TPTP Problem File: NUM924^4.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : NUM924^4 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 102, 5000 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : s2s_5000_thf_l102 [Bla11]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.50 v7.5.0, 1.00 v7.2.0, 0.75 v7.1.0, 0.67 v6.4.0, 0.83 v6.3.0, 0.80 v6.2.0, 0.86 v5.5.0, 0.83 v5.4.0, 1.00 v5.3.0
% Syntax : Number of formulae : 5606 (2149 unt; 394 typ; 0 def)
% Number of atoms : 11934 (4190 equ; 67 cnn)
% Maximal formula atoms : 14 ( 2 avg)
% Number of connectives : 51084 ( 909 ~; 172 |; 756 &;43850 @)
% ( 908 <=>;4480 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 1115 (1115 >; 0 *; 0 +; 0 <<)
% Number of symbols : 382 ( 374 usr; 30 con; 0-8 aty)
% ( 0 !!; 9 ??; 0 @@+; 0 @@-)
% Number of variables : 11278 ( 263 ^;10747 !; 268 ?;11278 :)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 20:21:09
%------------------------------------------------------------------------------
%----Should-be-implicit typings (20)
thf(ty_ty_tc__Code____Numeral__Ocode____numeral,type,
code_code_numeral: $tType ).
thf(ty_ty_tc__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_ty_tc__Int__Oint,type,
int: $tType ).
thf(ty_ty_tc__Limits__Ofilter_Itc__Complex__Ocomplex_J,type,
filter_complex: $tType ).
thf(ty_ty_tc__Limits__Ofilter_Itc__Nat__Onat_J,type,
filter_nat: $tType ).
thf(ty_ty_tc__Limits__Ofilter_Itc__RealDef__Oreal_J,type,
filter_real: $tType ).
thf(ty_ty_tc__List__Olist_Itc__Int__Oint_J,type,
list_int: $tType ).
thf(ty_ty_tc__Nat__Onat,type,
nat: $tType ).
thf(ty_ty_tc__Quickcheck____Narrowing__Ocode____int,type,
quickcheck_code_int: $tType ).
thf(ty_ty_tc__Rat__Orat,type,
rat: $tType ).
thf(ty_ty_tc__RealDef__Oreal,type,
real: $tType ).
thf(ty_ty_tc__prod_I_062_Itc__Int__Oint_M_Eo_J_M_062_Itc__Int__Oint_M_Eo_J_J,type,
produc975137661_int_o: $tType ).
thf(ty_ty_tc__prod_Itc__Code____Numeral__Ocode____numeral_Mtc__Code____Numeral__Ocod,type,
produc1359518119umeral: $tType ).
thf(ty_ty_tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
product_prod_int_int: $tType ).
thf(ty_ty_tc__prod_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_J,type,
produc393999548nt_int: $tType ).
thf(ty_ty_tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_ty_tc__prod_Itc__Quickcheck____Narrowing__Ocode____int_Mtc__Quickcheck____Nar,type,
produc167071911de_int: $tType ).
thf(ty_ty_tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J,type,
produc914805421l_real: $tType ).
thf(ty_ty_tc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc__Int__Oin,type,
produc1137372701nt_int: $tType ).
thf(ty_ty_tc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__prod_Itc__Nat__Ona,type,
produc1322466333at_nat: $tType ).
%----Explicit typings (376)
thf(sy_c_All,type,
all: ( nat > $o ) > $o ).
thf(sy_c_Archimedean__Field_Oceiling_000tc__RealDef__Oreal,type,
archim856651990g_real: real > int ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_000tc__Rat__Orat,type,
archim791455193or_rat: rat > int ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_000tc__RealDef__Oreal,type,
archim1246769320r_real: real > int ).
thf(sy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_000_062_Itc__Int__Oint_M_E,type,
big_co1971440592_o_nat: ( ( int > $o ) > nat ) > ( ( int > $o ) > $o ) > nat ).
thf(sy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_000tc__Int__Oint_000tc__In,type,
big_co230513141nt_int: ( int > int ) > ( int > $o ) > int ).
thf(sy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_000tc__Nat__Onat_000tc__In,type,
big_co1024481617at_int: ( nat > int ) > ( nat > $o ) > int ).
thf(sy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_000tc__Nat__Onat_000tc__Na,type,
big_co387207925at_nat: ( nat > nat ) > ( nat > $o ) > nat ).
thf(sy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_000tc__Nat__Onat_000tc__Re,type,
big_co604158596t_real: ( nat > real ) > ( nat > $o ) > real ).
thf(sy_c_Big__Operators_Ocomm__monoid__mult__class_Osetprod_000tc__Int__Oint_000tc__,type,
big_co1548731110nt_int: ( int > int ) > ( int > $o ) > int ).
thf(sy_c_Big__Operators_Ocomm__monoid__mult__class_Osetprod_000tc__Nat__Onat_000tc__,type,
big_co1705425894at_nat: ( nat > nat ) > ( nat > $o ) > nat ).
thf(sy_c_BijectionRel_ObijR_000tc__Int__Oint_000tc__Int__Oint,type,
bijR_int_int: ( int > int > $o ) > produc975137661_int_o > $o ).
thf(sy_c_Code__Numeral_OSuc__code__numeral,type,
code_S1047413653umeral: code_code_numeral > code_code_numeral ).
thf(sy_c_Code__Numeral_Ocode__numeral_Ocode__numeral__size,type,
code_c271388182l_size: code_code_numeral > nat ).
thf(sy_c_Code__Numeral_Odiv__mod__code__numeral,type,
code_d418564891umeral: code_code_numeral > code_code_numeral > produc1359518119umeral ).
thf(sy_c_Code__Numeral_Oint__of,type,
code_int_of: code_code_numeral > int ).
thf(sy_c_Code__Numeral_Onat__of__aux,type,
code_nat_of_aux: code_code_numeral > nat > nat ).
thf(sy_c_Complete__Lattice_OSup__class_OSup_000_062_Itc__Int__Oint_M_Eo_J,type,
comple1092985777_int_o: ( ( int > $o ) > $o ) > int > $o ).
thf(sy_c_Complete__Lattice_OSup__class_OSup_000tc__RealDef__Oreal,type,
comple124823625p_real: ( real > $o ) > real ).
thf(sy_c_Complex_OIm,type,
im: complex > real ).
thf(sy_c_Complex_ORe,type,
re: complex > real ).
thf(sy_c_Complex_Oarg,type,
arg: complex > real ).
thf(sy_c_Complex_Ocis,type,
cis: real > complex ).
thf(sy_c_Complex_Ocnj,type,
cnj: complex > complex ).
thf(sy_c_Complex_Ocomplex_OComplex,type,
complex_1: real > real > complex ).
thf(sy_c_Complex_Ocomplex_Ocomplex__size,type,
complex_size: complex > nat ).
thf(sy_c_Complex_Oexpi,type,
expi: complex > complex ).
thf(sy_c_Complex_Oii,type,
ii: complex ).
thf(sy_c_Complex_Orcis,type,
rcis: real > real > complex ).
thf(sy_c_Deriv_OBolzano__bisect,type,
bolzano_bisect: ( produc914805421l_real > $o ) > real > real > nat > produc914805421l_real ).
thf(sy_c_Deriv_Oderiv_000tc__RealDef__Oreal,type,
deriv_real: ( real > real ) > real > real > $o ).
thf(sy_c_Divides_Oadjust,type,
adjust: int > product_prod_int_int > product_prod_int_int ).
thf(sy_c_Divides_Odiv__class_Odiv_000tc__Code____Numeral__Ocode____numeral,type,
div_di1218280263umeral: code_code_numeral > code_code_numeral > code_code_numeral ).
thf(sy_c_Divides_Odiv__class_Odiv_000tc__Int__Oint,type,
div_div_int: int > int > int ).
thf(sy_c_Divides_Odiv__class_Odiv_000tc__Nat__Onat,type,
div_div_nat: nat > nat > nat ).
thf(sy_c_Divides_Odiv__class_Odiv_000tc__Quickcheck____Narrowing__Ocode____int,type,
div_di1430059507de_int: quickcheck_code_int > quickcheck_code_int > quickcheck_code_int ).
thf(sy_c_Divides_Odiv__class_Omod_000tc__Code____Numeral__Ocode____numeral,type,
div_mo1740067990umeral: code_code_numeral > code_code_numeral > code_code_numeral ).
thf(sy_c_Divides_Odiv__class_Omod_000tc__Int__Oint,type,
div_mod_int: int > int > int ).
thf(sy_c_Divides_Odiv__class_Omod_000tc__Nat__Onat,type,
div_mod_nat: nat > nat > nat ).
thf(sy_c_Divides_Odiv__class_Omod_000tc__Quickcheck____Narrowing__Ocode____int,type,
div_mo231679042de_int: quickcheck_code_int > quickcheck_code_int > quickcheck_code_int ).
thf(sy_c_Divides_Odivmod__int,type,
divmod_int: int > int > product_prod_int_int ).
thf(sy_c_Divides_Odivmod__int__rel,type,
divmod_int_rel: int > int > product_prod_int_int > $o ).
thf(sy_c_Divides_Odivmod__nat,type,
divmod_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Divides_Odivmod__nat__rel,type,
divmod_nat_rel: nat > nat > product_prod_nat_nat > $o ).
thf(sy_c_Divides_OnegDivAlg,type,
negDivAlg: int > int > product_prod_int_int ).
thf(sy_c_Divides_OnegDivAlg__rel,type,
negDivAlg_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_Divides_OnegateSnd,type,
negateSnd: product_prod_int_int > product_prod_int_int ).
thf(sy_c_Divides_Opdivmod,type,
pdivmod: int > int > product_prod_int_int ).
thf(sy_c_Divides_OposDivAlg,type,
posDivAlg: int > int > product_prod_int_int ).
thf(sy_c_Divides_OposDivAlg__rel,type,
posDivAlg_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_EulerFermat_OBnorRset,type,
bnorRset: int > int > int > $o ).
thf(sy_c_EulerFermat_ORRset2norRR,type,
rRset2norRR: ( int > $o ) > int > int > int ).
thf(sy_c_EulerFermat_ORsetR,type,
rsetR: int > ( int > $o ) > $o ).
thf(sy_c_EulerFermat_Ois__RRset,type,
is_RRset: ( int > $o ) > int > $o ).
thf(sy_c_EulerFermat_OnoXRRset,type,
noXRRset: int > int > int > $o ).
thf(sy_c_EulerFermat_OnorRRset,type,
norRRset: int > int > $o ).
thf(sy_c_EulerFermat_Ophi,type,
phi: int > nat ).
thf(sy_c_EulerFermat_Ozcongm,type,
zcongm: int > int > int > $o ).
thf(sy_c_Euler_OMultInvPair,type,
multInvPair: int > int > int > int > $o ).
thf(sy_c_Euler_OSetS,type,
setS: int > int > ( int > $o ) > $o ).
thf(sy_c_EvenOdd_OzEven,type,
zEven: int > $o ).
thf(sy_c_EvenOdd_OzOdd,type,
zOdd: int > $o ).
thf(sy_c_Fact_Ofact__class_Ofact_000tc__Int__Oint,type,
fact_fact_int: int > int ).
thf(sy_c_Fact_Ofact__class_Ofact_000tc__Nat__Onat,type,
fact_fact_nat: nat > nat ).
thf(sy_c_Fields_Oinverse__class_Odivide_000tc__Complex__Ocomplex,type,
invers1025623611omplex: complex > complex > complex ).
thf(sy_c_Fields_Oinverse__class_Odivide_000tc__Rat__Orat,type,
inverse_divide_rat: rat > rat > rat ).
thf(sy_c_Fields_Oinverse__class_Odivide_000tc__RealDef__Oreal,type,
inverse_divide_real: real > real > real ).
thf(sy_c_Fields_Oinverse__class_Oinverse_000tc__Complex__Ocomplex,type,
invers1449016382omplex: complex > complex ).
thf(sy_c_Fields_Oinverse__class_Oinverse_000tc__Rat__Orat,type,
inverse_inverse_rat: rat > rat ).
thf(sy_c_Fields_Oinverse__class_Oinverse_000tc__RealDef__Oreal,type,
inverse_inverse_real: real > real ).
thf(sy_c_Finite__Set_Ocard_000_062_Itc__Int__Oint_M_Eo_J,type,
finite_card_int_o: ( ( int > $o ) > $o ) > nat ).
thf(sy_c_Finite__Set_Ocard_000tc__Int__Oint,type,
finite_card_int: ( int > $o ) > nat ).
thf(sy_c_Finite__Set_Ocard_000tc__Nat__Onat,type,
finite_card_nat: ( nat > $o ) > nat ).
thf(sy_c_Finite__Set_Ofinite_000_062_Itc__Int__Oint_M_Eo_J,type,
finite_finite_int_o: ( ( int > $o ) > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__Int__Oint,type,
finite_finite_int: ( int > $o ) > $o ).
thf(sy_c_Finite__Set_Ofinite_000tc__Nat__Onat,type,
finite_finite_nat: ( nat > $o ) > $o ).
thf(sy_c_FunDef_Opair__leq,type,
pair_leq: produc1322466333at_nat > $o ).
thf(sy_c_FunDef_Opair__less,type,
pair_less: produc1322466333at_nat > $o ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > product_prod_int_int ).
thf(sy_c_GCD_Ogcd__class_Ogcd_000tc__Int__Oint,type,
gcd_gcd_int: int > int > int ).
thf(sy_c_GCD_Ogcd__class_Ogcd_000tc__Nat__Onat,type,
gcd_gcd_nat: nat > nat > nat ).
thf(sy_c_Groups_Oabs__class_Oabs_000tc__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Oabs__class_Oabs_000tc__Rat__Orat,type,
abs_abs_rat: rat > rat ).
thf(sy_c_Groups_Oabs__class_Oabs_000tc__RealDef__Oreal,type,
abs_abs_real: real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__Code____Numeral__Ocode____numeral,type,
minus_1690775515umeral: code_code_numeral > code_code_numeral > code_code_numeral ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__Complex__Ocomplex,type,
minus_minus_complex: complex > complex > complex ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__Quickcheck____Narrowing__Ocode____int,type,
minus_534354567de_int: quickcheck_code_int > quickcheck_code_int > quickcheck_code_int ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__Rat__Orat,type,
minus_minus_rat: rat > rat > rat ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__RealDef__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Oone__class_Oone_000tc__Code____Numeral__Ocode____numeral,type,
one_on1645066479umeral: code_code_numeral ).
thf(sy_c_Groups_Oone__class_Oone_000tc__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_000tc__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_000tc__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_000tc__Quickcheck____Narrowing__Ocode____int,type,
one_on1684967323de_int: quickcheck_code_int ).
thf(sy_c_Groups_Oone__class_Oone_000tc__Rat__Orat,type,
one_one_rat: rat ).
thf(sy_c_Groups_Oone__class_Oone_000tc__RealDef__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Code____Numeral__Ocode____numeral,type,
plus_p1627245867umeral: code_code_numeral > code_code_numeral > code_code_numeral ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Complex__Ocomplex,type,
plus_plus_complex: complex > complex > complex ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Quickcheck____Narrowing__Ocode____int,type,
plus_p1446045655de_int: quickcheck_code_int > quickcheck_code_int > quickcheck_code_int ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Rat__Orat,type,
plus_plus_rat: rat > rat > rat ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__RealDef__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Osgn__class_Osgn_000tc__Complex__Ocomplex,type,
sgn_sgn_complex: complex > complex ).
thf(sy_c_Groups_Osgn__class_Osgn_000tc__Int__Oint,type,
sgn_sgn_int: int > int ).
thf(sy_c_Groups_Osgn__class_Osgn_000tc__Rat__Orat,type,
sgn_sgn_rat: rat > rat ).
thf(sy_c_Groups_Osgn__class_Osgn_000tc__RealDef__Oreal,type,
sgn_sgn_real: real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_000tc__Code____Numeral__Ocode____numeral,type,
times_1655362735umeral: code_code_numeral > code_code_numeral > code_code_numeral ).
thf(sy_c_Groups_Otimes__class_Otimes_000tc__Complex__Ocomplex,type,
times_times_complex: complex > complex > complex ).
thf(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_000tc__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_000tc__Quickcheck____Narrowing__Ocode____int,type,
times_123202395de_int: quickcheck_code_int > quickcheck_code_int > quickcheck_code_int ).
thf(sy_c_Groups_Otimes__class_Otimes_000tc__Rat__Orat,type,
times_times_rat: rat > rat > rat ).
thf(sy_c_Groups_Otimes__class_Otimes_000tc__RealDef__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_000tc__Complex__Ocomplex,type,
uminus473333897omplex: complex > complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_000tc__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_000tc__Rat__Orat,type,
uminus_uminus_rat: rat > rat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_000tc__RealDef__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__Code____Numeral__Ocode____numeral,type,
zero_z126310315umeral: code_code_numeral ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__Quickcheck____Narrowing__Ocode____int,type,
zero_z891286103de_int: quickcheck_code_int ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__Rat__Orat,type,
zero_zero_rat: rat ).
thf(sy_c_Groups_Ozero__class_Ozero_000tc__RealDef__Oreal,type,
zero_zero_real: real ).
thf(sy_c_HOL_OThe_000tc__Int__Oint,type,
the_int: ( int > $o ) > int ).
thf(sy_c_HOL_OThe_000tc__RealDef__Oreal,type,
the_real: ( real > $o ) > real ).
thf(sy_c_HOL_OThe_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
the_Pr2103884470nt_int: ( product_prod_int_int > $o ) > product_prod_int_int ).
thf(sy_c_HOL_OThe_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
the_Pr588456374at_nat: ( product_prod_nat_nat > $o ) > product_prod_nat_nat ).
thf(sy_c_Hilbert__Choice_OEps_000tc__Int__Oint,type,
hilbert_Eps_int: ( int > $o ) > int ).
thf(sy_c_Hilbert__Choice_OEps_000tc__RealDef__Oreal,type,
hilbert_Eps_real: ( real > $o ) > real ).
thf(sy_c_If_000tc__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_000tc__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_000tc__RealDef__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
if_Pro1731782967nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).
thf(sy_c_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J,type,
if_Pro313124157l_real: $o > produc914805421l_real > produc914805421l_real > produc914805421l_real ).
thf(sy_c_Int2_OMultInv,type,
multInv: int > int > int ).
thf(sy_c_IntFact_Od22set,type,
d22set: int > int > $o ).
thf(sy_c_IntFact_Ozfact,type,
zfact: int > int ).
thf(sy_c_IntPrimes_Oxzgcd,type,
xzgcd: int > int > produc393999548nt_int ).
thf(sy_c_IntPrimes_Oxzgcda,type,
xzgcda: int > int > int > int > int > int > int > int > produc393999548nt_int ).
thf(sy_c_IntPrimes_Ozcong,type,
zcong: int > int > int > $o ).
thf(sy_c_IntPrimes_Ozprime,type,
zprime: int > $o ).
thf(sy_c_Int_OBit0,type,
bit0: int > int ).
thf(sy_c_Int_OBit1,type,
bit1: int > int ).
thf(sy_c_Int_OMin,type,
min: int ).
thf(sy_c_Int_OPls,type,
pls: int ).
thf(sy_c_Int_Oint__ge__less__than,type,
int_ge_less_than: int > product_prod_int_int > $o ).
thf(sy_c_Int_Oint__ge__less__than2,type,
int_ge_less_than2: int > product_prod_int_int > $o ).
thf(sy_c_Int_Oiszero_000tc__Int__Oint,type,
iszero_int: int > $o ).
thf(sy_c_Int_Oiszero_000tc__Rat__Orat,type,
iszero_rat: rat > $o ).
thf(sy_c_Int_Onat,type,
nat_1: int > nat ).
thf(sy_c_Int_Onat__aux,type,
nat_aux: int > nat > nat ).
thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Code____Numeral__Ocode____numeral,type,
number1443263063umeral: int > code_code_numeral ).
thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Complex__Ocomplex,type,
number528085621omplex: int > complex ).
thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Int__Oint,type,
number_number_of_int: int > int ).
thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Nat__Onat,type,
number_number_of_nat: int > nat ).
thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Quickcheck____Narrowing__Ocode____int,type,
number1226105091de_int: int > quickcheck_code_int ).
thf(sy_c_Int_Onumber__class_Onumber__of_000tc__Rat__Orat,type,
number_number_of_rat: int > rat ).
thf(sy_c_Int_Onumber__class_Onumber__of_000tc__RealDef__Oreal,type,
number267125858f_real: int > real ).
thf(sy_c_Int_Opred,type,
pred: int > int ).
thf(sy_c_Int_Oring__1__class_OInts_000tc__RealDef__Oreal,type,
ring_1_Ints_real: real > $o ).
thf(sy_c_Int_Oring__1__class_Oof__int_000tc__Complex__Ocomplex,type,
ring_11397209091omplex: int > complex ).
thf(sy_c_Int_Oring__1__class_Oof__int_000tc__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_Int_Oring__1__class_Oof__int_000tc__Rat__Orat,type,
ring_1_of_int_rat: int > rat ).
thf(sy_c_Int_Oring__1__class_Oof__int_000tc__RealDef__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_Int_Osucc,type,
succ: int > int ).
thf(sy_c_Lazy__Sequence_Osmall__lazy_H__rel,type,
lazy_small_lazy_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_Legacy__GCD_Ozgcd,type,
legacy_zgcd: int > int > int ).
thf(sy_c_Lim_OisCont_000tc__Complex__Ocomplex_000tc__Complex__Ocomplex,type,
isCont156215680omplex: ( complex > complex ) > complex > $o ).
thf(sy_c_Lim_OisCont_000tc__Complex__Ocomplex_000tc__RealDef__Oreal,type,
isCont_complex_real: ( complex > real ) > complex > $o ).
thf(sy_c_Lim_OisCont_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
isCont_real_real: ( real > real ) > real > $o ).
thf(sy_c_Limits_Oat_000tc__Complex__Ocomplex,type,
at_complex: complex > filter_complex ).
thf(sy_c_Limits_Oat_000tc__RealDef__Oreal,type,
at_real: real > filter_real ).
thf(sy_c_Limits_Osequentially,type,
sequentially: filter_nat ).
thf(sy_c_Limits_Otendsto_000tc__Complex__Ocomplex_000tc__Complex__Ocomplex,type,
tendst1507391555omplex: ( complex > complex ) > complex > filter_complex > $o ).
thf(sy_c_Limits_Otendsto_000tc__Complex__Ocomplex_000tc__RealDef__Oreal,type,
tendsto_complex_real: ( complex > real ) > real > filter_complex > $o ).
thf(sy_c_Limits_Otendsto_000tc__Nat__Onat_000tc__Complex__Ocomplex,type,
tendsto_nat_complex: ( nat > complex ) > complex > filter_nat > $o ).
thf(sy_c_Limits_Otendsto_000tc__Nat__Onat_000tc__RealDef__Oreal,type,
tendsto_nat_real: ( nat > real ) > real > filter_nat > $o ).
thf(sy_c_Limits_Otendsto_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
tendsto_real_real: ( real > real ) > real > filter_real > $o ).
thf(sy_c_Limits_Otrivial__limit_000tc__Nat__Onat,type,
trivial_limit_nat: filter_nat > $o ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_Log_Olog,type,
log: real > real > real ).
thf(sy_c_Log_Opowr,type,
powr: real > real > real ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Onat_Onat__case_000_Eo,type,
nat_case_o: $o > ( nat > $o ) > nat > $o ).
thf(sy_c_Nat_Onat_Onat__case_000tc__Nat__Onat,type,
nat_case_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Onat_Onat__size,type,
nat_size: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Code____Numeral__Ocode____numeral,type,
semiri1619134803umeral: nat > code_code_numeral ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Complex__Ocomplex,type,
semiri2020571505omplex: nat > complex ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Int__Oint,type,
semiri1621563631at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Nat__Onat,type,
semiri984289939at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Quickcheck____Narrowing__Ocode____i,type,
semiri1424489471de_int: nat > quickcheck_code_int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__Rat__Orat,type,
semiri151668891at_rat: nat > rat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_000tc__RealDef__Oreal,type,
semiri132038758t_real: nat > real ).
thf(sy_c_Nat_Osize__class_Osize_000tc__Code____Numeral__Ocode____numeral,type,
size_s945831648umeral: code_code_numeral > nat ).
thf(sy_c_Nat_Osize__class_Osize_000tc__Complex__Ocomplex,type,
size_size_complex: complex > nat ).
thf(sy_c_Nat_Osize__class_Osize_000tc__List__Olist_Itc__Int__Oint_J,type,
size_size_list_int: list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_000tc__Nat__Onat,type,
size_size_nat: nat > nat ).
thf(sy_c_Nat__Numeral_Oneg,type,
nat_neg: int > $o ).
thf(sy_c_Nat__Transfer_Ois__nat,type,
nat_is_nat: int > $o ).
thf(sy_c_Nat__Transfer_Onat__set,type,
nat_nat_set: ( int > $o ) > $o ).
thf(sy_c_Nat__Transfer_Otransfer__morphism_000tc__Int__Oint_000tc__Nat__Onat,type,
nat_tr876908586nt_nat: ( int > nat ) > ( int > $o ) > $o ).
thf(sy_c_Nat__Transfer_Otransfer__morphism_000tc__Nat__Onat_000tc__Int__Oint,type,
nat_tr160667106at_int: ( nat > int ) > ( nat > $o ) > $o ).
thf(sy_c_Nat__Transfer_Otsub,type,
nat_tsub: int > int > int ).
thf(sy_c_Nitpick_OFrac,type,
frac: product_prod_int_int > $o ).
thf(sy_c_Nitpick_Oint__gcd,type,
int_gcd: int > int > int ).
thf(sy_c_Nitpick_Oint__lcm,type,
int_lcm: int > int > int ).
thf(sy_c_Nitpick_Onat__gcd,type,
nat_gcd: nat > nat > nat ).
thf(sy_c_Nitpick_Onat__gcd__rel,type,
nat_gcd_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Nitpick_Onat__lcm,type,
nat_lcm: nat > nat > nat ).
thf(sy_c_Nitpick_Onorm__frac,type,
norm_frac: int > int > product_prod_int_int ).
thf(sy_c_Nitpick_Onorm__frac__rel,type,
norm_frac_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_NthRoot_Oroot,type,
root: nat > real > real ).
thf(sy_c_NthRoot_Osqrt,type,
sqrt: real > real ).
thf(sy_c_Orderings_Oord__class_Oless_000_062_Itc__Int__Oint_M_Eo_J,type,
ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000_062_Itc__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__Code____Numeral__Ocode____numeral,type,
ord_le1304079648umeral: code_code_numeral > code_code_numeral > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__Quickcheck____Narrowing__Ocode____int,type,
ord_le1860547276de_int: quickcheck_code_int > quickcheck_code_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__Rat__Orat,type,
ord_less_rat: rat > rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_000tc__RealDef__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Int__Oint_M_Eo_J,type,
ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000_062_Itc__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Code____Numeral__Ocode____numeral,type,
ord_le565307924umeral: code_code_numeral > code_code_numeral > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Quickcheck____Narrowing__Ocode____in,type,
ord_le258702272de_int: quickcheck_code_int > quickcheck_code_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Rat__Orat,type,
ord_less_eq_rat: rat > rat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__RealDef__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Omax_000tc__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omin_000tc__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Parity_Oeven__odd__class_Oeven_000tc__Int__Oint,type,
even_odd_even_int: int > $o ).
thf(sy_c_Parity_Oeven__odd__class_Oeven_000tc__Nat__Onat,type,
even_odd_even_nat: nat > $o ).
thf(sy_c_Power_Opower__class_Opower_000tc__Code____Numeral__Ocode____numeral,type,
power_2100829034umeral: code_code_numeral > nat > code_code_numeral ).
thf(sy_c_Power_Opower__class_Opower_000tc__Complex__Ocomplex,type,
power_power_complex: complex > nat > complex ).
thf(sy_c_Power_Opower__class_Opower_000tc__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_000tc__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_000tc__Quickcheck____Narrowing__Ocode____int,type,
power_881366806de_int: quickcheck_code_int > nat > quickcheck_code_int ).
thf(sy_c_Power_Opower__class_Opower_000tc__Rat__Orat,type,
power_power_rat: rat > nat > rat ).
thf(sy_c_Power_Opower__class_Opower_000tc__RealDef__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Primes_Ocoprime,type,
coprime: nat > nat > $o ).
thf(sy_c_Primes_Ofact,type,
fact: nat > nat ).
thf(sy_c_Primes_Oprime,type,
prime: nat > $o ).
thf(sy_c_Product__Type_OPair_000_062_Itc__Int__Oint_M_Eo_J_000_062_Itc__Int__Oint_M_,type,
produc398918003_int_o: ( int > $o ) > ( int > $o ) > produc975137661_int_o ).
thf(sy_c_Product__Type_OPair_000tc__Code____Numeral__Ocode____numeral_000tc__Code___,type,
produc2136830103umeral: code_code_numeral > code_code_numeral > produc1359518119umeral ).
thf(sy_c_Product__Type_OPair_000tc__Int__Oint_000tc__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(sy_c_Product__Type_OPair_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__O,type,
produc282740534nt_int: int > product_prod_int_int > produc393999548nt_int ).
thf(sy_c_Product__Type_OPair_000tc__Nat__Onat_000tc__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_000tc__Quickcheck____Narrowing__Ocode____int_000tc__Qui,type,
produc1318306967de_int: quickcheck_code_int > quickcheck_code_int > produc167071911de_int ).
thf(sy_c_Product__Type_OPair_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
produc865579683l_real: real > real > produc914805421l_real ).
thf(sy_c_Product__Type_OPair_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod,type,
produc883642259nt_int: product_prod_int_int > product_prod_int_int > produc1137372701nt_int ).
thf(sy_c_Product__Type_OPair_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__prod,type,
produc494345619at_nat: product_prod_nat_nat > product_prod_nat_nat > produc1322466333at_nat ).
thf(sy_c_Product__Type_Oapsnd_000tc__Int__Oint_000tc__Int__Oint_000tc__Int__Oint,type,
produc713050258nt_int: ( int > int ) > product_prod_int_int > product_prod_int_int ).
thf(sy_c_Product__Type_Ofst_000tc__Int__Oint_000tc__Int__Oint,type,
product_fst_int_int: product_prod_int_int > int ).
thf(sy_c_Product__Type_Ofst_000tc__Nat__Onat_000tc__Nat__Onat,type,
product_fst_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Ofst_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
produc1935615926l_real: produc914805421l_real > real ).
thf(sy_c_Product__Type_Oprod_Oprod__case_000tc__Int__Oint_000tc__Int__Oint_000_Eo,type,
produc450523309_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Product__Type_Oprod_Oprod__case_000tc__Int__Oint_000tc__Int__Oint_000tc__In,type,
produc1298267108nt_int: ( int > int > int ) > product_prod_int_int > int ).
thf(sy_c_Product__Type_Oprod_Oprod__case_000tc__Int__Oint_000tc__Int__Oint_000tc__pr,type,
produc1518849193nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
thf(sy_c_Product__Type_Oprod_Oprod__case_000tc__Nat__Onat_000tc__Nat__Onat_000_Eo,type,
produc1038563245_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_Product__Type_Oprod_Oprod__case_000tc__Nat__Onat_000tc__Nat__Onat_000tc__pr,type,
produc1391996073at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_Oprod_Oprod__case_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
produc595218619l_real: ( real > real > produc914805421l_real ) > produc914805421l_real > produc914805421l_real ).
thf(sy_c_Product__Type_Oprod_Oprod__case_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
produc141074865_int_o: ( product_prod_int_int > product_prod_int_int > $o ) > produc1137372701nt_int > $o ).
thf(sy_c_Product__Type_Osnd_000tc__Int__Oint_000tc__Int__Oint,type,
product_snd_int_int: product_prod_int_int > int ).
thf(sy_c_Product__Type_Osnd_000tc__Nat__Onat_000tc__Nat__Onat,type,
product_snd_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Osnd_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
produc556554744l_real: produc914805421l_real > real ).
thf(sy_c_Quickcheck__Narrowing_Oaround__zero,type,
quickc666637781d_zero: int > list_int ).
thf(sy_c_Quickcheck__Narrowing_Oaround__zero__rel,type,
quickc1265749348ro_rel: int > int > $o ).
thf(sy_c_Quickcheck__Narrowing_Odiv__mod__code__int,type,
quickc495462417de_int: quickcheck_code_int > quickcheck_code_int > produc167071911de_int ).
thf(sy_c_Quickcheck__Narrowing_Oint__of,type,
quickcheck_int_of: quickcheck_code_int > int ).
thf(sy_c_Quickcheck__Narrowing_Onat__of,type,
quickcheck_nat_of: quickcheck_code_int > nat ).
thf(sy_c_Quickcheck__Narrowing_Oof__int,type,
quickcheck_of_int: int > quickcheck_code_int ).
thf(sy_c_RComplete_Onatceiling,type,
natceiling: real > nat ).
thf(sy_c_RComplete_Onatfloor,type,
natfloor: real > nat ).
thf(sy_c_Rat_OFract,type,
fract: int > int > rat ).
thf(sy_c_Rat_OFrct,type,
frct: product_prod_int_int > rat ).
thf(sy_c_Rat_Ofield__char__0__class_ORats_000tc__RealDef__Oreal,type,
field_1210416355s_real: real > $o ).
thf(sy_c_Rat_Onormalize,type,
normalize: product_prod_int_int > product_prod_int_int ).
thf(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > product_prod_int_int ).
thf(sy_c_Rat_Oratrel,type,
ratrel: produc1137372701nt_int > $o ).
thf(sy_c_RealDef_ORatreal,type,
ratreal: rat > real ).
thf(sy_c_RealDef_Oreal_000tc__Int__Oint,type,
real_int: int > real ).
thf(sy_c_RealDef_Oreal_000tc__Nat__Onat,type,
real_nat: nat > real ).
thf(sy_c_RealDef_Ovanishes,type,
vanishes: ( nat > rat ) > $o ).
thf(sy_c_RealVector_Odist__class_Odist_000tc__Complex__Ocomplex,type,
dist_dist_complex: complex > complex > real ).
thf(sy_c_RealVector_Odist__class_Odist_000tc__RealDef__Oreal,type,
dist_dist_real: real > real > real ).
thf(sy_c_RealVector_Onorm__class_Onorm_000tc__Complex__Ocomplex,type,
norm_norm_complex: complex > real ).
thf(sy_c_RealVector_Onorm__class_Onorm_000tc__RealDef__Oreal,type,
norm_norm_real: real > real ).
thf(sy_c_RealVector_Oof__real_000tc__Complex__Ocomplex,type,
of_real_complex: real > complex ).
thf(sy_c_RealVector_OscaleR__class_OscaleR_000tc__Complex__Ocomplex,type,
scaleR1652505878omplex: real > complex > complex ).
thf(sy_c_RealVector_OscaleR__class_OscaleR_000tc__RealDef__Oreal,type,
scaleR_scaleR_real: real > real > real ).
thf(sy_c_Residues_OLegendre,type,
legendre: int > int > int ).
thf(sy_c_Residues_OQuadRes,type,
quadRes: int > int > $o ).
thf(sy_c_Residues_OResSet,type,
resSet: int > ( int > $o ) > $o ).
thf(sy_c_Residues_OSR,type,
sr: int > int > $o ).
thf(sy_c_Residues_OSRStar,type,
sRStar: int > int > $o ).
thf(sy_c_Residues_OStandardRes,type,
standardRes: int > int > int ).
thf(sy_c_Rings_Odvd__class_Odvd_000tc__Code____Numeral__Ocode____numeral,type,
dvd_dv174992974umeral: code_code_numeral > code_code_numeral > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_000tc__Complex__Ocomplex,type,
dvd_dvd_complex: complex > complex > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_000tc__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_000tc__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_000tc__Quickcheck____Narrowing__Ocode____int,type,
dvd_dv1760642554de_int: quickcheck_code_int > quickcheck_code_int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_000tc__Rat__Orat,type,
dvd_dvd_rat: rat > rat > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_000tc__RealDef__Oreal,type,
dvd_dvd_real: real > real > $o ).
thf(sy_c_SEQ_OBseq_000tc__RealDef__Oreal,type,
bseq_real: ( nat > real ) > $o ).
thf(sy_c_SEQ_OCauchy_000tc__Complex__Ocomplex,type,
cauchy_complex: ( nat > complex ) > $o ).
thf(sy_c_SEQ_OCauchy_000tc__RealDef__Oreal,type,
cauchy_real: ( nat > real ) > $o ).
thf(sy_c_SEQ_Omonoseq_000tc__RealDef__Oreal,type,
monoseq_real: ( nat > real ) > $o ).
thf(sy_c_SMT_Oz3div,type,
z3div: int > int > int ).
thf(sy_c_SMT_Oz3mod,type,
z3mod: int > int > int ).
thf(sy_c_Series_Osuminf_000tc__Complex__Ocomplex,type,
suminf_complex: ( nat > complex ) > complex ).
thf(sy_c_Series_Osuminf_000tc__RealDef__Oreal,type,
suminf_real: ( nat > real ) > real ).
thf(sy_c_Series_Osummable_000tc__Complex__Ocomplex,type,
summable_complex: ( nat > complex ) > $o ).
thf(sy_c_Series_Osummable_000tc__RealDef__Oreal,type,
summable_real: ( nat > real ) > $o ).
thf(sy_c_Series_Osums_000tc__Complex__Ocomplex,type,
sums_complex: ( nat > complex ) > complex > $o ).
thf(sy_c_Series_Osums_000tc__RealDef__Oreal,type,
sums_real: ( nat > real ) > real > $o ).
thf(sy_c_SetInterval_Oord__class_OatLeastAtMost_000tc__Int__Oint,type,
ord_at875362053st_int: int > int > int > $o ).
thf(sy_c_SetInterval_Oord__class_OatLeastAtMost_000tc__Nat__Onat,type,
ord_at238088361st_nat: nat > nat > nat > $o ).
thf(sy_c_SetInterval_Oord__class_OatLeastAtMost_000tc__RealDef__Oreal,type,
ord_at1589558736t_real: real > real > real > $o ).
thf(sy_c_SetInterval_Oord__class_OatLeastLessThan_000tc__Int__Oint,type,
ord_at641636577an_int: int > int > int > $o ).
thf(sy_c_SetInterval_Oord__class_OatLeastLessThan_000tc__Nat__Onat,type,
ord_at4362885an_nat: nat > nat > nat > $o ).
thf(sy_c_SetInterval_Oord__class_OatLeastLessThan_000tc__RealDef__Oreal,type,
ord_at1496968948n_real: real > real > real > $o ).
thf(sy_c_SetInterval_Oord__class_OatMost_000tc__Nat__Onat,type,
ord_atMost_nat: nat > nat > $o ).
thf(sy_c_SetInterval_Oord__class_OgreaterThanLessThan_000tc__Int__Oint,type,
ord_gr1297742076an_int: int > int > int > $o ).
thf(sy_c_SetInterval_Oord__class_OgreaterThanLessThan_000tc__Nat__Onat,type,
ord_gr660468384an_nat: nat > nat > nat > $o ).
thf(sy_c_SetInterval_Oord__class_OgreaterThanLessThan_000tc__RealDef__Oreal,type,
ord_gr788844697n_real: real > real > real > $o ).
thf(sy_c_SetInterval_Oord__class_OlessThan_000tc__Nat__Onat,type,
ord_lessThan_nat: nat > nat > $o ).
thf(sy_c_SetInterval_Oord__class_OlessThan_000tc__RealDef__Oreal,type,
ord_lessThan_real: real > real > $o ).
thf(sy_c_Set_OCollect_000tc__Int__Oint,type,
collect_int: ( int > $o ) > int > $o ).
thf(sy_c_Set_OCollect_000tc__Nat__Onat,type,
collect_nat: ( nat > $o ) > nat > $o ).
thf(sy_c_Set_OCollect_000tc__RealDef__Oreal,type,
collect_real: ( real > $o ) > real > $o ).
thf(sy_c_Set_OCollect_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
collec1347809874nt_int: ( product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Set_OCollect_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
collec1979865426at_nat: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_Set_OCollect_000tc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__pro,type,
collec50511176nt_int: ( produc1137372701nt_int > $o ) > produc1137372701nt_int > $o ).
thf(sy_c_Set_Oimage_000tc__Int__Oint_000_062_Itc__Int__Oint_M_Eo_J,type,
image_int_int_o: ( int > int > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_Set_Oimage_000tc__Int__Oint_000tc__Int__Oint,type,
image_int_int: ( int > int ) > ( int > $o ) > int > $o ).
thf(sy_c_Set_Oimage_000tc__Int__Oint_000tc__Nat__Onat,type,
image_int_nat: ( int > nat ) > ( int > $o ) > nat > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Int__Oint,type,
image_nat_int: ( nat > int ) > ( nat > $o ) > int > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Transcendental_Oarccos,type,
arccos: real > real ).
thf(sy_c_Transcendental_Oarcsin,type,
arcsin: real > real ).
thf(sy_c_Transcendental_Oarctan,type,
arctan: real > real ).
thf(sy_c_Transcendental_Ocos,type,
cos: real > real ).
thf(sy_c_Transcendental_Ocos__coeff,type,
cos_coeff: nat > real ).
thf(sy_c_Transcendental_Odiffs_000tc__RealDef__Oreal,type,
diffs_real: ( nat > real ) > nat > real ).
thf(sy_c_Transcendental_Oexp_000tc__RealDef__Oreal,type,
exp_real: real > real ).
thf(sy_c_Transcendental_Oln,type,
ln: real > real ).
thf(sy_c_Transcendental_Opi,type,
pi: real ).
thf(sy_c_Transcendental_Osin,type,
sin: real > real ).
thf(sy_c_Transcendental_Osin__coeff,type,
sin_coeff: nat > real ).
thf(sy_c_Transcendental_Otan,type,
tan: real > real ).
thf(sy_c_TwoSquares__Mirabelle__xzcihllart_Ois__sum2sq,type,
twoSqu1152398899sum2sq: int > $o ).
thf(sy_c_TwoSquares__Mirabelle__xzcihllart_Osum2sq,type,
twoSqu2072599593sum2sq: product_prod_int_int > int ).
thf(sy_c_Wellfounded_Oaccp_000tc__Int__Oint,type,
accp_int: ( int > int > $o ) > int > $o ).
thf(sy_c_Wellfounded_Oaccp_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
accp_P2006205492nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Wellfounded_Oaccp_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
accp_P490777396at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_nat: product_prod_nat_nat > $o ).
thf(sy_c_WilsonRuss_Oinv,type,
inv: int > int > int ).
thf(sy_c_WilsonRuss_Owset,type,
wset: int > int > int > $o ).
thf(sy_c_member_000_062_Itc__Int__Oint_M_Eo_J,type,
member_int_o: ( int > $o ) > ( ( int > $o ) > $o ) > $o ).
thf(sy_c_member_000tc__Int__Oint,type,
member_int: int > ( int > $o ) > $o ).
thf(sy_c_member_000tc__Nat__Onat,type,
member_nat: nat > ( nat > $o ) > $o ).
thf(sy_c_member_000tc__RealDef__Oreal,type,
member_real: real > ( real > $o ) > $o ).
thf(sy_c_member_000tc__prod_I_062_Itc__Int__Oint_M_Eo_J_M_062_Itc__Int__Oint_M_Eo_J_,type,
member1329254762_int_o: produc975137661_int_o > ( produc975137661_int_o > $o ) > $o ).
thf(sy_c_member_000tc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__prod_Itc_,type,
member2143287562nt_int: produc1137372701nt_int > ( produc1137372701nt_int > $o ) > $o ).
thf(sy_c_member_000tc__prod_Itc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_Mtc__prod_Itc_,type,
member180897546at_nat: produc1322466333at_nat > ( produc1322466333at_nat > $o ) > $o ).
thf(sy_v_m,type,
m: int ).
thf(sy_v_s1____,type,
s1: int ).
thf(sy_v_s____,type,
s: int ).
thf(sy_v_t____,type,
t: int ).
%----Relevant facts (5196)
thf(fact_0__096t_A_060_A0_096,axiom,
ord_less_int @ t @ zero_zero_int ).
thf(fact_1_calculation_I1_J,axiom,
ord_less_int @ t @ one_one_int ).
thf(fact_2__096_I4_A_K_Am_A_L_A1_J_A_K_At_A_060_A_I4_A_K_Am_A_L_A1_J_A_K_A0_096,axiom,
ord_less_int @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ t ) @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ zero_zero_int ) ).
thf(fact_3_t,axiom,
( ( plus_plus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ t ) ) ).
thf(fact_4_calculation_I2_J,axiom,
( ( t = zero_zero_int )
=> ( ( plus_plus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_int )
= zero_zero_int ) ) ).
thf(fact_5__096_126_A1_A_060_061_At_096,axiom,
~ ( ord_less_eq_int @ one_one_int @ t ) ).
thf(fact_6_p0,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_7_not__sum__power2__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_int ) ).
thf(fact_8_not__sum__power2__lt__zero,axiom,
! [X: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_real ) ).
thf(fact_9_not__sum__power2__lt__zero,axiom,
! [X: rat,Y: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_rat @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_rat ) ).
thf(fact_10_sum__power2__gt__zero__iff,axiom,
! [X_4: int,Y_3: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X_4 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_3 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
<=> ( ( X_4 != zero_zero_int )
| ( Y_3 != zero_zero_int ) ) ) ).
thf(fact_11_sum__power2__gt__zero__iff,axiom,
! [X_4: real,Y_3: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X_4 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_3 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
<=> ( ( X_4 != zero_zero_real )
| ( Y_3 != zero_zero_real ) ) ) ).
thf(fact_12_sum__power2__gt__zero__iff,axiom,
! [X_4: rat,Y_3: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X_4 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_rat @ Y_3 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
<=> ( ( X_4 != zero_zero_rat )
| ( Y_3 != zero_zero_rat ) ) ) ).
thf(fact_13_sum__power2__eq__zero__iff,axiom,
! [X_68: int,Y_51: int] :
( ( ( plus_plus_int @ ( power_power_int @ X_68 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_51 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_int )
<=> ( ( X_68 = zero_zero_int )
& ( Y_51 = zero_zero_int ) ) ) ).
thf(fact_14_sum__power2__eq__zero__iff,axiom,
! [X_68: real,Y_51: real] :
( ( ( plus_plus_real @ ( power_power_real @ X_68 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_51 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_real )
<=> ( ( X_68 = zero_zero_real )
& ( Y_51 = zero_zero_real ) ) ) ).
thf(fact_15_sum__power2__eq__zero__iff,axiom,
! [X_68: rat,Y_51: rat] :
( ( ( plus_plus_rat @ ( power_power_rat @ X_68 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_rat @ Y_51 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_rat )
<=> ( ( X_68 = zero_zero_rat )
& ( Y_51 = zero_zero_rat ) ) ) ).
thf(fact_16_power2__less__0,axiom,
! [A_290: int] :
~ ( ord_less_int @ ( power_power_int @ A_290 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_int ) ).
thf(fact_17_power2__less__0,axiom,
! [A_290: real] :
~ ( ord_less_real @ ( power_power_real @ A_290 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_real ) ).
thf(fact_18_power2__less__0,axiom,
! [A_290: rat] :
~ ( ord_less_rat @ ( power_power_rat @ A_290 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_rat ) ).
thf(fact_19_zero__less__power2,axiom,
! [A_289: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A_289 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( A_289 != zero_zero_int ) ) ).
thf(fact_20_zero__less__power2,axiom,
! [A_289: real] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A_289 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( A_289 != zero_zero_real ) ) ).
thf(fact_21_zero__less__power2,axiom,
! [A_289: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A_289 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( A_289 != zero_zero_rat ) ) ).
thf(fact_22_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_int ) ).
thf(fact_23_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_nat ) ).
thf(fact_24_one__power2,axiom,
( ( power_power_real @ one_one_real @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_real ) ).
thf(fact_25_one__power2,axiom,
( ( power_2100829034umeral @ one_on1645066479umeral @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_on1645066479umeral ) ).
thf(fact_26_one__power2,axiom,
( ( power_power_complex @ one_one_complex @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_complex ) ).
thf(fact_27_one__power2,axiom,
( ( power_881366806de_int @ one_on1684967323de_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_on1684967323de_int ) ).
thf(fact_28_one__power2,axiom,
( ( power_power_rat @ one_one_rat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_rat ) ).
thf(fact_29_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_int ) ).
thf(fact_30_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_nat ) ).
thf(fact_31_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_real ) ).
thf(fact_32_zero__power2,axiom,
( ( power_2100829034umeral @ zero_z126310315umeral @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_z126310315umeral ) ).
thf(fact_33_zero__power2,axiom,
( ( power_power_complex @ zero_zero_complex @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_complex ) ).
thf(fact_34_zero__power2,axiom,
( ( power_881366806de_int @ zero_z891286103de_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_z891286103de_int ) ).
thf(fact_35_zero__power2,axiom,
( ( power_power_rat @ zero_zero_rat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_rat ) ).
thf(fact_36_zero__eq__power2,axiom,
! [A_288: int] :
( ( ( power_power_int @ A_288 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_int )
<=> ( A_288 = zero_zero_int ) ) ).
thf(fact_37_zero__eq__power2,axiom,
! [A_288: real] :
( ( ( power_power_real @ A_288 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_real )
<=> ( A_288 = zero_zero_real ) ) ).
thf(fact_38_zero__eq__power2,axiom,
! [A_288: complex] :
( ( ( power_power_complex @ A_288 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_complex )
<=> ( A_288 = zero_zero_complex ) ) ).
thf(fact_39_zero__eq__power2,axiom,
! [A_288: rat] :
( ( ( power_power_rat @ A_288 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_rat )
<=> ( A_288 = zero_zero_rat ) ) ).
thf(fact_40_add__special_I2_J,axiom,
! [W_20: int] :
( ( plus_plus_int @ one_one_int @ ( number_number_of_int @ W_20 ) )
= ( number_number_of_int @ ( plus_plus_int @ ( bit1 @ pls ) @ W_20 ) ) ) ).
thf(fact_41_add__special_I2_J,axiom,
! [W_20: int] :
( ( plus_plus_real @ one_one_real @ ( number267125858f_real @ W_20 ) )
= ( number267125858f_real @ ( plus_plus_int @ ( bit1 @ pls ) @ W_20 ) ) ) ).
thf(fact_42_add__special_I2_J,axiom,
! [W_20: int] :
( ( plus_plus_complex @ one_one_complex @ ( number528085621omplex @ W_20 ) )
= ( number528085621omplex @ ( plus_plus_int @ ( bit1 @ pls ) @ W_20 ) ) ) ).
thf(fact_43_add__special_I2_J,axiom,
! [W_20: int] :
( ( plus_plus_rat @ one_one_rat @ ( number_number_of_rat @ W_20 ) )
= ( number_number_of_rat @ ( plus_plus_int @ ( bit1 @ pls ) @ W_20 ) ) ) ).
thf(fact_44_add__special_I3_J,axiom,
! [V_22: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_22 ) @ one_one_int )
= ( number_number_of_int @ ( plus_plus_int @ V_22 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_45_add__special_I3_J,axiom,
! [V_22: int] :
( ( plus_plus_real @ ( number267125858f_real @ V_22 ) @ one_one_real )
= ( number267125858f_real @ ( plus_plus_int @ V_22 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_46_add__special_I3_J,axiom,
! [V_22: int] :
( ( plus_plus_complex @ ( number528085621omplex @ V_22 ) @ one_one_complex )
= ( number528085621omplex @ ( plus_plus_int @ V_22 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_47_add__special_I3_J,axiom,
! [V_22: int] :
( ( plus_plus_rat @ ( number_number_of_rat @ V_22 ) @ one_one_rat )
= ( number_number_of_rat @ ( plus_plus_int @ V_22 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_48_t__l__p,axiom,
ord_less_int @ t @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_49__096_B_Bthesis_O_A_I_B_Bt_O_As_A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_L_A1_,axiom,
~ ! [T_1: int] :
( ( plus_plus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_int )
!= ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ T_1 ) ) ).
thf(fact_50_p,axiom,
zprime @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_51_qf1pt,axiom,
twoSqu1152398899sum2sq @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ t ) ).
thf(fact_52_zle__refl,axiom,
! [W: int] : ( ord_less_eq_int @ W @ W ) ).
thf(fact_53_number__of__is__id,axiom,
! [K_1: int] :
( ( number_number_of_int @ K_1 )
= K_1 ) ).
thf(fact_54_zmult__commute,axiom,
! [Z_1: int,W: int] :
( ( times_times_int @ Z_1 @ W )
= ( times_times_int @ W @ Z_1 ) ) ).
thf(fact_55_zle__linear,axiom,
! [Z_1: int,W: int] :
( ( ord_less_eq_int @ Z_1 @ W )
| ( ord_less_eq_int @ W @ Z_1 ) ) ).
thf(fact_56_times__numeral__code_I5_J,axiom,
! [V: int,W: int] :
( ( times_times_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) )
= ( number_number_of_int @ ( times_times_int @ V @ W ) ) ) ).
thf(fact_57_less__eq__number__of__int__code,axiom,
! [K_1: int,L: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ K_1 ) @ ( number_number_of_int @ L ) )
<=> ( ord_less_eq_int @ K_1 @ L ) ) ).
thf(fact_58_le__number__of,axiom,
! [X_67: int,Y_50: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_67 ) @ ( number_number_of_int @ Y_50 ) )
<=> ( ord_less_eq_int @ X_67 @ Y_50 ) ) ).
thf(fact_59_le__number__of,axiom,
! [X_67: int,Y_50: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_67 ) @ ( number267125858f_real @ Y_50 ) )
<=> ( ord_less_eq_int @ X_67 @ Y_50 ) ) ).
thf(fact_60_le__number__of,axiom,
! [X_67: int,Y_50: int] :
( ( ord_less_eq_rat @ ( number_number_of_rat @ X_67 ) @ ( number_number_of_rat @ Y_50 ) )
<=> ( ord_less_eq_int @ X_67 @ Y_50 ) ) ).
thf(fact_61_zmult__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] :
( ( times_times_int @ ( times_times_int @ Z1 @ Z2 ) @ Z3 )
= ( times_times_int @ Z1 @ ( times_times_int @ Z2 @ Z3 ) ) ) ).
thf(fact_62_zle__trans,axiom,
! [K_1: int,I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K_1 )
=> ( ord_less_eq_int @ I @ K_1 ) ) ) ).
thf(fact_63_zle__antisym,axiom,
! [Z_1: int,W: int] :
( ( ord_less_eq_int @ Z_1 @ W )
=> ( ( ord_less_eq_int @ W @ Z_1 )
=> ( Z_1 = W ) ) ) ).
thf(fact_64_zpower__zadd__distrib,axiom,
! [X: int,Y: nat,Z_1: nat] :
( ( power_power_int @ X @ ( plus_plus_nat @ Y @ Z_1 ) )
= ( times_times_int @ ( power_power_int @ X @ Y ) @ ( power_power_int @ X @ Z_1 ) ) ) ).
thf(fact_65_less__eq__int__code_I16_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_eq_int @ ( bit1 @ K1 ) @ ( bit1 @ K2 ) )
<=> ( ord_less_eq_int @ K1 @ K2 ) ) ).
thf(fact_66_rel__simps_I34_J,axiom,
! [K_1: int,L: int] :
( ( ord_less_eq_int @ ( bit1 @ K_1 ) @ ( bit1 @ L ) )
<=> ( ord_less_eq_int @ K_1 @ L ) ) ).
thf(fact_67_rel__simps_I19_J,axiom,
ord_less_eq_int @ pls @ pls ).
thf(fact_68_less__eq__int__code_I13_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_eq_int @ ( bit0 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_eq_int @ K1 @ K2 ) ) ).
