2924_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_2925_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_2926_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_2927_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_2928_div__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% div_minus1_right
thf(fact_2929_div__mult__mult1__if,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ( C = zero_z3403309356797280102nteger )
       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
          = zero_z3403309356797280102nteger ) )
      & ( ( C != zero_z3403309356797280102nteger )
       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_2930_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( C = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
          = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_2931_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( C = zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
          = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_2932_div__mult__mult1__if,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( ( C = zero_z2226904508553997617atural )
       => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) )
          = zero_z2226904508553997617atural ) )
      & ( ( C != zero_z2226904508553997617atural )
       => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) )
          = ( divide5121882707175180666atural @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_2933_div__mult__mult2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_2934_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_2935_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_2936_div__mult__mult2,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) )
        = ( divide5121882707175180666atural @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_2937_div__mult__mult1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_2938_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_2939_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_2940_div__mult__mult1,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) )
        = ( divide5121882707175180666atural @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_2941_bits__div__by__1,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ one_one_Code_integer )
      = A ) ).

% bits_div_by_1
thf(fact_2942_bits__div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% bits_div_by_1
thf(fact_2943_bits__div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% bits_div_by_1
thf(fact_2944_bits__div__by__1,axiom,
    ! [A: code_natural] :
      ( ( divide5121882707175180666atural @ A @ one_one_Code_natural )
      = A ) ).

% bits_div_by_1
thf(fact_2945_div__less__dividend,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ) ) ).

% div_less_dividend
thf(fact_2946_div__eq__dividend__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ( divide_divide_nat @ M2 @ N )
          = M2 )
        = ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_2947_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_2948_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_2949_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_2950_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_2951_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_2952_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_2953_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_2954_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_2955_power__minus_H,axiom,
    ! [X: int,N: nat] :
      ( ( nO_MATCH_int_int @ one_one_int @ X )
     => ( ( power_power_int @ ( uminus_uminus_int @ X ) @ N )
        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ X @ N ) ) ) ) ).

% power_minus'
thf(fact_2956_power__minus_H,axiom,
    ! [X: rat,N: nat] :
      ( ( nO_MATCH_rat_rat @ one_one_rat @ X )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ N )
        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ X @ N ) ) ) ) ).

% power_minus'
thf(fact_2957_power__minus_H,axiom,
    ! [X: code_integer,N: nat] :
      ( ( nO_MAT8252062027627875367nteger @ one_one_Code_integer @ X )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ N )
        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ X @ N ) ) ) ) ).

% power_minus'
thf(fact_2958_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_2959_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_2960_count__list_Osimps_I2_J,axiom,
    ! [X: int,Y: int,Xs: list_int] :
      ( ( ( X = Y )
       => ( ( count_list_int @ ( cons_int @ X @ Xs ) @ Y )
          = ( plus_plus_nat @ ( count_list_int @ Xs @ Y ) @ one_one_nat ) ) )
      & ( ( X != Y )
       => ( ( count_list_int @ ( cons_int @ X @ Xs ) @ Y )
          = ( count_list_int @ Xs @ Y ) ) ) ) ).

% count_list.simps(2)
thf(fact_2961_count__list_Osimps_I2_J,axiom,
    ! [X: nat,Y: nat,Xs: list_nat] :
      ( ( ( X = Y )
       => ( ( count_list_nat @ ( cons_nat @ X @ Xs ) @ Y )
          = ( plus_plus_nat @ ( count_list_nat @ Xs @ Y ) @ one_one_nat ) ) )
      & ( ( X != Y )
       => ( ( count_list_nat @ ( cons_nat @ X @ Xs ) @ Y )
          = ( count_list_nat @ Xs @ Y ) ) ) ) ).

% count_list.simps(2)
thf(fact_2962_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_2963_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_2964_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_2965_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_2966_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: code_natural,A: code_natural,B: code_natural] :
      ( ( nO_MAT7518554365073749349atural @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
        = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_2967_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_2968_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_2969_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_2970_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_2971_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: code_natural,A: code_natural,B: code_natural] :
      ( ( nO_MAT7890207925310593985atural @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ A @ B ) @ C )
        = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) ) ) ) ).

% distrib_right_NO_MATCH
thf(fact_2972_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_2973_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_2974_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_2975_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_2976_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: code_natural,B: code_natural,C: code_natural] :
      ( ( nO_MAT7518554365073749349atural @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_2397367101498566445atural @ A @ ( plus_p4538020629002901425atural @ B @ C ) )
        = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ B ) @ ( times_2397367101498566445atural @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_2977_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_2978_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_2979_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_2980_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_2981_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: code_natural,B: code_natural,C: code_natural] :
      ( ( nO_MAT7890207925310593985atural @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_2397367101498566445atural @ A @ ( plus_p4538020629002901425atural @ B @ C ) )
        = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ B ) @ ( times_2397367101498566445atural @ A @ C ) ) ) ) ).

% distrib_left_NO_MATCH
thf(fact_2982_power__minus__mult,axiom,
    ! [N: nat,A: assn] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_assn @ ( power_power_assn @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_assn @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_2983_power__minus__mult,axiom,
    ! [N: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_nat @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_2984_power__minus__mult,axiom,
    ! [N: nat,A: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_int @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_2985_power__minus__mult,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_2986_power__minus__mult,axiom,
    ! [N: nat,A: code_natural] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_2397367101498566445atural @ ( power_7079662738309270450atural @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_7079662738309270450atural @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_2987_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_2988_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_2989_diff__self,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% diff_self
thf(fact_2990_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_2991_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_2992_diff__0__right,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% diff_0_right
thf(fact_2993_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_2994_zero__diff,axiom,
    ! [A: code_natural] :
      ( ( minus_7197305767214868737atural @ zero_z2226904508553997617atural @ A )
      = zero_z2226904508553997617atural ) ).

% zero_diff
thf(fact_2995_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_2996_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_2997_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_2998_diff__zero,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% diff_zero
thf(fact_2999_diff__zero,axiom,
    ! [A: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ zero_z2226904508553997617atural )
      = A ) ).

% diff_zero
thf(fact_3000_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_3001_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_3002_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_3003_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_3004_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ A )
      = zero_z2226904508553997617atural ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3005_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V2 @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3006_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V2 ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V2 @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3007_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V2 @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3008_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V2 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Z ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V2 @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3009_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V2 ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ W ) @ Z ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( times_times_num @ V2 @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3010_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_3011_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_3012_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_3013_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_3014_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ M2 ) @ ( numera5444537566228673987atural @ N ) )
      = ( numera5444537566228673987atural @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_3015_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_3016_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_3017_add__diff__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_3018_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_3019_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_3020_diff__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_3021_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_3022_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_3023_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_3024_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_3025_add__diff__cancel__left,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ C @ A ) @ ( plus_p4538020629002901425atural @ C @ B ) )
      = ( minus_7197305767214868737atural @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_3026_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_3027_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_3028_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_3029_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_3030_add__diff__cancel__left_H,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_3031_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_3032_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_3033_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_3034_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_3035_add__diff__cancel__right,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ A @ C ) @ ( plus_p4538020629002901425atural @ B @ C ) )
      = ( minus_7197305767214868737atural @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_3036_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_3037_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_3038_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_3039_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_3040_add__diff__cancel__right_H,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ ( plus_p4538020629002901425atural @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_3041_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_3042_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_3043_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% minus_diff_eq
thf(fact_3044_power__add__numeral2,axiom,
    ! [A: assn,M2: num,N: num,B: assn] :
      ( ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_3045_power__add__numeral2,axiom,
    ! [A: nat,M2: num,N: num,B: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_3046_power__add__numeral2,axiom,
    ! [A: int,M2: num,N: num,B: int] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_3047_power__add__numeral2,axiom,
    ! [A: code_integer,M2: num,N: num,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_3048_power__add__numeral2,axiom,
    ! [A: code_natural,M2: num,N: num,B: code_natural] :
      ( ( times_2397367101498566445atural @ ( power_7079662738309270450atural @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_2397367101498566445atural @ ( power_7079662738309270450atural @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_2397367101498566445atural @ ( power_7079662738309270450atural @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_3049_power__add__numeral,axiom,
    ! [A: assn,M2: num,N: num] :
      ( ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_assn @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_assn @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_3050_power__add__numeral,axiom,
    ! [A: nat,M2: num,N: num] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_3051_power__add__numeral,axiom,
    ! [A: int,M2: num,N: num] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_3052_power__add__numeral,axiom,
    ! [A: code_integer,M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_3053_power__add__numeral,axiom,
    ! [A: code_natural,M2: num,N: num] :
      ( ( times_2397367101498566445atural @ ( power_7079662738309270450atural @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_7079662738309270450atural @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_7079662738309270450atural @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_3054_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_3055_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_3056_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_3057_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_3058_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_3059_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_3060_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_3061_diff__add__zero,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ ( plus_p4538020629002901425atural @ A @ B ) )
      = zero_z2226904508553997617atural ) ).

% diff_add_zero
thf(fact_3062_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_3063_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_3064_diff__numeral__special_I9_J,axiom,
    ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ one_one_Code_integer )
    = zero_z3403309356797280102nteger ) ).

% diff_numeral_special(9)
thf(fact_3065_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V2: num] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V2 ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V2 ) ) ) ) ).

% distrib_right_numeral
thf(fact_3066_distrib__right__numeral,axiom,
    ! [A: int,B: int,V2: num] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V2 ) )
      = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V2 ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V2 ) ) ) ) ).

% distrib_right_numeral
thf(fact_3067_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V2: num] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V2 ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V2 ) ) ) ) ).

% distrib_right_numeral
thf(fact_3068_distrib__right__numeral,axiom,
    ! [A: code_integer,B: code_integer,V2: num] :
      ( ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( numera6620942414471956472nteger @ V2 ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ V2 ) ) @ ( times_3573771949741848930nteger @ B @ ( numera6620942414471956472nteger @ V2 ) ) ) ) ).

% distrib_right_numeral
thf(fact_3069_distrib__right__numeral,axiom,
    ! [A: code_natural,B: code_natural,V2: num] :
      ( ( times_2397367101498566445atural @ ( plus_p4538020629002901425atural @ A @ B ) @ ( numera5444537566228673987atural @ V2 ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ ( numera5444537566228673987atural @ V2 ) ) @ ( times_2397367101498566445atural @ B @ ( numera5444537566228673987atural @ V2 ) ) ) ) ).

% distrib_right_numeral
thf(fact_3070_distrib__left__numeral,axiom,
    ! [V2: num,B: nat,C: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_3071_distrib__left__numeral,axiom,
    ! [V2: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V2 ) @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_3072_distrib__left__numeral,axiom,
    ! [V2: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_3073_distrib__left__numeral,axiom,
    ! [V2: num,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V2 ) @ ( plus_p5714425477246183910nteger @ B @ C ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V2 ) @ B ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V2 ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_3074_distrib__left__numeral,axiom,
    ! [V2: num,B: code_natural,C: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V2 ) @ ( plus_p4538020629002901425atural @ B @ C ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V2 ) @ B ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ V2 ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_3075_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V2 ) @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3076_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3077_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V2 ) @ ( minus_8373710615458151222nteger @ B @ C ) )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V2 ) @ B ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V2 ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3078_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V2: num] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V2 ) )
      = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V2 ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V2 ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_3079_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V2: num] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V2 ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V2 ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V2 ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_3080_left__diff__distrib__numeral,axiom,
    ! [A: code_integer,B: code_integer,V2: num] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ ( numera6620942414471956472nteger @ V2 ) )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ V2 ) ) @ ( times_3573771949741848930nteger @ B @ ( numera6620942414471956472nteger @ V2 ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_3081_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_3082_diff__0,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A )
      = ( uminus_uminus_rat @ A ) ) ).

% diff_0
thf(fact_3083_diff__0,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% diff_0
thf(fact_3084_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_3085_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_3086_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_3087_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_3088_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_3089_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_3090_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_3091_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_3092_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_3093_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_3094_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_3095_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_3096_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_3097_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_3098_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_3099_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_3100_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_3101_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
        = A )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_3102_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
            = B ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_3103_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_3104_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_3105_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_3106_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_3107_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_3108_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_3109_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_3110_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
        = A )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_3111_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
            = B ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_3112_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_3113_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_3114_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_3115_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_3116_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_3117_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_3118_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_3119_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ ( minus_7197305767214868737atural @ A @ C ) @ B )
      = ( minus_7197305767214868737atural @ ( minus_7197305767214868737atural @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_3120_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_3121_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_3122_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_3123_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_3124_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_3125_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_3126_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_3127_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_3128_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_3129_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_3130_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_3131_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_3132_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: rat,Z4: rat] : Y4 = Z4 )
    = ( ^ [A2: rat,B2: rat] :
          ( ( minus_minus_rat @ A2 @ B2 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3133_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: int,Z4: int] : Y4 = Z4 )
    = ( ^ [A2: int,B2: int] :
          ( ( minus_minus_int @ A2 @ B2 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3134_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: code_integer,Z4: code_integer] : Y4 = Z4 )
    = ( ^ [A2: code_integer,B2: code_integer] :
          ( ( minus_8373710615458151222nteger @ A2 @ B2 )
          = zero_z3403309356797280102nteger ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3135_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_3136_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_3137_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_3138_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_3139_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_3140_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_3141_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_3142_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_3143_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_3144_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_3145_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_3146_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_3147_group__cancel_Osub1,axiom,
    ! [A3: rat,K: rat,A: rat,B: rat] :
      ( ( A3
        = ( plus_plus_rat @ K @ A ) )
     => ( ( minus_minus_rat @ A3 @ B )
        = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_3148_group__cancel_Osub1,axiom,
    ! [A3: int,K: int,A: int,B: int] :
      ( ( A3
        = ( plus_plus_int @ K @ A ) )
     => ( ( minus_minus_int @ A3 @ B )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_3149_group__cancel_Osub1,axiom,
    ! [A3: code_integer,K: code_integer,A: code_integer,B: code_integer] :
      ( ( A3
        = ( plus_p5714425477246183910nteger @ K @ A ) )
     => ( ( minus_8373710615458151222nteger @ A3 @ B )
        = ( plus_p5714425477246183910nteger @ K @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_3150_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_3151_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_3152_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_3153_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_3154_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_3155_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_3156_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_3157_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_3158_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_3159_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_3160_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_3161_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_3162_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_3163_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_3164_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_3165_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_3166_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_3167_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_3168_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_3169_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_3170_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_3171_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_3172_add__implies__diff,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( ( plus_p4538020629002901425atural @ C @ B )
        = A )
     => ( C
        = ( minus_7197305767214868737atural @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_3173_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_3174_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_3175_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_3176_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_3177_diff__diff__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( minus_7197305767214868737atural @ ( minus_7197305767214868737atural @ A @ B ) @ C )
      = ( minus_7197305767214868737atural @ A @ ( plus_p4538020629002901425atural @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_3178_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
      = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_3179_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_3180_right__diff__distrib_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_3181_right__diff__distrib_H,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( times_2397367101498566445atural @ A @ ( minus_7197305767214868737atural @ B @ C ) )
      = ( minus_7197305767214868737atural @ ( times_2397367101498566445atural @ A @ B ) @ ( times_2397367101498566445atural @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_3182_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
      = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_3183_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] :
      ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
      = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_3184_left__diff__distrib_H,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ C ) @ A )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_3185_left__diff__distrib_H,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( times_2397367101498566445atural @ ( minus_7197305767214868737atural @ B @ C ) @ A )
      = ( minus_7197305767214868737atural @ ( times_2397367101498566445atural @ B @ A ) @ ( times_2397367101498566445atural @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_3186_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_3187_right__diff__distrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_3188_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_3189_left__diff__distrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
      = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_3190_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_3191_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_3192_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_3193_left__diff__distrib__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 @ ( minus_minus_int @ A @ B ) @ C )
        = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_3194_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y ) @ C )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_3195_left__diff__distrib__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 @ ( minus_minus_int @ A @ B ) @ C )
        = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_3196_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y ) @ C )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_3197_left__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,C: int,A: int,B: int] :
      ( ( nO_MATCH_rat_int @ ( divide_divide_rat @ X @ Y ) @ C )
     => ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
        = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_3198_left__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT243659986023775650nteger @ ( divide_divide_rat @ X @ Y ) @ C )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_3199_left__diff__distrib__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,C: int,A: int,B: int] :
      ( ( nO_MAT4469370267861770177al_int @ ( divide5121882707175180666atural @ X @ Y ) @ C )
     => ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
        = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_3200_left__diff__distrib__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,C: code_integer,A: code_integer,B: code_integer] :
      ( ( nO_MAT2693162092599435762nteger @ ( divide5121882707175180666atural @ X @ Y ) @ C )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) ) ) ) ).

% left_diff_distrib_NO_MATCH
thf(fact_3201_right__diff__distrib__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 @ ( minus_minus_int @ B @ C ) )
        = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_3202_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT8694959213317031834nteger @ ( divide_divide_nat @ X @ Y ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_3203_right__diff__distrib__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 @ ( minus_minus_int @ B @ C ) )
        = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_3204_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT9066612773553876470nteger @ ( divide_divide_int @ X @ Y ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_3205_right__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,A: int,B: int,C: int] :
      ( ( nO_MATCH_rat_int @ ( divide_divide_rat @ X @ Y ) @ A )
     => ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
        = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_3206_right__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT243659986023775650nteger @ ( divide_divide_rat @ X @ Y ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_3207_right__diff__distrib__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,A: int,B: int,C: int] :
      ( ( nO_MAT4469370267861770177al_int @ ( divide5121882707175180666atural @ X @ Y ) @ A )
     => ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
        = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_3208_right__diff__distrib__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,A: code_integer,B: code_integer,C: code_integer] :
      ( ( nO_MAT2693162092599435762nteger @ ( divide5121882707175180666atural @ X @ Y ) @ A )
     => ( ( times_3573771949741848930nteger @ A @ ( minus_8373710615458151222nteger @ B @ C ) )
        = ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% right_diff_distrib_NO_MATCH
thf(fact_3209_le__iff__diff__le__0,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A2: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% le_iff_diff_le_0
thf(fact_3210_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A2: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_3211_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_3212_less__iff__diff__less__0,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A2: code_integer,B2: code_integer] : ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ zero_z3403309356797280102nteger ) ) ) ).

% less_iff_diff_less_0
thf(fact_3213_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A2: rat,B2: rat] : ( ord_less_rat @ ( minus_minus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_3214_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_3215_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_3216_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_3217_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_3218_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_3219_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_3220_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_3221_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_3222_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_3223_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_3224_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_3225_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_3226_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_3227_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_3228_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_3229_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_3230_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_3231_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_3232_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_3233_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_3234_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_3235_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_3236_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_3237_eq__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_3238_eq__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_3239_eq__add__iff1,axiom,
    ! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E2 ) @ C )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
      = ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ E2 ) @ C )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_3240_eq__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
      = ( C
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_3241_eq__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_3242_eq__add__iff2,axiom,
    ! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E2 ) @ C )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
      = ( C
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ E2 ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_3243_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_3244_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_3245_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_3246_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3247_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A2: rat,B2: rat] : ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3248_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A2: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_3249_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A2: int,B2: int] : ( plus_plus_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3250_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A2: rat,B2: rat] : ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3251_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A2: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_3252_group__cancel_Osub2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3
        = ( plus_plus_int @ K @ B ) )
     => ( ( minus_minus_int @ A @ B3 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_3253_group__cancel_Osub2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3
        = ( plus_plus_rat @ K @ B ) )
     => ( ( minus_minus_rat @ A @ B3 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_3254_group__cancel_Osub2,axiom,
    ! [B3: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B3
        = ( plus_p5714425477246183910nteger @ K @ B ) )
     => ( ( minus_8373710615458151222nteger @ A @ B3 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_3255_diff__eq,axiom,
    ( minus_1356011639430497352at_nat
    = ( ^ [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ X2 @ ( uminus6524753893492686040at_nat @ Y2 ) ) ) ) ).

% diff_eq
thf(fact_3256_diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [X2: set_nat,Y2: set_nat] : ( inf_inf_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ Y2 ) ) ) ) ).

% diff_eq
thf(fact_3257_diff__eq,axiom,
    ( minus_711738161318947805_int_o
    = ( ^ [X2: product_prod_int_int > $o,Y2: product_prod_int_int > $o] : ( inf_in3604695632404883862_int_o @ X2 @ ( uminus7117520113953359693_int_o @ Y2 ) ) ) ) ).

% diff_eq
thf(fact_3258_diff__eq,axiom,
    ( minus_minus_assn
    = ( ^ [X2: assn,Y2: assn] : ( inf_inf_assn @ X2 @ ( uminus_uminus_assn @ Y2 ) ) ) ) ).

% diff_eq
thf(fact_3259_one__le__numeral,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) ) ).

% one_le_numeral
thf(fact_3260_one__le__numeral,axiom,
    ! [N: num] : ( ord_le1926595141338095240atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ N ) ) ).

% one_le_numeral
thf(fact_3261_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).

% one_le_numeral
thf(fact_3262_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% one_le_numeral
thf(fact_3263_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).

% one_le_numeral
thf(fact_3264_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_3265_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_3266_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_3267_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer ) ).

% not_numeral_less_one
thf(fact_3268_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_le5570908160329646204atural @ ( numera5444537566228673987atural @ N ) @ one_one_Code_natural ) ).

% not_numeral_less_one
thf(fact_3269_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_3270_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_3271_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_3272_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_3273_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_3274_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_3275_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_3276_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_Code_integer
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_3277_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ N )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_3278_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ N )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_3279_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ N )
     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% numeral_neq_neg_one
thf(fact_3280_le__add__iff1,axiom,
    ! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
      = ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ E2 ) @ C ) @ D2 ) ) ).

% le_add_iff1
thf(fact_3281_le__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D2 ) ) ).

% le_add_iff1
thf(fact_3282_le__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D2 ) ) ).

% le_add_iff1
thf(fact_3283_le__add__iff2,axiom,
    ! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
      = ( ord_le3102999989581377725nteger @ C @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ E2 ) @ D2 ) ) ) ).

% le_add_iff2
thf(fact_3284_le__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D2 ) ) ) ).

% le_add_iff2
thf(fact_3285_le__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D2 ) ) ) ).

% le_add_iff2
thf(fact_3286_less__add__iff1,axiom,
    ! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
      = ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ E2 ) @ C ) @ D2 ) ) ).

% less_add_iff1
thf(fact_3287_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D2 ) ) ).

% less_add_iff1
thf(fact_3288_less__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D2 ) ) ).

% less_add_iff1
thf(fact_3289_less__add__iff2,axiom,
    ! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ E2 ) @ C ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ E2 ) @ D2 ) )
      = ( ord_le6747313008572928689nteger @ C @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ E2 ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_3290_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D2 ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_3291_less__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_3292_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_3293_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_3294_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_3295_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_3296_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_3297_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_3298_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_3299_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_3300_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_3301_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_3302_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_3303_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_3304_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_3305_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_3306_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_3307_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_3308_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_3309_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_3310_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_3311_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_3312_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_3313_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_3314_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( numeral_numeral_rat @ W ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_3315_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ( numeral_numeral_rat @ W )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_3316_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_3317_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_3318_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_3319_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_3320_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_3321_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_3322_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_3323_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_3324_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_3325_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_3326_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_3327_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_3328_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_3329_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_3330_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_3331_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X2: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_3332_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X2: int] : ( minus_minus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_3333_dbl__dec__def,axiom,
    ( neg_nu7757733837767384882nteger
    = ( ^ [X2: code_integer] : ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ X2 @ X2 ) @ one_one_Code_integer ) ) ) ).

% dbl_dec_def
thf(fact_3334_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_3335_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_3336_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_3337_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_3338_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_3339_count__list_Osimps_I1_J,axiom,
    ! [Y: produc6575502325842934193n_assn] :
      ( ( count_2530312006313534765n_assn @ nil_Pr5671120429643327159n_assn @ Y )
      = zero_zero_nat ) ).

% count_list.simps(1)
thf(fact_3340_count__list_Osimps_I1_J,axiom,
    ! [Y: int] :
      ( ( count_list_int @ nil_int @ Y )
      = zero_zero_nat ) ).

% count_list.simps(1)
thf(fact_3341_count__list_Osimps_I1_J,axiom,
    ! [Y: nat] :
      ( ( count_list_nat @ nil_nat @ Y )
      = zero_zero_nat ) ).

% count_list.simps(1)
thf(fact_3342_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_3343_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_3344_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_3345_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_3346_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).

% mult_eq_if
thf(fact_3347_scaling__mono,axiom,
    ! [U: rat,V2: rat,R2: rat,S3: rat] :
      ( ( ord_less_eq_rat @ U @ V2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
       => ( ( ord_less_eq_rat @ R2 @ S3 )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V2 @ U ) ) @ S3 ) ) @ V2 ) ) ) ) ).

% scaling_mono
thf(fact_3348_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_3349_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_3350_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_3351_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_3352_power__diff__power__eq,axiom,
    ! [A: code_integer,N: nat,M2: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M2 ) @ ( power_8256067586552552935nteger @ A @ N ) )
            = ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M2 ) @ ( power_8256067586552552935nteger @ A @ N ) )
            = ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3353_power__diff__power__eq,axiom,
    ! [A: nat,N: nat,M2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
            = ( power_power_nat @ A @ ( minus_minus_nat @ M2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3354_power__diff__power__eq,axiom,
    ! [A: int,N: nat,M2: nat] :
      ( ( A != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
            = ( power_power_int @ A @ ( minus_minus_nat @ M2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3355_power__diff__power__eq,axiom,
    ! [A: code_natural,N: nat,M2: nat] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide5121882707175180666atural @ ( power_7079662738309270450atural @ A @ M2 ) @ ( power_7079662738309270450atural @ A @ N ) )
            = ( power_7079662738309270450atural @ A @ ( minus_minus_nat @ M2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide5121882707175180666atural @ ( power_7079662738309270450atural @ A @ M2 ) @ ( power_7079662738309270450atural @ A @ N ) )
            = ( divide5121882707175180666atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ A @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3356_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P6: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P6 @ ( power_power_rat @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3357_power__eq__if,axiom,
    ( power_power_assn
    = ( ^ [P6: assn,M3: nat] : ( if_assn @ ( M3 = zero_zero_nat ) @ one_one_assn @ ( times_times_assn @ P6 @ ( power_power_assn @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3358_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P6: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3359_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P6: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P6 @ ( power_power_int @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3360_power__eq__if,axiom,
    ( power_8256067586552552935nteger
    = ( ^ [P6: code_integer,M3: nat] : ( if_Code_integer @ ( M3 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ P6 @ ( power_8256067586552552935nteger @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3361_power__eq__if,axiom,
    ( power_7079662738309270450atural
    = ( ^ [P6: code_natural,M3: nat] : ( if_Code_natural @ ( M3 = zero_zero_nat ) @ one_one_Code_natural @ ( times_2397367101498566445atural @ P6 @ ( power_7079662738309270450atural @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3362_semiring__norm_I170_J,axiom,
    ! [V2: num,W: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_3363_semiring__norm_I170_J,axiom,
    ! [V2: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_3364_semiring__norm_I170_J,axiom,
    ! [V2: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V2 ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_3365_semiring__norm_I171_J,axiom,
    ! [V2: num,W: num,Y: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V2 ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_3366_semiring__norm_I171_J,axiom,
    ! [V2: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_3367_semiring__norm_I171_J,axiom,
    ! [V2: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V2 ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_3368_semiring__norm_I172_J,axiom,
    ! [V2: num,W: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V2 @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_3369_semiring__norm_I172_J,axiom,
    ! [V2: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V2 @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_3370_semiring__norm_I172_J,axiom,
    ! [V2: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V2 ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V2 @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_3371_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_3372_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_3373_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_3374_numeral__times__minus__swap,axiom,
    ! [W: num,X: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
      = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_3375_numeral__times__minus__swap,axiom,
    ! [W: num,X: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
      = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_3376_numeral__times__minus__swap,axiom,
    ! [W: num,X: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X ) )
      = ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_3377_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: int,Xs: list_int] :
      ( ( ( ( Begin = zero_zero_nat )
          & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_int @ Begin @ End @ ( cons_int @ X @ Xs ) )
          = ( cons_int @ X @ ( slice_int @ Begin @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_int @ Begin @ End @ ( cons_int @ X @ Xs ) )
          = ( slice_int @ ( minus_minus_nat @ Begin @ one_one_nat ) @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) ) ).

% slice_Cons
thf(fact_3378_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: nat,Xs: list_nat] :
      ( ( ( ( Begin = zero_zero_nat )
          & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_nat @ Begin @ End @ ( cons_nat @ X @ Xs ) )
          = ( cons_nat @ X @ ( slice_nat @ Begin @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_nat @ Begin @ End @ ( cons_nat @ X @ Xs ) )
          = ( slice_nat @ ( minus_minus_nat @ Begin @ one_one_nat ) @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) ) ).

% slice_Cons
thf(fact_3379_inf__period_I1_J,axiom,
    ! [P2: int > $o,D: int,Q2: int > $o] :
      ( ! [X3: int,K2: int] :
          ( ( P2 @ X3 )
          = ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
     => ( ! [X3: int,K2: int] :
            ( ( Q2 @ X3 )
            = ( Q2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
       => ! [X4: int,K3: int] :
            ( ( ( P2 @ X4 )
              & ( Q2 @ X4 ) )
            = ( ( P2 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) )
              & ( Q2 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3380_inf__period_I1_J,axiom,
    ! [P2: code_integer > $o,D: code_integer,Q2: code_integer > $o] :
      ( ! [X3: code_integer,K2: code_integer] :
          ( ( P2 @ X3 )
          = ( P2 @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K2 @ D ) ) ) )
     => ( ! [X3: code_integer,K2: code_integer] :
            ( ( Q2 @ X3 )
            = ( Q2 @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K2 @ D ) ) ) )
       => ! [X4: code_integer,K3: code_integer] :
            ( ( ( P2 @ X4 )
              & ( Q2 @ X4 ) )
            = ( ( P2 @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K3 @ D ) ) )
              & ( Q2 @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K3 @ D ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3381_inf__period_I2_J,axiom,
    ! [P2: int > $o,D: int,Q2: int > $o] :
      ( ! [X3: int,K2: int] :
          ( ( P2 @ X3 )
          = ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
     => ( ! [X3: int,K2: int] :
            ( ( Q2 @ X3 )
            = ( Q2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
       => ! [X4: int,K3: int] :
            ( ( ( P2 @ X4 )
              | ( Q2 @ X4 ) )
            = ( ( P2 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) )
              | ( Q2 @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3382_inf__period_I2_J,axiom,
    ! [P2: code_integer > $o,D: code_integer,Q2: code_integer > $o] :
      ( ! [X3: code_integer,K2: code_integer] :
          ( ( P2 @ X3 )
          = ( P2 @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K2 @ D ) ) ) )
     => ( ! [X3: code_integer,K2: code_integer] :
            ( ( Q2 @ X3 )
            = ( Q2 @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K2 @ D ) ) ) )
       => ! [X4: code_integer,K3: code_integer] :
            ( ( ( P2 @ X4 )
              | ( Q2 @ X4 ) )
            = ( ( P2 @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K3 @ D ) ) )
              | ( Q2 @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K3 @ D ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3383_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_3384_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_3385_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_3386_Diff__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( minus_1356011639430497352at_nat @ B3 @ A3 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Diff_disjoint
thf(fact_3387_Diff__disjoint,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ A3 ) )
      = bot_bot_set_nat ) ).

% Diff_disjoint
thf(fact_3388_Diff__disjoint,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( inf_inf_set_int @ A3 @ ( minus_minus_set_int @ B3 @ A3 ) )
      = bot_bot_set_int ) ).