thf(fact_69_rel__simps_I31_J,axiom,
! [K_1: int,L: int] :
( ( ord_less_eq_int @ ( bit0 @ K_1 ) @ ( bit0 @ L ) )
<=> ( ord_less_eq_int @ K_1 @ L ) ) ).
thf(fact_70_zless__le,axiom,
! [Z_1: int,W: int] :
( ( ord_less_int @ Z_1 @ W )
<=> ( ( ord_less_eq_int @ Z_1 @ W )
& ( Z_1 != W ) ) ) ).
thf(fact_71_zadd__left__mono,axiom,
! [K_1: int,I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ord_less_eq_int @ ( plus_plus_int @ K_1 @ I ) @ ( plus_plus_int @ K_1 @ J ) ) ) ).
thf(fact_72_eq__number__of__0,axiom,
! [V: int] :
( ( ( number_number_of_nat @ V )
= zero_zero_nat )
<=> ( ord_less_eq_int @ V @ pls ) ) ).
thf(fact_73_eq__0__number__of,axiom,
! [V: int] :
( ( zero_zero_nat
= ( number_number_of_nat @ V ) )
<=> ( ord_less_eq_int @ V @ pls ) ) ).
thf(fact_74_semiring__mult__number__of,axiom,
! [V_21: int,V_20: int] :
( ( ord_less_eq_int @ pls @ V_20 )
=> ( ( ord_less_eq_int @ pls @ V_21 )
=> ( ( times_times_int @ ( number_number_of_int @ V_20 ) @ ( number_number_of_int @ V_21 ) )
= ( number_number_of_int @ ( times_times_int @ V_20 @ V_21 ) ) ) ) ) ).
thf(fact_75_semiring__mult__number__of,axiom,
! [V_21: int,V_20: int] :
( ( ord_less_eq_int @ pls @ V_20 )
=> ( ( ord_less_eq_int @ pls @ V_21 )
=> ( ( times_times_nat @ ( number_number_of_nat @ V_20 ) @ ( number_number_of_nat @ V_21 ) )
= ( number_number_of_nat @ ( times_times_int @ V_20 @ V_21 ) ) ) ) ) ).
thf(fact_76_semiring__mult__number__of,axiom,
! [V_21: int,V_20: int] :
( ( ord_less_eq_int @ pls @ V_20 )
=> ( ( ord_less_eq_int @ pls @ V_21 )
=> ( ( times_times_real @ ( number267125858f_real @ V_20 ) @ ( number267125858f_real @ V_21 ) )
= ( number267125858f_real @ ( times_times_int @ V_20 @ V_21 ) ) ) ) ) ).
thf(fact_77_semiring__mult__number__of,axiom,
! [V_21: int,V_20: int] :
( ( ord_less_eq_int @ pls @ V_20 )
=> ( ( ord_less_eq_int @ pls @ V_21 )
=> ( ( times_times_complex @ ( number528085621omplex @ V_20 ) @ ( number528085621omplex @ V_21 ) )
= ( number528085621omplex @ ( times_times_int @ V_20 @ V_21 ) ) ) ) ) ).
thf(fact_78_semiring__mult__number__of,axiom,
! [V_21: int,V_20: int] :
( ( ord_less_eq_int @ pls @ V_20 )
=> ( ( ord_less_eq_int @ pls @ V_21 )
=> ( ( times_times_rat @ ( number_number_of_rat @ V_20 ) @ ( number_number_of_rat @ V_21 ) )
= ( number_number_of_rat @ ( times_times_int @ V_20 @ V_21 ) ) ) ) ) ).
thf(fact_79_mult__number__of__left,axiom,
! [V_19: int,W_19: int,Z_15: int] :
( ( times_times_int @ ( number_number_of_int @ V_19 ) @ ( times_times_int @ ( number_number_of_int @ W_19 ) @ Z_15 ) )
= ( times_times_int @ ( number_number_of_int @ ( times_times_int @ V_19 @ W_19 ) ) @ Z_15 ) ) ).
thf(fact_80_mult__number__of__left,axiom,
! [V_19: int,W_19: int,Z_15: real] :
( ( times_times_real @ ( number267125858f_real @ V_19 ) @ ( times_times_real @ ( number267125858f_real @ W_19 ) @ Z_15 ) )
= ( times_times_real @ ( number267125858f_real @ ( times_times_int @ V_19 @ W_19 ) ) @ Z_15 ) ) ).
thf(fact_81_mult__number__of__left,axiom,
! [V_19: int,W_19: int,Z_15: complex] :
( ( times_times_complex @ ( number528085621omplex @ V_19 ) @ ( times_times_complex @ ( number528085621omplex @ W_19 ) @ Z_15 ) )
= ( times_times_complex @ ( number528085621omplex @ ( times_times_int @ V_19 @ W_19 ) ) @ Z_15 ) ) ).
thf(fact_82_mult__number__of__left,axiom,
! [V_19: int,W_19: int,Z_15: rat] :
( ( times_times_rat @ ( number_number_of_rat @ V_19 ) @ ( times_times_rat @ ( number_number_of_rat @ W_19 ) @ Z_15 ) )
= ( times_times_rat @ ( number_number_of_rat @ ( times_times_int @ V_19 @ W_19 ) ) @ Z_15 ) ) ).
thf(fact_83_arith__simps_I32_J,axiom,
! [V_18: int,W_18: int] :
( ( times_times_int @ ( number_number_of_int @ V_18 ) @ ( number_number_of_int @ W_18 ) )
= ( number_number_of_int @ ( times_times_int @ V_18 @ W_18 ) ) ) ).
thf(fact_84_arith__simps_I32_J,axiom,
! [V_18: int,W_18: int] :
( ( times_times_real @ ( number267125858f_real @ V_18 ) @ ( number267125858f_real @ W_18 ) )
= ( number267125858f_real @ ( times_times_int @ V_18 @ W_18 ) ) ) ).
thf(fact_85_arith__simps_I32_J,axiom,
! [V_18: int,W_18: int] :
( ( times_times_complex @ ( number528085621omplex @ V_18 ) @ ( number528085621omplex @ W_18 ) )
= ( number528085621omplex @ ( times_times_int @ V_18 @ W_18 ) ) ) ).
thf(fact_86_arith__simps_I32_J,axiom,
! [V_18: int,W_18: int] :
( ( times_times_rat @ ( number_number_of_rat @ V_18 ) @ ( number_number_of_rat @ W_18 ) )
= ( number_number_of_rat @ ( times_times_int @ V_18 @ W_18 ) ) ) ).
thf(fact_87_number__of__mult,axiom,
! [V_17: int,W_17: int] :
( ( number_number_of_int @ ( times_times_int @ V_17 @ W_17 ) )
= ( times_times_int @ ( number_number_of_int @ V_17 ) @ ( number_number_of_int @ W_17 ) ) ) ).
thf(fact_88_number__of__mult,axiom,
! [V_17: int,W_17: int] :
( ( number267125858f_real @ ( times_times_int @ V_17 @ W_17 ) )
= ( times_times_real @ ( number267125858f_real @ V_17 ) @ ( number267125858f_real @ W_17 ) ) ) ).
thf(fact_89_number__of__mult,axiom,
! [V_17: int,W_17: int] :
( ( number528085621omplex @ ( times_times_int @ V_17 @ W_17 ) )
= ( times_times_complex @ ( number528085621omplex @ V_17 ) @ ( number528085621omplex @ W_17 ) ) ) ).
thf(fact_90_number__of__mult,axiom,
! [V_17: int,W_17: int] :
( ( number_number_of_rat @ ( times_times_int @ V_17 @ W_17 ) )
= ( times_times_rat @ ( number_number_of_rat @ V_17 ) @ ( number_number_of_rat @ W_17 ) ) ) ).
thf(fact_91_sum__squares__le__zero__iff,axiom,
! [X_66: int,Y_49: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X_66 @ X_66 ) @ ( times_times_int @ Y_49 @ Y_49 ) ) @ zero_zero_int )
<=> ( ( X_66 = zero_zero_int )
& ( Y_49 = zero_zero_int ) ) ) ).
thf(fact_92_sum__squares__le__zero__iff,axiom,
! [X_66: real,Y_49: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X_66 @ X_66 ) @ ( times_times_real @ Y_49 @ Y_49 ) ) @ zero_zero_real )
<=> ( ( X_66 = zero_zero_real )
& ( Y_49 = zero_zero_real ) ) ) ).
thf(fact_93_sum__squares__le__zero__iff,axiom,
! [X_66: rat,Y_49: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X_66 @ X_66 ) @ ( times_times_rat @ Y_49 @ Y_49 ) ) @ zero_zero_rat )
<=> ( ( X_66 = zero_zero_rat )
& ( Y_49 = zero_zero_rat ) ) ) ).
thf(fact_94_sum__squares__ge__zero,axiom,
! [X_65: int,Y_48: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X_65 @ X_65 ) @ ( times_times_int @ Y_48 @ Y_48 ) ) ) ).
thf(fact_95_sum__squares__ge__zero,axiom,
! [X_65: real,Y_48: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X_65 @ X_65 ) @ ( times_times_real @ Y_48 @ Y_48 ) ) ) ).
thf(fact_96_sum__squares__ge__zero,axiom,
! [X_65: rat,Y_48: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X_65 @ X_65 ) @ ( times_times_rat @ Y_48 @ Y_48 ) ) ) ).
thf(fact_97_le__special_I3_J,axiom,
! [X_64: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_64 ) @ zero_zero_int )
<=> ( ord_less_eq_int @ X_64 @ pls ) ) ).
thf(fact_98_le__special_I3_J,axiom,
! [X_64: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_64 ) @ zero_zero_real )
<=> ( ord_less_eq_int @ X_64 @ pls ) ) ).
thf(fact_99_le__special_I3_J,axiom,
! [X_64: int] :
( ( ord_less_eq_rat @ ( number_number_of_rat @ X_64 ) @ zero_zero_rat )
<=> ( ord_less_eq_int @ X_64 @ pls ) ) ).
thf(fact_100_le__special_I1_J,axiom,
! [Y_47: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ Y_47 ) )
<=> ( ord_less_eq_int @ pls @ Y_47 ) ) ).
thf(fact_101_le__special_I1_J,axiom,
! [Y_47: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( number267125858f_real @ Y_47 ) )
<=> ( ord_less_eq_int @ pls @ Y_47 ) ) ).
thf(fact_102_le__special_I1_J,axiom,
! [Y_47: int] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( number_number_of_rat @ Y_47 ) )
<=> ( ord_less_eq_int @ pls @ Y_47 ) ) ).
thf(fact_103_less__0__number__of,axiom,
! [V: int] :
( ( ord_less_nat @ zero_zero_nat @ ( number_number_of_nat @ V ) )
<=> ( ord_less_int @ pls @ V ) ) ).
thf(fact_104_le__number__of__eq__not__less,axiom,
! [V_16: int,W_16: int] :
( ( ord_le565307924umeral @ ( number1443263063umeral @ V_16 ) @ ( number1443263063umeral @ W_16 ) )
<=> ~ ( ord_le1304079648umeral @ ( number1443263063umeral @ W_16 ) @ ( number1443263063umeral @ V_16 ) ) ) ).
thf(fact_105_le__number__of__eq__not__less,axiom,
! [V_16: int,W_16: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ V_16 ) @ ( number_number_of_int @ W_16 ) )
<=> ~ ( ord_less_int @ ( number_number_of_int @ W_16 ) @ ( number_number_of_int @ V_16 ) ) ) ).
thf(fact_106_le__number__of__eq__not__less,axiom,
! [V_16: int,W_16: int] :
( ( ord_less_eq_nat @ ( number_number_of_nat @ V_16 ) @ ( number_number_of_nat @ W_16 ) )
<=> ~ ( ord_less_nat @ ( number_number_of_nat @ W_16 ) @ ( number_number_of_nat @ V_16 ) ) ) ).
thf(fact_107_le__number__of__eq__not__less,axiom,
! [V_16: int,W_16: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ V_16 ) @ ( number267125858f_real @ W_16 ) )
<=> ~ ( ord_less_real @ ( number267125858f_real @ W_16 ) @ ( number267125858f_real @ V_16 ) ) ) ).
thf(fact_108_le__number__of__eq__not__less,axiom,
! [V_16: int,W_16: int] :
( ( ord_le258702272de_int @ ( number1226105091de_int @ V_16 ) @ ( number1226105091de_int @ W_16 ) )
<=> ~ ( ord_le1860547276de_int @ ( number1226105091de_int @ W_16 ) @ ( number1226105091de_int @ V_16 ) ) ) ).
thf(fact_109_le__number__of__eq__not__less,axiom,
! [V_16: int,W_16: int] :
( ( ord_less_eq_rat @ ( number_number_of_rat @ V_16 ) @ ( number_number_of_rat @ W_16 ) )
<=> ~ ( ord_less_rat @ ( number_number_of_rat @ W_16 ) @ ( number_number_of_rat @ V_16 ) ) ) ).
thf(fact_110_rel__simps_I22_J,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ pls @ ( bit1 @ K_1 ) )
<=> ( ord_less_eq_int @ pls @ K_1 ) ) ).
thf(fact_111_less__eq__int__code_I14_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_eq_int @ ( bit0 @ K1 ) @ ( bit1 @ K2 ) )
<=> ( ord_less_eq_int @ K1 @ K2 ) ) ).
thf(fact_112_rel__simps_I32_J,axiom,
! [K_1: int,L: int] :
( ( ord_less_eq_int @ ( bit0 @ K_1 ) @ ( bit1 @ L ) )
<=> ( ord_less_eq_int @ K_1 @ L ) ) ).
thf(fact_113_rel__simps_I27_J,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ ( bit0 @ K_1 ) @ pls )
<=> ( ord_less_eq_int @ K_1 @ pls ) ) ).
thf(fact_114_rel__simps_I21_J,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ pls @ ( bit0 @ K_1 ) )
<=> ( ord_less_eq_int @ pls @ K_1 ) ) ).
thf(fact_115_zadd__zless__mono,axiom,
! [Z_3: int,Z_1: int,W_15: int,W: int] :
( ( ord_less_int @ W_15 @ W )
=> ( ( ord_less_eq_int @ Z_3 @ Z_1 )
=> ( ord_less_int @ ( plus_plus_int @ W_15 @ Z_3 ) @ ( plus_plus_int @ W @ Z_1 ) ) ) ) ).
thf(fact_116_nat__number__of__Pls,axiom,
( ( number_number_of_nat @ pls )
= zero_zero_nat ) ).
thf(fact_117_semiring__norm_I113_J,axiom,
( zero_zero_nat
= ( number_number_of_nat @ pls ) ) ).
thf(fact_118_le__special_I4_J,axiom,
! [X_63: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_63 ) @ one_one_int )
<=> ( ord_less_eq_int @ X_63 @ ( bit1 @ pls ) ) ) ).
thf(fact_119_le__special_I4_J,axiom,
! [X_63: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_63 ) @ one_one_real )
<=> ( ord_less_eq_int @ X_63 @ ( bit1 @ pls ) ) ) ).
thf(fact_120_le__special_I4_J,axiom,
! [X_63: int] :
( ( ord_less_eq_rat @ ( number_number_of_rat @ X_63 ) @ one_one_rat )
<=> ( ord_less_eq_int @ X_63 @ ( bit1 @ pls ) ) ) ).
thf(fact_121_le__special_I2_J,axiom,
! [Y_46: int] :
( ( ord_less_eq_int @ one_one_int @ ( number_number_of_int @ Y_46 ) )
<=> ( ord_less_eq_int @ ( bit1 @ pls ) @ Y_46 ) ) ).
thf(fact_122_le__special_I2_J,axiom,
! [Y_46: int] :
( ( ord_less_eq_real @ one_one_real @ ( number267125858f_real @ Y_46 ) )
<=> ( ord_less_eq_int @ ( bit1 @ pls ) @ Y_46 ) ) ).
thf(fact_123_le__special_I2_J,axiom,
! [Y_46: int] :
( ( ord_less_eq_rat @ one_one_rat @ ( number_number_of_rat @ Y_46 ) )
<=> ( ord_less_eq_int @ ( bit1 @ pls ) @ Y_46 ) ) ).
thf(fact_124_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_125_mult__Pls,axiom,
! [W: int] :
( ( times_times_int @ pls @ W )
= pls ) ).
thf(fact_126_mult__Bit0,axiom,
! [K_1: int,L: int] :
( ( times_times_int @ ( bit0 @ K_1 ) @ L )
= ( bit0 @ ( times_times_int @ K_1 @ L ) ) ) ).
thf(fact_127_less__number__of__int__code,axiom,
! [K_1: int,L: int] :
( ( ord_less_int @ ( number_number_of_int @ K_1 ) @ ( number_number_of_int @ L ) )
<=> ( ord_less_int @ K_1 @ L ) ) ).
thf(fact_128_zmult__1__right,axiom,
! [Z_1: int] :
( ( times_times_int @ Z_1 @ one_one_int )
= Z_1 ) ).
thf(fact_129_zmult__1,axiom,
! [Z_1: int] :
( ( times_times_int @ one_one_int @ Z_1 )
= Z_1 ) ).
thf(fact_130_plus__numeral__code_I9_J,axiom,
! [V: int,W: int] :
( ( plus_plus_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) )
= ( number_number_of_int @ ( plus_plus_int @ V @ W ) ) ) ).
thf(fact_131_zadd__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z2 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
thf(fact_132_zadd__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
thf(fact_133_rel__simps_I29_J,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ ( bit1 @ K_1 ) @ pls )
<=> ( ord_less_int @ K_1 @ pls ) ) ).
thf(fact_134_rel__simps_I5_J,axiom,
! [K_1: int] :
( ( ord_less_int @ pls @ ( bit1 @ K_1 ) )
<=> ( ord_less_eq_int @ pls @ K_1 ) ) ).
thf(fact_135_less__eq__int__code_I15_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_eq_int @ ( bit1 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_136_rel__simps_I33_J,axiom,
! [K_1: int,L: int] :
( ( ord_less_eq_int @ ( bit1 @ K_1 ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K_1 @ L ) ) ).
thf(fact_137_less__int__code_I14_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit0 @ K1 ) @ ( bit1 @ K2 ) )
<=> ( ord_less_eq_int @ K1 @ K2 ) ) ).
thf(fact_138_mem__def,axiom,
! [X_62: int,A_287: int > $o] :
( ( member_int @ X_62 @ A_287 )
<=> ( A_287 @ X_62 ) ) ).
thf(fact_139_mem__def,axiom,
! [X_62: produc1322466333at_nat,A_287: produc1322466333at_nat > $o] :
( ( member180897546at_nat @ X_62 @ A_287 )
<=> ( A_287 @ X_62 ) ) ).
thf(fact_140_mem__def,axiom,
! [X_62: int > $o,A_287: ( int > $o ) > $o] :
( ( member_int_o @ X_62 @ A_287 )
<=> ( A_287 @ X_62 ) ) ).
thf(fact_141_mem__def,axiom,
! [X_62: nat,A_287: nat > $o] :
( ( member_nat @ X_62 @ A_287 )
<=> ( A_287 @ X_62 ) ) ).
thf(fact_142_mem__def,axiom,
! [X_62: real,A_287: real > $o] :
( ( member_real @ X_62 @ A_287 )
<=> ( A_287 @ X_62 ) ) ).
thf(fact_143_mem__def,axiom,
! [X_62: produc1137372701nt_int,A_287: produc1137372701nt_int > $o] :
( ( member2143287562nt_int @ X_62 @ A_287 )
<=> ( A_287 @ X_62 ) ) ).
thf(fact_144_mem__def,axiom,
! [X_62: produc975137661_int_o,A_287: produc975137661_int_o > $o] :
( ( member1329254762_int_o @ X_62 @ A_287 )
<=> ( A_287 @ X_62 ) ) ).
thf(fact_145_Collect__def,axiom,
! [P_9: int > $o] :
( ( collect_int @ P_9 )
= P_9 ) ).
thf(fact_146_Collect__def,axiom,
! [P_9: nat > $o] :
( ( collect_nat @ P_9 )
= P_9 ) ).
thf(fact_147_Collect__def,axiom,
! [P_9: produc1137372701nt_int > $o] :
( ( collec50511176nt_int @ P_9 )
= P_9 ) ).
thf(fact_148_Collect__def,axiom,
! [P_9: product_prod_int_int > $o] :
( ( collec1347809874nt_int @ P_9 )
= P_9 ) ).
thf(fact_149_Collect__def,axiom,
! [P_9: real > $o] :
( ( collect_real @ P_9 )
= P_9 ) ).
thf(fact_150_Collect__def,axiom,
! [P_9: product_prod_nat_nat > $o] :
( ( collec1979865426at_nat @ P_9 )
= P_9 ) ).
thf(fact_151_rel__simps_I15_J,axiom,
! [K_1: int,L: int] :
( ( ord_less_int @ ( bit0 @ K_1 ) @ ( bit1 @ L ) )
<=> ( ord_less_eq_int @ K_1 @ L ) ) ).
thf(fact_152_less__nat__number__of,axiom,
! [V: int,V_1: int] :
( ( ord_less_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
<=> ( ( ( ord_less_int @ V @ V_1 )
=> ( ord_less_int @ pls @ V_1 ) )
& ( ord_less_int @ V @ V_1 ) ) ) ).
thf(fact_153_int__one__le__iff__zero__less,axiom,
! [Z_1: int] :
( ( ord_less_eq_int @ one_one_int @ Z_1 )
<=> ( ord_less_int @ zero_zero_int @ Z_1 ) ) ).
thf(fact_154_nat__numeral__1__eq__1,axiom,
( ( number_number_of_nat @ ( bit1 @ pls ) )
= one_one_nat ) ).
thf(fact_155_Numeral1__eq1__nat,axiom,
( one_one_nat
= ( number_number_of_nat @ ( bit1 @ pls ) ) ) ).
thf(fact_156_zless__imp__add1__zle,axiom,
! [W: int,Z_1: int] :
( ( ord_less_int @ W @ Z_1 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z_1 ) ) ).
thf(fact_157_add1__zle__eq,axiom,
! [W: int,Z_1: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z_1 )
<=> ( ord_less_int @ W @ Z_1 ) ) ).
thf(fact_158_zle__add1__eq__le,axiom,
! [W: int,Z_1: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z_1 @ one_one_int ) )
<=> ( ord_less_eq_int @ W @ Z_1 ) ) ).
thf(fact_159_semiring__add__number__of,axiom,
! [V_15: int,V_14: int] :
( ( ord_less_eq_int @ pls @ V_14 )
=> ( ( ord_less_eq_int @ pls @ V_15 )
=> ( ( plus_plus_int @ ( number_number_of_int @ V_14 ) @ ( number_number_of_int @ V_15 ) )
= ( number_number_of_int @ ( plus_plus_int @ V_14 @ V_15 ) ) ) ) ) ).
thf(fact_160_semiring__add__number__of,axiom,
! [V_15: int,V_14: int] :
( ( ord_less_eq_int @ pls @ V_14 )
=> ( ( ord_less_eq_int @ pls @ V_15 )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V_14 ) @ ( number_number_of_nat @ V_15 ) )
= ( number_number_of_nat @ ( plus_plus_int @ V_14 @ V_15 ) ) ) ) ) ).
thf(fact_161_semiring__add__number__of,axiom,
! [V_15: int,V_14: int] :
( ( ord_less_eq_int @ pls @ V_14 )
=> ( ( ord_less_eq_int @ pls @ V_15 )
=> ( ( plus_plus_real @ ( number267125858f_real @ V_14 ) @ ( number267125858f_real @ V_15 ) )
= ( number267125858f_real @ ( plus_plus_int @ V_14 @ V_15 ) ) ) ) ) ).
thf(fact_162_semiring__add__number__of,axiom,
! [V_15: int,V_14: int] :
( ( ord_less_eq_int @ pls @ V_14 )
=> ( ( ord_less_eq_int @ pls @ V_15 )
=> ( ( plus_plus_complex @ ( number528085621omplex @ V_14 ) @ ( number528085621omplex @ V_15 ) )
= ( number528085621omplex @ ( plus_plus_int @ V_14 @ V_15 ) ) ) ) ) ).
thf(fact_163_semiring__add__number__of,axiom,
! [V_15: int,V_14: int] :
( ( ord_less_eq_int @ pls @ V_14 )
=> ( ( ord_less_eq_int @ pls @ V_15 )
=> ( ( plus_plus_rat @ ( number_number_of_rat @ V_14 ) @ ( number_number_of_rat @ V_15 ) )
= ( number_number_of_rat @ ( plus_plus_int @ V_14 @ V_15 ) ) ) ) ) ).
thf(fact_164_add__nat__number__of,axiom,
! [V_1: int,V: int] :
( ( ( ord_less_int @ V @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
= ( number_number_of_nat @ V_1 ) ) )
& ( ~ ( ord_less_int @ V @ pls )
=> ( ( ( ord_less_int @ V_1 @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
= ( number_number_of_nat @ V ) ) )
& ( ~ ( ord_less_int @ V_1 @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
= ( number_number_of_nat @ ( plus_plus_int @ V @ V_1 ) ) ) ) ) ) ) ).
thf(fact_165_le__imp__0__less,axiom,
! [Z_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z_1 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z_1 ) ) ) ).
thf(fact_166_eq__number__of,axiom,
! [X_61: int,Y_45: int] :
( ( ( number_number_of_int @ X_61 )
= ( number_number_of_int @ Y_45 ) )
<=> ( X_61 = Y_45 ) ) ).
thf(fact_167_eq__number__of,axiom,
! [X_61: int,Y_45: int] :
( ( ( number267125858f_real @ X_61 )
= ( number267125858f_real @ Y_45 ) )
<=> ( X_61 = Y_45 ) ) ).
thf(fact_168_eq__number__of,axiom,
! [X_61: int,Y_45: int] :
( ( ( number528085621omplex @ X_61 )
= ( number528085621omplex @ Y_45 ) )
<=> ( X_61 = Y_45 ) ) ).
thf(fact_169_eq__number__of,axiom,
! [X_61: int,Y_45: int] :
( ( ( number_number_of_rat @ X_61 )
= ( number_number_of_rat @ Y_45 ) )
<=> ( X_61 = Y_45 ) ) ).
thf(fact_170_number__of__reorient,axiom,
! [W_14: int,X_60: nat] :
( ( ( number_number_of_nat @ W_14 )
= X_60 )
<=> ( X_60
= ( number_number_of_nat @ W_14 ) ) ) ).
thf(fact_171_number__of__reorient,axiom,
! [W_14: int,X_60: int] :
( ( ( number_number_of_int @ W_14 )
= X_60 )
<=> ( X_60
= ( number_number_of_int @ W_14 ) ) ) ).
thf(fact_172_number__of__reorient,axiom,
! [W_14: int,X_60: real] :
( ( ( number267125858f_real @ W_14 )
= X_60 )
<=> ( X_60
= ( number267125858f_real @ W_14 ) ) ) ).
thf(fact_173_number__of__reorient,axiom,
! [W_14: int,X_60: code_code_numeral] :
( ( ( number1443263063umeral @ W_14 )
= X_60 )
<=> ( X_60
= ( number1443263063umeral @ W_14 ) ) ) ).
thf(fact_174_number__of__reorient,axiom,
! [W_14: int,X_60: complex] :
( ( ( number528085621omplex @ W_14 )
= X_60 )
<=> ( X_60
= ( number528085621omplex @ W_14 ) ) ) ).
thf(fact_175_number__of__reorient,axiom,
! [W_14: int,X_60: quickcheck_code_int] :
( ( ( number1226105091de_int @ W_14 )
= X_60 )
<=> ( X_60
= ( number1226105091de_int @ W_14 ) ) ) ).
thf(fact_176_number__of__reorient,axiom,
! [W_14: int,X_60: rat] :
( ( ( number_number_of_rat @ W_14 )
= X_60 )
<=> ( X_60
= ( number_number_of_rat @ W_14 ) ) ) ).
thf(fact_177_rel__simps_I51_J,axiom,
! [K_1: int,L: int] :
( ( ( bit1 @ K_1 )
= ( bit1 @ L ) )
<=> ( K_1 = L ) ) ).
thf(fact_178_rel__simps_I48_J,axiom,
! [K_1: int,L: int] :
( ( ( bit0 @ K_1 )
= ( bit0 @ L ) )
<=> ( K_1 = L ) ) ).
thf(fact_179_zless__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
thf(fact_180_sum__squares__eq__zero__iff,axiom,
! [X_59: int,Y_44: int] :
( ( ( plus_plus_int @ ( times_times_int @ X_59 @ X_59 ) @ ( times_times_int @ Y_44 @ Y_44 ) )
= zero_zero_int )
<=> ( ( X_59 = zero_zero_int )
& ( Y_44 = zero_zero_int ) ) ) ).
thf(fact_181_sum__squares__eq__zero__iff,axiom,
! [X_59: real,Y_44: real] :
( ( ( plus_plus_real @ ( times_times_real @ X_59 @ X_59 ) @ ( times_times_real @ Y_44 @ Y_44 ) )
= zero_zero_real )
<=> ( ( X_59 = zero_zero_real )
& ( Y_44 = zero_zero_real ) ) ) ).
thf(fact_182_sum__squares__eq__zero__iff,axiom,
! [X_59: rat,Y_44: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ X_59 @ X_59 ) @ ( times_times_rat @ Y_44 @ Y_44 ) )
= zero_zero_rat )
<=> ( ( X_59 = zero_zero_rat )
& ( Y_44 = zero_zero_rat ) ) ) ).
thf(fact_183_left__distrib__number__of,axiom,
! [A_286: code_code_numeral,B_204: code_code_numeral,V_13: int] :
( ( times_1655362735umeral @ ( plus_p1627245867umeral @ A_286 @ B_204 ) @ ( number1443263063umeral @ V_13 ) )
= ( plus_p1627245867umeral @ ( times_1655362735umeral @ A_286 @ ( number1443263063umeral @ V_13 ) ) @ ( times_1655362735umeral @ B_204 @ ( number1443263063umeral @ V_13 ) ) ) ) ).
thf(fact_184_left__distrib__number__of,axiom,
! [A_286: int,B_204: int,V_13: int] :
( ( times_times_int @ ( plus_plus_int @ A_286 @ B_204 ) @ ( number_number_of_int @ V_13 ) )
= ( plus_plus_int @ ( times_times_int @ A_286 @ ( number_number_of_int @ V_13 ) ) @ ( times_times_int @ B_204 @ ( number_number_of_int @ V_13 ) ) ) ) ).
thf(fact_185_left__distrib__number__of,axiom,
! [A_286: nat,B_204: nat,V_13: int] :
( ( times_times_nat @ ( plus_plus_nat @ A_286 @ B_204 ) @ ( number_number_of_nat @ V_13 ) )
= ( plus_plus_nat @ ( times_times_nat @ A_286 @ ( number_number_of_nat @ V_13 ) ) @ ( times_times_nat @ B_204 @ ( number_number_of_nat @ V_13 ) ) ) ) ).
thf(fact_186_left__distrib__number__of,axiom,
! [A_286: real,B_204: real,V_13: int] :
( ( times_times_real @ ( plus_plus_real @ A_286 @ B_204 ) @ ( number267125858f_real @ V_13 ) )
= ( plus_plus_real @ ( times_times_real @ A_286 @ ( number267125858f_real @ V_13 ) ) @ ( times_times_real @ B_204 @ ( number267125858f_real @ V_13 ) ) ) ) ).
thf(fact_187_left__distrib__number__of,axiom,
! [A_286: complex,B_204: complex,V_13: int] :
( ( times_times_complex @ ( plus_plus_complex @ A_286 @ B_204 ) @ ( number528085621omplex @ V_13 ) )
= ( plus_plus_complex @ ( times_times_complex @ A_286 @ ( number528085621omplex @ V_13 ) ) @ ( times_times_complex @ B_204 @ ( number528085621omplex @ V_13 ) ) ) ) ).
thf(fact_188_left__distrib__number__of,axiom,
! [A_286: quickcheck_code_int,B_204: quickcheck_code_int,V_13: int] :
( ( times_123202395de_int @ ( plus_p1446045655de_int @ A_286 @ B_204 ) @ ( number1226105091de_int @ V_13 ) )
= ( plus_p1446045655de_int @ ( times_123202395de_int @ A_286 @ ( number1226105091de_int @ V_13 ) ) @ ( times_123202395de_int @ B_204 @ ( number1226105091de_int @ V_13 ) ) ) ) ).
thf(fact_189_left__distrib__number__of,axiom,
! [A_286: rat,B_204: rat,V_13: int] :
( ( times_times_rat @ ( plus_plus_rat @ A_286 @ B_204 ) @ ( number_number_of_rat @ V_13 ) )
= ( plus_plus_rat @ ( times_times_rat @ A_286 @ ( number_number_of_rat @ V_13 ) ) @ ( times_times_rat @ B_204 @ ( number_number_of_rat @ V_13 ) ) ) ) ).
thf(fact_190_right__distrib__number__of,axiom,
! [V_12: int,B_203: code_code_numeral,C_123: code_code_numeral] :
( ( times_1655362735umeral @ ( number1443263063umeral @ V_12 ) @ ( plus_p1627245867umeral @ B_203 @ C_123 ) )
= ( plus_p1627245867umeral @ ( times_1655362735umeral @ ( number1443263063umeral @ V_12 ) @ B_203 ) @ ( times_1655362735umeral @ ( number1443263063umeral @ V_12 ) @ C_123 ) ) ) ).
thf(fact_191_right__distrib__number__of,axiom,
! [V_12: int,B_203: int,C_123: int] :
( ( times_times_int @ ( number_number_of_int @ V_12 ) @ ( plus_plus_int @ B_203 @ C_123 ) )
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ V_12 ) @ B_203 ) @ ( times_times_int @ ( number_number_of_int @ V_12 ) @ C_123 ) ) ) ).
thf(fact_192_right__distrib__number__of,axiom,
! [V_12: int,B_203: nat,C_123: nat] :
( ( times_times_nat @ ( number_number_of_nat @ V_12 ) @ ( plus_plus_nat @ B_203 @ C_123 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ V_12 ) @ B_203 ) @ ( times_times_nat @ ( number_number_of_nat @ V_12 ) @ C_123 ) ) ) ).
thf(fact_193_right__distrib__number__of,axiom,
! [V_12: int,B_203: real,C_123: real] :
( ( times_times_real @ ( number267125858f_real @ V_12 ) @ ( plus_plus_real @ B_203 @ C_123 ) )
= ( plus_plus_real @ ( times_times_real @ ( number267125858f_real @ V_12 ) @ B_203 ) @ ( times_times_real @ ( number267125858f_real @ V_12 ) @ C_123 ) ) ) ).
thf(fact_194_right__distrib__number__of,axiom,
! [V_12: int,B_203: complex,C_123: complex] :
( ( times_times_complex @ ( number528085621omplex @ V_12 ) @ ( plus_plus_complex @ B_203 @ C_123 ) )
= ( plus_plus_complex @ ( times_times_complex @ ( number528085621omplex @ V_12 ) @ B_203 ) @ ( times_times_complex @ ( number528085621omplex @ V_12 ) @ C_123 ) ) ) ).
thf(fact_195_right__distrib__number__of,axiom,
! [V_12: int,B_203: quickcheck_code_int,C_123: quickcheck_code_int] :
( ( times_123202395de_int @ ( number1226105091de_int @ V_12 ) @ ( plus_p1446045655de_int @ B_203 @ C_123 ) )
= ( plus_p1446045655de_int @ ( times_123202395de_int @ ( number1226105091de_int @ V_12 ) @ B_203 ) @ ( times_123202395de_int @ ( number1226105091de_int @ V_12 ) @ C_123 ) ) ) ).
thf(fact_196_right__distrib__number__of,axiom,
! [V_12: int,B_203: rat,C_123: rat] :
( ( times_times_rat @ ( number_number_of_rat @ V_12 ) @ ( plus_plus_rat @ B_203 @ C_123 ) )
= ( plus_plus_rat @ ( times_times_rat @ ( number_number_of_rat @ V_12 ) @ B_203 ) @ ( times_times_rat @ ( number_number_of_rat @ V_12 ) @ C_123 ) ) ) ).
thf(fact_197_zadd__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] :
( ( plus_plus_int @ ( plus_plus_int @ Z1 @ Z2 ) @ Z3 )
= ( plus_plus_int @ Z1 @ ( plus_plus_int @ Z2 @ Z3 ) ) ) ).
thf(fact_198_zadd__left__commute,axiom,
! [X: int,Y: int,Z_1: int] :
( ( plus_plus_int @ X @ ( plus_plus_int @ Y @ Z_1 ) )
= ( plus_plus_int @ Y @ ( plus_plus_int @ X @ Z_1 ) ) ) ).
thf(fact_199_zadd__commute,axiom,
! [Z_1: int,W: int] :
( ( plus_plus_int @ Z_1 @ W )
= ( plus_plus_int @ W @ Z_1 ) ) ).
thf(fact_200_zero__is__num__zero,axiom,
( zero_zero_int
= ( number_number_of_int @ pls ) ) ).
thf(fact_201_zmult__zless__mono2,axiom,
! [K_1: int,I: int,J: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K_1 )
=> ( ord_less_int @ ( times_times_int @ K_1 @ I ) @ ( times_times_int @ K_1 @ J ) ) ) ) ).
thf(fact_202_power2__eq__imp__eq,axiom,
! [X_58: int,Y_43: int] :
( ( ( power_power_int @ X_58 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_int @ Y_43 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ X_58 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_43 )
=> ( X_58 = Y_43 ) ) ) ) ).
thf(fact_203_power2__eq__imp__eq,axiom,
! [X_58: nat,Y_43: nat] :
( ( ( power_power_nat @ X_58 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_nat @ Y_43 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X_58 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_43 )
=> ( X_58 = Y_43 ) ) ) ) ).
thf(fact_204_power2__eq__imp__eq,axiom,
! [X_58: real,Y_43: real] :
( ( ( power_power_real @ X_58 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_real @ Y_43 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X_58 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_43 )
=> ( X_58 = Y_43 ) ) ) ) ).
thf(fact_205_power2__eq__imp__eq,axiom,
! [X_58: code_code_numeral,Y_43: code_code_numeral] :
( ( ( power_2100829034umeral @ X_58 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_2100829034umeral @ Y_43 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ X_58 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ Y_43 )
=> ( X_58 = Y_43 ) ) ) ) ).
thf(fact_206_power2__eq__imp__eq,axiom,
! [X_58: quickcheck_code_int,Y_43: quickcheck_code_int] :
( ( ( power_881366806de_int @ X_58 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_881366806de_int @ Y_43 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ X_58 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ Y_43 )
=> ( X_58 = Y_43 ) ) ) ) ).
thf(fact_207_power2__eq__imp__eq,axiom,
! [X_58: rat,Y_43: rat] :
( ( ( power_power_rat @ X_58 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_rat @ Y_43 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ X_58 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y_43 )
=> ( X_58 = Y_43 ) ) ) ) ).
thf(fact_208_power2__le__imp__le,axiom,
! [X_57: int,Y_42: int] :
( ( ord_less_eq_int @ ( power_power_int @ X_57 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_42 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_42 )
=> ( ord_less_eq_int @ X_57 @ Y_42 ) ) ) ).
thf(fact_209_power2__le__imp__le,axiom,
! [X_57: nat,Y_42: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X_57 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_nat @ Y_42 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_42 )
=> ( ord_less_eq_nat @ X_57 @ Y_42 ) ) ) ).
thf(fact_210_power2__le__imp__le,axiom,
! [X_57: real,Y_42: real] :
( ( ord_less_eq_real @ ( power_power_real @ X_57 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_42 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_42 )
=> ( ord_less_eq_real @ X_57 @ Y_42 ) ) ) ).
thf(fact_211_power2__le__imp__le,axiom,
! [X_57: code_code_numeral,Y_42: code_code_numeral] :
( ( ord_le565307924umeral @ ( power_2100829034umeral @ X_57 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_2100829034umeral @ Y_42 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ Y_42 )
=> ( ord_le565307924umeral @ X_57 @ Y_42 ) ) ) ).
thf(fact_212_power2__le__imp__le,axiom,
! [X_57: quickcheck_code_int,Y_42: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( power_881366806de_int @ X_57 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_881366806de_int @ Y_42 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ Y_42 )
=> ( ord_le258702272de_int @ X_57 @ Y_42 ) ) ) ).
thf(fact_213_power2__le__imp__le,axiom,
! [X_57: rat,Y_42: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ X_57 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_rat @ Y_42 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y_42 )
=> ( ord_less_eq_rat @ X_57 @ Y_42 ) ) ) ).
thf(fact_214_zero__le__power2,axiom,
! [A_285: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A_285 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_215_zero__le__power2,axiom,
! [A_285: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A_285 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_216_zero__le__power2,axiom,
! [A_285: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A_285 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_217_power2__less__imp__less,axiom,
! [X_56: int,Y_41: int] :
( ( ord_less_int @ ( power_power_int @ X_56 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_41 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_41 )
=> ( ord_less_int @ X_56 @ Y_41 ) ) ) ).
thf(fact_218_power2__less__imp__less,axiom,
! [X_56: nat,Y_41: nat] :
( ( ord_less_nat @ ( power_power_nat @ X_56 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_nat @ Y_41 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_41 )
=> ( ord_less_nat @ X_56 @ Y_41 ) ) ) ).
thf(fact_219_power2__less__imp__less,axiom,
! [X_56: real,Y_41: real] :
( ( ord_less_real @ ( power_power_real @ X_56 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_41 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_41 )
=> ( ord_less_real @ X_56 @ Y_41 ) ) ) ).
thf(fact_220_power2__less__imp__less,axiom,
! [X_56: code_code_numeral,Y_41: code_code_numeral] :
( ( ord_le1304079648umeral @ ( power_2100829034umeral @ X_56 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_2100829034umeral @ Y_41 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ Y_41 )
=> ( ord_le1304079648umeral @ X_56 @ Y_41 ) ) ) ).
thf(fact_221_power2__less__imp__less,axiom,
! [X_56: quickcheck_code_int,Y_41: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( power_881366806de_int @ X_56 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_881366806de_int @ Y_41 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ Y_41 )
=> ( ord_le1860547276de_int @ X_56 @ Y_41 ) ) ) ).
thf(fact_222_power2__less__imp__less,axiom,
! [X_56: rat,Y_41: rat] :
( ( ord_less_rat @ ( power_power_rat @ X_56 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_rat @ Y_41 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y_41 )
=> ( ord_less_rat @ X_56 @ Y_41 ) ) ) ).
thf(fact_223_sum__power2__le__zero__iff,axiom,
! [X_55: int,Y_40: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X_55 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_40 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_int )
<=> ( ( X_55 = zero_zero_int )
& ( Y_40 = zero_zero_int ) ) ) ).
thf(fact_224_sum__power2__le__zero__iff,axiom,
! [X_55: real,Y_40: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X_55 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_40 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_real )
<=> ( ( X_55 = zero_zero_real )
& ( Y_40 = zero_zero_real ) ) ) ).
thf(fact_225_sum__power2__le__zero__iff,axiom,
! [X_55: rat,Y_40: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X_55 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_rat @ Y_40 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_rat )
<=> ( ( X_55 = zero_zero_rat )
& ( Y_40 = zero_zero_rat ) ) ) ).
thf(fact_226_sum__power2__ge__zero,axiom,
! [X_54: int,Y_39: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X_54 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_39 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_227_sum__power2__ge__zero,axiom,
! [X_54: real,Y_39: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X_54 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_39 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_228_sum__power2__ge__zero,axiom,
! [X_54: rat,Y_39: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X_54 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_rat @ Y_39 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_229_sum__squares__gt__zero__iff,axiom,
! [X_53: int,Y_38: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X_53 @ X_53 ) @ ( times_times_int @ Y_38 @ Y_38 ) ) )
<=> ( ( X_53 != zero_zero_int )
| ( Y_38 != zero_zero_int ) ) ) ).
thf(fact_230_sum__squares__gt__zero__iff,axiom,
! [X_53: real,Y_38: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X_53 @ X_53 ) @ ( times_times_real @ Y_38 @ Y_38 ) ) )
<=> ( ( X_53 != zero_zero_real )
| ( Y_38 != zero_zero_real ) ) ) ).
thf(fact_231_sum__squares__gt__zero__iff,axiom,
! [X_53: rat,Y_38: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X_53 @ X_53 ) @ ( times_times_rat @ Y_38 @ Y_38 ) ) )
<=> ( ( X_53 != zero_zero_rat )
| ( Y_38 != zero_zero_rat ) ) ) ).
thf(fact_232_not__sum__squares__lt__zero,axiom,
! [X_52: int,Y_37: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X_52 @ X_52 ) @ ( times_times_int @ Y_37 @ Y_37 ) ) @ zero_zero_int ) ).
thf(fact_233_not__sum__squares__lt__zero,axiom,
! [X_52: real,Y_37: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X_52 @ X_52 ) @ ( times_times_real @ Y_37 @ Y_37 ) ) @ zero_zero_real ) ).
thf(fact_234_not__sum__squares__lt__zero,axiom,
! [X_52: rat,Y_37: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X_52 @ X_52 ) @ ( times_times_rat @ Y_37 @ Y_37 ) ) @ zero_zero_rat ) ).
thf(fact_235_mult__numeral__1,axiom,
! [A_284: int] :
( ( times_times_int @ ( number_number_of_int @ ( bit1 @ pls ) ) @ A_284 )
= A_284 ) ).
thf(fact_236_mult__numeral__1,axiom,
! [A_284: real] :
( ( times_times_real @ ( number267125858f_real @ ( bit1 @ pls ) ) @ A_284 )
= A_284 ) ).
thf(fact_237_mult__numeral__1,axiom,
! [A_284: complex] :
( ( times_times_complex @ ( number528085621omplex @ ( bit1 @ pls ) ) @ A_284 )
= A_284 ) ).
thf(fact_238_mult__numeral__1,axiom,
! [A_284: rat] :
( ( times_times_rat @ ( number_number_of_rat @ ( bit1 @ pls ) ) @ A_284 )
= A_284 ) ).
thf(fact_239_mult__numeral__1__right,axiom,
! [A_283: int] :
( ( times_times_int @ A_283 @ ( number_number_of_int @ ( bit1 @ pls ) ) )
= A_283 ) ).
thf(fact_240_mult__numeral__1__right,axiom,
! [A_283: real] :
( ( times_times_real @ A_283 @ ( number267125858f_real @ ( bit1 @ pls ) ) )
= A_283 ) ).
thf(fact_241_mult__numeral__1__right,axiom,
! [A_283: complex] :
( ( times_times_complex @ A_283 @ ( number528085621omplex @ ( bit1 @ pls ) ) )
= A_283 ) ).
thf(fact_242_mult__numeral__1__right,axiom,
! [A_283: rat] :
( ( times_times_rat @ A_283 @ ( number_number_of_rat @ ( bit1 @ pls ) ) )
= A_283 ) ).
thf(fact_243_one__is__num__one,axiom,
( one_one_int
= ( number_number_of_int @ ( bit1 @ pls ) ) ) ).
thf(fact_244_mult__Bit1,axiom,
! [K_1: int,L: int] :
( ( times_times_int @ ( bit1 @ K_1 ) @ L )
= ( plus_plus_int @ ( bit0 @ ( times_times_int @ K_1 @ L ) ) @ L ) ) ).
thf(fact_245_pos__zmult__eq__1__iff,axiom,
! [N: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
<=> ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
thf(fact_246_double__eq__0__iff,axiom,
! [A_282: int] :
( ( ( plus_plus_int @ A_282 @ A_282 )
= zero_zero_int )
<=> ( A_282 = zero_zero_int ) ) ).
thf(fact_247_double__eq__0__iff,axiom,
! [A_282: real] :
( ( ( plus_plus_real @ A_282 @ A_282 )
= zero_zero_real )
<=> ( A_282 = zero_zero_real ) ) ).
thf(fact_248_double__eq__0__iff,axiom,
! [A_282: rat] :
( ( ( plus_plus_rat @ A_282 @ A_282 )
= zero_zero_rat )
<=> ( A_282 = zero_zero_rat ) ) ).
thf(fact_249_rel__simps_I46_J,axiom,
! [K_1: int] :
( ( bit1 @ K_1 )
!= pls ) ).
thf(fact_250_rel__simps_I39_J,axiom,
! [L: int] :
( pls
!= ( bit1 @ L ) ) ).
thf(fact_251_rel__simps_I50_J,axiom,
! [K_1: int,L: int] :
( ( bit1 @ K_1 )
!= ( bit0 @ L ) ) ).
thf(fact_252_rel__simps_I49_J,axiom,
! [K_1: int,L: int] :
( ( bit0 @ K_1 )
!= ( bit1 @ L ) ) ).
thf(fact_253_rel__simps_I44_J,axiom,
! [K_1: int] :
( ( ( bit0 @ K_1 )
= pls )
<=> ( K_1 = pls ) ) ).
thf(fact_254_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls
= ( bit0 @ L ) )
<=> ( pls = L ) ) ).
thf(fact_255_Bit0__Pls,axiom,
( ( bit0 @ pls )
= pls ) ).
thf(fact_256_Pls__def,axiom,
pls = zero_zero_int ).
thf(fact_257_less__int__code_I16_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit1 @ K1 ) @ ( bit1 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_258_rel__simps_I17_J,axiom,
! [K_1: int,L: int] :
( ( ord_less_int @ ( bit1 @ K_1 ) @ ( bit1 @ L ) )
<=> ( ord_less_int @ K_1 @ L ) ) ).
thf(fact_259_rel__simps_I2_J,axiom,
~ ( ord_less_int @ pls @ pls ) ).
thf(fact_260_less__int__code_I13_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit0 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_261_rel__simps_I14_J,axiom,
! [K_1: int,L: int] :
( ( ord_less_int @ ( bit0 @ K_1 ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K_1 @ L ) ) ).
thf(fact_262_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
thf(fact_263_add__Pls__right,axiom,
! [K_1: int] :
( ( plus_plus_int @ K_1 @ pls )
= K_1 ) ).
thf(fact_264_add__Pls,axiom,
! [K_1: int] :
( ( plus_plus_int @ pls @ K_1 )
= K_1 ) ).
thf(fact_265_add__Bit0__Bit0,axiom,
! [K_1: int,L: int] :
( ( plus_plus_int @ ( bit0 @ K_1 ) @ ( bit0 @ L ) )
= ( bit0 @ ( plus_plus_int @ K_1 @ L ) ) ) ).
thf(fact_266_Bit0__def,axiom,
! [K_1: int] :
( ( bit0 @ K_1 )
= ( plus_plus_int @ K_1 @ K_1 ) ) ).
thf(fact_267_zadd__0__right,axiom,
! [Z_1: int] :
( ( plus_plus_int @ Z_1 @ zero_zero_int )
= Z_1 ) ).
thf(fact_268_zadd__0,axiom,
! [Z_1: int] :
( ( plus_plus_int @ zero_zero_int @ Z_1 )
= Z_1 ) ).
thf(fact_269_zadd__strict__right__mono,axiom,
! [K_1: int,I: int,J: int] :
( ( ord_less_int @ I @ J )
=> ( ord_less_int @ ( plus_plus_int @ I @ K_1 ) @ ( plus_plus_int @ J @ K_1 ) ) ) ).
thf(fact_270_double__number__of__Bit0,axiom,
! [W_13: int] :
( ( times_times_int @ ( plus_plus_int @ one_one_int @ one_one_int ) @ ( number_number_of_int @ W_13 ) )
= ( number_number_of_int @ ( bit0 @ W_13 ) ) ) ).
thf(fact_271_double__number__of__Bit0,axiom,
! [W_13: int] :
( ( times_times_real @ ( plus_plus_real @ one_one_real @ one_one_real ) @ ( number267125858f_real @ W_13 ) )
= ( number267125858f_real @ ( bit0 @ W_13 ) ) ) ).
thf(fact_272_double__number__of__Bit0,axiom,
! [W_13: int] :
( ( times_times_complex @ ( plus_plus_complex @ one_one_complex @ one_one_complex ) @ ( number528085621omplex @ W_13 ) )
= ( number528085621omplex @ ( bit0 @ W_13 ) ) ) ).
thf(fact_273_double__number__of__Bit0,axiom,
! [W_13: int] :
( ( times_times_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) @ ( number_number_of_rat @ W_13 ) )
= ( number_number_of_rat @ ( bit0 @ W_13 ) ) ) ).
thf(fact_274_power3__eq__cube,axiom,
! [A_281: code_code_numeral] :
( ( power_2100829034umeral @ A_281 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_1655362735umeral @ ( times_1655362735umeral @ A_281 @ A_281 ) @ A_281 ) ) ).
thf(fact_275_power3__eq__cube,axiom,
! [A_281: quickcheck_code_int] :
( ( power_881366806de_int @ A_281 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_123202395de_int @ ( times_123202395de_int @ A_281 @ A_281 ) @ A_281 ) ) ).
thf(fact_276_power3__eq__cube,axiom,
! [A_281: rat] :
( ( power_power_rat @ A_281 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_times_rat @ ( times_times_rat @ A_281 @ A_281 ) @ A_281 ) ) ).
thf(fact_277_power3__eq__cube,axiom,
! [A_281: int] :
( ( power_power_int @ A_281 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_times_int @ ( times_times_int @ A_281 @ A_281 ) @ A_281 ) ) ).
thf(fact_278_power3__eq__cube,axiom,
! [A_281: nat] :
( ( power_power_nat @ A_281 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_times_nat @ ( times_times_nat @ A_281 @ A_281 ) @ A_281 ) ) ).
thf(fact_279_power3__eq__cube,axiom,
! [A_281: real] :
( ( power_power_real @ A_281 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_times_real @ ( times_times_real @ A_281 @ A_281 ) @ A_281 ) ) ).
thf(fact_280_power3__eq__cube,axiom,
! [A_281: complex] :
( ( power_power_complex @ A_281 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_times_complex @ ( times_times_complex @ A_281 @ A_281 ) @ A_281 ) ) ).
thf(fact_281_semiring__mult__2,axiom,
! [Z_14: int] :
( ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_14 )
= ( plus_plus_int @ Z_14 @ Z_14 ) ) ).
thf(fact_282_semiring__mult__2,axiom,
! [Z_14: nat] :
( ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_14 )
= ( plus_plus_nat @ Z_14 @ Z_14 ) ) ).
thf(fact_283_semiring__mult__2,axiom,
! [Z_14: real] :
( ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_14 )
= ( plus_plus_real @ Z_14 @ Z_14 ) ) ).
thf(fact_284_semiring__mult__2,axiom,
! [Z_14: complex] :
( ( times_times_complex @ ( number528085621omplex @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_14 )
= ( plus_plus_complex @ Z_14 @ Z_14 ) ) ).
thf(fact_285_semiring__mult__2,axiom,
! [Z_14: rat] :
( ( times_times_rat @ ( number_number_of_rat @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_14 )
= ( plus_plus_rat @ Z_14 @ Z_14 ) ) ).
thf(fact_286_mult__2,axiom,
! [Z_13: int] :
( ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_13 )
= ( plus_plus_int @ Z_13 @ Z_13 ) ) ).
thf(fact_287_mult__2,axiom,
! [Z_13: real] :
( ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_13 )
= ( plus_plus_real @ Z_13 @ Z_13 ) ) ).
thf(fact_288_mult__2,axiom,
! [Z_13: complex] :
( ( times_times_complex @ ( number528085621omplex @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_13 )
= ( plus_plus_complex @ Z_13 @ Z_13 ) ) ).
thf(fact_289_mult__2,axiom,
! [Z_13: rat] :
( ( times_times_rat @ ( number_number_of_rat @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_13 )
= ( plus_plus_rat @ Z_13 @ Z_13 ) ) ).