% Diff_disjoint
thf(fact_3389_inter__compl__diff__conv,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( uminus6524753893492686040at_nat @ B3 ) )
      = ( minus_1356011639430497352at_nat @ A3 @ B3 ) ) ).

% inter_compl_diff_conv
thf(fact_3390_inter__compl__diff__conv,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ B3 ) )
      = ( minus_minus_set_nat @ A3 @ B3 ) ) ).

% inter_compl_diff_conv
thf(fact_3391_Diff__Compl,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A3 @ ( uminus6524753893492686040at_nat @ B3 ) )
      = ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ).

% Diff_Compl
thf(fact_3392_Diff__Compl,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ B3 ) )
      = ( inf_inf_set_nat @ A3 @ B3 ) ) ).

% Diff_Compl
thf(fact_3393_slice__Nil,axiom,
    ! [Begin: nat,End: nat] :
      ( ( slice_7964966981652229398n_assn @ Begin @ End @ nil_Pr5671120429643327159n_assn )
      = nil_Pr5671120429643327159n_assn ) ).

% slice_Nil
thf(fact_3394_slice__Nil,axiom,
    ! [Begin: nat,End: nat] :
      ( ( slice_int @ Begin @ End @ nil_int )
      = nil_int ) ).

% slice_Nil
thf(fact_3395_slice__Nil,axiom,
    ! [Begin: nat,End: nat] :
      ( ( slice_nat @ Begin @ End @ nil_nat )
      = nil_nat ) ).

% slice_Nil
thf(fact_3396_slice__eq__bounds__empty,axiom,
    ! [I2: nat,Xs: list_P8527749157015355191n_assn] :
      ( ( slice_7964966981652229398n_assn @ I2 @ I2 @ Xs )
      = nil_Pr5671120429643327159n_assn ) ).

% slice_eq_bounds_empty
thf(fact_3397_slice__eq__bounds__empty,axiom,
    ! [I2: nat,Xs: list_int] :
      ( ( slice_int @ I2 @ I2 @ Xs )
      = nil_int ) ).

% slice_eq_bounds_empty
thf(fact_3398_slice__eq__bounds__empty,axiom,
    ! [I2: nat,Xs: list_nat] :
      ( ( slice_nat @ I2 @ I2 @ Xs )
      = nil_nat ) ).

% slice_eq_bounds_empty
thf(fact_3399_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_3400_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_3401_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_3402_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_3403_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_3404_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_3405_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_3406_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_3407_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_3408_Int__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ C2 )
      = ( inf_in2572325071724192079at_nat @ A3 @ ( minus_1356011639430497352at_nat @ B3 @ C2 ) ) ) ).

% Int_Diff
thf(fact_3409_Int__Diff,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ C2 )
      = ( inf_inf_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ C2 ) ) ) ).

% Int_Diff
thf(fact_3410_Diff__Int2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) @ ( inf_in2572325071724192079at_nat @ B3 @ C2 ) )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) @ B3 ) ) ).

% Diff_Int2
thf(fact_3411_Diff__Int2,axiom,
    ! [A3: set_nat,C2: set_nat,B3: set_nat] :
      ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C2 ) @ ( inf_inf_set_nat @ B3 @ C2 ) )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C2 ) @ B3 ) ) ).

% Diff_Int2
thf(fact_3412_Diff__Diff__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A3 @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
      = ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ).

% Diff_Diff_Int
thf(fact_3413_Diff__Diff__Int,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( minus_minus_set_nat @ A3 @ B3 ) )
      = ( inf_inf_set_nat @ A3 @ B3 ) ) ).

% Diff_Diff_Int
thf(fact_3414_Diff__Int__distrib,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ C2 @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ C2 @ A3 ) @ ( inf_in2572325071724192079at_nat @ C2 @ B3 ) ) ) ).

% Diff_Int_distrib
thf(fact_3415_Diff__Int__distrib,axiom,
    ! [C2: set_nat,A3: set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ C2 @ ( minus_minus_set_nat @ A3 @ B3 ) )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ C2 @ A3 ) @ ( inf_inf_set_nat @ C2 @ B3 ) ) ) ).

% Diff_Int_distrib
thf(fact_3416_Diff__Int__distrib2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) @ C2 )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) @ ( inf_in2572325071724192079at_nat @ B3 @ C2 ) ) ) ).

% Diff_Int_distrib2
thf(fact_3417_Diff__Int__distrib2,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ C2 )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C2 ) @ ( inf_inf_set_nat @ B3 @ C2 ) ) ) ).

% Diff_Int_distrib2
thf(fact_3418_disjoint__alt__simp3,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) @ A3 )
      = ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
       != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp3
thf(fact_3419_disjoint__alt__simp3,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ A3 )
      = ( ( inf_inf_set_nat @ A3 @ B3 )
       != bot_bot_set_nat ) ) ).

% disjoint_alt_simp3
thf(fact_3420_disjoint__alt__simp3,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_set_int @ ( minus_minus_set_int @ A3 @ B3 ) @ A3 )
      = ( ( inf_inf_set_int @ A3 @ B3 )
       != bot_bot_set_int ) ) ).

% disjoint_alt_simp3
thf(fact_3421_disjoint__alt__simp1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ A3 @ B3 )
        = A3 )
      = ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp1
thf(fact_3422_disjoint__alt__simp1,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( minus_minus_set_nat @ A3 @ B3 )
        = A3 )
      = ( ( inf_inf_set_nat @ A3 @ B3 )
        = bot_bot_set_nat ) ) ).

% disjoint_alt_simp1
thf(fact_3423_disjoint__alt__simp1,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ( minus_minus_set_int @ A3 @ B3 )
        = A3 )
      = ( ( inf_inf_set_int @ A3 @ B3 )
        = bot_bot_set_int ) ) ).

% disjoint_alt_simp1
thf(fact_3424_disjoint__alt__simp2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ A3 @ B3 )
       != A3 )
      = ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
       != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp2
thf(fact_3425_disjoint__alt__simp2,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( minus_minus_set_nat @ A3 @ B3 )
       != A3 )
      = ( ( inf_inf_set_nat @ A3 @ B3 )
       != bot_bot_set_nat ) ) ).

% disjoint_alt_simp2
thf(fact_3426_disjoint__alt__simp2,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ( minus_minus_set_int @ A3 @ B3 )
       != A3 )
      = ( ( inf_inf_set_int @ A3 @ B3 )
       != bot_bot_set_int ) ) ).

% disjoint_alt_simp2
thf(fact_3427_Diff__triv,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat )
     => ( ( minus_1356011639430497352at_nat @ A3 @ B3 )
        = A3 ) ) ).

% Diff_triv
thf(fact_3428_Diff__triv,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B3 )
        = bot_bot_set_nat )
     => ( ( minus_minus_set_nat @ A3 @ B3 )
        = A3 ) ) ).

% Diff_triv
thf(fact_3429_Diff__triv,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ( inf_inf_set_int @ A3 @ B3 )
        = bot_bot_set_int )
     => ( ( minus_minus_set_int @ A3 @ B3 )
        = A3 ) ) ).

% Diff_triv
thf(fact_3430_Int__Diff__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Int_Diff_disjoint
thf(fact_3431_Int__Diff__disjoint,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ A3 @ B3 ) )
      = bot_bot_set_nat ) ).

% Int_Diff_disjoint
thf(fact_3432_Int__Diff__disjoint,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ A3 @ B3 ) @ ( minus_minus_set_int @ A3 @ B3 ) )
      = bot_bot_set_int ) ).

% Int_Diff_disjoint
thf(fact_3433_Diff__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A3 @ ( sup_su6327502436637775413at_nat @ B3 @ C2 ) )
      = ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) @ ( minus_1356011639430497352at_nat @ A3 @ C2 ) ) ) ).

% Diff_Un
thf(fact_3434_Diff__Un,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C2 ) )
      = ( inf_inf_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ A3 @ C2 ) ) ) ).

% Diff_Un
thf(fact_3435_Diff__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B3 @ C2 ) )
      = ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) @ ( minus_1356011639430497352at_nat @ A3 @ C2 ) ) ) ).

% Diff_Int
thf(fact_3436_Diff__Int,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C2 ) )
      = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ A3 @ C2 ) ) ) ).

% Diff_Int
thf(fact_3437_Int__Diff__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
      = A3 ) ).

% Int_Diff_Un
thf(fact_3438_Int__Diff__Un,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ A3 @ B3 ) )
      = A3 ) ).

% Int_Diff_Un
thf(fact_3439_Un__Diff__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
      = A3 ) ).

% Un_Diff_Int
thf(fact_3440_Un__Diff__Int,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ ( inf_inf_set_nat @ A3 @ B3 ) )
      = A3 ) ).

% Un_Diff_Int
thf(fact_3441_minus__assn__def,axiom,
    ( minus_minus_assn
    = ( ^ [A2: assn,B2: assn] : ( inf_inf_assn @ A2 @ ( uminus_uminus_assn @ B2 ) ) ) ) ).

% minus_assn_def
thf(fact_3442_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_nat @ ( inc @ X ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).

% numeral_inc
thf(fact_3443_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_int @ ( inc @ X ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).

% numeral_inc
thf(fact_3444_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_rat @ ( inc @ X ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).

% numeral_inc
thf(fact_3445_numeral__inc,axiom,
    ! [X: num] :
      ( ( numera6620942414471956472nteger @ ( inc @ X ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer ) ) ).

% numeral_inc
thf(fact_3446_numeral__inc,axiom,
    ! [X: num] :
      ( ( numera5444537566228673987atural @ ( inc @ X ) )
      = ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ X ) @ one_one_Code_natural ) ) ).

% numeral_inc
thf(fact_3447_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_3448_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_3449_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_3450_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_3451_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_3452_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_3453_nth__Cons__pos,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
        = ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_3454_nth__Cons__pos,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
        = ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_3455_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_3456_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_3457_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_3458_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_3459_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ N ) )
      = ( numera5444537566228673987atural @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_3460_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_3461_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
      = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_3462_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
      = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_3463_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_3464_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_3465_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_3466_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_3467_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_3468_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_3469_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_3470_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_3471_Suc__diff__1,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
        = N ) ) ).

% Suc_diff_1
thf(fact_3472_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_nat @ N )
        = one_one_nat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_3473_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_int @ N )
        = one_one_int )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_3474_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_rat @ N )
        = one_one_rat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_3475_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera6620942414471956472nteger @ N )
        = one_one_Code_integer )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_3476_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera5444537566228673987atural @ N )
        = one_one_Code_natural )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_3477_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_3478_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_3479_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_3480_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_Code_integer
        = ( numera6620942414471956472nteger @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_3481_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_Code_natural
        = ( numera5444537566228673987atural @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_3482_nth__Cons__0,axiom,
    ! [X: int,Xs: list_int] :
      ( ( nth_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_3483_nth__Cons__0,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_3484_nth__Cons__Suc,axiom,
    ! [X: int,Xs: list_int,N: nat] :
      ( ( nth_int @ ( cons_int @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_int @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_3485_nth__Cons__Suc,axiom,
    ! [X: nat,Xs: list_nat,N: nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_nat @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_3486_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_3487_enumerate__simps_I2_J,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( ( enumerate_int @ N @ ( cons_int @ X @ Xs ) )
      = ( cons_P2335045147070616083at_int @ ( product_Pair_nat_int @ N @ X ) @ ( enumerate_int @ ( suc @ N ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_3488_enumerate__simps_I2_J,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( ( enumerate_nat @ N @ ( cons_nat @ X @ Xs ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X ) @ ( enumerate_nat @ ( suc @ N ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_3489_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_3490_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_le1926595141338095240atural @ ( numera5444537566228673987atural @ N ) @ one_one_Code_natural )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_3491_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_3492_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_3493_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_3494_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_3495_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_3496_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_3497_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_3498_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_le5570908160329646204atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_3499_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_3500_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_3501_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_3502_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_int @ one_one_int )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_3503_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_rat @ one_one_rat )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_3504_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_3505_Suc__diff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_eq_nat @ one_one_nat @ M2 )
       => ( ( suc @ ( minus_minus_nat @ N @ M2 ) )
          = ( minus_minus_nat @ N @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ).

% Suc_diff
thf(fact_3506_nth__Cons__numeral,axiom,
    ! [X: int,Xs: list_int,V2: num] :
      ( ( nth_int @ ( cons_int @ X @ Xs ) @ ( numeral_numeral_nat @ V2 ) )
      = ( nth_int @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V2 ) @ one_one_nat ) ) ) ).

% nth_Cons_numeral
thf(fact_3507_nth__Cons__numeral,axiom,
    ! [X: nat,Xs: list_nat,V2: num] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( numeral_numeral_nat @ V2 ) )
      = ( nth_nat @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V2 ) @ one_one_nat ) ) ) ).

% nth_Cons_numeral
thf(fact_3508_power__Suc,axiom,
    ! [A: assn,N: nat] :
      ( ( power_power_assn @ A @ ( suc @ N ) )
      = ( times_times_assn @ A @ ( power_power_assn @ A @ N ) ) ) ).

% power_Suc
thf(fact_3509_power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).

% power_Suc
thf(fact_3510_power__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ N ) )
      = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).

% power_Suc
thf(fact_3511_power__Suc,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_Suc
thf(fact_3512_power__Suc,axiom,
    ! [A: code_natural,N: nat] :
      ( ( power_7079662738309270450atural @ A @ ( suc @ N ) )
      = ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ A @ N ) ) ) ).

% power_Suc
thf(fact_3513_power__Suc2,axiom,
    ! [A: assn,N: nat] :
      ( ( power_power_assn @ A @ ( suc @ N ) )
      = ( times_times_assn @ ( power_power_assn @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_3514_power__Suc2,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_3515_power__Suc2,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ N ) )
      = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_3516_power__Suc2,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_3517_power__Suc2,axiom,
    ! [A: code_natural,N: nat] :
      ( ( power_7079662738309270450atural @ A @ ( suc @ N ) )
      = ( times_2397367101498566445atural @ ( power_7079662738309270450atural @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_3518_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_3519_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_3520_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_3521_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_3522_diff__Suc__eq__diff__pred,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ M2 @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_3523_mult__numeral__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3524_mult__numeral__1__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3525_mult__numeral__1__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3526_mult__numeral__1__right,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3527_mult__numeral__1__right,axiom,
    ! [A: code_natural] :
      ( ( times_2397367101498566445atural @ A @ ( numera5444537566228673987atural @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3528_mult__numeral__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3529_mult__numeral__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3530_mult__numeral__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3531_mult__numeral__1,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3532_mult__numeral__1,axiom,
    ! [A: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3533_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_3534_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_3535_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_3536_numeral__One,axiom,
    ( ( numera6620942414471956472nteger @ one )
    = one_one_Code_integer ) ).

% numeral_One
thf(fact_3537_numeral__One,axiom,
    ( ( numera5444537566228673987atural @ one )
    = one_one_Code_natural ) ).

% numeral_One
thf(fact_3538_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_3539_gen__length__code_I2_J,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( ( gen_length_int @ N @ ( cons_int @ X @ Xs ) )
      = ( gen_length_int @ ( suc @ N ) @ Xs ) ) ).

% gen_length_code(2)
thf(fact_3540_gen__length__code_I2_J,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( ( gen_length_nat @ N @ ( cons_nat @ X @ Xs ) )
      = ( gen_length_nat @ ( suc @ N ) @ Xs ) ) ).

% gen_length_code(2)
thf(fact_3541_power__gt1,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_3542_power__gt1,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_3543_power__gt1,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_3544_power__gt1,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_3545_nat__induct__non__zero,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P2 @ one_one_nat )
       => ( ! [N4: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N4 )
             => ( ( P2 @ N4 )
               => ( P2 @ ( suc @ N4 ) ) ) )
         => ( P2 @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_3546_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_3547_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
      = ( uminus_uminus_rat @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_3548_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_3549_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_3550_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] :
      ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
      = ( uminus_uminus_rat @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_3551_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] :
      ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_3552_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_3553_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_3554_uminus__numeral__One,axiom,
    ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% uminus_numeral_One
thf(fact_3555_nth__Cons_H,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
          = ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_3556_nth__Cons_H,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
          = ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_3557_power__Suc__le__self,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_3558_power__Suc__le__self,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ord_le1926595141338095240atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le1926595141338095240atural @ A @ one_one_Code_natural )
       => ( ord_le1926595141338095240atural @ ( power_7079662738309270450atural @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_3559_power__Suc__le__self,axiom,
    ! [A: rat,N: 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 @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_3560_power__Suc__le__self,axiom,
    ! [A: nat,N: 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 @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_3561_power__Suc__le__self,axiom,
    ! [A: int,N: 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 @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_3562_power__Suc__less__one,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ord_le5570908160329646204atural @ zero_z2226904508553997617atural @ A )
     => ( ( ord_le5570908160329646204atural @ A @ one_one_Code_natural )
       => ( ord_le5570908160329646204atural @ ( power_7079662738309270450atural @ A @ ( suc @ N ) ) @ one_one_Code_natural ) ) ) ).

% power_Suc_less_one
thf(fact_3563_power__Suc__less__one,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N ) ) @ one_one_Code_integer ) ) ) ).

% power_Suc_less_one
thf(fact_3564_power__Suc__less__one,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).

% power_Suc_less_one
thf(fact_3565_power__Suc__less__one,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_3566_power__Suc__less__one,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_3567_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
        = ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_3568_Suc__pred_H,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( N
        = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_3569_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).

% add_eq_if
thf(fact_3570_Suc__n__minus__m__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_nat @ one_one_nat @ M2 )
       => ( ( suc @ ( minus_minus_nat @ N @ M2 ) )
          = ( minus_minus_nat @ N @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ).

% Suc_n_minus_m_eq
thf(fact_3571_nth__non__equal__first__eq,axiom,
    ! [X: int,Y: int,Xs: list_int,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
          = Y )
        = ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_3572_nth__non__equal__first__eq,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
          = Y )
        = ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_3573_one__div__numeral,axiom,
    ! [N: num] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( produc8508995932063986495nteger @ ( unique3479559517661332726nteger @ one @ N ) ) ) ).

% one_div_numeral
thf(fact_3574_one__div__numeral,axiom,
    ! [N: num] :
      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ N ) ) ) ).

% one_div_numeral
thf(fact_3575_one__div__numeral,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( product_fst_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ).

% one_div_numeral
thf(fact_3576_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_int ) ).

% power_minus1_even
thf(fact_3577_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_rat ) ).

% power_minus1_even
thf(fact_3578_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_Code_integer ) ).

% power_minus1_even
thf(fact_3579_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% power_minus1_odd
thf(fact_3580_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_rat @ one_one_rat ) ) ).

% power_minus1_odd
thf(fact_3581_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% power_minus1_odd
thf(fact_3582_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_3583_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_3584_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_3585_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_3586_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_3587_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_3588_diff__numeral__special_I7_J,axiom,
    ! [N: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( neg_numeral_sub_int @ N @ one ) ) ).

% diff_numeral_special(7)
thf(fact_3589_diff__numeral__special_I7_J,axiom,
    ! [N: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( neg_numeral_sub_rat @ N @ one ) ) ).

% diff_numeral_special(7)
thf(fact_3590_diff__numeral__special_I7_J,axiom,
    ! [N: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( neg_nu5755505904847501662nteger @ N @ one ) ) ).

% diff_numeral_special(7)
thf(fact_3591_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_3592_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_3593_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_3594_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_3595_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_3596_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_3597_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_3598_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_3599_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_3600_one__add__one,axiom,
    ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_3601_one__add__one,axiom,
    ( ( plus_p4538020629002901425atural @ one_one_Code_natural @ one_one_Code_natural )
    = ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_3602_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_3603_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_3604_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_3605_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_3606_diff__numeral__special_I1_J,axiom,
    ! [N: num] :
      ( ( minus_minus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( neg_numeral_sub_int @ one @ N ) ) ).

% diff_numeral_special(1)
thf(fact_3607_diff__numeral__special_I1_J,axiom,
    ! [N: num] :
      ( ( minus_minus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
      = ( neg_numeral_sub_rat @ one @ N ) ) ).

% diff_numeral_special(1)
thf(fact_3608_diff__numeral__special_I1_J,axiom,
    ! [N: num] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( neg_nu5755505904847501662nteger @ one @ N ) ) ).

% diff_numeral_special(1)
thf(fact_3609_one__div__two__eq__zero,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% one_div_two_eq_zero
thf(fact_3610_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_3611_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_3612_one__div__two__eq__zero,axiom,
    ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
    = zero_z2226904508553997617atural ) ).

% one_div_two_eq_zero
thf(fact_3613_bits__1__div__2,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% bits_1_div_2
thf(fact_3614_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_3615_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_3616_bits__1__div__2,axiom,
    ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
    = zero_z2226904508553997617atural ) ).

% bits_1_div_2
thf(fact_3617_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_3618_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_3619_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_3620_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_3621_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_3622_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_3623_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_3624_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_3625_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_3626_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_3627_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_3628_add__neg__numeral__special_I4_J,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( neg_numeral_sub_int @ N @ one ) ) ).

% add_neg_numeral_special(4)
thf(fact_3629_add__neg__numeral__special_I4_J,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
      = ( neg_numeral_sub_rat @ N @ one ) ) ).

% add_neg_numeral_special(4)
thf(fact_3630_add__neg__numeral__special_I4_J,axiom,
    ! [N: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
      = ( neg_nu5755505904847501662nteger @ N @ one ) ) ).

% add_neg_numeral_special(4)
thf(fact_3631_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_3632_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_3633_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_3634_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_3635_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_3636_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_3637_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_3638_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_3639_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% minus_1_div_exp_eq_int
thf(fact_3640_int__bit__induct,axiom,
    ! [P2: int > $o,K: int] :
      ( ( P2 @ zero_zero_int )
     => ( ( P2 @ ( uminus_uminus_int @ one_one_int ) )
       => ( ! [K2: int] :
              ( ( P2 @ K2 )
             => ( ( K2 != zero_zero_int )
               => ( P2 @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
         => ( ! [K2: int] :
                ( ( P2 @ K2 )
               => ( ( K2
                   != ( uminus_uminus_int @ one_one_int ) )
                 => ( P2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
           => ( P2 @ K ) ) ) ) ) ).

% int_bit_induct
thf(fact_3641_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_3642_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_3643_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_3644_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_3645_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_3646_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_3647_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_3648_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_3649_mult__2__right,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ Z @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( plus_p5714425477246183910nteger @ Z @ Z ) ) ).

% mult_2_right
thf(fact_3650_mult__2__right,axiom,
    ! [Z: code_natural] :
      ( ( times_2397367101498566445atural @ Z @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
      = ( plus_p4538020629002901425atural @ Z @ Z ) ) ).

% mult_2_right
thf(fact_3651_mult__2,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2
thf(fact_3652_mult__2,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2
thf(fact_3653_mult__2,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2
thf(fact_3654_mult__2,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Z )
      = ( plus_p5714425477246183910nteger @ Z @ Z ) ) ).

% mult_2
thf(fact_3655_mult__2,axiom,
    ! [Z: code_natural] :
      ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ Z )
      = ( plus_p4538020629002901425atural @ Z @ Z ) ) ).

% mult_2
thf(fact_3656_power2__eq__square,axiom,
    ! [A: assn] :
      ( ( power_power_assn @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_assn @ A @ A ) ) ).

% power2_eq_square
thf(fact_3657_power2__eq__square,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ A @ A ) ) ).

% power2_eq_square
thf(fact_3658_power2__eq__square,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_int @ A @ A ) ) ).

% power2_eq_square
thf(fact_3659_power2__eq__square,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_3573771949741848930nteger @ A @ A ) ) ).

% power2_eq_square
thf(fact_3660_power2__eq__square,axiom,
    ! [A: code_natural] :
      ( ( power_7079662738309270450atural @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_2397367101498566445atural @ A @ A ) ) ).

% power2_eq_square
thf(fact_3661_power4__eq__xxxx,axiom,
    ! [X: assn] :
      ( ( power_power_assn @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_3662_power4__eq__xxxx,axiom,
    ! [X: nat] :
      ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_3663_power4__eq__xxxx,axiom,
    ! [X: int] :
      ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_3664_power4__eq__xxxx,axiom,
    ! [X: code_integer] :
      ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_3665_power4__eq__xxxx,axiom,
    ! [X: code_natural] :
      ( ( power_7079662738309270450atural @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_2397367101498566445atural @ ( times_2397367101498566445atural @ ( times_2397367101498566445atural @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_3666_one__power2,axiom,
    ( ( power_8256067586552552935nteger @ one_one_Code_integer @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% one_power2
thf(fact_3667_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_3668_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_3669_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_3670_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_3671_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_3672_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_3673_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_3674_nat__induct2,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ( P2 @ one_one_nat )
       => ( ! [N4: nat] :
              ( ( P2 @ N4 )
             => ( P2 @ ( plus_plus_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P2 @ N ) ) ) ) ).

% nat_induct2
thf(fact_3675_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_3676_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_3677_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_3678_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_3679_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_3680_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_3681_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_3682_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_3683_minus__power__mult__self,axiom,
    ! [A: int,N: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
      = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_3684_minus__power__mult__self,axiom,
    ! [A: rat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
      = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_3685_minus__power__mult__self,axiom,
    ! [A: code_integer,N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
      = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_3686_power__odd__eq,axiom,
    ! [A: assn,N: nat] :
      ( ( power_power_assn @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_assn @ A @ ( power_power_assn @ ( power_power_assn @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_3687_power__odd__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_3688_power__odd__eq,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_3689_power__odd__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_3690_power__odd__eq,axiom,
    ! [A: code_natural,N: nat] :
      ( ( power_7079662738309270450atural @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ ( power_7079662738309270450atural @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_3691_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 )
       => ? [N4: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N4 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_3692_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 )
       => ? [N4: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N4 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_3693_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q4: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ 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 @ Q4 @ R2 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_3694_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q4: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ 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 @ Q4 @ R2 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_3695_divmod__step__eq,axiom,
    ! [L: num,R2: code_integer,Q4: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_3696_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_3697_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_3698_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_3699_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_3700_unset__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se8260200283734997820nteger @ zero_zero_nat @ A )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_3701_unset__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_3702_unset__bit__0,axiom,
    ! [A: code_natural] :
      ( ( bit_se7083795435491715335atural @ zero_zero_nat @ A )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_3703_unset__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_3704_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_3705_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_3706_dbl__simps_I4_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_3707_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_3708_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_3709_dbl__simps_I3_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_3710_even__succ__div__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
          = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_3711_even__succ__div__exp,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
          = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_3712_even__succ__div__exp,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
          = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_3713_even__succ__div__exp,axiom,
    ! [A: code_natural,N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
          = ( divide5121882707175180666atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_3714_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M2: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M2 )
        | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
          = zero_z3403309356797280102nteger )
        | ( ( ord_less_eq_nat @ M2 @ N )
          & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_3715_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M2 )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M2 @ N )
          & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_3716_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M2 )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M2 @ N )
          & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_3717_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_natural,M2: nat,N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M2 )
        | ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N )
          = zero_z2226904508553997617atural )
        | ( ( ord_less_eq_nat @ M2 @ N )
          & ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_3718_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_3719_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_3720_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_3721_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_3722_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_3723_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_3724_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_3725_dvd__times__right__cancel__iff,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ B @ A ) @ ( times_2397367101498566445atural @ C @ A ) )
        = ( dvd_dvd_Code_natural @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_3726_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_3727_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_3728_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_3729_dvd__times__left__cancel__iff,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A @ B ) @ ( times_2397367101498566445atural @ A @ C ) )
        = ( dvd_dvd_Code_natural @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_3730_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_3731_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_3732_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_3733_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_3734_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_3735_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_3736_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_3737_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_3738_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_3739_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_3740_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ ( plus_p4538020629002901425atural @ B @ ( times_2397367101498566445atural @ C @ A ) ) )
      = ( dvd_dvd_Code_natural @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3741_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_3742_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_3743_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_3744_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_3745_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ C @ A ) @ B ) )
      = ( dvd_dvd_Code_natural @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3746_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_3747_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_3748_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_3749_unit__prod,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 @ ( times_2397367101498566445atural @ A @ B ) @ one_one_Code_natural ) ) ) ).

% unit_prod
thf(fact_3750_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self2_is_0
thf(fact_3751_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self2_is_0
thf(fact_3752_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% mod_mult_self2_is_0
thf(fact_3753_mod__mult__self2__is__0,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ B )
      = zero_z2226904508553997617atural ) ).

% mod_mult_self2_is_0
thf(fact_3754_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self1_is_0
thf(fact_3755_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self1_is_0
thf(fact_3756_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
      = zero_zero_int ) ).

% mod_mult_self1_is_0
thf(fact_3757_mod__mult__self1__is__0,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ B @ A ) @ B )
      = zero_z2226904508553997617atural ) ).

% mod_mult_self1_is_0
thf(fact_3758_mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% mod_by_1
thf(fact_3759_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_3760_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_3761_mod__by__1,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural @ A @ one_one_Code_natural )
      = zero_z2226904508553997617atural ) ).

% mod_by_1
thf(fact_3762_bits__mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
thf(fact_3763_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_3764_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_3765_bits__mod__by__1,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural @ A @ one_one_Code_natural )
      = zero_z2226904508553997617atural ) ).

% bits_mod_by_1
thf(fact_3766_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_3767_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_3768_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_3769_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_3770_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_3771_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_3772_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_3773_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_3774_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_3775_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_3776_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_3777_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_3778_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_3779_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_3780_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_3781_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_3782_dvd__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_3783_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_3784_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_3785_dvd__div__mult__self,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ B )
     => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_3786_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_3787_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_3788_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_3789_dvd__mult__div__cancel,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ B )
     => ( ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_3790_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_3791_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_3792_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_3793_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_3794_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_3795_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_3796_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_3797_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_3798_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_3799_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_3800_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_3801_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_3802_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_3803_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_3804_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_3805_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_3806_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_3807_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_3808_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_3809_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_3810_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_3811_mod__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_3812_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K )
       != one_one_nat )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
        = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
thf(fact_3813_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_3814_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_3815_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_3816_even__mult__iff,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( times_2397367101498566445atural @ A @ B ) )
      = ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_3817_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_3818_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_3819_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_3820_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_3821_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_3822_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_3823_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_3824_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_3825_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_3826_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_3827_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_3828_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_3829_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_3830_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_3831_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_3832_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_3833_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_3834_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_3835_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_3836_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_3837_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_3838_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_3839_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_3840_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_3841_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
        = one_one_int ) ) ).

% neg_one_even_power
thf(fact_3842_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
        = one_one_rat ) ) ).

% neg_one_even_power
thf(fact_3843_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
        = one_one_Code_integer ) ) ).

% neg_one_even_power
thf(fact_3844_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% neg_one_odd_power
thf(fact_3845_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% neg_one_odd_power
thf(fact_3846_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% neg_one_odd_power
thf(fact_3847_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_3848_one__mod__numeral,axiom,
    ! [N: num] :
      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( produc6174133586879617921nteger @ ( unique3479559517661332726nteger @ one @ N ) ) ) ).

% one_mod_numeral
thf(fact_3849_one__mod__numeral,axiom,
    ! [N: num] :
      ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ N ) ) ) ).

% one_mod_numeral
thf(fact_3850_one__mod__numeral,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ).

% one_mod_numeral
thf(fact_3851_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_3852_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_3853_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_3854_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_3855_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_3856_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_3857_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_3858_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_3859_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_3860_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_3861_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_3862_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_3863_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% odd_two_times_div_two_nat
thf(fact_3864_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_3865_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_3866_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_3867_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_3868_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3869_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3870_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3871_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) @ one_one_Code_natural ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3872_even__succ__mod__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
          = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_3873_even__succ__mod__exp,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
          = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_3874_even__succ__mod__exp,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
          = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_3875_even__succ__mod__exp,axiom,
    ! [A: code_natural,N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
          = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_3876_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_3877_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
        = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).