thf(fact_290_semiring__mult__2__right,axiom,
! [Z_12: int] :
( ( times_times_int @ Z_12 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ Z_12 @ Z_12 ) ) ).
thf(fact_291_semiring__mult__2__right,axiom,
! [Z_12: nat] :
( ( times_times_nat @ Z_12 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_nat @ Z_12 @ Z_12 ) ) ).
thf(fact_292_semiring__mult__2__right,axiom,
! [Z_12: real] :
( ( times_times_real @ Z_12 @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_real @ Z_12 @ Z_12 ) ) ).
thf(fact_293_semiring__mult__2__right,axiom,
! [Z_12: complex] :
( ( times_times_complex @ Z_12 @ ( number528085621omplex @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_complex @ Z_12 @ Z_12 ) ) ).
thf(fact_294_semiring__mult__2__right,axiom,
! [Z_12: rat] :
( ( times_times_rat @ Z_12 @ ( number_number_of_rat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_rat @ Z_12 @ Z_12 ) ) ).
thf(fact_295_mult__2__right,axiom,
! [Z_11: int] :
( ( times_times_int @ Z_11 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ Z_11 @ Z_11 ) ) ).
thf(fact_296_mult__2__right,axiom,
! [Z_11: real] :
( ( times_times_real @ Z_11 @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_real @ Z_11 @ Z_11 ) ) ).
thf(fact_297_mult__2__right,axiom,
! [Z_11: complex] :
( ( times_times_complex @ Z_11 @ ( number528085621omplex @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_complex @ Z_11 @ Z_11 ) ) ).
thf(fact_298_mult__2__right,axiom,
! [Z_11: rat] :
( ( times_times_rat @ Z_11 @ ( number_number_of_rat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_rat @ Z_11 @ Z_11 ) ) ).
thf(fact_299_power2__eq__square,axiom,
! [A_280: code_code_numeral] :
( ( power_2100829034umeral @ A_280 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_1655362735umeral @ A_280 @ A_280 ) ) ).
thf(fact_300_power2__eq__square,axiom,
! [A_280: quickcheck_code_int] :
( ( power_881366806de_int @ A_280 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_123202395de_int @ A_280 @ A_280 ) ) ).
thf(fact_301_power2__eq__square,axiom,
! [A_280: rat] :
( ( power_power_rat @ A_280 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_rat @ A_280 @ A_280 ) ) ).
thf(fact_302_power2__eq__square,axiom,
! [A_280: int] :
( ( power_power_int @ A_280 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_int @ A_280 @ A_280 ) ) ).
thf(fact_303_power2__eq__square,axiom,
! [A_280: nat] :
( ( power_power_nat @ A_280 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_nat @ A_280 @ A_280 ) ) ).
thf(fact_304_power2__eq__square,axiom,
! [A_280: real] :
( ( power_power_real @ A_280 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_real @ A_280 @ A_280 ) ) ).
thf(fact_305_power2__eq__square,axiom,
! [A_280: complex] :
( ( power_power_complex @ A_280 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_complex @ A_280 @ A_280 ) ) ).
thf(fact_306_even__less__0__iff,axiom,
! [A_279: int] :
( ( ord_less_int @ ( plus_plus_int @ A_279 @ A_279 ) @ zero_zero_int )
<=> ( ord_less_int @ A_279 @ zero_zero_int ) ) ).
thf(fact_307_even__less__0__iff,axiom,
! [A_279: real] :
( ( ord_less_real @ ( plus_plus_real @ A_279 @ A_279 ) @ zero_zero_real )
<=> ( ord_less_real @ A_279 @ zero_zero_real ) ) ).
thf(fact_308_even__less__0__iff,axiom,
! [A_279: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A_279 @ A_279 ) @ zero_zero_rat )
<=> ( ord_less_rat @ A_279 @ zero_zero_rat ) ) ).
thf(fact_309_semiring__numeral__0__eq__0,axiom,
( ( number_number_of_int @ pls )
= zero_zero_int ) ).
thf(fact_310_semiring__numeral__0__eq__0,axiom,
( ( number_number_of_nat @ pls )
= zero_zero_nat ) ).
thf(fact_311_semiring__numeral__0__eq__0,axiom,
( ( number267125858f_real @ pls )
= zero_zero_real ) ).
thf(fact_312_semiring__numeral__0__eq__0,axiom,
( ( number528085621omplex @ pls )
= zero_zero_complex ) ).
thf(fact_313_semiring__numeral__0__eq__0,axiom,
( ( number_number_of_rat @ pls )
= zero_zero_rat ) ).
thf(fact_314_number__of__Pls,axiom,
( ( number_number_of_int @ pls )
= zero_zero_int ) ).
thf(fact_315_number__of__Pls,axiom,
( ( number267125858f_real @ pls )
= zero_zero_real ) ).
thf(fact_316_number__of__Pls,axiom,
( ( number528085621omplex @ pls )
= zero_zero_complex ) ).
thf(fact_317_number__of__Pls,axiom,
( ( number_number_of_rat @ pls )
= zero_zero_rat ) ).
thf(fact_318_semiring__norm_I112_J,axiom,
( zero_zero_int
= ( number_number_of_int @ pls ) ) ).
thf(fact_319_semiring__norm_I112_J,axiom,
( zero_zero_real
= ( number267125858f_real @ pls ) ) ).
thf(fact_320_semiring__norm_I112_J,axiom,
( zero_zero_complex
= ( number528085621omplex @ pls ) ) ).
thf(fact_321_semiring__norm_I112_J,axiom,
( zero_zero_rat
= ( number_number_of_rat @ pls ) ) ).
thf(fact_322_add__numeral__0,axiom,
! [A_278: int] :
( ( plus_plus_int @ ( number_number_of_int @ pls ) @ A_278 )
= A_278 ) ).
thf(fact_323_add__numeral__0,axiom,
! [A_278: real] :
( ( plus_plus_real @ ( number267125858f_real @ pls ) @ A_278 )
= A_278 ) ).
thf(fact_324_add__numeral__0,axiom,
! [A_278: complex] :
( ( plus_plus_complex @ ( number528085621omplex @ pls ) @ A_278 )
= A_278 ) ).
thf(fact_325_add__numeral__0,axiom,
! [A_278: rat] :
( ( plus_plus_rat @ ( number_number_of_rat @ pls ) @ A_278 )
= A_278 ) ).
thf(fact_326_add__numeral__0__right,axiom,
! [A_277: int] :
( ( plus_plus_int @ A_277 @ ( number_number_of_int @ pls ) )
= A_277 ) ).
thf(fact_327_add__numeral__0__right,axiom,
! [A_277: real] :
( ( plus_plus_real @ A_277 @ ( number267125858f_real @ pls ) )
= A_277 ) ).
thf(fact_328_add__numeral__0__right,axiom,
! [A_277: complex] :
( ( plus_plus_complex @ A_277 @ ( number528085621omplex @ pls ) )
= A_277 ) ).
thf(fact_329_add__numeral__0__right,axiom,
! [A_277: rat] :
( ( plus_plus_rat @ A_277 @ ( number_number_of_rat @ pls ) )
= A_277 ) ).
thf(fact_330_less__number__of,axiom,
! [X_51: int,Y_36: int] :
( ( ord_less_int @ ( number_number_of_int @ X_51 ) @ ( number_number_of_int @ Y_36 ) )
<=> ( ord_less_int @ X_51 @ Y_36 ) ) ).
thf(fact_331_less__number__of,axiom,
! [X_51: int,Y_36: int] :
( ( ord_less_real @ ( number267125858f_real @ X_51 ) @ ( number267125858f_real @ Y_36 ) )
<=> ( ord_less_int @ X_51 @ Y_36 ) ) ).
thf(fact_332_less__number__of,axiom,
! [X_51: int,Y_36: int] :
( ( ord_less_rat @ ( number_number_of_rat @ X_51 ) @ ( number_number_of_rat @ Y_36 ) )
<=> ( ord_less_int @ X_51 @ Y_36 ) ) ).
thf(fact_333_add__number__of__left,axiom,
! [V_11: int,W_12: int,Z_10: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_11 ) @ ( plus_plus_int @ ( number_number_of_int @ W_12 ) @ Z_10 ) )
= ( plus_plus_int @ ( number_number_of_int @ ( plus_plus_int @ V_11 @ W_12 ) ) @ Z_10 ) ) ).
thf(fact_334_add__number__of__left,axiom,
! [V_11: int,W_12: int,Z_10: real] :
( ( plus_plus_real @ ( number267125858f_real @ V_11 ) @ ( plus_plus_real @ ( number267125858f_real @ W_12 ) @ Z_10 ) )
= ( plus_plus_real @ ( number267125858f_real @ ( plus_plus_int @ V_11 @ W_12 ) ) @ Z_10 ) ) ).
thf(fact_335_add__number__of__left,axiom,
! [V_11: int,W_12: int,Z_10: complex] :
( ( plus_plus_complex @ ( number528085621omplex @ V_11 ) @ ( plus_plus_complex @ ( number528085621omplex @ W_12 ) @ Z_10 ) )
= ( plus_plus_complex @ ( number528085621omplex @ ( plus_plus_int @ V_11 @ W_12 ) ) @ Z_10 ) ) ).
thf(fact_336_add__number__of__left,axiom,
! [V_11: int,W_12: int,Z_10: rat] :
( ( plus_plus_rat @ ( number_number_of_rat @ V_11 ) @ ( plus_plus_rat @ ( number_number_of_rat @ W_12 ) @ Z_10 ) )
= ( plus_plus_rat @ ( number_number_of_rat @ ( plus_plus_int @ V_11 @ W_12 ) ) @ Z_10 ) ) ).
thf(fact_337_add__number__of__eq,axiom,
! [V_10: int,W_11: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_10 ) @ ( number_number_of_int @ W_11 ) )
= ( number_number_of_int @ ( plus_plus_int @ V_10 @ W_11 ) ) ) ).
thf(fact_338_add__number__of__eq,axiom,
! [V_10: int,W_11: int] :
( ( plus_plus_real @ ( number267125858f_real @ V_10 ) @ ( number267125858f_real @ W_11 ) )
= ( number267125858f_real @ ( plus_plus_int @ V_10 @ W_11 ) ) ) ).
thf(fact_339_add__number__of__eq,axiom,
! [V_10: int,W_11: int] :
( ( plus_plus_complex @ ( number528085621omplex @ V_10 ) @ ( number528085621omplex @ W_11 ) )
= ( number528085621omplex @ ( plus_plus_int @ V_10 @ W_11 ) ) ) ).
thf(fact_340_add__number__of__eq,axiom,
! [V_10: int,W_11: int] :
( ( plus_plus_rat @ ( number_number_of_rat @ V_10 ) @ ( number_number_of_rat @ W_11 ) )
= ( number_number_of_rat @ ( plus_plus_int @ V_10 @ W_11 ) ) ) ).
thf(fact_341_number__of__add,axiom,
! [V_9: int,W_10: int] :
( ( number_number_of_int @ ( plus_plus_int @ V_9 @ W_10 ) )
= ( plus_plus_int @ ( number_number_of_int @ V_9 ) @ ( number_number_of_int @ W_10 ) ) ) ).
thf(fact_342_number__of__add,axiom,
! [V_9: int,W_10: int] :
( ( number267125858f_real @ ( plus_plus_int @ V_9 @ W_10 ) )
= ( plus_plus_real @ ( number267125858f_real @ V_9 ) @ ( number267125858f_real @ W_10 ) ) ) ).
thf(fact_343_number__of__add,axiom,
! [V_9: int,W_10: int] :
( ( number528085621omplex @ ( plus_plus_int @ V_9 @ W_10 ) )
= ( plus_plus_complex @ ( number528085621omplex @ V_9 ) @ ( number528085621omplex @ W_10 ) ) ) ).
thf(fact_344_number__of__add,axiom,
! [V_9: int,W_10: int] :
( ( number_number_of_rat @ ( plus_plus_int @ V_9 @ W_10 ) )
= ( plus_plus_rat @ ( number_number_of_rat @ V_9 ) @ ( number_number_of_rat @ W_10 ) ) ) ).
thf(fact_345_rel__simps_I12_J,axiom,
! [K_1: int] :
( ( ord_less_int @ ( bit1 @ K_1 ) @ pls )
<=> ( ord_less_int @ K_1 @ pls ) ) ).
thf(fact_346_less__int__code_I15_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit1 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_347_rel__simps_I16_J,axiom,
! [K_1: int,L: int] :
( ( ord_less_int @ ( bit1 @ K_1 ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K_1 @ L ) ) ).
thf(fact_348_bin__less__0__simps_I4_J,axiom,
! [W: int] :
( ( ord_less_int @ ( bit1 @ W ) @ zero_zero_int )
<=> ( ord_less_int @ W @ zero_zero_int ) ) ).
thf(fact_349_rel__simps_I10_J,axiom,
! [K_1: int] :
( ( ord_less_int @ ( bit0 @ K_1 ) @ pls )
<=> ( ord_less_int @ K_1 @ pls ) ) ).
thf(fact_350_rel__simps_I4_J,axiom,
! [K_1: int] :
( ( ord_less_int @ pls @ ( bit0 @ K_1 ) )
<=> ( ord_less_int @ pls @ K_1 ) ) ).
thf(fact_351_bin__less__0__simps_I1_J,axiom,
~ ( ord_less_int @ pls @ zero_zero_int ) ).
thf(fact_352_bin__less__0__simps_I3_J,axiom,
! [W: int] :
( ( ord_less_int @ ( bit0 @ W ) @ zero_zero_int )
<=> ( ord_less_int @ W @ zero_zero_int ) ) ).
thf(fact_353_add__Bit1__Bit0,axiom,
! [K_1: int,L: int] :
( ( plus_plus_int @ ( bit1 @ K_1 ) @ ( bit0 @ L ) )
= ( bit1 @ ( plus_plus_int @ K_1 @ L ) ) ) ).
thf(fact_354_add__Bit0__Bit1,axiom,
! [K_1: int,L: int] :
( ( plus_plus_int @ ( bit0 @ K_1 ) @ ( bit1 @ L ) )
= ( bit1 @ ( plus_plus_int @ K_1 @ L ) ) ) ).
thf(fact_355_int__0__less__1,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
thf(fact_356_Bit1__def,axiom,
! [K_1: int] :
( ( bit1 @ K_1 )
= ( plus_plus_int @ ( plus_plus_int @ one_one_int @ K_1 ) @ K_1 ) ) ).
thf(fact_357_odd__nonzero,axiom,
! [Z_1: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z_1 ) @ Z_1 )
!= zero_zero_int ) ).
thf(fact_358_zless__add1__eq,axiom,
! [W: int,Z_1: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z_1 @ one_one_int ) )
<=> ( ( ord_less_int @ W @ Z_1 )
| ( W = Z_1 ) ) ) ).
thf(fact_359_power2__eq__square__number__of,axiom,
! [W_9: int] :
( ( power_2100829034umeral @ ( number1443263063umeral @ W_9 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_1655362735umeral @ ( number1443263063umeral @ W_9 ) @ ( number1443263063umeral @ W_9 ) ) ) ).
thf(fact_360_power2__eq__square__number__of,axiom,
! [W_9: int] :
( ( power_881366806de_int @ ( number1226105091de_int @ W_9 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_123202395de_int @ ( number1226105091de_int @ W_9 ) @ ( number1226105091de_int @ W_9 ) ) ) ).
thf(fact_361_power2__eq__square__number__of,axiom,
! [W_9: int] :
( ( power_power_rat @ ( number_number_of_rat @ W_9 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_rat @ ( number_number_of_rat @ W_9 ) @ ( number_number_of_rat @ W_9 ) ) ) ).
thf(fact_362_power2__eq__square__number__of,axiom,
! [W_9: int] :
( ( power_power_int @ ( number_number_of_int @ W_9 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_int @ ( number_number_of_int @ W_9 ) @ ( number_number_of_int @ W_9 ) ) ) ).
thf(fact_363_power2__eq__square__number__of,axiom,
! [W_9: int] :
( ( power_power_nat @ ( number_number_of_nat @ W_9 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_nat @ ( number_number_of_nat @ W_9 ) @ ( number_number_of_nat @ W_9 ) ) ) ).
thf(fact_364_power2__eq__square__number__of,axiom,
! [W_9: int] :
( ( power_power_real @ ( number267125858f_real @ W_9 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_real @ ( number267125858f_real @ W_9 ) @ ( number267125858f_real @ W_9 ) ) ) ).
thf(fact_365_power2__eq__square__number__of,axiom,
! [W_9: int] :
( ( power_power_complex @ ( number528085621omplex @ W_9 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_complex @ ( number528085621omplex @ W_9 ) @ ( number528085621omplex @ W_9 ) ) ) ).
thf(fact_366_power2__sum,axiom,
! [X_50: int,Y_35: int] :
( ( power_power_int @ ( plus_plus_int @ X_50 @ Y_35 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X_50 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_35 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_50 ) @ Y_35 ) ) ) ).
thf(fact_367_power2__sum,axiom,
! [X_50: nat,Y_35: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X_50 @ Y_35 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X_50 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_nat @ Y_35 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_50 ) @ Y_35 ) ) ) ).
thf(fact_368_power2__sum,axiom,
! [X_50: real,Y_35: real] :
( ( power_power_real @ ( plus_plus_real @ X_50 @ Y_35 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X_50 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_35 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_50 ) @ Y_35 ) ) ) ).
thf(fact_369_power2__sum,axiom,
! [X_50: complex,Y_35: complex] :
( ( power_power_complex @ ( plus_plus_complex @ X_50 @ Y_35 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X_50 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_complex @ Y_35 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( number528085621omplex @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_50 ) @ Y_35 ) ) ) ).
thf(fact_370_power2__sum,axiom,
! [X_50: rat,Y_35: rat] :
( ( power_power_rat @ ( plus_plus_rat @ X_50 @ Y_35 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X_50 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_rat @ Y_35 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( number_number_of_rat @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_50 ) @ Y_35 ) ) ) ).
thf(fact_371_number__of__Bit0,axiom,
! [W_8: int] :
( ( number_number_of_int @ ( bit0 @ W_8 ) )
= ( plus_plus_int @ ( plus_plus_int @ zero_zero_int @ ( number_number_of_int @ W_8 ) ) @ ( number_number_of_int @ W_8 ) ) ) ).
thf(fact_372_number__of__Bit0,axiom,
! [W_8: int] :
( ( number267125858f_real @ ( bit0 @ W_8 ) )
= ( plus_plus_real @ ( plus_plus_real @ zero_zero_real @ ( number267125858f_real @ W_8 ) ) @ ( number267125858f_real @ W_8 ) ) ) ).
thf(fact_373_number__of__Bit0,axiom,
! [W_8: int] :
( ( number528085621omplex @ ( bit0 @ W_8 ) )
= ( plus_plus_complex @ ( plus_plus_complex @ zero_zero_complex @ ( number528085621omplex @ W_8 ) ) @ ( number528085621omplex @ W_8 ) ) ) ).
thf(fact_374_number__of__Bit0,axiom,
! [W_8: int] :
( ( number_number_of_rat @ ( bit0 @ W_8 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ zero_zero_rat @ ( number_number_of_rat @ W_8 ) ) @ ( number_number_of_rat @ W_8 ) ) ) ).
thf(fact_375_number__of__Bit1,axiom,
! [W_7: int] :
( ( number_number_of_int @ ( bit1 @ W_7 ) )
= ( plus_plus_int @ ( plus_plus_int @ one_one_int @ ( number_number_of_int @ W_7 ) ) @ ( number_number_of_int @ W_7 ) ) ) ).
thf(fact_376_number__of__Bit1,axiom,
! [W_7: int] :
( ( number267125858f_real @ ( bit1 @ W_7 ) )
= ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( number267125858f_real @ W_7 ) ) @ ( number267125858f_real @ W_7 ) ) ) ).
thf(fact_377_number__of__Bit1,axiom,
! [W_7: int] :
( ( number528085621omplex @ ( bit1 @ W_7 ) )
= ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ ( number528085621omplex @ W_7 ) ) @ ( number528085621omplex @ W_7 ) ) ) ).
thf(fact_378_number__of__Bit1,axiom,
! [W_7: int] :
( ( number_number_of_rat @ ( bit1 @ W_7 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ ( number_number_of_rat @ W_7 ) ) @ ( number_number_of_rat @ W_7 ) ) ) ).
thf(fact_379_semiring__numeral__1__eq__1,axiom,
( ( number_number_of_int @ ( bit1 @ pls ) )
= one_one_int ) ).
thf(fact_380_semiring__numeral__1__eq__1,axiom,
( ( number_number_of_nat @ ( bit1 @ pls ) )
= one_one_nat ) ).
thf(fact_381_semiring__numeral__1__eq__1,axiom,
( ( number267125858f_real @ ( bit1 @ pls ) )
= one_one_real ) ).
thf(fact_382_semiring__numeral__1__eq__1,axiom,
( ( number528085621omplex @ ( bit1 @ pls ) )
= one_one_complex ) ).
thf(fact_383_semiring__numeral__1__eq__1,axiom,
( ( number_number_of_rat @ ( bit1 @ pls ) )
= one_one_rat ) ).
thf(fact_384_numeral__1__eq__1,axiom,
( ( number_number_of_int @ ( bit1 @ pls ) )
= one_one_int ) ).
thf(fact_385_numeral__1__eq__1,axiom,
( ( number267125858f_real @ ( bit1 @ pls ) )
= one_one_real ) ).
thf(fact_386_numeral__1__eq__1,axiom,
( ( number528085621omplex @ ( bit1 @ pls ) )
= one_one_complex ) ).
thf(fact_387_numeral__1__eq__1,axiom,
( ( number_number_of_rat @ ( bit1 @ pls ) )
= one_one_rat ) ).
thf(fact_388_semiring__norm_I110_J,axiom,
( one_one_int
= ( number_number_of_int @ ( bit1 @ pls ) ) ) ).
thf(fact_389_semiring__norm_I110_J,axiom,
( one_one_real
= ( number267125858f_real @ ( bit1 @ pls ) ) ) ).
thf(fact_390_semiring__norm_I110_J,axiom,
( one_one_complex
= ( number528085621omplex @ ( bit1 @ pls ) ) ) ).
thf(fact_391_semiring__norm_I110_J,axiom,
( one_one_rat
= ( number_number_of_rat @ ( bit1 @ pls ) ) ) ).
thf(fact_392_odd__less__0,axiom,
! [Z_1: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z_1 ) @ Z_1 ) @ zero_zero_int )
<=> ( ord_less_int @ Z_1 @ zero_zero_int ) ) ).
thf(fact_393_less__special_I3_J,axiom,
! [X_49: int] :
( ( ord_less_int @ ( number_number_of_int @ X_49 ) @ zero_zero_int )
<=> ( ord_less_int @ X_49 @ pls ) ) ).
thf(fact_394_less__special_I3_J,axiom,
! [X_49: int] :
( ( ord_less_real @ ( number267125858f_real @ X_49 ) @ zero_zero_real )
<=> ( ord_less_int @ X_49 @ pls ) ) ).
thf(fact_395_less__special_I3_J,axiom,
! [X_49: int] :
( ( ord_less_rat @ ( number_number_of_rat @ X_49 ) @ zero_zero_rat )
<=> ( ord_less_int @ X_49 @ pls ) ) ).
thf(fact_396_less__special_I1_J,axiom,
! [Y_34: int] :
( ( ord_less_int @ zero_zero_int @ ( number_number_of_int @ Y_34 ) )
<=> ( ord_less_int @ pls @ Y_34 ) ) ).
thf(fact_397_less__special_I1_J,axiom,
! [Y_34: int] :
( ( ord_less_real @ zero_zero_real @ ( number267125858f_real @ Y_34 ) )
<=> ( ord_less_int @ pls @ Y_34 ) ) ).
thf(fact_398_less__special_I1_J,axiom,
! [Y_34: int] :
( ( ord_less_rat @ zero_zero_rat @ ( number_number_of_rat @ Y_34 ) )
<=> ( ord_less_int @ pls @ Y_34 ) ) ).
thf(fact_399_semiring__one__add__one__is__two,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_400_semiring__one__add__one__is__two,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_401_semiring__one__add__one__is__two,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_402_semiring__one__add__one__is__two,axiom,
( ( plus_plus_complex @ one_one_complex @ one_one_complex )
= ( number528085621omplex @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_403_semiring__one__add__one__is__two,axiom,
( ( plus_plus_rat @ one_one_rat @ one_one_rat )
= ( number_number_of_rat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_404_one__add__one__is__two,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_405_one__add__one__is__two,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_406_one__add__one__is__two,axiom,
( ( plus_plus_complex @ one_one_complex @ one_one_complex )
= ( number528085621omplex @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_407_one__add__one__is__two,axiom,
( ( plus_plus_rat @ one_one_rat @ one_one_rat )
= ( number_number_of_rat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_408_less__special_I4_J,axiom,
! [X_48: int] :
( ( ord_less_int @ ( number_number_of_int @ X_48 ) @ one_one_int )
<=> ( ord_less_int @ X_48 @ ( bit1 @ pls ) ) ) ).
thf(fact_409_less__special_I4_J,axiom,
! [X_48: int] :
( ( ord_less_real @ ( number267125858f_real @ X_48 ) @ one_one_real )
<=> ( ord_less_int @ X_48 @ ( bit1 @ pls ) ) ) ).
thf(fact_410_less__special_I4_J,axiom,
! [X_48: int] :
( ( ord_less_rat @ ( number_number_of_rat @ X_48 ) @ one_one_rat )
<=> ( ord_less_int @ X_48 @ ( bit1 @ pls ) ) ) ).
thf(fact_411_less__special_I2_J,axiom,
! [Y_33: int] :
( ( ord_less_int @ one_one_int @ ( number_number_of_int @ Y_33 ) )
<=> ( ord_less_int @ ( bit1 @ pls ) @ Y_33 ) ) ).
thf(fact_412_less__special_I2_J,axiom,
! [Y_33: int] :
( ( ord_less_real @ one_one_real @ ( number267125858f_real @ Y_33 ) )
<=> ( ord_less_int @ ( bit1 @ pls ) @ Y_33 ) ) ).
thf(fact_413_less__special_I2_J,axiom,
! [Y_33: int] :
( ( ord_less_rat @ one_one_rat @ ( number_number_of_rat @ Y_33 ) )
<=> ( ord_less_int @ ( bit1 @ pls ) @ Y_33 ) ) ).
thf(fact_414__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_L_A1_096,axiom,
dvd_dvd_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( plus_plus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_int ) ).
thf(fact_415_zadd__power3,axiom,
! [A: int,B: int] :
( ( power_power_int @ ( plus_plus_int @ A @ B ) @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ B ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) @ A ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_416_zadd__power2,axiom,
! [A: int,B: int] :
( ( power_power_int @ ( plus_plus_int @ A @ B ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) @ B ) ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_417_int__pos__lt__two__imp__zero__or__one,axiom,
! [X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
=> ( ( X = zero_zero_int )
| ( X = one_one_int ) ) ) ) ).
thf(fact_418_s0p,axiom,
( ( ord_less_eq_int @ zero_zero_int @ s )
& ( ord_less_int @ s @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
& ( zcong @ s1 @ s @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ) ) ).
thf(fact_419_cube__square,axiom,
! [A: int] :
( ( times_times_int @ A @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( power_power_int @ A @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_420_power2__ge__self,axiom,
! [X: int] : ( ord_less_eq_int @ X @ ( power_power_int @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_421_self__quotient__aux1,axiom,
! [R_1: int,Q: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( A
= ( plus_plus_int @ R_1 @ ( times_times_int @ A @ Q ) ) )
=> ( ( ord_less_int @ R_1 @ A )
=> ( ord_less_eq_int @ one_one_int @ Q ) ) ) ) ).
thf(fact_422_self__quotient__aux2,axiom,
! [R_1: int,Q: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( A
= ( plus_plus_int @ R_1 @ ( times_times_int @ A @ Q ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ord_less_eq_int @ Q @ one_one_int ) ) ) ) ).
thf(fact_423_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J,axiom,
ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_424_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_47: code_code_numeral,N_74: nat] :
( ( power_2100829034umeral @ X_47 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_74 ) )
= ( times_1655362735umeral @ ( power_2100829034umeral @ X_47 @ N_74 ) @ ( power_2100829034umeral @ X_47 @ N_74 ) ) ) ).
thf(fact_425_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_47: quickcheck_code_int,N_74: nat] :
( ( power_881366806de_int @ X_47 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_74 ) )
= ( times_123202395de_int @ ( power_881366806de_int @ X_47 @ N_74 ) @ ( power_881366806de_int @ X_47 @ N_74 ) ) ) ).
thf(fact_426_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_47: rat,N_74: nat] :
( ( power_power_rat @ X_47 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_74 ) )
= ( times_times_rat @ ( power_power_rat @ X_47 @ N_74 ) @ ( power_power_rat @ X_47 @ N_74 ) ) ) ).
thf(fact_427_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_47: int,N_74: nat] :
( ( power_power_int @ X_47 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_74 ) )
= ( times_times_int @ ( power_power_int @ X_47 @ N_74 ) @ ( power_power_int @ X_47 @ N_74 ) ) ) ).
thf(fact_428_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_47: nat,N_74: nat] :
( ( power_power_nat @ X_47 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_74 ) )
= ( times_times_nat @ ( power_power_nat @ X_47 @ N_74 ) @ ( power_power_nat @ X_47 @ N_74 ) ) ) ).
thf(fact_429_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_47: real,N_74: nat] :
( ( power_power_real @ X_47 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_74 ) )
= ( times_times_real @ ( power_power_real @ X_47 @ N_74 ) @ ( power_power_real @ X_47 @ N_74 ) ) ) ).
thf(fact_430_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_47: complex,N_74: nat] :
( ( power_power_complex @ X_47 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_74 ) )
= ( times_times_complex @ ( power_power_complex @ X_47 @ N_74 ) @ ( power_power_complex @ X_47 @ N_74 ) ) ) ).
thf(fact_431_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_46: code_code_numeral] :
( ( times_1655362735umeral @ X_46 @ X_46 )
= ( power_2100829034umeral @ X_46 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_432_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_46: quickcheck_code_int] :
( ( times_123202395de_int @ X_46 @ X_46 )
= ( power_881366806de_int @ X_46 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_433_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_46: rat] :
( ( times_times_rat @ X_46 @ X_46 )
= ( power_power_rat @ X_46 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_434_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_46: int] :
( ( times_times_int @ X_46 @ X_46 )
= ( power_power_int @ X_46 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_435_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_46: nat] :
( ( times_times_nat @ X_46 @ X_46 )
= ( power_power_nat @ X_46 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_436_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_46: real] :
( ( times_times_real @ X_46 @ X_46 )
= ( power_power_real @ X_46 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_437_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_46: complex] :
( ( times_times_complex @ X_46 @ X_46 )
= ( power_power_complex @ X_46 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_438__096_091s_A_094_A2_A_061_As1_A_094_A2_093_A_Imod_A4_A_K_Am_A_L_A1_J_096,axiom,
zcong @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ s1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_439__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s1,axiom,
? [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
& ( ord_less_int @ X_1 @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
& ( zcong @ s1 @ X_1 @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
& ! [Y_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Y_1 )
& ( ord_less_int @ Y_1 @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
& ( zcong @ s1 @ Y_1 @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ) )
=> ( Y_1 = X_1 ) ) ) ).
thf(fact_440__096_B_Bthesis_O_A_I_B_Bs_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_,axiom,
~ ! [S_2: int] :
~ ( ( ord_less_eq_int @ zero_zero_int @ S_2 )
& ( ord_less_int @ S_2 @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
& ( zcong @ s1 @ S_2 @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ) ) ).
thf(fact_441_s1,axiom,
zcong @ ( power_power_int @ s1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of_int @ min ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_442_zero__less__power__nat__eq,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
<=> ( ( N = zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ X ) ) ) ).
thf(fact_443_zprime__zdvd__power,axiom,
! [A: int,N: nat,P_3: int] :
( ( zprime @ P_3 )
=> ( ( dvd_dvd_int @ P_3 @ ( power_power_int @ A @ N ) )
=> ( dvd_dvd_int @ P_3 @ A ) ) ) ).
thf(fact_444_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_45: quickcheck_code_int,P_8: nat,Q_10: nat] :
( ( power_881366806de_int @ ( power_881366806de_int @ X_45 @ P_8 ) @ Q_10 )
= ( power_881366806de_int @ X_45 @ ( times_times_nat @ P_8 @ Q_10 ) ) ) ).
thf(fact_445_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_45: code_code_numeral,P_8: nat,Q_10: nat] :
( ( power_2100829034umeral @ ( power_2100829034umeral @ X_45 @ P_8 ) @ Q_10 )
= ( power_2100829034umeral @ X_45 @ ( times_times_nat @ P_8 @ Q_10 ) ) ) ).
thf(fact_446_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_45: rat,P_8: nat,Q_10: nat] :
( ( power_power_rat @ ( power_power_rat @ X_45 @ P_8 ) @ Q_10 )
= ( power_power_rat @ X_45 @ ( times_times_nat @ P_8 @ Q_10 ) ) ) ).
thf(fact_447_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_45: int,P_8: nat,Q_10: nat] :
( ( power_power_int @ ( power_power_int @ X_45 @ P_8 ) @ Q_10 )
= ( power_power_int @ X_45 @ ( times_times_nat @ P_8 @ Q_10 ) ) ) ).
thf(fact_448_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_45: nat,P_8: nat,Q_10: nat] :
( ( power_power_nat @ ( power_power_nat @ X_45 @ P_8 ) @ Q_10 )
= ( power_power_nat @ X_45 @ ( times_times_nat @ P_8 @ Q_10 ) ) ) ).
thf(fact_449_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_45: real,P_8: nat,Q_10: nat] :
( ( power_power_real @ ( power_power_real @ X_45 @ P_8 ) @ Q_10 )
= ( power_power_real @ X_45 @ ( times_times_nat @ P_8 @ Q_10 ) ) ) ).
thf(fact_450_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_45: complex,P_8: nat,Q_10: nat] :
( ( power_power_complex @ ( power_power_complex @ X_45 @ P_8 ) @ Q_10 )
= ( power_power_complex @ X_45 @ ( times_times_nat @ P_8 @ Q_10 ) ) ) ).
thf(fact_451_zprime__power__zdvd__cancel__right,axiom,
! [N: nat,A: int,B: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ~ ( dvd_dvd_int @ P_3 @ B )
=> ( ( dvd_dvd_int @ ( power_power_int @ P_3 @ N ) @ ( times_times_int @ A @ B ) )
=> ( dvd_dvd_int @ ( power_power_int @ P_3 @ N ) @ A ) ) ) ) ).
thf(fact_452_zprime__power__zdvd__cancel__left,axiom,
! [N: nat,B: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ~ ( dvd_dvd_int @ P_3 @ A )
=> ( ( dvd_dvd_int @ ( power_power_int @ P_3 @ N ) @ ( times_times_int @ A @ B ) )
=> ( dvd_dvd_int @ ( power_power_int @ P_3 @ N ) @ B ) ) ) ) ).
thf(fact_453_zpower__zpower,axiom,
! [X: int,Y: nat,Z_1: nat] :
( ( power_power_int @ ( power_power_int @ X @ Y ) @ Z_1 )
= ( power_power_int @ X @ ( times_times_nat @ Y @ Z_1 ) ) ) ).
thf(fact_454_zdvd__not__zless,axiom,
! [N: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_int @ M @ N )
=> ~ ( dvd_dvd_int @ N @ M ) ) ) ).
thf(fact_455_zdvd__antisym__nonneg,axiom,
! [N: int,M: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( dvd_dvd_int @ M @ N )
=> ( ( dvd_dvd_int @ N @ M )
=> ( M = N ) ) ) ) ) ).
thf(fact_456_zdvd__mult__cancel,axiom,
! [K_1: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K_1 @ M ) @ ( times_times_int @ K_1 @ N ) )
=> ( ( K_1 != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
thf(fact_457_zdvd__reduce,axiom,
! [K_1: int,N: int,M: int] :
( ( dvd_dvd_int @ K_1 @ ( plus_plus_int @ N @ ( times_times_int @ K_1 @ M ) ) )
<=> ( dvd_dvd_int @ K_1 @ N ) ) ).
thf(fact_458_zdvd__period,axiom,
! [C: int,X: int,T: int,A: int,D: int] :
( ( dvd_dvd_int @ A @ D )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
<=> ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
thf(fact_459_zprime__2,axiom,
zprime @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_460_zdvd__imp__le,axiom,
! [Z_1: int,N: int] :
( ( dvd_dvd_int @ Z_1 @ N )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ Z_1 @ N ) ) ) ).
thf(fact_461_is__mult__sum2sq,axiom,
! [Y: int,X: int] :
( ( twoSqu1152398899sum2sq @ X )
=> ( ( twoSqu1152398899sum2sq @ Y )
=> ( twoSqu1152398899sum2sq @ ( times_times_int @ X @ Y ) ) ) ) ).
thf(fact_462_le__nat__number__of,axiom,
! [V: int,V_1: int] :
( ( ord_less_eq_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
<=> ( ~ ( ord_less_eq_int @ V @ V_1 )
=> ( ord_less_eq_int @ V @ pls ) ) ) ).
thf(fact_463_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: code_code_numeral,Ly_4: code_code_numeral,Rx_6: code_code_numeral,Ry_4: code_code_numeral] :
( ( times_1655362735umeral @ ( times_1655362735umeral @ Lx_6 @ Ly_4 ) @ ( times_1655362735umeral @ Rx_6 @ Ry_4 ) )
= ( times_1655362735umeral @ ( times_1655362735umeral @ Lx_6 @ Rx_6 ) @ ( times_1655362735umeral @ Ly_4 @ Ry_4 ) ) ) ).
thf(fact_464_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: int,Ly_4: int,Rx_6: int,Ry_4: int] :
( ( times_times_int @ ( times_times_int @ Lx_6 @ Ly_4 ) @ ( times_times_int @ Rx_6 @ Ry_4 ) )
= ( times_times_int @ ( times_times_int @ Lx_6 @ Rx_6 ) @ ( times_times_int @ Ly_4 @ Ry_4 ) ) ) ).
thf(fact_465_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: nat,Ly_4: nat,Rx_6: nat,Ry_4: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_6 @ Ly_4 ) @ ( times_times_nat @ Rx_6 @ Ry_4 ) )
= ( times_times_nat @ ( times_times_nat @ Lx_6 @ Rx_6 ) @ ( times_times_nat @ Ly_4 @ Ry_4 ) ) ) ).
thf(fact_466_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: real,Ly_4: real,Rx_6: real,Ry_4: real] :
( ( times_times_real @ ( times_times_real @ Lx_6 @ Ly_4 ) @ ( times_times_real @ Rx_6 @ Ry_4 ) )
= ( times_times_real @ ( times_times_real @ Lx_6 @ Rx_6 ) @ ( times_times_real @ Ly_4 @ Ry_4 ) ) ) ).
thf(fact_467_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: complex,Ly_4: complex,Rx_6: complex,Ry_4: complex] :
( ( times_times_complex @ ( times_times_complex @ Lx_6 @ Ly_4 ) @ ( times_times_complex @ Rx_6 @ Ry_4 ) )
= ( times_times_complex @ ( times_times_complex @ Lx_6 @ Rx_6 ) @ ( times_times_complex @ Ly_4 @ Ry_4 ) ) ) ).
thf(fact_468_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: quickcheck_code_int,Ly_4: quickcheck_code_int,Rx_6: quickcheck_code_int,Ry_4: quickcheck_code_int] :
( ( times_123202395de_int @ ( times_123202395de_int @ Lx_6 @ Ly_4 ) @ ( times_123202395de_int @ Rx_6 @ Ry_4 ) )
= ( times_123202395de_int @ ( times_123202395de_int @ Lx_6 @ Rx_6 ) @ ( times_123202395de_int @ Ly_4 @ Ry_4 ) ) ) ).
thf(fact_469_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx_6: rat,Ly_4: rat,Rx_6: rat,Ry_4: rat] :
( ( times_times_rat @ ( times_times_rat @ Lx_6 @ Ly_4 ) @ ( times_times_rat @ Rx_6 @ Ry_4 ) )
= ( times_times_rat @ ( times_times_rat @ Lx_6 @ Rx_6 ) @ ( times_times_rat @ Ly_4 @ Ry_4 ) ) ) ).
thf(fact_470_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: code_code_numeral,Ly_3: code_code_numeral,Rx_5: code_code_numeral,Ry_3: code_code_numeral] :
( ( times_1655362735umeral @ ( times_1655362735umeral @ Lx_5 @ Ly_3 ) @ ( times_1655362735umeral @ Rx_5 @ Ry_3 ) )
= ( times_1655362735umeral @ Rx_5 @ ( times_1655362735umeral @ ( times_1655362735umeral @ Lx_5 @ Ly_3 ) @ Ry_3 ) ) ) ).
thf(fact_471_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: int,Ly_3: int,Rx_5: int,Ry_3: int] :
( ( times_times_int @ ( times_times_int @ Lx_5 @ Ly_3 ) @ ( times_times_int @ Rx_5 @ Ry_3 ) )
= ( times_times_int @ Rx_5 @ ( times_times_int @ ( times_times_int @ Lx_5 @ Ly_3 ) @ Ry_3 ) ) ) ).
thf(fact_472_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: nat,Ly_3: nat,Rx_5: nat,Ry_3: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_5 @ Ly_3 ) @ ( times_times_nat @ Rx_5 @ Ry_3 ) )
= ( times_times_nat @ Rx_5 @ ( times_times_nat @ ( times_times_nat @ Lx_5 @ Ly_3 ) @ Ry_3 ) ) ) ).
thf(fact_473_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: real,Ly_3: real,Rx_5: real,Ry_3: real] :
( ( times_times_real @ ( times_times_real @ Lx_5 @ Ly_3 ) @ ( times_times_real @ Rx_5 @ Ry_3 ) )
= ( times_times_real @ Rx_5 @ ( times_times_real @ ( times_times_real @ Lx_5 @ Ly_3 ) @ Ry_3 ) ) ) ).
thf(fact_474_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: complex,Ly_3: complex,Rx_5: complex,Ry_3: complex] :
( ( times_times_complex @ ( times_times_complex @ Lx_5 @ Ly_3 ) @ ( times_times_complex @ Rx_5 @ Ry_3 ) )
= ( times_times_complex @ Rx_5 @ ( times_times_complex @ ( times_times_complex @ Lx_5 @ Ly_3 ) @ Ry_3 ) ) ) ).
thf(fact_475_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: quickcheck_code_int,Ly_3: quickcheck_code_int,Rx_5: quickcheck_code_int,Ry_3: quickcheck_code_int] :
( ( times_123202395de_int @ ( times_123202395de_int @ Lx_5 @ Ly_3 ) @ ( times_123202395de_int @ Rx_5 @ Ry_3 ) )
= ( times_123202395de_int @ Rx_5 @ ( times_123202395de_int @ ( times_123202395de_int @ Lx_5 @ Ly_3 ) @ Ry_3 ) ) ) ).
thf(fact_476_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_5: rat,Ly_3: rat,Rx_5: rat,Ry_3: rat] :
( ( times_times_rat @ ( times_times_rat @ Lx_5 @ Ly_3 ) @ ( times_times_rat @ Rx_5 @ Ry_3 ) )
= ( times_times_rat @ Rx_5 @ ( times_times_rat @ ( times_times_rat @ Lx_5 @ Ly_3 ) @ Ry_3 ) ) ) ).
thf(fact_477_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: code_code_numeral,Ly_2: code_code_numeral,Rx_4: code_code_numeral,Ry_2: code_code_numeral] :
( ( times_1655362735umeral @ ( times_1655362735umeral @ Lx_4 @ Ly_2 ) @ ( times_1655362735umeral @ Rx_4 @ Ry_2 ) )
= ( times_1655362735umeral @ Lx_4 @ ( times_1655362735umeral @ Ly_2 @ ( times_1655362735umeral @ Rx_4 @ Ry_2 ) ) ) ) ).
thf(fact_478_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: int,Ly_2: int,Rx_4: int,Ry_2: int] :
( ( times_times_int @ ( times_times_int @ Lx_4 @ Ly_2 ) @ ( times_times_int @ Rx_4 @ Ry_2 ) )
= ( times_times_int @ Lx_4 @ ( times_times_int @ Ly_2 @ ( times_times_int @ Rx_4 @ Ry_2 ) ) ) ) ).
thf(fact_479_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: nat,Ly_2: nat,Rx_4: nat,Ry_2: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_4 @ Ly_2 ) @ ( times_times_nat @ Rx_4 @ Ry_2 ) )
= ( times_times_nat @ Lx_4 @ ( times_times_nat @ Ly_2 @ ( times_times_nat @ Rx_4 @ Ry_2 ) ) ) ) ).
thf(fact_480_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: real,Ly_2: real,Rx_4: real,Ry_2: real] :
( ( times_times_real @ ( times_times_real @ Lx_4 @ Ly_2 ) @ ( times_times_real @ Rx_4 @ Ry_2 ) )
= ( times_times_real @ Lx_4 @ ( times_times_real @ Ly_2 @ ( times_times_real @ Rx_4 @ Ry_2 ) ) ) ) ).
thf(fact_481_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: complex,Ly_2: complex,Rx_4: complex,Ry_2: complex] :
( ( times_times_complex @ ( times_times_complex @ Lx_4 @ Ly_2 ) @ ( times_times_complex @ Rx_4 @ Ry_2 ) )
= ( times_times_complex @ Lx_4 @ ( times_times_complex @ Ly_2 @ ( times_times_complex @ Rx_4 @ Ry_2 ) ) ) ) ).
thf(fact_482_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: quickcheck_code_int,Ly_2: quickcheck_code_int,Rx_4: quickcheck_code_int,Ry_2: quickcheck_code_int] :
( ( times_123202395de_int @ ( times_123202395de_int @ Lx_4 @ Ly_2 ) @ ( times_123202395de_int @ Rx_4 @ Ry_2 ) )
= ( times_123202395de_int @ Lx_4 @ ( times_123202395de_int @ Ly_2 @ ( times_123202395de_int @ Rx_4 @ Ry_2 ) ) ) ) ).
thf(fact_483_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_4: rat,Ly_2: rat,Rx_4: rat,Ry_2: rat] :
( ( times_times_rat @ ( times_times_rat @ Lx_4 @ Ly_2 ) @ ( times_times_rat @ Rx_4 @ Ry_2 ) )
= ( times_times_rat @ Lx_4 @ ( times_times_rat @ Ly_2 @ ( times_times_rat @ Rx_4 @ Ry_2 ) ) ) ) ).
thf(fact_484_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: code_code_numeral,Ly_1: code_code_numeral,Rx_3: code_code_numeral] :
( ( times_1655362735umeral @ ( times_1655362735umeral @ Lx_3 @ Ly_1 ) @ Rx_3 )
= ( times_1655362735umeral @ ( times_1655362735umeral @ Lx_3 @ Rx_3 ) @ Ly_1 ) ) ).
thf(fact_485_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: int,Ly_1: int,Rx_3: int] :
( ( times_times_int @ ( times_times_int @ Lx_3 @ Ly_1 ) @ Rx_3 )
= ( times_times_int @ ( times_times_int @ Lx_3 @ Rx_3 ) @ Ly_1 ) ) ).
thf(fact_486_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: nat,Ly_1: nat,Rx_3: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_3 @ Ly_1 ) @ Rx_3 )
= ( times_times_nat @ ( times_times_nat @ Lx_3 @ Rx_3 ) @ Ly_1 ) ) ).
thf(fact_487_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: real,Ly_1: real,Rx_3: real] :
( ( times_times_real @ ( times_times_real @ Lx_3 @ Ly_1 ) @ Rx_3 )
= ( times_times_real @ ( times_times_real @ Lx_3 @ Rx_3 ) @ Ly_1 ) ) ).
thf(fact_488_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: complex,Ly_1: complex,Rx_3: complex] :
( ( times_times_complex @ ( times_times_complex @ Lx_3 @ Ly_1 ) @ Rx_3 )
= ( times_times_complex @ ( times_times_complex @ Lx_3 @ Rx_3 ) @ Ly_1 ) ) ).
thf(fact_489_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: quickcheck_code_int,Ly_1: quickcheck_code_int,Rx_3: quickcheck_code_int] :
( ( times_123202395de_int @ ( times_123202395de_int @ Lx_3 @ Ly_1 ) @ Rx_3 )
= ( times_123202395de_int @ ( times_123202395de_int @ Lx_3 @ Rx_3 ) @ Ly_1 ) ) ).
thf(fact_490_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: rat,Ly_1: rat,Rx_3: rat] :
( ( times_times_rat @ ( times_times_rat @ Lx_3 @ Ly_1 ) @ Rx_3 )
= ( times_times_rat @ ( times_times_rat @ Lx_3 @ Rx_3 ) @ Ly_1 ) ) ).
thf(fact_491_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: code_code_numeral,Ly: code_code_numeral,Rx_2: code_code_numeral] :
( ( times_1655362735umeral @ ( times_1655362735umeral @ Lx_2 @ Ly ) @ Rx_2 )
= ( times_1655362735umeral @ Lx_2 @ ( times_1655362735umeral @ Ly @ Rx_2 ) ) ) ).
thf(fact_492_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: int,Ly: int,Rx_2: int] :
( ( times_times_int @ ( times_times_int @ Lx_2 @ Ly ) @ Rx_2 )
= ( times_times_int @ Lx_2 @ ( times_times_int @ Ly @ Rx_2 ) ) ) ).
thf(fact_493_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: nat,Ly: nat,Rx_2: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_2 @ Ly ) @ Rx_2 )
= ( times_times_nat @ Lx_2 @ ( times_times_nat @ Ly @ Rx_2 ) ) ) ).
thf(fact_494_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: real,Ly: real,Rx_2: real] :
( ( times_times_real @ ( times_times_real @ Lx_2 @ Ly ) @ Rx_2 )
= ( times_times_real @ Lx_2 @ ( times_times_real @ Ly @ Rx_2 ) ) ) ).
thf(fact_495_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: complex,Ly: complex,Rx_2: complex] :
( ( times_times_complex @ ( times_times_complex @ Lx_2 @ Ly ) @ Rx_2 )
= ( times_times_complex @ Lx_2 @ ( times_times_complex @ Ly @ Rx_2 ) ) ) ).
thf(fact_496_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: quickcheck_code_int,Ly: quickcheck_code_int,Rx_2: quickcheck_code_int] :
( ( times_123202395de_int @ ( times_123202395de_int @ Lx_2 @ Ly ) @ Rx_2 )
= ( times_123202395de_int @ Lx_2 @ ( times_123202395de_int @ Ly @ Rx_2 ) ) ) ).
thf(fact_497_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_2: rat,Ly: rat,Rx_2: rat] :
( ( times_times_rat @ ( times_times_rat @ Lx_2 @ Ly ) @ Rx_2 )
= ( times_times_rat @ Lx_2 @ ( times_times_rat @ Ly @ Rx_2 ) ) ) ).
thf(fact_498_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: code_code_numeral,Rx_1: code_code_numeral,Ry_1: code_code_numeral] :
( ( times_1655362735umeral @ Lx_1 @ ( times_1655362735umeral @ Rx_1 @ Ry_1 ) )
= ( times_1655362735umeral @ ( times_1655362735umeral @ Lx_1 @ Rx_1 ) @ Ry_1 ) ) ).
thf(fact_499_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: int,Rx_1: int,Ry_1: int] :
( ( times_times_int @ Lx_1 @ ( times_times_int @ Rx_1 @ Ry_1 ) )
= ( times_times_int @ ( times_times_int @ Lx_1 @ Rx_1 ) @ Ry_1 ) ) ).
thf(fact_500_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: nat,Rx_1: nat,Ry_1: nat] :
( ( times_times_nat @ Lx_1 @ ( times_times_nat @ Rx_1 @ Ry_1 ) )
= ( times_times_nat @ ( times_times_nat @ Lx_1 @ Rx_1 ) @ Ry_1 ) ) ).
thf(fact_501_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: real,Rx_1: real,Ry_1: real] :
( ( times_times_real @ Lx_1 @ ( times_times_real @ Rx_1 @ Ry_1 ) )
= ( times_times_real @ ( times_times_real @ Lx_1 @ Rx_1 ) @ Ry_1 ) ) ).
thf(fact_502_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: complex,Rx_1: complex,Ry_1: complex] :
( ( times_times_complex @ Lx_1 @ ( times_times_complex @ Rx_1 @ Ry_1 ) )
= ( times_times_complex @ ( times_times_complex @ Lx_1 @ Rx_1 ) @ Ry_1 ) ) ).
thf(fact_503_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: quickcheck_code_int,Rx_1: quickcheck_code_int,Ry_1: quickcheck_code_int] :
( ( times_123202395de_int @ Lx_1 @ ( times_123202395de_int @ Rx_1 @ Ry_1 ) )
= ( times_123202395de_int @ ( times_123202395de_int @ Lx_1 @ Rx_1 ) @ Ry_1 ) ) ).
thf(fact_504_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_1: rat,Rx_1: rat,Ry_1: rat] :
( ( times_times_rat @ Lx_1 @ ( times_times_rat @ Rx_1 @ Ry_1 ) )
= ( times_times_rat @ ( times_times_rat @ Lx_1 @ Rx_1 ) @ Ry_1 ) ) ).
thf(fact_505_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: code_code_numeral,Rx: code_code_numeral,Ry: code_code_numeral] :
( ( times_1655362735umeral @ Lx @ ( times_1655362735umeral @ Rx @ Ry ) )
= ( times_1655362735umeral @ Rx @ ( times_1655362735umeral @ Lx @ Ry ) ) ) ).
thf(fact_506_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: int,Rx: int,Ry: int] :
( ( times_times_int @ Lx @ ( times_times_int @ Rx @ Ry ) )
= ( times_times_int @ Rx @ ( times_times_int @ Lx @ Ry ) ) ) ).
thf(fact_507_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: nat,Rx: nat,Ry: nat] :
( ( times_times_nat @ Lx @ ( times_times_nat @ Rx @ Ry ) )
= ( times_times_nat @ Rx @ ( times_times_nat @ Lx @ Ry ) ) ) ).
thf(fact_508_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: real,Rx: real,Ry: real] :
( ( times_times_real @ Lx @ ( times_times_real @ Rx @ Ry ) )
= ( times_times_real @ Rx @ ( times_times_real @ Lx @ Ry ) ) ) ).
thf(fact_509_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: complex,Rx: complex,Ry: complex] :
( ( times_times_complex @ Lx @ ( times_times_complex @ Rx @ Ry ) )
= ( times_times_complex @ Rx @ ( times_times_complex @ Lx @ Ry ) ) ) ).
thf(fact_510_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: quickcheck_code_int,Rx: quickcheck_code_int,Ry: quickcheck_code_int] :
( ( times_123202395de_int @ Lx @ ( times_123202395de_int @ Rx @ Ry ) )
= ( times_123202395de_int @ Rx @ ( times_123202395de_int @ Lx @ Ry ) ) ) ).
thf(fact_511_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx: rat,Rx: rat,Ry: rat] :
( ( times_times_rat @ Lx @ ( times_times_rat @ Rx @ Ry ) )
= ( times_times_rat @ Rx @ ( times_times_rat @ Lx @ Ry ) ) ) ).
thf(fact_512_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_276: code_code_numeral,B_202: code_code_numeral] :
( ( times_1655362735umeral @ A_276 @ B_202 )
= ( times_1655362735umeral @ B_202 @ A_276 ) ) ).
thf(fact_513_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_276: int,B_202: int] :
( ( times_times_int @ A_276 @ B_202 )
= ( times_times_int @ B_202 @ A_276 ) ) ).