% minus_mod_int_eq
thf(fact_3878_zmod__minus1,axiom,
    ! [B: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
        = ( minus_minus_int @ B @ one_one_int ) ) ) ).

% zmod_minus1
thf(fact_3879_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_3880_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ one_one_int @ K )
       => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).

% int_div_less_self
thf(fact_3881_div__eq__minus1,axiom,
    ! [B: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% div_eq_minus1
thf(fact_3882_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B )
            = zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B )
           != zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus1_eq_if
thf(fact_3883_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B )
            = zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B )
           != zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus2_eq_if
thf(fact_3884_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_3885_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_3886_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_3887_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_3888_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_3889_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_3890_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_3891_mod__mult__right__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_3892_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_3893_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_3894_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_3895_mod__mult__left__eq,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ C ) @ B ) @ C )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_3896_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_3897_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
      = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_3898_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] :
      ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
      = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_3899_mult__mod__right,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( times_2397367101498566445atural @ C @ ( modulo8411746178871703098atural @ A @ B ) )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ C @ A ) @ ( times_2397367101498566445atural @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_3900_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_3901_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
      = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_3902_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_3903_mod__mult__mult2,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ C ) )
      = ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_3904_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B6: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A5 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B6 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A5 @ B6 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_3905_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A5: nat,B: nat,B6: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A5 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B6 @ C ) )
       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( times_times_nat @ A5 @ B6 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_3906_mod__mult__cong,axiom,
    ! [A: int,C: int,A5: int,B: int,B6: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A5 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B6 @ C ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( times_times_int @ A5 @ B6 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_3907_mod__mult__cong,axiom,
    ! [A: code_natural,C: code_natural,A5: code_natural,B: code_natural,B6: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A @ C )
        = ( modulo8411746178871703098atural @ A5 @ C ) )
     => ( ( ( modulo8411746178871703098atural @ B @ C )
          = ( modulo8411746178871703098atural @ B6 @ C ) )
       => ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C )
          = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A5 @ B6 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_3908_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_3909_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_3910_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_3911_mod__mult__eq,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ C ) @ ( modulo8411746178871703098atural @ B @ C ) ) @ C )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_3912_dvd__triv__right,axiom,
    ! [A: assn,B: assn] : ( dvd_dvd_assn @ A @ ( times_times_assn @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3913_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3914_dvd__triv__right,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3915_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3916_dvd__triv__right,axiom,
    ! [A: code_natural,B: code_natural] : ( dvd_dvd_Code_natural @ A @ ( times_2397367101498566445atural @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3917_dvd__mult__right,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( dvd_dvd_assn @ ( times_times_assn @ A @ B ) @ C )
     => ( dvd_dvd_assn @ B @ C ) ) ).

% dvd_mult_right
thf(fact_3918_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
     => ( dvd_dvd_nat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_3919_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
     => ( dvd_dvd_int @ B @ C ) ) ).

% dvd_mult_right
thf(fact_3920_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
     => ( dvd_dvd_Code_integer @ B @ C ) ) ).

% dvd_mult_right
thf(fact_3921_dvd__mult__right,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A @ B ) @ C )
     => ( dvd_dvd_Code_natural @ B @ C ) ) ).

% dvd_mult_right
thf(fact_3922_mult__dvd__mono,axiom,
    ! [A: assn,B: assn,C: assn,D2: assn] :
      ( ( dvd_dvd_assn @ A @ B )
     => ( ( dvd_dvd_assn @ C @ D2 )
       => ( dvd_dvd_assn @ ( times_times_assn @ A @ C ) @ ( times_times_assn @ B @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_3923_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ C @ D2 )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_3924_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ C @ D2 )
       => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_3925_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ C @ D2 )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_3926_mult__dvd__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ B )
     => ( ( dvd_dvd_Code_natural @ C @ D2 )
       => ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_3927_dvd__triv__left,axiom,
    ! [A: assn,B: assn] : ( dvd_dvd_assn @ A @ ( times_times_assn @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3928_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3929_dvd__triv__left,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3930_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3931_dvd__triv__left,axiom,
    ! [A: code_natural,B: code_natural] : ( dvd_dvd_Code_natural @ A @ ( times_2397367101498566445atural @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3932_dvd__mult__left,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( dvd_dvd_assn @ ( times_times_assn @ A @ B ) @ C )
     => ( dvd_dvd_assn @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3933_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
     => ( dvd_dvd_nat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3934_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
     => ( dvd_dvd_int @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3935_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
     => ( dvd_dvd_Code_integer @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3936_dvd__mult__left,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A @ B ) @ C )
     => ( dvd_dvd_Code_natural @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3937_dvd__mult2,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( dvd_dvd_assn @ A @ B )
     => ( dvd_dvd_assn @ A @ ( times_times_assn @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_3938_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_3939_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_3940_dvd__mult2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_3941_dvd__mult2,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ B )
     => ( dvd_dvd_Code_natural @ A @ ( times_2397367101498566445atural @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_3942_dvd__mult,axiom,
    ! [A: assn,C: assn,B: assn] :
      ( ( dvd_dvd_assn @ A @ C )
     => ( dvd_dvd_assn @ A @ ( times_times_assn @ B @ C ) ) ) ).

% dvd_mult
thf(fact_3943_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_3944_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% dvd_mult
thf(fact_3945_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ C )
     => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% dvd_mult
thf(fact_3946_dvd__mult,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ C )
     => ( dvd_dvd_Code_natural @ A @ ( times_2397367101498566445atural @ B @ C ) ) ) ).

% dvd_mult
thf(fact_3947_dvd__def,axiom,
    ( dvd_dvd_assn
    = ( ^ [B2: assn,A2: assn] :
        ? [K4: assn] :
          ( A2
          = ( times_times_assn @ B2 @ K4 ) ) ) ) ).

% dvd_def
thf(fact_3948_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B2: nat,A2: nat] :
        ? [K4: nat] :
          ( A2
          = ( times_times_nat @ B2 @ K4 ) ) ) ) ).

% dvd_def
thf(fact_3949_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B2: int,A2: int] :
        ? [K4: int] :
          ( A2
          = ( times_times_int @ B2 @ K4 ) ) ) ) ).

% dvd_def
thf(fact_3950_dvd__def,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [B2: code_integer,A2: code_integer] :
        ? [K4: code_integer] :
          ( A2
          = ( times_3573771949741848930nteger @ B2 @ K4 ) ) ) ) ).

% dvd_def
thf(fact_3951_dvd__def,axiom,
    ( dvd_dvd_Code_natural
    = ( ^ [B2: code_natural,A2: code_natural] :
        ? [K4: code_natural] :
          ( A2
          = ( times_2397367101498566445atural @ B2 @ K4 ) ) ) ) ).

% dvd_def
thf(fact_3952_dvdI,axiom,
    ! [A: assn,B: assn,K: assn] :
      ( ( A
        = ( times_times_assn @ B @ K ) )
     => ( dvd_dvd_assn @ B @ A ) ) ).

% dvdI
thf(fact_3953_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A
        = ( times_times_nat @ B @ K ) )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% dvdI
thf(fact_3954_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A
        = ( times_times_int @ B @ K ) )
     => ( dvd_dvd_int @ B @ A ) ) ).

% dvdI
thf(fact_3955_dvdI,axiom,
    ! [A: code_integer,B: code_integer,K: code_integer] :
      ( ( A
        = ( times_3573771949741848930nteger @ B @ K ) )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% dvdI
thf(fact_3956_dvdI,axiom,
    ! [A: code_natural,B: code_natural,K: code_natural] :
      ( ( A
        = ( times_2397367101498566445atural @ B @ K ) )
     => ( dvd_dvd_Code_natural @ B @ A ) ) ).

% dvdI
thf(fact_3957_dvdE,axiom,
    ! [B: assn,A: assn] :
      ( ( dvd_dvd_assn @ B @ A )
     => ~ ! [K2: assn] :
            ( A
           != ( times_times_assn @ B @ K2 ) ) ) ).

% dvdE
thf(fact_3958_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ~ ! [K2: nat] :
            ( A
           != ( times_times_nat @ B @ K2 ) ) ) ).

% dvdE
thf(fact_3959_dvdE,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ~ ! [K2: int] :
            ( A
           != ( times_times_int @ B @ K2 ) ) ) ).

% dvdE
thf(fact_3960_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ~ ! [K2: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ B @ K2 ) ) ) ).

% dvdE
thf(fact_3961_dvdE,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ A )
     => ~ ! [K2: code_natural] :
            ( A
           != ( times_2397367101498566445atural @ B @ K2 ) ) ) ).

% dvdE
thf(fact_3962_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_3963_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_3964_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_3965_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_3966_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_3967_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_3968_one__dvd,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).

% one_dvd
thf(fact_3969_one__dvd,axiom,
    ! [A: assn] : ( dvd_dvd_assn @ one_one_assn @ A ) ).

% one_dvd
thf(fact_3970_one__dvd,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).

% one_dvd
thf(fact_3971_one__dvd,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).

% one_dvd
thf(fact_3972_one__dvd,axiom,
    ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).

% one_dvd
thf(fact_3973_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_3974_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_3975_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_3976_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_3977_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B @ C ) )
     => ~ ! [D3: code_integer] :
            ( B
           != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D3 ) ) ) ) ).

% mod_eqE
thf(fact_3978_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
     => ~ ! [D3: int] :
            ( B
           != ( plus_plus_int @ A @ ( times_times_int @ C @ D3 ) ) ) ) ).

% mod_eqE
thf(fact_3979_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_3980_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_3981_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_3982_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_3983_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).

% not_is_unit_0
thf(fact_3984_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_natural @ zero_z2226904508553997617atural @ one_one_Code_natural ) ).

% not_is_unit_0
thf(fact_3985_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_3986_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_3987_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_3988_unit__mult__right__cancel,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( ( times_2397367101498566445atural @ B @ A )
          = ( times_2397367101498566445atural @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_3989_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_3990_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_3991_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_3992_unit__mult__left__cancel,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( ( times_2397367101498566445atural @ A @ B )
          = ( times_2397367101498566445atural @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_3993_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_3994_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_3995_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_3996_mult__unit__dvd__iff_H,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A @ B ) @ C )
        = ( dvd_dvd_Code_natural @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_3997_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_3998_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_3999_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_4000_dvd__mult__unit__iff_H,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( dvd_dvd_Code_natural @ A @ ( times_2397367101498566445atural @ B @ C ) )
        = ( dvd_dvd_Code_natural @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_4001_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_4002_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_4003_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_4004_mult__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 @ ( times_2397367101498566445atural @ A @ B ) @ C )
        = ( dvd_dvd_Code_natural @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_4005_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_4006_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_4007_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_4008_dvd__mult__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 @ ( times_2397367101498566445atural @ C @ B ) )
        = ( dvd_dvd_Code_natural @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_4009_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_4010_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_4011_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_4012_is__unit__mult__iff,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A @ B ) @ one_one_Code_natural )
      = ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
        & ( dvd_dvd_Code_natural @ B @ one_one_Code_natural ) ) ) ).

% is_unit_mult_iff
thf(fact_4013_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_4014_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
        = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_4015_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
        = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_4016_dvd__div__mult,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ B )
     => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ B @ C ) @ A )
        = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_4017_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_4018_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_4019_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_4020_div__mult__swap,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ B )
     => ( ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ C ) )
        = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_4021_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
          = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_4022_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
          = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_4023_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
          = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_4024_div__div__eq__right,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ C @ B )
     => ( ( dvd_dvd_Code_natural @ B @ A )
       => ( ( divide5121882707175180666atural @ A @ ( divide5121882707175180666atural @ B @ C ) )
          = ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_4025_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
     => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_4026_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_4027_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_4028_dvd__div__mult2__eq,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ B @ C ) @ A )
     => ( ( divide5121882707175180666atural @ A @ ( times_2397367101498566445atural @ B @ C ) )
        = ( divide5121882707175180666atural @ ( divide5121882707175180666atural @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_4029_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
     => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_4030_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
     => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_4031_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
     => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_4032_dvd__mult__imp__div,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A @ C ) @ B )
     => ( dvd_dvd_Code_natural @ A @ ( divide5121882707175180666atural @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_4033_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D2: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( dvd_dvd_Code_integer @ D2 @ C )
       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D2 ) )
          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D2 ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_4034_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D2: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( dvd_dvd_nat @ D2 @ C )
       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D2 ) )
          = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_4035_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D2: int,C: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( dvd_dvd_int @ D2 @ C )
       => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D2 ) )
          = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_4036_div__mult__div__if__dvd,axiom,
    ! [B: code_natural,A: code_natural,D2: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ A )
     => ( ( dvd_dvd_Code_natural @ D2 @ C )
       => ( ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ ( divide5121882707175180666atural @ C @ D2 ) )
          = ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ C ) @ ( times_2397367101498566445atural @ B @ D2 ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_4037_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_4038_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_4039_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_4040_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_4041_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_4042_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_4043_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_4044_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_4045_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_4046_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_4047_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_4048_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_4049_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_4050_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_4051_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_4052_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_4053_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_4054_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_4055_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_4056_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_4057_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_4058_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_4059_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_4060_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_4061_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_4062_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_4063_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_4064_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_4065_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_4066_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_4067_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_4068_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_4069_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_4070_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_4071_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_4072_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_4073_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_4074_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_4075_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_4076_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_4077_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_4078_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_4079_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_4080_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_4081_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_4082_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_4083_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_4084_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_4085_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_4086_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_4087_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_4088_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_4089_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_4090_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_4091_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_4092_minus__div__mult__eq__mod,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_4093_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_4094_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
      = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_4095_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
      = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_4096_minus__mod__eq__div__mult,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) )
      = ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_4097_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_4098_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
      = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_4099_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
      = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_4100_minus__mod__eq__mult__div,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) )
      = ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_4101_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_4102_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_4103_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_4104_minus__mult__div__eq__mod,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_4105_unity__coeff__ex,axiom,
    ! [P2: rat > $o,L: rat] :
      ( ( ? [X2: rat] : ( P2 @ ( times_times_rat @ L @ X2 ) ) )
      = ( ? [X2: rat] :
            ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X2 @ zero_zero_rat ) )
            & ( P2 @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4106_unity__coeff__ex,axiom,
    ! [P2: nat > $o,L: nat] :
      ( ( ? [X2: nat] : ( P2 @ ( times_times_nat @ L @ X2 ) ) )
      = ( ? [X2: nat] :
            ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
            & ( P2 @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4107_unity__coeff__ex,axiom,
    ! [P2: int > $o,L: int] :
      ( ( ? [X2: int] : ( P2 @ ( times_times_int @ L @ X2 ) ) )
      = ( ? [X2: int] :
            ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X2 @ zero_zero_int ) )
            & ( P2 @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4108_unity__coeff__ex,axiom,
    ! [P2: code_integer > $o,L: code_integer] :
      ( ( ? [X2: code_integer] : ( P2 @ ( times_3573771949741848930nteger @ L @ X2 ) ) )
      = ( ? [X2: code_integer] :
            ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X2 @ zero_z3403309356797280102nteger ) )
            & ( P2 @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4109_unity__coeff__ex,axiom,
    ! [P2: code_natural > $o,L: code_natural] :
      ( ( ? [X2: code_natural] : ( P2 @ ( times_2397367101498566445atural @ L @ X2 ) ) )
      = ( ? [X2: code_natural] :
            ( ( dvd_dvd_Code_natural @ L @ ( plus_p4538020629002901425atural @ X2 @ zero_z2226904508553997617atural ) )
            & ( P2 @ X2 ) ) ) ) ).

% unity_coeff_ex
thf(fact_4110_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_4111_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_4112_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_4113_unit__dvdE,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ~ ( ( A != zero_z2226904508553997617atural )
         => ! [C3: code_natural] :
              ( B
             != ( times_2397367101498566445atural @ A @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_4114_inf__period_I3_J,axiom,
    ! [D2: rat,D: rat,T3: rat] :
      ( ( dvd_dvd_rat @ D2 @ D )
     => ! [X4: rat,K3: rat] :
          ( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X4 @ T3 ) )
          = ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D ) ) @ T3 ) ) ) ) ).

% inf_period(3)
thf(fact_4115_inf__period_I3_J,axiom,
    ! [D2: int,D: int,T3: int] :
      ( ( dvd_dvd_int @ D2 @ D )
     => ! [X4: int,K3: int] :
          ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ T3 ) )
          = ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) @ T3 ) ) ) ) ).

% inf_period(3)
thf(fact_4116_inf__period_I3_J,axiom,
    ! [D2: code_integer,D: code_integer,T3: code_integer] :
      ( ( dvd_dvd_Code_integer @ D2 @ D )
     => ! [X4: code_integer,K3: code_integer] :
          ( ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X4 @ T3 ) )
          = ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K3 @ D ) ) @ T3 ) ) ) ) ).

% inf_period(3)
thf(fact_4117_inf__period_I4_J,axiom,
    ! [D2: rat,D: rat,T3: rat] :
      ( ( dvd_dvd_rat @ D2 @ D )
     => ! [X4: rat,K3: rat] :
          ( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X4 @ T3 ) ) )
          = ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K3 @ D ) ) @ T3 ) ) ) ) ) ).

% inf_period(4)
thf(fact_4118_inf__period_I4_J,axiom,
    ! [D2: int,D: int,T3: int] :
      ( ( dvd_dvd_int @ D2 @ D )
     => ! [X4: int,K3: int] :
          ( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ T3 ) ) )
          = ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D ) ) @ T3 ) ) ) ) ) ).

% inf_period(4)
thf(fact_4119_inf__period_I4_J,axiom,
    ! [D2: code_integer,D: code_integer,T3: code_integer] :
      ( ( dvd_dvd_Code_integer @ D2 @ D )
     => ! [X4: code_integer,K3: code_integer] :
          ( ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ X4 @ T3 ) ) )
          = ( ~ ( dvd_dvd_Code_integer @ D2 @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X4 @ ( times_3573771949741848930nteger @ K3 @ D ) ) @ T3 ) ) ) ) ) ).

% inf_period(4)
thf(fact_4120_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ A @ B )
       => ( ( ( divide6298287555418463151nteger @ B @ A )
            = C )
          = ( B
            = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_4121_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ A @ B )
       => ( ( ( divide_divide_nat @ B @ A )
            = C )
          = ( B
            = ( times_times_nat @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_4122_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ A @ B )
       => ( ( ( divide_divide_int @ B @ A )
            = C )
          = ( B
            = ( times_times_int @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_4123_dvd__div__eq__mult,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ A @ B )
       => ( ( ( divide5121882707175180666atural @ B @ A )
            = C )
          = ( B
            = ( times_2397367101498566445atural @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_4124_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
          = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_4125_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
          = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_4126_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
          = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_4127_div__dvd__iff__mult,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ B @ A )
       => ( ( dvd_dvd_Code_natural @ ( divide5121882707175180666atural @ A @ B ) @ C )
          = ( dvd_dvd_Code_natural @ A @ ( times_2397367101498566445atural @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_4128_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
          = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_4129_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
          = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_4130_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
          = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_4131_dvd__div__iff__mult,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ C @ B )
       => ( ( dvd_dvd_Code_natural @ A @ ( divide5121882707175180666atural @ B @ C ) )
          = ( dvd_dvd_Code_natural @ ( times_2397367101498566445atural @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_4132_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( ( dvd_dvd_Code_integer @ A @ B )
         => ( ( dvd_dvd_Code_integer @ C @ D2 )
           => ( ( ( divide6298287555418463151nteger @ B @ A )
                = ( divide6298287555418463151nteger @ D2 @ C ) )
              = ( ( times_3573771949741848930nteger @ B @ C )
                = ( times_3573771949741848930nteger @ A @ D2 ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_4133_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( ( dvd_dvd_nat @ A @ B )
         => ( ( dvd_dvd_nat @ C @ D2 )
           => ( ( ( divide_divide_nat @ B @ A )
                = ( divide_divide_nat @ D2 @ C ) )
              = ( ( times_times_nat @ B @ C )
                = ( times_times_nat @ A @ D2 ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_4134_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D2: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( ( dvd_dvd_int @ A @ B )
         => ( ( dvd_dvd_int @ C @ D2 )
           => ( ( ( divide_divide_int @ B @ A )
                = ( divide_divide_int @ D2 @ C ) )
              = ( ( times_times_int @ B @ C )
                = ( times_times_int @ A @ D2 ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_4135_dvd__div__div__eq__mult,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural,D2: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( C != zero_z2226904508553997617atural )
       => ( ( dvd_dvd_Code_natural @ A @ B )
         => ( ( dvd_dvd_Code_natural @ C @ D2 )
           => ( ( ( divide5121882707175180666atural @ B @ A )
                = ( divide5121882707175180666atural @ D2 @ C ) )
              = ( ( times_2397367101498566445atural @ B @ C )
                = ( times_2397367101498566445atural @ A @ D2 ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_4136_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_4137_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_4138_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_4139_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_4140_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_4141_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_4142_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_4143_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_4144_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_4145_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_4146_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_4147_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_4148_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_4149_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_4150_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_4151_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_4152_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_4153_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_4154_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_4155_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_4156_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_4157_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_4158_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_4159_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_4160_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_4161_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_4162_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_4163_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_4164_is__unit__power__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_4165_is__unit__power__iff,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_4166_is__unit__power__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_4167_power__dvd__imp__le,axiom,
    ! [I2: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I2 @ M2 ) @ ( power_power_nat @ I2 @ N ) )
     => ( ( ord_less_nat @ one_one_nat @ I2 )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_dvd_imp_le
thf(fact_4168_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B5: code_integer] :
              ( ( B5 != zero_z3403309356797280102nteger )
             => ( ( dvd_dvd_Code_integer @ B5 @ one_one_Code_integer )
               => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
                    = B5 )
                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B5 )
                      = A )
                   => ( ( ( times_3573771949741848930nteger @ A @ B5 )
                        = one_one_Code_integer )
                     => ( ( divide6298287555418463151nteger @ C @ A )
                       != ( times_3573771949741848930nteger @ C @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4169_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [B5: nat] :
              ( ( B5 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B5 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A )
                    = B5 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B5 )
                      = A )
                   => ( ( ( times_times_nat @ A @ B5 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C @ A )
                       != ( times_times_nat @ C @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4170_is__unitE,axiom,
    ! [A: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [B5: int] :
              ( ( B5 != zero_zero_int )
             => ( ( dvd_dvd_int @ B5 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A )
                    = B5 )
                 => ( ( ( divide_divide_int @ one_one_int @ B5 )
                      = A )
                   => ( ( ( times_times_int @ A @ B5 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C @ A )
                       != ( times_times_int @ C @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4171_is__unitE,axiom,
    ! [A: code_natural,C: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ one_one_Code_natural )
     => ~ ( ( A != zero_z2226904508553997617atural )
         => ! [B5: code_natural] :
              ( ( B5 != zero_z2226904508553997617atural )
             => ( ( dvd_dvd_Code_natural @ B5 @ one_one_Code_natural )
               => ( ( ( divide5121882707175180666atural @ one_one_Code_natural @ A )
                    = B5 )
                 => ( ( ( divide5121882707175180666atural @ one_one_Code_natural @ B5 )
                      = A )
                   => ( ( ( times_2397367101498566445atural @ A @ B5 )
                        = one_one_Code_natural )
                     => ( ( divide5121882707175180666atural @ C @ A )
                       != ( times_2397367101498566445atural @ C @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4172_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_4173_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_4174_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_4175_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_4176_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_4177_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_4178_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_4179_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_4180_evenE,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: nat] :
            ( A
           != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) ) ) ).

% evenE
thf(fact_4181_evenE,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: int] :
            ( A
           != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) ) ) ).

% evenE
thf(fact_4182_evenE,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B5 ) ) ) ).

% evenE
thf(fact_4183_evenE,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: code_natural] :
            ( A
           != ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B5 ) ) ) ).

% evenE
thf(fact_4184_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_4185_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_4186_odd__one,axiom,
    ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).

% odd_one
thf(fact_4187_odd__one,axiom,
    ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ one_one_Code_natural ) ).

% odd_one
thf(fact_4188_dvd__power__iff,axiom,
    ! [X: code_integer,M2: nat,N: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M2 ) @ ( power_8256067586552552935nteger @ X @ N ) )
        = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4189_dvd__power__iff,axiom,
    ! [X: code_natural,M2: nat,N: nat] :
      ( ( X != zero_z2226904508553997617atural )
     => ( ( dvd_dvd_Code_natural @ ( power_7079662738309270450atural @ X @ M2 ) @ ( power_7079662738309270450atural @ X @ N ) )
        = ( ( dvd_dvd_Code_natural @ X @ one_one_Code_natural )
          | ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4190_dvd__power__iff,axiom,
    ! [X: int,M2: nat,N: nat] :
      ( ( X != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X @ M2 ) @ ( power_power_int @ X @ N ) )
        = ( ( dvd_dvd_int @ X @ one_one_int )
          | ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4191_dvd__power__iff,axiom,
    ! [X: nat,M2: nat,N: nat] :
      ( ( X != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M2 ) @ ( power_power_nat @ X @ N ) )
        = ( ( dvd_dvd_nat @ X @ one_one_nat )
          | ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_4192_dvd__power,axiom,
    ! [N: nat,X: code_integer] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_Code_integer ) )
     => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).

% dvd_power
thf(fact_4193_dvd__power,axiom,
    ! [N: nat,X: rat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_rat ) )
     => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).

% dvd_power
thf(fact_4194_dvd__power,axiom,
    ! [N: nat,X: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_int ) )
     => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).

% dvd_power
thf(fact_4195_dvd__power,axiom,
    ! [N: nat,X: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_nat ) )
     => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).

% dvd_power
thf(fact_4196_dvd__mult__cancel2,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M2 ) @ M2 )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_4197_dvd__mult__cancel1,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M2 @ N ) @ M2 )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_4198_set__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_4199_set__bit__Suc,axiom,
    ! [N: nat,A: code_natural] :
      ( ( bit_se1617098188084679374atural @ ( suc @ N ) @ A )
      = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se1617098188084679374atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_4200_set__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_4201_set__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_4202_unset__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_4203_unset__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_4204_unset__bit__Suc,axiom,
    ! [N: nat,A: code_natural] :
      ( ( bit_se7083795435491715335atural @ ( suc @ N ) @ A )
      = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se7083795435491715335atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_4205_unset__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_4206_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_4207_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_4208_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_4209_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_4210_even__two__times__div__two,axiom,
    ! [A: 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 ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_4211_even__two__times__div__two,axiom,
    ! [A: 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 ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_4212_even__two__times__div__two,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_4213_Suc__times__mod__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M2 @ N ) ) @ M2 )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_4214_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
          = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_4215_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
          = ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_4216_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
          = ( modulo_modulo_int @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_4217_oddE,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: nat] :
            ( A
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_4218_oddE,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: int] :
            ( A
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_4219_oddE,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: code_integer] :
            ( A
           != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B5 ) @ one_one_Code_integer ) ) ) ).

% oddE
thf(fact_4220_oddE,axiom,
    ! [A: code_natural] :
      ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: code_natural] :
            ( A
           != ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B5 ) @ one_one_Code_natural ) ) ) ).

% oddE
thf(fact_4221_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
          = one_one_int ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% minus_one_power_iff
thf(fact_4222_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
          = one_one_rat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% minus_one_power_iff
thf(fact_4223_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
          = one_one_Code_integer ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% minus_one_power_iff
thf(fact_4224_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_4225_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_4226_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_4227_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4228_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
        = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4229_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
        = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4230_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: code_natural] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
        = ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4231_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_4232_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_4233_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_4234_divmod__digit__1_I2_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 ) ) )
         => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_4235_divmod__digit__1_I2_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 ) ) )
         => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_4236_divmod__digit__1_I2_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 ) ) )
         => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_4237_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: 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 ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% even_mask_div_iff'
thf(fact_4238_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: 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 ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% even_mask_div_iff'
thf(fact_4239_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: 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 ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% even_mask_div_iff'
thf(fact_4240_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: 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 ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% even_mask_div_iff'
thf(fact_4241_even__mask__div__iff,axiom,
    ! [M2: nat,N: 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 ) ) @ N ) ) )
      = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
          = zero_z3403309356797280102nteger )
        | ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% even_mask_div_iff
thf(fact_4242_even__mask__div__iff,axiom,
    ! [M2: nat,N: 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 ) ) @ N ) ) )
      = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          = zero_zero_nat )
        | ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% even_mask_div_iff
thf(fact_4243_even__mask__div__iff,axiom,
    ! [M2: nat,N: 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 ) ) @ N ) ) )
      = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
          = zero_zero_int )
        | ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% even_mask_div_iff
thf(fact_4244_even__mask__div__iff,axiom,
    ! [M2: nat,N: 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 ) ) @ N ) ) )
      = ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N )
          = zero_z2226904508553997617atural )
        | ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% even_mask_div_iff
thf(fact_4245_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_4246_verit__le__mono__div,axiom,
    ! [A3: nat,B3: nat,N: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A3 @ N )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B3 @ N )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B3 @ N ) ) ) ) ).

% verit_le_mono_div
thf(fact_4247_verit__le__mono__div__int,axiom,
    ! [A3: int,B3: int,N: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A3 @ N )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B3 @ N )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B3 @ N ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_4248_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
      = ( ord_less_int @ W @ Z ) ) ).

% zle_diff1_eq
thf(fact_4249_one__div__minus__numeral,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% one_div_minus_numeral
thf(fact_4250_minus__one__div__numeral,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% minus_one_div_numeral
thf(fact_4251_one__mod__minus__numeral,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).

% one_mod_minus_numeral
thf(fact_4252_minus__one__mod__numeral,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% minus_one_mod_numeral
thf(fact_4253_int__less__induct,axiom,
    ! [I2: int,K: int,P2: int > $o] :
      ( ( ord_less_int @ I2 @ K )
     => ( ( P2 @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ I3 @ K )
             => ( ( P2 @ I3 )
               => ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P2 @ I2 ) ) ) ) ).

% int_less_induct
thf(fact_4254_int__le__induct,axiom,
    ! [I2: int,K: int,P2: int > $o] :
      ( ( ord_less_eq_int @ I2 @ K )
     => ( ( P2 @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P2 @ I3 )
               => ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P2 @ I2 ) ) ) ) ).

% int_le_induct
thf(fact_4255_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_4256_int__gr__induct,axiom,
    ! [K: int,I2: int,P2: int > $o] :
      ( ( ord_less_int @ K @ I2 )
     => ( ( P2 @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ K @ I3 )
             => ( ( P2 @ I3 )
               => ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P2 @ I2 ) ) ) ) ).

% int_gr_induct
thf(fact_4257_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ Z )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% int_one_le_iff_zero_less
thf(fact_4258_pos__zmult__eq__1__iff,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ M2 )
     => ( ( ( times_times_int @ M2 @ N )
          = one_one_int )
        = ( ( M2 = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_4259_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_4260_odd__nonzero,axiom,
    ! [Z: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_4261_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_4262_int__ge__induct,axiom,
    ! [K: int,I2: int,P2: int > $o] :
      ( ( ord_less_eq_int @ K @ I2 )
     => ( ( P2 @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ K @ I3 )
             => ( ( P2 @ I3 )
               => ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P2 @ I2 ) ) ) ) ).

% int_ge_induct
thf(fact_4263_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_4264_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_4265_int__induct,axiom,
    ! [P2: int > $o,K: int,I2: int] :
      ( ( P2 @ K )
     => ( ! [I3: int] :
            ( ( ord_less_eq_int @ K @ I3 )
           => ( ( P2 @ I3 )
             => ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P2 @ I3 )
               => ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P2 @ I2 ) ) ) ) ).