thf(fact_514_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_276: nat,B_202: nat] :
( ( times_times_nat @ A_276 @ B_202 )
= ( times_times_nat @ B_202 @ A_276 ) ) ).
thf(fact_515_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_276: real,B_202: real] :
( ( times_times_real @ A_276 @ B_202 )
= ( times_times_real @ B_202 @ A_276 ) ) ).
thf(fact_516_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_276: complex,B_202: complex] :
( ( times_times_complex @ A_276 @ B_202 )
= ( times_times_complex @ B_202 @ A_276 ) ) ).
thf(fact_517_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_276: quickcheck_code_int,B_202: quickcheck_code_int] :
( ( times_123202395de_int @ A_276 @ B_202 )
= ( times_123202395de_int @ B_202 @ A_276 ) ) ).
thf(fact_518_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_276: rat,B_202: rat] :
( ( times_times_rat @ A_276 @ B_202 )
= ( times_times_rat @ B_202 @ A_276 ) ) ).
thf(fact_519_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_275: code_code_numeral,C_122: code_code_numeral] :
( ( plus_p1627245867umeral @ A_275 @ C_122 )
= ( plus_p1627245867umeral @ C_122 @ A_275 ) ) ).
thf(fact_520_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_275: int,C_122: int] :
( ( plus_plus_int @ A_275 @ C_122 )
= ( plus_plus_int @ C_122 @ A_275 ) ) ).
thf(fact_521_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_275: nat,C_122: nat] :
( ( plus_plus_nat @ A_275 @ C_122 )
= ( plus_plus_nat @ C_122 @ A_275 ) ) ).
thf(fact_522_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_275: real,C_122: real] :
( ( plus_plus_real @ A_275 @ C_122 )
= ( plus_plus_real @ C_122 @ A_275 ) ) ).
thf(fact_523_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_275: complex,C_122: complex] :
( ( plus_plus_complex @ A_275 @ C_122 )
= ( plus_plus_complex @ C_122 @ A_275 ) ) ).
thf(fact_524_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_275: quickcheck_code_int,C_122: quickcheck_code_int] :
( ( plus_p1446045655de_int @ A_275 @ C_122 )
= ( plus_p1446045655de_int @ C_122 @ A_275 ) ) ).
thf(fact_525_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_275: rat,C_122: rat] :
( ( plus_plus_rat @ A_275 @ C_122 )
= ( plus_plus_rat @ C_122 @ A_275 ) ) ).
thf(fact_526_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_274: code_code_numeral,C_121: code_code_numeral,D_38: code_code_numeral] :
( ( plus_p1627245867umeral @ A_274 @ ( plus_p1627245867umeral @ C_121 @ D_38 ) )
= ( plus_p1627245867umeral @ C_121 @ ( plus_p1627245867umeral @ A_274 @ D_38 ) ) ) ).
thf(fact_527_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_274: int,C_121: int,D_38: int] :
( ( plus_plus_int @ A_274 @ ( plus_plus_int @ C_121 @ D_38 ) )
= ( plus_plus_int @ C_121 @ ( plus_plus_int @ A_274 @ D_38 ) ) ) ).
thf(fact_528_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_274: nat,C_121: nat,D_38: nat] :
( ( plus_plus_nat @ A_274 @ ( plus_plus_nat @ C_121 @ D_38 ) )
= ( plus_plus_nat @ C_121 @ ( plus_plus_nat @ A_274 @ D_38 ) ) ) ).
thf(fact_529_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_274: real,C_121: real,D_38: real] :
( ( plus_plus_real @ A_274 @ ( plus_plus_real @ C_121 @ D_38 ) )
= ( plus_plus_real @ C_121 @ ( plus_plus_real @ A_274 @ D_38 ) ) ) ).
thf(fact_530_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_274: complex,C_121: complex,D_38: complex] :
( ( plus_plus_complex @ A_274 @ ( plus_plus_complex @ C_121 @ D_38 ) )
= ( plus_plus_complex @ C_121 @ ( plus_plus_complex @ A_274 @ D_38 ) ) ) ).
thf(fact_531_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_274: quickcheck_code_int,C_121: quickcheck_code_int,D_38: quickcheck_code_int] :
( ( plus_p1446045655de_int @ A_274 @ ( plus_p1446045655de_int @ C_121 @ D_38 ) )
= ( plus_p1446045655de_int @ C_121 @ ( plus_p1446045655de_int @ A_274 @ D_38 ) ) ) ).
thf(fact_532_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_274: rat,C_121: rat,D_38: rat] :
( ( plus_plus_rat @ A_274 @ ( plus_plus_rat @ C_121 @ D_38 ) )
= ( plus_plus_rat @ C_121 @ ( plus_plus_rat @ A_274 @ D_38 ) ) ) ).
thf(fact_533_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_273: code_code_numeral,C_120: code_code_numeral,D_37: code_code_numeral] :
( ( plus_p1627245867umeral @ A_273 @ ( plus_p1627245867umeral @ C_120 @ D_37 ) )
= ( plus_p1627245867umeral @ ( plus_p1627245867umeral @ A_273 @ C_120 ) @ D_37 ) ) ).
thf(fact_534_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_273: int,C_120: int,D_37: int] :
( ( plus_plus_int @ A_273 @ ( plus_plus_int @ C_120 @ D_37 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_273 @ C_120 ) @ D_37 ) ) ).
thf(fact_535_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_273: nat,C_120: nat,D_37: nat] :
( ( plus_plus_nat @ A_273 @ ( plus_plus_nat @ C_120 @ D_37 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_273 @ C_120 ) @ D_37 ) ) ).
thf(fact_536_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_273: real,C_120: real,D_37: real] :
( ( plus_plus_real @ A_273 @ ( plus_plus_real @ C_120 @ D_37 ) )
= ( plus_plus_real @ ( plus_plus_real @ A_273 @ C_120 ) @ D_37 ) ) ).
thf(fact_537_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_273: complex,C_120: complex,D_37: complex] :
( ( plus_plus_complex @ A_273 @ ( plus_plus_complex @ C_120 @ D_37 ) )
= ( plus_plus_complex @ ( plus_plus_complex @ A_273 @ C_120 ) @ D_37 ) ) ).
thf(fact_538_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_273: quickcheck_code_int,C_120: quickcheck_code_int,D_37: quickcheck_code_int] :
( ( plus_p1446045655de_int @ A_273 @ ( plus_p1446045655de_int @ C_120 @ D_37 ) )
= ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_273 @ C_120 ) @ D_37 ) ) ).
thf(fact_539_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_273: rat,C_120: rat,D_37: rat] :
( ( plus_plus_rat @ A_273 @ ( plus_plus_rat @ C_120 @ D_37 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ A_273 @ C_120 ) @ D_37 ) ) ).
thf(fact_540_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_272: code_code_numeral,B_201: code_code_numeral,C_119: code_code_numeral] :
( ( plus_p1627245867umeral @ ( plus_p1627245867umeral @ A_272 @ B_201 ) @ C_119 )
= ( plus_p1627245867umeral @ A_272 @ ( plus_p1627245867umeral @ B_201 @ C_119 ) ) ) ).
thf(fact_541_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_272: int,B_201: int,C_119: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_272 @ B_201 ) @ C_119 )
= ( plus_plus_int @ A_272 @ ( plus_plus_int @ B_201 @ C_119 ) ) ) ).
thf(fact_542_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_272: nat,B_201: nat,C_119: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_272 @ B_201 ) @ C_119 )
= ( plus_plus_nat @ A_272 @ ( plus_plus_nat @ B_201 @ C_119 ) ) ) ).
thf(fact_543_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_272: real,B_201: real,C_119: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_272 @ B_201 ) @ C_119 )
= ( plus_plus_real @ A_272 @ ( plus_plus_real @ B_201 @ C_119 ) ) ) ).
thf(fact_544_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_272: complex,B_201: complex,C_119: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A_272 @ B_201 ) @ C_119 )
= ( plus_plus_complex @ A_272 @ ( plus_plus_complex @ B_201 @ C_119 ) ) ) ).
thf(fact_545_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_272: quickcheck_code_int,B_201: quickcheck_code_int,C_119: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_272 @ B_201 ) @ C_119 )
= ( plus_p1446045655de_int @ A_272 @ ( plus_p1446045655de_int @ B_201 @ C_119 ) ) ) ).
thf(fact_546_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_272: rat,B_201: rat,C_119: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A_272 @ B_201 ) @ C_119 )
= ( plus_plus_rat @ A_272 @ ( plus_plus_rat @ B_201 @ C_119 ) ) ) ).
thf(fact_547_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_271: code_code_numeral,B_200: code_code_numeral,C_118: code_code_numeral] :
( ( plus_p1627245867umeral @ ( plus_p1627245867umeral @ A_271 @ B_200 ) @ C_118 )
= ( plus_p1627245867umeral @ ( plus_p1627245867umeral @ A_271 @ C_118 ) @ B_200 ) ) ).
thf(fact_548_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_271: int,B_200: int,C_118: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_271 @ B_200 ) @ C_118 )
= ( plus_plus_int @ ( plus_plus_int @ A_271 @ C_118 ) @ B_200 ) ) ).
thf(fact_549_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_271: nat,B_200: nat,C_118: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_271 @ B_200 ) @ C_118 )
= ( plus_plus_nat @ ( plus_plus_nat @ A_271 @ C_118 ) @ B_200 ) ) ).
thf(fact_550_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_271: real,B_200: real,C_118: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_271 @ B_200 ) @ C_118 )
= ( plus_plus_real @ ( plus_plus_real @ A_271 @ C_118 ) @ B_200 ) ) ).
thf(fact_551_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_271: complex,B_200: complex,C_118: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A_271 @ B_200 ) @ C_118 )
= ( plus_plus_complex @ ( plus_plus_complex @ A_271 @ C_118 ) @ B_200 ) ) ).
thf(fact_552_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_271: quickcheck_code_int,B_200: quickcheck_code_int,C_118: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_271 @ B_200 ) @ C_118 )
= ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_271 @ C_118 ) @ B_200 ) ) ).
thf(fact_553_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_271: rat,B_200: rat,C_118: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A_271 @ B_200 ) @ C_118 )
= ( plus_plus_rat @ ( plus_plus_rat @ A_271 @ C_118 ) @ B_200 ) ) ).
thf(fact_554_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_270: code_code_numeral,B_199: code_code_numeral,C_117: code_code_numeral,D_36: code_code_numeral] :
( ( plus_p1627245867umeral @ ( plus_p1627245867umeral @ A_270 @ B_199 ) @ ( plus_p1627245867umeral @ C_117 @ D_36 ) )
= ( plus_p1627245867umeral @ ( plus_p1627245867umeral @ A_270 @ C_117 ) @ ( plus_p1627245867umeral @ B_199 @ D_36 ) ) ) ).
thf(fact_555_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_270: int,B_199: int,C_117: int,D_36: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_270 @ B_199 ) @ ( plus_plus_int @ C_117 @ D_36 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_270 @ C_117 ) @ ( plus_plus_int @ B_199 @ D_36 ) ) ) ).
thf(fact_556_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_270: nat,B_199: nat,C_117: nat,D_36: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_270 @ B_199 ) @ ( plus_plus_nat @ C_117 @ D_36 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_270 @ C_117 ) @ ( plus_plus_nat @ B_199 @ D_36 ) ) ) ).
thf(fact_557_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_270: real,B_199: real,C_117: real,D_36: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_270 @ B_199 ) @ ( plus_plus_real @ C_117 @ D_36 ) )
= ( plus_plus_real @ ( plus_plus_real @ A_270 @ C_117 ) @ ( plus_plus_real @ B_199 @ D_36 ) ) ) ).
thf(fact_558_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_270: complex,B_199: complex,C_117: complex,D_36: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A_270 @ B_199 ) @ ( plus_plus_complex @ C_117 @ D_36 ) )
= ( plus_plus_complex @ ( plus_plus_complex @ A_270 @ C_117 ) @ ( plus_plus_complex @ B_199 @ D_36 ) ) ) ).
thf(fact_559_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_270: quickcheck_code_int,B_199: quickcheck_code_int,C_117: quickcheck_code_int,D_36: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_270 @ B_199 ) @ ( plus_p1446045655de_int @ C_117 @ D_36 ) )
= ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_270 @ C_117 ) @ ( plus_p1446045655de_int @ B_199 @ D_36 ) ) ) ).
thf(fact_560_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_270: rat,B_199: rat,C_117: rat,D_36: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A_270 @ B_199 ) @ ( plus_plus_rat @ C_117 @ D_36 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ A_270 @ C_117 ) @ ( plus_plus_rat @ B_199 @ D_36 ) ) ) ).
thf(fact_561_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_44: quickcheck_code_int] :
( ( power_881366806de_int @ X_44 @ one_one_nat )
= X_44 ) ).
thf(fact_562_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_44: code_code_numeral] :
( ( power_2100829034umeral @ X_44 @ one_one_nat )
= X_44 ) ).
thf(fact_563_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_44: rat] :
( ( power_power_rat @ X_44 @ one_one_nat )
= X_44 ) ).
thf(fact_564_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_44: int] :
( ( power_power_int @ X_44 @ one_one_nat )
= X_44 ) ).
thf(fact_565_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_44: nat] :
( ( power_power_nat @ X_44 @ one_one_nat )
= X_44 ) ).
thf(fact_566_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_44: real] :
( ( power_power_real @ X_44 @ one_one_nat )
= X_44 ) ).
thf(fact_567_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_44: complex] :
( ( power_power_complex @ X_44 @ one_one_nat )
= X_44 ) ).
thf(fact_568_zero__less__power__nat__eq__number__of,axiom,
! [X: nat,W: int] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ ( number_number_of_nat @ W ) ) )
<=> ( ( ( number_number_of_nat @ W )
= zero_zero_nat )
| ( ord_less_nat @ zero_zero_nat @ X ) ) ) ).
thf(fact_569_nat__mult__2__right,axiom,
! [Z_1: nat] :
( ( times_times_nat @ Z_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_nat @ Z_1 @ Z_1 ) ) ).
thf(fact_570_nat__mult__2,axiom,
! [Z_1: nat] :
( ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_1 )
= ( plus_plus_nat @ Z_1 @ Z_1 ) ) ).
thf(fact_571_mult__nat__number__of,axiom,
! [V_1: int,V: int] :
( ( ( ord_less_int @ V @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_int @ V @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
= ( number_number_of_nat @ ( times_times_int @ V @ V_1 ) ) ) ) ) ).
thf(fact_572_nat__number__of__mult__left,axiom,
! [V_1: int,K_1: nat,V: int] :
( ( ( ord_less_int @ V @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V ) @ ( times_times_nat @ ( number_number_of_nat @ V_1 ) @ K_1 ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_int @ V @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V ) @ ( times_times_nat @ ( number_number_of_nat @ V_1 ) @ K_1 ) )
= ( times_times_nat @ ( number_number_of_nat @ ( times_times_int @ V @ V_1 ) ) @ K_1 ) ) ) ) ).
thf(fact_573_power__even__eq,axiom,
! [A_269: quickcheck_code_int,N_73: nat] :
( ( power_881366806de_int @ A_269 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_73 ) )
= ( power_881366806de_int @ ( power_881366806de_int @ A_269 @ N_73 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_574_power__even__eq,axiom,
! [A_269: code_code_numeral,N_73: nat] :
( ( power_2100829034umeral @ A_269 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_73 ) )
= ( power_2100829034umeral @ ( power_2100829034umeral @ A_269 @ N_73 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_575_power__even__eq,axiom,
! [A_269: rat,N_73: nat] :
( ( power_power_rat @ A_269 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_73 ) )
= ( power_power_rat @ ( power_power_rat @ A_269 @ N_73 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_576_power__even__eq,axiom,
! [A_269: int,N_73: nat] :
( ( power_power_int @ A_269 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_73 ) )
= ( power_power_int @ ( power_power_int @ A_269 @ N_73 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_577_power__even__eq,axiom,
! [A_269: nat,N_73: nat] :
( ( power_power_nat @ A_269 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_73 ) )
= ( power_power_nat @ ( power_power_nat @ A_269 @ N_73 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_578_power__even__eq,axiom,
! [A_269: real,N_73: nat] :
( ( power_power_real @ A_269 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_73 ) )
= ( power_power_real @ ( power_power_real @ A_269 @ N_73 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_579_power__even__eq,axiom,
! [A_269: complex,N_73: nat] :
( ( power_power_complex @ A_269 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_73 ) )
= ( power_power_complex @ ( power_power_complex @ A_269 @ N_73 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_580_even__power__le__0__imp__0,axiom,
! [A_268: int,K_8: nat] :
( ( ord_less_eq_int @ ( power_power_int @ A_268 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_8 ) ) @ zero_zero_int )
=> ( A_268 = zero_zero_int ) ) ).
thf(fact_581_even__power__le__0__imp__0,axiom,
! [A_268: real,K_8: nat] :
( ( ord_less_eq_real @ ( power_power_real @ A_268 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_8 ) ) @ zero_zero_real )
=> ( A_268 = zero_zero_real ) ) ).
thf(fact_582_even__power__le__0__imp__0,axiom,
! [A_268: rat,K_8: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A_268 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_8 ) ) @ zero_zero_rat )
=> ( A_268 = zero_zero_rat ) ) ).
thf(fact_583_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_267: int] :
( ( times_times_int @ zero_zero_int @ A_267 )
= zero_zero_int ) ).
thf(fact_584_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_267: nat] :
( ( times_times_nat @ zero_zero_nat @ A_267 )
= zero_zero_nat ) ).
thf(fact_585_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_267: real] :
( ( times_times_real @ zero_zero_real @ A_267 )
= zero_zero_real ) ).
thf(fact_586_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_267: code_code_numeral] :
( ( times_1655362735umeral @ zero_z126310315umeral @ A_267 )
= zero_z126310315umeral ) ).
thf(fact_587_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_267: complex] :
( ( times_times_complex @ zero_zero_complex @ A_267 )
= zero_zero_complex ) ).
thf(fact_588_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_267: quickcheck_code_int] :
( ( times_123202395de_int @ zero_z891286103de_int @ A_267 )
= zero_z891286103de_int ) ).
thf(fact_589_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_267: rat] :
( ( times_times_rat @ zero_zero_rat @ A_267 )
= zero_zero_rat ) ).
thf(fact_590_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_266: int] :
( ( times_times_int @ A_266 @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_591_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_266: nat] :
( ( times_times_nat @ A_266 @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_592_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_266: real] :
( ( times_times_real @ A_266 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_593_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_266: code_code_numeral] :
( ( times_1655362735umeral @ A_266 @ zero_z126310315umeral )
= zero_z126310315umeral ) ).
thf(fact_594_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_266: complex] :
( ( times_times_complex @ A_266 @ zero_zero_complex )
= zero_zero_complex ) ).
thf(fact_595_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_266: quickcheck_code_int] :
( ( times_123202395de_int @ A_266 @ zero_z891286103de_int )
= zero_z891286103de_int ) ).
thf(fact_596_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_266: rat] :
( ( times_times_rat @ A_266 @ zero_zero_rat )
= zero_zero_rat ) ).
thf(fact_597_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_265: code_code_numeral] :
( ( plus_p1627245867umeral @ zero_z126310315umeral @ A_265 )
= A_265 ) ).
thf(fact_598_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_265: int] :
( ( plus_plus_int @ zero_zero_int @ A_265 )
= A_265 ) ).
thf(fact_599_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_265: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_265 )
= A_265 ) ).
thf(fact_600_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_265: real] :
( ( plus_plus_real @ zero_zero_real @ A_265 )
= A_265 ) ).
thf(fact_601_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_265: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A_265 )
= A_265 ) ).
thf(fact_602_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_265: quickcheck_code_int] :
( ( plus_p1446045655de_int @ zero_z891286103de_int @ A_265 )
= A_265 ) ).
thf(fact_603_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_265: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A_265 )
= A_265 ) ).
thf(fact_604_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_264: code_code_numeral] :
( ( plus_p1627245867umeral @ A_264 @ zero_z126310315umeral )
= A_264 ) ).
thf(fact_605_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_264: int] :
( ( plus_plus_int @ A_264 @ zero_zero_int )
= A_264 ) ).
thf(fact_606_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_264: nat] :
( ( plus_plus_nat @ A_264 @ zero_zero_nat )
= A_264 ) ).
thf(fact_607_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_264: real] :
( ( plus_plus_real @ A_264 @ zero_zero_real )
= A_264 ) ).
thf(fact_608_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_264: complex] :
( ( plus_plus_complex @ A_264 @ zero_zero_complex )
= A_264 ) ).
thf(fact_609_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_264: quickcheck_code_int] :
( ( plus_p1446045655de_int @ A_264 @ zero_z891286103de_int )
= A_264 ) ).
thf(fact_610_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_264: rat] :
( ( plus_plus_rat @ A_264 @ zero_zero_rat )
= A_264 ) ).
thf(fact_611_add__0__iff,axiom,
! [B_198: int,A_263: int] :
( ( B_198
= ( plus_plus_int @ B_198 @ A_263 ) )
<=> ( A_263 = zero_zero_int ) ) ).
thf(fact_612_add__0__iff,axiom,
! [B_198: nat,A_263: nat] :
( ( B_198
= ( plus_plus_nat @ B_198 @ A_263 ) )
<=> ( A_263 = zero_zero_nat ) ) ).
thf(fact_613_add__0__iff,axiom,
! [B_198: real,A_263: real] :
( ( B_198
= ( plus_plus_real @ B_198 @ A_263 ) )
<=> ( A_263 = zero_zero_real ) ) ).
thf(fact_614_add__0__iff,axiom,
! [B_198: complex,A_263: complex] :
( ( B_198
= ( plus_plus_complex @ B_198 @ A_263 ) )
<=> ( A_263 = zero_zero_complex ) ) ).
thf(fact_615_add__0__iff,axiom,
! [B_198: rat,A_263: rat] :
( ( B_198
= ( plus_plus_rat @ B_198 @ A_263 ) )
<=> ( A_263 = zero_zero_rat ) ) ).
thf(fact_616_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_43: code_code_numeral,Y_32: code_code_numeral,Z_9: code_code_numeral] :
( ( times_1655362735umeral @ X_43 @ ( plus_p1627245867umeral @ Y_32 @ Z_9 ) )
= ( plus_p1627245867umeral @ ( times_1655362735umeral @ X_43 @ Y_32 ) @ ( times_1655362735umeral @ X_43 @ Z_9 ) ) ) ).
thf(fact_617_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_43: int,Y_32: int,Z_9: int] :
( ( times_times_int @ X_43 @ ( plus_plus_int @ Y_32 @ Z_9 ) )
= ( plus_plus_int @ ( times_times_int @ X_43 @ Y_32 ) @ ( times_times_int @ X_43 @ Z_9 ) ) ) ).
thf(fact_618_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_43: nat,Y_32: nat,Z_9: nat] :
( ( times_times_nat @ X_43 @ ( plus_plus_nat @ Y_32 @ Z_9 ) )
= ( plus_plus_nat @ ( times_times_nat @ X_43 @ Y_32 ) @ ( times_times_nat @ X_43 @ Z_9 ) ) ) ).
thf(fact_619_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_43: real,Y_32: real,Z_9: real] :
( ( times_times_real @ X_43 @ ( plus_plus_real @ Y_32 @ Z_9 ) )
= ( plus_plus_real @ ( times_times_real @ X_43 @ Y_32 ) @ ( times_times_real @ X_43 @ Z_9 ) ) ) ).
thf(fact_620_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_43: complex,Y_32: complex,Z_9: complex] :
( ( times_times_complex @ X_43 @ ( plus_plus_complex @ Y_32 @ Z_9 ) )
= ( plus_plus_complex @ ( times_times_complex @ X_43 @ Y_32 ) @ ( times_times_complex @ X_43 @ Z_9 ) ) ) ).
thf(fact_621_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_43: quickcheck_code_int,Y_32: quickcheck_code_int,Z_9: quickcheck_code_int] :
( ( times_123202395de_int @ X_43 @ ( plus_p1446045655de_int @ Y_32 @ Z_9 ) )
= ( plus_p1446045655de_int @ ( times_123202395de_int @ X_43 @ Y_32 ) @ ( times_123202395de_int @ X_43 @ Z_9 ) ) ) ).
thf(fact_622_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_43: rat,Y_32: rat,Z_9: rat] :
( ( times_times_rat @ X_43 @ ( plus_plus_rat @ Y_32 @ Z_9 ) )
= ( plus_plus_rat @ ( times_times_rat @ X_43 @ Y_32 ) @ ( times_times_rat @ X_43 @ Z_9 ) ) ) ).
thf(fact_623_crossproduct__noteq,axiom,
! [C_116: int,D_35: int,A_262: int,B_197: int] :
( ( ( A_262 != B_197 )
& ( C_116 != D_35 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ A_262 @ C_116 ) @ ( times_times_int @ B_197 @ D_35 ) )
!= ( plus_plus_int @ ( times_times_int @ A_262 @ D_35 ) @ ( times_times_int @ B_197 @ C_116 ) ) ) ) ).
thf(fact_624_crossproduct__noteq,axiom,
! [C_116: nat,D_35: nat,A_262: nat,B_197: nat] :
( ( ( A_262 != B_197 )
& ( C_116 != D_35 ) )
<=> ( ( plus_plus_nat @ ( times_times_nat @ A_262 @ C_116 ) @ ( times_times_nat @ B_197 @ D_35 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A_262 @ D_35 ) @ ( times_times_nat @ B_197 @ C_116 ) ) ) ) ).
thf(fact_625_crossproduct__noteq,axiom,
! [C_116: real,D_35: real,A_262: real,B_197: real] :
( ( ( A_262 != B_197 )
& ( C_116 != D_35 ) )
<=> ( ( plus_plus_real @ ( times_times_real @ A_262 @ C_116 ) @ ( times_times_real @ B_197 @ D_35 ) )
!= ( plus_plus_real @ ( times_times_real @ A_262 @ D_35 ) @ ( times_times_real @ B_197 @ C_116 ) ) ) ) ).
thf(fact_626_crossproduct__noteq,axiom,
! [C_116: complex,D_35: complex,A_262: complex,B_197: complex] :
( ( ( A_262 != B_197 )
& ( C_116 != D_35 ) )
<=> ( ( plus_plus_complex @ ( times_times_complex @ A_262 @ C_116 ) @ ( times_times_complex @ B_197 @ D_35 ) )
!= ( plus_plus_complex @ ( times_times_complex @ A_262 @ D_35 ) @ ( times_times_complex @ B_197 @ C_116 ) ) ) ) ).
thf(fact_627_crossproduct__noteq,axiom,
! [C_116: rat,D_35: rat,A_262: rat,B_197: rat] :
( ( ( A_262 != B_197 )
& ( C_116 != D_35 ) )
<=> ( ( plus_plus_rat @ ( times_times_rat @ A_262 @ C_116 ) @ ( times_times_rat @ B_197 @ D_35 ) )
!= ( plus_plus_rat @ ( times_times_rat @ A_262 @ D_35 ) @ ( times_times_rat @ B_197 @ C_116 ) ) ) ) ).
thf(fact_628_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_261: code_code_numeral,B_196: code_code_numeral,C_115: code_code_numeral] :
( ( times_1655362735umeral @ ( plus_p1627245867umeral @ A_261 @ B_196 ) @ C_115 )
= ( plus_p1627245867umeral @ ( times_1655362735umeral @ A_261 @ C_115 ) @ ( times_1655362735umeral @ B_196 @ C_115 ) ) ) ).
thf(fact_629_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_261: int,B_196: int,C_115: int] :
( ( times_times_int @ ( plus_plus_int @ A_261 @ B_196 ) @ C_115 )
= ( plus_plus_int @ ( times_times_int @ A_261 @ C_115 ) @ ( times_times_int @ B_196 @ C_115 ) ) ) ).
thf(fact_630_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_261: nat,B_196: nat,C_115: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_261 @ B_196 ) @ C_115 )
= ( plus_plus_nat @ ( times_times_nat @ A_261 @ C_115 ) @ ( times_times_nat @ B_196 @ C_115 ) ) ) ).
thf(fact_631_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_261: real,B_196: real,C_115: real] :
( ( times_times_real @ ( plus_plus_real @ A_261 @ B_196 ) @ C_115 )
= ( plus_plus_real @ ( times_times_real @ A_261 @ C_115 ) @ ( times_times_real @ B_196 @ C_115 ) ) ) ).
thf(fact_632_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_261: complex,B_196: complex,C_115: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A_261 @ B_196 ) @ C_115 )
= ( plus_plus_complex @ ( times_times_complex @ A_261 @ C_115 ) @ ( times_times_complex @ B_196 @ C_115 ) ) ) ).
thf(fact_633_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_261: quickcheck_code_int,B_196: quickcheck_code_int,C_115: quickcheck_code_int] :
( ( times_123202395de_int @ ( plus_p1446045655de_int @ A_261 @ B_196 ) @ C_115 )
= ( plus_p1446045655de_int @ ( times_123202395de_int @ A_261 @ C_115 ) @ ( times_123202395de_int @ B_196 @ C_115 ) ) ) ).
thf(fact_634_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_261: rat,B_196: rat,C_115: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A_261 @ B_196 ) @ C_115 )
= ( plus_plus_rat @ ( times_times_rat @ A_261 @ C_115 ) @ ( times_times_rat @ B_196 @ C_115 ) ) ) ).
thf(fact_635_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_260: code_code_numeral,M_35: code_code_numeral,B_195: code_code_numeral] :
( ( plus_p1627245867umeral @ ( times_1655362735umeral @ A_260 @ M_35 ) @ ( times_1655362735umeral @ B_195 @ M_35 ) )
= ( times_1655362735umeral @ ( plus_p1627245867umeral @ A_260 @ B_195 ) @ M_35 ) ) ).
thf(fact_636_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_260: int,M_35: int,B_195: int] :
( ( plus_plus_int @ ( times_times_int @ A_260 @ M_35 ) @ ( times_times_int @ B_195 @ M_35 ) )
= ( times_times_int @ ( plus_plus_int @ A_260 @ B_195 ) @ M_35 ) ) ).
thf(fact_637_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_260: nat,M_35: nat,B_195: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_260 @ M_35 ) @ ( times_times_nat @ B_195 @ M_35 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_260 @ B_195 ) @ M_35 ) ) ).
thf(fact_638_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_260: real,M_35: real,B_195: real] :
( ( plus_plus_real @ ( times_times_real @ A_260 @ M_35 ) @ ( times_times_real @ B_195 @ M_35 ) )
= ( times_times_real @ ( plus_plus_real @ A_260 @ B_195 ) @ M_35 ) ) ).
thf(fact_639_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_260: complex,M_35: complex,B_195: complex] :
( ( plus_plus_complex @ ( times_times_complex @ A_260 @ M_35 ) @ ( times_times_complex @ B_195 @ M_35 ) )
= ( times_times_complex @ ( plus_plus_complex @ A_260 @ B_195 ) @ M_35 ) ) ).
thf(fact_640_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_260: quickcheck_code_int,M_35: quickcheck_code_int,B_195: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( times_123202395de_int @ A_260 @ M_35 ) @ ( times_123202395de_int @ B_195 @ M_35 ) )
= ( times_123202395de_int @ ( plus_p1446045655de_int @ A_260 @ B_195 ) @ M_35 ) ) ).
thf(fact_641_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_260: rat,M_35: rat,B_195: rat] :
( ( plus_plus_rat @ ( times_times_rat @ A_260 @ M_35 ) @ ( times_times_rat @ B_195 @ M_35 ) )
= ( times_times_rat @ ( plus_plus_rat @ A_260 @ B_195 ) @ M_35 ) ) ).
thf(fact_642_crossproduct__eq,axiom,
! [W_6: int,Y_31: int,X_42: int,Z_8: int] :
( ( ( plus_plus_int @ ( times_times_int @ W_6 @ Y_31 ) @ ( times_times_int @ X_42 @ Z_8 ) )
= ( plus_plus_int @ ( times_times_int @ W_6 @ Z_8 ) @ ( times_times_int @ X_42 @ Y_31 ) ) )
<=> ( ( W_6 = X_42 )
| ( Y_31 = Z_8 ) ) ) ).
thf(fact_643_crossproduct__eq,axiom,
! [W_6: nat,Y_31: nat,X_42: nat,Z_8: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W_6 @ Y_31 ) @ ( times_times_nat @ X_42 @ Z_8 ) )
= ( plus_plus_nat @ ( times_times_nat @ W_6 @ Z_8 ) @ ( times_times_nat @ X_42 @ Y_31 ) ) )
<=> ( ( W_6 = X_42 )
| ( Y_31 = Z_8 ) ) ) ).
thf(fact_644_crossproduct__eq,axiom,
! [W_6: real,Y_31: real,X_42: real,Z_8: real] :
( ( ( plus_plus_real @ ( times_times_real @ W_6 @ Y_31 ) @ ( times_times_real @ X_42 @ Z_8 ) )
= ( plus_plus_real @ ( times_times_real @ W_6 @ Z_8 ) @ ( times_times_real @ X_42 @ Y_31 ) ) )
<=> ( ( W_6 = X_42 )
| ( Y_31 = Z_8 ) ) ) ).
thf(fact_645_crossproduct__eq,axiom,
! [W_6: complex,Y_31: complex,X_42: complex,Z_8: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ W_6 @ Y_31 ) @ ( times_times_complex @ X_42 @ Z_8 ) )
= ( plus_plus_complex @ ( times_times_complex @ W_6 @ Z_8 ) @ ( times_times_complex @ X_42 @ Y_31 ) ) )
<=> ( ( W_6 = X_42 )
| ( Y_31 = Z_8 ) ) ) ).
thf(fact_646_crossproduct__eq,axiom,
! [W_6: rat,Y_31: rat,X_42: rat,Z_8: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ W_6 @ Y_31 ) @ ( times_times_rat @ X_42 @ Z_8 ) )
= ( plus_plus_rat @ ( times_times_rat @ W_6 @ Z_8 ) @ ( times_times_rat @ X_42 @ Y_31 ) ) )
<=> ( ( W_6 = X_42 )
| ( Y_31 = Z_8 ) ) ) ).
thf(fact_647_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_259: int] :
( ( times_times_int @ one_one_int @ A_259 )
= A_259 ) ).
thf(fact_648_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_259: nat] :
( ( times_times_nat @ one_one_nat @ A_259 )
= A_259 ) ).
thf(fact_649_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_259: real] :
( ( times_times_real @ one_one_real @ A_259 )
= A_259 ) ).
thf(fact_650_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_259: code_code_numeral] :
( ( times_1655362735umeral @ one_on1645066479umeral @ A_259 )
= A_259 ) ).
thf(fact_651_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_259: complex] :
( ( times_times_complex @ one_one_complex @ A_259 )
= A_259 ) ).
thf(fact_652_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_259: quickcheck_code_int] :
( ( times_123202395de_int @ one_on1684967323de_int @ A_259 )
= A_259 ) ).
thf(fact_653_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_259: rat] :
( ( times_times_rat @ one_one_rat @ A_259 )
= A_259 ) ).
thf(fact_654_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_258: int] :
( ( times_times_int @ A_258 @ one_one_int )
= A_258 ) ).
thf(fact_655_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_258: nat] :
( ( times_times_nat @ A_258 @ one_one_nat )
= A_258 ) ).
thf(fact_656_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_258: real] :
( ( times_times_real @ A_258 @ one_one_real )
= A_258 ) ).
thf(fact_657_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_258: code_code_numeral] :
( ( times_1655362735umeral @ A_258 @ one_on1645066479umeral )
= A_258 ) ).
thf(fact_658_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_258: complex] :
( ( times_times_complex @ A_258 @ one_one_complex )
= A_258 ) ).
thf(fact_659_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_258: quickcheck_code_int] :
( ( times_123202395de_int @ A_258 @ one_on1684967323de_int )
= A_258 ) ).
thf(fact_660_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_258: rat] :
( ( times_times_rat @ A_258 @ one_one_rat )
= A_258 ) ).
thf(fact_661_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_41: code_code_numeral,Y_30: code_code_numeral,Q_9: nat] :
( ( power_2100829034umeral @ ( times_1655362735umeral @ X_41 @ Y_30 ) @ Q_9 )
= ( times_1655362735umeral @ ( power_2100829034umeral @ X_41 @ Q_9 ) @ ( power_2100829034umeral @ Y_30 @ Q_9 ) ) ) ).
thf(fact_662_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_41: quickcheck_code_int,Y_30: quickcheck_code_int,Q_9: nat] :
( ( power_881366806de_int @ ( times_123202395de_int @ X_41 @ Y_30 ) @ Q_9 )
= ( times_123202395de_int @ ( power_881366806de_int @ X_41 @ Q_9 ) @ ( power_881366806de_int @ Y_30 @ Q_9 ) ) ) ).
thf(fact_663_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_41: rat,Y_30: rat,Q_9: nat] :
( ( power_power_rat @ ( times_times_rat @ X_41 @ Y_30 ) @ Q_9 )
= ( times_times_rat @ ( power_power_rat @ X_41 @ Q_9 ) @ ( power_power_rat @ Y_30 @ Q_9 ) ) ) ).
thf(fact_664_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_41: int,Y_30: int,Q_9: nat] :
( ( power_power_int @ ( times_times_int @ X_41 @ Y_30 ) @ Q_9 )
= ( times_times_int @ ( power_power_int @ X_41 @ Q_9 ) @ ( power_power_int @ Y_30 @ Q_9 ) ) ) ).
thf(fact_665_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_41: nat,Y_30: nat,Q_9: nat] :
( ( power_power_nat @ ( times_times_nat @ X_41 @ Y_30 ) @ Q_9 )
= ( times_times_nat @ ( power_power_nat @ X_41 @ Q_9 ) @ ( power_power_nat @ Y_30 @ Q_9 ) ) ) ).
thf(fact_666_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_41: real,Y_30: real,Q_9: nat] :
( ( power_power_real @ ( times_times_real @ X_41 @ Y_30 ) @ Q_9 )
= ( times_times_real @ ( power_power_real @ X_41 @ Q_9 ) @ ( power_power_real @ Y_30 @ Q_9 ) ) ) ).
thf(fact_667_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_41: complex,Y_30: complex,Q_9: nat] :
( ( power_power_complex @ ( times_times_complex @ X_41 @ Y_30 ) @ Q_9 )
= ( times_times_complex @ ( power_power_complex @ X_41 @ Q_9 ) @ ( power_power_complex @ Y_30 @ Q_9 ) ) ) ).
thf(fact_668_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_40: code_code_numeral,P_7: nat,Q_8: nat] :
( ( times_1655362735umeral @ ( power_2100829034umeral @ X_40 @ P_7 ) @ ( power_2100829034umeral @ X_40 @ Q_8 ) )
= ( power_2100829034umeral @ X_40 @ ( plus_plus_nat @ P_7 @ Q_8 ) ) ) ).
thf(fact_669_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_40: quickcheck_code_int,P_7: nat,Q_8: nat] :
( ( times_123202395de_int @ ( power_881366806de_int @ X_40 @ P_7 ) @ ( power_881366806de_int @ X_40 @ Q_8 ) )
= ( power_881366806de_int @ X_40 @ ( plus_plus_nat @ P_7 @ Q_8 ) ) ) ).
thf(fact_670_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_40: rat,P_7: nat,Q_8: nat] :
( ( times_times_rat @ ( power_power_rat @ X_40 @ P_7 ) @ ( power_power_rat @ X_40 @ Q_8 ) )
= ( power_power_rat @ X_40 @ ( plus_plus_nat @ P_7 @ Q_8 ) ) ) ).
thf(fact_671_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_40: int,P_7: nat,Q_8: nat] :
( ( times_times_int @ ( power_power_int @ X_40 @ P_7 ) @ ( power_power_int @ X_40 @ Q_8 ) )
= ( power_power_int @ X_40 @ ( plus_plus_nat @ P_7 @ Q_8 ) ) ) ).
thf(fact_672_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_40: nat,P_7: nat,Q_8: nat] :
( ( times_times_nat @ ( power_power_nat @ X_40 @ P_7 ) @ ( power_power_nat @ X_40 @ Q_8 ) )
= ( power_power_nat @ X_40 @ ( plus_plus_nat @ P_7 @ Q_8 ) ) ) ).
thf(fact_673_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_40: real,P_7: nat,Q_8: nat] :
( ( times_times_real @ ( power_power_real @ X_40 @ P_7 ) @ ( power_power_real @ X_40 @ Q_8 ) )
= ( power_power_real @ X_40 @ ( plus_plus_nat @ P_7 @ Q_8 ) ) ) ).
thf(fact_674_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_40: complex,P_7: nat,Q_8: nat] :
( ( times_times_complex @ ( power_power_complex @ X_40 @ P_7 ) @ ( power_power_complex @ X_40 @ Q_8 ) )
= ( power_power_complex @ X_40 @ ( plus_plus_nat @ P_7 @ Q_8 ) ) ) ).
thf(fact_675_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_39: int] :
( ( power_power_int @ X_39 @ zero_zero_nat )
= one_one_int ) ).
thf(fact_676_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_39: nat] :
( ( power_power_nat @ X_39 @ zero_zero_nat )
= one_one_nat ) ).
thf(fact_677_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_39: real] :
( ( power_power_real @ X_39 @ zero_zero_nat )
= one_one_real ) ).
thf(fact_678_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_39: code_code_numeral] :
( ( power_2100829034umeral @ X_39 @ zero_zero_nat )
= one_on1645066479umeral ) ).
thf(fact_679_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_39: complex] :
( ( power_power_complex @ X_39 @ zero_zero_nat )
= one_one_complex ) ).
thf(fact_680_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_39: quickcheck_code_int] :
( ( power_881366806de_int @ X_39 @ zero_zero_nat )
= one_on1684967323de_int ) ).
thf(fact_681_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_39: rat] :
( ( power_power_rat @ X_39 @ zero_zero_nat )
= one_one_rat ) ).
thf(fact_682_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
thf(fact_683_zero__le__even__power_H,axiom,
! [A_257: int,N_72: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A_257 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_72 ) ) ) ).
thf(fact_684_zero__le__even__power_H,axiom,
! [A_257: real,N_72: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A_257 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_72 ) ) ) ).
thf(fact_685_zero__le__even__power_H,axiom,
! [A_257: rat,N_72: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A_257 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_72 ) ) ) ).
thf(fact_686_add__scale__eq__noteq,axiom,
! [C_114: int,D_34: int,A_256: int,B_194: int,R_6: int] :
( ( R_6 != zero_zero_int )
=> ( ( ( A_256 = B_194 )
& ( C_114 != D_34 ) )
=> ( ( plus_plus_int @ A_256 @ ( times_times_int @ R_6 @ C_114 ) )
!= ( plus_plus_int @ B_194 @ ( times_times_int @ R_6 @ D_34 ) ) ) ) ) ).
thf(fact_687_add__scale__eq__noteq,axiom,
! [C_114: nat,D_34: nat,A_256: nat,B_194: nat,R_6: nat] :
( ( R_6 != zero_zero_nat )
=> ( ( ( A_256 = B_194 )
& ( C_114 != D_34 ) )
=> ( ( plus_plus_nat @ A_256 @ ( times_times_nat @ R_6 @ C_114 ) )
!= ( plus_plus_nat @ B_194 @ ( times_times_nat @ R_6 @ D_34 ) ) ) ) ) ).
thf(fact_688_add__scale__eq__noteq,axiom,
! [C_114: real,D_34: real,A_256: real,B_194: real,R_6: real] :
( ( R_6 != zero_zero_real )
=> ( ( ( A_256 = B_194 )
& ( C_114 != D_34 ) )
=> ( ( plus_plus_real @ A_256 @ ( times_times_real @ R_6 @ C_114 ) )
!= ( plus_plus_real @ B_194 @ ( times_times_real @ R_6 @ D_34 ) ) ) ) ) ).
thf(fact_689_add__scale__eq__noteq,axiom,
! [C_114: complex,D_34: complex,A_256: complex,B_194: complex,R_6: complex] :
( ( R_6 != zero_zero_complex )
=> ( ( ( A_256 = B_194 )
& ( C_114 != D_34 ) )
=> ( ( plus_plus_complex @ A_256 @ ( times_times_complex @ R_6 @ C_114 ) )
!= ( plus_plus_complex @ B_194 @ ( times_times_complex @ R_6 @ D_34 ) ) ) ) ) ).
thf(fact_690_add__scale__eq__noteq,axiom,
! [C_114: rat,D_34: rat,A_256: rat,B_194: rat,R_6: rat] :
( ( R_6 != zero_zero_rat )
=> ( ( ( A_256 = B_194 )
& ( C_114 != D_34 ) )
=> ( ( plus_plus_rat @ A_256 @ ( times_times_rat @ R_6 @ C_114 ) )
!= ( plus_plus_rat @ B_194 @ ( times_times_rat @ R_6 @ D_34 ) ) ) ) ) ).
thf(fact_691_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_34: int] :
( ( plus_plus_int @ M_34 @ M_34 )
= ( times_times_int @ ( plus_plus_int @ one_one_int @ one_one_int ) @ M_34 ) ) ).
thf(fact_692_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_34: nat] :
( ( plus_plus_nat @ M_34 @ M_34 )
= ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) @ M_34 ) ) ).
thf(fact_693_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_34: real] :
( ( plus_plus_real @ M_34 @ M_34 )
= ( times_times_real @ ( plus_plus_real @ one_one_real @ one_one_real ) @ M_34 ) ) ).
thf(fact_694_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_34: code_code_numeral] :
( ( plus_p1627245867umeral @ M_34 @ M_34 )
= ( times_1655362735umeral @ ( plus_p1627245867umeral @ one_on1645066479umeral @ one_on1645066479umeral ) @ M_34 ) ) ).
thf(fact_695_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_34: complex] :
( ( plus_plus_complex @ M_34 @ M_34 )
= ( times_times_complex @ ( plus_plus_complex @ one_one_complex @ one_one_complex ) @ M_34 ) ) ).
thf(fact_696_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_34: quickcheck_code_int] :
( ( plus_p1446045655de_int @ M_34 @ M_34 )
= ( times_123202395de_int @ ( plus_p1446045655de_int @ one_on1684967323de_int @ one_on1684967323de_int ) @ M_34 ) ) ).
thf(fact_697_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_34: rat] :
( ( plus_plus_rat @ M_34 @ M_34 )
= ( times_times_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) @ M_34 ) ) ).
thf(fact_698_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_33: int,A_255: int] :
( ( plus_plus_int @ M_33 @ ( times_times_int @ A_255 @ M_33 ) )
= ( times_times_int @ ( plus_plus_int @ A_255 @ one_one_int ) @ M_33 ) ) ).
thf(fact_699_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_33: nat,A_255: nat] :
( ( plus_plus_nat @ M_33 @ ( times_times_nat @ A_255 @ M_33 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_255 @ one_one_nat ) @ M_33 ) ) ).
thf(fact_700_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_33: real,A_255: real] :
( ( plus_plus_real @ M_33 @ ( times_times_real @ A_255 @ M_33 ) )
= ( times_times_real @ ( plus_plus_real @ A_255 @ one_one_real ) @ M_33 ) ) ).
thf(fact_701_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_33: code_code_numeral,A_255: code_code_numeral] :
( ( plus_p1627245867umeral @ M_33 @ ( times_1655362735umeral @ A_255 @ M_33 ) )
= ( times_1655362735umeral @ ( plus_p1627245867umeral @ A_255 @ one_on1645066479umeral ) @ M_33 ) ) ).
thf(fact_702_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_33: complex,A_255: complex] :
( ( plus_plus_complex @ M_33 @ ( times_times_complex @ A_255 @ M_33 ) )
= ( times_times_complex @ ( plus_plus_complex @ A_255 @ one_one_complex ) @ M_33 ) ) ).
thf(fact_703_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_33: quickcheck_code_int,A_255: quickcheck_code_int] :
( ( plus_p1446045655de_int @ M_33 @ ( times_123202395de_int @ A_255 @ M_33 ) )
= ( times_123202395de_int @ ( plus_p1446045655de_int @ A_255 @ one_on1684967323de_int ) @ M_33 ) ) ).
thf(fact_704_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_33: rat,A_255: rat] :
( ( plus_plus_rat @ M_33 @ ( times_times_rat @ A_255 @ M_33 ) )
= ( times_times_rat @ ( plus_plus_rat @ A_255 @ one_one_rat ) @ M_33 ) ) ).
thf(fact_705_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_254: int,M_32: int] :
( ( plus_plus_int @ ( times_times_int @ A_254 @ M_32 ) @ M_32 )
= ( times_times_int @ ( plus_plus_int @ A_254 @ one_one_int ) @ M_32 ) ) ).
thf(fact_706_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_254: nat,M_32: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_254 @ M_32 ) @ M_32 )
= ( times_times_nat @ ( plus_plus_nat @ A_254 @ one_one_nat ) @ M_32 ) ) ).
thf(fact_707_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_254: real,M_32: real] :
( ( plus_plus_real @ ( times_times_real @ A_254 @ M_32 ) @ M_32 )
= ( times_times_real @ ( plus_plus_real @ A_254 @ one_one_real ) @ M_32 ) ) ).
thf(fact_708_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_254: code_code_numeral,M_32: code_code_numeral] :
( ( plus_p1627245867umeral @ ( times_1655362735umeral @ A_254 @ M_32 ) @ M_32 )
= ( times_1655362735umeral @ ( plus_p1627245867umeral @ A_254 @ one_on1645066479umeral ) @ M_32 ) ) ).
thf(fact_709_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_254: complex,M_32: complex] :
( ( plus_plus_complex @ ( times_times_complex @ A_254 @ M_32 ) @ M_32 )
= ( times_times_complex @ ( plus_plus_complex @ A_254 @ one_one_complex ) @ M_32 ) ) ).
thf(fact_710_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_254: quickcheck_code_int,M_32: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( times_123202395de_int @ A_254 @ M_32 ) @ M_32 )
= ( times_123202395de_int @ ( plus_p1446045655de_int @ A_254 @ one_on1684967323de_int ) @ M_32 ) ) ).
thf(fact_711_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_254: rat,M_32: rat] :
( ( plus_plus_rat @ ( times_times_rat @ A_254 @ M_32 ) @ M_32 )
= ( times_times_rat @ ( plus_plus_rat @ A_254 @ one_one_rat ) @ M_32 ) ) ).
thf(fact_712_power__eq__0__iff__number__of,axiom,
! [A_253: int,W_5: int] :
( ( ( power_power_int @ A_253 @ ( number_number_of_nat @ W_5 ) )
= zero_zero_int )
<=> ( ( A_253 = zero_zero_int )
& ( ( number_number_of_nat @ W_5 )
!= zero_zero_nat ) ) ) ).
thf(fact_713_power__eq__0__iff__number__of,axiom,
! [A_253: nat,W_5: int] :
( ( ( power_power_nat @ A_253 @ ( number_number_of_nat @ W_5 ) )
= zero_zero_nat )
<=> ( ( A_253 = zero_zero_nat )
& ( ( number_number_of_nat @ W_5 )
!= zero_zero_nat ) ) ) ).
thf(fact_714_power__eq__0__iff__number__of,axiom,
! [A_253: real,W_5: int] :
( ( ( power_power_real @ A_253 @ ( number_number_of_nat @ W_5 ) )
= zero_zero_real )
<=> ( ( A_253 = zero_zero_real )
& ( ( number_number_of_nat @ W_5 )
!= zero_zero_nat ) ) ) ).
thf(fact_715_power__eq__0__iff__number__of,axiom,
! [A_253: code_code_numeral,W_5: int] :
( ( ( power_2100829034umeral @ A_253 @ ( number_number_of_nat @ W_5 ) )
= zero_z126310315umeral )
<=> ( ( A_253 = zero_z126310315umeral )
& ( ( number_number_of_nat @ W_5 )
!= zero_zero_nat ) ) ) ).
thf(fact_716_power__eq__0__iff__number__of,axiom,
! [A_253: complex,W_5: int] :
( ( ( power_power_complex @ A_253 @ ( number_number_of_nat @ W_5 ) )
= zero_zero_complex )
<=> ( ( A_253 = zero_zero_complex )
& ( ( number_number_of_nat @ W_5 )
!= zero_zero_nat ) ) ) ).
thf(fact_717_power__eq__0__iff__number__of,axiom,
! [A_253: quickcheck_code_int,W_5: int] :
( ( ( power_881366806de_int @ A_253 @ ( number_number_of_nat @ W_5 ) )
= zero_z891286103de_int )
<=> ( ( A_253 = zero_z891286103de_int )
& ( ( number_number_of_nat @ W_5 )
!= zero_zero_nat ) ) ) ).
thf(fact_718_power__eq__0__iff__number__of,axiom,
! [A_253: rat,W_5: int] :
( ( ( power_power_rat @ A_253 @ ( number_number_of_nat @ W_5 ) )
= zero_zero_rat )
<=> ( ( A_253 = zero_zero_rat )
& ( ( number_number_of_nat @ W_5 )
!= zero_zero_nat ) ) ) ).
thf(fact_719_pos__zmult__pos,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
thf(fact_720_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
thf(fact_721_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ X @ Y ) ) ) ) ).
thf(fact_722_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ X @ Y ) ) ) ) ).
thf(fact_723_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [N: nat,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ X @ N ) ) ) ).
thf(fact_724_power__0__left__number__of,axiom,
! [W_4: int] :
( ( ( ( number_number_of_nat @ W_4 )
= zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ W_4 ) )
= one_one_int ) )
& ( ( ( number_number_of_nat @ W_4 )
!= zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ W_4 ) )
= zero_zero_int ) ) ) ).
thf(fact_725_power__0__left__number__of,axiom,
! [W_4: int] :
( ( ( ( number_number_of_nat @ W_4 )
= zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ W_4 ) )
= one_one_nat ) )
& ( ( ( number_number_of_nat @ W_4 )
!= zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ W_4 ) )
= zero_zero_nat ) ) ) ).
thf(fact_726_power__0__left__number__of,axiom,
! [W_4: int] :
( ( ( ( number_number_of_nat @ W_4 )
= zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ ( number_number_of_nat @ W_4 ) )
= one_one_real ) )
& ( ( ( number_number_of_nat @ W_4 )
!= zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ ( number_number_of_nat @ W_4 ) )
= zero_zero_real ) ) ) ).
thf(fact_727_power__0__left__number__of,axiom,
! [W_4: int] :
( ( ( ( number_number_of_nat @ W_4 )
= zero_zero_nat )
=> ( ( power_2100829034umeral @ zero_z126310315umeral @ ( number_number_of_nat @ W_4 ) )
= one_on1645066479umeral ) )
& ( ( ( number_number_of_nat @ W_4 )
!= zero_zero_nat )
=> ( ( power_2100829034umeral @ zero_z126310315umeral @ ( number_number_of_nat @ W_4 ) )
= zero_z126310315umeral ) ) ) ).
thf(fact_728_power__0__left__number__of,axiom,
! [W_4: int] :
( ( ( ( number_number_of_nat @ W_4 )
= zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ ( number_number_of_nat @ W_4 ) )
= one_one_complex ) )
& ( ( ( number_number_of_nat @ W_4 )
!= zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ ( number_number_of_nat @ W_4 ) )
= zero_zero_complex ) ) ) ).
thf(fact_729_power__0__left__number__of,axiom,
! [W_4: int] :
( ( ( ( number_number_of_nat @ W_4 )
= zero_zero_nat )
=> ( ( power_881366806de_int @ zero_z891286103de_int @ ( number_number_of_nat @ W_4 ) )
= one_on1684967323de_int ) )
& ( ( ( number_number_of_nat @ W_4 )
!= zero_zero_nat )
=> ( ( power_881366806de_int @ zero_z891286103de_int @ ( number_number_of_nat @ W_4 ) )
= zero_z891286103de_int ) ) ) ).