% int_induct
thf(fact_4266_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M2: int,N: int] :
      ( ( ( times_times_int @ M2 @ N )
        = one_one_int )
     => ( ( M2 = one_one_int )
        | ( M2
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
thf(fact_4267_zmult__eq__1__iff,axiom,
    ! [M2: int,N: int] :
      ( ( ( times_times_int @ M2 @ N )
        = one_one_int )
      = ( ( ( M2 = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M2
            = ( uminus_uminus_int @ one_one_int ) )
          & ( N
            = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zmult_eq_1_iff
thf(fact_4268_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_4269_flip__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_4270_flip__bit__Suc,axiom,
    ! [N: nat,A: code_natural] :
      ( ( bit_se168947363167071951atural @ ( suc @ N ) @ A )
      = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se168947363167071951atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_4271_flip__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_4272_flip__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_4273_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N3: nat,A2: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_4274_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,A2: int] : ( if_int @ ( N3 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_4275_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: 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 @ Q4 @ 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 @ Q4 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_4276_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: int,R2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ 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 @ Q4 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_4277_signed__take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_4278_signed__take__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_4279_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] :
      ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% signed_take_bit_of_minus_1
thf(fact_4280_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% signed_take_bit_of_minus_1
thf(fact_4281_signed__take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% signed_take_bit_Suc_1
thf(fact_4282_signed__take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_Suc_1
thf(fact_4283_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_4284_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_4285_zminus1__lemma,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ 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 @ Q4 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q4 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R2 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_4286_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( 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_4287_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_4288_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_4289_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_4290_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_4291_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_4292_take__bit__rec,axiom,
    ( bit_se1745604003318907178nteger
    = ( ^ [N3: nat,A2: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_4293_take__bit__rec,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N3: nat,A2: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_4294_take__bit__rec,axiom,
    ( bit_se569199155075624693atural
    = ( ^ [N3: nat,A2: code_natural] : ( if_Code_natural @ ( N3 = zero_zero_nat ) @ zero_z2226904508553997617atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( modulo8411746178871703098atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_4295_take__bit__rec,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N3: nat,A2: int] : ( if_int @ ( N3 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_4296_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_4297_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_4298_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_4299_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n8403883297036319079atural @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_4300_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_4301_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I4: int,J2: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J2 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J2 @ one_one_int ) @ ( cons_int @ J2 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_4302_push__bit__numeral__minus__1,axiom,
    ! [N: num] :
      ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).

% push_bit_numeral_minus_1
thf(fact_4303_push__bit__numeral__minus__1,axiom,
    ! [N: num] :
      ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ N ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N ) ) ) ) ).

% push_bit_numeral_minus_1
thf(fact_4304_of__bool__eq_I2_J,axiom,
    ( ( zero_n356916108424825756nteger @ $true )
    = one_one_Code_integer ) ).

% of_bool_eq(2)
thf(fact_4305_of__bool__eq_I2_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $true )
    = one_one_rat ) ).

% of_bool_eq(2)
thf(fact_4306_of__bool__eq_I2_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $true )
    = one_one_nat ) ).

% of_bool_eq(2)
thf(fact_4307_of__bool__eq_I2_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $true )
    = one_one_int ) ).

% of_bool_eq(2)
thf(fact_4308_of__bool__eq__1__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n356916108424825756nteger @ P2 )
        = one_one_Code_integer )
      = P2 ) ).

% of_bool_eq_1_iff
thf(fact_4309_of__bool__eq__1__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P2 )
        = one_one_rat )
      = P2 ) ).

% of_bool_eq_1_iff
thf(fact_4310_of__bool__eq__1__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P2 )
        = one_one_nat )
      = P2 ) ).

% of_bool_eq_1_iff
thf(fact_4311_of__bool__eq__1__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P2 )
        = one_one_int )
      = P2 ) ).

% of_bool_eq_1_iff
thf(fact_4312_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% take_bit_Suc_1
thf(fact_4313_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_Suc_1
thf(fact_4314_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
      = one_one_int ) ).

% take_bit_Suc_1
thf(fact_4315_of__bool__less__one__iff,axiom,
    ! [P2: $o] :
      ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P2 ) @ one_one_Code_integer )
      = ~ P2 ) ).

% of_bool_less_one_iff
thf(fact_4316_of__bool__less__one__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) @ one_one_rat )
      = ~ P2 ) ).

% of_bool_less_one_iff
thf(fact_4317_of__bool__less__one__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ one_one_nat )
      = ~ P2 ) ).

% of_bool_less_one_iff
thf(fact_4318_of__bool__less__one__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ one_one_int )
      = ~ P2 ) ).

% of_bool_less_one_iff
thf(fact_4319_and_Oleft__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
      = A ) ).

% and.left_neutral
thf(fact_4320_and_Oleft__neutral,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
      = A ) ).

% and.left_neutral
thf(fact_4321_and_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = A ) ).

% and.right_neutral
thf(fact_4322_and_Oright__neutral,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = A ) ).

% and.right_neutral
thf(fact_4323_bit_Oconj__one__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se3949692690581998587nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = X ) ).

% bit.conj_one_right
thf(fact_4324_bit_Oconj__one__right,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ X @ ( uminus_uminus_int @ one_one_int ) )
      = X ) ).

% bit.conj_one_right
thf(fact_4325_of__bool__not__iff,axiom,
    ! [P2: $o] :
      ( ( zero_n2052037380579107095ol_rat @ ~ P2 )
      = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) ) ) ).

% of_bool_not_iff
thf(fact_4326_of__bool__not__iff,axiom,
    ! [P2: $o] :
      ( ( zero_n356916108424825756nteger @ ~ P2 )
      = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P2 ) ) ) ).

% of_bool_not_iff
thf(fact_4327_of__bool__not__iff,axiom,
    ! [P2: $o] :
      ( ( zero_n2684676970156552555ol_int @ ~ P2 )
      = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P2 ) ) ) ).

% of_bool_not_iff
thf(fact_4328_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_4329_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_4330_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_4331_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
        = zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_4332_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
        = zero_z3403309356797280102nteger )
      = ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_4333_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se569199155075624693atural @ N @ one_one_Code_natural )
        = zero_z2226904508553997617atural )
      = ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_4334_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
        = zero_zero_int )
      = ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_4335_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_4336_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_4337_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_4338_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_4339_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_4340_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_4341_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_4342_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_4343_sgn__mult__self__eq,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ A ) )
      = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).

% sgn_mult_self_eq
thf(fact_4344_sgn__mult__self__eq,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
      = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).

% sgn_mult_self_eq
thf(fact_4345_sgn__mult__self__eq,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ A ) )
      = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).

% sgn_mult_self_eq
thf(fact_4346_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = zero_z3403309356797280102nteger ) ).

% and_numerals(1)
thf(fact_4347_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se2773287842338716102atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = zero_z2226904508553997617atural ) ).

% and_numerals(1)
thf(fact_4348_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_4349_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_4350_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% and_numerals(5)
thf(fact_4351_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ one_one_Code_natural )
      = zero_z2226904508553997617atural ) ).

% and_numerals(5)
thf(fact_4352_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_4353_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_4354_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_4355_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_4356_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_4357_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_4358_take__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_1
thf(fact_4359_take__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_1
thf(fact_4360_take__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_1
thf(fact_4361_and__minus__numerals_I6_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
      = one_one_int ) ).

% and_minus_numerals(6)
thf(fact_4362_and__minus__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = one_one_int ) ).

% and_minus_numerals(2)
thf(fact_4363_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_4364_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_4365_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_4366_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_4367_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_4368_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_4369_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_4370_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_4371_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2773287842338716102atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_4372_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_4373_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_4374_push__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se7788150548672797655nteger @ ( suc @ N ) @ A )
      = ( bit_se7788150548672797655nteger @ N @ ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% push_bit_Suc
thf(fact_4375_push__bit__Suc,axiom,
    ! [N: nat,A: code_natural] :
      ( ( bit_se6611745700429515170atural @ ( suc @ N ) @ A )
      = ( bit_se6611745700429515170atural @ N @ ( times_2397367101498566445atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% push_bit_Suc
thf(fact_4376_push__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ ( suc @ N ) @ A )
      = ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% push_bit_Suc
thf(fact_4377_push__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se547839408752420682it_nat @ ( suc @ N ) @ A )
      = ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% push_bit_Suc
thf(fact_4378_push__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se7788150548672797655nteger @ N @ one_one_Code_integer )
      = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ).

% push_bit_of_1
thf(fact_4379_push__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se6611745700429515170atural @ N @ one_one_Code_natural )
      = ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ).

% push_bit_of_1
thf(fact_4380_push__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se545348938243370406it_int @ N @ one_one_int )
      = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% push_bit_of_1
thf(fact_4381_push__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se547839408752420682it_nat @ N @ one_one_nat )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% push_bit_of_1
thf(fact_4382_and__minus__numerals_I5_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_minus_numerals(5)
thf(fact_4383_and__minus__numerals_I1_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = zero_zero_int ) ).

% and_minus_numerals(1)
thf(fact_4384_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_4385_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_4386_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_4387_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_4388_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_4389_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_4390_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_4391_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_4392_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_4393_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n8403883297036319079atural @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_4394_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_4395_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_4396_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_4397_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide5121882707175180666atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n8403883297036319079atural @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_4398_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_4399_take__bit__of__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( bit_se2925701944663578781it_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_4400_take__bit__of__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( bit_se1745604003318907178nteger @ M2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N @ M2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_4401_take__bit__of__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( bit_se569199155075624693atural @ M2 @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ ( ord_less_nat @ N @ M2 ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_4402_take__bit__of__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( bit_se2923211474154528505it_int @ M2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_4403_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_4404_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_4405_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se569199155075624693atural @ N @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
      = ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_4406_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_4407_zmod__numeral__Bit1,axiom,
    ! [V2: num,W: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V2 ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V2 ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).

% zmod_numeral_Bit1
thf(fact_4408_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_bit1
thf(fact_4409_of__bool__conj,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( zero_n2687167440665602831ol_nat
        @ ( P2
          & Q2 ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q2 ) ) ) ).

% of_bool_conj
thf(fact_4410_of__bool__conj,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( zero_n356916108424825756nteger
        @ ( P2
          & Q2 ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P2 ) @ ( zero_n356916108424825756nteger @ Q2 ) ) ) ).

% of_bool_conj
thf(fact_4411_of__bool__conj,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( zero_n8403883297036319079atural
        @ ( P2
          & Q2 ) )
      = ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ P2 ) @ ( zero_n8403883297036319079atural @ Q2 ) ) ) ).

% of_bool_conj
thf(fact_4412_of__bool__conj,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( zero_n2684676970156552555ol_int
        @ ( P2
          & Q2 ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q2 ) ) ) ).

% of_bool_conj
thf(fact_4413_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_4414_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_4415_of__bool__less__eq__one,axiom,
    ! [P2: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P2 ) @ one_one_Code_integer ) ).

% of_bool_less_eq_one
thf(fact_4416_of__bool__less__eq__one,axiom,
    ! [P2: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) @ one_one_rat ) ).

% of_bool_less_eq_one
thf(fact_4417_of__bool__less__eq__one,axiom,
    ! [P2: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ one_one_nat ) ).

% of_bool_less_eq_one
thf(fact_4418_of__bool__less__eq__one,axiom,
    ! [P2: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ one_one_int ) ).

% of_bool_less_eq_one
thf(fact_4419_of__bool__def,axiom,
    ( zero_n2687167440665602831ol_nat
    = ( ^ [P6: $o] : ( if_nat @ P6 @ one_one_nat @ zero_zero_nat ) ) ) ).

% of_bool_def
thf(fact_4420_of__bool__def,axiom,
    ( zero_n2052037380579107095ol_rat
    = ( ^ [P6: $o] : ( if_rat @ P6 @ one_one_rat @ zero_zero_rat ) ) ) ).

% of_bool_def
thf(fact_4421_of__bool__def,axiom,
    ( zero_n356916108424825756nteger
    = ( ^ [P6: $o] : ( if_Code_integer @ P6 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).

% of_bool_def
thf(fact_4422_of__bool__def,axiom,
    ( zero_n8403883297036319079atural
    = ( ^ [P6: $o] : ( if_Code_natural @ P6 @ one_one_Code_natural @ zero_z2226904508553997617atural ) ) ) ).

% of_bool_def
thf(fact_4423_of__bool__def,axiom,
    ( zero_n2684676970156552555ol_int
    = ( ^ [P6: $o] : ( if_int @ P6 @ one_one_int @ zero_zero_int ) ) ) ).

% of_bool_def
thf(fact_4424_split__of__bool,axiom,
    ! [P2: nat > $o,P5: $o] :
      ( ( P2 @ ( zero_n2687167440665602831ol_nat @ P5 ) )
      = ( ( P5
         => ( P2 @ one_one_nat ) )
        & ( ~ P5
         => ( P2 @ zero_zero_nat ) ) ) ) ).

% split_of_bool
thf(fact_4425_split__of__bool,axiom,
    ! [P2: rat > $o,P5: $o] :
      ( ( P2 @ ( zero_n2052037380579107095ol_rat @ P5 ) )
      = ( ( P5
         => ( P2 @ one_one_rat ) )
        & ( ~ P5
         => ( P2 @ zero_zero_rat ) ) ) ) ).

% split_of_bool
thf(fact_4426_split__of__bool,axiom,
    ! [P2: code_integer > $o,P5: $o] :
      ( ( P2 @ ( zero_n356916108424825756nteger @ P5 ) )
      = ( ( P5
         => ( P2 @ one_one_Code_integer ) )
        & ( ~ P5
         => ( P2 @ zero_z3403309356797280102nteger ) ) ) ) ).

% split_of_bool
thf(fact_4427_split__of__bool,axiom,
    ! [P2: code_natural > $o,P5: $o] :
      ( ( P2 @ ( zero_n8403883297036319079atural @ P5 ) )
      = ( ( P5
         => ( P2 @ one_one_Code_natural ) )
        & ( ~ P5
         => ( P2 @ zero_z2226904508553997617atural ) ) ) ) ).

% split_of_bool
thf(fact_4428_split__of__bool,axiom,
    ! [P2: int > $o,P5: $o] :
      ( ( P2 @ ( zero_n2684676970156552555ol_int @ P5 ) )
      = ( ( P5
         => ( P2 @ one_one_int ) )
        & ( ~ P5
         => ( P2 @ zero_zero_int ) ) ) ) ).

% split_of_bool
thf(fact_4429_split__of__bool__asm,axiom,
    ! [P2: nat > $o,P5: $o] :
      ( ( P2 @ ( zero_n2687167440665602831ol_nat @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P2 @ one_one_nat ) )
            | ( ~ P5
              & ~ ( P2 @ zero_zero_nat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4430_split__of__bool__asm,axiom,
    ! [P2: rat > $o,P5: $o] :
      ( ( P2 @ ( zero_n2052037380579107095ol_rat @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P2 @ one_one_rat ) )
            | ( ~ P5
              & ~ ( P2 @ zero_zero_rat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4431_split__of__bool__asm,axiom,
    ! [P2: code_integer > $o,P5: $o] :
      ( ( P2 @ ( zero_n356916108424825756nteger @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P2 @ one_one_Code_integer ) )
            | ( ~ P5
              & ~ ( P2 @ zero_z3403309356797280102nteger ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4432_split__of__bool__asm,axiom,
    ! [P2: code_natural > $o,P5: $o] :
      ( ( P2 @ ( zero_n8403883297036319079atural @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P2 @ one_one_Code_natural ) )
            | ( ~ P5
              & ~ ( P2 @ zero_z2226904508553997617atural ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4433_split__of__bool__asm,axiom,
    ! [P2: int > $o,P5: $o] :
      ( ( P2 @ ( zero_n2684676970156552555ol_int @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P2 @ one_one_int ) )
            | ( ~ P5
              & ~ ( P2 @ zero_zero_int ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4434_and_Ocomm__monoid__axioms,axiom,
    comm_m1654649897431166824nteger @ bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ).

% and.comm_monoid_axioms
thf(fact_4435_and_Ocomm__monoid__axioms,axiom,
    comm_monoid_int @ bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) ).

% and.comm_monoid_axioms
thf(fact_4436_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
       != zero_zero_int )
     => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
        = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).

% take_bit_decr_eq
thf(fact_4437_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).

% numeral_Bit1
thf(fact_4438_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).

% numeral_Bit1
thf(fact_4439_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).

% numeral_Bit1
thf(fact_4440_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ ( bit1 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ N ) ) @ one_one_Code_integer ) ) ).

% numeral_Bit1
thf(fact_4441_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numera5444537566228673987atural @ ( bit1 @ N ) )
      = ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( numera5444537566228673987atural @ N ) @ ( numera5444537566228673987atural @ N ) ) @ one_one_Code_natural ) ) ).

% numeral_Bit1
thf(fact_4442_and_Omonoid__axioms,axiom,
    monoid_Code_integer @ bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ).

% and.monoid_axioms
thf(fact_4443_and_Omonoid__axioms,axiom,
    monoid_int @ bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) ).

% and.monoid_axioms
thf(fact_4444_and_Osemilattice__neutr__axioms,axiom,
    semila6106414119144251899nteger @ bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ).

% and.semilattice_neutr_axioms
thf(fact_4445_and_Osemilattice__neutr__axioms,axiom,
    semila9079005292280841162tr_int @ bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) ).

% and.semilattice_neutr_axioms
thf(fact_4446_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_Suc_bit1
thf(fact_4447_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( numera6620942414471956472nteger @ ( bit1 @ K ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ).

% take_bit_Suc_bit1
thf(fact_4448_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se569199155075624693atural @ ( suc @ N ) @ ( numera5444537566228673987atural @ ( bit1 @ K ) ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ one_one_Code_natural ) ) ).

% take_bit_Suc_bit1
thf(fact_4449_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_bit1
thf(fact_4450_power3__eq__cube,axiom,
    ! [A: assn] :
      ( ( power_power_assn @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_assn @ ( times_times_assn @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_4451_power3__eq__cube,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_4452_power3__eq__cube,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_4453_power3__eq__cube,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_4454_power3__eq__cube,axiom,
    ! [A: code_natural] :
      ( ( power_7079662738309270450atural @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_2397367101498566445atural @ ( times_2397367101498566445atural @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_4455_push__bit__double,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se7788150548672797655nteger @ N @ ( times_3573771949741848930nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
      = ( times_3573771949741848930nteger @ ( bit_se7788150548672797655nteger @ N @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% push_bit_double
thf(fact_4456_push__bit__double,axiom,
    ! [N: nat,A: code_natural] :
      ( ( bit_se6611745700429515170atural @ N @ ( times_2397367101498566445atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
      = ( times_2397367101498566445atural @ ( bit_se6611745700429515170atural @ N @ A ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% push_bit_double
thf(fact_4457_push__bit__double,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ N @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( bit_se545348938243370406it_int @ N @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% push_bit_double
thf(fact_4458_push__bit__double,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se547839408752420682it_nat @ N @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( bit_se547839408752420682it_nat @ N @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% push_bit_double
thf(fact_4459_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_bit1
thf(fact_4460_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_4461_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) )
      = ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_4462_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se569199155075624693atural @ ( suc @ N ) @ ( numera5444537566228673987atural @ ( bit0 @ K ) ) )
      = ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_4463_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_4464_and__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_4465_and__one__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se2773287842338716102atural @ A @ one_one_Code_natural )
      = ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_4466_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_4467_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_4468_one__and__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_4469_one__and__eq,axiom,
    ! [A: code_natural] :
      ( ( bit_se2773287842338716102atural @ one_one_Code_natural @ A )
      = ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_4470_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_4471_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_4472_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bitM @ N ) )
      = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).

% numeral_BitM
thf(fact_4473_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bitM @ N ) )
      = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N ) ) @ one_one_rat ) ) ).

% numeral_BitM
thf(fact_4474_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ ( bitM @ N ) )
      = ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ one_one_Code_integer ) ) ).

% numeral_BitM
thf(fact_4475_push__bit__eq__mult,axiom,
    ( bit_se7788150548672797655nteger
    = ( ^ [N3: nat,A2: code_integer] : ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% push_bit_eq_mult
thf(fact_4476_push__bit__eq__mult,axiom,
    ( bit_se6611745700429515170atural
    = ( ^ [N3: nat,A2: code_natural] : ( times_2397367101498566445atural @ A2 @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% push_bit_eq_mult
thf(fact_4477_push__bit__eq__mult,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N3: nat,A2: int] : ( times_times_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% push_bit_eq_mult
thf(fact_4478_push__bit__eq__mult,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N3: nat,A2: nat] : ( times_times_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% push_bit_eq_mult
thf(fact_4479_bits__induct,axiom,
    ! [P2: code_integer > $o,A: code_integer] :
      ( ! [A6: code_integer] :
          ( ( ( divide6298287555418463151nteger @ A6 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = A6 )
         => ( P2 @ A6 ) )
     => ( ! [A6: code_integer,B5: $o] :
            ( ( P2 @ A6 )
           => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B5 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A6 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = A6 )
             => ( P2 @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B5 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A6 ) ) ) ) )
       => ( P2 @ A ) ) ) ).

% bits_induct
thf(fact_4480_bits__induct,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ! [A6: nat] :
          ( ( ( divide_divide_nat @ A6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = A6 )
         => ( P2 @ A6 ) )
     => ( ! [A6: nat,B5: $o] :
            ( ( P2 @ A6 )
           => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B5 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A6 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                = A6 )
             => ( P2 @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B5 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A6 ) ) ) ) )
       => ( P2 @ A ) ) ) ).

% bits_induct
thf(fact_4481_bits__induct,axiom,
    ! [P2: code_natural > $o,A: code_natural] :
      ( ! [A6: code_natural] :
          ( ( ( divide5121882707175180666atural @ A6 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
            = A6 )
         => ( P2 @ A6 ) )
     => ( ! [A6: code_natural,B5: $o] :
            ( ( P2 @ A6 )
           => ( ( ( divide5121882707175180666atural @ ( plus_p4538020629002901425atural @ ( zero_n8403883297036319079atural @ B5 ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A6 ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
                = A6 )
             => ( P2 @ ( plus_p4538020629002901425atural @ ( zero_n8403883297036319079atural @ B5 ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A6 ) ) ) ) )
       => ( P2 @ A ) ) ) ).

% bits_induct
thf(fact_4482_bits__induct,axiom,
    ! [P2: int > $o,A: int] :
      ( ! [A6: int] :
          ( ( ( divide_divide_int @ A6 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = A6 )
         => ( P2 @ A6 ) )
     => ( ! [A6: int,B5: $o] :
            ( ( P2 @ A6 )
           => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B5 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A6 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = A6 )
             => ( P2 @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B5 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A6 ) ) ) ) )
       => ( P2 @ A ) ) ) ).

% bits_induct
thf(fact_4483_push__bit__minus__one,axiom,
    ! [N: nat] :
      ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% push_bit_minus_one
thf(fact_4484_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M2 @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% exp_mod_exp
thf(fact_4485_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M2 @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% exp_mod_exp
thf(fact_4486_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo8411746178871703098atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ ( ord_less_nat @ M2 @ N ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% exp_mod_exp
thf(fact_4487_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M2 @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% exp_mod_exp
thf(fact_4488_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
       != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
     => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
        = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).

% take_bit_incr_eq
thf(fact_4489_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_Code_integer ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_4490_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_4491_take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_4492_take__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( 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_4493_take__bit__Suc,axiom,
    ! [N: nat,A: code_natural] :
      ( ( bit_se569199155075624693atural @ ( suc @ N ) @ A )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_4494_take__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( 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_4495_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_4496_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_4497_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se1745604003318907178nteger @ N @ A )
            = zero_z3403309356797280102nteger ) )
        & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se1745604003318907178nteger @ N @ A )
            = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_4498_stable__imp__take__bit__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2925701944663578781it_nat @ N @ A )
            = zero_zero_nat ) )
        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2925701944663578781it_nat @ N @ A )
            = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_4499_stable__imp__take__bit__eq,axiom,
    ! [A: code_natural,N: nat] :
      ( ( ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se569199155075624693atural @ N @ A )
            = zero_z2226904508553997617atural ) )
        & ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se569199155075624693atural @ N @ A )
            = ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) @ one_one_Code_natural ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_4500_stable__imp__take__bit__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2923211474154528505it_int @ N @ A )
            = zero_zero_int ) )
        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2923211474154528505it_int @ N @ A )
            = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_4501_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger
        @ ( zero_n356916108424825756nteger
          @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 )
             != zero_z3403309356797280102nteger )
            & ( ord_less_eq_nat @ N @ M2 ) ) )
        @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4502_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat
        @ ( zero_n2687167440665602831ol_nat
          @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
             != zero_zero_nat )
            & ( ord_less_eq_nat @ N @ M2 ) ) )
        @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4503_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide5121882707175180666atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( times_2397367101498566445atural
        @ ( zero_n8403883297036319079atural
          @ ( ( ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 )
             != zero_z2226904508553997617atural )
            & ( ord_less_eq_nat @ N @ M2 ) ) )
        @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4504_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int
        @ ( zero_n2684676970156552555ol_int
          @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 )
             != zero_zero_int )
            & ( ord_less_eq_nat @ N @ M2 ) ) )
        @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4505_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_4506_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_4507_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_4508_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K4: int,L3: int] :
          ( if_int
          @ ( ( member_int2 @ K4 @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int2 @ L3 @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( 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_4509_and__int_Oelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
        = Y )
     => ( ( ( ( member_int2 @ X @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int2 @ Xa @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( 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_int2 @ X @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int2 @ Xa @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( 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_4510_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_4511_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_4512_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_4513_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_4514_upto__rec__numeral_I4_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( 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 @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_4515_upto__rec__numeral_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
          = ( 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 @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_4516_Int__insert__right__if1,axiom,
    ! [A: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ A @ A3 )
     => ( ( inf_in1906310914598751387_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ A @ B3 ) )
        = ( insert5175938949040314269_nat_o @ A @ ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_4517_Int__insert__right__if1,axiom,
    ! [A: int,A3: set_int,B3: set_int] :
      ( ( member_int2 @ A @ A3 )
     => ( ( inf_inf_set_int @ A3 @ ( insert_int2 @ A @ B3 ) )
        = ( insert_int2 @ A @ ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_4518_Int__insert__right__if1,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A3 )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A @ B3 ) )
        = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_4519_Int__insert__right__if1,axiom,
    ! [A: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat2 @ A @ A3 )
     => ( ( inf_inf_set_nat @ A3 @ ( insert_nat2 @ A @ B3 ) )
        = ( insert_nat2 @ A @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_4520_Int__insert__right__if0,axiom,
    ! [A: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ A @ A3 )
     => ( ( inf_in1906310914598751387_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ A @ B3 ) )
        = ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) ) ) ).

% Int_insert_right_if0
thf(fact_4521_Int__insert__right__if0,axiom,
    ! [A: int,A3: set_int,B3: set_int] :
      ( ~ ( member_int2 @ A @ A3 )
     => ( ( inf_inf_set_int @ A3 @ ( insert_int2 @ A @ B3 ) )
        = ( inf_inf_set_int @ A3 @ B3 ) ) ) ).

% Int_insert_right_if0
thf(fact_4522_Int__insert__right__if0,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ A @ A3 )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A @ B3 ) )
        = ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ).

% Int_insert_right_if0
thf(fact_4523_Int__insert__right__if0,axiom,
    ! [A: nat,A3: set_nat,B3: set_nat] :
      ( ~ ( member_nat2 @ A @ A3 )
     => ( ( inf_inf_set_nat @ A3 @ ( insert_nat2 @ A @ B3 ) )
        = ( inf_inf_set_nat @ A3 @ B3 ) ) ) ).

% Int_insert_right_if0
thf(fact_4524_insert__inter__insert,axiom,
    ! [A: int,A3: set_int,B3: set_int] :
      ( ( inf_inf_set_int @ ( insert_int2 @ A @ A3 ) @ ( insert_int2 @ A @ B3 ) )
      = ( insert_int2 @ A @ ( inf_inf_set_int @ A3 @ B3 ) ) ) ).

% insert_inter_insert
thf(fact_4525_insert__inter__insert,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A3 ) @ ( insert8211810215607154385at_nat @ A @ B3 ) )
      = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ).

% insert_inter_insert
thf(fact_4526_insert__inter__insert,axiom,
    ! [A: nat,A3: set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ A3 ) @ ( insert_nat2 @ A @ B3 ) )
      = ( insert_nat2 @ A @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ).

% insert_inter_insert
thf(fact_4527_Int__insert__left__if1,axiom,
    ! [A: produc3658429121746597890et_nat > $o,C2: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ A @ C2 )
     => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A @ B3 ) @ C2 )
        = ( insert5175938949040314269_nat_o @ A @ ( inf_in1906310914598751387_nat_o @ B3 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_4528_Int__insert__left__if1,axiom,
    ! [A: int,C2: set_int,B3: set_int] :
      ( ( member_int2 @ A @ C2 )
     => ( ( inf_inf_set_int @ ( insert_int2 @ A @ B3 ) @ C2 )
        = ( insert_int2 @ A @ ( inf_inf_set_int @ B3 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_4529_Int__insert__left__if1,axiom,
    ! [A: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ C2 )
     => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B3 ) @ C2 )
        = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ B3 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_4530_Int__insert__left__if1,axiom,
    ! [A: nat,C2: set_nat,B3: set_nat] :
      ( ( member_nat2 @ A @ C2 )
     => ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B3 ) @ C2 )
        = ( insert_nat2 @ A @ ( inf_inf_set_nat @ B3 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_4531_Int__insert__left__if0,axiom,
    ! [A: produc3658429121746597890et_nat > $o,C2: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ A @ C2 )
     => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A @ B3 ) @ C2 )
        = ( inf_in1906310914598751387_nat_o @ B3 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_4532_Int__insert__left__if0,axiom,
    ! [A: int,C2: set_int,B3: set_int] :
      ( ~ ( member_int2 @ A @ C2 )
     => ( ( inf_inf_set_int @ ( insert_int2 @ A @ B3 ) @ C2 )
        = ( inf_inf_set_int @ B3 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_4533_Int__insert__left__if0,axiom,
    ! [A: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ A @ C2 )
     => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B3 ) @ C2 )
        = ( inf_in2572325071724192079at_nat @ B3 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_4534_Int__insert__left__if0,axiom,
    ! [A: nat,C2: set_nat,B3: set_nat] :
      ( ~ ( member_nat2 @ A @ C2 )
     => ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B3 ) @ C2 )
        = ( inf_inf_set_nat @ B3 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_4535_insert__disjoint_I1_J,axiom,
    ! [A: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A @ A3 ) @ B3 )
        = bot_bo7824918357723582233_nat_o )
      = ( ~ ( member6576561426505652726_nat_o @ A @ B3 )
        & ( ( inf_in1906310914598751387_nat_o @ A3 @ B3 )
          = bot_bo7824918357723582233_nat_o ) ) ) ).

% insert_disjoint(1)
thf(fact_4536_insert__disjoint_I1_J,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A3 ) @ B3 )
        = bot_bo2099793752762293965at_nat )
      = ( ~ ( member8440522571783428010at_nat @ A @ B3 )
        & ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_4537_insert__disjoint_I1_J,axiom,
    ! [A: nat,A3: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ A3 ) @ B3 )
        = bot_bot_set_nat )
      = ( ~ ( member_nat2 @ A @ B3 )
        & ( ( inf_inf_set_nat @ A3 @ B3 )
          = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_4538_insert__disjoint_I1_J,axiom,
    ! [A: int,A3: set_int,B3: set_int] :
      ( ( ( inf_inf_set_int @ ( insert_int2 @ A @ A3 ) @ B3 )
        = bot_bot_set_int )
      = ( ~ ( member_int2 @ A @ B3 )
        & ( ( inf_inf_set_int @ A3 @ B3 )
          = bot_bot_set_int ) ) ) ).