thf(fact_730_power__0__left__number__of,axiom,
! [W_4: int] :
( ( ( ( number_number_of_nat @ W_4 )
= zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ ( number_number_of_nat @ W_4 ) )
= one_one_rat ) )
& ( ( ( number_number_of_nat @ W_4 )
!= zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ ( number_number_of_nat @ W_4 ) )
= zero_zero_rat ) ) ) ).
thf(fact_731_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,
ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) ).
thf(fact_732_q__pos__lemma,axiom,
! [B_5: int,Q_6: int,R_3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_5 @ Q_6 ) @ R_3 ) )
=> ( ( ord_less_int @ R_3 @ B_5 )
=> ( ( ord_less_int @ zero_zero_int @ B_5 )
=> ( ord_less_eq_int @ zero_zero_int @ Q_6 ) ) ) ) ).
thf(fact_733_q__neg__lemma,axiom,
! [B_5: int,Q_6: int,R_3: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_5 @ Q_6 ) @ R_3 ) @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_3 )
=> ( ( ord_less_int @ zero_zero_int @ B_5 )
=> ( ord_less_eq_int @ Q_6 @ zero_zero_int ) ) ) ) ).
thf(fact_734_unique__quotient__lemma,axiom,
! [B: int,Q_6: int,R_3: int,Q: int,R_1: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q_6 ) @ R_3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R_1 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_3 )
=> ( ( ord_less_int @ R_3 @ B )
=> ( ( ord_less_int @ R_1 @ B )
=> ( ord_less_eq_int @ Q_6 @ Q ) ) ) ) ) ).
thf(fact_735_zdiv__mono2__lemma,axiom,
! [B: int,Q: int,R_1: int,B_5: int,Q_6: int,R_3: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R_1 )
= ( plus_plus_int @ ( times_times_int @ B_5 @ Q_6 ) @ R_3 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_5 @ Q_6 ) @ R_3 ) )
=> ( ( ord_less_int @ R_3 @ B_5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ( ord_less_int @ zero_zero_int @ B_5 )
=> ( ( ord_less_eq_int @ B_5 @ B )
=> ( ord_less_eq_int @ Q @ Q_6 ) ) ) ) ) ) ) ).
thf(fact_736_unique__quotient__lemma__neg,axiom,
! [B: int,Q_6: int,R_3: int,Q: int,R_1: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q_6 ) @ R_3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R_1 ) )
=> ( ( ord_less_eq_int @ R_1 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R_1 )
=> ( ( ord_less_int @ B @ R_3 )
=> ( ord_less_eq_int @ Q @ Q_6 ) ) ) ) ) ).
thf(fact_737_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q: int,R_1: int,B_5: int,Q_6: int,R_3: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R_1 )
= ( plus_plus_int @ ( times_times_int @ B_5 @ Q_6 ) @ R_3 ) )
=> ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_5 @ Q_6 ) @ R_3 ) @ zero_zero_int )
=> ( ( ord_less_int @ R_1 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_3 )
=> ( ( ord_less_int @ zero_zero_int @ B_5 )
=> ( ( ord_less_eq_int @ B_5 @ B )
=> ( ord_less_eq_int @ Q_6 @ Q ) ) ) ) ) ) ) ).
thf(fact_738_quartic__square__square,axiom,
! [X: int] :
( ( power_power_int @ ( power_power_int @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_int @ X @ ( number_number_of_nat @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_739_s,axiom,
zcong @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of_int @ min ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_740_Euler_Oaux____1,axiom,
! [Y: int,X: int,P_3: int] :
( ~ ( zcong @ X @ zero_zero_int @ P_3 )
=> ( ( zcong @ ( power_power_int @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ X @ P_3 )
=> ~ ( dvd_dvd_int @ P_3 @ Y ) ) ) ).
thf(fact_741_zprime__def,axiom,
! [P_3: int] :
( ( zprime @ P_3 )
<=> ( ( ord_less_int @ one_one_int @ P_3 )
& ! [M_2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ M_2 )
& ( dvd_dvd_int @ M_2 @ P_3 ) )
=> ( ( M_2 = one_one_int )
| ( M_2 = P_3 ) ) ) ) ) ).
thf(fact_742_prime__g__5,axiom,
! [P_3: int] :
( ( zprime @ P_3 )
=> ( ( P_3
!= ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
=> ( ( P_3
!= ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) )
=> ( ord_less_eq_int @ ( number_number_of_int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P_3 ) ) ) ) ).
thf(fact_743_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_744_power__strict__mono,axiom,
! [N_71: nat,A_252: int,B_193: int] :
( ( ord_less_int @ A_252 @ B_193 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_252 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_71 )
=> ( ord_less_int @ ( power_power_int @ A_252 @ N_71 ) @ ( power_power_int @ B_193 @ N_71 ) ) ) ) ) ).
thf(fact_745_power__strict__mono,axiom,
! [N_71: nat,A_252: nat,B_193: nat] :
( ( ord_less_nat @ A_252 @ B_193 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_252 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_71 )
=> ( ord_less_nat @ ( power_power_nat @ A_252 @ N_71 ) @ ( power_power_nat @ B_193 @ N_71 ) ) ) ) ) ).
thf(fact_746_power__strict__mono,axiom,
! [N_71: nat,A_252: real,B_193: real] :
( ( ord_less_real @ A_252 @ B_193 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_252 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_71 )
=> ( ord_less_real @ ( power_power_real @ A_252 @ N_71 ) @ ( power_power_real @ B_193 @ N_71 ) ) ) ) ) ).
thf(fact_747_power__strict__mono,axiom,
! [N_71: nat,A_252: code_code_numeral,B_193: code_code_numeral] :
( ( ord_le1304079648umeral @ A_252 @ B_193 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_252 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_71 )
=> ( ord_le1304079648umeral @ ( power_2100829034umeral @ A_252 @ N_71 ) @ ( power_2100829034umeral @ B_193 @ N_71 ) ) ) ) ) ).
thf(fact_748_power__strict__mono,axiom,
! [N_71: nat,A_252: quickcheck_code_int,B_193: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_252 @ B_193 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_252 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_71 )
=> ( ord_le1860547276de_int @ ( power_881366806de_int @ A_252 @ N_71 ) @ ( power_881366806de_int @ B_193 @ N_71 ) ) ) ) ) ).
thf(fact_749_power__strict__mono,axiom,
! [N_71: nat,A_252: rat,B_193: rat] :
( ( ord_less_rat @ A_252 @ B_193 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A_252 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_71 )
=> ( ord_less_rat @ ( power_power_rat @ A_252 @ N_71 ) @ ( power_power_rat @ B_193 @ N_71 ) ) ) ) ) ).
thf(fact_750_convex__bound__lt,axiom,
! [V_8: int,U_3: int,Y_29: int,X_38: int,A_251: int] :
( ( ord_less_int @ X_38 @ A_251 )
=> ( ( ord_less_int @ Y_29 @ A_251 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U_3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ V_8 )
=> ( ( ( plus_plus_int @ U_3 @ V_8 )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U_3 @ X_38 ) @ ( times_times_int @ V_8 @ Y_29 ) ) @ A_251 ) ) ) ) ) ) ).
thf(fact_751_convex__bound__lt,axiom,
! [V_8: real,U_3: real,Y_29: real,X_38: real,A_251: real] :
( ( ord_less_real @ X_38 @ A_251 )
=> ( ( ord_less_real @ Y_29 @ A_251 )
=> ( ( ord_less_eq_real @ zero_zero_real @ U_3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ V_8 )
=> ( ( ( plus_plus_real @ U_3 @ V_8 )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U_3 @ X_38 ) @ ( times_times_real @ V_8 @ Y_29 ) ) @ A_251 ) ) ) ) ) ) ).
thf(fact_752_convex__bound__lt,axiom,
! [V_8: rat,U_3: rat,Y_29: rat,X_38: rat,A_251: rat] :
( ( ord_less_rat @ X_38 @ A_251 )
=> ( ( ord_less_rat @ Y_29 @ A_251 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ U_3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ V_8 )
=> ( ( ( plus_plus_rat @ U_3 @ V_8 )
= one_one_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U_3 @ X_38 ) @ ( times_times_rat @ V_8 @ Y_29 ) ) @ A_251 ) ) ) ) ) ) ).
thf(fact_753_zpower__zdvd__prop2,axiom,
! [Y: int,N: nat,P_3: int] :
( ( zprime @ P_3 )
=> ( ( dvd_dvd_int @ P_3 @ ( power_power_int @ Y @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( dvd_dvd_int @ P_3 @ Y ) ) ) ) ).
thf(fact_754_zprime__zdvd__zmult,axiom,
! [N: int,P_3: int,M: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( zprime @ P_3 )
=> ( ( dvd_dvd_int @ P_3 @ ( times_times_int @ M @ N ) )
=> ( ( dvd_dvd_int @ P_3 @ M )
| ( dvd_dvd_int @ P_3 @ N ) ) ) ) ) ).
thf(fact_755_zcong__zprime__prod__zero__contra,axiom,
! [B: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
& ~ ( zcong @ B @ zero_zero_int @ P_3 ) )
=> ~ ( zcong @ ( times_times_int @ A @ B ) @ zero_zero_int @ P_3 ) ) ) ) ).
thf(fact_756_zcong__zprime__prod__zero,axiom,
! [B: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( zcong @ ( times_times_int @ A @ B ) @ zero_zero_int @ P_3 )
=> ( ( zcong @ A @ zero_zero_int @ P_3 )
| ( zcong @ B @ zero_zero_int @ P_3 ) ) ) ) ) ).
thf(fact_757_dvd__0__right,axiom,
! [A_250: real] : ( dvd_dvd_real @ A_250 @ zero_zero_real ) ).
thf(fact_758_dvd__0__right,axiom,
! [A_250: code_code_numeral] : ( dvd_dv174992974umeral @ A_250 @ zero_z126310315umeral ) ).
thf(fact_759_dvd__0__right,axiom,
! [A_250: complex] : ( dvd_dvd_complex @ A_250 @ zero_zero_complex ) ).
thf(fact_760_dvd__0__right,axiom,
! [A_250: quickcheck_code_int] : ( dvd_dv1760642554de_int @ A_250 @ zero_z891286103de_int ) ).
thf(fact_761_dvd__0__right,axiom,
! [A_250: rat] : ( dvd_dvd_rat @ A_250 @ zero_zero_rat ) ).
thf(fact_762_dvd__0__right,axiom,
! [A_250: int] : ( dvd_dvd_int @ A_250 @ zero_zero_int ) ).
thf(fact_763_dvd__0__right,axiom,
! [A_250: nat] : ( dvd_dvd_nat @ A_250 @ zero_zero_nat ) ).
thf(fact_764__096_B_Bthesis_O_A_I_B_Bs1_O_A_091s1_A_094_A2_A_061_A_N1_093_A_Imod_A4_,axiom,
~ ! [S1: int] :
~ ( zcong @ ( power_power_int @ S1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of_int @ min ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ) ).
thf(fact_765__096Legendre_A_N1_A_I4_A_K_Am_A_L_A1_J_A_061_A1_096,axiom,
( ( legendre @ ( number_number_of_int @ min ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
= one_one_int ) ).
thf(fact_766_rel__simps_I47_J,axiom,
! [K_1: int] :
( ( ( bit1 @ K_1 )
= min )
<=> ( K_1 = min ) ) ).
thf(fact_767_rel__simps_I43_J,axiom,
! [L: int] :
( ( min
= ( bit1 @ L ) )
<=> ( min = L ) ) ).
thf(fact_768_Bit1__Min,axiom,
( ( bit1 @ min )
= min ) ).
thf(fact_769_rel__simps_I37_J,axiom,
pls != min ).
thf(fact_770_rel__simps_I40_J,axiom,
min != pls ).
thf(fact_771_rel__simps_I45_J,axiom,
! [K_1: int] :
( ( bit0 @ K_1 )
!= min ) ).
thf(fact_772_rel__simps_I42_J,axiom,
! [L: int] :
( min
!= ( bit0 @ L ) ) ).
thf(fact_773_rel__simps_I7_J,axiom,
~ ( ord_less_int @ min @ min ) ).
thf(fact_774_rel__simps_I24_J,axiom,
ord_less_eq_int @ min @ min ).
thf(fact_775_rel__simps_I13_J,axiom,
! [K_1: int] :
( ( ord_less_int @ ( bit1 @ K_1 ) @ min )
<=> ( ord_less_int @ K_1 @ min ) ) ).
thf(fact_776_rel__simps_I9_J,axiom,
! [K_1: int] :
( ( ord_less_int @ min @ ( bit1 @ K_1 ) )
<=> ( ord_less_int @ min @ K_1 ) ) ).
thf(fact_777_rel__simps_I3_J,axiom,
~ ( ord_less_int @ pls @ min ) ).
thf(fact_778_rel__simps_I6_J,axiom,
ord_less_int @ min @ pls ).
thf(fact_779_rel__simps_I8_J,axiom,
! [K_1: int] :
( ( ord_less_int @ min @ ( bit0 @ K_1 ) )
<=> ( ord_less_int @ min @ K_1 ) ) ).
thf(fact_780_bin__less__0__simps_I2_J,axiom,
ord_less_int @ min @ zero_zero_int ).
thf(fact_781_rel__simps_I26_J,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ min @ ( bit1 @ K_1 ) )
<=> ( ord_less_eq_int @ min @ K_1 ) ) ).
thf(fact_782_rel__simps_I30_J,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ ( bit1 @ K_1 ) @ min )
<=> ( ord_less_eq_int @ K_1 @ min ) ) ).
thf(fact_783_rel__simps_I23_J,axiom,
ord_less_eq_int @ min @ pls ).
thf(fact_784_rel__simps_I20_J,axiom,
~ ( ord_less_eq_int @ pls @ min ) ).
thf(fact_785_rel__simps_I28_J,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ ( bit0 @ K_1 ) @ min )
<=> ( ord_less_eq_int @ K_1 @ min ) ) ).
thf(fact_786_eq__number__of__Pls__Min,axiom,
( ( number_number_of_int @ pls )
!= ( number_number_of_int @ min ) ) ).
thf(fact_787_rel__simps_I11_J,axiom,
! [K_1: int] :
( ( ord_less_int @ ( bit0 @ K_1 ) @ min )
<=> ( ord_less_eq_int @ K_1 @ min ) ) ).
thf(fact_788_rel__simps_I25_J,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ min @ ( bit0 @ K_1 ) )
<=> ( ord_less_int @ min @ K_1 ) ) ).
thf(fact_789_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
<=> ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( number_number_of_int @ min ) )
& ( N
= ( number_number_of_int @ min ) ) ) ) ) ).
thf(fact_790_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( number_number_of_int @ min ) ) ) ) ).
thf(fact_791_linorder__neqE__linordered__idom,axiom,
! [X_37: int,Y_28: int] :
( ( X_37 != Y_28 )
=> ( ~ ( ord_less_int @ X_37 @ Y_28 )
=> ( ord_less_int @ Y_28 @ X_37 ) ) ) ).
thf(fact_792_linorder__neqE__linordered__idom,axiom,
! [X_37: real,Y_28: real] :
( ( X_37 != Y_28 )
=> ( ~ ( ord_less_real @ X_37 @ Y_28 )
=> ( ord_less_real @ Y_28 @ X_37 ) ) ) ).
thf(fact_793_linorder__neqE__linordered__idom,axiom,
! [X_37: rat,Y_28: rat] :
( ( X_37 != Y_28 )
=> ( ~ ( ord_less_rat @ X_37 @ Y_28 )
=> ( ord_less_rat @ Y_28 @ X_37 ) ) ) ).
thf(fact_794_dvd__refl,axiom,
! [A_249: quickcheck_code_int] : ( dvd_dv1760642554de_int @ A_249 @ A_249 ) ).
thf(fact_795_dvd__refl,axiom,
! [A_249: code_code_numeral] : ( dvd_dv174992974umeral @ A_249 @ A_249 ) ).
thf(fact_796_dvd__refl,axiom,
! [A_249: rat] : ( dvd_dvd_rat @ A_249 @ A_249 ) ).
thf(fact_797_dvd__refl,axiom,
! [A_249: real] : ( dvd_dvd_real @ A_249 @ A_249 ) ).
thf(fact_798_dvd__refl,axiom,
! [A_249: complex] : ( dvd_dvd_complex @ A_249 @ A_249 ) ).
thf(fact_799_dvd__refl,axiom,
! [A_249: int] : ( dvd_dvd_int @ A_249 @ A_249 ) ).
thf(fact_800_dvd__refl,axiom,
! [A_249: nat] : ( dvd_dvd_nat @ A_249 @ A_249 ) ).
thf(fact_801_dvd__trans,axiom,
! [C_113: quickcheck_code_int,A_248: quickcheck_code_int,B_192: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ A_248 @ B_192 )
=> ( ( dvd_dv1760642554de_int @ B_192 @ C_113 )
=> ( dvd_dv1760642554de_int @ A_248 @ C_113 ) ) ) ).
thf(fact_802_dvd__trans,axiom,
! [C_113: code_code_numeral,A_248: code_code_numeral,B_192: code_code_numeral] :
( ( dvd_dv174992974umeral @ A_248 @ B_192 )
=> ( ( dvd_dv174992974umeral @ B_192 @ C_113 )
=> ( dvd_dv174992974umeral @ A_248 @ C_113 ) ) ) ).
thf(fact_803_dvd__trans,axiom,
! [C_113: rat,A_248: rat,B_192: rat] :
( ( dvd_dvd_rat @ A_248 @ B_192 )
=> ( ( dvd_dvd_rat @ B_192 @ C_113 )
=> ( dvd_dvd_rat @ A_248 @ C_113 ) ) ) ).
thf(fact_804_dvd__trans,axiom,
! [C_113: real,A_248: real,B_192: real] :
( ( dvd_dvd_real @ A_248 @ B_192 )
=> ( ( dvd_dvd_real @ B_192 @ C_113 )
=> ( dvd_dvd_real @ A_248 @ C_113 ) ) ) ).
thf(fact_805_dvd__trans,axiom,
! [C_113: complex,A_248: complex,B_192: complex] :
( ( dvd_dvd_complex @ A_248 @ B_192 )
=> ( ( dvd_dvd_complex @ B_192 @ C_113 )
=> ( dvd_dvd_complex @ A_248 @ C_113 ) ) ) ).
thf(fact_806_dvd__trans,axiom,
! [C_113: int,A_248: int,B_192: int] :
( ( dvd_dvd_int @ A_248 @ B_192 )
=> ( ( dvd_dvd_int @ B_192 @ C_113 )
=> ( dvd_dvd_int @ A_248 @ C_113 ) ) ) ).
thf(fact_807_dvd__trans,axiom,
! [C_113: nat,A_248: nat,B_192: nat] :
( ( dvd_dvd_nat @ A_248 @ B_192 )
=> ( ( dvd_dvd_nat @ B_192 @ C_113 )
=> ( dvd_dvd_nat @ A_248 @ C_113 ) ) ) ).
thf(fact_808_zcong__sym,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
<=> ( zcong @ B @ A @ M ) ) ).
thf(fact_809_zcong__refl,axiom,
! [K_1: int,M: int] : ( zcong @ K_1 @ K_1 @ M ) ).
thf(fact_810_zcong__eq__trans,axiom,
! [D: int,C: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( B = C )
=> ( ( zcong @ C @ D @ M )
=> ( zcong @ A @ D @ M ) ) ) ) ).
thf(fact_811_zcong__trans,axiom,
! [C: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ B @ C @ M )
=> ( zcong @ A @ C @ M ) ) ) ).
thf(fact_812_zcong__neg__1__impl__ne__1,axiom,
! [X: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zcong @ X @ ( number_number_of_int @ min ) @ P_3 )
=> ~ ( zcong @ X @ one_one_int @ P_3 ) ) ) ).
thf(fact_813_order__le__neq__implies__less,axiom,
! [X_36: code_code_numeral,Y_27: code_code_numeral] :
( ( ord_le565307924umeral @ X_36 @ Y_27 )
=> ( ( X_36 != Y_27 )
=> ( ord_le1304079648umeral @ X_36 @ Y_27 ) ) ) ).
thf(fact_814_order__le__neq__implies__less,axiom,
! [X_36: int,Y_27: int] :
( ( ord_less_eq_int @ X_36 @ Y_27 )
=> ( ( X_36 != Y_27 )
=> ( ord_less_int @ X_36 @ Y_27 ) ) ) ).
thf(fact_815_order__le__neq__implies__less,axiom,
! [X_36: nat,Y_27: nat] :
( ( ord_less_eq_nat @ X_36 @ Y_27 )
=> ( ( X_36 != Y_27 )
=> ( ord_less_nat @ X_36 @ Y_27 ) ) ) ).
thf(fact_816_order__le__neq__implies__less,axiom,
! [X_36: real,Y_27: real] :
( ( ord_less_eq_real @ X_36 @ Y_27 )
=> ( ( X_36 != Y_27 )
=> ( ord_less_real @ X_36 @ Y_27 ) ) ) ).
thf(fact_817_order__le__neq__implies__less,axiom,
! [X_36: quickcheck_code_int,Y_27: quickcheck_code_int] :
( ( ord_le258702272de_int @ X_36 @ Y_27 )
=> ( ( X_36 != Y_27 )
=> ( ord_le1860547276de_int @ X_36 @ Y_27 ) ) ) ).
thf(fact_818_order__le__neq__implies__less,axiom,
! [X_36: nat > $o,Y_27: nat > $o] :
( ( ord_less_eq_nat_o @ X_36 @ Y_27 )
=> ( ( X_36 != Y_27 )
=> ( ord_less_nat_o @ X_36 @ Y_27 ) ) ) ).
thf(fact_819_order__le__neq__implies__less,axiom,
! [X_36: int > $o,Y_27: int > $o] :
( ( ord_less_eq_int_o @ X_36 @ Y_27 )
=> ( ( X_36 != Y_27 )
=> ( ord_less_int_o @ X_36 @ Y_27 ) ) ) ).
thf(fact_820_order__le__neq__implies__less,axiom,
! [X_36: rat,Y_27: rat] :
( ( ord_less_eq_rat @ X_36 @ Y_27 )
=> ( ( X_36 != Y_27 )
=> ( ord_less_rat @ X_36 @ Y_27 ) ) ) ).
thf(fact_821_divisors__zero,axiom,
! [A_247: int,B_191: int] :
( ( ( times_times_int @ A_247 @ B_191 )
= zero_zero_int )
=> ( ( A_247 = zero_zero_int )
| ( B_191 = zero_zero_int ) ) ) ).
thf(fact_822_divisors__zero,axiom,
! [A_247: nat,B_191: nat] :
( ( ( times_times_nat @ A_247 @ B_191 )
= zero_zero_nat )
=> ( ( A_247 = zero_zero_nat )
| ( B_191 = zero_zero_nat ) ) ) ).
thf(fact_823_divisors__zero,axiom,
! [A_247: real,B_191: real] :
( ( ( times_times_real @ A_247 @ B_191 )
= zero_zero_real )
=> ( ( A_247 = zero_zero_real )
| ( B_191 = zero_zero_real ) ) ) ).
thf(fact_824_divisors__zero,axiom,
! [A_247: code_code_numeral,B_191: code_code_numeral] :
( ( ( times_1655362735umeral @ A_247 @ B_191 )
= zero_z126310315umeral )
=> ( ( A_247 = zero_z126310315umeral )
| ( B_191 = zero_z126310315umeral ) ) ) ).
thf(fact_825_divisors__zero,axiom,
! [A_247: complex,B_191: complex] :
( ( ( times_times_complex @ A_247 @ B_191 )
= zero_zero_complex )
=> ( ( A_247 = zero_zero_complex )
| ( B_191 = zero_zero_complex ) ) ) ).
thf(fact_826_divisors__zero,axiom,
! [A_247: quickcheck_code_int,B_191: quickcheck_code_int] :
( ( ( times_123202395de_int @ A_247 @ B_191 )
= zero_z891286103de_int )
=> ( ( A_247 = zero_z891286103de_int )
| ( B_191 = zero_z891286103de_int ) ) ) ).
thf(fact_827_divisors__zero,axiom,
! [A_247: rat,B_191: rat] :
( ( ( times_times_rat @ A_247 @ B_191 )
= zero_zero_rat )
=> ( ( A_247 = zero_zero_rat )
| ( B_191 = zero_zero_rat ) ) ) ).
thf(fact_828_no__zero__divisors,axiom,
! [B_190: int,A_246: int] :
( ( A_246 != zero_zero_int )
=> ( ( B_190 != zero_zero_int )
=> ( ( times_times_int @ A_246 @ B_190 )
!= zero_zero_int ) ) ) ).
thf(fact_829_no__zero__divisors,axiom,
! [B_190: nat,A_246: nat] :
( ( A_246 != zero_zero_nat )
=> ( ( B_190 != zero_zero_nat )
=> ( ( times_times_nat @ A_246 @ B_190 )
!= zero_zero_nat ) ) ) ).
thf(fact_830_no__zero__divisors,axiom,
! [B_190: real,A_246: real] :
( ( A_246 != zero_zero_real )
=> ( ( B_190 != zero_zero_real )
=> ( ( times_times_real @ A_246 @ B_190 )
!= zero_zero_real ) ) ) ).
thf(fact_831_no__zero__divisors,axiom,
! [B_190: code_code_numeral,A_246: code_code_numeral] :
( ( A_246 != zero_z126310315umeral )
=> ( ( B_190 != zero_z126310315umeral )
=> ( ( times_1655362735umeral @ A_246 @ B_190 )
!= zero_z126310315umeral ) ) ) ).
thf(fact_832_no__zero__divisors,axiom,
! [B_190: complex,A_246: complex] :
( ( A_246 != zero_zero_complex )
=> ( ( B_190 != zero_zero_complex )
=> ( ( times_times_complex @ A_246 @ B_190 )
!= zero_zero_complex ) ) ) ).
thf(fact_833_no__zero__divisors,axiom,
! [B_190: quickcheck_code_int,A_246: quickcheck_code_int] :
( ( A_246 != zero_z891286103de_int )
=> ( ( B_190 != zero_z891286103de_int )
=> ( ( times_123202395de_int @ A_246 @ B_190 )
!= zero_z891286103de_int ) ) ) ).
thf(fact_834_no__zero__divisors,axiom,
! [B_190: rat,A_246: rat] :
( ( A_246 != zero_zero_rat )
=> ( ( B_190 != zero_zero_rat )
=> ( ( times_times_rat @ A_246 @ B_190 )
!= zero_zero_rat ) ) ) ).
thf(fact_835_mult__eq__0__iff,axiom,
! [A_245: int,B_189: int] :
( ( ( times_times_int @ A_245 @ B_189 )
= zero_zero_int )
<=> ( ( A_245 = zero_zero_int )
| ( B_189 = zero_zero_int ) ) ) ).
thf(fact_836_mult__eq__0__iff,axiom,
! [A_245: real,B_189: real] :
( ( ( times_times_real @ A_245 @ B_189 )
= zero_zero_real )
<=> ( ( A_245 = zero_zero_real )
| ( B_189 = zero_zero_real ) ) ) ).
thf(fact_837_mult__eq__0__iff,axiom,
! [A_245: complex,B_189: complex] :
( ( ( times_times_complex @ A_245 @ B_189 )
= zero_zero_complex )
<=> ( ( A_245 = zero_zero_complex )
| ( B_189 = zero_zero_complex ) ) ) ).
thf(fact_838_mult__eq__0__iff,axiom,
! [A_245: rat,B_189: rat] :
( ( ( times_times_rat @ A_245 @ B_189 )
= zero_zero_rat )
<=> ( ( A_245 = zero_zero_rat )
| ( B_189 = zero_zero_rat ) ) ) ).
thf(fact_839_mult__zero__right,axiom,
! [A_244: int] :
( ( times_times_int @ A_244 @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_840_mult__zero__right,axiom,
! [A_244: nat] :
( ( times_times_nat @ A_244 @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_841_mult__zero__right,axiom,
! [A_244: real] :
( ( times_times_real @ A_244 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_842_mult__zero__right,axiom,
! [A_244: code_code_numeral] :
( ( times_1655362735umeral @ A_244 @ zero_z126310315umeral )
= zero_z126310315umeral ) ).
thf(fact_843_mult__zero__right,axiom,
! [A_244: complex] :
( ( times_times_complex @ A_244 @ zero_zero_complex )
= zero_zero_complex ) ).
thf(fact_844_mult__zero__right,axiom,
! [A_244: quickcheck_code_int] :
( ( times_123202395de_int @ A_244 @ zero_z891286103de_int )
= zero_z891286103de_int ) ).
thf(fact_845_mult__zero__right,axiom,
! [A_244: rat] :
( ( times_times_rat @ A_244 @ zero_zero_rat )
= zero_zero_rat ) ).
thf(fact_846_mult__zero__left,axiom,
! [A_243: int] :
( ( times_times_int @ zero_zero_int @ A_243 )
= zero_zero_int ) ).
thf(fact_847_mult__zero__left,axiom,
! [A_243: nat] :
( ( times_times_nat @ zero_zero_nat @ A_243 )
= zero_zero_nat ) ).
thf(fact_848_mult__zero__left,axiom,
! [A_243: real] :
( ( times_times_real @ zero_zero_real @ A_243 )
= zero_zero_real ) ).
thf(fact_849_mult__zero__left,axiom,
! [A_243: code_code_numeral] :
( ( times_1655362735umeral @ zero_z126310315umeral @ A_243 )
= zero_z126310315umeral ) ).
thf(fact_850_mult__zero__left,axiom,
! [A_243: complex] :
( ( times_times_complex @ zero_zero_complex @ A_243 )
= zero_zero_complex ) ).
thf(fact_851_mult__zero__left,axiom,
! [A_243: quickcheck_code_int] :
( ( times_123202395de_int @ zero_z891286103de_int @ A_243 )
= zero_z891286103de_int ) ).
thf(fact_852_mult__zero__left,axiom,
! [A_243: rat] :
( ( times_times_rat @ zero_zero_rat @ A_243 )
= zero_zero_rat ) ).
thf(fact_853_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
thf(fact_854_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
thf(fact_855_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
thf(fact_856_zero__neq__one,axiom,
zero_z126310315umeral != one_on1645066479umeral ).
thf(fact_857_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
thf(fact_858_zero__neq__one,axiom,
zero_z891286103de_int != one_on1684967323de_int ).
thf(fact_859_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
thf(fact_860_one__neq__zero,axiom,
one_one_int != zero_zero_int ).
thf(fact_861_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
thf(fact_862_one__neq__zero,axiom,
one_one_real != zero_zero_real ).
thf(fact_863_one__neq__zero,axiom,
one_on1645066479umeral != zero_z126310315umeral ).
thf(fact_864_one__neq__zero,axiom,
one_one_complex != zero_zero_complex ).
thf(fact_865_one__neq__zero,axiom,
one_on1684967323de_int != zero_z891286103de_int ).
thf(fact_866_one__neq__zero,axiom,
one_one_rat != zero_zero_rat ).
thf(fact_867_combine__common__factor,axiom,
! [A_242: code_code_numeral,E_8: code_code_numeral,B_188: code_code_numeral,C_112: code_code_numeral] :
( ( plus_p1627245867umeral @ ( times_1655362735umeral @ A_242 @ E_8 ) @ ( plus_p1627245867umeral @ ( times_1655362735umeral @ B_188 @ E_8 ) @ C_112 ) )
= ( plus_p1627245867umeral @ ( times_1655362735umeral @ ( plus_p1627245867umeral @ A_242 @ B_188 ) @ E_8 ) @ C_112 ) ) ).
thf(fact_868_combine__common__factor,axiom,
! [A_242: int,E_8: int,B_188: int,C_112: int] :
( ( plus_plus_int @ ( times_times_int @ A_242 @ E_8 ) @ ( plus_plus_int @ ( times_times_int @ B_188 @ E_8 ) @ C_112 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A_242 @ B_188 ) @ E_8 ) @ C_112 ) ) ).
thf(fact_869_combine__common__factor,axiom,
! [A_242: nat,E_8: nat,B_188: nat,C_112: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_242 @ E_8 ) @ ( plus_plus_nat @ ( times_times_nat @ B_188 @ E_8 ) @ C_112 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A_242 @ B_188 ) @ E_8 ) @ C_112 ) ) ).
thf(fact_870_combine__common__factor,axiom,
! [A_242: real,E_8: real,B_188: real,C_112: real] :
( ( plus_plus_real @ ( times_times_real @ A_242 @ E_8 ) @ ( plus_plus_real @ ( times_times_real @ B_188 @ E_8 ) @ C_112 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A_242 @ B_188 ) @ E_8 ) @ C_112 ) ) ).
thf(fact_871_combine__common__factor,axiom,
! [A_242: complex,E_8: complex,B_188: complex,C_112: complex] :
( ( plus_plus_complex @ ( times_times_complex @ A_242 @ E_8 ) @ ( plus_plus_complex @ ( times_times_complex @ B_188 @ E_8 ) @ C_112 ) )
= ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A_242 @ B_188 ) @ E_8 ) @ C_112 ) ) ).
thf(fact_872_combine__common__factor,axiom,
! [A_242: quickcheck_code_int,E_8: quickcheck_code_int,B_188: quickcheck_code_int,C_112: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( times_123202395de_int @ A_242 @ E_8 ) @ ( plus_p1446045655de_int @ ( times_123202395de_int @ B_188 @ E_8 ) @ C_112 ) )
= ( plus_p1446045655de_int @ ( times_123202395de_int @ ( plus_p1446045655de_int @ A_242 @ B_188 ) @ E_8 ) @ C_112 ) ) ).
thf(fact_873_combine__common__factor,axiom,
! [A_242: rat,E_8: rat,B_188: rat,C_112: rat] :
( ( plus_plus_rat @ ( times_times_rat @ A_242 @ E_8 ) @ ( plus_plus_rat @ ( times_times_rat @ B_188 @ E_8 ) @ C_112 ) )
= ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A_242 @ B_188 ) @ E_8 ) @ C_112 ) ) ).
thf(fact_874_comm__semiring__class_Odistrib,axiom,
! [A_241: code_code_numeral,B_187: code_code_numeral,C_111: code_code_numeral] :
( ( times_1655362735umeral @ ( plus_p1627245867umeral @ A_241 @ B_187 ) @ C_111 )
= ( plus_p1627245867umeral @ ( times_1655362735umeral @ A_241 @ C_111 ) @ ( times_1655362735umeral @ B_187 @ C_111 ) ) ) ).
thf(fact_875_comm__semiring__class_Odistrib,axiom,
! [A_241: int,B_187: int,C_111: int] :
( ( times_times_int @ ( plus_plus_int @ A_241 @ B_187 ) @ C_111 )
= ( plus_plus_int @ ( times_times_int @ A_241 @ C_111 ) @ ( times_times_int @ B_187 @ C_111 ) ) ) ).
thf(fact_876_comm__semiring__class_Odistrib,axiom,
! [A_241: nat,B_187: nat,C_111: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_241 @ B_187 ) @ C_111 )
= ( plus_plus_nat @ ( times_times_nat @ A_241 @ C_111 ) @ ( times_times_nat @ B_187 @ C_111 ) ) ) ).
thf(fact_877_comm__semiring__class_Odistrib,axiom,
! [A_241: real,B_187: real,C_111: real] :
( ( times_times_real @ ( plus_plus_real @ A_241 @ B_187 ) @ C_111 )
= ( plus_plus_real @ ( times_times_real @ A_241 @ C_111 ) @ ( times_times_real @ B_187 @ C_111 ) ) ) ).
thf(fact_878_comm__semiring__class_Odistrib,axiom,
! [A_241: complex,B_187: complex,C_111: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A_241 @ B_187 ) @ C_111 )
= ( plus_plus_complex @ ( times_times_complex @ A_241 @ C_111 ) @ ( times_times_complex @ B_187 @ C_111 ) ) ) ).
thf(fact_879_comm__semiring__class_Odistrib,axiom,
! [A_241: quickcheck_code_int,B_187: quickcheck_code_int,C_111: quickcheck_code_int] :
( ( times_123202395de_int @ ( plus_p1446045655de_int @ A_241 @ B_187 ) @ C_111 )
= ( plus_p1446045655de_int @ ( times_123202395de_int @ A_241 @ C_111 ) @ ( times_123202395de_int @ B_187 @ C_111 ) ) ) ).
thf(fact_880_comm__semiring__class_Odistrib,axiom,
! [A_241: rat,B_187: rat,C_111: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A_241 @ B_187 ) @ C_111 )
= ( plus_plus_rat @ ( times_times_rat @ A_241 @ C_111 ) @ ( times_times_rat @ B_187 @ C_111 ) ) ) ).
thf(fact_881_field__power__not__zero,axiom,
! [N_70: nat,A_240: int] :
( ( A_240 != zero_zero_int )
=> ( ( power_power_int @ A_240 @ N_70 )
!= zero_zero_int ) ) ).
thf(fact_882_field__power__not__zero,axiom,
! [N_70: nat,A_240: real] :
( ( A_240 != zero_zero_real )
=> ( ( power_power_real @ A_240 @ N_70 )
!= zero_zero_real ) ) ).
thf(fact_883_field__power__not__zero,axiom,
! [N_70: nat,A_240: complex] :
( ( A_240 != zero_zero_complex )
=> ( ( power_power_complex @ A_240 @ N_70 )
!= zero_zero_complex ) ) ).
thf(fact_884_field__power__not__zero,axiom,
! [N_70: nat,A_240: rat] :
( ( A_240 != zero_zero_rat )
=> ( ( power_power_rat @ A_240 @ N_70 )
!= zero_zero_rat ) ) ).
thf(fact_885_dvd__0__left,axiom,
! [A_239: real] :
( ( dvd_dvd_real @ zero_zero_real @ A_239 )
=> ( A_239 = zero_zero_real ) ) ).
thf(fact_886_dvd__0__left,axiom,
! [A_239: code_code_numeral] :
( ( dvd_dv174992974umeral @ zero_z126310315umeral @ A_239 )
=> ( A_239 = zero_z126310315umeral ) ) ).
thf(fact_887_dvd__0__left,axiom,
! [A_239: complex] :
( ( dvd_dvd_complex @ zero_zero_complex @ A_239 )
=> ( A_239 = zero_zero_complex ) ) ).
thf(fact_888_dvd__0__left,axiom,
! [A_239: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ zero_z891286103de_int @ A_239 )
=> ( A_239 = zero_z891286103de_int ) ) ).
thf(fact_889_dvd__0__left,axiom,
! [A_239: rat] :
( ( dvd_dvd_rat @ zero_zero_rat @ A_239 )
=> ( A_239 = zero_zero_rat ) ) ).
thf(fact_890_dvd__0__left,axiom,
! [A_239: int] :
( ( dvd_dvd_int @ zero_zero_int @ A_239 )
=> ( A_239 = zero_zero_int ) ) ).
thf(fact_891_dvd__0__left,axiom,
! [A_239: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A_239 )
=> ( A_239 = zero_zero_nat ) ) ).
thf(fact_892_power__mult__distrib,axiom,
! [A_238: code_code_numeral,B_186: code_code_numeral,N_69: nat] :
( ( power_2100829034umeral @ ( times_1655362735umeral @ A_238 @ B_186 ) @ N_69 )
= ( times_1655362735umeral @ ( power_2100829034umeral @ A_238 @ N_69 ) @ ( power_2100829034umeral @ B_186 @ N_69 ) ) ) ).
thf(fact_893_power__mult__distrib,axiom,
! [A_238: quickcheck_code_int,B_186: quickcheck_code_int,N_69: nat] :
( ( power_881366806de_int @ ( times_123202395de_int @ A_238 @ B_186 ) @ N_69 )
= ( times_123202395de_int @ ( power_881366806de_int @ A_238 @ N_69 ) @ ( power_881366806de_int @ B_186 @ N_69 ) ) ) ).
thf(fact_894_power__mult__distrib,axiom,
! [A_238: rat,B_186: rat,N_69: nat] :
( ( power_power_rat @ ( times_times_rat @ A_238 @ B_186 ) @ N_69 )
= ( times_times_rat @ ( power_power_rat @ A_238 @ N_69 ) @ ( power_power_rat @ B_186 @ N_69 ) ) ) ).
thf(fact_895_power__mult__distrib,axiom,
! [A_238: int,B_186: int,N_69: nat] :
( ( power_power_int @ ( times_times_int @ A_238 @ B_186 ) @ N_69 )
= ( times_times_int @ ( power_power_int @ A_238 @ N_69 ) @ ( power_power_int @ B_186 @ N_69 ) ) ) ).
thf(fact_896_power__mult__distrib,axiom,
! [A_238: nat,B_186: nat,N_69: nat] :
( ( power_power_nat @ ( times_times_nat @ A_238 @ B_186 ) @ N_69 )
= ( times_times_nat @ ( power_power_nat @ A_238 @ N_69 ) @ ( power_power_nat @ B_186 @ N_69 ) ) ) ).
thf(fact_897_power__mult__distrib,axiom,
! [A_238: real,B_186: real,N_69: nat] :
( ( power_power_real @ ( times_times_real @ A_238 @ B_186 ) @ N_69 )
= ( times_times_real @ ( power_power_real @ A_238 @ N_69 ) @ ( power_power_real @ B_186 @ N_69 ) ) ) ).
thf(fact_898_power__mult__distrib,axiom,
! [A_238: complex,B_186: complex,N_69: nat] :
( ( power_power_complex @ ( times_times_complex @ A_238 @ B_186 ) @ N_69 )
= ( times_times_complex @ ( power_power_complex @ A_238 @ N_69 ) @ ( power_power_complex @ B_186 @ N_69 ) ) ) ).
thf(fact_899_power__commutes,axiom,
! [A_237: code_code_numeral,N_68: nat] :
( ( times_1655362735umeral @ ( power_2100829034umeral @ A_237 @ N_68 ) @ A_237 )
= ( times_1655362735umeral @ A_237 @ ( power_2100829034umeral @ A_237 @ N_68 ) ) ) ).
thf(fact_900_power__commutes,axiom,
! [A_237: quickcheck_code_int,N_68: nat] :
( ( times_123202395de_int @ ( power_881366806de_int @ A_237 @ N_68 ) @ A_237 )
= ( times_123202395de_int @ A_237 @ ( power_881366806de_int @ A_237 @ N_68 ) ) ) ).
thf(fact_901_power__commutes,axiom,
! [A_237: rat,N_68: nat] :
( ( times_times_rat @ ( power_power_rat @ A_237 @ N_68 ) @ A_237 )
= ( times_times_rat @ A_237 @ ( power_power_rat @ A_237 @ N_68 ) ) ) ).
thf(fact_902_power__commutes,axiom,
! [A_237: int,N_68: nat] :
( ( times_times_int @ ( power_power_int @ A_237 @ N_68 ) @ A_237 )
= ( times_times_int @ A_237 @ ( power_power_int @ A_237 @ N_68 ) ) ) ).
thf(fact_903_power__commutes,axiom,
! [A_237: nat,N_68: nat] :
( ( times_times_nat @ ( power_power_nat @ A_237 @ N_68 ) @ A_237 )
= ( times_times_nat @ A_237 @ ( power_power_nat @ A_237 @ N_68 ) ) ) ).
thf(fact_904_power__commutes,axiom,
! [A_237: real,N_68: nat] :
( ( times_times_real @ ( power_power_real @ A_237 @ N_68 ) @ A_237 )
= ( times_times_real @ A_237 @ ( power_power_real @ A_237 @ N_68 ) ) ) ).
thf(fact_905_power__commutes,axiom,
! [A_237: complex,N_68: nat] :
( ( times_times_complex @ ( power_power_complex @ A_237 @ N_68 ) @ A_237 )
= ( times_times_complex @ A_237 @ ( power_power_complex @ A_237 @ N_68 ) ) ) ).
thf(fact_906_dvd__triv__left,axiom,
! [A_236: code_code_numeral,B_185: code_code_numeral] : ( dvd_dv174992974umeral @ A_236 @ ( times_1655362735umeral @ A_236 @ B_185 ) ) ).
thf(fact_907_dvd__triv__left,axiom,
! [A_236: real,B_185: real] : ( dvd_dvd_real @ A_236 @ ( times_times_real @ A_236 @ B_185 ) ) ).
thf(fact_908_dvd__triv__left,axiom,
! [A_236: complex,B_185: complex] : ( dvd_dvd_complex @ A_236 @ ( times_times_complex @ A_236 @ B_185 ) ) ).
thf(fact_909_dvd__triv__left,axiom,
! [A_236: quickcheck_code_int,B_185: quickcheck_code_int] : ( dvd_dv1760642554de_int @ A_236 @ ( times_123202395de_int @ A_236 @ B_185 ) ) ).
thf(fact_910_dvd__triv__left,axiom,
! [A_236: rat,B_185: rat] : ( dvd_dvd_rat @ A_236 @ ( times_times_rat @ A_236 @ B_185 ) ) ).
thf(fact_911_dvd__triv__left,axiom,
! [A_236: int,B_185: int] : ( dvd_dvd_int @ A_236 @ ( times_times_int @ A_236 @ B_185 ) ) ).
thf(fact_912_dvd__triv__left,axiom,
! [A_236: nat,B_185: nat] : ( dvd_dvd_nat @ A_236 @ ( times_times_nat @ A_236 @ B_185 ) ) ).
thf(fact_913_dvd__triv__right,axiom,
! [A_235: code_code_numeral,B_184: code_code_numeral] : ( dvd_dv174992974umeral @ A_235 @ ( times_1655362735umeral @ B_184 @ A_235 ) ) ).
thf(fact_914_dvd__triv__right,axiom,
! [A_235: real,B_184: real] : ( dvd_dvd_real @ A_235 @ ( times_times_real @ B_184 @ A_235 ) ) ).
thf(fact_915_dvd__triv__right,axiom,
! [A_235: complex,B_184: complex] : ( dvd_dvd_complex @ A_235 @ ( times_times_complex @ B_184 @ A_235 ) ) ).
thf(fact_916_dvd__triv__right,axiom,
! [A_235: quickcheck_code_int,B_184: quickcheck_code_int] : ( dvd_dv1760642554de_int @ A_235 @ ( times_123202395de_int @ B_184 @ A_235 ) ) ).
thf(fact_917_dvd__triv__right,axiom,
! [A_235: rat,B_184: rat] : ( dvd_dvd_rat @ A_235 @ ( times_times_rat @ B_184 @ A_235 ) ) ).
thf(fact_918_dvd__triv__right,axiom,
! [A_235: int,B_184: int] : ( dvd_dvd_int @ A_235 @ ( times_times_int @ B_184 @ A_235 ) ) ).
thf(fact_919_dvd__triv__right,axiom,
! [A_235: nat,B_184: nat] : ( dvd_dvd_nat @ A_235 @ ( times_times_nat @ B_184 @ A_235 ) ) ).
thf(fact_920_dvd__mult2,axiom,
! [C_110: code_code_numeral,A_234: code_code_numeral,B_183: code_code_numeral] :
( ( dvd_dv174992974umeral @ A_234 @ B_183 )
=> ( dvd_dv174992974umeral @ A_234 @ ( times_1655362735umeral @ B_183 @ C_110 ) ) ) ).
thf(fact_921_dvd__mult2,axiom,
! [C_110: real,A_234: real,B_183: real] :
( ( dvd_dvd_real @ A_234 @ B_183 )
=> ( dvd_dvd_real @ A_234 @ ( times_times_real @ B_183 @ C_110 ) ) ) ).
thf(fact_922_dvd__mult2,axiom,
! [C_110: complex,A_234: complex,B_183: complex] :
( ( dvd_dvd_complex @ A_234 @ B_183 )
=> ( dvd_dvd_complex @ A_234 @ ( times_times_complex @ B_183 @ C_110 ) ) ) ).
thf(fact_923_dvd__mult2,axiom,
! [C_110: quickcheck_code_int,A_234: quickcheck_code_int,B_183: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ A_234 @ B_183 )
=> ( dvd_dv1760642554de_int @ A_234 @ ( times_123202395de_int @ B_183 @ C_110 ) ) ) ).
thf(fact_924_dvd__mult2,axiom,
! [C_110: rat,A_234: rat,B_183: rat] :
( ( dvd_dvd_rat @ A_234 @ B_183 )
=> ( dvd_dvd_rat @ A_234 @ ( times_times_rat @ B_183 @ C_110 ) ) ) ).
thf(fact_925_dvd__mult2,axiom,
! [C_110: int,A_234: int,B_183: int] :
( ( dvd_dvd_int @ A_234 @ B_183 )
=> ( dvd_dvd_int @ A_234 @ ( times_times_int @ B_183 @ C_110 ) ) ) ).
thf(fact_926_dvd__mult2,axiom,
! [C_110: nat,A_234: nat,B_183: nat] :
( ( dvd_dvd_nat @ A_234 @ B_183 )
=> ( dvd_dvd_nat @ A_234 @ ( times_times_nat @ B_183 @ C_110 ) ) ) ).
thf(fact_927_dvd__mult,axiom,
! [B_182: code_code_numeral,A_233: code_code_numeral,C_109: code_code_numeral] :
( ( dvd_dv174992974umeral @ A_233 @ C_109 )
=> ( dvd_dv174992974umeral @ A_233 @ ( times_1655362735umeral @ B_182 @ C_109 ) ) ) ).
thf(fact_928_dvd__mult,axiom,
! [B_182: real,A_233: real,C_109: real] :
( ( dvd_dvd_real @ A_233 @ C_109 )
=> ( dvd_dvd_real @ A_233 @ ( times_times_real @ B_182 @ C_109 ) ) ) ).
thf(fact_929_dvd__mult,axiom,
! [B_182: complex,A_233: complex,C_109: complex] :
( ( dvd_dvd_complex @ A_233 @ C_109 )
=> ( dvd_dvd_complex @ A_233 @ ( times_times_complex @ B_182 @ C_109 ) ) ) ).
thf(fact_930_dvd__mult,axiom,
! [B_182: quickcheck_code_int,A_233: quickcheck_code_int,C_109: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ A_233 @ C_109 )
=> ( dvd_dv1760642554de_int @ A_233 @ ( times_123202395de_int @ B_182 @ C_109 ) ) ) ).
thf(fact_931_dvd__mult,axiom,
! [B_182: rat,A_233: rat,C_109: rat] :
( ( dvd_dvd_rat @ A_233 @ C_109 )
=> ( dvd_dvd_rat @ A_233 @ ( times_times_rat @ B_182 @ C_109 ) ) ) ).
thf(fact_932_dvd__mult,axiom,
! [B_182: int,A_233: int,C_109: int] :
( ( dvd_dvd_int @ A_233 @ C_109 )
=> ( dvd_dvd_int @ A_233 @ ( times_times_int @ B_182 @ C_109 ) ) ) ).
thf(fact_933_dvd__mult,axiom,
! [B_182: nat,A_233: nat,C_109: nat] :
( ( dvd_dvd_nat @ A_233 @ C_109 )
=> ( dvd_dvd_nat @ A_233 @ ( times_times_nat @ B_182 @ C_109 ) ) ) ).
thf(fact_934_mult__dvd__mono,axiom,
! [C_108: code_code_numeral,D_33: code_code_numeral,A_232: code_code_numeral,B_181: code_code_numeral] :
( ( dvd_dv174992974umeral @ A_232 @ B_181 )
=> ( ( dvd_dv174992974umeral @ C_108 @ D_33 )
=> ( dvd_dv174992974umeral @ ( times_1655362735umeral @ A_232 @ C_108 ) @ ( times_1655362735umeral @ B_181 @ D_33 ) ) ) ) ).
thf(fact_935_mult__dvd__mono,axiom,
! [C_108: real,D_33: real,A_232: real,B_181: real] :
( ( dvd_dvd_real @ A_232 @ B_181 )
=> ( ( dvd_dvd_real @ C_108 @ D_33 )
=> ( dvd_dvd_real @ ( times_times_real @ A_232 @ C_108 ) @ ( times_times_real @ B_181 @ D_33 ) ) ) ) ).
thf(fact_936_mult__dvd__mono,axiom,
! [C_108: complex,D_33: complex,A_232: complex,B_181: complex] :
( ( dvd_dvd_complex @ A_232 @ B_181 )
=> ( ( dvd_dvd_complex @ C_108 @ D_33 )
=> ( dvd_dvd_complex @ ( times_times_complex @ A_232 @ C_108 ) @ ( times_times_complex @ B_181 @ D_33 ) ) ) ) ).
thf(fact_937_mult__dvd__mono,axiom,
! [C_108: quickcheck_code_int,D_33: quickcheck_code_int,A_232: quickcheck_code_int,B_181: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ A_232 @ B_181 )
=> ( ( dvd_dv1760642554de_int @ C_108 @ D_33 )
=> ( dvd_dv1760642554de_int @ ( times_123202395de_int @ A_232 @ C_108 ) @ ( times_123202395de_int @ B_181 @ D_33 ) ) ) ) ).
thf(fact_938_mult__dvd__mono,axiom,
! [C_108: rat,D_33: rat,A_232: rat,B_181: rat] :
( ( dvd_dvd_rat @ A_232 @ B_181 )
=> ( ( dvd_dvd_rat @ C_108 @ D_33 )
=> ( dvd_dvd_rat @ ( times_times_rat @ A_232 @ C_108 ) @ ( times_times_rat @ B_181 @ D_33 ) ) ) ) ).
thf(fact_939_mult__dvd__mono,axiom,
! [C_108: int,D_33: int,A_232: int,B_181: int] :
( ( dvd_dvd_int @ A_232 @ B_181 )
=> ( ( dvd_dvd_int @ C_108 @ D_33 )
=> ( dvd_dvd_int @ ( times_times_int @ A_232 @ C_108 ) @ ( times_times_int @ B_181 @ D_33 ) ) ) ) ).
thf(fact_940_mult__dvd__mono,axiom,
! [C_108: nat,D_33: nat,A_232: nat,B_181: nat] :
( ( dvd_dvd_nat @ A_232 @ B_181 )
=> ( ( dvd_dvd_nat @ C_108 @ D_33 )
=> ( dvd_dvd_nat @ ( times_times_nat @ A_232 @ C_108 ) @ ( times_times_nat @ B_181 @ D_33 ) ) ) ) ).
thf(fact_941_dvdI,axiom,
! [A_231: code_code_numeral,B_180: code_code_numeral,K_7: code_code_numeral] :
( ( A_231
= ( times_1655362735umeral @ B_180 @ K_7 ) )
=> ( dvd_dv174992974umeral @ B_180 @ A_231 ) ) ).
thf(fact_942_dvdI,axiom,
! [A_231: real,B_180: real,K_7: real] :
( ( A_231
= ( times_times_real @ B_180 @ K_7 ) )
=> ( dvd_dvd_real @ B_180 @ A_231 ) ) ).
thf(fact_943_dvdI,axiom,
! [A_231: complex,B_180: complex,K_7: complex] :
( ( A_231
= ( times_times_complex @ B_180 @ K_7 ) )
=> ( dvd_dvd_complex @ B_180 @ A_231 ) ) ).
thf(fact_944_dvdI,axiom,
! [A_231: quickcheck_code_int,B_180: quickcheck_code_int,K_7: quickcheck_code_int] :
( ( A_231
= ( times_123202395de_int @ B_180 @ K_7 ) )
=> ( dvd_dv1760642554de_int @ B_180 @ A_231 ) ) ).
thf(fact_945_dvdI,axiom,
! [A_231: rat,B_180: rat,K_7: rat] :
( ( A_231
= ( times_times_rat @ B_180 @ K_7 ) )
=> ( dvd_dvd_rat @ B_180 @ A_231 ) ) ).
thf(fact_946_dvdI,axiom,
! [A_231: int,B_180: int,K_7: int] :
( ( A_231
= ( times_times_int @ B_180 @ K_7 ) )
=> ( dvd_dvd_int @ B_180 @ A_231 ) ) ).
thf(fact_947_dvdI,axiom,
! [A_231: nat,B_180: nat,K_7: nat] :
( ( A_231
= ( times_times_nat @ B_180 @ K_7 ) )
=> ( dvd_dvd_nat @ B_180 @ A_231 ) ) ).
thf(fact_948_dvd__mult__left,axiom,
! [A_230: code_code_numeral,B_179: code_code_numeral,C_107: code_code_numeral] :
( ( dvd_dv174992974umeral @ ( times_1655362735umeral @ A_230 @ B_179 ) @ C_107 )
=> ( dvd_dv174992974umeral @ A_230 @ C_107 ) ) ).
thf(fact_949_dvd__mult__left,axiom,
! [A_230: real,B_179: real,C_107: real] :
( ( dvd_dvd_real @ ( times_times_real @ A_230 @ B_179 ) @ C_107 )
=> ( dvd_dvd_real @ A_230 @ C_107 ) ) ).
thf(fact_950_dvd__mult__left,axiom,
! [A_230: complex,B_179: complex,C_107: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A_230 @ B_179 ) @ C_107 )
=> ( dvd_dvd_complex @ A_230 @ C_107 ) ) ).
thf(fact_951_dvd__mult__left,axiom,
! [A_230: quickcheck_code_int,B_179: quickcheck_code_int,C_107: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ ( times_123202395de_int @ A_230 @ B_179 ) @ C_107 )
=> ( dvd_dv1760642554de_int @ A_230 @ C_107 ) ) ).
thf(fact_952_dvd__mult__left,axiom,
! [A_230: rat,B_179: rat,C_107: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A_230 @ B_179 ) @ C_107 )
=> ( dvd_dvd_rat @ A_230 @ C_107 ) ) ).
thf(fact_953_dvd__mult__left,axiom,
! [A_230: int,B_179: int,C_107: int] :
( ( dvd_dvd_int @ ( times_times_int @ A_230 @ B_179 ) @ C_107 )
=> ( dvd_dvd_int @ A_230 @ C_107 ) ) ).
thf(fact_954_dvd__mult__left,axiom,
! [A_230: nat,B_179: nat,C_107: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A_230 @ B_179 ) @ C_107 )
=> ( dvd_dvd_nat @ A_230 @ C_107 ) ) ).
thf(fact_955_dvd__mult__right,axiom,
! [A_229: code_code_numeral,B_178: code_code_numeral,C_106: code_code_numeral] :
( ( dvd_dv174992974umeral @ ( times_1655362735umeral @ A_229 @ B_178 ) @ C_106 )
=> ( dvd_dv174992974umeral @ B_178 @ C_106 ) ) ).
thf(fact_956_dvd__mult__right,axiom,
! [A_229: real,B_178: real,C_106: real] :
( ( dvd_dvd_real @ ( times_times_real @ A_229 @ B_178 ) @ C_106 )
=> ( dvd_dvd_real @ B_178 @ C_106 ) ) ).
thf(fact_957_dvd__mult__right,axiom,
! [A_229: complex,B_178: complex,C_106: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A_229 @ B_178 ) @ C_106 )
=> ( dvd_dvd_complex @ B_178 @ C_106 ) ) ).
thf(fact_958_dvd__mult__right,axiom,
! [A_229: quickcheck_code_int,B_178: quickcheck_code_int,C_106: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ ( times_123202395de_int @ A_229 @ B_178 ) @ C_106 )
=> ( dvd_dv1760642554de_int @ B_178 @ C_106 ) ) ).
thf(fact_959_dvd__mult__right,axiom,
! [A_229: rat,B_178: rat,C_106: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A_229 @ B_178 ) @ C_106 )
=> ( dvd_dvd_rat @ B_178 @ C_106 ) ) ).
thf(fact_960_dvd__mult__right,axiom,
! [A_229: int,B_178: int,C_106: int] :
( ( dvd_dvd_int @ ( times_times_int @ A_229 @ B_178 ) @ C_106 )
=> ( dvd_dvd_int @ B_178 @ C_106 ) ) ).
thf(fact_961_dvd__mult__right,axiom,
! [A_229: nat,B_178: nat,C_106: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A_229 @ B_178 ) @ C_106 )
=> ( dvd_dvd_nat @ B_178 @ C_106 ) ) ).
thf(fact_962_dvd__add,axiom,
! [C_105: code_code_numeral,A_228: code_code_numeral,B_177: code_code_numeral] :
( ( dvd_dv174992974umeral @ A_228 @ B_177 )
=> ( ( dvd_dv174992974umeral @ A_228 @ C_105 )
=> ( dvd_dv174992974umeral @ A_228 @ ( plus_p1627245867umeral @ B_177 @ C_105 ) ) ) ) ).
thf(fact_963_dvd__add,axiom,
! [C_105: real,A_228: real,B_177: real] :
( ( dvd_dvd_real @ A_228 @ B_177 )
=> ( ( dvd_dvd_real @ A_228 @ C_105 )
=> ( dvd_dvd_real @ A_228 @ ( plus_plus_real @ B_177 @ C_105 ) ) ) ) ).
thf(fact_964_dvd__add,axiom,
! [C_105: complex,A_228: complex,B_177: complex] :
( ( dvd_dvd_complex @ A_228 @ B_177 )
=> ( ( dvd_dvd_complex @ A_228 @ C_105 )
=> ( dvd_dvd_complex @ A_228 @ ( plus_plus_complex @ B_177 @ C_105 ) ) ) ) ).
thf(fact_965_dvd__add,axiom,
! [C_105: quickcheck_code_int,A_228: quickcheck_code_int,B_177: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ A_228 @ B_177 )
=> ( ( dvd_dv1760642554de_int @ A_228 @ C_105 )
=> ( dvd_dv1760642554de_int @ A_228 @ ( plus_p1446045655de_int @ B_177 @ C_105 ) ) ) ) ).
thf(fact_966_dvd__add,axiom,
! [C_105: rat,A_228: rat,B_177: rat] :
( ( dvd_dvd_rat @ A_228 @ B_177 )
=> ( ( dvd_dvd_rat @ A_228 @ C_105 )
=> ( dvd_dvd_rat @ A_228 @ ( plus_plus_rat @ B_177 @ C_105 ) ) ) ) ).
thf(fact_967_dvd__add,axiom,
! [C_105: int,A_228: int,B_177: int] :
( ( dvd_dvd_int @ A_228 @ B_177 )
=> ( ( dvd_dvd_int @ A_228 @ C_105 )
=> ( dvd_dvd_int @ A_228 @ ( plus_plus_int @ B_177 @ C_105 ) ) ) ) ).
thf(fact_968_dvd__add,axiom,
! [C_105: nat,A_228: nat,B_177: nat] :
( ( dvd_dvd_nat @ A_228 @ B_177 )
=> ( ( dvd_dvd_nat @ A_228 @ C_105 )
=> ( dvd_dvd_nat @ A_228 @ ( plus_plus_nat @ B_177 @ C_105 ) ) ) ) ).
thf(fact_969_power__one,axiom,
! [N_67: nat] :
( ( power_power_int @ one_one_int @ N_67 )
= one_one_int ) ).
thf(fact_970_power__one,axiom,
! [N_67: nat] :
( ( power_power_nat @ one_one_nat @ N_67 )
= one_one_nat ) ).
thf(fact_971_power__one,axiom,
! [N_67: nat] :
( ( power_power_real @ one_one_real @ N_67 )
= one_one_real ) ).
thf(fact_972_power__one,axiom,
! [N_67: nat] :
( ( power_2100829034umeral @ one_on1645066479umeral @ N_67 )
= one_on1645066479umeral ) ).
thf(fact_973_power__one,axiom,
! [N_67: nat] :
( ( power_power_complex @ one_one_complex @ N_67 )
= one_one_complex ) ).
thf(fact_974_power__one,axiom,
! [N_67: nat] :
( ( power_881366806de_int @ one_on1684967323de_int @ N_67 )
= one_on1684967323de_int ) ).
thf(fact_975_power__one,axiom,
! [N_67: nat] :
( ( power_power_rat @ one_one_rat @ N_67 )
= one_one_rat ) ).
thf(fact_976_one__dvd,axiom,
! [A_227: real] : ( dvd_dvd_real @ one_one_real @ A_227 ) ).
thf(fact_977_one__dvd,axiom,
! [A_227: code_code_numeral] : ( dvd_dv174992974umeral @ one_on1645066479umeral @ A_227 ) ).
thf(fact_978_one__dvd,axiom,
! [A_227: complex] : ( dvd_dvd_complex @ one_one_complex @ A_227 ) ).
thf(fact_979_one__dvd,axiom,
! [A_227: quickcheck_code_int] : ( dvd_dv1760642554de_int @ one_on1684967323de_int @ A_227 ) ).
thf(fact_980_one__dvd,axiom,
! [A_227: rat] : ( dvd_dvd_rat @ one_one_rat @ A_227 ) ).
thf(fact_981_one__dvd,axiom,
! [A_227: int] : ( dvd_dvd_int @ one_one_int @ A_227 ) ).
thf(fact_982_one__dvd,axiom,
! [A_227: nat] : ( dvd_dvd_nat @ one_one_nat @ A_227 ) ).
thf(fact_983_dvd__power__same,axiom,
! [N_66: nat,X_35: quickcheck_code_int,Y_26: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ X_35 @ Y_26 )
=> ( dvd_dv1760642554de_int @ ( power_881366806de_int @ X_35 @ N_66 ) @ ( power_881366806de_int @ Y_26 @ N_66 ) ) ) ).
thf(fact_984_dvd__power__same,axiom,
! [N_66: nat,X_35: code_code_numeral,Y_26: code_code_numeral] :
( ( dvd_dv174992974umeral @ X_35 @ Y_26 )
=> ( dvd_dv174992974umeral @ ( power_2100829034umeral @ X_35 @ N_66 ) @ ( power_2100829034umeral @ Y_26 @ N_66 ) ) ) ).
thf(fact_985_dvd__power__same,axiom,
! [N_66: nat,X_35: rat,Y_26: rat] :
( ( dvd_dvd_rat @ X_35 @ Y_26 )
=> ( dvd_dvd_rat @ ( power_power_rat @ X_35 @ N_66 ) @ ( power_power_rat @ Y_26 @ N_66 ) ) ) ).
thf(fact_986_dvd__power__same,axiom,
! [N_66: nat,X_35: real,Y_26: real] :
( ( dvd_dvd_real @ X_35 @ Y_26 )
=> ( dvd_dvd_real @ ( power_power_real @ X_35 @ N_66 ) @ ( power_power_real @ Y_26 @ N_66 ) ) ) ).
thf(fact_987_dvd__power__same,axiom,
! [N_66: nat,X_35: complex,Y_26: complex] :
( ( dvd_dvd_complex @ X_35 @ Y_26 )
=> ( dvd_dvd_complex @ ( power_power_complex @ X_35 @ N_66 ) @ ( power_power_complex @ Y_26 @ N_66 ) ) ) ).
thf(fact_988_dvd__power__same,axiom,
! [N_66: nat,X_35: int,Y_26: int] :
( ( dvd_dvd_int @ X_35 @ Y_26 )
=> ( dvd_dvd_int @ ( power_power_int @ X_35 @ N_66 ) @ ( power_power_int @ Y_26 @ N_66 ) ) ) ).
thf(fact_989_dvd__power__same,axiom,
! [N_66: nat,X_35: nat,Y_26: nat] :
( ( dvd_dvd_nat @ X_35 @ Y_26 )
=> ( dvd_dvd_nat @ ( power_power_nat @ X_35 @ N_66 ) @ ( power_power_nat @ Y_26 @ N_66 ) ) ) ).
thf(fact_990_power__le__dvd,axiom,
! [M_31: nat,A_226: quickcheck_code_int,N_65: nat,B_176: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ ( power_881366806de_int @ A_226 @ N_65 ) @ B_176 )
=> ( ( ord_less_eq_nat @ M_31 @ N_65 )
=> ( dvd_dv1760642554de_int @ ( power_881366806de_int @ A_226 @ M_31 ) @ B_176 ) ) ) ).
thf(fact_991_power__le__dvd,axiom,
! [M_31: nat,A_226: code_code_numeral,N_65: nat,B_176: code_code_numeral] :
( ( dvd_dv174992974umeral @ ( power_2100829034umeral @ A_226 @ N_65 ) @ B_176 )
=> ( ( ord_less_eq_nat @ M_31 @ N_65 )
=> ( dvd_dv174992974umeral @ ( power_2100829034umeral @ A_226 @ M_31 ) @ B_176 ) ) ) ).
thf(fact_992_power__le__dvd,axiom,
! [M_31: nat,A_226: rat,N_65: nat,B_176: rat] :
( ( dvd_dvd_rat @ ( power_power_rat @ A_226 @ N_65 ) @ B_176 )
=> ( ( ord_less_eq_nat @ M_31 @ N_65 )
=> ( dvd_dvd_rat @ ( power_power_rat @ A_226 @ M_31 ) @ B_176 ) ) ) ).
thf(fact_993_power__le__dvd,axiom,
! [M_31: nat,A_226: real,N_65: nat,B_176: real] :
( ( dvd_dvd_real @ ( power_power_real @ A_226 @ N_65 ) @ B_176 )
=> ( ( ord_less_eq_nat @ M_31 @ N_65 )
=> ( dvd_dvd_real @ ( power_power_real @ A_226 @ M_31 ) @ B_176 ) ) ) ).
thf(fact_994_power__le__dvd,axiom,
! [M_31: nat,A_226: complex,N_65: nat,B_176: complex] :
( ( dvd_dvd_complex @ ( power_power_complex @ A_226 @ N_65 ) @ B_176 )
=> ( ( ord_less_eq_nat @ M_31 @ N_65 )
=> ( dvd_dvd_complex @ ( power_power_complex @ A_226 @ M_31 ) @ B_176 ) ) ) ).
thf(fact_995_power__le__dvd,axiom,
! [M_31: nat,A_226: int,N_65: nat,B_176: int] :
( ( dvd_dvd_int @ ( power_power_int @ A_226 @ N_65 ) @ B_176 )
=> ( ( ord_less_eq_nat @ M_31 @ N_65 )
=> ( dvd_dvd_int @ ( power_power_int @ A_226 @ M_31 ) @ B_176 ) ) ) ).
thf(fact_996_power__le__dvd,axiom,
! [M_31: nat,A_226: nat,N_65: nat,B_176: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A_226 @ N_65 ) @ B_176 )
=> ( ( ord_less_eq_nat @ M_31 @ N_65 )
=> ( dvd_dvd_nat @ ( power_power_nat @ A_226 @ M_31 ) @ B_176 ) ) ) ).
thf(fact_997_dvd__power__le,axiom,
! [N_64: nat,M_30: nat,X_34: quickcheck_code_int,Y_25: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ X_34 @ Y_25 )
=> ( ( ord_less_eq_nat @ N_64 @ M_30 )
=> ( dvd_dv1760642554de_int @ ( power_881366806de_int @ X_34 @ N_64 ) @ ( power_881366806de_int @ Y_25 @ M_30 ) ) ) ) ).
thf(fact_998_dvd__power__le,axiom,
! [N_64: nat,M_30: nat,X_34: code_code_numeral,Y_25: code_code_numeral] :
( ( dvd_dv174992974umeral @ X_34 @ Y_25 )
=> ( ( ord_less_eq_nat @ N_64 @ M_30 )
=> ( dvd_dv174992974umeral @ ( power_2100829034umeral @ X_34 @ N_64 ) @ ( power_2100829034umeral @ Y_25 @ M_30 ) ) ) ) ).
thf(fact_999_dvd__power__le,axiom,
! [N_64: nat,M_30: nat,X_34: rat,Y_25: rat] :
( ( dvd_dvd_rat @ X_34 @ Y_25 )
=> ( ( ord_less_eq_nat @ N_64 @ M_30 )
=> ( dvd_dvd_rat @ ( power_power_rat @ X_34 @ N_64 ) @ ( power_power_rat @ Y_25 @ M_30 ) ) ) ) ).
thf(fact_1000_dvd__power__le,axiom,
! [N_64: nat,M_30: nat,X_34: real,Y_25: real] :
( ( dvd_dvd_real @ X_34 @ Y_25 )
=> ( ( ord_less_eq_nat @ N_64 @ M_30 )
=> ( dvd_dvd_real @ ( power_power_real @ X_34 @ N_64 ) @ ( power_power_real @ Y_25 @ M_30 ) ) ) ) ).
thf(fact_1001_dvd__power__le,axiom,
! [N_64: nat,M_30: nat,X_34: complex,Y_25: complex] :
( ( dvd_dvd_complex @ X_34 @ Y_25 )
=> ( ( ord_less_eq_nat @ N_64 @ M_30 )
=> ( dvd_dvd_complex @ ( power_power_complex @ X_34 @ N_64 ) @ ( power_power_complex @ Y_25 @ M_30 ) ) ) ) ).
thf(fact_1002_dvd__power__le,axiom,
! [N_64: nat,M_30: nat,X_34: int,Y_25: int] :
( ( dvd_dvd_int @ X_34 @ Y_25 )
=> ( ( ord_less_eq_nat @ N_64 @ M_30 )
=> ( dvd_dvd_int @ ( power_power_int @ X_34 @ N_64 ) @ ( power_power_int @ Y_25 @ M_30 ) ) ) ) ).
thf(fact_1003_dvd__power__le,axiom,
! [N_64: nat,M_30: nat,X_34: nat,Y_25: nat] :
( ( dvd_dvd_nat @ X_34 @ Y_25 )
=> ( ( ord_less_eq_nat @ N_64 @ M_30 )
=> ( dvd_dvd_nat @ ( power_power_nat @ X_34 @ N_64 ) @ ( power_power_nat @ Y_25 @ M_30 ) ) ) ) ).
thf(fact_1004_le__imp__power__dvd,axiom,
! [A_225: quickcheck_code_int,M_29: nat,N_63: nat] :
( ( ord_less_eq_nat @ M_29 @ N_63 )
=> ( dvd_dv1760642554de_int @ ( power_881366806de_int @ A_225 @ M_29 ) @ ( power_881366806de_int @ A_225 @ N_63 ) ) ) ).
thf(fact_1005_le__imp__power__dvd,axiom,
! [A_225: code_code_numeral,M_29: nat,N_63: nat] :
( ( ord_less_eq_nat @ M_29 @ N_63 )
=> ( dvd_dv174992974umeral @ ( power_2100829034umeral @ A_225 @ M_29 ) @ ( power_2100829034umeral @ A_225 @ N_63 ) ) ) ).
thf(fact_1006_le__imp__power__dvd,axiom,
! [A_225: rat,M_29: nat,N_63: nat] :
( ( ord_less_eq_nat @ M_29 @ N_63 )
=> ( dvd_dvd_rat @ ( power_power_rat @ A_225 @ M_29 ) @ ( power_power_rat @ A_225 @ N_63 ) ) ) ).
thf(fact_1007_le__imp__power__dvd,axiom,
! [A_225: real,M_29: nat,N_63: nat] :
( ( ord_less_eq_nat @ M_29 @ N_63 )
=> ( dvd_dvd_real @ ( power_power_real @ A_225 @ M_29 ) @ ( power_power_real @ A_225 @ N_63 ) ) ) ).
thf(fact_1008_le__imp__power__dvd,axiom,
! [A_225: complex,M_29: nat,N_63: nat] :
( ( ord_less_eq_nat @ M_29 @ N_63 )
=> ( dvd_dvd_complex @ ( power_power_complex @ A_225 @ M_29 ) @ ( power_power_complex @ A_225 @ N_63 ) ) ) ).
thf(fact_1009_le__imp__power__dvd,axiom,
! [A_225: int,M_29: nat,N_63: nat] :
( ( ord_less_eq_nat @ M_29 @ N_63 )
=> ( dvd_dvd_int @ ( power_power_int @ A_225 @ M_29 ) @ ( power_power_int @ A_225 @ N_63 ) ) ) ).
thf(fact_1010_le__imp__power__dvd,axiom,
! [A_225: nat,M_29: nat,N_63: nat] :
( ( ord_less_eq_nat @ M_29 @ N_63 )
=> ( dvd_dvd_nat @ ( power_power_nat @ A_225 @ M_29 ) @ ( power_power_nat @ A_225 @ N_63 ) ) ) ).
thf(fact_1011_nat__power__less__imp__less,axiom,
! [M: nat,N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_1012_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
thf(fact_1013_power__mult,axiom,
! [A_224: quickcheck_code_int,M_28: nat,N_62: nat] :
( ( power_881366806de_int @ A_224 @ ( times_times_nat @ M_28 @ N_62 ) )
= ( power_881366806de_int @ ( power_881366806de_int @ A_224 @ M_28 ) @ N_62 ) ) ).
thf(fact_1014_power__mult,axiom,
! [A_224: code_code_numeral,M_28: nat,N_62: nat] :
( ( power_2100829034umeral @ A_224 @ ( times_times_nat @ M_28 @ N_62 ) )
= ( power_2100829034umeral @ ( power_2100829034umeral @ A_224 @ M_28 ) @ N_62 ) ) ).
thf(fact_1015_power__mult,axiom,
! [A_224: rat,M_28: nat,N_62: nat] :
( ( power_power_rat @ A_224 @ ( times_times_nat @ M_28 @ N_62 ) )
= ( power_power_rat @ ( power_power_rat @ A_224 @ M_28 ) @ N_62 ) ) ).
thf(fact_1016_power__mult,axiom,
! [A_224: int,M_28: nat,N_62: nat] :
( ( power_power_int @ A_224 @ ( times_times_nat @ M_28 @ N_62 ) )
= ( power_power_int @ ( power_power_int @ A_224 @ M_28 ) @ N_62 ) ) ).
thf(fact_1017_power__mult,axiom,
! [A_224: nat,M_28: nat,N_62: nat] :
( ( power_power_nat @ A_224 @ ( times_times_nat @ M_28 @ N_62 ) )
= ( power_power_nat @ ( power_power_nat @ A_224 @ M_28 ) @ N_62 ) ) ).
thf(fact_1018_power__mult,axiom,
! [A_224: real,M_28: nat,N_62: nat] :
( ( power_power_real @ A_224 @ ( times_times_nat @ M_28 @ N_62 ) )
= ( power_power_real @ ( power_power_real @ A_224 @ M_28 ) @ N_62 ) ) ).
thf(fact_1019_power__mult,axiom,
! [A_224: complex,M_28: nat,N_62: nat] :
( ( power_power_complex @ A_224 @ ( times_times_nat @ M_28 @ N_62 ) )
= ( power_power_complex @ ( power_power_complex @ A_224 @ M_28 ) @ N_62 ) ) ).
thf(fact_1020_Euler_Oaux2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ A @ C )
=> ( ( ord_less_int @ B @ C )
=> ( ( ord_less_eq_int @ A @ B )
| ( ord_less_eq_int @ B @ A ) ) ) ) ).
thf(fact_1021_power__one__right,axiom,
! [A_223: quickcheck_code_int] :
( ( power_881366806de_int @ A_223 @ one_one_nat )
= A_223 ) ).
thf(fact_1022_power__one__right,axiom,
! [A_223: code_code_numeral] :
( ( power_2100829034umeral @ A_223 @ one_one_nat )
= A_223 ) ).
thf(fact_1023_power__one__right,axiom,
! [A_223: rat] :
( ( power_power_rat @ A_223 @ one_one_nat )
= A_223 ) ).
thf(fact_1024_power__one__right,axiom,
! [A_223: int] :
( ( power_power_int @ A_223 @ one_one_nat )
= A_223 ) ).
thf(fact_1025_power__one__right,axiom,
! [A_223: nat] :
( ( power_power_nat @ A_223 @ one_one_nat )
= A_223 ) ).
thf(fact_1026_power__one__right,axiom,
! [A_223: real] :
( ( power_power_real @ A_223 @ one_one_nat )
= A_223 ) ).
thf(fact_1027_power__one__right,axiom,
! [A_223: complex] :
( ( power_power_complex @ A_223 @ one_one_nat )
= A_223 ) ).
thf(fact_1028_zcong__id,axiom,
! [M: int] : ( zcong @ M @ zero_zero_int @ M ) ).
thf(fact_1029_IntPrimes_Ozcong__zero,axiom,
! [A: int,B: int] :
( ( zcong @ A @ B @ zero_zero_int )
<=> ( A = B ) ) ).
thf(fact_1030_zcong__1,axiom,
! [A: int,B: int] : ( zcong @ A @ B @ one_one_int ) ).
thf(fact_1031_zcong__zmult__self,axiom,
! [A: int,M: int,B: int] : ( zcong @ ( times_times_int @ A @ M ) @ ( times_times_int @ B @ M ) @ M ) ).
thf(fact_1032_zcong__zmult__prop1,axiom,
! [C: int,D: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ C @ ( times_times_int @ A @ D ) @ M )
<=> ( zcong @ C @ ( times_times_int @ B @ D ) @ M ) ) ) ).
thf(fact_1033_zcong__zmult__prop2,axiom,
! [C: int,D: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ C @ ( times_times_int @ D @ A ) @ M )
<=> ( zcong @ C @ ( times_times_int @ D @ B ) @ M ) ) ) ).
thf(fact_1034_zcong__scalar,axiom,
! [K_1: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( zcong @ ( times_times_int @ A @ K_1 ) @ ( times_times_int @ B @ K_1 ) @ M ) ) ).
thf(fact_1035_zcong__scalar2,axiom,
! [K_1: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( zcong @ ( times_times_int @ K_1 @ A ) @ ( times_times_int @ K_1 @ B ) @ M ) ) ).
thf(fact_1036_zcong__zmult,axiom,
! [C: int,D: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ C @ D @ M )
=> ( zcong @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) @ M ) ) ) ).
thf(fact_1037_zcong__zadd,axiom,
! [C: int,D: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ C @ D @ M )
=> ( zcong @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) @ M ) ) ) ).
thf(fact_1038_zcong__shift,axiom,
! [C: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( zcong @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) @ M ) ) ).
thf(fact_1039_zcong__zpower,axiom,
! [Z_1: nat,X: int,Y: int,M: int] :
( ( zcong @ X @ Y @ M )
=> ( zcong @ ( power_power_int @ X @ Z_1 ) @ ( power_power_int @ Y @ Z_1 ) @ M ) ) ).
thf(fact_1040_power__m1__even,axiom,
! [N_61: nat] :
( ( power_power_int @ ( number_number_of_int @ min ) @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_61 ) )
= one_one_int ) ).
thf(fact_1041_power__m1__even,axiom,
! [N_61: nat] :
( ( power_power_real @ ( number267125858f_real @ min ) @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_61 ) )
= one_one_real ) ).
thf(fact_1042_power__m1__even,axiom,
! [N_61: nat] :
( ( power_power_complex @ ( number528085621omplex @ min ) @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_61 ) )
= one_one_complex ) ).
thf(fact_1043_power__m1__even,axiom,
! [N_61: nat] :
( ( power_power_rat @ ( number_number_of_rat @ min ) @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_61 ) )
= one_one_rat ) ).
thf(fact_1044_split__mult__neg__le,axiom,
! [B_175: int,A_222: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A_222 )
& ( ord_less_eq_int @ B_175 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A_222 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B_175 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A_222 @ B_175 ) @ zero_zero_int ) ) ).
thf(fact_1045_split__mult__neg__le,axiom,
! [B_175: nat,A_222: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A_222 )
& ( ord_less_eq_nat @ B_175 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A_222 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B_175 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_222 @ B_175 ) @ zero_zero_nat ) ) ).
thf(fact_1046_split__mult__neg__le,axiom,
! [B_175: real,A_222: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A_222 )
& ( ord_less_eq_real @ B_175 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A_222 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B_175 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A_222 @ B_175 ) @ zero_zero_real ) ) ).
thf(fact_1047_split__mult__neg__le,axiom,
! [B_175: code_code_numeral,A_222: code_code_numeral] :
( ( ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_222 )
& ( ord_le565307924umeral @ B_175 @ zero_z126310315umeral ) )
| ( ( ord_le565307924umeral @ A_222 @ zero_z126310315umeral )
& ( ord_le565307924umeral @ zero_z126310315umeral @ B_175 ) ) )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ A_222 @ B_175 ) @ zero_z126310315umeral ) ) ).
thf(fact_1048_split__mult__neg__le,axiom,
! [B_175: quickcheck_code_int,A_222: quickcheck_code_int] :
( ( ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_222 )
& ( ord_le258702272de_int @ B_175 @ zero_z891286103de_int ) )
| ( ( ord_le258702272de_int @ A_222 @ zero_z891286103de_int )
& ( ord_le258702272de_int @ zero_z891286103de_int @ B_175 ) ) )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ A_222 @ B_175 ) @ zero_z891286103de_int ) ) ).
thf(fact_1049_split__mult__neg__le,axiom,
! [B_175: rat,A_222: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A_222 )
& ( ord_less_eq_rat @ B_175 @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A_222 @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B_175 ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_222 @ B_175 ) @ zero_zero_rat ) ) ).
thf(fact_1050_split__mult__pos__le,axiom,
! [B_174: int,A_221: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A_221 )
& ( ord_less_eq_int @ zero_zero_int @ B_174 ) )
| ( ( ord_less_eq_int @ A_221 @ zero_zero_int )
& ( ord_less_eq_int @ B_174 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_221 @ B_174 ) ) ) ).
thf(fact_1051_split__mult__pos__le,axiom,
! [B_174: real,A_221: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A_221 )
& ( ord_less_eq_real @ zero_zero_real @ B_174 ) )
| ( ( ord_less_eq_real @ A_221 @ zero_zero_real )
& ( ord_less_eq_real @ B_174 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_221 @ B_174 ) ) ) ).
thf(fact_1052_split__mult__pos__le,axiom,
! [B_174: rat,A_221: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A_221 )
& ( ord_less_eq_rat @ zero_zero_rat @ B_174 ) )
| ( ( ord_less_eq_rat @ A_221 @ zero_zero_rat )
& ( ord_less_eq_rat @ B_174 @ zero_zero_rat ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A_221 @ B_174 ) ) ) ).
thf(fact_1053_mult__mono,axiom,
! [C_104: int,D_32: int,A_220: int,B_173: int] :
( ( ord_less_eq_int @ A_220 @ B_173 )
=> ( ( ord_less_eq_int @ C_104 @ D_32 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_173 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_104 )
=> ( ord_less_eq_int @ ( times_times_int @ A_220 @ C_104 ) @ ( times_times_int @ B_173 @ D_32 ) ) ) ) ) ) ).
thf(fact_1054_mult__mono,axiom,
! [C_104: nat,D_32: nat,A_220: nat,B_173: nat] :
( ( ord_less_eq_nat @ A_220 @ B_173 )
=> ( ( ord_less_eq_nat @ C_104 @ D_32 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_173 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_104 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_220 @ C_104 ) @ ( times_times_nat @ B_173 @ D_32 ) ) ) ) ) ) ).
thf(fact_1055_mult__mono,axiom,
! [C_104: real,D_32: real,A_220: real,B_173: real] :
( ( ord_less_eq_real @ A_220 @ B_173 )
=> ( ( ord_less_eq_real @ C_104 @ D_32 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_173 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_104 )
=> ( ord_less_eq_real @ ( times_times_real @ A_220 @ C_104 ) @ ( times_times_real @ B_173 @ D_32 ) ) ) ) ) ) ).
thf(fact_1056_mult__mono,axiom,
! [C_104: code_code_numeral,D_32: code_code_numeral,A_220: code_code_numeral,B_173: code_code_numeral] :
( ( ord_le565307924umeral @ A_220 @ B_173 )
=> ( ( ord_le565307924umeral @ C_104 @ D_32 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_173 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_104 )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ A_220 @ C_104 ) @ ( times_1655362735umeral @ B_173 @ D_32 ) ) ) ) ) ) ).
thf(fact_1057_mult__mono,axiom,
! [C_104: quickcheck_code_int,D_32: quickcheck_code_int,A_220: quickcheck_code_int,B_173: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_220 @ B_173 )
=> ( ( ord_le258702272de_int @ C_104 @ D_32 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_173 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_104 )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ A_220 @ C_104 ) @ ( times_123202395de_int @ B_173 @ D_32 ) ) ) ) ) ) ).
thf(fact_1058_mult__mono,axiom,
! [C_104: rat,D_32: rat,A_220: rat,B_173: rat] :
( ( ord_less_eq_rat @ A_220 @ B_173 )
=> ( ( ord_less_eq_rat @ C_104 @ D_32 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_173 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_104 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_220 @ C_104 ) @ ( times_times_rat @ B_173 @ D_32 ) ) ) ) ) ) ).
thf(fact_1059_mult__mono_H,axiom,
! [C_103: int,D_31: int,A_219: int,B_172: int] :
( ( ord_less_eq_int @ A_219 @ B_172 )
=> ( ( ord_less_eq_int @ C_103 @ D_31 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_219 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_103 )
=> ( ord_less_eq_int @ ( times_times_int @ A_219 @ C_103 ) @ ( times_times_int @ B_172 @ D_31 ) ) ) ) ) ) ).
thf(fact_1060_mult__mono_H,axiom,
! [C_103: nat,D_31: nat,A_219: nat,B_172: nat] :
( ( ord_less_eq_nat @ A_219 @ B_172 )
=> ( ( ord_less_eq_nat @ C_103 @ D_31 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_219 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_103 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_219 @ C_103 ) @ ( times_times_nat @ B_172 @ D_31 ) ) ) ) ) ) ).
thf(fact_1061_mult__mono_H,axiom,
! [C_103: real,D_31: real,A_219: real,B_172: real] :
( ( ord_less_eq_real @ A_219 @ B_172 )
=> ( ( ord_less_eq_real @ C_103 @ D_31 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_219 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_103 )
=> ( ord_less_eq_real @ ( times_times_real @ A_219 @ C_103 ) @ ( times_times_real @ B_172 @ D_31 ) ) ) ) ) ) ).
thf(fact_1062_mult__mono_H,axiom,
! [C_103: code_code_numeral,D_31: code_code_numeral,A_219: code_code_numeral,B_172: code_code_numeral] :
( ( ord_le565307924umeral @ A_219 @ B_172 )
=> ( ( ord_le565307924umeral @ C_103 @ D_31 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_219 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_103 )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ A_219 @ C_103 ) @ ( times_1655362735umeral @ B_172 @ D_31 ) ) ) ) ) ) ).
thf(fact_1063_mult__mono_H,axiom,
! [C_103: quickcheck_code_int,D_31: quickcheck_code_int,A_219: quickcheck_code_int,B_172: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_219 @ B_172 )
=> ( ( ord_le258702272de_int @ C_103 @ D_31 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_219 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_103 )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ A_219 @ C_103 ) @ ( times_123202395de_int @ B_172 @ D_31 ) ) ) ) ) ) ).
thf(fact_1064_mult__mono_H,axiom,
! [C_103: rat,D_31: rat,A_219: rat,B_172: rat] :
( ( ord_less_eq_rat @ A_219 @ B_172 )
=> ( ( ord_less_eq_rat @ C_103 @ D_31 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A_219 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_103 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_219 @ C_103 ) @ ( times_times_rat @ B_172 @ D_31 ) ) ) ) ) ) ).
thf(fact_1065_mult__left__mono__neg,axiom,
! [C_102: int,B_171: int,A_218: int] :
( ( ord_less_eq_int @ B_171 @ A_218 )
=> ( ( ord_less_eq_int @ C_102 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C_102 @ A_218 ) @ ( times_times_int @ C_102 @ B_171 ) ) ) ) ).
thf(fact_1066_mult__left__mono__neg,axiom,
! [C_102: real,B_171: real,A_218: real] :
( ( ord_less_eq_real @ B_171 @ A_218 )
=> ( ( ord_less_eq_real @ C_102 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C_102 @ A_218 ) @ ( times_times_real @ C_102 @ B_171 ) ) ) ) ).
thf(fact_1067_mult__left__mono__neg,axiom,
! [C_102: rat,B_171: rat,A_218: rat] :
( ( ord_less_eq_rat @ B_171 @ A_218 )
=> ( ( ord_less_eq_rat @ C_102 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ C_102 @ A_218 ) @ ( times_times_rat @ C_102 @ B_171 ) ) ) ) ).
thf(fact_1068_mult__right__mono__neg,axiom,
! [C_101: int,B_170: int,A_217: int] :
( ( ord_less_eq_int @ B_170 @ A_217 )
=> ( ( ord_less_eq_int @ C_101 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A_217 @ C_101 ) @ ( times_times_int @ B_170 @ C_101 ) ) ) ) ).
thf(fact_1069_mult__right__mono__neg,axiom,
! [C_101: real,B_170: real,A_217: real] :
( ( ord_less_eq_real @ B_170 @ A_217 )
=> ( ( ord_less_eq_real @ C_101 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A_217 @ C_101 ) @ ( times_times_real @ B_170 @ C_101 ) ) ) ) ).
thf(fact_1070_mult__right__mono__neg,axiom,
! [C_101: rat,B_170: rat,A_217: rat] :
( ( ord_less_eq_rat @ B_170 @ A_217 )
=> ( ( ord_less_eq_rat @ C_101 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_217 @ C_101 ) @ ( times_times_rat @ B_170 @ C_101 ) ) ) ) ).
thf(fact_1071_comm__mult__left__mono,axiom,
! [C_100: int,A_216: int,B_169: int] :
( ( ord_less_eq_int @ A_216 @ B_169 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_100 )
=> ( ord_less_eq_int @ ( times_times_int @ C_100 @ A_216 ) @ ( times_times_int @ C_100 @ B_169 ) ) ) ) ).
thf(fact_1072_comm__mult__left__mono,axiom,
! [C_100: nat,A_216: nat,B_169: nat] :
( ( ord_less_eq_nat @ A_216 @ B_169 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_100 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C_100 @ A_216 ) @ ( times_times_nat @ C_100 @ B_169 ) ) ) ) ).
thf(fact_1073_comm__mult__left__mono,axiom,
! [C_100: real,A_216: real,B_169: real] :
( ( ord_less_eq_real @ A_216 @ B_169 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_100 )
=> ( ord_less_eq_real @ ( times_times_real @ C_100 @ A_216 ) @ ( times_times_real @ C_100 @ B_169 ) ) ) ) ).
thf(fact_1074_comm__mult__left__mono,axiom,
! [C_100: code_code_numeral,A_216: code_code_numeral,B_169: code_code_numeral] :
( ( ord_le565307924umeral @ A_216 @ B_169 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_100 )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ C_100 @ A_216 ) @ ( times_1655362735umeral @ C_100 @ B_169 ) ) ) ) ).
thf(fact_1075_comm__mult__left__mono,axiom,
! [C_100: quickcheck_code_int,A_216: quickcheck_code_int,B_169: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_216 @ B_169 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_100 )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ C_100 @ A_216 ) @ ( times_123202395de_int @ C_100 @ B_169 ) ) ) ) ).
thf(fact_1076_comm__mult__left__mono,axiom,
! [C_100: rat,A_216: rat,B_169: rat] :
( ( ord_less_eq_rat @ A_216 @ B_169 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_100 )
=> ( ord_less_eq_rat @ ( times_times_rat @ C_100 @ A_216 ) @ ( times_times_rat @ C_100 @ B_169 ) ) ) ) ).
thf(fact_1077_mult__left__mono,axiom,
! [C_99: int,A_215: int,B_168: int] :
( ( ord_less_eq_int @ A_215 @ B_168 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_99 )
=> ( ord_less_eq_int @ ( times_times_int @ C_99 @ A_215 ) @ ( times_times_int @ C_99 @ B_168 ) ) ) ) ).
thf(fact_1078_mult__left__mono,axiom,
! [C_99: nat,A_215: nat,B_168: nat] :
( ( ord_less_eq_nat @ A_215 @ B_168 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_99 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C_99 @ A_215 ) @ ( times_times_nat @ C_99 @ B_168 ) ) ) ) ).
thf(fact_1079_mult__left__mono,axiom,
! [C_99: real,A_215: real,B_168: real] :
( ( ord_less_eq_real @ A_215 @ B_168 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_99 )
=> ( ord_less_eq_real @ ( times_times_real @ C_99 @ A_215 ) @ ( times_times_real @ C_99 @ B_168 ) ) ) ) ).
thf(fact_1080_mult__left__mono,axiom,
! [C_99: code_code_numeral,A_215: code_code_numeral,B_168: code_code_numeral] :
( ( ord_le565307924umeral @ A_215 @ B_168 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_99 )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ C_99 @ A_215 ) @ ( times_1655362735umeral @ C_99 @ B_168 ) ) ) ) ).
thf(fact_1081_mult__left__mono,axiom,
! [C_99: quickcheck_code_int,A_215: quickcheck_code_int,B_168: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_215 @ B_168 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_99 )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ C_99 @ A_215 ) @ ( times_123202395de_int @ C_99 @ B_168 ) ) ) ) ).
thf(fact_1082_mult__left__mono,axiom,
! [C_99: rat,A_215: rat,B_168: rat] :
( ( ord_less_eq_rat @ A_215 @ B_168 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_99 )
=> ( ord_less_eq_rat @ ( times_times_rat @ C_99 @ A_215 ) @ ( times_times_rat @ C_99 @ B_168 ) ) ) ) ).
thf(fact_1083_mult__right__mono,axiom,
! [C_98: int,A_214: int,B_167: int] :
( ( ord_less_eq_int @ A_214 @ B_167 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_98 )
=> ( ord_less_eq_int @ ( times_times_int @ A_214 @ C_98 ) @ ( times_times_int @ B_167 @ C_98 ) ) ) ) ).
thf(fact_1084_mult__right__mono,axiom,
! [C_98: nat,A_214: nat,B_167: nat] :
( ( ord_less_eq_nat @ A_214 @ B_167 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_98 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_214 @ C_98 ) @ ( times_times_nat @ B_167 @ C_98 ) ) ) ) ).
thf(fact_1085_mult__right__mono,axiom,
! [C_98: real,A_214: real,B_167: real] :
( ( ord_less_eq_real @ A_214 @ B_167 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_98 )
=> ( ord_less_eq_real @ ( times_times_real @ A_214 @ C_98 ) @ ( times_times_real @ B_167 @ C_98 ) ) ) ) ).
thf(fact_1086_mult__right__mono,axiom,
! [C_98: code_code_numeral,A_214: code_code_numeral,B_167: code_code_numeral] :
( ( ord_le565307924umeral @ A_214 @ B_167 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_98 )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ A_214 @ C_98 ) @ ( times_1655362735umeral @ B_167 @ C_98 ) ) ) ) ).
thf(fact_1087_mult__right__mono,axiom,
! [C_98: quickcheck_code_int,A_214: quickcheck_code_int,B_167: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_214 @ B_167 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_98 )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ A_214 @ C_98 ) @ ( times_123202395de_int @ B_167 @ C_98 ) ) ) ) ).
thf(fact_1088_mult__right__mono,axiom,
! [C_98: rat,A_214: rat,B_167: rat] :
( ( ord_less_eq_rat @ A_214 @ B_167 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_98 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_214 @ C_98 ) @ ( times_times_rat @ B_167 @ C_98 ) ) ) ) ).
thf(fact_1089_mult__nonpos__nonpos,axiom,
! [B_166: int,A_213: int] :
( ( ord_less_eq_int @ A_213 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B_166 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_213 @ B_166 ) ) ) ) ).
thf(fact_1090_mult__nonpos__nonpos,axiom,
! [B_166: real,A_213: real] :
( ( ord_less_eq_real @ A_213 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B_166 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_213 @ B_166 ) ) ) ) ).
thf(fact_1091_mult__nonpos__nonpos,axiom,
! [B_166: rat,A_213: rat] :
( ( ord_less_eq_rat @ A_213 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B_166 @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A_213 @ B_166 ) ) ) ) ).
thf(fact_1092_mult__nonpos__nonneg,axiom,
! [B_165: int,A_212: int] :
( ( ord_less_eq_int @ A_212 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_165 )
=> ( ord_less_eq_int @ ( times_times_int @ A_212 @ B_165 ) @ zero_zero_int ) ) ) ).
thf(fact_1093_mult__nonpos__nonneg,axiom,
! [B_165: nat,A_212: nat] :
( ( ord_less_eq_nat @ A_212 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_165 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_212 @ B_165 ) @ zero_zero_nat ) ) ) ).
thf(fact_1094_mult__nonpos__nonneg,axiom,
! [B_165: real,A_212: real] :
( ( ord_less_eq_real @ A_212 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_165 )
=> ( ord_less_eq_real @ ( times_times_real @ A_212 @ B_165 ) @ zero_zero_real ) ) ) ).
thf(fact_1095_mult__nonpos__nonneg,axiom,
! [B_165: code_code_numeral,A_212: code_code_numeral] :
( ( ord_le565307924umeral @ A_212 @ zero_z126310315umeral )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_165 )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ A_212 @ B_165 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1096_mult__nonpos__nonneg,axiom,
! [B_165: quickcheck_code_int,A_212: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_212 @ zero_z891286103de_int )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_165 )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ A_212 @ B_165 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1097_mult__nonpos__nonneg,axiom,
! [B_165: rat,A_212: rat] :
( ( ord_less_eq_rat @ A_212 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_165 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_212 @ B_165 ) @ zero_zero_rat ) ) ) ).
thf(fact_1098_mult__nonneg__nonpos2,axiom,
! [B_164: int,A_211: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_211 )
=> ( ( ord_less_eq_int @ B_164 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B_164 @ A_211 ) @ zero_zero_int ) ) ) ).
thf(fact_1099_mult__nonneg__nonpos2,axiom,
! [B_164: nat,A_211: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_211 )
=> ( ( ord_less_eq_nat @ B_164 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B_164 @ A_211 ) @ zero_zero_nat ) ) ) ).
thf(fact_1100_mult__nonneg__nonpos2,axiom,
! [B_164: real,A_211: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_211 )
=> ( ( ord_less_eq_real @ B_164 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B_164 @ A_211 ) @ zero_zero_real ) ) ) ).
thf(fact_1101_mult__nonneg__nonpos2,axiom,
! [B_164: code_code_numeral,A_211: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_211 )
=> ( ( ord_le565307924umeral @ B_164 @ zero_z126310315umeral )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ B_164 @ A_211 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1102_mult__nonneg__nonpos2,axiom,
! [B_164: quickcheck_code_int,A_211: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_211 )
=> ( ( ord_le258702272de_int @ B_164 @ zero_z891286103de_int )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ B_164 @ A_211 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1103_mult__nonneg__nonpos2,axiom,
! [B_164: rat,A_211: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_211 )
=> ( ( ord_less_eq_rat @ B_164 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ B_164 @ A_211 ) @ zero_zero_rat ) ) ) ).
thf(fact_1104_mult__nonneg__nonpos,axiom,
! [B_163: int,A_210: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_210 )
=> ( ( ord_less_eq_int @ B_163 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A_210 @ B_163 ) @ zero_zero_int ) ) ) ).
thf(fact_1105_mult__nonneg__nonpos,axiom,
! [B_163: nat,A_210: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_210 )
=> ( ( ord_less_eq_nat @ B_163 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_210 @ B_163 ) @ zero_zero_nat ) ) ) ).
thf(fact_1106_mult__nonneg__nonpos,axiom,
! [B_163: real,A_210: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_210 )
=> ( ( ord_less_eq_real @ B_163 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A_210 @ B_163 ) @ zero_zero_real ) ) ) ).
thf(fact_1107_mult__nonneg__nonpos,axiom,
! [B_163: code_code_numeral,A_210: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_210 )
=> ( ( ord_le565307924umeral @ B_163 @ zero_z126310315umeral )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ A_210 @ B_163 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1108_mult__nonneg__nonpos,axiom,
! [B_163: quickcheck_code_int,A_210: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_210 )
=> ( ( ord_le258702272de_int @ B_163 @ zero_z891286103de_int )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ A_210 @ B_163 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1109_mult__nonneg__nonpos,axiom,
! [B_163: rat,A_210: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_210 )
=> ( ( ord_less_eq_rat @ B_163 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_210 @ B_163 ) @ zero_zero_rat ) ) ) ).
thf(fact_1110_mult__nonneg__nonneg,axiom,
! [B_162: int,A_209: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_209 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_162 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_209 @ B_162 ) ) ) ) ).
thf(fact_1111_mult__nonneg__nonneg,axiom,
! [B_162: nat,A_209: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_209 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_162 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A_209 @ B_162 ) ) ) ) ).
thf(fact_1112_mult__nonneg__nonneg,axiom,
! [B_162: real,A_209: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_209 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_162 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_209 @ B_162 ) ) ) ) ).
thf(fact_1113_mult__nonneg__nonneg,axiom,
! [B_162: code_code_numeral,A_209: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_209 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_162 )
=> ( ord_le565307924umeral @ zero_z126310315umeral @ ( times_1655362735umeral @ A_209 @ B_162 ) ) ) ) ).
thf(fact_1114_mult__nonneg__nonneg,axiom,
! [B_162: quickcheck_code_int,A_209: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_209 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_162 )
=> ( ord_le258702272de_int @ zero_z891286103de_int @ ( times_123202395de_int @ A_209 @ B_162 ) ) ) ) ).
thf(fact_1115_mult__nonneg__nonneg,axiom,
! [B_162: rat,A_209: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_209 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_162 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A_209 @ B_162 ) ) ) ) ).
thf(fact_1116_mult__le__0__iff,axiom,
! [A_208: int,B_161: int] :
( ( ord_less_eq_int @ ( times_times_int @ A_208 @ B_161 ) @ zero_zero_int )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ A_208 )
& ( ord_less_eq_int @ B_161 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A_208 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B_161 ) ) ) ) ).
thf(fact_1117_mult__le__0__iff,axiom,
! [A_208: real,B_161: real] :
( ( ord_less_eq_real @ ( times_times_real @ A_208 @ B_161 ) @ zero_zero_real )
<=> ( ( ( ord_less_eq_real @ zero_zero_real @ A_208 )
& ( ord_less_eq_real @ B_161 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A_208 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B_161 ) ) ) ) ).
thf(fact_1118_mult__le__0__iff,axiom,
! [A_208: rat,B_161: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A_208 @ B_161 ) @ zero_zero_rat )
<=> ( ( ( ord_less_eq_rat @ zero_zero_rat @ A_208 )
& ( ord_less_eq_rat @ B_161 @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A_208 @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B_161 ) ) ) ) ).
thf(fact_1119_zero__le__mult__iff,axiom,
! [A_207: int,B_160: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_207 @ B_160 ) )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ A_207 )
& ( ord_less_eq_int @ zero_zero_int @ B_160 ) )
| ( ( ord_less_eq_int @ A_207 @ zero_zero_int )
& ( ord_less_eq_int @ B_160 @ zero_zero_int ) ) ) ) ).
thf(fact_1120_zero__le__mult__iff,axiom,
! [A_207: real,B_160: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_207 @ B_160 ) )
<=> ( ( ( ord_less_eq_real @ zero_zero_real @ A_207 )
& ( ord_less_eq_real @ zero_zero_real @ B_160 ) )
| ( ( ord_less_eq_real @ A_207 @ zero_zero_real )
& ( ord_less_eq_real @ B_160 @ zero_zero_real ) ) ) ) ).
thf(fact_1121_zero__le__mult__iff,axiom,
! [A_207: rat,B_160: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A_207 @ B_160 ) )
<=> ( ( ( ord_less_eq_rat @ zero_zero_rat @ A_207 )
& ( ord_less_eq_rat @ zero_zero_rat @ B_160 ) )
| ( ( ord_less_eq_rat @ A_207 @ zero_zero_rat )
& ( ord_less_eq_rat @ B_160 @ zero_zero_rat ) ) ) ) ).
thf(fact_1122_zero__le__square,axiom,
! [A_206: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_206 @ A_206 ) ) ).
thf(fact_1123_zero__le__square,axiom,
! [A_206: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_206 @ A_206 ) ) ).
thf(fact_1124_zero__le__square,axiom,
! [A_206: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A_206 @ A_206 ) ) ).
thf(fact_1125_mult__strict__left__mono__neg,axiom,
! [C_97: int,B_159: int,A_205: int] :
( ( ord_less_int @ B_159 @ A_205 )
=> ( ( ord_less_int @ C_97 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C_97 @ A_205 ) @ ( times_times_int @ C_97 @ B_159 ) ) ) ) ).
thf(fact_1126_mult__strict__left__mono__neg,axiom,
! [C_97: real,B_159: real,A_205: real] :
( ( ord_less_real @ B_159 @ A_205 )
=> ( ( ord_less_real @ C_97 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C_97 @ A_205 ) @ ( times_times_real @ C_97 @ B_159 ) ) ) ) ).
thf(fact_1127_mult__strict__left__mono__neg,axiom,
! [C_97: rat,B_159: rat,A_205: rat] :
( ( ord_less_rat @ B_159 @ A_205 )
=> ( ( ord_less_rat @ C_97 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ C_97 @ A_205 ) @ ( times_times_rat @ C_97 @ B_159 ) ) ) ) ).
thf(fact_1128_mult__strict__right__mono__neg,axiom,
! [C_96: int,B_158: int,A_204: int] :
( ( ord_less_int @ B_158 @ A_204 )
=> ( ( ord_less_int @ C_96 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A_204 @ C_96 ) @ ( times_times_int @ B_158 @ C_96 ) ) ) ) ).
thf(fact_1129_mult__strict__right__mono__neg,axiom,
! [C_96: real,B_158: real,A_204: real] :
( ( ord_less_real @ B_158 @ A_204 )
=> ( ( ord_less_real @ C_96 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A_204 @ C_96 ) @ ( times_times_real @ B_158 @ C_96 ) ) ) ) ).
thf(fact_1130_mult__strict__right__mono__neg,axiom,
! [C_96: rat,B_158: rat,A_204: rat] :
( ( ord_less_rat @ B_158 @ A_204 )
=> ( ( ord_less_rat @ C_96 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A_204 @ C_96 ) @ ( times_times_rat @ B_158 @ C_96 ) ) ) ) ).
thf(fact_1131_comm__mult__strict__left__mono,axiom,
! [C_95: int,A_203: int,B_157: int] :
( ( ord_less_int @ A_203 @ B_157 )
=> ( ( ord_less_int @ zero_zero_int @ C_95 )
=> ( ord_less_int @ ( times_times_int @ C_95 @ A_203 ) @ ( times_times_int @ C_95 @ B_157 ) ) ) ) ).
thf(fact_1132_comm__mult__strict__left__mono,axiom,
! [C_95: nat,A_203: nat,B_157: nat] :
( ( ord_less_nat @ A_203 @ B_157 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_95 )
=> ( ord_less_nat @ ( times_times_nat @ C_95 @ A_203 ) @ ( times_times_nat @ C_95 @ B_157 ) ) ) ) ).
thf(fact_1133_comm__mult__strict__left__mono,axiom,
! [C_95: real,A_203: real,B_157: real] :
( ( ord_less_real @ A_203 @ B_157 )
=> ( ( ord_less_real @ zero_zero_real @ C_95 )
=> ( ord_less_real @ ( times_times_real @ C_95 @ A_203 ) @ ( times_times_real @ C_95 @ B_157 ) ) ) ) ).
thf(fact_1134_comm__mult__strict__left__mono,axiom,
! [C_95: code_code_numeral,A_203: code_code_numeral,B_157: code_code_numeral] :
( ( ord_le1304079648umeral @ A_203 @ B_157 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ C_95 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ C_95 @ A_203 ) @ ( times_1655362735umeral @ C_95 @ B_157 ) ) ) ) ).