% insert_disjoint(1)
thf(fact_4539_insert__disjoint_I2_J,axiom,
    ! [A: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( bot_bo7824918357723582233_nat_o
        = ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A @ A3 ) @ B3 ) )
      = ( ~ ( member6576561426505652726_nat_o @ A @ B3 )
        & ( bot_bo7824918357723582233_nat_o
          = ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_4540_insert__disjoint_I2_J,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A3 ) @ B3 ) )
      = ( ~ ( member8440522571783428010at_nat @ A @ B3 )
        & ( bot_bo2099793752762293965at_nat
          = ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_4541_insert__disjoint_I2_J,axiom,
    ! [A: nat,A3: set_nat,B3: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ ( insert_nat2 @ A @ A3 ) @ B3 ) )
      = ( ~ ( member_nat2 @ A @ B3 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_4542_insert__disjoint_I2_J,axiom,
    ! [A: int,A3: set_int,B3: set_int] :
      ( ( bot_bot_set_int
        = ( inf_inf_set_int @ ( insert_int2 @ A @ A3 ) @ B3 ) )
      = ( ~ ( member_int2 @ A @ B3 )
        & ( bot_bot_set_int
          = ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_4543_disjoint__insert_I1_J,axiom,
    ! [B3: set_Pr4532377907799695533_nat_o,A: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o] :
      ( ( ( inf_in1906310914598751387_nat_o @ B3 @ ( insert5175938949040314269_nat_o @ A @ A3 ) )
        = bot_bo7824918357723582233_nat_o )
      = ( ~ ( member6576561426505652726_nat_o @ A @ B3 )
        & ( ( inf_in1906310914598751387_nat_o @ B3 @ A3 )
          = bot_bo7824918357723582233_nat_o ) ) ) ).

% disjoint_insert(1)
thf(fact_4544_disjoint__insert_I1_J,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ B3 @ ( insert8211810215607154385at_nat @ A @ A3 ) )
        = bot_bo2099793752762293965at_nat )
      = ( ~ ( member8440522571783428010at_nat @ A @ B3 )
        & ( ( inf_in2572325071724192079at_nat @ B3 @ A3 )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_4545_disjoint__insert_I1_J,axiom,
    ! [B3: set_nat,A: nat,A3: set_nat] :
      ( ( ( inf_inf_set_nat @ B3 @ ( insert_nat2 @ A @ A3 ) )
        = bot_bot_set_nat )
      = ( ~ ( member_nat2 @ A @ B3 )
        & ( ( inf_inf_set_nat @ B3 @ A3 )
          = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_4546_disjoint__insert_I1_J,axiom,
    ! [B3: set_int,A: int,A3: set_int] :
      ( ( ( inf_inf_set_int @ B3 @ ( insert_int2 @ A @ A3 ) )
        = bot_bot_set_int )
      = ( ~ ( member_int2 @ A @ B3 )
        & ( ( inf_inf_set_int @ B3 @ A3 )
          = bot_bot_set_int ) ) ) ).

% disjoint_insert(1)
thf(fact_4547_disjoint__insert_I2_J,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( bot_bo7824918357723582233_nat_o
        = ( inf_in1906310914598751387_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ B @ B3 ) ) )
      = ( ~ ( member6576561426505652726_nat_o @ B @ A3 )
        & ( bot_bo7824918357723582233_nat_o
          = ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_4548_disjoint__insert_I2_J,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ B @ B3 ) ) )
      = ( ~ ( member8440522571783428010at_nat @ B @ A3 )
        & ( bot_bo2099793752762293965at_nat
          = ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_4549_disjoint__insert_I2_J,axiom,
    ! [A3: set_nat,B: nat,B3: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ A3 @ ( insert_nat2 @ B @ B3 ) ) )
      = ( ~ ( member_nat2 @ B @ A3 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_4550_disjoint__insert_I2_J,axiom,
    ! [A3: set_int,B: int,B3: set_int] :
      ( ( bot_bot_set_int
        = ( inf_inf_set_int @ A3 @ ( insert_int2 @ B @ B3 ) ) )
      = ( ~ ( member_int2 @ B @ A3 )
        & ( bot_bot_set_int
          = ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_4551_upto__Nil,axiom,
    ! [I2: int,J: int] :
      ( ( ( upto @ I2 @ J )
        = nil_int )
      = ( ord_less_int @ J @ I2 ) ) ).

% upto_Nil
thf(fact_4552_upto__Nil2,axiom,
    ! [I2: int,J: int] :
      ( ( nil_int
        = ( upto @ I2 @ J ) )
      = ( ord_less_int @ J @ I2 ) ) ).

% upto_Nil2
thf(fact_4553_upto__empty,axiom,
    ! [J: int,I2: int] :
      ( ( ord_less_int @ J @ I2 )
     => ( ( upto @ I2 @ J )
        = nil_int ) ) ).

% upto_empty
thf(fact_4554_upto__single,axiom,
    ! [I2: int] :
      ( ( upto @ I2 @ I2 )
      = ( cons_int @ I2 @ nil_int ) ) ).

% upto_single
thf(fact_4555_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_4556_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_4557_upto__rec__numeral_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_4558_upto__rec__numeral_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_4559_Int__insert__right,axiom,
    ! [A: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( ( member6576561426505652726_nat_o @ A @ A3 )
       => ( ( inf_in1906310914598751387_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ A @ B3 ) )
          = ( insert5175938949040314269_nat_o @ A @ ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) ) ) )
      & ( ~ ( member6576561426505652726_nat_o @ A @ A3 )
       => ( ( inf_in1906310914598751387_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ A @ B3 ) )
          = ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) ) ) ) ).

% Int_insert_right
thf(fact_4560_Int__insert__right,axiom,
    ! [A: int,A3: set_int,B3: set_int] :
      ( ( ( member_int2 @ A @ A3 )
       => ( ( inf_inf_set_int @ A3 @ ( insert_int2 @ A @ B3 ) )
          = ( insert_int2 @ A @ ( inf_inf_set_int @ A3 @ B3 ) ) ) )
      & ( ~ ( member_int2 @ A @ A3 )
       => ( ( inf_inf_set_int @ A3 @ ( insert_int2 @ A @ B3 ) )
          = ( inf_inf_set_int @ A3 @ B3 ) ) ) ) ).

% Int_insert_right
thf(fact_4561_Int__insert__right,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A @ B3 ) )
          = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ A @ A3 )
       => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A @ B3 ) )
          = ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ).

% Int_insert_right
thf(fact_4562_Int__insert__right,axiom,
    ! [A: nat,A3: set_nat,B3: set_nat] :
      ( ( ( member_nat2 @ A @ A3 )
       => ( ( inf_inf_set_nat @ A3 @ ( insert_nat2 @ A @ B3 ) )
          = ( insert_nat2 @ A @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) )
      & ( ~ ( member_nat2 @ A @ A3 )
       => ( ( inf_inf_set_nat @ A3 @ ( insert_nat2 @ A @ B3 ) )
          = ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).

% Int_insert_right
thf(fact_4563_Int__insert__left,axiom,
    ! [A: produc3658429121746597890et_nat > $o,C2: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( ( member6576561426505652726_nat_o @ A @ C2 )
       => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A @ B3 ) @ C2 )
          = ( insert5175938949040314269_nat_o @ A @ ( inf_in1906310914598751387_nat_o @ B3 @ C2 ) ) ) )
      & ( ~ ( member6576561426505652726_nat_o @ A @ C2 )
       => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A @ B3 ) @ C2 )
          = ( inf_in1906310914598751387_nat_o @ B3 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_4564_Int__insert__left,axiom,
    ! [A: int,C2: set_int,B3: set_int] :
      ( ( ( member_int2 @ A @ C2 )
       => ( ( inf_inf_set_int @ ( insert_int2 @ A @ B3 ) @ C2 )
          = ( insert_int2 @ A @ ( inf_inf_set_int @ B3 @ C2 ) ) ) )
      & ( ~ ( member_int2 @ A @ C2 )
       => ( ( inf_inf_set_int @ ( insert_int2 @ A @ B3 ) @ C2 )
          = ( inf_inf_set_int @ B3 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_4565_Int__insert__left,axiom,
    ! [A: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ A @ C2 )
       => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B3 ) @ C2 )
          = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ B3 @ C2 ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ A @ C2 )
       => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B3 ) @ C2 )
          = ( inf_in2572325071724192079at_nat @ B3 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_4566_Int__insert__left,axiom,
    ! [A: nat,C2: set_nat,B3: set_nat] :
      ( ( ( member_nat2 @ A @ C2 )
       => ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B3 ) @ C2 )
          = ( insert_nat2 @ A @ ( inf_inf_set_nat @ B3 @ C2 ) ) ) )
      & ( ~ ( member_nat2 @ A @ C2 )
       => ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B3 ) @ C2 )
          = ( inf_inf_set_nat @ B3 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_4567_upto__code,axiom,
    ( upto
    = ( ^ [I4: int,J2: int] : ( upto_aux @ I4 @ J2 @ nil_int ) ) ) ).

% upto_code
thf(fact_4568_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K4: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K4 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_4569_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_4570_upto_Osimps,axiom,
    ( upto
    = ( ^ [I4: int,J2: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J2 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J2 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_4571_upto__rec1,axiom,
    ! [I2: int,J: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( upto @ I2 @ J )
        = ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) ) ).

% upto_rec1
thf(fact_4572_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_4573_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ L ) @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) )
      = ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( pred_numeral @ L ) @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_4574_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se569199155075624693atural @ ( numeral_numeral_nat @ L ) @ ( numera5444537566228673987atural @ ( bit0 @ K ) ) )
      = ( times_2397367101498566445atural @ ( bit_se569199155075624693atural @ ( pred_numeral @ L ) @ ( numera5444537566228673987atural @ K ) ) @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_4575_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_4576_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
     => ( ( ( ( member_int2 @ K @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int2 @ L @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( 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_int2 @ K @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int2 @ L @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( 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_4577_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_int2 @ X @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int2 @ Xa @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( 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_int2 @ X @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int2 @ Xa @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( 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_4578_mask__numeral,axiom,
    ! [N: num] :
      ( ( bit_se2119862282449309892nteger @ ( numeral_numeral_nat @ N ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ ( pred_numeral @ N ) ) ) ) ) ).

% mask_numeral
thf(fact_4579_mask__numeral,axiom,
    ! [N: num] :
      ( ( bit_se943457434206027407atural @ ( numeral_numeral_nat @ N ) )
      = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se943457434206027407atural @ ( pred_numeral @ N ) ) ) ) ) ).

% mask_numeral
thf(fact_4580_mask__numeral,axiom,
    ! [N: num] :
      ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N ) ) ) ) ) ).

% mask_numeral
thf(fact_4581_mask__numeral,axiom,
    ! [N: num] :
      ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N ) ) ) ) ) ).

% mask_numeral
thf(fact_4582_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = ( bit_se2000444600071755411sk_int @ N ) )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).

% take_bit_eq_mask_iff_exp_dvd
thf(fact_4583_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_4584_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_4585_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_4586_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_4587_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_4588_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_4589_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_4590_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_4591_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_4592_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_4593_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_4594_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_4595_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_4596_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_4597_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_4598_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_4599_mask__Suc__0,axiom,
    ( ( bit_se2119862282449309892nteger @ ( suc @ zero_zero_nat ) )
    = one_one_Code_integer ) ).

% mask_Suc_0
thf(fact_4600_mask__Suc__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% mask_Suc_0
thf(fact_4601_mask__Suc__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% mask_Suc_0
thf(fact_4602_take__bit__minus__one__eq__mask,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( bit_se2119862282449309892nteger @ N ) ) ).

% take_bit_minus_one_eq_mask
thf(fact_4603_take__bit__minus__one__eq__mask,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( bit_se2000444600071755411sk_int @ N ) ) ).

% take_bit_minus_one_eq_mask
thf(fact_4604_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ).

% xor_numerals(3)
thf(fact_4605_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ).

% xor_numerals(3)
thf(fact_4606_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% xor_numerals(3)
thf(fact_4607_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% xor_numerals(3)
thf(fact_4608_xor__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) ) ).

% xor_numerals(1)
thf(fact_4609_xor__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se2046307713759805098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = ( numera5444537566228673987atural @ ( bit1 @ Y ) ) ) ).

% xor_numerals(1)
thf(fact_4610_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_4611_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_4612_xor__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) ) ).

% xor_numerals(2)
thf(fact_4613_xor__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se2046307713759805098atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = ( numera5444537566228673987atural @ ( bit0 @ Y ) ) ) ).

% xor_numerals(2)
thf(fact_4614_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_4615_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_4616_xor__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( bit1 @ X ) ) ) ).

% xor_numerals(5)
thf(fact_4617_xor__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( bit1 @ X ) ) ) ).

% xor_numerals(5)
thf(fact_4618_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_4619_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_4620_xor__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( bit0 @ X ) ) ) ).

% xor_numerals(8)
thf(fact_4621_xor__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( bit0 @ X ) ) ) ).

% xor_numerals(8)
thf(fact_4622_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_4623_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_4624_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ).

% xor_numerals(7)
thf(fact_4625_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se2046307713759805098atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ).

% xor_numerals(7)
thf(fact_4626_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% xor_numerals(7)
thf(fact_4627_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% xor_numerals(7)
thf(fact_4628_flip__bit__nat__def,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [M3: nat,N3: nat] : ( bit_se6528837805403552850or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).

% flip_bit_nat_def
thf(fact_4629_flip__bit__int__def,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N3: nat,K4: int] : ( bit_se6526347334894502574or_int @ K4 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).

% flip_bit_int_def
thf(fact_4630_flip__bit__eq__xor,axiom,
    ( bit_se1345352211410354436nteger
    = ( ^ [N3: nat,A2: code_integer] : ( bit_se3222712562003087583nteger @ A2 @ ( bit_se7788150548672797655nteger @ N3 @ one_one_Code_integer ) ) ) ) ).

% flip_bit_eq_xor
thf(fact_4631_flip__bit__eq__xor,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N3: nat,A2: int] : ( bit_se6526347334894502574or_int @ A2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).

% flip_bit_eq_xor
thf(fact_4632_flip__bit__eq__xor,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [N3: nat,A2: nat] : ( bit_se6528837805403552850or_nat @ A2 @ ( bit_se547839408752420682it_nat @ N3 @ one_one_nat ) ) ) ) ).

% flip_bit_eq_xor
thf(fact_4633_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = ( bit_se2000444600071755411sk_int @ N ) )
      = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
        = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
thf(fact_4634_mask__half__int,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% mask_half_int
thf(fact_4635_mask__nat__def,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).

% mask_nat_def
thf(fact_4636_mask__int__def,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).

% mask_int_def
thf(fact_4637_mask__eq__exp__minus__1,axiom,
    ( bit_se2119862282449309892nteger
    = ( ^ [N3: nat] : ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) @ one_one_Code_integer ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_4638_mask__eq__exp__minus__1,axiom,
    ( bit_se943457434206027407atural
    = ( ^ [N3: nat] : ( minus_7197305767214868737atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N3 ) @ one_one_Code_natural ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_4639_mask__eq__exp__minus__1,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_4640_mask__eq__exp__minus__1,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_4641_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P2: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [K2: int,L4: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L4 ) )
           => ( ( ~ ( ( member_int2 @ K2 @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int2 @ L4 @ ( insert_int2 @ zero_zero_int @ ( insert_int2 @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P2 @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P2 @ K2 @ L4 ) ) )
       => ( P2 @ A0 @ A1 ) ) ) ).

% and_int.pinduct
thf(fact_4642_upto_Opsimps,axiom,
    ! [I2: int,J: int] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J ) )
     => ( ( ( ord_less_eq_int @ I2 @ J )
         => ( ( upto @ I2 @ J )
            = ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) )
        & ( ~ ( ord_less_eq_int @ I2 @ J )
         => ( ( upto @ I2 @ J )
            = nil_int ) ) ) ) ).

% upto.psimps
thf(fact_4643_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_4644_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P2: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [I3: int,J3: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J3 ) )
           => ( ( ( ord_less_eq_int @ I3 @ J3 )
               => ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) )
             => ( P2 @ I3 @ J3 ) ) )
       => ( P2 @ A0 @ A1 ) ) ) ).

% upto.pinduct
thf(fact_4645_nat0__intermed__int__val,axiom,
    ! [N: nat,F2: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F2 @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F2 @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F2 @ N ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N )
              & ( ( F2 @ I3 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_4646_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_4647_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_4648_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_4649_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_4650_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_4651_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_4652_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_4653_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_4654_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_4655_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_4656_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_4657_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_4658_and__not__numerals_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( 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 @ N ) ) ) ) ) ) ).

% and_not_numerals(8)
thf(fact_4659_abs__idempotent,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_idempotent
thf(fact_4660_abs__idempotent,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_idempotent
thf(fact_4661_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_4662_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_4663_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_4664_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_4665_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_4666_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_4667_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_4668_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_4669_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_4670_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_4671_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_4672_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_4673_abs__mult__self__eq,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
      = ( times_times_int @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_4674_abs__mult__self__eq,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
      = ( times_3573771949741848930nteger @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_4675_abs__1,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_1
thf(fact_4676_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_4677_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_4678_abs__minus__cancel,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus_cancel
thf(fact_4679_abs__minus__cancel,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus_cancel
thf(fact_4680_abs__minus__cancel,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus_cancel
thf(fact_4681_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_4682_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_4683_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_4684_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_4685_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_4686_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_4687_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_4688_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_4689_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_4690_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_4691_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_4692_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_4693_abs__neg__one,axiom,
    ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
    = one_one_int ) ).

% abs_neg_one
thf(fact_4694_abs__neg__one,axiom,
    ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = one_one_rat ) ).

% abs_neg_one
thf(fact_4695_abs__neg__one,axiom,
    ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = one_one_Code_integer ) ).

% abs_neg_one
thf(fact_4696_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_4697_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_4698_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_4699_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_4700_zdvd1__eq,axiom,
    ! [X: int] :
      ( ( dvd_dvd_int @ X @ one_one_int )
      = ( ( abs_abs_int @ X )
        = one_one_int ) ) ).

% zdvd1_eq
thf(fact_4701_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_nonpos
thf(fact_4702_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_4703_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_4704_bit_Ocompl__zero,axiom,
    ( ( bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.compl_zero
thf(fact_4705_bit_Ocompl__zero,axiom,
    ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.compl_zero
thf(fact_4706_bit_Ocompl__one,axiom,
    ( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% bit.compl_one
thf(fact_4707_bit_Ocompl__one,axiom,
    ( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ one_one_int ) )
    = zero_zero_int ) ).

% bit.compl_one
thf(fact_4708_or__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) ) ).

% or_numerals(2)
thf(fact_4709_or__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se9127793120404214118atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit1 @ Y ) ) )
      = ( numera5444537566228673987atural @ ( bit1 @ Y ) ) ) ).

% or_numerals(2)
thf(fact_4710_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_4711_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_4712_or__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( bit1 @ X ) ) ) ).

% or_numerals(8)
thf(fact_4713_or__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit1 @ X ) ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( bit1 @ X ) ) ) ).

% or_numerals(8)
thf(fact_4714_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_4715_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_4716_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_4717_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_4718_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_4719_or__minus__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).

% or_minus_numerals(2)
thf(fact_4720_or__minus__numerals_I6_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).

% or_minus_numerals(6)
thf(fact_4721_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_4722_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_4723_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_4724_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_4725_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_4726_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_4727_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_4728_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_4729_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_4730_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_4731_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_4732_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_4733_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
      = ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
thf(fact_4734_or__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ).

% or_numerals(3)
thf(fact_4735_or__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ X ) @ ( numera5444537566228673987atural @ Y ) ) ) ) ).

% or_numerals(3)
thf(fact_4736_or__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% or_numerals(3)
thf(fact_4737_or__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% or_numerals(3)
thf(fact_4738_or__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se1080825931792720795nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( bit1 @ X ) ) ) ).

% or_numerals(5)
thf(fact_4739_or__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se9127793120404214118atural @ ( numera5444537566228673987atural @ ( bit0 @ X ) ) @ one_one_Code_natural )
      = ( numera5444537566228673987atural @ ( bit1 @ X ) ) ) ).

% or_numerals(5)
thf(fact_4740_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_4741_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_4742_or__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) ) ).

% or_numerals(1)
thf(fact_4743_or__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se9127793120404214118atural @ one_one_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ Y ) ) )
      = ( numera5444537566228673987atural @ ( bit1 @ Y ) ) ) ).

% or_numerals(1)
thf(fact_4744_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_4745_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_4746_push__bit__minus__one__eq__not__mask,axiom,
    ! [N: nat] :
      ( ( bit_se7788150548672797655nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N ) ) ) ).

% push_bit_minus_one_eq_not_mask
thf(fact_4747_push__bit__minus__one__eq__not__mask,axiom,
    ! [N: nat] :
      ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).

% push_bit_minus_one_eq_not_mask
thf(fact_4748_not__one__eq,axiom,
    ( ( bit_ri7632146776885996613nteger @ one_one_Code_integer )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% not_one_eq
thf(fact_4749_not__one__eq,axiom,
    ( ( bit_ri7919022796975470100ot_int @ one_one_int )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% not_one_eq
thf(fact_4750_or__minus__minus__numerals,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).

% or_minus_minus_numerals
thf(fact_4751_and__minus__minus__numerals,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).

% and_minus_minus_numerals
thf(fact_4752_or__not__numerals_I1_J,axiom,
    ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
    = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).

% or_not_numerals(1)
thf(fact_4753_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_4754_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_4755_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_4756_abs__ge__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_self
thf(fact_4757_abs__ge__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).

% abs_ge_self
thf(fact_4758_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_4759_abs__mult,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
      = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_mult
thf(fact_4760_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_mult
thf(fact_4761_abs__one,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_one
thf(fact_4762_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_4763_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_4764_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_4765_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_4766_set__bit__nat__def,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [M3: nat,N3: nat] : ( bit_se1412395901928357646or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).

% set_bit_nat_def
thf(fact_4767_or__not__numerals_I4_J,axiom,
    ! [M2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).

% or_not_numerals(4)
thf(fact_4768_or__not__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).

% or_not_numerals(2)
thf(fact_4769_or__not__numerals_I3_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).

% or_not_numerals(3)
thf(fact_4770_or__not__numerals_I7_J,axiom,
    ! [M2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).

% or_not_numerals(7)
thf(fact_4771_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_4772_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_4773_abs__ge__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_zero
thf(fact_4774_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_4775_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_4776_abs__not__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_4777_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_4778_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_4779_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_4780_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_4781_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_4782_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_4783_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_4784_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_4785_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_4786_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_4787_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_4788_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_4789_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_4790_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_4791_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_4792_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_4793_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_4794_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D2 )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_4795_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
     => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D2 )
       => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_4796_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
     => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D2 )
       => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_4797_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_4798_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_4799_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_4800_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_4801_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_4802_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_4803_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_4804_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_4805_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_4806_abs__ge__minus__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_minus_self
thf(fact_4807_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_4808_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_4809_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_rat
    = ( ^ [K4: rat] : ( times_times_rat @ K4 @ ( sgn_sgn_rat @ K4 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_4810_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_int
    = ( ^ [K4: int] : ( times_times_int @ K4 @ ( sgn_sgn_int @ K4 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_4811_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_Code_integer
    = ( ^ [K4: code_integer] : ( times_3573771949741848930nteger @ K4 @ ( sgn_sgn_Code_integer @ K4 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_4812_abs__mult__sgn,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( sgn_sgn_rat @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_4813_abs__mult__sgn,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( abs_abs_int @ A ) @ ( sgn_sgn_int @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_4814_abs__mult__sgn,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_4815_sgn__mult__abs,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( abs_abs_rat @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_4816_sgn__mult__abs,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( abs_abs_int @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_4817_sgn__mult__abs,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_4818_mult__sgn__abs,axiom,
    ! [X: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ X ) @ ( abs_abs_rat @ X ) )
      = X ) ).

% mult_sgn_abs
thf(fact_4819_mult__sgn__abs,axiom,
    ! [X: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ X ) @ ( abs_abs_int @ X ) )
      = X ) ).

% mult_sgn_abs
thf(fact_4820_mult__sgn__abs,axiom,
    ! [X: code_integer] :
      ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ X ) @ ( abs_abs_Code_integer @ X ) )
      = X ) ).

% mult_sgn_abs
thf(fact_4821_abs__zmult__eq__1,axiom,
    ! [M2: int,N: int] :
      ( ( ( abs_abs_int @ ( times_times_int @ M2 @ N ) )
        = one_one_int )
     => ( ( abs_abs_int @ M2 )
        = one_one_int ) ) ).

% abs_zmult_eq_1
thf(fact_4822_set__bit__int__def,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N3: nat,K4: int] : ( bit_se1409905431419307370or_int @ K4 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).

% set_bit_int_def
thf(fact_4823_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_4824_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_4825_or__not__numerals_I5_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( 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 @ N ) ) ) ) ) ) ).

% or_not_numerals(5)
thf(fact_4826_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E ) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_4827_abs__mult__pos,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X )
        = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_4828_abs__mult__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
        = ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_4829_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
        = ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_4830_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
        & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
          | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
     => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
        = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_4831_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
        & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
          | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
     => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
        = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_4832_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          | ( ord_less_eq_int @ A @ zero_zero_int ) )
        & ( ( ord_less_eq_int @ zero_zero_int @ B )
          | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
        = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_4833_abs__minus__le__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_4834_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_4835_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_4836_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_4837_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_4838_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_4839_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_4840_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_4841_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_4842_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_4843_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_4844_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_neg
thf(fact_4845_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_4846_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_4847_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_4848_minus__eq__not__plus__1,axiom,
    ( uminus1351360451143612070nteger
    = ( ^ [A2: code_integer] : ( plus_p5714425477246183910nteger @ ( bit_ri7632146776885996613nteger @ A2 ) @ one_one_Code_integer ) ) ) ).

% minus_eq_not_plus_1
thf(fact_4849_minus__eq__not__plus__1,axiom,
    ( uminus_uminus_int
    = ( ^ [A2: int] : ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A2 ) @ one_one_int ) ) ) ).

% minus_eq_not_plus_1
thf(fact_4850_minus__eq__not__minus__1,axiom,
    ( uminus1351360451143612070nteger
    = ( ^ [A2: code_integer] : ( bit_ri7632146776885996613nteger @ ( minus_8373710615458151222nteger @ A2 @ one_one_Code_integer ) ) ) ) ).

% minus_eq_not_minus_1
thf(fact_4851_minus__eq__not__minus__1,axiom,
    ( uminus_uminus_int
    = ( ^ [A2: int] : ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A2 @ one_one_int ) ) ) ) ).

% minus_eq_not_minus_1
thf(fact_4852_not__eq__complement,axiom,
    ( bit_ri7632146776885996613nteger
    = ( ^ [A2: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A2 ) @ one_one_Code_integer ) ) ) ).

% not_eq_complement
thf(fact_4853_not__eq__complement,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [A2: int] : ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ one_one_int ) ) ) ).

% not_eq_complement
thf(fact_4854_not__int__def,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K4: int] : ( minus_minus_int @ ( uminus_uminus_int @ K4 ) @ one_one_int ) ) ) ).

% not_int_def
thf(fact_4855_and__not__numerals_I1_J,axiom,
    ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
    = zero_zero_int ) ).

% and_not_numerals(1)
thf(fact_4856_set__bit__eq__or,axiom,
    ( bit_se2793503036327961859nteger
    = ( ^ [N3: nat,A2: code_integer] : ( bit_se1080825931792720795nteger @ A2 @ ( bit_se7788150548672797655nteger @ N3 @ one_one_Code_integer ) ) ) ) ).

% set_bit_eq_or
thf(fact_4857_set__bit__eq__or,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N3: nat,A2: int] : ( bit_se1409905431419307370or_int @ A2 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).

% set_bit_eq_or
thf(fact_4858_set__bit__eq__or,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [N3: nat,A2: nat] : ( bit_se1412395901928357646or_nat @ A2 @ ( bit_se547839408752420682it_nat @ N3 @ one_one_nat ) ) ) ) ).

% set_bit_eq_or
thf(fact_4859_or__not__numerals_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( 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 @ N ) ) ) ) ) ) ).

% or_not_numerals(8)
thf(fact_4860_or__not__numerals_I9_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( 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 @ N ) ) ) ) ) ) ).

% or_not_numerals(9)
thf(fact_4861_unset__bit__int__def,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N3: nat,K4: int] : ( bit_se725231765392027082nd_int @ K4 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ) ).

% unset_bit_int_def
thf(fact_4862_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_4863_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_4864_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_4865_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_4866_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_4867_and__not__numerals_I4_J,axiom,
    ! [M2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( bit0 @ M2 ) ) ) ).

% and_not_numerals(4)
thf(fact_4868_and__not__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = one_one_int ) ).

% and_not_numerals(2)
thf(fact_4869_zdvd__mult__cancel1,axiom,
    ! [M2: int,N: int] :
      ( ( M2 != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ M2 @ N ) @ M2 )
        = ( ( abs_abs_int @ N )
          = one_one_int ) ) ) ).

% zdvd_mult_cancel1
thf(fact_4870_unset__bit__eq__and__not,axiom,
    ( bit_se8260200283734997820nteger
    = ( ^ [N3: nat,A2: code_integer] : ( bit_se3949692690581998587nteger @ A2 @ ( bit_ri7632146776885996613nteger @ ( bit_se7788150548672797655nteger @ N3 @ one_one_Code_integer ) ) ) ) ) ).

% unset_bit_eq_and_not
thf(fact_4871_unset__bit__eq__and__not,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N3: nat,A2: int] : ( bit_se725231765392027082nd_int @ A2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ) ).

% unset_bit_eq_and_not
thf(fact_4872_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_4873_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_4874_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_4875_and__not__numerals_I7_J,axiom,
    ! [M2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( bit0 @ M2 ) ) ) ).

% and_not_numerals(7)
thf(fact_4876_and__not__numerals_I3_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = zero_zero_int ) ).

% and_not_numerals(3)
thf(fact_4877_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_4878_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_4879_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_4880_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_4881_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_4882_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_4883_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_4884_nat__intermed__int__val,axiom,
    ! [M2: nat,N: nat,F2: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ( ord_less_eq_nat @ M2 @ I3 )
            & ( ord_less_nat @ I3 @ N ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( ord_less_eq_int @ ( F2 @ M2 ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F2 @ N ) )
           => ? [I3: nat] :
                ( ( ord_less_eq_nat @ M2 @ I3 )
                & ( ord_less_eq_nat @ I3 @ N )
                & ( ( F2 @ I3 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_4885_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_4886_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_4887_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_4888_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_4889_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_4890_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_4891_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_4892_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_4893_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_4894_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_4895_mask__Suc__double,axiom,
    ! [N: nat] :
      ( ( bit_se2119862282449309892nteger @ ( suc @ N ) )
      = ( bit_se1080825931792720795nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N ) ) ) ) ).

% mask_Suc_double
thf(fact_4896_mask__Suc__double,axiom,
    ! [N: nat] :
      ( ( bit_se943457434206027407atural @ ( suc @ N ) )
      = ( bit_se9127793120404214118atural @ one_one_Code_natural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se943457434206027407atural @ N ) ) ) ) ).

% mask_Suc_double
thf(fact_4897_mask__Suc__double,axiom,
    ! [N: nat] :
      ( ( bit_se2000444600071755411sk_int @ ( suc @ N ) )
      = ( bit_se1409905431419307370or_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) ) ) ) ).

% mask_Suc_double
thf(fact_4898_mask__Suc__double,axiom,
    ! [N: nat] :
      ( ( bit_se2002935070580805687sk_nat @ ( suc @ N ) )
      = ( bit_se1412395901928357646or_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ) ).

% mask_Suc_double
thf(fact_4899_nat__ivt__aux,axiom,
    ! [N: nat,F2: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F2 @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F2 @ N ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N )
              & ( ( F2 @ I3 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_4900_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_4901_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_4902_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_4903_or__minus__numerals_I5_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(5)
thf(fact_4904_or__minus__numerals_I1_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(1)
thf(fact_4905_product__nth,axiom,
    ! [N: nat,Xs: list_h2745205591237434579t_unit,Ys: list_set_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s7011499401410611007t_unit @ Xs ) @ ( size_s3254054031482475050et_nat @ Ys ) ) )
     => ( ( nth_Pr7683458668995092411et_nat @ ( produc7481864959766642638et_nat @ Xs @ Ys ) @ N )
        = ( produc7507926704131184380et_nat @ ( nth_he7452997726511547988t_unit @ Xs @ ( divide_divide_nat @ N @ ( size_s3254054031482475050et_nat @ Ys ) ) ) @ ( nth_set_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_s3254054031482475050et_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_4906_product__nth,axiom,
    ! [N: nat,Xs: list_P7985473006766602707_nat_o,Ys: list_P9062070895058802706et_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s8908918004648414271_nat_o @ Xs ) @ ( size_s2864271983995610790et_nat @ Ys ) ) )
     => ( ( nth_Pr3610913089060847927et_nat @ ( produc2866688925348378442et_nat @ Xs @ Ys ) @ N )
        = ( produc5001842942810119800et_nat @ ( nth_Pr6840583978799713108_nat_o @ Xs @ ( divide_divide_nat @ N @ ( size_s2864271983995610790et_nat @ Ys ) ) ) @ ( nth_Pr7683458668995092411et_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_s2864271983995610790et_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_4907_product__nth,axiom,
    ! [N: nat,Xs: list_P7985473006766602707_nat_o,Ys: list_P2321686559999237006et_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s8908918004648414271_nat_o @ Xs ) @ ( size_s6841928198497307170et_nat @ Ys ) ) )
     => ( ( nth_Pr7590663101126261939et_nat @ ( produc7119410244218060998et_nat @ Xs @ Ys ) @ N )
        = ( produc2245416461498447860et_nat @ ( nth_Pr6840583978799713108_nat_o @ Xs @ ( divide_divide_nat @ N @ ( size_s6841928198497307170et_nat @ Ys ) ) ) @ ( nth_Pr3610913089060847927et_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6841928198497307170et_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_4908_product__nth,axiom,
    ! [N: nat,Xs: list_int,Ys: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr4439495888332055232nt_int @ ( product_int_int @ Xs @ Ys ) @ N )
        = ( product_Pair_int_int @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_4909_product__nth,axiom,
    ! [N: nat,Xs: list_int,Ys: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr8617346907841251940nt_nat @ ( product_int_nat @ Xs @ Ys ) @ N )
        = ( product_Pair_int_nat @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_4910_product__nth,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr3440142176431000676at_int @ ( product_nat_int @ Xs @ Ys ) @ N )
        = ( product_Pair_nat_int @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_4911_product__nth,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( product_nat_nat @ Xs @ Ys ) @ N )
        = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_4912_drop__bit__rec,axiom,
    ( bit_se3928097537394005634nteger
    = ( ^ [N3: nat,A2: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ A2 @ ( bit_se3928097537394005634nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% drop_bit_rec
thf(fact_4913_drop__bit__rec,axiom,
    ( bit_se8570568707652914677it_nat
    = ( ^ [N3: nat,A2: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ A2 @ ( bit_se8570568707652914677it_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% drop_bit_rec
thf(fact_4914_drop__bit__rec,axiom,
    ( bit_se2751692689150723149atural
    = ( ^ [N3: nat,A2: code_natural] : ( if_Code_natural @ ( N3 = zero_zero_nat ) @ A2 @ ( bit_se2751692689150723149atural @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% drop_bit_rec
thf(fact_4915_drop__bit__rec,axiom,
    ( bit_se8568078237143864401it_int
    = ( ^ [N3: nat,A2: int] : ( if_int @ ( N3 = zero_zero_nat ) @ A2 @ ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% drop_bit_rec
thf(fact_4916_bit__rec,axiom,
    ( bit_se9216721137139052372nteger
    = ( ^ [A2: code_integer,N3: nat] :
          ( ( ( N3 = zero_zero_nat )
           => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) )
          & ( ( N3 != zero_zero_nat )
           => ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_4917_bit__rec,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [A2: nat,N3: nat] :
          ( ( ( N3 = zero_zero_nat )
           => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) )
          & ( ( N3 != zero_zero_nat )
           => ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_4918_bit__rec,axiom,
    ( bit_se8040316288895769887atural
    = ( ^ [A2: code_natural,N3: nat] :
          ( ( ( N3 = zero_zero_nat )
           => ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A2 ) )
          & ( ( N3 != zero_zero_nat )
           => ( bit_se8040316288895769887atural @ ( divide5121882707175180666atural @ A2 @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_4919_bit__rec,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [A2: int,N3: nat] :
          ( ( ( N3 = zero_zero_nat )
           => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
          & ( ( N3 != zero_zero_nat )
           => ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_4920_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B: nat,N: 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 ) ) @ N )
        = ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_4921_bit__sum__mult__2__cases,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( ! [J3: nat] :
          ~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J3 ) )
     => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ N )
        = ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_4922_bit__sum__mult__2__cases,axiom,
    ! [A: code_natural,B: code_natural,N: nat] :
      ( ! [J3: nat] :
          ~ ( bit_se8040316288895769887atural @ A @ ( suc @ J3 ) )
     => ( ( bit_se8040316288895769887atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B ) ) @ N )
        = ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se8040316288895769887atural @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_4923_bit__sum__mult__2__cases,axiom,
    ! [A: int,B: int,N: 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 ) ) @ N )
        = ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_4924_max_Oright__idem,axiom,
    ! [A: int,B: int] :
      ( ( ord_max_int @ ( ord_max_int @ A @ B ) @ B )
      = ( ord_max_int @ A @ B ) ) ).

% max.right_idem
thf(fact_4925_max_Oleft__idem,axiom,
    ! [A: int,B: int] :
      ( ( ord_max_int @ A @ ( ord_max_int @ A @ B ) )
      = ( ord_max_int @ A @ B ) ) ).

% max.left_idem
thf(fact_4926_max_Oidem,axiom,
    ! [A: int] :
      ( ( ord_max_int @ A @ A )
      = A ) ).

% max.idem
thf(fact_4927_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_4928_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_4929_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_4930_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_4931_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_4932_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_4933_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_4934_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_4935_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_4936_length__map,axiom,
    ! [F2: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn] :
      ( ( size_size_list_assn @ ( map_Pr8991440229025900053n_assn @ F2 @ Xs ) )
      = ( size_s6829681357464350627n_assn @ Xs ) ) ).

% length_map
thf(fact_4937_length__map,axiom,
    ! [F2: int > int,Xs: list_int] :
      ( ( size_size_list_int @ ( map_int_int @ F2 @ Xs ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_map
thf(fact_4938_length__map,axiom,
    ! [F2: nat > int,Xs: list_nat] :
      ( ( size_size_list_int @ ( map_nat_int @ F2 @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_4939_length__map,axiom,
    ! [F2: int > nat,Xs: list_int] :
      ( ( size_size_list_nat @ ( map_int_nat @ F2 @ Xs ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_map
thf(fact_4940_length__map,axiom,
    ! [F2: nat > nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( map_nat_nat @ F2 @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_4941_max__less__iff__conj,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z )
      = ( ( ord_le6747313008572928689nteger @ X @ Z )
        & ( ord_le6747313008572928689nteger @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_4942_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ( ord_less_rat @ X @ Z )
        & ( ord_less_rat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_4943_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z )
      = ( ( ord_less_nat @ X @ Z )
        & ( ord_less_nat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_4944_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ( ord_less_int @ X @ Z )
        & ( ord_less_int @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_4945_max_Oabsorb4,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_max_Code_integer @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4946_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4947_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4948_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4949_max_Oabsorb3,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_max_Code_integer @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4950_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4951_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4952_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4953_drop__bit__minus__one,axiom,
    ! [N: nat] :
      ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% drop_bit_minus_one
thf(fact_4954_zip__eq__zip__same__len,axiom,
    ! [A: list_int,B: list_int,A5: list_int,B6: list_int] :
      ( ( ( size_size_list_int @ A )
        = ( size_size_list_int @ B ) )
     => ( ( ( size_size_list_int @ A5 )
          = ( size_size_list_int @ B6 ) )
       => ( ( ( zip_int_int @ A @ B )
            = ( zip_int_int @ A5 @ B6 ) )
          = ( ( A = A5 )
            & ( B = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_4955_zip__eq__zip__same__len,axiom,
    ! [A: list_int,B: list_nat,A5: list_int,B6: list_nat] :
      ( ( ( size_size_list_int @ A )
        = ( size_size_list_nat @ B ) )
     => ( ( ( size_size_list_int @ A5 )
          = ( size_size_list_nat @ B6 ) )
       => ( ( ( zip_int_nat @ A @ B )
            = ( zip_int_nat @ A5 @ B6 ) )
          = ( ( A = A5 )
            & ( B = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_4956_zip__eq__zip__same__len,axiom,
    ! [A: list_nat,B: list_int,A5: list_nat,B6: list_int] :
      ( ( ( size_size_list_nat @ A )
        = ( size_size_list_int @ B ) )
     => ( ( ( size_size_list_nat @ A5 )
          = ( size_size_list_int @ B6 ) )
       => ( ( ( zip_nat_int @ A @ B )
            = ( zip_nat_int @ A5 @ B6 ) )
          = ( ( A = A5 )
            & ( B = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_4957_zip__eq__zip__same__len,axiom,
    ! [A: list_nat,B: list_nat,A5: list_nat,B6: list_nat] :
      ( ( ( size_size_list_nat @ A )
        = ( size_size_list_nat @ B ) )
     => ( ( ( size_size_list_nat @ A5 )
          = ( size_size_list_nat @ B6 ) )
       => ( ( ( zip_nat_nat @ A @ B )
            = ( zip_nat_nat @ A5 @ B6 ) )
          = ( ( A = A5 )
            & ( B = B6 ) ) ) ) ) ).

% zip_eq_zip_same_len
thf(fact_4958_length__enumerate,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( size_s2970893825323803983at_int @ ( enumerate_int @ N @ Xs ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_enumerate
thf(fact_4959_length__enumerate,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_enumerate
thf(fact_4960_length__rotate1,axiom,
    ! [Xs: list_int] :
      ( ( size_size_list_int @ ( rotate1_int @ Xs ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_rotate1
thf(fact_4961_length__rotate1,axiom,
    ! [Xs: list_nat] :
      ( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_rotate1
thf(fact_4962_length__0__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( ( size_s6829681357464350627n_assn @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% length_0_conv
thf(fact_4963_length__0__conv,axiom,
    ! [Xs: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_int ) ) ).

% length_0_conv
thf(fact_4964_length__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_nat ) ) ).

% length_0_conv
thf(fact_4965_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_4966_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_4967_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_4968_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_4969_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_4970_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_4971_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_4972_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_4973_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_4974_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_4975_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_4976_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_4977_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_4978_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_4979_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_4980_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_4981_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_4982_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_4983_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_4984_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_4985_length__splice,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int @ ( splice_int @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_splice
thf(fact_4986_length__splice,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat @ ( splice_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_splice
thf(fact_4987_length__product,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_s5157815400016825771nt_int @ ( product_int_int @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_4988_length__product,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( ( size_s7647898544948552527nt_nat @ ( product_int_nat @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_4989_length__product,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( ( size_s2970893825323803983at_int @ ( product_nat_int @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_4990_length__product,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( product_nat_nat @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_4991_length__greater__0__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s6829681357464350627n_assn @ Xs ) )
      = ( Xs != nil_Pr5671120429643327159n_assn ) ) ).

% length_greater_0_conv
thf(fact_4992_length__greater__0__conv,axiom,
    ! [Xs: list_int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) )
      = ( Xs != nil_int ) ) ).

% length_greater_0_conv
thf(fact_4993_length__greater__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
      = ( Xs != nil_nat ) ) ).

% length_greater_0_conv
thf(fact_4994_nth__map,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn,F2: produc6575502325842934193n_assn > assn] :
      ( ( ord_less_nat @ N @ ( size_s6829681357464350627n_assn @ Xs ) )
     => ( ( nth_assn @ ( map_Pr8991440229025900053n_assn @ F2 @ Xs ) @ N )
        = ( F2 @ ( nth_Pr1769885009046257848n_assn @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_4995_nth__map,axiom,
    ! [N: nat,Xs: list_int,F2: int > nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( ( nth_nat @ ( map_int_nat @ F2 @ Xs ) @ N )
        = ( F2 @ ( nth_int @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_4996_nth__map,axiom,
    ! [N: nat,Xs: list_int,F2: int > int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( ( nth_int @ ( map_int_int @ F2 @ Xs ) @ N )
        = ( F2 @ ( nth_int @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_4997_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F2: nat > int] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_int @ ( map_nat_int @ F2 @ Xs ) @ N )
        = ( F2 @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_4998_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F2: nat > nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( map_nat_nat @ F2 @ Xs ) @ N )
        = ( F2 @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_4999_drop__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se3928097537394005634nteger @ N @ one_one_Code_integer )
      = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).

% drop_bit_of_1
thf(fact_5000_drop__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se8570568707652914677it_nat @ N @ one_one_nat )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% drop_bit_of_1
thf(fact_5001_drop__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se8568078237143864401it_int @ N @ one_one_int )
      = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).

% drop_bit_of_1
thf(fact_5002_map__fst__zip,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( map_Pr4307537318738222789at_int @ product_fst_int_nat @ ( zip_int_nat @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_5003_map__fst__zip,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( map_Pr6125295803703450217nt_nat @ product_fst_nat_int @ ( zip_nat_int @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_5004_map__fst__zip,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ ( zip_nat_nat @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_5005_map__fst__zip,axiom,
    ! [Xs: list_h2745205591237434579t_unit,Ys: list_set_nat] :
      ( ( ( size_s7011499401410611007t_unit @ Xs )
        = ( size_s3254054031482475050et_nat @ Ys ) )
     => ( ( map_Pr7042224834074979589t_unit @ produc1824681642469235216et_nat @ ( zip_he6711929980790699996et_nat @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_5006_map__fst__zip,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ ( zip_int_int @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_5007_map__fst__zip,axiom,
    ! [Xs: list_assn,Ys: list_assn] :
      ( ( ( size_size_list_assn @ Xs )
        = ( size_size_list_assn @ Ys ) )
     => ( ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ ( zip_assn_assn @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_5008_map__snd__zip,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( map_Pr4310027789247273065at_nat @ product_snd_int_nat @ ( zip_int_nat @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_5009_map__snd__zip,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( map_Pr6122805333194399941nt_int @ product_snd_nat_int @ ( zip_nat_int @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_5010_map__snd__zip,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat @ ( zip_nat_nat @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_5011_map__snd__zip,axiom,
    ! [Xs: list_h2745205591237434579t_unit,Ys: list_set_nat] :
      ( ( ( size_s7011499401410611007t_unit @ Xs )
        = ( size_s3254054031482475050et_nat @ Ys ) )
     => ( ( map_Pr4871828589360411470et_nat @ produc8586169260539613262et_nat @ ( zip_he6711929980790699996et_nat @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_5012_map__snd__zip,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ ( zip_int_int @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_5013_map__snd__zip,axiom,
    ! [Xs: list_assn,Ys: list_assn] :
      ( ( ( size_size_list_assn @ Xs )
        = ( size_size_list_assn @ Ys ) )
     => ( ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn @ ( zip_assn_assn @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_5014_rotate1__length01,axiom,
    ! [Xs: list_int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
     => ( ( rotate1_int @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_5015_rotate1__length01,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
     => ( ( rotate1_nat @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_5016_length__ge__1__conv,axiom,
    ! [L: list_P8527749157015355191n_assn] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_s6829681357464350627n_assn @ L ) )
      = ( L != nil_Pr5671120429643327159n_assn ) ) ).

% length_ge_1_conv
thf(fact_5017_length__ge__1__conv,axiom,
    ! [L: list_int] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_int @ L ) )
      = ( L != nil_int ) ) ).

% length_ge_1_conv
thf(fact_5018_length__ge__1__conv,axiom,
    ! [L: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_nat @ L ) )
      = ( L != nil_nat ) ) ).

% length_ge_1_conv
thf(fact_5019_nth__zip,axiom,
    ! [I2: nat,Xs: list_h2745205591237434579t_unit,Ys: list_set_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s7011499401410611007t_unit @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Ys ) )
       => ( ( nth_Pr7683458668995092411et_nat @ ( zip_he6711929980790699996et_nat @ Xs @ Ys ) @ I2 )
          = ( produc7507926704131184380et_nat @ ( nth_he7452997726511547988t_unit @ Xs @ I2 ) @ ( nth_set_nat @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_5020_nth__zip,axiom,
    ! [I2: nat,Xs: list_P7985473006766602707_nat_o,Ys: list_P9062070895058802706et_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s8908918004648414271_nat_o @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_s2864271983995610790et_nat @ Ys ) )
       => ( ( nth_Pr3610913089060847927et_nat @ ( zip_Pr7134870689397686104et_nat @ Xs @ Ys ) @ I2 )
          = ( produc5001842942810119800et_nat @ ( nth_Pr6840583978799713108_nat_o @ Xs @ I2 ) @ ( nth_Pr7683458668995092411et_nat @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_5021_nth__zip,axiom,
    ! [I2: nat,Xs: list_P7985473006766602707_nat_o,Ys: list_P2321686559999237006et_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s8908918004648414271_nat_o @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_s6841928198497307170et_nat @ Ys ) )
       => ( ( nth_Pr7590663101126261939et_nat @ ( zip_Pr8136144321567152340et_nat @ Xs @ Ys ) @ I2 )
          = ( produc2245416461498447860et_nat @ ( nth_Pr6840583978799713108_nat_o @ Xs @ I2 ) @ ( nth_Pr3610913089060847927et_nat @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_5022_nth__zip,axiom,
    ! [I2: nat,Xs: list_int,Ys: list_int] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Ys ) )
       => ( ( nth_Pr4439495888332055232nt_int @ ( zip_int_int @ Xs @ Ys ) @ I2 )
          = ( product_Pair_int_int @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_5023_nth__zip,axiom,
    ! [I2: nat,Xs: list_int,Ys: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr8617346907841251940nt_nat @ ( zip_int_nat @ Xs @ Ys ) @ I2 )
          = ( product_Pair_int_nat @ ( nth_int @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_5024_nth__zip,axiom,
    ! [I2: nat,Xs: list_nat,Ys: list_int] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Ys ) )
       => ( ( nth_Pr3440142176431000676at_int @ ( zip_nat_int @ Xs @ Ys ) @ I2 )
          = ( product_Pair_nat_int @ ( nth_nat @ Xs @ I2 ) @ ( nth_int @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_5025_nth__zip,axiom,
    ! [I2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I2 )
          = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_5026_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs )
       != ( size_size_list_int @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_5027_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_5028_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_int] :
      ( ( size_size_list_int @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_5029_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] :
      ( ( size_size_list_nat @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_5030_max_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_max_int @ B @ ( ord_max_int @ A @ C ) )
      = ( ord_max_int @ A @ ( ord_max_int @ B @ C ) ) ) ).

% max.left_commute
thf(fact_5031_max_Ocommute,axiom,
    ( ord_max_int
    = ( ^ [A2: int,B2: int] : ( ord_max_int @ B2 @ A2 ) ) ) ).

% max.commute
thf(fact_5032_max_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_max_int @ ( ord_max_int @ A @ B ) @ C )
      = ( ord_max_int @ A @ ( ord_max_int @ B @ C ) ) ) ).

% max.assoc
thf(fact_5033_pair__list__split,axiom,
    ! [L: list_P5707943133018811711nt_int] :
      ~ ! [L1: list_int,L2: list_int] :
          ( ( L
            = ( zip_int_int @ L1 @ L2 ) )
         => ( ( ( size_size_list_int @ L1 )
              = ( size_size_list_int @ L2 ) )
           => ( ( size_s5157815400016825771nt_int @ L )
             != ( size_size_list_int @ L2 ) ) ) ) ).

% pair_list_split
thf(fact_5034_pair__list__split,axiom,
    ! [L: list_P8198026277950538467nt_nat] :
      ~ ! [L1: list_int,L2: list_nat] :
          ( ( L
            = ( zip_int_nat @ L1 @ L2 ) )
         => ( ( ( size_size_list_int @ L1 )
              = ( size_size_list_nat @ L2 ) )
           => ( ( size_s7647898544948552527nt_nat @ L )
             != ( size_size_list_nat @ L2 ) ) ) ) ).

% pair_list_split
thf(fact_5035_pair__list__split,axiom,
    ! [L: list_P3521021558325789923at_int] :
      ~ ! [L1: list_nat,L2: list_int] :
          ( ( L
            = ( zip_nat_int @ L1 @ L2 ) )
         => ( ( ( size_size_list_nat @ L1 )
              = ( size_size_list_int @ L2 ) )
           => ( ( size_s2970893825323803983at_int @ L )
             != ( size_size_list_int @ L2 ) ) ) ) ).

% pair_list_split
thf(fact_5036_pair__list__split,axiom,
    ! [L: list_P6011104703257516679at_nat] :
      ~ ! [L1: list_nat,L2: list_nat] :
          ( ( L
            = ( zip_nat_nat @ L1 @ L2 ) )
         => ( ( ( size_size_list_nat @ L1 )
              = ( size_size_list_nat @ L2 ) )
           => ( ( size_s5460976970255530739at_nat @ L )
             != ( size_size_list_nat @ L2 ) ) ) ) ).

% pair_list_split
thf(fact_5037_bit__iff__and__drop__bit__eq__1,axiom,
    ( bit_se9216721137139052372nteger
    = ( ^ [A2: code_integer,N3: nat] :
          ( ( bit_se3949692690581998587nteger @ ( bit_se3928097537394005634nteger @ N3 @ A2 ) @ one_one_Code_integer )
          = one_one_Code_integer ) ) ) ).

% bit_iff_and_drop_bit_eq_1
thf(fact_5038_bit__iff__and__drop__bit__eq__1,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [A2: nat,N3: nat] :
          ( ( bit_se727722235901077358nd_nat @ ( bit_se8570568707652914677it_nat @ N3 @ A2 ) @ one_one_nat )
          = one_one_nat ) ) ) ).

% bit_iff_and_drop_bit_eq_1
thf(fact_5039_bit__iff__and__drop__bit__eq__1,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [A2: int,N3: nat] :
          ( ( bit_se725231765392027082nd_int @ ( bit_se8568078237143864401it_int @ N3 @ A2 ) @ one_one_int )
          = one_one_int ) ) ) ).

% bit_iff_and_drop_bit_eq_1
thf(fact_5040_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ B )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_5041_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_5042_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C @ B )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_5043_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_5044_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_5045_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_5046_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_max_rat @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_5047_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_max_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_5048_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_max_int @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_5049_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( ord_max_rat @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_5050_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_max_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_5051_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_max_int @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_5052_le__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_eq_rat @ Z @ X )
        | ( ord_less_eq_rat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_5053_le__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_eq_nat @ Z @ X )
        | ( ord_less_eq_nat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_5054_le__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_eq_int @ Z @ X )
        | ( ord_less_eq_int @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_5055_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded2
thf(fact_5056_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded2
thf(fact_5057_max_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded2
thf(fact_5058_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded1
thf(fact_5059_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded1
thf(fact_5060_max_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded1
thf(fact_5061_max_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A2: rat] :
          ( A2
          = ( ord_max_rat @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_5062_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( A2
          = ( ord_max_nat @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_5063_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( A2
          = ( ord_max_int @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_5064_max_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ A )
       => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_5065_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_5066_max_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ A )
       => ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_5067_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_rat @ B @ A )
         => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_5068_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_5069_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C @ A ) ) ) ).

% max.boundedE
thf(fact_5070_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( ord_max_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% max.orderI
thf(fact_5071_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% max.orderI
thf(fact_5072_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_max_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% max.orderI
thf(fact_5073_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( ord_max_rat @ A @ B ) ) ) ).

% max.orderE
thf(fact_5074_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( ord_max_nat @ A @ B ) ) ) ).

% max.orderE
thf(fact_5075_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( ord_max_int @ A @ B ) ) ) ).

% max.orderE
thf(fact_5076_max_Omono,axiom,
    ! [C: rat,A: rat,D2: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ( ord_less_eq_rat @ D2 @ B )
       => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D2 ) @ ( ord_max_rat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_5077_max_Omono,axiom,
    ! [C: nat,A: nat,D2: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D2 @ B )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D2 ) @ ( ord_max_nat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_5078_max_Omono,axiom,
    ! [C: int,A: int,D2: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ( ord_less_eq_int @ D2 @ B )
       => ( ord_less_eq_int @ ( ord_max_int @ C @ D2 ) @ ( ord_max_int @ A @ B ) ) ) ) ).

% max.mono
thf(fact_5079_max_Ostrict__coboundedI2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ B )
     => ( ord_le6747313008572928689nteger @ C @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_5080_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ B )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_5081_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C @ B )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_5082_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_int @ C @ B )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_5083_max_Ostrict__coboundedI1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ A )
     => ( ord_le6747313008572928689nteger @ C @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_5084_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ A )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_5085_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_5086_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ A )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_5087_max_Ostrict__order__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [B2: code_integer,A2: code_integer] :
          ( ( A2
            = ( ord_max_Code_integer @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5088_max_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( A2
            = ( ord_max_rat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5089_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( A2
            = ( ord_max_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5090_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A2: int] :
          ( ( A2
            = ( ord_max_int @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5091_max_Ostrict__boundedE,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_max_Code_integer @ B @ C ) @ A )
     => ~ ( ( ord_le6747313008572928689nteger @ B @ A )
         => ~ ( ord_le6747313008572928689nteger @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_5092_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_rat @ B @ A )
         => ~ ( ord_less_rat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_5093_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_5094_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
     => ~ ( ( ord_less_int @ B @ A )
         => ~ ( ord_less_int @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_5095_less__max__iff__disj,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z @ ( ord_max_Code_integer @ X @ Y ) )
      = ( ( ord_le6747313008572928689nteger @ Z @ X )
        | ( ord_le6747313008572928689nteger @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_5096_less__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_rat @ Z @ X )
        | ( ord_less_rat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_5097_less__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ Z @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Z @ X )
        | ( ord_less_nat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_5098_less__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ Z @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_int @ Z @ X )
        | ( ord_less_int @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_5099_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_5100_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_5101_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_5102_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_5103_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_5104_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_5105_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_5106_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_5107_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_5108_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_5109_sup__max,axiom,
    sup_sup_int = ord_max_int ).

% sup_max
thf(fact_5110_length__induct,axiom,
    ! [P2: list_int > $o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys5: list_int] :
              ( ( ord_less_nat @ ( size_size_list_int @ Ys5 ) @ ( size_size_list_int @ Xs2 ) )
             => ( P2 @ Ys5 ) )
         => ( P2 @ Xs2 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_5111_length__induct,axiom,
    ! [P2: list_nat > $o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys5: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys5 ) @ ( size_size_list_nat @ Xs2 ) )
             => ( P2 @ Ys5 ) )
         => ( P2 @ Xs2 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_5112_finite__maxlen,axiom,
    ! [M: set_list_int] :
      ( ( finite3922522038869484883st_int @ M )
     => ? [N4: nat] :
        ! [X4: list_int] :
          ( ( member_list_int @ X4 @ M )
         => ( ord_less_nat @ ( size_size_list_int @ X4 ) @ N4 ) ) ) ).

% finite_maxlen
thf(fact_5113_finite__maxlen,axiom,
    ! [M: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M )
     => ? [N4: nat] :
        ! [X4: list_nat] :
          ( ( member_list_nat @ X4 @ M )
         => ( ord_less_nat @ ( size_size_list_nat @ X4 ) @ N4 ) ) ) ).

% finite_maxlen
thf(fact_5114_map__eq__imp__length__eq,axiom,
    ! [F2: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn,G3: produc6575502325842934193n_assn > assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( map_Pr8991440229025900053n_assn @ F2 @ Xs )
        = ( map_Pr8991440229025900053n_assn @ G3 @ Ys ) )
     => ( ( size_s6829681357464350627n_assn @ Xs )
        = ( size_s6829681357464350627n_assn @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5115_map__eq__imp__length__eq,axiom,
    ! [F2: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn,G3: int > assn,Ys: list_int] :
      ( ( ( map_Pr8991440229025900053n_assn @ F2 @ Xs )
        = ( map_int_assn @ G3 @ Ys ) )
     => ( ( size_s6829681357464350627n_assn @ Xs )
        = ( size_size_list_int @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5116_map__eq__imp__length__eq,axiom,
    ! [F2: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn,G3: nat > assn,Ys: list_nat] :
      ( ( ( map_Pr8991440229025900053n_assn @ F2 @ Xs )
        = ( map_nat_assn @ G3 @ Ys ) )
     => ( ( size_s6829681357464350627n_assn @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5117_map__eq__imp__length__eq,axiom,
    ! [F2: int > assn,Xs: list_int,G3: produc6575502325842934193n_assn > assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( map_int_assn @ F2 @ Xs )
        = ( map_Pr8991440229025900053n_assn @ G3 @ Ys ) )
     => ( ( size_size_list_int @ Xs )
        = ( size_s6829681357464350627n_assn @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5118_map__eq__imp__length__eq,axiom,
    ! [F2: int > nat,Xs: list_int,G3: nat > nat,Ys: list_nat] :
      ( ( ( map_int_nat @ F2 @ Xs )
        = ( map_nat_nat @ G3 @ Ys ) )
     => ( ( size_size_list_int @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5119_map__eq__imp__length__eq,axiom,
    ! [F2: nat > assn,Xs: list_nat,G3: produc6575502325842934193n_assn > assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( map_nat_assn @ F2 @ Xs )
        = ( map_Pr8991440229025900053n_assn @ G3 @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s6829681357464350627n_assn @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5120_map__eq__imp__length__eq,axiom,
    ! [F2: nat > nat,Xs: list_nat,G3: int > nat,Ys: list_int] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = ( map_int_nat @ G3 @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5121_map__eq__imp__length__eq,axiom,
    ! [F2: nat > nat,Xs: list_nat,G3: nat > nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = ( map_nat_nat @ G3 @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5122_max_Oabel__semigroup__axioms,axiom,
    abel_semigroup_int @ ord_max_int ).

% max.abel_semigroup_axioms
thf(fact_5123_zip__inj,axiom,
    ! [A: list_int,B: list_int,A5: list_int,B6: list_int] :
      ( ( ( size_size_list_int @ A )
        = ( size_size_list_int @ B ) )
     => ( ( ( size_size_list_int @ A5 )
          = ( size_size_list_int @ B6 ) )
       => ( ( ( zip_int_int @ A @ B )
            = ( zip_int_int @ A5 @ B6 ) )
         => ( ( A = A5 )
            & ( B = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_5124_zip__inj,axiom,
    ! [A: list_int,B: list_nat,A5: list_int,B6: list_nat] :
      ( ( ( size_size_list_int @ A )
        = ( size_size_list_nat @ B ) )
     => ( ( ( size_size_list_int @ A5 )
          = ( size_size_list_nat @ B6 ) )
       => ( ( ( zip_int_nat @ A @ B )
            = ( zip_int_nat @ A5 @ B6 ) )
         => ( ( A = A5 )
            & ( B = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_5125_zip__inj,axiom,
    ! [A: list_nat,B: list_int,A5: list_nat,B6: list_int] :
      ( ( ( size_size_list_nat @ A )
        = ( size_size_list_int @ B ) )
     => ( ( ( size_size_list_nat @ A5 )
          = ( size_size_list_int @ B6 ) )
       => ( ( ( zip_nat_int @ A @ B )
            = ( zip_nat_int @ A5 @ B6 ) )
         => ( ( A = A5 )
            & ( B = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_5126_zip__inj,axiom,
    ! [A: list_nat,B: list_nat,A5: list_nat,B6: list_nat] :
      ( ( ( size_size_list_nat @ A )
        = ( size_size_list_nat @ B ) )
     => ( ( ( size_size_list_nat @ A5 )
          = ( size_size_list_nat @ B6 ) )
       => ( ( ( zip_nat_nat @ A @ B )
            = ( zip_nat_nat @ A5 @ B6 ) )
         => ( ( A = A5 )
            & ( B = B6 ) ) ) ) ) ).