thf(fact_1135_comm__mult__strict__left__mono,axiom,
! [C_95: quickcheck_code_int,A_203: quickcheck_code_int,B_157: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_203 @ B_157 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ C_95 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ C_95 @ A_203 ) @ ( times_123202395de_int @ C_95 @ B_157 ) ) ) ) ).
thf(fact_1136_comm__mult__strict__left__mono,axiom,
! [C_95: rat,A_203: rat,B_157: rat] :
( ( ord_less_rat @ A_203 @ B_157 )
=> ( ( ord_less_rat @ zero_zero_rat @ C_95 )
=> ( ord_less_rat @ ( times_times_rat @ C_95 @ A_203 ) @ ( times_times_rat @ C_95 @ B_157 ) ) ) ) ).
thf(fact_1137_mult__strict__left__mono,axiom,
! [C_94: int,A_202: int,B_156: int] :
( ( ord_less_int @ A_202 @ B_156 )
=> ( ( ord_less_int @ zero_zero_int @ C_94 )
=> ( ord_less_int @ ( times_times_int @ C_94 @ A_202 ) @ ( times_times_int @ C_94 @ B_156 ) ) ) ) ).
thf(fact_1138_mult__strict__left__mono,axiom,
! [C_94: nat,A_202: nat,B_156: nat] :
( ( ord_less_nat @ A_202 @ B_156 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_94 )
=> ( ord_less_nat @ ( times_times_nat @ C_94 @ A_202 ) @ ( times_times_nat @ C_94 @ B_156 ) ) ) ) ).
thf(fact_1139_mult__strict__left__mono,axiom,
! [C_94: real,A_202: real,B_156: real] :
( ( ord_less_real @ A_202 @ B_156 )
=> ( ( ord_less_real @ zero_zero_real @ C_94 )
=> ( ord_less_real @ ( times_times_real @ C_94 @ A_202 ) @ ( times_times_real @ C_94 @ B_156 ) ) ) ) ).
thf(fact_1140_mult__strict__left__mono,axiom,
! [C_94: code_code_numeral,A_202: code_code_numeral,B_156: code_code_numeral] :
( ( ord_le1304079648umeral @ A_202 @ B_156 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ C_94 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ C_94 @ A_202 ) @ ( times_1655362735umeral @ C_94 @ B_156 ) ) ) ) ).
thf(fact_1141_mult__strict__left__mono,axiom,
! [C_94: quickcheck_code_int,A_202: quickcheck_code_int,B_156: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_202 @ B_156 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ C_94 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ C_94 @ A_202 ) @ ( times_123202395de_int @ C_94 @ B_156 ) ) ) ) ).
thf(fact_1142_mult__strict__left__mono,axiom,
! [C_94: rat,A_202: rat,B_156: rat] :
( ( ord_less_rat @ A_202 @ B_156 )
=> ( ( ord_less_rat @ zero_zero_rat @ C_94 )
=> ( ord_less_rat @ ( times_times_rat @ C_94 @ A_202 ) @ ( times_times_rat @ C_94 @ B_156 ) ) ) ) ).
thf(fact_1143_mult__strict__right__mono,axiom,
! [C_93: int,A_201: int,B_155: int] :
( ( ord_less_int @ A_201 @ B_155 )
=> ( ( ord_less_int @ zero_zero_int @ C_93 )
=> ( ord_less_int @ ( times_times_int @ A_201 @ C_93 ) @ ( times_times_int @ B_155 @ C_93 ) ) ) ) ).
thf(fact_1144_mult__strict__right__mono,axiom,
! [C_93: nat,A_201: nat,B_155: nat] :
( ( ord_less_nat @ A_201 @ B_155 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_93 )
=> ( ord_less_nat @ ( times_times_nat @ A_201 @ C_93 ) @ ( times_times_nat @ B_155 @ C_93 ) ) ) ) ).
thf(fact_1145_mult__strict__right__mono,axiom,
! [C_93: real,A_201: real,B_155: real] :
( ( ord_less_real @ A_201 @ B_155 )
=> ( ( ord_less_real @ zero_zero_real @ C_93 )
=> ( ord_less_real @ ( times_times_real @ A_201 @ C_93 ) @ ( times_times_real @ B_155 @ C_93 ) ) ) ) ).
thf(fact_1146_mult__strict__right__mono,axiom,
! [C_93: code_code_numeral,A_201: code_code_numeral,B_155: code_code_numeral] :
( ( ord_le1304079648umeral @ A_201 @ B_155 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ C_93 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_201 @ C_93 ) @ ( times_1655362735umeral @ B_155 @ C_93 ) ) ) ) ).
thf(fact_1147_mult__strict__right__mono,axiom,
! [C_93: quickcheck_code_int,A_201: quickcheck_code_int,B_155: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_201 @ B_155 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ C_93 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_201 @ C_93 ) @ ( times_123202395de_int @ B_155 @ C_93 ) ) ) ) ).
thf(fact_1148_mult__strict__right__mono,axiom,
! [C_93: rat,A_201: rat,B_155: rat] :
( ( ord_less_rat @ A_201 @ B_155 )
=> ( ( ord_less_rat @ zero_zero_rat @ C_93 )
=> ( ord_less_rat @ ( times_times_rat @ A_201 @ C_93 ) @ ( times_times_rat @ B_155 @ C_93 ) ) ) ) ).
thf(fact_1149_mult__neg__neg,axiom,
! [B_154: int,A_200: int] :
( ( ord_less_int @ A_200 @ zero_zero_int )
=> ( ( ord_less_int @ B_154 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_200 @ B_154 ) ) ) ) ).
thf(fact_1150_mult__neg__neg,axiom,
! [B_154: real,A_200: real] :
( ( ord_less_real @ A_200 @ zero_zero_real )
=> ( ( ord_less_real @ B_154 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A_200 @ B_154 ) ) ) ) ).
thf(fact_1151_mult__neg__neg,axiom,
! [B_154: rat,A_200: rat] :
( ( ord_less_rat @ A_200 @ zero_zero_rat )
=> ( ( ord_less_rat @ B_154 @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A_200 @ B_154 ) ) ) ) ).
thf(fact_1152_mult__neg__pos,axiom,
! [B_153: int,A_199: int] :
( ( ord_less_int @ A_199 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B_153 )
=> ( ord_less_int @ ( times_times_int @ A_199 @ B_153 ) @ zero_zero_int ) ) ) ).
thf(fact_1153_mult__neg__pos,axiom,
! [B_153: nat,A_199: nat] :
( ( ord_less_nat @ A_199 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B_153 )
=> ( ord_less_nat @ ( times_times_nat @ A_199 @ B_153 ) @ zero_zero_nat ) ) ) ).
thf(fact_1154_mult__neg__pos,axiom,
! [B_153: real,A_199: real] :
( ( ord_less_real @ A_199 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B_153 )
=> ( ord_less_real @ ( times_times_real @ A_199 @ B_153 ) @ zero_zero_real ) ) ) ).
thf(fact_1155_mult__neg__pos,axiom,
! [B_153: code_code_numeral,A_199: code_code_numeral] :
( ( ord_le1304079648umeral @ A_199 @ zero_z126310315umeral )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ B_153 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_199 @ B_153 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1156_mult__neg__pos,axiom,
! [B_153: quickcheck_code_int,A_199: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_199 @ zero_z891286103de_int )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ B_153 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_199 @ B_153 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1157_mult__neg__pos,axiom,
! [B_153: rat,A_199: rat] :
( ( ord_less_rat @ A_199 @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ B_153 )
=> ( ord_less_rat @ ( times_times_rat @ A_199 @ B_153 ) @ zero_zero_rat ) ) ) ).
thf(fact_1158_mult__less__cancel__left__neg,axiom,
! [A_198: int,B_152: int,C_92: int] :
( ( ord_less_int @ C_92 @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C_92 @ A_198 ) @ ( times_times_int @ C_92 @ B_152 ) )
<=> ( ord_less_int @ B_152 @ A_198 ) ) ) ).
thf(fact_1159_mult__less__cancel__left__neg,axiom,
! [A_198: real,B_152: real,C_92: real] :
( ( ord_less_real @ C_92 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C_92 @ A_198 ) @ ( times_times_real @ C_92 @ B_152 ) )
<=> ( ord_less_real @ B_152 @ A_198 ) ) ) ).
thf(fact_1160_mult__less__cancel__left__neg,axiom,
! [A_198: rat,B_152: rat,C_92: rat] :
( ( ord_less_rat @ C_92 @ zero_zero_rat )
=> ( ( ord_less_rat @ ( times_times_rat @ C_92 @ A_198 ) @ ( times_times_rat @ C_92 @ B_152 ) )
<=> ( ord_less_rat @ B_152 @ A_198 ) ) ) ).
thf(fact_1161_zero__less__mult__pos2,axiom,
! [B_151: int,A_197: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B_151 @ A_197 ) )
=> ( ( ord_less_int @ zero_zero_int @ A_197 )
=> ( ord_less_int @ zero_zero_int @ B_151 ) ) ) ).
thf(fact_1162_zero__less__mult__pos2,axiom,
! [B_151: nat,A_197: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B_151 @ A_197 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A_197 )
=> ( ord_less_nat @ zero_zero_nat @ B_151 ) ) ) ).
thf(fact_1163_zero__less__mult__pos2,axiom,
! [B_151: real,A_197: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B_151 @ A_197 ) )
=> ( ( ord_less_real @ zero_zero_real @ A_197 )
=> ( ord_less_real @ zero_zero_real @ B_151 ) ) ) ).
thf(fact_1164_zero__less__mult__pos2,axiom,
! [B_151: code_code_numeral,A_197: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ ( times_1655362735umeral @ B_151 @ A_197 ) )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_197 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ B_151 ) ) ) ).
thf(fact_1165_zero__less__mult__pos2,axiom,
! [B_151: quickcheck_code_int,A_197: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ ( times_123202395de_int @ B_151 @ A_197 ) )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_197 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ B_151 ) ) ) ).
thf(fact_1166_zero__less__mult__pos2,axiom,
! [B_151: rat,A_197: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B_151 @ A_197 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A_197 )
=> ( ord_less_rat @ zero_zero_rat @ B_151 ) ) ) ).
thf(fact_1167_zero__less__mult__pos,axiom,
! [A_196: int,B_150: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_196 @ B_150 ) )
=> ( ( ord_less_int @ zero_zero_int @ A_196 )
=> ( ord_less_int @ zero_zero_int @ B_150 ) ) ) ).
thf(fact_1168_zero__less__mult__pos,axiom,
! [A_196: nat,B_150: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A_196 @ B_150 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A_196 )
=> ( ord_less_nat @ zero_zero_nat @ B_150 ) ) ) ).
thf(fact_1169_zero__less__mult__pos,axiom,
! [A_196: real,B_150: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A_196 @ B_150 ) )
=> ( ( ord_less_real @ zero_zero_real @ A_196 )
=> ( ord_less_real @ zero_zero_real @ B_150 ) ) ) ).
thf(fact_1170_zero__less__mult__pos,axiom,
! [A_196: code_code_numeral,B_150: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ ( times_1655362735umeral @ A_196 @ B_150 ) )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_196 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ B_150 ) ) ) ).
thf(fact_1171_zero__less__mult__pos,axiom,
! [A_196: quickcheck_code_int,B_150: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ ( times_123202395de_int @ A_196 @ B_150 ) )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_196 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ B_150 ) ) ) ).
thf(fact_1172_zero__less__mult__pos,axiom,
! [A_196: rat,B_150: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A_196 @ B_150 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A_196 )
=> ( ord_less_rat @ zero_zero_rat @ B_150 ) ) ) ).
thf(fact_1173_mult__pos__neg2,axiom,
! [B_149: int,A_195: int] :
( ( ord_less_int @ zero_zero_int @ A_195 )
=> ( ( ord_less_int @ B_149 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B_149 @ A_195 ) @ zero_zero_int ) ) ) ).
thf(fact_1174_mult__pos__neg2,axiom,
! [B_149: nat,A_195: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_195 )
=> ( ( ord_less_nat @ B_149 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B_149 @ A_195 ) @ zero_zero_nat ) ) ) ).
thf(fact_1175_mult__pos__neg2,axiom,
! [B_149: real,A_195: real] :
( ( ord_less_real @ zero_zero_real @ A_195 )
=> ( ( ord_less_real @ B_149 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B_149 @ A_195 ) @ zero_zero_real ) ) ) ).
thf(fact_1176_mult__pos__neg2,axiom,
! [B_149: code_code_numeral,A_195: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_195 )
=> ( ( ord_le1304079648umeral @ B_149 @ zero_z126310315umeral )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ B_149 @ A_195 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1177_mult__pos__neg2,axiom,
! [B_149: quickcheck_code_int,A_195: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_195 )
=> ( ( ord_le1860547276de_int @ B_149 @ zero_z891286103de_int )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ B_149 @ A_195 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1178_mult__pos__neg2,axiom,
! [B_149: rat,A_195: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_195 )
=> ( ( ord_less_rat @ B_149 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ B_149 @ A_195 ) @ zero_zero_rat ) ) ) ).
thf(fact_1179_mult__pos__neg,axiom,
! [B_148: int,A_194: int] :
( ( ord_less_int @ zero_zero_int @ A_194 )
=> ( ( ord_less_int @ B_148 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A_194 @ B_148 ) @ zero_zero_int ) ) ) ).
thf(fact_1180_mult__pos__neg,axiom,
! [B_148: nat,A_194: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_194 )
=> ( ( ord_less_nat @ B_148 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A_194 @ B_148 ) @ zero_zero_nat ) ) ) ).
thf(fact_1181_mult__pos__neg,axiom,
! [B_148: real,A_194: real] :
( ( ord_less_real @ zero_zero_real @ A_194 )
=> ( ( ord_less_real @ B_148 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A_194 @ B_148 ) @ zero_zero_real ) ) ) ).
thf(fact_1182_mult__pos__neg,axiom,
! [B_148: code_code_numeral,A_194: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_194 )
=> ( ( ord_le1304079648umeral @ B_148 @ zero_z126310315umeral )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_194 @ B_148 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1183_mult__pos__neg,axiom,
! [B_148: quickcheck_code_int,A_194: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_194 )
=> ( ( ord_le1860547276de_int @ B_148 @ zero_z891286103de_int )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_194 @ B_148 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1184_mult__pos__neg,axiom,
! [B_148: rat,A_194: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_194 )
=> ( ( ord_less_rat @ B_148 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A_194 @ B_148 ) @ zero_zero_rat ) ) ) ).
thf(fact_1185_mult__pos__pos,axiom,
! [B_147: int,A_193: int] :
( ( ord_less_int @ zero_zero_int @ A_193 )
=> ( ( ord_less_int @ zero_zero_int @ B_147 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_193 @ B_147 ) ) ) ) ).
thf(fact_1186_mult__pos__pos,axiom,
! [B_147: nat,A_193: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_193 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_147 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A_193 @ B_147 ) ) ) ) ).
thf(fact_1187_mult__pos__pos,axiom,
! [B_147: real,A_193: real] :
( ( ord_less_real @ zero_zero_real @ A_193 )
=> ( ( ord_less_real @ zero_zero_real @ B_147 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A_193 @ B_147 ) ) ) ) ).
thf(fact_1188_mult__pos__pos,axiom,
! [B_147: code_code_numeral,A_193: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_193 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ B_147 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ ( times_1655362735umeral @ A_193 @ B_147 ) ) ) ) ).
thf(fact_1189_mult__pos__pos,axiom,
! [B_147: quickcheck_code_int,A_193: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_193 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ B_147 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ ( times_123202395de_int @ A_193 @ B_147 ) ) ) ) ).
thf(fact_1190_mult__pos__pos,axiom,
! [B_147: rat,A_193: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_193 )
=> ( ( ord_less_rat @ zero_zero_rat @ B_147 )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A_193 @ B_147 ) ) ) ) ).
thf(fact_1191_mult__less__cancel__left__pos,axiom,
! [A_192: int,B_146: int,C_91: int] :
( ( ord_less_int @ zero_zero_int @ C_91 )
=> ( ( ord_less_int @ ( times_times_int @ C_91 @ A_192 ) @ ( times_times_int @ C_91 @ B_146 ) )
<=> ( ord_less_int @ A_192 @ B_146 ) ) ) ).
thf(fact_1192_mult__less__cancel__left__pos,axiom,
! [A_192: real,B_146: real,C_91: real] :
( ( ord_less_real @ zero_zero_real @ C_91 )
=> ( ( ord_less_real @ ( times_times_real @ C_91 @ A_192 ) @ ( times_times_real @ C_91 @ B_146 ) )
<=> ( ord_less_real @ A_192 @ B_146 ) ) ) ).
thf(fact_1193_mult__less__cancel__left__pos,axiom,
! [A_192: rat,B_146: rat,C_91: rat] :
( ( ord_less_rat @ zero_zero_rat @ C_91 )
=> ( ( ord_less_rat @ ( times_times_rat @ C_91 @ A_192 ) @ ( times_times_rat @ C_91 @ B_146 ) )
<=> ( ord_less_rat @ A_192 @ B_146 ) ) ) ).
thf(fact_1194_mult__less__cancel__left__disj,axiom,
! [C_90: int,A_191: int,B_145: int] :
( ( ord_less_int @ ( times_times_int @ C_90 @ A_191 ) @ ( times_times_int @ C_90 @ B_145 ) )
<=> ( ( ( ord_less_int @ zero_zero_int @ C_90 )
& ( ord_less_int @ A_191 @ B_145 ) )
| ( ( ord_less_int @ C_90 @ zero_zero_int )
& ( ord_less_int @ B_145 @ A_191 ) ) ) ) ).
thf(fact_1195_mult__less__cancel__left__disj,axiom,
! [C_90: real,A_191: real,B_145: real] :
( ( ord_less_real @ ( times_times_real @ C_90 @ A_191 ) @ ( times_times_real @ C_90 @ B_145 ) )
<=> ( ( ( ord_less_real @ zero_zero_real @ C_90 )
& ( ord_less_real @ A_191 @ B_145 ) )
| ( ( ord_less_real @ C_90 @ zero_zero_real )
& ( ord_less_real @ B_145 @ A_191 ) ) ) ) ).
thf(fact_1196_mult__less__cancel__left__disj,axiom,
! [C_90: rat,A_191: rat,B_145: rat] :
( ( ord_less_rat @ ( times_times_rat @ C_90 @ A_191 ) @ ( times_times_rat @ C_90 @ B_145 ) )
<=> ( ( ( ord_less_rat @ zero_zero_rat @ C_90 )
& ( ord_less_rat @ A_191 @ B_145 ) )
| ( ( ord_less_rat @ C_90 @ zero_zero_rat )
& ( ord_less_rat @ B_145 @ A_191 ) ) ) ) ).
thf(fact_1197_mult__less__cancel__right__disj,axiom,
! [A_190: int,C_89: int,B_144: int] :
( ( ord_less_int @ ( times_times_int @ A_190 @ C_89 ) @ ( times_times_int @ B_144 @ C_89 ) )
<=> ( ( ( ord_less_int @ zero_zero_int @ C_89 )
& ( ord_less_int @ A_190 @ B_144 ) )
| ( ( ord_less_int @ C_89 @ zero_zero_int )
& ( ord_less_int @ B_144 @ A_190 ) ) ) ) ).
thf(fact_1198_mult__less__cancel__right__disj,axiom,
! [A_190: real,C_89: real,B_144: real] :
( ( ord_less_real @ ( times_times_real @ A_190 @ C_89 ) @ ( times_times_real @ B_144 @ C_89 ) )
<=> ( ( ( ord_less_real @ zero_zero_real @ C_89 )
& ( ord_less_real @ A_190 @ B_144 ) )
| ( ( ord_less_real @ C_89 @ zero_zero_real )
& ( ord_less_real @ B_144 @ A_190 ) ) ) ) ).
thf(fact_1199_mult__less__cancel__right__disj,axiom,
! [A_190: rat,C_89: rat,B_144: rat] :
( ( ord_less_rat @ ( times_times_rat @ A_190 @ C_89 ) @ ( times_times_rat @ B_144 @ C_89 ) )
<=> ( ( ( ord_less_rat @ zero_zero_rat @ C_89 )
& ( ord_less_rat @ A_190 @ B_144 ) )
| ( ( ord_less_rat @ C_89 @ zero_zero_rat )
& ( ord_less_rat @ B_144 @ A_190 ) ) ) ) ).
thf(fact_1200_not__square__less__zero,axiom,
! [A_189: int] :
~ ( ord_less_int @ ( times_times_int @ A_189 @ A_189 ) @ zero_zero_int ) ).
thf(fact_1201_not__square__less__zero,axiom,
! [A_189: real] :
~ ( ord_less_real @ ( times_times_real @ A_189 @ A_189 ) @ zero_zero_real ) ).
thf(fact_1202_not__square__less__zero,axiom,
! [A_189: rat] :
~ ( ord_less_rat @ ( times_times_rat @ A_189 @ A_189 ) @ zero_zero_rat ) ).
thf(fact_1203_pos__add__strict,axiom,
! [B_143: code_code_numeral,C_88: code_code_numeral,A_188: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_188 )
=> ( ( ord_le1304079648umeral @ B_143 @ C_88 )
=> ( ord_le1304079648umeral @ B_143 @ ( plus_p1627245867umeral @ A_188 @ C_88 ) ) ) ) ).
thf(fact_1204_pos__add__strict,axiom,
! [B_143: int,C_88: int,A_188: int] :
( ( ord_less_int @ zero_zero_int @ A_188 )
=> ( ( ord_less_int @ B_143 @ C_88 )
=> ( ord_less_int @ B_143 @ ( plus_plus_int @ A_188 @ C_88 ) ) ) ) ).
thf(fact_1205_pos__add__strict,axiom,
! [B_143: nat,C_88: nat,A_188: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_188 )
=> ( ( ord_less_nat @ B_143 @ C_88 )
=> ( ord_less_nat @ B_143 @ ( plus_plus_nat @ A_188 @ C_88 ) ) ) ) ).
thf(fact_1206_pos__add__strict,axiom,
! [B_143: real,C_88: real,A_188: real] :
( ( ord_less_real @ zero_zero_real @ A_188 )
=> ( ( ord_less_real @ B_143 @ C_88 )
=> ( ord_less_real @ B_143 @ ( plus_plus_real @ A_188 @ C_88 ) ) ) ) ).
thf(fact_1207_pos__add__strict,axiom,
! [B_143: quickcheck_code_int,C_88: quickcheck_code_int,A_188: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_188 )
=> ( ( ord_le1860547276de_int @ B_143 @ C_88 )
=> ( ord_le1860547276de_int @ B_143 @ ( plus_p1446045655de_int @ A_188 @ C_88 ) ) ) ) ).
thf(fact_1208_pos__add__strict,axiom,
! [B_143: rat,C_88: rat,A_188: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_188 )
=> ( ( ord_less_rat @ B_143 @ C_88 )
=> ( ord_less_rat @ B_143 @ ( plus_plus_rat @ A_188 @ C_88 ) ) ) ) ).
thf(fact_1209_zero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
thf(fact_1210_zero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
thf(fact_1211_zero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
thf(fact_1212_zero__le__one,axiom,
ord_le565307924umeral @ zero_z126310315umeral @ one_on1645066479umeral ).
thf(fact_1213_zero__le__one,axiom,
ord_le258702272de_int @ zero_z891286103de_int @ one_on1684967323de_int ).
thf(fact_1214_zero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
thf(fact_1215_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
thf(fact_1216_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
thf(fact_1217_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
thf(fact_1218_not__one__le__zero,axiom,
~ ( ord_le565307924umeral @ one_on1645066479umeral @ zero_z126310315umeral ) ).
thf(fact_1219_not__one__le__zero,axiom,
~ ( ord_le258702272de_int @ one_on1684967323de_int @ zero_z891286103de_int ) ).
thf(fact_1220_not__one__le__zero,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
thf(fact_1221_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
thf(fact_1222_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
thf(fact_1223_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
thf(fact_1224_zero__less__one,axiom,
ord_le1304079648umeral @ zero_z126310315umeral @ one_on1645066479umeral ).
thf(fact_1225_zero__less__one,axiom,
ord_le1860547276de_int @ zero_z891286103de_int @ one_on1684967323de_int ).
thf(fact_1226_zero__less__one,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
thf(fact_1227_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
thf(fact_1228_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
thf(fact_1229_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
thf(fact_1230_not__one__less__zero,axiom,
~ ( ord_le1304079648umeral @ one_on1645066479umeral @ zero_z126310315umeral ) ).
thf(fact_1231_not__one__less__zero,axiom,
~ ( ord_le1860547276de_int @ one_on1684967323de_int @ zero_z891286103de_int ) ).
thf(fact_1232_not__one__less__zero,axiom,
~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
thf(fact_1233_power__mono,axiom,
! [N_60: nat,A_187: int,B_142: int] :
( ( ord_less_eq_int @ A_187 @ B_142 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_187 )
=> ( ord_less_eq_int @ ( power_power_int @ A_187 @ N_60 ) @ ( power_power_int @ B_142 @ N_60 ) ) ) ) ).
thf(fact_1234_power__mono,axiom,
! [N_60: nat,A_187: nat,B_142: nat] :
( ( ord_less_eq_nat @ A_187 @ B_142 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_187 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A_187 @ N_60 ) @ ( power_power_nat @ B_142 @ N_60 ) ) ) ) ).
thf(fact_1235_power__mono,axiom,
! [N_60: nat,A_187: real,B_142: real] :
( ( ord_less_eq_real @ A_187 @ B_142 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_187 )
=> ( ord_less_eq_real @ ( power_power_real @ A_187 @ N_60 ) @ ( power_power_real @ B_142 @ N_60 ) ) ) ) ).
thf(fact_1236_power__mono,axiom,
! [N_60: nat,A_187: code_code_numeral,B_142: code_code_numeral] :
( ( ord_le565307924umeral @ A_187 @ B_142 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_187 )
=> ( ord_le565307924umeral @ ( power_2100829034umeral @ A_187 @ N_60 ) @ ( power_2100829034umeral @ B_142 @ N_60 ) ) ) ) ).
thf(fact_1237_power__mono,axiom,
! [N_60: nat,A_187: quickcheck_code_int,B_142: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_187 @ B_142 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_187 )
=> ( ord_le258702272de_int @ ( power_881366806de_int @ A_187 @ N_60 ) @ ( power_881366806de_int @ B_142 @ N_60 ) ) ) ) ).
thf(fact_1238_power__mono,axiom,
! [N_60: nat,A_187: rat,B_142: rat] :
( ( ord_less_eq_rat @ A_187 @ B_142 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A_187 )
=> ( ord_less_eq_rat @ ( power_power_rat @ A_187 @ N_60 ) @ ( power_power_rat @ B_142 @ N_60 ) ) ) ) ).
thf(fact_1239_zero__le__power,axiom,
! [N_59: nat,A_186: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_186 )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A_186 @ N_59 ) ) ) ).
thf(fact_1240_zero__le__power,axiom,
! [N_59: nat,A_186: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_186 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A_186 @ N_59 ) ) ) ).
thf(fact_1241_zero__le__power,axiom,
! [N_59: nat,A_186: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_186 )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A_186 @ N_59 ) ) ) ).
thf(fact_1242_zero__le__power,axiom,
! [N_59: nat,A_186: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_186 )
=> ( ord_le565307924umeral @ zero_z126310315umeral @ ( power_2100829034umeral @ A_186 @ N_59 ) ) ) ).
thf(fact_1243_zero__le__power,axiom,
! [N_59: nat,A_186: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_186 )
=> ( ord_le258702272de_int @ zero_z891286103de_int @ ( power_881366806de_int @ A_186 @ N_59 ) ) ) ).
thf(fact_1244_zero__le__power,axiom,
! [N_59: nat,A_186: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_186 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A_186 @ N_59 ) ) ) ).
thf(fact_1245_less__1__mult,axiom,
! [N_58: int,M_27: int] :
( ( ord_less_int @ one_one_int @ M_27 )
=> ( ( ord_less_int @ one_one_int @ N_58 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M_27 @ N_58 ) ) ) ) ).
thf(fact_1246_less__1__mult,axiom,
! [N_58: nat,M_27: nat] :
( ( ord_less_nat @ one_one_nat @ M_27 )
=> ( ( ord_less_nat @ one_one_nat @ N_58 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M_27 @ N_58 ) ) ) ) ).
thf(fact_1247_less__1__mult,axiom,
! [N_58: real,M_27: real] :
( ( ord_less_real @ one_one_real @ M_27 )
=> ( ( ord_less_real @ one_one_real @ N_58 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M_27 @ N_58 ) ) ) ) ).
thf(fact_1248_less__1__mult,axiom,
! [N_58: code_code_numeral,M_27: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ M_27 )
=> ( ( ord_le1304079648umeral @ one_on1645066479umeral @ N_58 )
=> ( ord_le1304079648umeral @ one_on1645066479umeral @ ( times_1655362735umeral @ M_27 @ N_58 ) ) ) ) ).
thf(fact_1249_less__1__mult,axiom,
! [N_58: quickcheck_code_int,M_27: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ M_27 )
=> ( ( ord_le1860547276de_int @ one_on1684967323de_int @ N_58 )
=> ( ord_le1860547276de_int @ one_on1684967323de_int @ ( times_123202395de_int @ M_27 @ N_58 ) ) ) ) ).
thf(fact_1250_less__1__mult,axiom,
! [N_58: rat,M_27: rat] :
( ( ord_less_rat @ one_one_rat @ M_27 )
=> ( ( ord_less_rat @ one_one_rat @ N_58 )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M_27 @ N_58 ) ) ) ) ).
thf(fact_1251_zero__less__power,axiom,
! [N_57: nat,A_185: int] :
( ( ord_less_int @ zero_zero_int @ A_185 )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A_185 @ N_57 ) ) ) ).
thf(fact_1252_zero__less__power,axiom,
! [N_57: nat,A_185: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_185 )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A_185 @ N_57 ) ) ) ).
thf(fact_1253_zero__less__power,axiom,
! [N_57: nat,A_185: real] :
( ( ord_less_real @ zero_zero_real @ A_185 )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A_185 @ N_57 ) ) ) ).
thf(fact_1254_zero__less__power,axiom,
! [N_57: nat,A_185: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_185 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ ( power_2100829034umeral @ A_185 @ N_57 ) ) ) ).
thf(fact_1255_zero__less__power,axiom,
! [N_57: nat,A_185: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_185 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ ( power_881366806de_int @ A_185 @ N_57 ) ) ) ).
thf(fact_1256_zero__less__power,axiom,
! [N_57: nat,A_185: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_185 )
=> ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A_185 @ N_57 ) ) ) ).
thf(fact_1257_less__add__one,axiom,
! [A_184: int] : ( ord_less_int @ A_184 @ ( plus_plus_int @ A_184 @ one_one_int ) ) ).
thf(fact_1258_less__add__one,axiom,
! [A_184: nat] : ( ord_less_nat @ A_184 @ ( plus_plus_nat @ A_184 @ one_one_nat ) ) ).
thf(fact_1259_less__add__one,axiom,
! [A_184: real] : ( ord_less_real @ A_184 @ ( plus_plus_real @ A_184 @ one_one_real ) ) ).
thf(fact_1260_less__add__one,axiom,
! [A_184: code_code_numeral] : ( ord_le1304079648umeral @ A_184 @ ( plus_p1627245867umeral @ A_184 @ one_on1645066479umeral ) ) ).
thf(fact_1261_less__add__one,axiom,
! [A_184: quickcheck_code_int] : ( ord_le1860547276de_int @ A_184 @ ( plus_p1446045655de_int @ A_184 @ one_on1684967323de_int ) ) ).
thf(fact_1262_less__add__one,axiom,
! [A_184: rat] : ( ord_less_rat @ A_184 @ ( plus_plus_rat @ A_184 @ one_one_rat ) ) ).
thf(fact_1263_dvd__mult__cancel__left,axiom,
! [C_87: real,A_183: real,B_141: real] :
( ( dvd_dvd_real @ ( times_times_real @ C_87 @ A_183 ) @ ( times_times_real @ C_87 @ B_141 ) )
<=> ( ( C_87 = zero_zero_real )
| ( dvd_dvd_real @ A_183 @ B_141 ) ) ) ).
thf(fact_1264_dvd__mult__cancel__left,axiom,
! [C_87: complex,A_183: complex,B_141: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ C_87 @ A_183 ) @ ( times_times_complex @ C_87 @ B_141 ) )
<=> ( ( C_87 = zero_zero_complex )
| ( dvd_dvd_complex @ A_183 @ B_141 ) ) ) ).
thf(fact_1265_dvd__mult__cancel__left,axiom,
! [C_87: rat,A_183: rat,B_141: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ C_87 @ A_183 ) @ ( times_times_rat @ C_87 @ B_141 ) )
<=> ( ( C_87 = zero_zero_rat )
| ( dvd_dvd_rat @ A_183 @ B_141 ) ) ) ).
thf(fact_1266_dvd__mult__cancel__left,axiom,
! [C_87: int,A_183: int,B_141: int] :
( ( dvd_dvd_int @ ( times_times_int @ C_87 @ A_183 ) @ ( times_times_int @ C_87 @ B_141 ) )
<=> ( ( C_87 = zero_zero_int )
| ( dvd_dvd_int @ A_183 @ B_141 ) ) ) ).
thf(fact_1267_dvd__mult__cancel__right,axiom,
! [A_182: real,C_86: real,B_140: real] :
( ( dvd_dvd_real @ ( times_times_real @ A_182 @ C_86 ) @ ( times_times_real @ B_140 @ C_86 ) )
<=> ( ( C_86 = zero_zero_real )
| ( dvd_dvd_real @ A_182 @ B_140 ) ) ) ).
thf(fact_1268_dvd__mult__cancel__right,axiom,
! [A_182: complex,C_86: complex,B_140: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A_182 @ C_86 ) @ ( times_times_complex @ B_140 @ C_86 ) )
<=> ( ( C_86 = zero_zero_complex )
| ( dvd_dvd_complex @ A_182 @ B_140 ) ) ) ).
thf(fact_1269_dvd__mult__cancel__right,axiom,
! [A_182: rat,C_86: rat,B_140: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A_182 @ C_86 ) @ ( times_times_rat @ B_140 @ C_86 ) )
<=> ( ( C_86 = zero_zero_rat )
| ( dvd_dvd_rat @ A_182 @ B_140 ) ) ) ).
thf(fact_1270_dvd__mult__cancel__right,axiom,
! [A_182: int,C_86: int,B_140: int] :
( ( dvd_dvd_int @ ( times_times_int @ A_182 @ C_86 ) @ ( times_times_int @ B_140 @ C_86 ) )
<=> ( ( C_86 = zero_zero_int )
| ( dvd_dvd_int @ A_182 @ B_140 ) ) ) ).
thf(fact_1271_power__increasing,axiom,
! [A_181: int,N_56: nat,N_55: nat] :
( ( ord_less_eq_nat @ N_56 @ N_55 )
=> ( ( ord_less_eq_int @ one_one_int @ A_181 )
=> ( ord_less_eq_int @ ( power_power_int @ A_181 @ N_56 ) @ ( power_power_int @ A_181 @ N_55 ) ) ) ) ).
thf(fact_1272_power__increasing,axiom,
! [A_181: nat,N_56: nat,N_55: nat] :
( ( ord_less_eq_nat @ N_56 @ N_55 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A_181 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A_181 @ N_56 ) @ ( power_power_nat @ A_181 @ N_55 ) ) ) ) ).
thf(fact_1273_power__increasing,axiom,
! [A_181: real,N_56: nat,N_55: nat] :
( ( ord_less_eq_nat @ N_56 @ N_55 )
=> ( ( ord_less_eq_real @ one_one_real @ A_181 )
=> ( ord_less_eq_real @ ( power_power_real @ A_181 @ N_56 ) @ ( power_power_real @ A_181 @ N_55 ) ) ) ) ).
thf(fact_1274_power__increasing,axiom,
! [A_181: code_code_numeral,N_56: nat,N_55: nat] :
( ( ord_less_eq_nat @ N_56 @ N_55 )
=> ( ( ord_le565307924umeral @ one_on1645066479umeral @ A_181 )
=> ( ord_le565307924umeral @ ( power_2100829034umeral @ A_181 @ N_56 ) @ ( power_2100829034umeral @ A_181 @ N_55 ) ) ) ) ).
thf(fact_1275_power__increasing,axiom,
! [A_181: quickcheck_code_int,N_56: nat,N_55: nat] :
( ( ord_less_eq_nat @ N_56 @ N_55 )
=> ( ( ord_le258702272de_int @ one_on1684967323de_int @ A_181 )
=> ( ord_le258702272de_int @ ( power_881366806de_int @ A_181 @ N_56 ) @ ( power_881366806de_int @ A_181 @ N_55 ) ) ) ) ).
thf(fact_1276_power__increasing,axiom,
! [A_181: rat,N_56: nat,N_55: nat] :
( ( ord_less_eq_nat @ N_56 @ N_55 )
=> ( ( ord_less_eq_rat @ one_one_rat @ A_181 )
=> ( ord_less_eq_rat @ ( power_power_rat @ A_181 @ N_56 ) @ ( power_power_rat @ A_181 @ N_55 ) ) ) ) ).
thf(fact_1277_one__le__power,axiom,
! [N_54: nat,A_180: int] :
( ( ord_less_eq_int @ one_one_int @ A_180 )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A_180 @ N_54 ) ) ) ).
thf(fact_1278_one__le__power,axiom,
! [N_54: nat,A_180: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A_180 )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A_180 @ N_54 ) ) ) ).
thf(fact_1279_one__le__power,axiom,
! [N_54: nat,A_180: real] :
( ( ord_less_eq_real @ one_one_real @ A_180 )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A_180 @ N_54 ) ) ) ).
thf(fact_1280_one__le__power,axiom,
! [N_54: nat,A_180: code_code_numeral] :
( ( ord_le565307924umeral @ one_on1645066479umeral @ A_180 )
=> ( ord_le565307924umeral @ one_on1645066479umeral @ ( power_2100829034umeral @ A_180 @ N_54 ) ) ) ).
thf(fact_1281_one__le__power,axiom,
! [N_54: nat,A_180: quickcheck_code_int] :
( ( ord_le258702272de_int @ one_on1684967323de_int @ A_180 )
=> ( ord_le258702272de_int @ one_on1684967323de_int @ ( power_881366806de_int @ A_180 @ N_54 ) ) ) ).
thf(fact_1282_one__le__power,axiom,
! [N_54: nat,A_180: rat] :
( ( ord_less_eq_rat @ one_one_rat @ A_180 )
=> ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A_180 @ N_54 ) ) ) ).
thf(fact_1283_power__inject__exp,axiom,
! [M_26: nat,N_53: nat,A_179: int] :
( ( ord_less_int @ one_one_int @ A_179 )
=> ( ( ( power_power_int @ A_179 @ M_26 )
= ( power_power_int @ A_179 @ N_53 ) )
<=> ( M_26 = N_53 ) ) ) ).
thf(fact_1284_power__inject__exp,axiom,
! [M_26: nat,N_53: nat,A_179: nat] :
( ( ord_less_nat @ one_one_nat @ A_179 )
=> ( ( ( power_power_nat @ A_179 @ M_26 )
= ( power_power_nat @ A_179 @ N_53 ) )
<=> ( M_26 = N_53 ) ) ) ).
thf(fact_1285_power__inject__exp,axiom,
! [M_26: nat,N_53: nat,A_179: real] :
( ( ord_less_real @ one_one_real @ A_179 )
=> ( ( ( power_power_real @ A_179 @ M_26 )
= ( power_power_real @ A_179 @ N_53 ) )
<=> ( M_26 = N_53 ) ) ) ).
thf(fact_1286_power__inject__exp,axiom,
! [M_26: nat,N_53: nat,A_179: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ A_179 )
=> ( ( ( power_2100829034umeral @ A_179 @ M_26 )
= ( power_2100829034umeral @ A_179 @ N_53 ) )
<=> ( M_26 = N_53 ) ) ) ).
thf(fact_1287_power__inject__exp,axiom,
! [M_26: nat,N_53: nat,A_179: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ A_179 )
=> ( ( ( power_881366806de_int @ A_179 @ M_26 )
= ( power_881366806de_int @ A_179 @ N_53 ) )
<=> ( M_26 = N_53 ) ) ) ).
thf(fact_1288_power__inject__exp,axiom,
! [M_26: nat,N_53: nat,A_179: rat] :
( ( ord_less_rat @ one_one_rat @ A_179 )
=> ( ( ( power_power_rat @ A_179 @ M_26 )
= ( power_power_rat @ A_179 @ N_53 ) )
<=> ( M_26 = N_53 ) ) ) ).
thf(fact_1289_power__eq__0__iff,axiom,
! [A_178: int,N_52: nat] :
( ( ( power_power_int @ A_178 @ N_52 )
= zero_zero_int )
<=> ( ( A_178 = zero_zero_int )
& ( N_52 != zero_zero_nat ) ) ) ).
thf(fact_1290_power__eq__0__iff,axiom,
! [A_178: nat,N_52: nat] :
( ( ( power_power_nat @ A_178 @ N_52 )
= zero_zero_nat )
<=> ( ( A_178 = zero_zero_nat )
& ( N_52 != zero_zero_nat ) ) ) ).
thf(fact_1291_power__eq__0__iff,axiom,
! [A_178: real,N_52: nat] :
( ( ( power_power_real @ A_178 @ N_52 )
= zero_zero_real )
<=> ( ( A_178 = zero_zero_real )
& ( N_52 != zero_zero_nat ) ) ) ).
thf(fact_1292_power__eq__0__iff,axiom,
! [A_178: code_code_numeral,N_52: nat] :
( ( ( power_2100829034umeral @ A_178 @ N_52 )
= zero_z126310315umeral )
<=> ( ( A_178 = zero_z126310315umeral )
& ( N_52 != zero_zero_nat ) ) ) ).
thf(fact_1293_power__eq__0__iff,axiom,
! [A_178: complex,N_52: nat] :
( ( ( power_power_complex @ A_178 @ N_52 )
= zero_zero_complex )
<=> ( ( A_178 = zero_zero_complex )
& ( N_52 != zero_zero_nat ) ) ) ).
thf(fact_1294_power__eq__0__iff,axiom,
! [A_178: quickcheck_code_int,N_52: nat] :
( ( ( power_881366806de_int @ A_178 @ N_52 )
= zero_z891286103de_int )
<=> ( ( A_178 = zero_z891286103de_int )
& ( N_52 != zero_zero_nat ) ) ) ).
thf(fact_1295_power__eq__0__iff,axiom,
! [A_178: rat,N_52: nat] :
( ( ( power_power_rat @ A_178 @ N_52 )
= zero_zero_rat )
<=> ( ( A_178 = zero_zero_rat )
& ( N_52 != zero_zero_nat ) ) ) ).
thf(fact_1296_power__0,axiom,
! [A_177: int] :
( ( power_power_int @ A_177 @ zero_zero_nat )
= one_one_int ) ).
thf(fact_1297_power__0,axiom,
! [A_177: nat] :
( ( power_power_nat @ A_177 @ zero_zero_nat )
= one_one_nat ) ).
thf(fact_1298_power__0,axiom,
! [A_177: real] :
( ( power_power_real @ A_177 @ zero_zero_nat )
= one_one_real ) ).
thf(fact_1299_power__0,axiom,
! [A_177: code_code_numeral] :
( ( power_2100829034umeral @ A_177 @ zero_zero_nat )
= one_on1645066479umeral ) ).
thf(fact_1300_power__0,axiom,
! [A_177: complex] :
( ( power_power_complex @ A_177 @ zero_zero_nat )
= one_one_complex ) ).
thf(fact_1301_power__0,axiom,
! [A_177: quickcheck_code_int] :
( ( power_881366806de_int @ A_177 @ zero_zero_nat )
= one_on1684967323de_int ) ).
thf(fact_1302_power__0,axiom,
! [A_177: rat] :
( ( power_power_rat @ A_177 @ zero_zero_nat )
= one_one_rat ) ).
thf(fact_1303_power__add,axiom,
! [A_176: code_code_numeral,M_25: nat,N_51: nat] :
( ( power_2100829034umeral @ A_176 @ ( plus_plus_nat @ M_25 @ N_51 ) )
= ( times_1655362735umeral @ ( power_2100829034umeral @ A_176 @ M_25 ) @ ( power_2100829034umeral @ A_176 @ N_51 ) ) ) ).
thf(fact_1304_power__add,axiom,
! [A_176: quickcheck_code_int,M_25: nat,N_51: nat] :
( ( power_881366806de_int @ A_176 @ ( plus_plus_nat @ M_25 @ N_51 ) )
= ( times_123202395de_int @ ( power_881366806de_int @ A_176 @ M_25 ) @ ( power_881366806de_int @ A_176 @ N_51 ) ) ) ).
thf(fact_1305_power__add,axiom,
! [A_176: rat,M_25: nat,N_51: nat] :
( ( power_power_rat @ A_176 @ ( plus_plus_nat @ M_25 @ N_51 ) )
= ( times_times_rat @ ( power_power_rat @ A_176 @ M_25 ) @ ( power_power_rat @ A_176 @ N_51 ) ) ) ).
thf(fact_1306_power__add,axiom,
! [A_176: int,M_25: nat,N_51: nat] :
( ( power_power_int @ A_176 @ ( plus_plus_nat @ M_25 @ N_51 ) )
= ( times_times_int @ ( power_power_int @ A_176 @ M_25 ) @ ( power_power_int @ A_176 @ N_51 ) ) ) ).
thf(fact_1307_power__add,axiom,
! [A_176: nat,M_25: nat,N_51: nat] :
( ( power_power_nat @ A_176 @ ( plus_plus_nat @ M_25 @ N_51 ) )
= ( times_times_nat @ ( power_power_nat @ A_176 @ M_25 ) @ ( power_power_nat @ A_176 @ N_51 ) ) ) ).
thf(fact_1308_power__add,axiom,
! [A_176: real,M_25: nat,N_51: nat] :
( ( power_power_real @ A_176 @ ( plus_plus_nat @ M_25 @ N_51 ) )
= ( times_times_real @ ( power_power_real @ A_176 @ M_25 ) @ ( power_power_real @ A_176 @ N_51 ) ) ) ).
thf(fact_1309_power__add,axiom,
! [A_176: complex,M_25: nat,N_51: nat] :
( ( power_power_complex @ A_176 @ ( plus_plus_nat @ M_25 @ N_51 ) )
= ( times_times_complex @ ( power_power_complex @ A_176 @ M_25 ) @ ( power_power_complex @ A_176 @ N_51 ) ) ) ).
thf(fact_1310_zcong__not,axiom,
! [B: int,M: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ M )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ A )
=> ~ ( zcong @ A @ B @ M ) ) ) ) ) ).
thf(fact_1311_zcong__not__zero,axiom,
! [M: int,X: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ M )
=> ~ ( zcong @ X @ zero_zero_int @ M ) ) ) ).
thf(fact_1312_zcong__less__eq,axiom,
! [M: int,Y: int,X: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ zero_zero_int @ Y )
=> ( ( ord_less_int @ zero_zero_int @ M )
=> ( ( zcong @ X @ Y @ M )
=> ( ( ord_less_int @ X @ M )
=> ( ( ord_less_int @ Y @ M )
=> ( X = Y ) ) ) ) ) ) ) ).
thf(fact_1313_zdvd__bounds,axiom,
! [N: int,M: int] :
( ( dvd_dvd_int @ N @ M )
=> ( ( ord_less_eq_int @ M @ zero_zero_int )
| ( ord_less_eq_int @ N @ M ) ) ) ).
thf(fact_1314_zcong__iff__lin,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
<=> ? [K: int] :
( B
= ( plus_plus_int @ A @ ( times_times_int @ M @ K ) ) ) ) ).
thf(fact_1315_zcong__eq__zdvd__prop,axiom,
! [X: int,P_3: int] :
( ( zcong @ X @ zero_zero_int @ P_3 )
<=> ( dvd_dvd_int @ P_3 @ X ) ) ).
thf(fact_1316_zcong__zero__equiv__div,axiom,
! [A: int,M: int] :
( ( zcong @ A @ zero_zero_int @ M )
<=> ( dvd_dvd_int @ M @ A ) ) ).
thf(fact_1317_zprime__zdvd__zmult__better,axiom,
! [M: int,N: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( dvd_dvd_int @ P_3 @ ( times_times_int @ M @ N ) )
=> ( ( dvd_dvd_int @ P_3 @ M )
| ( dvd_dvd_int @ P_3 @ N ) ) ) ) ).
thf(fact_1318_mult__left__le__imp__le,axiom,
! [C_85: int,A_175: int,B_139: int] :
( ( ord_less_eq_int @ ( times_times_int @ C_85 @ A_175 ) @ ( times_times_int @ C_85 @ B_139 ) )
=> ( ( ord_less_int @ zero_zero_int @ C_85 )
=> ( ord_less_eq_int @ A_175 @ B_139 ) ) ) ).
thf(fact_1319_mult__left__le__imp__le,axiom,
! [C_85: nat,A_175: nat,B_139: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C_85 @ A_175 ) @ ( times_times_nat @ C_85 @ B_139 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C_85 )
=> ( ord_less_eq_nat @ A_175 @ B_139 ) ) ) ).
thf(fact_1320_mult__left__le__imp__le,axiom,
! [C_85: real,A_175: real,B_139: real] :
( ( ord_less_eq_real @ ( times_times_real @ C_85 @ A_175 ) @ ( times_times_real @ C_85 @ B_139 ) )
=> ( ( ord_less_real @ zero_zero_real @ C_85 )
=> ( ord_less_eq_real @ A_175 @ B_139 ) ) ) ).
thf(fact_1321_mult__left__le__imp__le,axiom,
! [C_85: code_code_numeral,A_175: code_code_numeral,B_139: code_code_numeral] :
( ( ord_le565307924umeral @ ( times_1655362735umeral @ C_85 @ A_175 ) @ ( times_1655362735umeral @ C_85 @ B_139 ) )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ C_85 )
=> ( ord_le565307924umeral @ A_175 @ B_139 ) ) ) ).
thf(fact_1322_mult__left__le__imp__le,axiom,
! [C_85: quickcheck_code_int,A_175: quickcheck_code_int,B_139: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( times_123202395de_int @ C_85 @ A_175 ) @ ( times_123202395de_int @ C_85 @ B_139 ) )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ C_85 )
=> ( ord_le258702272de_int @ A_175 @ B_139 ) ) ) ).
thf(fact_1323_mult__left__le__imp__le,axiom,
! [C_85: rat,A_175: rat,B_139: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C_85 @ A_175 ) @ ( times_times_rat @ C_85 @ B_139 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C_85 )
=> ( ord_less_eq_rat @ A_175 @ B_139 ) ) ) ).
thf(fact_1324_mult__right__le__imp__le,axiom,
! [A_174: int,C_84: int,B_138: int] :
( ( ord_less_eq_int @ ( times_times_int @ A_174 @ C_84 ) @ ( times_times_int @ B_138 @ C_84 ) )
=> ( ( ord_less_int @ zero_zero_int @ C_84 )
=> ( ord_less_eq_int @ A_174 @ B_138 ) ) ) ).
thf(fact_1325_mult__right__le__imp__le,axiom,
! [A_174: nat,C_84: nat,B_138: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A_174 @ C_84 ) @ ( times_times_nat @ B_138 @ C_84 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C_84 )
=> ( ord_less_eq_nat @ A_174 @ B_138 ) ) ) ).
thf(fact_1326_mult__right__le__imp__le,axiom,
! [A_174: real,C_84: real,B_138: real] :
( ( ord_less_eq_real @ ( times_times_real @ A_174 @ C_84 ) @ ( times_times_real @ B_138 @ C_84 ) )
=> ( ( ord_less_real @ zero_zero_real @ C_84 )
=> ( ord_less_eq_real @ A_174 @ B_138 ) ) ) ).
thf(fact_1327_mult__right__le__imp__le,axiom,
! [A_174: code_code_numeral,C_84: code_code_numeral,B_138: code_code_numeral] :
( ( ord_le565307924umeral @ ( times_1655362735umeral @ A_174 @ C_84 ) @ ( times_1655362735umeral @ B_138 @ C_84 ) )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ C_84 )
=> ( ord_le565307924umeral @ A_174 @ B_138 ) ) ) ).
thf(fact_1328_mult__right__le__imp__le,axiom,
! [A_174: quickcheck_code_int,C_84: quickcheck_code_int,B_138: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( times_123202395de_int @ A_174 @ C_84 ) @ ( times_123202395de_int @ B_138 @ C_84 ) )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ C_84 )
=> ( ord_le258702272de_int @ A_174 @ B_138 ) ) ) ).
thf(fact_1329_mult__right__le__imp__le,axiom,
! [A_174: rat,C_84: rat,B_138: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A_174 @ C_84 ) @ ( times_times_rat @ B_138 @ C_84 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C_84 )
=> ( ord_less_eq_rat @ A_174 @ B_138 ) ) ) ).
thf(fact_1330_mult__less__imp__less__left,axiom,
! [C_83: int,A_173: int,B_137: int] :
( ( ord_less_int @ ( times_times_int @ C_83 @ A_173 ) @ ( times_times_int @ C_83 @ B_137 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_83 )
=> ( ord_less_int @ A_173 @ B_137 ) ) ) ).
thf(fact_1331_mult__less__imp__less__left,axiom,
! [C_83: nat,A_173: nat,B_137: nat] :
( ( ord_less_nat @ ( times_times_nat @ C_83 @ A_173 ) @ ( times_times_nat @ C_83 @ B_137 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_83 )
=> ( ord_less_nat @ A_173 @ B_137 ) ) ) ).
thf(fact_1332_mult__less__imp__less__left,axiom,
! [C_83: real,A_173: real,B_137: real] :
( ( ord_less_real @ ( times_times_real @ C_83 @ A_173 ) @ ( times_times_real @ C_83 @ B_137 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_83 )
=> ( ord_less_real @ A_173 @ B_137 ) ) ) ).
thf(fact_1333_mult__less__imp__less__left,axiom,
! [C_83: code_code_numeral,A_173: code_code_numeral,B_137: code_code_numeral] :
( ( ord_le1304079648umeral @ ( times_1655362735umeral @ C_83 @ A_173 ) @ ( times_1655362735umeral @ C_83 @ B_137 ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_83 )
=> ( ord_le1304079648umeral @ A_173 @ B_137 ) ) ) ).
thf(fact_1334_mult__less__imp__less__left,axiom,
! [C_83: quickcheck_code_int,A_173: quickcheck_code_int,B_137: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( times_123202395de_int @ C_83 @ A_173 ) @ ( times_123202395de_int @ C_83 @ B_137 ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_83 )
=> ( ord_le1860547276de_int @ A_173 @ B_137 ) ) ) ).
thf(fact_1335_mult__less__imp__less__left,axiom,
! [C_83: rat,A_173: rat,B_137: rat] :
( ( ord_less_rat @ ( times_times_rat @ C_83 @ A_173 ) @ ( times_times_rat @ C_83 @ B_137 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_83 )
=> ( ord_less_rat @ A_173 @ B_137 ) ) ) ).
thf(fact_1336_mult__left__less__imp__less,axiom,
! [C_82: int,A_172: int,B_136: int] :
( ( ord_less_int @ ( times_times_int @ C_82 @ A_172 ) @ ( times_times_int @ C_82 @ B_136 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_82 )
=> ( ord_less_int @ A_172 @ B_136 ) ) ) ).
thf(fact_1337_mult__left__less__imp__less,axiom,
! [C_82: nat,A_172: nat,B_136: nat] :
( ( ord_less_nat @ ( times_times_nat @ C_82 @ A_172 ) @ ( times_times_nat @ C_82 @ B_136 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_82 )
=> ( ord_less_nat @ A_172 @ B_136 ) ) ) ).