% zip_inj
thf(fact_5127_max_Osemilattice__axioms,axiom,
    semilattice_int @ ord_max_int ).

% max.semilattice_axioms
thf(fact_5128_max_Osemigroup__axioms,axiom,
    semigroup_int @ ord_max_int ).

% max.semigroup_axioms
thf(fact_5129_bit__1__iff,axiom,
    ! [N: nat] :
      ( ( bit_se9216721137139052372nteger @ one_one_Code_integer @ N )
      = ( N = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_5130_bit__1__iff,axiom,
    ! [N: nat] :
      ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N )
      = ( N = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_5131_bit__1__iff,axiom,
    ! [N: nat] :
      ( ( bit_se1146084159140164899it_int @ one_one_int @ N )
      = ( N = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_5132_not__bit__1__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se9216721137139052372nteger @ one_one_Code_integer @ ( suc @ N ) ) ).

% not_bit_1_Suc
thf(fact_5133_not__bit__1__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N ) ) ).

% not_bit_1_Suc
thf(fact_5134_not__bit__1__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N ) ) ).

% not_bit_1_Suc
thf(fact_5135_bit__numeral__simps_I1_J,axiom,
    ! [N: num] :
      ~ ( bit_se9216721137139052372nteger @ one_one_Code_integer @ ( numeral_numeral_nat @ N ) ) ).

% bit_numeral_simps(1)
thf(fact_5136_bit__numeral__simps_I1_J,axiom,
    ! [N: num] :
      ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% bit_numeral_simps(1)
thf(fact_5137_bit__numeral__simps_I1_J,axiom,
    ! [N: num] :
      ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( numeral_numeral_nat @ N ) ) ).

% bit_numeral_simps(1)
thf(fact_5138_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_int,Ws: list_int,P2: list_int > list_int > list_int > list_int > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( ( size_size_list_int @ Zs )
            = ( size_size_list_int @ Ws ) )
         => ( ( P2 @ nil_int @ nil_int @ nil_int @ nil_int )
           => ( ! [X3: int,Xs2: list_int,Y3: int,Ys3: list_int,Z2: int,Zs3: list_int,W3: int,Ws2: list_int] :
                  ( ( ( size_size_list_int @ Xs2 )
                    = ( size_size_list_int @ Ys3 ) )
                 => ( ( ( size_size_list_int @ Ys3 )
                      = ( size_size_list_int @ Zs3 ) )
                   => ( ( ( size_size_list_int @ Zs3 )
                        = ( size_size_list_int @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) @ ( cons_int @ Z2 @ Zs3 ) @ ( cons_int @ W3 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5139_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_int,Ws: list_nat,P2: list_int > list_int > list_int > list_nat > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( ( size_size_list_int @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P2 @ nil_int @ nil_int @ nil_int @ nil_nat )
           => ( ! [X3: int,Xs2: list_int,Y3: int,Ys3: list_int,Z2: int,Zs3: list_int,W3: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_int @ Xs2 )
                    = ( size_size_list_int @ Ys3 ) )
                 => ( ( ( size_size_list_int @ Ys3 )
                      = ( size_size_list_int @ Zs3 ) )
                   => ( ( ( size_size_list_int @ Zs3 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) @ ( cons_int @ Z2 @ Zs3 ) @ ( cons_nat @ W3 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5140_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_nat,Ws: list_int,P2: list_int > list_int > list_nat > list_int > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_int @ Ws ) )
         => ( ( P2 @ nil_int @ nil_int @ nil_nat @ nil_int )
           => ( ! [X3: int,Xs2: list_int,Y3: int,Ys3: list_int,Z2: nat,Zs3: list_nat,W3: int,Ws2: list_int] :
                  ( ( ( size_size_list_int @ Xs2 )
                    = ( size_size_list_int @ Ys3 ) )
                 => ( ( ( size_size_list_int @ Ys3 )
                      = ( size_size_list_nat @ Zs3 ) )
                   => ( ( ( size_size_list_nat @ Zs3 )
                        = ( size_size_list_int @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) @ ( cons_nat @ Z2 @ Zs3 ) @ ( cons_int @ W3 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5141_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_nat,Ws: list_nat,P2: list_int > list_int > list_nat > list_nat > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P2 @ nil_int @ nil_int @ nil_nat @ nil_nat )
           => ( ! [X3: int,Xs2: list_int,Y3: int,Ys3: list_int,Z2: nat,Zs3: list_nat,W3: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_int @ Xs2 )
                    = ( size_size_list_int @ Ys3 ) )
                 => ( ( ( size_size_list_int @ Ys3 )
                      = ( size_size_list_nat @ Zs3 ) )
                   => ( ( ( size_size_list_nat @ Zs3 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) @ ( cons_nat @ Z2 @ Zs3 ) @ ( cons_nat @ W3 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5142_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_nat,Zs: list_int,Ws: list_int,P2: list_int > list_nat > list_int > list_int > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( ( size_size_list_int @ Zs )
            = ( size_size_list_int @ Ws ) )
         => ( ( P2 @ nil_int @ nil_nat @ nil_int @ nil_int )
           => ( ! [X3: int,Xs2: list_int,Y3: nat,Ys3: list_nat,Z2: int,Zs3: list_int,W3: int,Ws2: list_int] :
                  ( ( ( size_size_list_int @ Xs2 )
                    = ( size_size_list_nat @ Ys3 ) )
                 => ( ( ( size_size_list_nat @ Ys3 )
                      = ( size_size_list_int @ Zs3 ) )
                   => ( ( ( size_size_list_int @ Zs3 )
                        = ( size_size_list_int @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) @ ( cons_int @ Z2 @ Zs3 ) @ ( cons_int @ W3 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5143_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_nat,Zs: list_int,Ws: list_nat,P2: list_int > list_nat > list_int > list_nat > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( ( size_size_list_int @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P2 @ nil_int @ nil_nat @ nil_int @ nil_nat )
           => ( ! [X3: int,Xs2: list_int,Y3: nat,Ys3: list_nat,Z2: int,Zs3: list_int,W3: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_int @ Xs2 )
                    = ( size_size_list_nat @ Ys3 ) )
                 => ( ( ( size_size_list_nat @ Ys3 )
                      = ( size_size_list_int @ Zs3 ) )
                   => ( ( ( size_size_list_int @ Zs3 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) @ ( cons_int @ Z2 @ Zs3 ) @ ( cons_nat @ W3 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5144_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_nat,Zs: list_nat,Ws: list_int,P2: list_int > list_nat > list_nat > list_int > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_int @ Ws ) )
         => ( ( P2 @ nil_int @ nil_nat @ nil_nat @ nil_int )
           => ( ! [X3: int,Xs2: list_int,Y3: nat,Ys3: list_nat,Z2: nat,Zs3: list_nat,W3: int,Ws2: list_int] :
                  ( ( ( size_size_list_int @ Xs2 )
                    = ( size_size_list_nat @ Ys3 ) )
                 => ( ( ( size_size_list_nat @ Ys3 )
                      = ( size_size_list_nat @ Zs3 ) )
                   => ( ( ( size_size_list_nat @ Zs3 )
                        = ( size_size_list_int @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) @ ( cons_nat @ Z2 @ Zs3 ) @ ( cons_int @ W3 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5145_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_nat,Zs: list_nat,Ws: list_nat,P2: list_int > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P2 @ nil_int @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X3: int,Xs2: list_int,Y3: nat,Ys3: list_nat,Z2: nat,Zs3: list_nat,W3: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_int @ Xs2 )
                    = ( size_size_list_nat @ Ys3 ) )
                 => ( ( ( size_size_list_nat @ Ys3 )
                      = ( size_size_list_nat @ Zs3 ) )
                   => ( ( ( size_size_list_nat @ Zs3 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_int @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys3 ) @ ( cons_nat @ Z2 @ Zs3 ) @ ( cons_nat @ W3 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5146_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_int,Ws: list_int,P2: list_nat > list_int > list_int > list_int > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( ( size_size_list_int @ Zs )
            = ( size_size_list_int @ Ws ) )
         => ( ( P2 @ nil_nat @ nil_int @ nil_int @ nil_int )
           => ( ! [X3: nat,Xs2: list_nat,Y3: int,Ys3: list_int,Z2: int,Zs3: list_int,W3: int,Ws2: list_int] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_int @ Ys3 ) )
                 => ( ( ( size_size_list_int @ Ys3 )
                      = ( size_size_list_int @ Zs3 ) )
                   => ( ( ( size_size_list_int @ Zs3 )
                        = ( size_size_list_int @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) @ ( cons_int @ Z2 @ Zs3 ) @ ( cons_int @ W3 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5147_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_int,Ws: list_nat,P2: list_nat > list_int > list_int > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_int @ Zs ) )
       => ( ( ( size_size_list_int @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P2 @ nil_nat @ nil_int @ nil_int @ nil_nat )
           => ( ! [X3: nat,Xs2: list_nat,Y3: int,Ys3: list_int,Z2: int,Zs3: list_int,W3: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_int @ Ys3 ) )
                 => ( ( ( size_size_list_int @ Ys3 )
                      = ( size_size_list_int @ Zs3 ) )
                   => ( ( ( size_size_list_int @ Zs3 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_int @ Y3 @ Ys3 ) @ ( cons_int @ Z2 @ Zs3 ) @ ( cons_nat @ W3 @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_5148_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% bit_not_int_iff'
thf(fact_5149_bit__minus__int__iff,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
      = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).

% bit_minus_int_iff
thf(fact_5150_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N4: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_eq_nat @ N4 @ M4 )
             => ( ( bit_se1146084159140164899it_int @ K @ M4 )
                = ( bit_se1146084159140164899it_int @ K @ N4 ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N4 )
             => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N4 @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K @ N4 ) ) ) ) ) ).

% int_bit_bound
thf(fact_5151_nat__numeral__diff__1,axiom,
    ! [V2: num] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ V2 ) @ one_one_nat )
      = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V2 ) @ one_one_int ) ) ) ).

% nat_numeral_diff_1
thf(fact_5152_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_5153_length__upto,axiom,
    ! [I2: int,J: int] :
      ( ( size_size_list_int @ ( upto @ I2 @ J ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I2 ) @ one_one_int ) ) ) ).

% length_upto
thf(fact_5154_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% one_less_nat_eq
thf(fact_5155_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I2: nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ns ) )
       => ( ord_less_eq_nat @ I2 @ ( nth_nat @ Ns @ I2 ) ) ) ) ).

% sorted_wrt_less_idx
thf(fact_5156_sorted__upto,axiom,
    ! [M2: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M2 @ N ) ) ).

% sorted_upto
thf(fact_5157_sorted__wrt__upto,axiom,
    ! [I2: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I2 @ J ) ) ).

% sorted_wrt_upto
thf(fact_5158_nat__one__as__int,axiom,
    ( one_one_nat
    = ( nat2 @ one_one_int ) ) ).

% nat_one_as_int
thf(fact_5159_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_5160_upto__rec2,axiom,
    ! [I2: int,J: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( upto @ I2 @ J )
        = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).

% upto_rec2
thf(fact_5161_normalize__denom__zero,axiom,
    ! [P5: int] :
      ( ( normalize @ ( product_Pair_int_int @ P5 @ zero_zero_int ) )
      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% normalize_denom_zero
thf(fact_5162_upt__conv__Nil,axiom,
    ! [J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ J @ I2 )
     => ( ( upt @ I2 @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_5163_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J: nat] :
      ( ( ( upt @ I2 @ J )
        = nil_nat )
      = ( ( J = zero_zero_nat )
        | ( ord_less_eq_nat @ J @ I2 ) ) ) ).

% upt_eq_Nil_conv
thf(fact_5164_length__upt,axiom,
    ! [I2: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I2 @ J ) )
      = ( minus_minus_nat @ J @ I2 ) ) ).

% length_upt
thf(fact_5165_upt__rec__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M2 ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_5166_nth__upt,axiom,
    ! [I2: nat,K: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J )
     => ( ( nth_nat @ ( upt @ I2 @ J ) @ K )
        = ( plus_plus_nat @ I2 @ K ) ) ) ).

% nth_upt
thf(fact_5167_upt__0,axiom,
    ! [I2: nat] :
      ( ( upt @ I2 @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_5168_map__Suc__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M2 @ N ) )
      = ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).

% map_Suc_upt
thf(fact_5169_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N: nat,Ns: list_nat,Q4: nat] :
      ( ( ( cons_nat @ M2 @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M2 @ Q4 ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M2 ) @ Q4 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_5170_upt__conv__Cons,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( upt @ I2 @ J )
        = ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_5171_upt__rec,axiom,
    ( upt
    = ( ^ [I4: nat,J2: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J2 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J2 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_5172_sorted__wrt__upt,axiom,
    ! [M2: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M2 @ N ) ) ).

% sorted_wrt_upt
thf(fact_5173_sorted__upt,axiom,
    ! [M2: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M2 @ N ) ) ).

% sorted_upt
thf(fact_5174_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( ( upt @ I2 @ J )
        = ( cons_nat @ X @ Xs ) )
      = ( ( ord_less_nat @ I2 @ J )
        & ( I2 = X )
        & ( ( upt @ ( plus_plus_nat @ I2 @ one_one_nat ) @ J )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_5175_upto__aux__def,axiom,
    ( upto_aux
    = ( ^ [I4: int,J2: int] : ( append_int @ ( upto @ I4 @ J2 ) ) ) ) ).

% upto_aux_def
thf(fact_5176_upto__split2,axiom,
    ! [I2: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I2 @ K )
          = ( append_int @ ( upto @ I2 @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).

% upto_split2
thf(fact_5177_upto__split1,axiom,
    ! [I2: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I2 @ K )
          = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).

% upto_split1
thf(fact_5178_upto__split3,axiom,
    ! [I2: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I2 @ K )
          = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).

% upto_split3
thf(fact_5179_bezw__0,axiom,
    ! [X: nat] :
      ( ( bezw @ X @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_5180_upt__Suc__append,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( upt @ I2 @ ( suc @ J ) )
        = ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_5181_upt__Suc,axiom,
    ! [I2: nat,J: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J )
       => ( ( upt @ I2 @ ( suc @ J ) )
          = ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I2 @ J )
       => ( ( upt @ I2 @ ( suc @ J ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_5182_upt__add__eq__append,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( upt @ I2 @ ( plus_plus_nat @ J @ K ) )
        = ( append_nat @ ( upt @ I2 @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_5183_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_5184_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X2: nat,Y2: nat] : ( if_Pro3027730157355071871nt_int @ ( Y2 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y2 ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_5185_nth__upto,axiom,
    ! [I2: int,K: nat,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) @ J )
     => ( ( nth_int @ ( upto @ I2 @ J ) @ K )
        = ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).

% nth_upto
thf(fact_5186_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_5187_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_5188_int__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).

% int_Suc
thf(fact_5189_Suc__as__int,axiom,
    ( suc
    = ( ^ [A2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_5190_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_5191_take__upt,axiom,
    ! [I2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ M2 ) @ N )
     => ( ( take_nat @ M2 @ ( upt @ I2 @ N ) )
        = ( upt @ I2 @ ( plus_plus_nat @ I2 @ M2 ) ) ) ) ).

% take_upt
thf(fact_5192_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_5193_dvd__fact,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( dvd_dvd_nat @ M2 @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% dvd_fact
thf(fact_5194_choose__two,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% choose_two
thf(fact_5195_binomial__n__n,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ N )
      = one_one_nat ) ).

% binomial_n_n
thf(fact_5196_binomial__n__0,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_5197_last__upt,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( last_nat @ ( upt @ I2 @ J ) )
        = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).

% last_upt
thf(fact_5198_choose__one,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ one_one_nat )
      = N ) ).

% choose_one
thf(fact_5199_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).

% binomial_absorb_comp
thf(fact_5200_binomial__absorption,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).

% binomial_absorption
thf(fact_5201_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( binomial @ N @ K )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_5202_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_5203_binomial__code,axiom,
    ( binomial
    = ( ^ [N3: nat,K4: nat] : ( if_nat @ ( ord_less_nat @ N3 @ K4 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K4 ) ) @ ( binomial @ N3 @ ( minus_minus_nat @ N3 @ K4 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N3 @ K4 ) @ one_one_nat ) @ N3 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K4 ) ) ) ) ) ) ).

% binomial_code
thf(fact_5204_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( binomial @ N @ ( suc @ K ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).

% binomial_addition_formula
thf(fact_5205_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I2: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I2 ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_5206_drop__upt,axiom,
    ! [M2: nat,I2: nat,J: nat] :
      ( ( drop_nat @ M2 @ ( upt @ I2 @ J ) )
      = ( upt @ ( plus_plus_nat @ I2 @ M2 ) @ J ) ) ).

% drop_upt
thf(fact_5207_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ M3 ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_5208_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ ( suc @ I2 ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J ) )
        = ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanLessThan
thf(fact_5209_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I2: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I2 ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_5210_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).

% sorted_list_of_set_atMost_Suc
thf(fact_5211_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).

% sorted_list_of_set_lessThan_Suc
thf(fact_5212_hd__upt,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( hd_nat @ ( upt @ I2 @ J ) )
        = I2 ) ) ).

% hd_upt
thf(fact_5213_distinct__upt,axiom,
    ! [I2: nat,J: nat] : ( distinct_nat @ ( upt @ I2 @ J ) ) ).

% distinct_upt
thf(fact_5214_distinct__upto,axiom,
    ! [I2: int,J: int] : ( distinct_int @ ( upto @ I2 @ J ) ) ).

% distinct_upto
thf(fact_5215_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N3 ) ) ) ) ).

% atLeast_upt
thf(fact_5216_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ ( suc @ M3 ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_5217_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N3 ) ) ) ) ) ).

% atMost_upto
thf(fact_5218_butlast__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( butlast_nat @ ( upt @ M2 @ N ) )
      = ( upt @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% butlast_upt
thf(fact_5219_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( suc @ I2 ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J ) )
        = ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanAtMost
thf(fact_5220_greaterThanLessThan__upto,axiom,
    ( set_or5832277885323065728an_int
    = ( ^ [I4: int,J2: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ ( minus_minus_int @ J2 @ one_one_int ) ) ) ) ) ).

% greaterThanLessThan_upto
thf(fact_5221_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_5222_cppi,axiom,
    ! [D: int,P2: int > $o,P3: int > $o,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( P2 @ X3 )
              = ( P3 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( ( member_int2 @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                 => ! [Xb2: int] :
                      ( ( member_int2 @ Xb2 @ A3 )
                     => ( X3
                       != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P2 @ X3 )
               => ( P2 @ ( plus_plus_int @ X3 @ D ) ) ) )
         => ( ! [X3: int,K2: int] :
                ( ( P3 @ X3 )
                = ( P3 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
           => ( ( ? [X5: int] : ( P2 @ X5 ) )
              = ( ? [X2: int] :
                    ( ( member_int2 @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                    & ( P3 @ X2 ) )
                | ? [X2: int] :
                    ( ( member_int2 @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                    & ? [Y2: int] :
                        ( ( member_int2 @ Y2 @ A3 )
                        & ( P2 @ ( minus_minus_int @ Y2 @ X2 ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_5223_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_5224_rat__one__code,axiom,
    ( ( quotient_of @ one_one_rat )
    = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).

% rat_one_code
thf(fact_5225_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_5226_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_5227_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_5228_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_5229_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_5230_sgn__rat__def,axiom,
    ( sgn_sgn_rat
    = ( ^ [A2: rat] : ( if_rat @ ( A2 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A2 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_rat_def
thf(fact_5231_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_5232_atLeastAtMost__upto,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I4: int,J2: int] : ( set_int2 @ ( upto @ I4 @ J2 ) ) ) ) ).

% atLeastAtMost_upto
thf(fact_5233_bset_I1_J,axiom,
    ! [D: int,B3: set_int,P2: int > $o,Q2: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int2 @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb2: int] :
                  ( ( member_int2 @ Xb2 @ B3 )
                 => ( X3
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P2 @ X3 )
           => ( P2 @ ( minus_minus_int @ X3 @ D ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int2 @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb2: int] :
                    ( ( member_int2 @ Xb2 @ B3 )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q2 @ X3 )
             => ( Q2 @ ( minus_minus_int @ X3 @ D ) ) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int2 @ Xb3 @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P2 @ X4 )
                & ( Q2 @ X4 ) )
             => ( ( P2 @ ( minus_minus_int @ X4 @ D ) )
                & ( Q2 @ ( minus_minus_int @ X4 @ D ) ) ) ) ) ) ) ).

% bset(1)
thf(fact_5234_bset_I2_J,axiom,
    ! [D: int,B3: set_int,P2: int > $o,Q2: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int2 @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb2: int] :
                  ( ( member_int2 @ Xb2 @ B3 )
                 => ( X3
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P2 @ X3 )
           => ( P2 @ ( minus_minus_int @ X3 @ D ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int2 @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb2: int] :
                    ( ( member_int2 @ Xb2 @ B3 )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q2 @ X3 )
             => ( Q2 @ ( minus_minus_int @ X3 @ D ) ) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int2 @ Xb3 @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P2 @ X4 )
                | ( Q2 @ X4 ) )
             => ( ( P2 @ ( minus_minus_int @ X4 @ D ) )
                | ( Q2 @ ( minus_minus_int @ X4 @ D ) ) ) ) ) ) ) ).

% bset(2)
thf(fact_5235_aset_I1_J,axiom,
    ! [D: int,A3: set_int,P2: int > $o,Q2: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int2 @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb2: int] :
                  ( ( member_int2 @ Xb2 @ A3 )
                 => ( X3
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P2 @ X3 )
           => ( P2 @ ( plus_plus_int @ X3 @ D ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int2 @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb2: int] :
                    ( ( member_int2 @ Xb2 @ A3 )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q2 @ X3 )
             => ( Q2 @ ( plus_plus_int @ X3 @ D ) ) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int2 @ Xb3 @ A3 )
                   => ( X4
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P2 @ X4 )
                & ( Q2 @ X4 ) )
             => ( ( P2 @ ( plus_plus_int @ X4 @ D ) )
                & ( Q2 @ ( plus_plus_int @ X4 @ D ) ) ) ) ) ) ) ).

% aset(1)
thf(fact_5236_aset_I2_J,axiom,
    ! [D: int,A3: set_int,P2: int > $o,Q2: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa2: int] :
              ( ( member_int2 @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb2: int] :
                  ( ( member_int2 @ Xb2 @ A3 )
                 => ( X3
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P2 @ X3 )
           => ( P2 @ ( plus_plus_int @ X3 @ D ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa2: int] :
                ( ( member_int2 @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb2: int] :
                    ( ( member_int2 @ Xb2 @ A3 )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q2 @ X3 )
             => ( Q2 @ ( plus_plus_int @ X3 @ D ) ) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int2 @ Xb3 @ A3 )
                   => ( X4
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P2 @ X4 )
                | ( Q2 @ X4 ) )
             => ( ( P2 @ ( plus_plus_int @ X4 @ D ) )
                | ( Q2 @ ( plus_plus_int @ X4 @ D ) ) ) ) ) ) ) ).

% aset(2)
thf(fact_5237_greaterThanAtMost__upto,axiom,
    ( set_or6656581121297822940st_int
    = ( ^ [I4: int,J2: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J2 ) ) ) ) ).

% greaterThanAtMost_upto
thf(fact_5238_bset_I9_J,axiom,
    ! [D2: int,D: int,B3: set_int,T3: int] :
      ( ( dvd_dvd_int @ D2 @ D )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int2 @ Xb3 @ B3 )
                 => ( X4
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ T3 ) )
           => ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X4 @ D ) @ T3 ) ) ) ) ) ).

% bset(9)
thf(fact_5239_bset_I10_J,axiom,
    ! [D2: int,D: int,B3: set_int,T3: int] :
      ( ( dvd_dvd_int @ D2 @ D )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int2 @ Xb3 @ B3 )
                 => ( X4
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ T3 ) )
           => ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X4 @ D ) @ T3 ) ) ) ) ) ).

% bset(10)
thf(fact_5240_aset_I9_J,axiom,
    ! [D2: int,D: int,A3: set_int,T3: int] :
      ( ( dvd_dvd_int @ D2 @ D )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int2 @ Xb3 @ A3 )
                 => ( X4
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ T3 ) )
           => ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( plus_plus_int @ X4 @ D ) @ T3 ) ) ) ) ) ).

% aset(9)
thf(fact_5241_aset_I10_J,axiom,
    ! [D2: int,D: int,A3: set_int,T3: int] :
      ( ( dvd_dvd_int @ D2 @ D )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int2 @ Xb3 @ A3 )
                 => ( X4
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X4 @ T3 ) )
           => ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( plus_plus_int @ X4 @ D ) @ T3 ) ) ) ) ) ).

% aset(10)
thf(fact_5242_atLeastAtMostPlus1__int__conv,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_eq_int @ M2 @ ( plus_plus_int @ one_one_int @ N ) )
     => ( ( set_or1266510415728281911st_int @ M2 @ ( plus_plus_int @ one_one_int @ N ) )
        = ( insert_int2 @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M2 @ N ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_5243_periodic__finite__ex,axiom,
    ! [D2: int,P2: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X3: int,K2: int] :
            ( ( P2 @ X3 )
            = ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D2 ) ) ) )
       => ( ( ? [X5: int] : ( P2 @ X5 ) )
          = ( ? [X2: int] :
                ( ( member_int2 @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
                & ( P2 @ X2 ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_5244_aset_I7_J,axiom,
    ! [D: int,A3: set_int,T3: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int2 @ Xb3 @ A3 )
                 => ( X4
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ T3 @ X4 )
           => ( ord_less_int @ T3 @ ( plus_plus_int @ X4 @ D ) ) ) ) ) ).

% aset(7)
thf(fact_5245_aset_I5_J,axiom,
    ! [D: int,T3: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int2 @ T3 @ A3 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int2 @ Xb3 @ A3 )
                   => ( X4
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ X4 @ T3 )
             => ( ord_less_int @ ( plus_plus_int @ X4 @ D ) @ T3 ) ) ) ) ) ).

% aset(5)
thf(fact_5246_aset_I4_J,axiom,
    ! [D: int,T3: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int2 @ T3 @ A3 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int2 @ Xb3 @ A3 )
                   => ( X4
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X4 != T3 )
             => ( ( plus_plus_int @ X4 @ D )
               != T3 ) ) ) ) ) ).

% aset(4)
thf(fact_5247_aset_I3_J,axiom,
    ! [D: int,T3: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int2 @ ( plus_plus_int @ T3 @ one_one_int ) @ A3 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int2 @ Xb3 @ A3 )
                   => ( X4
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X4 = T3 )
             => ( ( plus_plus_int @ X4 @ D )
                = T3 ) ) ) ) ) ).

% aset(3)
thf(fact_5248_bset_I7_J,axiom,
    ! [D: int,T3: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int2 @ T3 @ B3 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int2 @ Xb3 @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ T3 @ X4 )
             => ( ord_less_int @ T3 @ ( minus_minus_int @ X4 @ D ) ) ) ) ) ) ).

% bset(7)
thf(fact_5249_bset_I5_J,axiom,
    ! [D: int,B3: set_int,T3: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int2 @ Xb3 @ B3 )
                 => ( X4
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ X4 @ T3 )
           => ( ord_less_int @ ( minus_minus_int @ X4 @ D ) @ T3 ) ) ) ) ).

% bset(5)
thf(fact_5250_bset_I4_J,axiom,
    ! [D: int,T3: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int2 @ T3 @ B3 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int2 @ Xb3 @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X4 != T3 )
             => ( ( minus_minus_int @ X4 @ D )
               != T3 ) ) ) ) ) ).

% bset(4)
thf(fact_5251_bset_I3_J,axiom,
    ! [D: int,T3: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int2 @ ( minus_minus_int @ T3 @ one_one_int ) @ B3 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int2 @ Xb3 @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X4 = T3 )
             => ( ( minus_minus_int @ X4 @ D )
                = T3 ) ) ) ) ) ).

% bset(3)
thf(fact_5252_rat__sgn__code,axiom,
    ! [P5: rat] :
      ( ( quotient_of @ ( sgn_sgn_rat @ P5 ) )
      = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P5 ) ) ) @ one_one_int ) ) ).

% rat_sgn_code
thf(fact_5253_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I4: int,J2: int] : ( if_set_int @ ( ord_less_int @ J2 @ I4 ) @ bot_bot_set_int @ ( insert_int2 @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J2 ) ) ) ) ) ).

% simp_from_to
thf(fact_5254_aset_I8_J,axiom,
    ! [D: int,A3: set_int,T3: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int2 @ Xb3 @ A3 )
                 => ( X4
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T3 @ X4 )
           => ( ord_less_eq_int @ T3 @ ( plus_plus_int @ X4 @ D ) ) ) ) ) ).

% aset(8)
thf(fact_5255_aset_I6_J,axiom,
    ! [D: int,T3: int,A3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int2 @ ( plus_plus_int @ T3 @ one_one_int ) @ A3 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int2 @ Xb3 @ A3 )
                   => ( X4
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X4 @ T3 )
             => ( ord_less_eq_int @ ( plus_plus_int @ X4 @ D ) @ T3 ) ) ) ) ) ).

% aset(6)
thf(fact_5256_bset_I8_J,axiom,
    ! [D: int,T3: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int2 @ ( minus_minus_int @ T3 @ one_one_int ) @ B3 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int2 @ Xb3 @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T3 @ X4 )
             => ( ord_less_eq_int @ T3 @ ( minus_minus_int @ X4 @ D ) ) ) ) ) ) ).

% bset(8)
thf(fact_5257_bset_I6_J,axiom,
    ! [D: int,B3: set_int,T3: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int2 @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int2 @ Xb3 @ B3 )
                 => ( X4
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X4 @ T3 )
           => ( ord_less_eq_int @ ( minus_minus_int @ X4 @ D ) @ T3 ) ) ) ) ).

% bset(6)
thf(fact_5258_cpmi,axiom,
    ! [D: int,P2: int > $o,P3: int > $o,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( P2 @ X3 )
              = ( P3 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( ( member_int2 @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                 => ! [Xb2: int] :
                      ( ( member_int2 @ Xb2 @ B3 )
                     => ( X3
                       != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P2 @ X3 )
               => ( P2 @ ( minus_minus_int @ X3 @ D ) ) ) )
         => ( ! [X3: int,K2: int] :
                ( ( P3 @ X3 )
                = ( P3 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
           => ( ( ? [X5: int] : ( P2 @ X5 ) )
              = ( ? [X2: int] :
                    ( ( member_int2 @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                    & ( P3 @ X2 ) )
                | ? [X2: int] :
                    ( ( member_int2 @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                    & ? [Y2: int] :
                        ( ( member_int2 @ Y2 @ B3 )
                        & ( P2 @ ( plus_plus_int @ Y2 @ X2 ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_5259_quotient__of__int,axiom,
    ! [A: int] :
      ( ( quotient_of @ ( of_int @ A ) )
      = ( product_Pair_int_int @ A @ one_one_int ) ) ).

% quotient_of_int
thf(fact_5260_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_5261_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_5262_all__nat__less,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N )
           => ( P2 @ M3 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
           => ( P2 @ X2 ) ) ) ) ).

% all_nat_less
thf(fact_5263_ex__nat__less,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N )
            & ( P2 @ M3 ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat2 @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
            & ( P2 @ X2 ) ) ) ) ).

% ex_nat_less
thf(fact_5264_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ N3 @ ( suc @ M3 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_5265_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_5266_zero__rat__def,axiom,
    ( zero_zero_rat
    = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).

% zero_rat_def
thf(fact_5267_one__rat__def,axiom,
    ( one_one_rat
    = ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).

% one_rat_def
thf(fact_5268_zero__rat_Otransfer,axiom,
    pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).

% zero_rat.transfer
thf(fact_5269_image__Suc__atMost,axiom,
    ! [N: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).

% image_Suc_atMost
thf(fact_5270_image__Suc__lessThan,axiom,
    ! [N: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).

% image_Suc_lessThan
thf(fact_5271_one__rat_Otransfer,axiom,
    pcr_rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ one_one_rat ).

% one_rat.transfer
thf(fact_5272_sorted__list__of__set__range,axiom,
    ! [M2: nat,N: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
      = ( upt @ M2 @ N ) ) ).

% sorted_list_of_set_range
thf(fact_5273_ex__nat__less__eq,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_nat @ M3 @ N )
            & ( P2 @ M3 ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat2 @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P2 @ X2 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_5274_all__nat__less__eq,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_nat @ M3 @ N )
           => ( P2 @ M3 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P2 @ X2 ) ) ) ) ).

% all_nat_less_eq
thf(fact_5275_atLeastLessThan__upt,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [I4: nat,J2: nat] : ( set_nat2 @ ( upt @ I4 @ J2 ) ) ) ) ).

% atLeastLessThan_upt
thf(fact_5276_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_5277_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_5278_atLeastLessThan__upto,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [I4: int,J2: int] : ( set_int2 @ ( upto @ I4 @ ( minus_minus_int @ J2 @ one_one_int ) ) ) ) ) ).

% atLeastLessThan_upto
thf(fact_5279_bool__assn__proper_I4_J,axiom,
    ! [P2: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] :
      ( ( proper @ P2 )
     => ( ( proper @ Q2 )
       => ( proper
          @ ^ [H: produc3658429121746597890et_nat] :
              ( ( P2 @ H )
              & ( Q2 @ H ) ) ) ) ) ).

% bool_assn_proper(4)
thf(fact_5280_bool__assn__proper_I3_J,axiom,
    ! [P2: produc3658429121746597890et_nat > $o,Q2: produc3658429121746597890et_nat > $o] :
      ( ( proper @ P2 )
     => ( ( proper @ Q2 )
       => ( proper
          @ ^ [H: produc3658429121746597890et_nat] :
              ( ( P2 @ H )
              | ( Q2 @ H ) ) ) ) ) ).

% bool_assn_proper(3)
thf(fact_5281_bool__assn__proper_I2_J,axiom,
    ( proper
    @ ^ [Uu2: produc3658429121746597890et_nat] : $false ) ).

% bool_assn_proper(2)
thf(fact_5282_tl__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( tl_nat @ ( upt @ M2 @ N ) )
      = ( upt @ ( suc @ M2 ) @ N ) ) ).

% tl_upt
thf(fact_5283_bool__assn__proper_I5_J,axiom,
    ! [P2: produc3658429121746597890et_nat > $o] :
      ( ( proper @ P2 )
     => ( proper
        @ ^ [H: produc3658429121746597890et_nat] :
            ( ( in_range @ H )
            & ~ ( P2 @ H ) ) ) ) ).

% bool_assn_proper(5)
thf(fact_5284_bot__assn__def,axiom,
    ( bot_bot_assn
    = ( abs_assn
      @ ^ [Uu2: produc3658429121746597890et_nat] : $false ) ) ).

% bot_assn_def
thf(fact_5285_sup__assn__def,axiom,
    ( sup_sup_assn
    = ( ^ [P: assn,Q: assn] :
          ( abs_assn
          @ ^ [H: produc3658429121746597890et_nat] :
              ( ( rep_assn @ P @ H )
              | ( rep_assn @ Q @ H ) ) ) ) ) ).

% sup_assn_def
thf(fact_5286_inf__assn__def,axiom,
    ( inf_inf_assn
    = ( ^ [P: assn,Q: assn] :
          ( abs_assn
          @ ^ [H: produc3658429121746597890et_nat] :
              ( ( rep_assn @ P @ H )
              & ( rep_assn @ Q @ H ) ) ) ) ) ).

% inf_assn_def
thf(fact_5287_map__add__upt,axiom,
    ! [N: nat,M2: nat] :
      ( ( map_nat_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
        @ ( upt @ zero_zero_nat @ M2 ) )
      = ( upt @ N @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% map_add_upt
thf(fact_5288_uminus__assn__def,axiom,
    ( uminus_uminus_assn
    = ( ^ [P: assn] :
          ( abs_assn
          @ ^ [H: produc3658429121746597890et_nat] :
              ( ( in_range @ H )
              & ~ ( rep_assn @ P @ H ) ) ) ) ) ).

% uminus_assn_def
thf(fact_5289_map__decr__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( upt @ M2 @ N ) ) ).

% map_decr_upt
thf(fact_5290_fact__div__fact,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) )
        = ( groups708209901874060359at_nat
          @ ^ [X2: nat] : X2
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) ) ).

% fact_div_fact
thf(fact_5291_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J2: nat] : ( binomial @ ( plus_plus_nat @ N @ J2 ) @ N )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M2 ) @ one_one_nat ) @ M2 ) ) ).

% choose_rising_sum(2)
thf(fact_5292_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J2: nat] : ( binomial @ ( plus_plus_nat @ N @ J2 ) @ N )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M2 ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_5293_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_5294_Sum__Ico__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M2 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_5295_Sum__Icc__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X2: nat] : X2
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M2 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_5296_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_5297_choose__linear__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( times_times_nat @ I4 @ ( binomial @ N @ I4 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% choose_linear_sum
thf(fact_5298_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L3: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q6: nat,R4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L3 ) @ R4 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ one_one_nat ) @ ( minus_minus_nat @ R4 @ ( numeral_numeral_nat @ L3 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ R4 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_5299_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L3: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q6: int,R4: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L3 ) @ R4 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ one_one_int ) @ ( minus_minus_int @ R4 @ ( numeral_numeral_int @ L3 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ R4 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_5300_Sum__Icc__int,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_eq_int @ M2 @ N )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X2: int] : X2
          @ ( set_or1266510415728281911st_int @ M2 @ N ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M2 @ ( minus_minus_int @ M2 @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_5301_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_5302_remdups__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( remdups_nat @ ( upt @ M2 @ N ) )
      = ( upt @ M2 @ N ) ) ).

% remdups_upt
thf(fact_5303_gcd__1__int,axiom,
    ! [M2: int] :
      ( ( gcd_gcd_int @ M2 @ one_one_int )
      = one_one_int ) ).

% gcd_1_int
thf(fact_5304_gcd__1__nat,axiom,
    ! [M2: nat] :
      ( ( gcd_gcd_nat @ M2 @ one_one_nat )
      = one_one_nat ) ).

% gcd_1_nat
thf(fact_5305_sort__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( linord738340561235409698at_nat
        @ ^ [X2: nat] : X2
        @ ( upt @ M2 @ N ) )
      = ( upt @ M2 @ N ) ) ).

% sort_upt
thf(fact_5306_sort__upto,axiom,
    ! [I2: int,J: int] :
      ( ( linord1735203802627413978nt_int
        @ ^ [X2: int] : X2
        @ ( upto @ I2 @ J ) )
      = ( upto @ I2 @ J ) ) ).

% sort_upto
thf(fact_5307_rat__inverse__code,axiom,
    ! [P5: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( A2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A2 ) @ B2 ) @ ( abs_abs_int @ A2 ) ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_inverse_code
thf(fact_5308_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_5309_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L3: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q6: code_integer,R4: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L3 ) @ R4 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R4 @ ( numera6620942414471956472nteger @ L3 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ R4 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_5310_inverse__rat_Otransfer,axiom,
    ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
    @ ^ [X2: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X2 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) )
    @ inverse_inverse_rat ) ).

% inverse_rat.transfer
thf(fact_5311_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_5312_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_5313_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_5314_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_5315_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_5316_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_5317_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
thf(fact_5318_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
thf(fact_5319_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_5320_integer__of__num_I3_J,axiom,
    ! [N: num] :
      ( ( code_integer_of_num @ ( bit1 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).

% integer_of_num(3)
thf(fact_5321_Fract_Orsp,axiom,
    ( bNF_re157797125943740599nt_int
    @ ^ [Y4: int,Z4: int] : Y4 = Z4
    @ ( bNF_re6250860962936578807nt_int
      @ ^ [Y4: int,Z4: int] : Y4 = Z4
      @ ratrel )
    @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A2 @ B2 ) )
    @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A2 @ B2 ) ) ) ).

% Fract.rsp
thf(fact_5322_one__integer__def,axiom,
    ( one_one_Code_integer
    = ( code_integer_of_int @ one_one_int ) ) ).

% one_integer_def
thf(fact_5323_integer__of__num__triv_I1_J,axiom,
    ( ( code_integer_of_num @ one )
    = one_one_Code_integer ) ).

% integer_of_num_triv(1)
thf(fact_5324_FI__def,axiom,
    ( fi
    = ( ^ [M3: list_P8527749157015355191n_assn,P6: assn,Q6: assn,Up: assn,Uq: assn,F3: assn] :
          ( ! [X2: produc6575502325842934193n_assn] :
              ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ M3 ) )
             => ( produc7274209992780475162assn_o @ entails @ X2 ) )
         => ( entails @ ( times_times_assn @ ( times_times_assn @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ M3 ) @ one_one_assn ) @ P6 ) @ Up ) @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn @ M3 ) @ one_one_assn ) @ Q6 ) @ Uq ) @ F3 ) ) ) ) ) ).

% FI_def
thf(fact_5325_inverse__rat_Orsp,axiom,
    ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
    @ ^ [X2: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X2 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) )
    @ ^ [X2: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X2 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) ) ) ).

% inverse_rat.rsp
thf(fact_5326_FI__RESULT__def,axiom,
    ( fI_RESULT
    = ( ^ [M5: list_P8527749157015355191n_assn,UP: assn,UQ: assn,F: assn] :
          ( ! [X2: produc6575502325842934193n_assn] :
              ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ M5 ) )
             => ( produc7274209992780475162assn_o @ entails @ X2 ) )
         => ( entails @ ( times_times_assn @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ M5 ) @ one_one_assn ) @ UP ) @ ( times_times_assn @ ( times_times_assn @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn @ M5 ) @ one_one_assn ) @ UQ ) @ F ) ) ) ) ) ).

% FI_RESULT_def
thf(fact_5327_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_5328_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_5329_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
            @ ^ [R4: code_integer,S4: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K4 ) @ R4 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ S4 ) ) @ ( S4 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_5330_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_5331_one__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ one_one_Code_integer )
    = one_one_int ) ).

% one_integer.rep_eq
thf(fact_5332_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
                @ ^ [R4: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R4 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ 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
                    @ ^ [R4: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R4 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L3 ) @ S4 ) ) )
                    @ ( code_divmod_abs @ K4 @ L3 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_5333_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
                  @ ^ [R4: code_integer,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R4 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L3 ) @ S4 ) ) )
                  @ ( code_divmod_abs @ K4 @ L3 ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_5334_card__length__sum__list,axiom,
    ! [M2: nat,N2: nat] :
      ( ( finite_card_list_nat
        @ ( collect_list_nat
          @ ^ [L3: list_nat] :
              ( ( ( size_size_list_nat @ L3 )
                = M2 )
              & ( ( groups4561878855575611511st_nat @ L3 )
                = N2 ) ) ) )
      = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ one_one_nat ) @ N2 ) ) ).

% card_length_sum_list
thf(fact_5335_card__length__sum__list__rec,axiom,
    ! [M2: nat,N2: 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 )
                  = N2 ) ) ) )
        = ( 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 )
                    = N2 ) ) ) )
          @ ( 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 )
                    = N2 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_5336_rat__number__expand_I5_J,axiom,
    ! [K: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) )
      = ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).

% rat_number_expand(5)
thf(fact_5337_Fract__of__int__eq,axiom,
    ! [K: int] :
      ( ( fract @ K @ one_one_int )
      = ( ring_1_of_int_rat @ K ) ) ).

% Fract_of_int_eq
thf(fact_5338_eq__rat_I2_J,axiom,
    ! [A: int] :
      ( ( fract @ A @ zero_zero_int )
      = ( fract @ zero_zero_int @ one_one_int ) ) ).

% eq_rat(2)
thf(fact_5339_Fract__of__nat__eq,axiom,
    ! [K: nat] :
      ( ( fract @ ( semiri1314217659103216013at_int @ K ) @ one_one_int )
      = ( semiri681578069525770553at_rat @ K ) ) ).

% Fract_of_nat_eq
thf(fact_5340_One__rat__def,axiom,
    ( one_one_rat
    = ( fract @ one_one_int @ one_one_int ) ) ).

% One_rat_def
thf(fact_5341_Zero__rat__def,axiom,
    ( zero_zero_rat
    = ( fract @ zero_zero_int @ one_one_int ) ) ).

% Zero_rat_def
thf(fact_5342_Fract__add__one,axiom,
    ! [N: int,M2: int] :
      ( ( N != zero_zero_int )
     => ( ( fract @ ( plus_plus_int @ M2 @ N ) @ N )
        = ( plus_plus_rat @ ( fract @ M2 @ N ) @ one_one_rat ) ) ) ).

% Fract_add_one
thf(fact_5343_rat__number__collapse_I3_J,axiom,
    ! [W: num] :
      ( ( fract @ ( numeral_numeral_int @ W ) @ one_one_int )
      = ( numeral_numeral_rat @ W ) ) ).

% rat_number_collapse(3)
thf(fact_5344_rat__number__expand_I3_J,axiom,
    ( numeral_numeral_rat
    = ( ^ [K4: num] : ( fract @ ( numeral_numeral_int @ K4 ) @ one_one_int ) ) ) ).

% rat_number_expand(3)
thf(fact_5345_one__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% one_le_Fract_iff
thf(fact_5346_Fract__le__one__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% Fract_le_one_iff
thf(fact_5347_Fract_Oabs__eq,axiom,
    ( fract
    = ( ^ [Xa4: int,X2: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X2 ) ) ) ) ) ).

% Fract.abs_eq
thf(fact_5348_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_5349_one__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% one_less_Fract_iff
thf(fact_5350_Fract__less__one__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
        = ( ord_less_int @ A @ B ) ) ) ).

% Fract_less_one_iff
thf(fact_5351_rat__number__collapse_I5_J,axiom,
    ( ( fract @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% rat_number_collapse(5)
thf(fact_5352_Fract_Otransfer,axiom,
    ( bNF_re3461391660133120880nt_rat
    @ ^ [Y4: int,Z4: int] : Y4 = Z4
    @ ( bNF_re2214769303045360666nt_rat
      @ ^ [Y4: int,Z4: int] : Y4 = Z4
      @ pcr_rat )
    @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A2 @ B2 ) )
    @ fract ) ).

% Fract.transfer
thf(fact_5353_rat__number__collapse_I4_J,axiom,
    ! [W: num] :
      ( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ one_one_int )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ).

% rat_number_collapse(4)
thf(fact_5354_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X2: rat] :
          ( the_int
          @ ^ [Z3: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X2 )
              & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_5355_abs__division__segment,axiom,
    ! [K: int] :
      ( ( abs_abs_int @ ( euclid3395696857347342551nt_int @ K ) )
      = one_one_int ) ).

% abs_division_segment
thf(fact_5356_division__segment__nat__def,axiom,
    ( euclid3398187327856392827nt_nat
    = ( ^ [N3: nat] : one_one_nat ) ) ).

% division_segment_nat_def
thf(fact_5357_division__segment__int__def,axiom,
    ( euclid3395696857347342551nt_int
    = ( ^ [K4: int] : ( if_int @ ( ord_less_eq_int @ zero_zero_int @ K4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% division_segment_int_def
thf(fact_5358_inverse__rat__def,axiom,
    ( inverse_inverse_rat
    = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
      @ ^ [X2: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int
          @ ( ( product_fst_int_int @ X2 )
            = zero_zero_int )
          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
          @ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) ) ) ) ).

% inverse_rat_def
thf(fact_5359_num__of__nat__One,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ one_one_nat )
     => ( ( num_of_nat @ N )
        = one ) ) ).

% num_of_nat_One
thf(fact_5360_one__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).

% one_int.transfer
thf(fact_5361_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_5362_integer__of__nat__1,axiom,
    ( ( code_integer_of_nat @ one_one_nat )
    = one_one_Code_integer ) ).

% integer_of_nat_1
thf(fact_5363_list__all__iff__all__interval__int,axiom,
    ! [P2: int > $o,I2: int,J: int] :
      ( ( list_all_int @ P2 @ ( upto @ I2 @ J ) )
      = ( all_interval_int @ P2 @ I2 @ J ) ) ).

% list_all_iff_all_interval_int
thf(fact_5364_list__ex__iff__not__all__inverval__int,axiom,
    ! [P2: int > $o,I2: int,J: int] :
      ( ( list_ex_int @ P2 @ ( upto @ I2 @ J ) )
      = ( ~ ( all_interval_int @ ( comp_o_o_int @ (~) @ P2 ) @ I2 @ J ) ) ) ).

% list_ex_iff_not_all_inverval_int
thf(fact_5365_all__interval__int__def,axiom,
    ( all_interval_int
    = ( ^ [P: int > $o,I4: int,J2: int] :
        ! [X2: int] :
          ( ( member_int2 @ X2 @ ( set_or1266510415728281911st_int @ I4 @ J2 ) )
         => ( P @ X2 ) ) ) ) ).

% all_interval_int_def
thf(fact_5366_unit__factor__simps_I2_J,axiom,
    ! [N: nat] :
      ( ( unit_f2748546683901255202or_nat @ ( suc @ N ) )
      = one_one_nat ) ).

% unit_factor_simps(2)
thf(fact_5367_unit__factor__nat__def,axiom,
    ( unit_f2748546683901255202or_nat
    = ( ^ [N3: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat @ one_one_nat ) ) ) ).

% unit_factor_nat_def
thf(fact_5368_list__all__iff__all__interval__nat,axiom,
    ! [P2: nat > $o,I2: nat,J: nat] :
      ( ( list_all_nat @ P2 @ ( upt @ I2 @ J ) )
      = ( all_interval_nat @ P2 @ I2 @ J ) ) ).

% list_all_iff_all_interval_nat
thf(fact_5369_list__ex__iff__not__all__inverval__nat,axiom,
    ! [P2: nat > $o,I2: nat,J: nat] :
      ( ( list_ex_nat @ P2 @ ( upt @ I2 @ J ) )
      = ( ~ ( all_interval_nat @ ( comp_o_o_nat @ (~) @ P2 ) @ I2 @ J ) ) ) ).

% list_ex_iff_not_all_inverval_nat
thf(fact_5370_Gcd__nat__eq__one,axiom,
    ! [N2: set_nat] :
      ( ( member_nat2 @ one_one_nat @ N2 )
     => ( ( gcd_Gcd_nat @ N2 )
        = one_one_nat ) ) ).

% Gcd_nat_eq_one
thf(fact_5371_Lcm__nat__empty,axiom,
    ( ( gcd_Lcm_nat @ bot_bot_set_nat )
    = one_one_nat ) ).

% Lcm_nat_empty
thf(fact_5372_all__interval__nat__def,axiom,
    ( all_interval_nat
    = ( ^ [P: nat > $o,I4: nat,J2: nat] :
        ! [X2: nat] :
          ( ( member_nat2 @ X2 @ ( set_or4665077453230672383an_nat @ I4 @ J2 ) )
         => ( P @ X2 ) ) ) ) ).

% all_interval_nat_def
thf(fact_5373_lcm__1__iff__int,axiom,
    ! [M2: int,N: int] :
      ( ( ( gcd_lcm_int @ M2 @ N )
        = one_one_int )
      = ( ( ( M2 = one_one_int )
          | ( M2
            = ( uminus_uminus_int @ one_one_int ) ) )
        & ( ( N = one_one_int )
          | ( N
            = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% lcm_1_iff_int
thf(fact_5374_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_5375_Lcm__int__set__eq__fold,axiom,
    ! [Xs: list_int] :
      ( ( gcd_Lcm_int @ ( set_int2 @ Xs ) )
      = ( fold_int_int @ gcd_lcm_int @ Xs @ one_one_int ) ) ).

% Lcm_int_set_eq_fold
thf(fact_5376_Lcm__nat__def,axiom,
    ( gcd_Lcm_nat
    = ( ^ [M5: set_nat] : ( if_nat @ ( finite_finite_nat @ M5 ) @ ( lattic7826324295020591184_F_nat @ gcd_lcm_nat @ one_one_nat @ M5 ) @ zero_zero_nat ) ) ) ).

% Lcm_nat_def
thf(fact_5377_coprime__diff__one__right__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% coprime_diff_one_right_nat
thf(fact_5378_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_5379_coprime__diff__one__left__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).

% coprime_diff_one_left_nat
thf(fact_5380_in__range_Osimps,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As2 ) )
      = ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ As2 )
           => ( ord_less_nat @ X2 @ ( lim_Product_unit @ H2 ) ) ) ) ) ).

% in_range.simps
thf(fact_5381_coprime__common__divisor__int,axiom,
    ! [A: int,B: int,X: int] :
      ( ( algebr932160517623751201me_int @ A @ B )
     => ( ( dvd_dvd_int @ X @ A )
       => ( ( dvd_dvd_int @ X @ B )
         => ( ( abs_abs_int @ X )
            = one_one_int ) ) ) ) ).

% coprime_common_divisor_int
thf(fact_5382_in__range_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( in_range @ X )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ! [X3: nat] :
                ( ( member_nat2 @ X3 @ As )
               => ( ord_less_nat @ X3 @ ( lim_Product_unit @ H3 ) ) ) ) ) ).

% in_range.elims(3)
thf(fact_5383_in__range_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( in_range @ X )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ~ ! [X4: nat] :
                  ( ( member_nat2 @ X4 @ As )
                 => ( ord_less_nat @ X4 @ ( lim_Product_unit @ H3 ) ) ) ) ) ).

% in_range.elims(2)
thf(fact_5384_in__range_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( in_range @ X )
        = Y )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H3 @ As ) )
           => ( Y
              = ( ~ ! [X2: nat] :
                      ( ( member_nat2 @ X2 @ As )
                     => ( ord_less_nat @ X2 @ ( lim_Product_unit @ H3 ) ) ) ) ) ) ) ).

% in_range.elims(1)
thf(fact_5385_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_5386_wait__rule,axiom,
    ! [N: nat] :
      ( hoare_8945653483474564448t_unit @ one_one_assn @ ( heap_Time_wait @ N )
      @ ^ [Uu2: product_unit] : one_one_assn ) ).

% wait_rule
thf(fact_5387_inf__unit__def,axiom,
    ( inf_inf_Product_unit
    = ( ^ [Uu3: product_unit,Uv: product_unit] : product_Unity ) ) ).

% inf_unit_def
thf(fact_5388_assert_H__rule,axiom,
    ! [P2: assn,Phi: $o] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P2 @ H3 )
         => Phi )
     => ( hoare_8945653483474564448t_unit @ P2 @ ( refine_Imp_assert @ Phi )
        @ ^ [Uu2: product_unit] : P2 ) ) ).

% assert'_rule
thf(fact_5389_natural__zero__minus__one,axiom,
    ( ( minus_7197305767214868737atural @ zero_z2226904508553997617atural @ one_one_Code_natural )
    = zero_z2226904508553997617atural ) ).

% natural_zero_minus_one
thf(fact_5390_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_5391_log_Osimps,axiom,
    ( log
    = ( ^ [B2: code_natural,I4: code_natural] :
          ( if_Code_natural
          @ ( ( ord_le1926595141338095240atural @ B2 @ one_one_Code_natural )
            | ( ord_le5570908160329646204atural @ I4 @ B2 ) )
          @ one_one_Code_natural
          @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( log @ B2 @ ( divide5121882707175180666atural @ I4 @ B2 ) ) ) ) ) ) ).

% log.simps
thf(fact_5392_next_Osimps,axiom,
    ! [V2: code_natural,W: code_natural] :
      ( ( next @ ( produc3574140220909816553atural @ V2 @ 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 @ V2 @ ( 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 @ V2 @ ( 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 @ V2 @ ( 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 @ V2 @ ( 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_5393_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
                @ ^ [V3: code_natural] : ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ V3 @ ( 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 )
          @ ^ [V3: code_natural] : ( produc6639722614265839536atural @ ( modulo8411746178871703098atural @ V3 @ K4 ) ) ) ) ) ).

% Random.range_def
thf(fact_5394_inc__shift__def,axiom,
    ( inc_shift
    = ( ^ [V3: code_natural,K4: code_natural] : ( if_Code_natural @ ( V3 = K4 ) @ one_one_Code_natural @ ( plus_p4538020629002901425atural @ K4 @ one_one_Code_natural ) ) ) ) ).

% inc_shift_def
thf(fact_5395_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_5396_one__natural__def,axiom,
    ( one_one_Code_natural
    = ( code_natural_of_nat @ one_one_nat ) ) ).

% one_natural_def
thf(fact_5397_Suc__natural__minus__one,axiom,
    ! [N: code_natural] :
      ( ( minus_7197305767214868737atural @ ( code_Suc @ N ) @ one_one_Code_natural )
      = N ) ).

% Suc_natural_minus_one
thf(fact_5398_Code__Numeral_Osub__code_I9_J,axiom,
    ! [M2: num,N: num] :
      ( ( code_sub @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
      = ( minus_8373710615458151222nteger @ ( code_dup @ ( code_sub @ M2 @ N ) ) @ one_one_Code_integer ) ) ).

% Code_Numeral.sub_code(9)
thf(fact_5399_Code__Numeral_Osub__code_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( code_sub @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( code_dup @ ( code_sub @ M2 @ N ) ) @ one_one_Code_integer ) ) ).

% Code_Numeral.sub_code(8)
thf(fact_5400_Int_Osub__code_I9_J,axiom,
    ! [M2: num,N: num] :
      ( ( sub @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
      = ( minus_minus_int @ ( dup @ ( sub @ M2 @ N ) ) @ one_one_int ) ) ).

% Int.sub_code(9)
thf(fact_5401_Int_Osub__code_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( sub @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( plus_plus_int @ ( dup @ ( sub @ M2 @ N ) ) @ one_one_int ) ) ).

% Int.sub_code(8)
thf(fact_5402_Fract__def,axiom,
    ( fract
    = ( map_fu7831380289885515383nt_rat @ id_int @ ( map_fu3424225382358772769nt_rat @ id_int @ abs_Rat )
      @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( B2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A2 @ B2 ) ) ) ) ).

% Fract_def
thf(fact_5403_in__range_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( in_range @ X )
     => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H3 @ As ) )
             => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H3 @ As ) )
               => ! [X3: nat] :
                    ( ( member_nat2 @ X3 @ As )
                   => ( ord_less_nat @ X3 @ ( lim_Product_unit @ H3 ) ) ) ) ) ) ) ).

% in_range.pelims(3)
thf(fact_5404_in__range_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( in_range @ X )
        = Y )
     => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H3 @ As ) )
             => ( ( Y
                  = ( ! [X2: nat] :
                        ( ( member_nat2 @ X2 @ As )
                       => ( ord_less_nat @ X2 @ ( lim_Product_unit @ H3 ) ) ) ) )
               => ~ ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ) ) ) ).

% in_range.pelims(1)
thf(fact_5405_in__range_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( in_range @ X )
     => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H3 @ As ) )
             => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H3 @ As ) )
               => ~ ! [X4: nat] :
                      ( ( member_nat2 @ X4 @ As )
                     => ( ord_less_nat @ X4 @ ( lim_Product_unit @ H3 ) ) ) ) ) ) ) ).

% in_range.pelims(2)
thf(fact_5406_one__assn__raw_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( one_assn_raw @ X )
     => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H3 @ As ) )
             => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H3 @ As ) )
               => ( As = bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(3)
thf(fact_5407_one__assn__raw_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( one_assn_raw @ X )
     => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H3 @ As ) )
             => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H3 @ As ) )
               => ( As != bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(2)
thf(fact_5408_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 )
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H3 @ As ) )
             => ( ( Y
                  = ( As = bot_bot_set_nat ) )
               => ~ ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H3 @ As ) ) ) ) ) ) ).

% one_assn_raw.pelims(1)
thf(fact_5409_Restr__natLeq,axiom,
    ! [N: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat
          @ ( collect_nat
            @ ^ [X2: nat] : ( ord_less_nat @ X2 @ N ) )
          @ ^ [Uu2: nat] :
              ( collect_nat
              @ ^ [X2: nat] : ( ord_less_nat @ X2 @ N ) ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ N )
              & ( ord_less_nat @ Y2 @ N )
              & ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ).

% Restr_natLeq
thf(fact_5410_Restr__natLeq2,axiom,
    ! [N: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
          @ ^ [Uu2: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ N )
              & ( ord_less_nat @ Y2 @ N )
              & ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ).

% Restr_natLeq2
thf(fact_5411_one__integer__code,axiom,
    ( one_one_Code_integer
    = ( code_Pos @ one ) ) ).

% one_integer_code
thf(fact_5412_one__natural_Orep__eq,axiom,
    ( ( code_nat_of_natural @ one_one_Code_natural )
    = one_one_nat ) ).

% one_natural.rep_eq
thf(fact_5413_nat__of__num__code_I1_J,axiom,
    ( ( nat_of_num @ one )
    = one_one_nat ) ).

% nat_of_num_code(1)
thf(fact_5414_one__int__code,axiom,
    ( one_one_int
    = ( pos @ one ) ) ).

% one_int_code
thf(fact_5415_one__natural_Otransfer,axiom,
    code_pcr_natural @ one_one_nat @ one_one_Code_natural ).

% one_natural.transfer
thf(fact_5416_one__integer_Otransfer,axiom,
    code_pcr_integer @ one_one_int @ one_one_Code_integer ).

% one_integer.transfer
thf(fact_5417_rat_Odomain__par,axiom,
    ! [DR1: int > $o,DR2: int > $o,P22: product_prod_int_int > $o] :
      ( ( ( domainp_int_int
          @ ^ [Y4: int,Z4: int] : Y4 = Z4 )
        = DR1 )
     => ( ( ( domainp_int_int
            @ ^ [Y4: int,Z4: int] : Y4 = Z4 )
          = DR2 )
       => ( ( bNF_re8699439704749558557nt_o_o
            @ ( basic_4387203522000727145nt_int
              @ ^ [Y4: int,Z4: int] : Y4 = Z4
              @ ^ [Y4: int,Z4: int] : Y4 = Z4 )
            @ ^ [Y4: $o,Z4: $o] : Y4 = Z4
            @ P22
            @ ^ [X2: product_prod_int_int] : ( ratrel @ X2 @ X2 ) )
         => ( ( domain9213661015745956397nt_rat @ pcr_rat )
            = ( inf_in3604695632404883862_int_o @ ( basic_1567116559311922317nt_int @ DR1 @ DR2 ) @ P22 ) ) ) ) ) ).

% rat.domain_par

% Helper facts (29)
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__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_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__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_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

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 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    fI_QUERY @ p @ q @ f ).

%------------------------------------------------------------------------------