thf(fact_1338_mult__left__less__imp__less,axiom,
! [C_82: real,A_172: real,B_136: real] :
( ( ord_less_real @ ( times_times_real @ C_82 @ A_172 ) @ ( times_times_real @ C_82 @ B_136 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_82 )
=> ( ord_less_real @ A_172 @ B_136 ) ) ) ).
thf(fact_1339_mult__left__less__imp__less,axiom,
! [C_82: code_code_numeral,A_172: code_code_numeral,B_136: code_code_numeral] :
( ( ord_le1304079648umeral @ ( times_1655362735umeral @ C_82 @ A_172 ) @ ( times_1655362735umeral @ C_82 @ B_136 ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_82 )
=> ( ord_le1304079648umeral @ A_172 @ B_136 ) ) ) ).
thf(fact_1340_mult__left__less__imp__less,axiom,
! [C_82: quickcheck_code_int,A_172: quickcheck_code_int,B_136: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( times_123202395de_int @ C_82 @ A_172 ) @ ( times_123202395de_int @ C_82 @ B_136 ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_82 )
=> ( ord_le1860547276de_int @ A_172 @ B_136 ) ) ) ).
thf(fact_1341_mult__left__less__imp__less,axiom,
! [C_82: rat,A_172: rat,B_136: rat] :
( ( ord_less_rat @ ( times_times_rat @ C_82 @ A_172 ) @ ( times_times_rat @ C_82 @ B_136 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_82 )
=> ( ord_less_rat @ A_172 @ B_136 ) ) ) ).
thf(fact_1342_mult__less__imp__less__right,axiom,
! [A_171: int,C_81: int,B_135: int] :
( ( ord_less_int @ ( times_times_int @ A_171 @ C_81 ) @ ( times_times_int @ B_135 @ C_81 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_81 )
=> ( ord_less_int @ A_171 @ B_135 ) ) ) ).
thf(fact_1343_mult__less__imp__less__right,axiom,
! [A_171: nat,C_81: nat,B_135: nat] :
( ( ord_less_nat @ ( times_times_nat @ A_171 @ C_81 ) @ ( times_times_nat @ B_135 @ C_81 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_81 )
=> ( ord_less_nat @ A_171 @ B_135 ) ) ) ).
thf(fact_1344_mult__less__imp__less__right,axiom,
! [A_171: real,C_81: real,B_135: real] :
( ( ord_less_real @ ( times_times_real @ A_171 @ C_81 ) @ ( times_times_real @ B_135 @ C_81 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_81 )
=> ( ord_less_real @ A_171 @ B_135 ) ) ) ).
thf(fact_1345_mult__less__imp__less__right,axiom,
! [A_171: code_code_numeral,C_81: code_code_numeral,B_135: code_code_numeral] :
( ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_171 @ C_81 ) @ ( times_1655362735umeral @ B_135 @ C_81 ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_81 )
=> ( ord_le1304079648umeral @ A_171 @ B_135 ) ) ) ).
thf(fact_1346_mult__less__imp__less__right,axiom,
! [A_171: quickcheck_code_int,C_81: quickcheck_code_int,B_135: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( times_123202395de_int @ A_171 @ C_81 ) @ ( times_123202395de_int @ B_135 @ C_81 ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_81 )
=> ( ord_le1860547276de_int @ A_171 @ B_135 ) ) ) ).
thf(fact_1347_mult__less__imp__less__right,axiom,
! [A_171: rat,C_81: rat,B_135: rat] :
( ( ord_less_rat @ ( times_times_rat @ A_171 @ C_81 ) @ ( times_times_rat @ B_135 @ C_81 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_81 )
=> ( ord_less_rat @ A_171 @ B_135 ) ) ) ).
thf(fact_1348_mult__right__less__imp__less,axiom,
! [A_170: int,C_80: int,B_134: int] :
( ( ord_less_int @ ( times_times_int @ A_170 @ C_80 ) @ ( times_times_int @ B_134 @ C_80 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_80 )
=> ( ord_less_int @ A_170 @ B_134 ) ) ) ).
thf(fact_1349_mult__right__less__imp__less,axiom,
! [A_170: nat,C_80: nat,B_134: nat] :
( ( ord_less_nat @ ( times_times_nat @ A_170 @ C_80 ) @ ( times_times_nat @ B_134 @ C_80 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_80 )
=> ( ord_less_nat @ A_170 @ B_134 ) ) ) ).
thf(fact_1350_mult__right__less__imp__less,axiom,
! [A_170: real,C_80: real,B_134: real] :
( ( ord_less_real @ ( times_times_real @ A_170 @ C_80 ) @ ( times_times_real @ B_134 @ C_80 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_80 )
=> ( ord_less_real @ A_170 @ B_134 ) ) ) ).
thf(fact_1351_mult__right__less__imp__less,axiom,
! [A_170: code_code_numeral,C_80: code_code_numeral,B_134: code_code_numeral] :
( ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_170 @ C_80 ) @ ( times_1655362735umeral @ B_134 @ C_80 ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_80 )
=> ( ord_le1304079648umeral @ A_170 @ B_134 ) ) ) ).
thf(fact_1352_mult__right__less__imp__less,axiom,
! [A_170: quickcheck_code_int,C_80: quickcheck_code_int,B_134: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( times_123202395de_int @ A_170 @ C_80 ) @ ( times_123202395de_int @ B_134 @ C_80 ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_80 )
=> ( ord_le1860547276de_int @ A_170 @ B_134 ) ) ) ).
thf(fact_1353_mult__right__less__imp__less,axiom,
! [A_170: rat,C_80: rat,B_134: rat] :
( ( ord_less_rat @ ( times_times_rat @ A_170 @ C_80 ) @ ( times_times_rat @ B_134 @ C_80 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_80 )
=> ( ord_less_rat @ A_170 @ B_134 ) ) ) ).
thf(fact_1354_mult__le__less__imp__less,axiom,
! [C_79: int,D_30: int,A_169: int,B_133: int] :
( ( ord_less_eq_int @ A_169 @ B_133 )
=> ( ( ord_less_int @ C_79 @ D_30 )
=> ( ( ord_less_int @ zero_zero_int @ A_169 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_79 )
=> ( ord_less_int @ ( times_times_int @ A_169 @ C_79 ) @ ( times_times_int @ B_133 @ D_30 ) ) ) ) ) ) ).
thf(fact_1355_mult__le__less__imp__less,axiom,
! [C_79: nat,D_30: nat,A_169: nat,B_133: nat] :
( ( ord_less_eq_nat @ A_169 @ B_133 )
=> ( ( ord_less_nat @ C_79 @ D_30 )
=> ( ( ord_less_nat @ zero_zero_nat @ A_169 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_79 )
=> ( ord_less_nat @ ( times_times_nat @ A_169 @ C_79 ) @ ( times_times_nat @ B_133 @ D_30 ) ) ) ) ) ) ).
thf(fact_1356_mult__le__less__imp__less,axiom,
! [C_79: real,D_30: real,A_169: real,B_133: real] :
( ( ord_less_eq_real @ A_169 @ B_133 )
=> ( ( ord_less_real @ C_79 @ D_30 )
=> ( ( ord_less_real @ zero_zero_real @ A_169 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_79 )
=> ( ord_less_real @ ( times_times_real @ A_169 @ C_79 ) @ ( times_times_real @ B_133 @ D_30 ) ) ) ) ) ) ).
thf(fact_1357_mult__le__less__imp__less,axiom,
! [C_79: code_code_numeral,D_30: code_code_numeral,A_169: code_code_numeral,B_133: code_code_numeral] :
( ( ord_le565307924umeral @ A_169 @ B_133 )
=> ( ( ord_le1304079648umeral @ C_79 @ D_30 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_169 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_79 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_169 @ C_79 ) @ ( times_1655362735umeral @ B_133 @ D_30 ) ) ) ) ) ) ).
thf(fact_1358_mult__le__less__imp__less,axiom,
! [C_79: quickcheck_code_int,D_30: quickcheck_code_int,A_169: quickcheck_code_int,B_133: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_169 @ B_133 )
=> ( ( ord_le1860547276de_int @ C_79 @ D_30 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_169 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_79 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_169 @ C_79 ) @ ( times_123202395de_int @ B_133 @ D_30 ) ) ) ) ) ) ).
thf(fact_1359_mult__le__less__imp__less,axiom,
! [C_79: rat,D_30: rat,A_169: rat,B_133: rat] :
( ( ord_less_eq_rat @ A_169 @ B_133 )
=> ( ( ord_less_rat @ C_79 @ D_30 )
=> ( ( ord_less_rat @ zero_zero_rat @ A_169 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_79 )
=> ( ord_less_rat @ ( times_times_rat @ A_169 @ C_79 ) @ ( times_times_rat @ B_133 @ D_30 ) ) ) ) ) ) ).
thf(fact_1360_mult__less__le__imp__less,axiom,
! [C_78: int,D_29: int,A_168: int,B_132: int] :
( ( ord_less_int @ A_168 @ B_132 )
=> ( ( ord_less_eq_int @ C_78 @ D_29 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_168 )
=> ( ( ord_less_int @ zero_zero_int @ C_78 )
=> ( ord_less_int @ ( times_times_int @ A_168 @ C_78 ) @ ( times_times_int @ B_132 @ D_29 ) ) ) ) ) ) ).
thf(fact_1361_mult__less__le__imp__less,axiom,
! [C_78: nat,D_29: nat,A_168: nat,B_132: nat] :
( ( ord_less_nat @ A_168 @ B_132 )
=> ( ( ord_less_eq_nat @ C_78 @ D_29 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_168 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_78 )
=> ( ord_less_nat @ ( times_times_nat @ A_168 @ C_78 ) @ ( times_times_nat @ B_132 @ D_29 ) ) ) ) ) ) ).
thf(fact_1362_mult__less__le__imp__less,axiom,
! [C_78: real,D_29: real,A_168: real,B_132: real] :
( ( ord_less_real @ A_168 @ B_132 )
=> ( ( ord_less_eq_real @ C_78 @ D_29 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_168 )
=> ( ( ord_less_real @ zero_zero_real @ C_78 )
=> ( ord_less_real @ ( times_times_real @ A_168 @ C_78 ) @ ( times_times_real @ B_132 @ D_29 ) ) ) ) ) ) ).
thf(fact_1363_mult__less__le__imp__less,axiom,
! [C_78: code_code_numeral,D_29: code_code_numeral,A_168: code_code_numeral,B_132: code_code_numeral] :
( ( ord_le1304079648umeral @ A_168 @ B_132 )
=> ( ( ord_le565307924umeral @ C_78 @ D_29 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_168 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ C_78 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_168 @ C_78 ) @ ( times_1655362735umeral @ B_132 @ D_29 ) ) ) ) ) ) ).
thf(fact_1364_mult__less__le__imp__less,axiom,
! [C_78: quickcheck_code_int,D_29: quickcheck_code_int,A_168: quickcheck_code_int,B_132: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_168 @ B_132 )
=> ( ( ord_le258702272de_int @ C_78 @ D_29 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_168 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ C_78 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_168 @ C_78 ) @ ( times_123202395de_int @ B_132 @ D_29 ) ) ) ) ) ) ).
thf(fact_1365_mult__less__le__imp__less,axiom,
! [C_78: rat,D_29: rat,A_168: rat,B_132: rat] :
( ( ord_less_rat @ A_168 @ B_132 )
=> ( ( ord_less_eq_rat @ C_78 @ D_29 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A_168 )
=> ( ( ord_less_rat @ zero_zero_rat @ C_78 )
=> ( ord_less_rat @ ( times_times_rat @ A_168 @ C_78 ) @ ( times_times_rat @ B_132 @ D_29 ) ) ) ) ) ) ).
thf(fact_1366_mult__strict__mono_H,axiom,
! [C_77: int,D_28: int,A_167: int,B_131: int] :
( ( ord_less_int @ A_167 @ B_131 )
=> ( ( ord_less_int @ C_77 @ D_28 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_167 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_77 )
=> ( ord_less_int @ ( times_times_int @ A_167 @ C_77 ) @ ( times_times_int @ B_131 @ D_28 ) ) ) ) ) ) ).
thf(fact_1367_mult__strict__mono_H,axiom,
! [C_77: nat,D_28: nat,A_167: nat,B_131: nat] :
( ( ord_less_nat @ A_167 @ B_131 )
=> ( ( ord_less_nat @ C_77 @ D_28 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_167 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_77 )
=> ( ord_less_nat @ ( times_times_nat @ A_167 @ C_77 ) @ ( times_times_nat @ B_131 @ D_28 ) ) ) ) ) ) ).
thf(fact_1368_mult__strict__mono_H,axiom,
! [C_77: real,D_28: real,A_167: real,B_131: real] :
( ( ord_less_real @ A_167 @ B_131 )
=> ( ( ord_less_real @ C_77 @ D_28 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_167 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_77 )
=> ( ord_less_real @ ( times_times_real @ A_167 @ C_77 ) @ ( times_times_real @ B_131 @ D_28 ) ) ) ) ) ) ).
thf(fact_1369_mult__strict__mono_H,axiom,
! [C_77: code_code_numeral,D_28: code_code_numeral,A_167: code_code_numeral,B_131: code_code_numeral] :
( ( ord_le1304079648umeral @ A_167 @ B_131 )
=> ( ( ord_le1304079648umeral @ C_77 @ D_28 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_167 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_77 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_167 @ C_77 ) @ ( times_1655362735umeral @ B_131 @ D_28 ) ) ) ) ) ) ).
thf(fact_1370_mult__strict__mono_H,axiom,
! [C_77: quickcheck_code_int,D_28: quickcheck_code_int,A_167: quickcheck_code_int,B_131: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_167 @ B_131 )
=> ( ( ord_le1860547276de_int @ C_77 @ D_28 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_167 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_77 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_167 @ C_77 ) @ ( times_123202395de_int @ B_131 @ D_28 ) ) ) ) ) ) ).
thf(fact_1371_mult__strict__mono_H,axiom,
! [C_77: rat,D_28: rat,A_167: rat,B_131: rat] :
( ( ord_less_rat @ A_167 @ B_131 )
=> ( ( ord_less_rat @ C_77 @ D_28 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A_167 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_77 )
=> ( ord_less_rat @ ( times_times_rat @ A_167 @ C_77 ) @ ( times_times_rat @ B_131 @ D_28 ) ) ) ) ) ) ).
thf(fact_1372_mult__strict__mono,axiom,
! [C_76: int,D_27: int,A_166: int,B_130: int] :
( ( ord_less_int @ A_166 @ B_130 )
=> ( ( ord_less_int @ C_76 @ D_27 )
=> ( ( ord_less_int @ zero_zero_int @ B_130 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_76 )
=> ( ord_less_int @ ( times_times_int @ A_166 @ C_76 ) @ ( times_times_int @ B_130 @ D_27 ) ) ) ) ) ) ).
thf(fact_1373_mult__strict__mono,axiom,
! [C_76: nat,D_27: nat,A_166: nat,B_130: nat] :
( ( ord_less_nat @ A_166 @ B_130 )
=> ( ( ord_less_nat @ C_76 @ D_27 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_130 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_76 )
=> ( ord_less_nat @ ( times_times_nat @ A_166 @ C_76 ) @ ( times_times_nat @ B_130 @ D_27 ) ) ) ) ) ) ).
thf(fact_1374_mult__strict__mono,axiom,
! [C_76: real,D_27: real,A_166: real,B_130: real] :
( ( ord_less_real @ A_166 @ B_130 )
=> ( ( ord_less_real @ C_76 @ D_27 )
=> ( ( ord_less_real @ zero_zero_real @ B_130 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_76 )
=> ( ord_less_real @ ( times_times_real @ A_166 @ C_76 ) @ ( times_times_real @ B_130 @ D_27 ) ) ) ) ) ) ).
thf(fact_1375_mult__strict__mono,axiom,
! [C_76: code_code_numeral,D_27: code_code_numeral,A_166: code_code_numeral,B_130: code_code_numeral] :
( ( ord_le1304079648umeral @ A_166 @ B_130 )
=> ( ( ord_le1304079648umeral @ C_76 @ D_27 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ B_130 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_76 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_166 @ C_76 ) @ ( times_1655362735umeral @ B_130 @ D_27 ) ) ) ) ) ) ).
thf(fact_1376_mult__strict__mono,axiom,
! [C_76: quickcheck_code_int,D_27: quickcheck_code_int,A_166: quickcheck_code_int,B_130: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_166 @ B_130 )
=> ( ( ord_le1860547276de_int @ C_76 @ D_27 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ B_130 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_76 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_166 @ C_76 ) @ ( times_123202395de_int @ B_130 @ D_27 ) ) ) ) ) ) ).
thf(fact_1377_mult__strict__mono,axiom,
! [C_76: rat,D_27: rat,A_166: rat,B_130: rat] :
( ( ord_less_rat @ A_166 @ B_130 )
=> ( ( ord_less_rat @ C_76 @ D_27 )
=> ( ( ord_less_rat @ zero_zero_rat @ B_130 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_76 )
=> ( ord_less_rat @ ( times_times_rat @ A_166 @ C_76 ) @ ( times_times_rat @ B_130 @ D_27 ) ) ) ) ) ) ).
thf(fact_1378_mult__le__cancel__left__neg,axiom,
! [A_165: int,B_129: int,C_75: int] :
( ( ord_less_int @ C_75 @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C_75 @ A_165 ) @ ( times_times_int @ C_75 @ B_129 ) )
<=> ( ord_less_eq_int @ B_129 @ A_165 ) ) ) ).
thf(fact_1379_mult__le__cancel__left__neg,axiom,
! [A_165: real,B_129: real,C_75: real] :
( ( ord_less_real @ C_75 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C_75 @ A_165 ) @ ( times_times_real @ C_75 @ B_129 ) )
<=> ( ord_less_eq_real @ B_129 @ A_165 ) ) ) ).
thf(fact_1380_mult__le__cancel__left__neg,axiom,
! [A_165: rat,B_129: rat,C_75: rat] :
( ( ord_less_rat @ C_75 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C_75 @ A_165 ) @ ( times_times_rat @ C_75 @ B_129 ) )
<=> ( ord_less_eq_rat @ B_129 @ A_165 ) ) ) ).
thf(fact_1381_mult__le__cancel__left__pos,axiom,
! [A_164: int,B_128: int,C_74: int] :
( ( ord_less_int @ zero_zero_int @ C_74 )
=> ( ( ord_less_eq_int @ ( times_times_int @ C_74 @ A_164 ) @ ( times_times_int @ C_74 @ B_128 ) )
<=> ( ord_less_eq_int @ A_164 @ B_128 ) ) ) ).
thf(fact_1382_mult__le__cancel__left__pos,axiom,
! [A_164: real,B_128: real,C_74: real] :
( ( ord_less_real @ zero_zero_real @ C_74 )
=> ( ( ord_less_eq_real @ ( times_times_real @ C_74 @ A_164 ) @ ( times_times_real @ C_74 @ B_128 ) )
<=> ( ord_less_eq_real @ A_164 @ B_128 ) ) ) ).
thf(fact_1383_mult__le__cancel__left__pos,axiom,
! [A_164: rat,B_128: rat,C_74: rat] :
( ( ord_less_rat @ zero_zero_rat @ C_74 )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C_74 @ A_164 ) @ ( times_times_rat @ C_74 @ B_128 ) )
<=> ( ord_less_eq_rat @ A_164 @ B_128 ) ) ) ).
thf(fact_1384_mult__left__le__one__le,axiom,
! [Y_24: int,X_33: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_33 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_24 )
=> ( ( ord_less_eq_int @ Y_24 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y_24 @ X_33 ) @ X_33 ) ) ) ) ).
thf(fact_1385_mult__left__le__one__le,axiom,
! [Y_24: real,X_33: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_33 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_24 )
=> ( ( ord_less_eq_real @ Y_24 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y_24 @ X_33 ) @ X_33 ) ) ) ) ).
thf(fact_1386_mult__left__le__one__le,axiom,
! [Y_24: rat,X_33: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X_33 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y_24 )
=> ( ( ord_less_eq_rat @ Y_24 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Y_24 @ X_33 ) @ X_33 ) ) ) ) ).
thf(fact_1387_mult__right__le__one__le,axiom,
! [Y_23: int,X_32: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_32 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_23 )
=> ( ( ord_less_eq_int @ Y_23 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X_32 @ Y_23 ) @ X_32 ) ) ) ) ).
thf(fact_1388_mult__right__le__one__le,axiom,
! [Y_23: real,X_32: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_32 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_23 )
=> ( ( ord_less_eq_real @ Y_23 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X_32 @ Y_23 ) @ X_32 ) ) ) ) ).
thf(fact_1389_mult__right__le__one__le,axiom,
! [Y_23: rat,X_32: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X_32 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y_23 )
=> ( ( ord_less_eq_rat @ Y_23 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ X_32 @ Y_23 ) @ X_32 ) ) ) ) ).
thf(fact_1390_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
thf(fact_1391_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
thf(fact_1392_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
thf(fact_1393_zero__less__two,axiom,
ord_le1304079648umeral @ zero_z126310315umeral @ ( plus_p1627245867umeral @ one_on1645066479umeral @ one_on1645066479umeral ) ).
thf(fact_1394_zero__less__two,axiom,
ord_le1860547276de_int @ zero_z891286103de_int @ ( plus_p1446045655de_int @ one_on1684967323de_int @ one_on1684967323de_int ) ).
thf(fact_1395_zero__less__two,axiom,
ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
thf(fact_1396_power__less__imp__less__base,axiom,
! [A_163: int,N_50: nat,B_127: int] :
( ( ord_less_int @ ( power_power_int @ A_163 @ N_50 ) @ ( power_power_int @ B_127 @ N_50 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_127 )
=> ( ord_less_int @ A_163 @ B_127 ) ) ) ).
thf(fact_1397_power__less__imp__less__base,axiom,
! [A_163: nat,N_50: nat,B_127: nat] :
( ( ord_less_nat @ ( power_power_nat @ A_163 @ N_50 ) @ ( power_power_nat @ B_127 @ N_50 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_127 )
=> ( ord_less_nat @ A_163 @ B_127 ) ) ) ).
thf(fact_1398_power__less__imp__less__base,axiom,
! [A_163: real,N_50: nat,B_127: real] :
( ( ord_less_real @ ( power_power_real @ A_163 @ N_50 ) @ ( power_power_real @ B_127 @ N_50 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_127 )
=> ( ord_less_real @ A_163 @ B_127 ) ) ) ).
thf(fact_1399_power__less__imp__less__base,axiom,
! [A_163: code_code_numeral,N_50: nat,B_127: code_code_numeral] :
( ( ord_le1304079648umeral @ ( power_2100829034umeral @ A_163 @ N_50 ) @ ( power_2100829034umeral @ B_127 @ N_50 ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_127 )
=> ( ord_le1304079648umeral @ A_163 @ B_127 ) ) ) ).
thf(fact_1400_power__less__imp__less__base,axiom,
! [A_163: quickcheck_code_int,N_50: nat,B_127: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( power_881366806de_int @ A_163 @ N_50 ) @ ( power_881366806de_int @ B_127 @ N_50 ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_127 )
=> ( ord_le1860547276de_int @ A_163 @ B_127 ) ) ) ).
thf(fact_1401_power__less__imp__less__base,axiom,
! [A_163: rat,N_50: nat,B_127: rat] :
( ( ord_less_rat @ ( power_power_rat @ A_163 @ N_50 ) @ ( power_power_rat @ B_127 @ N_50 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_127 )
=> ( ord_less_rat @ A_163 @ B_127 ) ) ) ).
thf(fact_1402_power__decreasing,axiom,
! [A_162: int,N_49: nat,N_48: nat] :
( ( ord_less_eq_nat @ N_49 @ N_48 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_162 )
=> ( ( ord_less_eq_int @ A_162 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A_162 @ N_48 ) @ ( power_power_int @ A_162 @ N_49 ) ) ) ) ) ).
thf(fact_1403_power__decreasing,axiom,
! [A_162: nat,N_49: nat,N_48: nat] :
( ( ord_less_eq_nat @ N_49 @ N_48 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_162 )
=> ( ( ord_less_eq_nat @ A_162 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A_162 @ N_48 ) @ ( power_power_nat @ A_162 @ N_49 ) ) ) ) ) ).
thf(fact_1404_power__decreasing,axiom,
! [A_162: real,N_49: nat,N_48: nat] :
( ( ord_less_eq_nat @ N_49 @ N_48 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_162 )
=> ( ( ord_less_eq_real @ A_162 @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A_162 @ N_48 ) @ ( power_power_real @ A_162 @ N_49 ) ) ) ) ) ).
thf(fact_1405_power__decreasing,axiom,
! [A_162: code_code_numeral,N_49: nat,N_48: nat] :
( ( ord_less_eq_nat @ N_49 @ N_48 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_162 )
=> ( ( ord_le565307924umeral @ A_162 @ one_on1645066479umeral )
=> ( ord_le565307924umeral @ ( power_2100829034umeral @ A_162 @ N_48 ) @ ( power_2100829034umeral @ A_162 @ N_49 ) ) ) ) ) ).
thf(fact_1406_power__decreasing,axiom,
! [A_162: quickcheck_code_int,N_49: nat,N_48: nat] :
( ( ord_less_eq_nat @ N_49 @ N_48 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_162 )
=> ( ( ord_le258702272de_int @ A_162 @ one_on1684967323de_int )
=> ( ord_le258702272de_int @ ( power_881366806de_int @ A_162 @ N_48 ) @ ( power_881366806de_int @ A_162 @ N_49 ) ) ) ) ) ).
thf(fact_1407_power__decreasing,axiom,
! [A_162: rat,N_49: nat,N_48: nat] :
( ( ord_less_eq_nat @ N_49 @ N_48 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A_162 )
=> ( ( ord_less_eq_rat @ A_162 @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A_162 @ N_48 ) @ ( power_power_rat @ A_162 @ N_49 ) ) ) ) ) ).
thf(fact_1408_power__le__imp__le__exp,axiom,
! [M_24: nat,N_47: nat,A_161: int] :
( ( ord_less_int @ one_one_int @ A_161 )
=> ( ( ord_less_eq_int @ ( power_power_int @ A_161 @ M_24 ) @ ( power_power_int @ A_161 @ N_47 ) )
=> ( ord_less_eq_nat @ M_24 @ N_47 ) ) ) ).
thf(fact_1409_power__le__imp__le__exp,axiom,
! [M_24: nat,N_47: nat,A_161: nat] :
( ( ord_less_nat @ one_one_nat @ A_161 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A_161 @ M_24 ) @ ( power_power_nat @ A_161 @ N_47 ) )
=> ( ord_less_eq_nat @ M_24 @ N_47 ) ) ) ).
thf(fact_1410_power__le__imp__le__exp,axiom,
! [M_24: nat,N_47: nat,A_161: real] :
( ( ord_less_real @ one_one_real @ A_161 )
=> ( ( ord_less_eq_real @ ( power_power_real @ A_161 @ M_24 ) @ ( power_power_real @ A_161 @ N_47 ) )
=> ( ord_less_eq_nat @ M_24 @ N_47 ) ) ) ).
thf(fact_1411_power__le__imp__le__exp,axiom,
! [M_24: nat,N_47: nat,A_161: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ A_161 )
=> ( ( ord_le565307924umeral @ ( power_2100829034umeral @ A_161 @ M_24 ) @ ( power_2100829034umeral @ A_161 @ N_47 ) )
=> ( ord_less_eq_nat @ M_24 @ N_47 ) ) ) ).
thf(fact_1412_power__le__imp__le__exp,axiom,
! [M_24: nat,N_47: nat,A_161: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ A_161 )
=> ( ( ord_le258702272de_int @ ( power_881366806de_int @ A_161 @ M_24 ) @ ( power_881366806de_int @ A_161 @ N_47 ) )
=> ( ord_less_eq_nat @ M_24 @ N_47 ) ) ) ).
thf(fact_1413_power__le__imp__le__exp,axiom,
! [M_24: nat,N_47: nat,A_161: rat] :
( ( ord_less_rat @ one_one_rat @ A_161 )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ A_161 @ M_24 ) @ ( power_power_rat @ A_161 @ N_47 ) )
=> ( ord_less_eq_nat @ M_24 @ N_47 ) ) ) ).
thf(fact_1414_power__increasing__iff,axiom,
! [X_31: nat,Y_22: nat,B_126: int] :
( ( ord_less_int @ one_one_int @ B_126 )
=> ( ( ord_less_eq_int @ ( power_power_int @ B_126 @ X_31 ) @ ( power_power_int @ B_126 @ Y_22 ) )
<=> ( ord_less_eq_nat @ X_31 @ Y_22 ) ) ) ).
thf(fact_1415_power__increasing__iff,axiom,
! [X_31: nat,Y_22: nat,B_126: nat] :
( ( ord_less_nat @ one_one_nat @ B_126 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B_126 @ X_31 ) @ ( power_power_nat @ B_126 @ Y_22 ) )
<=> ( ord_less_eq_nat @ X_31 @ Y_22 ) ) ) ).
thf(fact_1416_power__increasing__iff,axiom,
! [X_31: nat,Y_22: nat,B_126: real] :
( ( ord_less_real @ one_one_real @ B_126 )
=> ( ( ord_less_eq_real @ ( power_power_real @ B_126 @ X_31 ) @ ( power_power_real @ B_126 @ Y_22 ) )
<=> ( ord_less_eq_nat @ X_31 @ Y_22 ) ) ) ).
thf(fact_1417_power__increasing__iff,axiom,
! [X_31: nat,Y_22: nat,B_126: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ B_126 )
=> ( ( ord_le565307924umeral @ ( power_2100829034umeral @ B_126 @ X_31 ) @ ( power_2100829034umeral @ B_126 @ Y_22 ) )
<=> ( ord_less_eq_nat @ X_31 @ Y_22 ) ) ) ).
thf(fact_1418_power__increasing__iff,axiom,
! [X_31: nat,Y_22: nat,B_126: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ B_126 )
=> ( ( ord_le258702272de_int @ ( power_881366806de_int @ B_126 @ X_31 ) @ ( power_881366806de_int @ B_126 @ Y_22 ) )
<=> ( ord_less_eq_nat @ X_31 @ Y_22 ) ) ) ).
thf(fact_1419_power__increasing__iff,axiom,
! [X_31: nat,Y_22: nat,B_126: rat] :
( ( ord_less_rat @ one_one_rat @ B_126 )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ B_126 @ X_31 ) @ ( power_power_rat @ B_126 @ Y_22 ) )
<=> ( ord_less_eq_nat @ X_31 @ Y_22 ) ) ) ).
thf(fact_1420_power__less__power__Suc,axiom,
! [N_46: nat,A_160: int] :
( ( ord_less_int @ one_one_int @ A_160 )
=> ( ord_less_int @ ( power_power_int @ A_160 @ N_46 ) @ ( times_times_int @ A_160 @ ( power_power_int @ A_160 @ N_46 ) ) ) ) ).
thf(fact_1421_power__less__power__Suc,axiom,
! [N_46: nat,A_160: nat] :
( ( ord_less_nat @ one_one_nat @ A_160 )
=> ( ord_less_nat @ ( power_power_nat @ A_160 @ N_46 ) @ ( times_times_nat @ A_160 @ ( power_power_nat @ A_160 @ N_46 ) ) ) ) ).
thf(fact_1422_power__less__power__Suc,axiom,
! [N_46: nat,A_160: real] :
( ( ord_less_real @ one_one_real @ A_160 )
=> ( ord_less_real @ ( power_power_real @ A_160 @ N_46 ) @ ( times_times_real @ A_160 @ ( power_power_real @ A_160 @ N_46 ) ) ) ) ).
thf(fact_1423_power__less__power__Suc,axiom,
! [N_46: nat,A_160: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ A_160 )
=> ( ord_le1304079648umeral @ ( power_2100829034umeral @ A_160 @ N_46 ) @ ( times_1655362735umeral @ A_160 @ ( power_2100829034umeral @ A_160 @ N_46 ) ) ) ) ).
thf(fact_1424_power__less__power__Suc,axiom,
! [N_46: nat,A_160: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ A_160 )
=> ( ord_le1860547276de_int @ ( power_881366806de_int @ A_160 @ N_46 ) @ ( times_123202395de_int @ A_160 @ ( power_881366806de_int @ A_160 @ N_46 ) ) ) ) ).
thf(fact_1425_power__less__power__Suc,axiom,
! [N_46: nat,A_160: rat] :
( ( ord_less_rat @ one_one_rat @ A_160 )
=> ( ord_less_rat @ ( power_power_rat @ A_160 @ N_46 ) @ ( times_times_rat @ A_160 @ ( power_power_rat @ A_160 @ N_46 ) ) ) ) ).
thf(fact_1426_power__gt1__lemma,axiom,
! [N_45: nat,A_159: int] :
( ( ord_less_int @ one_one_int @ A_159 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A_159 @ ( power_power_int @ A_159 @ N_45 ) ) ) ) ).
thf(fact_1427_power__gt1__lemma,axiom,
! [N_45: nat,A_159: nat] :
( ( ord_less_nat @ one_one_nat @ A_159 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A_159 @ ( power_power_nat @ A_159 @ N_45 ) ) ) ) ).
thf(fact_1428_power__gt1__lemma,axiom,
! [N_45: nat,A_159: real] :
( ( ord_less_real @ one_one_real @ A_159 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A_159 @ ( power_power_real @ A_159 @ N_45 ) ) ) ) ).
thf(fact_1429_power__gt1__lemma,axiom,
! [N_45: nat,A_159: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ A_159 )
=> ( ord_le1304079648umeral @ one_on1645066479umeral @ ( times_1655362735umeral @ A_159 @ ( power_2100829034umeral @ A_159 @ N_45 ) ) ) ) ).
thf(fact_1430_power__gt1__lemma,axiom,
! [N_45: nat,A_159: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ A_159 )
=> ( ord_le1860547276de_int @ one_on1684967323de_int @ ( times_123202395de_int @ A_159 @ ( power_881366806de_int @ A_159 @ N_45 ) ) ) ) ).
thf(fact_1431_power__gt1__lemma,axiom,
! [N_45: nat,A_159: rat] :
( ( ord_less_rat @ one_one_rat @ A_159 )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A_159 @ ( power_power_rat @ A_159 @ N_45 ) ) ) ) ).
thf(fact_1432_power__0__left,axiom,
! [N_44: nat] :
( ( ( N_44 = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N_44 )
= one_one_int ) )
& ( ( N_44 != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N_44 )
= zero_zero_int ) ) ) ).
thf(fact_1433_power__0__left,axiom,
! [N_44: nat] :
( ( ( N_44 = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N_44 )
= one_one_nat ) )
& ( ( N_44 != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N_44 )
= zero_zero_nat ) ) ) ).
thf(fact_1434_power__0__left,axiom,
! [N_44: nat] :
( ( ( N_44 = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N_44 )
= one_one_real ) )
& ( ( N_44 != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N_44 )
= zero_zero_real ) ) ) ).
thf(fact_1435_power__0__left,axiom,
! [N_44: nat] :
( ( ( N_44 = zero_zero_nat )
=> ( ( power_2100829034umeral @ zero_z126310315umeral @ N_44 )
= one_on1645066479umeral ) )
& ( ( N_44 != zero_zero_nat )
=> ( ( power_2100829034umeral @ zero_z126310315umeral @ N_44 )
= zero_z126310315umeral ) ) ) ).
thf(fact_1436_power__0__left,axiom,
! [N_44: nat] :
( ( ( N_44 = zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N_44 )
= one_one_complex ) )
& ( ( N_44 != zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N_44 )
= zero_zero_complex ) ) ) ).
thf(fact_1437_power__0__left,axiom,
! [N_44: nat] :
( ( ( N_44 = zero_zero_nat )
=> ( ( power_881366806de_int @ zero_z891286103de_int @ N_44 )
= one_on1684967323de_int ) )
& ( ( N_44 != zero_zero_nat )
=> ( ( power_881366806de_int @ zero_z891286103de_int @ N_44 )
= zero_z891286103de_int ) ) ) ).
thf(fact_1438_power__0__left,axiom,
! [N_44: nat] :
( ( ( N_44 = zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N_44 )
= one_one_rat ) )
& ( ( N_44 != zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N_44 )
= zero_zero_rat ) ) ) ).
thf(fact_1439_power__strict__increasing,axiom,
! [A_158: int,N_43: nat,N_42: nat] :
( ( ord_less_nat @ N_43 @ N_42 )
=> ( ( ord_less_int @ one_one_int @ A_158 )
=> ( ord_less_int @ ( power_power_int @ A_158 @ N_43 ) @ ( power_power_int @ A_158 @ N_42 ) ) ) ) ).
thf(fact_1440_power__strict__increasing,axiom,
! [A_158: nat,N_43: nat,N_42: nat] :
( ( ord_less_nat @ N_43 @ N_42 )
=> ( ( ord_less_nat @ one_one_nat @ A_158 )
=> ( ord_less_nat @ ( power_power_nat @ A_158 @ N_43 ) @ ( power_power_nat @ A_158 @ N_42 ) ) ) ) ).
thf(fact_1441_power__strict__increasing,axiom,
! [A_158: real,N_43: nat,N_42: nat] :
( ( ord_less_nat @ N_43 @ N_42 )
=> ( ( ord_less_real @ one_one_real @ A_158 )
=> ( ord_less_real @ ( power_power_real @ A_158 @ N_43 ) @ ( power_power_real @ A_158 @ N_42 ) ) ) ) ).
thf(fact_1442_power__strict__increasing,axiom,
! [A_158: code_code_numeral,N_43: nat,N_42: nat] :
( ( ord_less_nat @ N_43 @ N_42 )
=> ( ( ord_le1304079648umeral @ one_on1645066479umeral @ A_158 )
=> ( ord_le1304079648umeral @ ( power_2100829034umeral @ A_158 @ N_43 ) @ ( power_2100829034umeral @ A_158 @ N_42 ) ) ) ) ).
thf(fact_1443_power__strict__increasing,axiom,
! [A_158: quickcheck_code_int,N_43: nat,N_42: nat] :
( ( ord_less_nat @ N_43 @ N_42 )
=> ( ( ord_le1860547276de_int @ one_on1684967323de_int @ A_158 )
=> ( ord_le1860547276de_int @ ( power_881366806de_int @ A_158 @ N_43 ) @ ( power_881366806de_int @ A_158 @ N_42 ) ) ) ) ).
thf(fact_1444_power__strict__increasing,axiom,
! [A_158: rat,N_43: nat,N_42: nat] :
( ( ord_less_nat @ N_43 @ N_42 )
=> ( ( ord_less_rat @ one_one_rat @ A_158 )
=> ( ord_less_rat @ ( power_power_rat @ A_158 @ N_43 ) @ ( power_power_rat @ A_158 @ N_42 ) ) ) ) ).
thf(fact_1445_power__less__imp__less__exp,axiom,
! [M_23: nat,N_41: nat,A_157: int] :
( ( ord_less_int @ one_one_int @ A_157 )
=> ( ( ord_less_int @ ( power_power_int @ A_157 @ M_23 ) @ ( power_power_int @ A_157 @ N_41 ) )
=> ( ord_less_nat @ M_23 @ N_41 ) ) ) ).
thf(fact_1446_power__less__imp__less__exp,axiom,
! [M_23: nat,N_41: nat,A_157: nat] :
( ( ord_less_nat @ one_one_nat @ A_157 )
=> ( ( ord_less_nat @ ( power_power_nat @ A_157 @ M_23 ) @ ( power_power_nat @ A_157 @ N_41 ) )
=> ( ord_less_nat @ M_23 @ N_41 ) ) ) ).
thf(fact_1447_power__less__imp__less__exp,axiom,
! [M_23: nat,N_41: nat,A_157: real] :
( ( ord_less_real @ one_one_real @ A_157 )
=> ( ( ord_less_real @ ( power_power_real @ A_157 @ M_23 ) @ ( power_power_real @ A_157 @ N_41 ) )
=> ( ord_less_nat @ M_23 @ N_41 ) ) ) ).
thf(fact_1448_power__less__imp__less__exp,axiom,
! [M_23: nat,N_41: nat,A_157: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ A_157 )
=> ( ( ord_le1304079648umeral @ ( power_2100829034umeral @ A_157 @ M_23 ) @ ( power_2100829034umeral @ A_157 @ N_41 ) )
=> ( ord_less_nat @ M_23 @ N_41 ) ) ) ).
thf(fact_1449_power__less__imp__less__exp,axiom,
! [M_23: nat,N_41: nat,A_157: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ A_157 )
=> ( ( ord_le1860547276de_int @ ( power_881366806de_int @ A_157 @ M_23 ) @ ( power_881366806de_int @ A_157 @ N_41 ) )
=> ( ord_less_nat @ M_23 @ N_41 ) ) ) ).
thf(fact_1450_power__less__imp__less__exp,axiom,
! [M_23: nat,N_41: nat,A_157: rat] :
( ( ord_less_rat @ one_one_rat @ A_157 )
=> ( ( ord_less_rat @ ( power_power_rat @ A_157 @ M_23 ) @ ( power_power_rat @ A_157 @ N_41 ) )
=> ( ord_less_nat @ M_23 @ N_41 ) ) ) ).
thf(fact_1451_power__strict__increasing__iff,axiom,
! [X_30: nat,Y_21: nat,B_125: int] :
( ( ord_less_int @ one_one_int @ B_125 )
=> ( ( ord_less_int @ ( power_power_int @ B_125 @ X_30 ) @ ( power_power_int @ B_125 @ Y_21 ) )
<=> ( ord_less_nat @ X_30 @ Y_21 ) ) ) ).
thf(fact_1452_power__strict__increasing__iff,axiom,
! [X_30: nat,Y_21: nat,B_125: nat] :
( ( ord_less_nat @ one_one_nat @ B_125 )
=> ( ( ord_less_nat @ ( power_power_nat @ B_125 @ X_30 ) @ ( power_power_nat @ B_125 @ Y_21 ) )
<=> ( ord_less_nat @ X_30 @ Y_21 ) ) ) ).
thf(fact_1453_power__strict__increasing__iff,axiom,
! [X_30: nat,Y_21: nat,B_125: real] :
( ( ord_less_real @ one_one_real @ B_125 )
=> ( ( ord_less_real @ ( power_power_real @ B_125 @ X_30 ) @ ( power_power_real @ B_125 @ Y_21 ) )
<=> ( ord_less_nat @ X_30 @ Y_21 ) ) ) ).
thf(fact_1454_power__strict__increasing__iff,axiom,
! [X_30: nat,Y_21: nat,B_125: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ B_125 )
=> ( ( ord_le1304079648umeral @ ( power_2100829034umeral @ B_125 @ X_30 ) @ ( power_2100829034umeral @ B_125 @ Y_21 ) )
<=> ( ord_less_nat @ X_30 @ Y_21 ) ) ) ).
thf(fact_1455_power__strict__increasing__iff,axiom,
! [X_30: nat,Y_21: nat,B_125: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ B_125 )
=> ( ( ord_le1860547276de_int @ ( power_881366806de_int @ B_125 @ X_30 ) @ ( power_881366806de_int @ B_125 @ Y_21 ) )
<=> ( ord_less_nat @ X_30 @ Y_21 ) ) ) ).
thf(fact_1456_power__strict__increasing__iff,axiom,
! [X_30: nat,Y_21: nat,B_125: rat] :
( ( ord_less_rat @ one_one_rat @ B_125 )
=> ( ( ord_less_rat @ ( power_power_rat @ B_125 @ X_30 ) @ ( power_power_rat @ B_125 @ Y_21 ) )
<=> ( ord_less_nat @ X_30 @ Y_21 ) ) ) ).
thf(fact_1457_four__x__squared,axiom,
! [X: real] :
( ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( power_power_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_1458_Int2_Ozcong__zero,axiom,
! [M: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ M )
=> ( ( zcong @ X @ zero_zero_int @ M )
=> ( X = zero_zero_int ) ) ) ) ).
thf(fact_1459_zcong__zless__0,axiom,
! [M: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ M )
=> ( ( zcong @ A @ zero_zero_int @ M )
=> ( A = zero_zero_int ) ) ) ) ).
thf(fact_1460_zcong__zless__imp__eq,axiom,
! [B: int,M: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ M )
=> ( ( zcong @ A @ B @ M )
=> ( A = B ) ) ) ) ) ) ).
thf(fact_1461_zpower__zdvd__prop1,axiom,
! [P_3: int,Y: int,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ P_3 @ Y )
=> ( dvd_dvd_int @ P_3 @ ( power_power_int @ Y @ N ) ) ) ) ).
thf(fact_1462_zcong__zpower__zmult,axiom,
! [Z_1: nat,X: int,Y: nat,P_3: int] :
( ( zcong @ ( power_power_int @ X @ Y ) @ one_one_int @ P_3 )
=> ( zcong @ ( power_power_int @ X @ ( times_times_nat @ Y @ Z_1 ) ) @ one_one_int @ P_3 ) ) ).
thf(fact_1463_zcong__zmult__prop3,axiom,
! [Y: int,X: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ~ ( zcong @ X @ zero_zero_int @ P_3 )
=> ( ~ ( zcong @ Y @ zero_zero_int @ P_3 )
=> ~ ( zcong @ ( times_times_int @ X @ Y ) @ zero_zero_int @ P_3 ) ) ) ) ).
thf(fact_1464_convex__bound__le,axiom,
! [V_7: int,U_2: int,Y_20: int,X_29: int,A_156: int] :
( ( ord_less_eq_int @ X_29 @ A_156 )
=> ( ( ord_less_eq_int @ Y_20 @ A_156 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U_2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ V_7 )
=> ( ( ( plus_plus_int @ U_2 @ V_7 )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U_2 @ X_29 ) @ ( times_times_int @ V_7 @ Y_20 ) ) @ A_156 ) ) ) ) ) ) ).
thf(fact_1465_convex__bound__le,axiom,
! [V_7: real,U_2: real,Y_20: real,X_29: real,A_156: real] :
( ( ord_less_eq_real @ X_29 @ A_156 )
=> ( ( ord_less_eq_real @ Y_20 @ A_156 )
=> ( ( ord_less_eq_real @ zero_zero_real @ U_2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ V_7 )
=> ( ( ( plus_plus_real @ U_2 @ V_7 )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U_2 @ X_29 ) @ ( times_times_real @ V_7 @ Y_20 ) ) @ A_156 ) ) ) ) ) ) ).
thf(fact_1466_convex__bound__le,axiom,
! [V_7: rat,U_2: rat,Y_20: rat,X_29: rat,A_156: rat] :
( ( ord_less_eq_rat @ X_29 @ A_156 )
=> ( ( ord_less_eq_rat @ Y_20 @ A_156 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ U_2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ V_7 )
=> ( ( ( plus_plus_rat @ U_2 @ V_7 )
= one_one_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U_2 @ X_29 ) @ ( times_times_rat @ V_7 @ Y_20 ) ) @ A_156 ) ) ) ) ) ) ).
thf(fact_1467_power__Suc__less,axiom,
! [N_40: nat,A_155: int] :
( ( ord_less_int @ zero_zero_int @ A_155 )
=> ( ( ord_less_int @ A_155 @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A_155 @ ( power_power_int @ A_155 @ N_40 ) ) @ ( power_power_int @ A_155 @ N_40 ) ) ) ) ).
thf(fact_1468_power__Suc__less,axiom,
! [N_40: nat,A_155: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_155 )
=> ( ( ord_less_nat @ A_155 @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A_155 @ ( power_power_nat @ A_155 @ N_40 ) ) @ ( power_power_nat @ A_155 @ N_40 ) ) ) ) ).
thf(fact_1469_power__Suc__less,axiom,
! [N_40: nat,A_155: real] :
( ( ord_less_real @ zero_zero_real @ A_155 )
=> ( ( ord_less_real @ A_155 @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A_155 @ ( power_power_real @ A_155 @ N_40 ) ) @ ( power_power_real @ A_155 @ N_40 ) ) ) ) ).
thf(fact_1470_power__Suc__less,axiom,
! [N_40: nat,A_155: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_155 )
=> ( ( ord_le1304079648umeral @ A_155 @ one_on1645066479umeral )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_155 @ ( power_2100829034umeral @ A_155 @ N_40 ) ) @ ( power_2100829034umeral @ A_155 @ N_40 ) ) ) ) ).
thf(fact_1471_power__Suc__less,axiom,
! [N_40: nat,A_155: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_155 )
=> ( ( ord_le1860547276de_int @ A_155 @ one_on1684967323de_int )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_155 @ ( power_881366806de_int @ A_155 @ N_40 ) ) @ ( power_881366806de_int @ A_155 @ N_40 ) ) ) ) ).
thf(fact_1472_power__Suc__less,axiom,
! [N_40: nat,A_155: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_155 )
=> ( ( ord_less_rat @ A_155 @ one_one_rat )
=> ( ord_less_rat @ ( times_times_rat @ A_155 @ ( power_power_rat @ A_155 @ N_40 ) ) @ ( power_power_rat @ A_155 @ N_40 ) ) ) ) ).
thf(fact_1473_power__strict__decreasing,axiom,
! [A_154: int,N_39: nat,N_38: nat] :
( ( ord_less_nat @ N_39 @ N_38 )
=> ( ( ord_less_int @ zero_zero_int @ A_154 )
=> ( ( ord_less_int @ A_154 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A_154 @ N_38 ) @ ( power_power_int @ A_154 @ N_39 ) ) ) ) ) ).
thf(fact_1474_power__strict__decreasing,axiom,
! [A_154: nat,N_39: nat,N_38: nat] :
( ( ord_less_nat @ N_39 @ N_38 )
=> ( ( ord_less_nat @ zero_zero_nat @ A_154 )
=> ( ( ord_less_nat @ A_154 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A_154 @ N_38 ) @ ( power_power_nat @ A_154 @ N_39 ) ) ) ) ) ).
thf(fact_1475_power__strict__decreasing,axiom,
! [A_154: real,N_39: nat,N_38: nat] :
( ( ord_less_nat @ N_39 @ N_38 )
=> ( ( ord_less_real @ zero_zero_real @ A_154 )
=> ( ( ord_less_real @ A_154 @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A_154 @ N_38 ) @ ( power_power_real @ A_154 @ N_39 ) ) ) ) ) ).
thf(fact_1476_power__strict__decreasing,axiom,
! [A_154: code_code_numeral,N_39: nat,N_38: nat] :
( ( ord_less_nat @ N_39 @ N_38 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_154 )
=> ( ( ord_le1304079648umeral @ A_154 @ one_on1645066479umeral )
=> ( ord_le1304079648umeral @ ( power_2100829034umeral @ A_154 @ N_38 ) @ ( power_2100829034umeral @ A_154 @ N_39 ) ) ) ) ) ).
thf(fact_1477_power__strict__decreasing,axiom,
! [A_154: quickcheck_code_int,N_39: nat,N_38: nat] :
( ( ord_less_nat @ N_39 @ N_38 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_154 )
=> ( ( ord_le1860547276de_int @ A_154 @ one_on1684967323de_int )
=> ( ord_le1860547276de_int @ ( power_881366806de_int @ A_154 @ N_38 ) @ ( power_881366806de_int @ A_154 @ N_39 ) ) ) ) ) ).
thf(fact_1478_power__strict__decreasing,axiom,
! [A_154: rat,N_39: nat,N_38: nat] :
( ( ord_less_nat @ N_39 @ N_38 )
=> ( ( ord_less_rat @ zero_zero_rat @ A_154 )
=> ( ( ord_less_rat @ A_154 @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A_154 @ N_38 ) @ ( power_power_rat @ A_154 @ N_39 ) ) ) ) ) ).
thf(fact_1479_power__eq__imp__eq__base,axiom,
! [A_153: int,N_37: nat,B_124: int] :
( ( ( power_power_int @ A_153 @ N_37 )
= ( power_power_int @ B_124 @ N_37 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_153 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_124 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_37 )
=> ( A_153 = B_124 ) ) ) ) ) ).
thf(fact_1480_power__eq__imp__eq__base,axiom,
! [A_153: nat,N_37: nat,B_124: nat] :
( ( ( power_power_nat @ A_153 @ N_37 )
= ( power_power_nat @ B_124 @ N_37 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_153 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_124 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_37 )
=> ( A_153 = B_124 ) ) ) ) ) ).
thf(fact_1481_power__eq__imp__eq__base,axiom,
! [A_153: real,N_37: nat,B_124: real] :
( ( ( power_power_real @ A_153 @ N_37 )
= ( power_power_real @ B_124 @ N_37 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_153 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_124 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_37 )
=> ( A_153 = B_124 ) ) ) ) ) ).
thf(fact_1482_power__eq__imp__eq__base,axiom,
! [A_153: code_code_numeral,N_37: nat,B_124: code_code_numeral] :
( ( ( power_2100829034umeral @ A_153 @ N_37 )
= ( power_2100829034umeral @ B_124 @ N_37 ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_153 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_124 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_37 )
=> ( A_153 = B_124 ) ) ) ) ) ).
thf(fact_1483_power__eq__imp__eq__base,axiom,
! [A_153: quickcheck_code_int,N_37: nat,B_124: quickcheck_code_int] :
( ( ( power_881366806de_int @ A_153 @ N_37 )
= ( power_881366806de_int @ B_124 @ N_37 ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_153 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_124 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_37 )
=> ( A_153 = B_124 ) ) ) ) ) ).
thf(fact_1484_power__eq__imp__eq__base,axiom,
! [A_153: rat,N_37: nat,B_124: rat] :
( ( ( power_power_rat @ A_153 @ N_37 )
= ( power_power_rat @ B_124 @ N_37 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A_153 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_124 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_37 )
=> ( A_153 = B_124 ) ) ) ) ) ).
thf(fact_1485_one__less__power,axiom,
! [N_36: nat,A_152: int] :
( ( ord_less_int @ one_one_int @ A_152 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_36 )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A_152 @ N_36 ) ) ) ) ).
thf(fact_1486_one__less__power,axiom,
! [N_36: nat,A_152: nat] :
( ( ord_less_nat @ one_one_nat @ A_152 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_36 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A_152 @ N_36 ) ) ) ) ).
thf(fact_1487_one__less__power,axiom,
! [N_36: nat,A_152: real] :
( ( ord_less_real @ one_one_real @ A_152 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_36 )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A_152 @ N_36 ) ) ) ) ).
thf(fact_1488_one__less__power,axiom,
! [N_36: nat,A_152: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ A_152 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_36 )
=> ( ord_le1304079648umeral @ one_on1645066479umeral @ ( power_2100829034umeral @ A_152 @ N_36 ) ) ) ) ).
thf(fact_1489_one__less__power,axiom,
! [N_36: nat,A_152: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ A_152 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_36 )
=> ( ord_le1860547276de_int @ one_on1684967323de_int @ ( power_881366806de_int @ A_152 @ N_36 ) ) ) ) ).
thf(fact_1490_one__less__power,axiom,
! [N_36: nat,A_152: rat] :
( ( ord_less_rat @ one_one_rat @ A_152 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_36 )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A_152 @ N_36 ) ) ) ) ).
thf(fact_1491_dvd__power,axiom,
! [X_28: real,N_35: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N_35 )
| ( X_28 = one_one_real ) )
=> ( dvd_dvd_real @ X_28 @ ( power_power_real @ X_28 @ N_35 ) ) ) ).
thf(fact_1492_dvd__power,axiom,
! [X_28: code_code_numeral,N_35: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N_35 )
| ( X_28 = one_on1645066479umeral ) )
=> ( dvd_dv174992974umeral @ X_28 @ ( power_2100829034umeral @ X_28 @ N_35 ) ) ) ).
thf(fact_1493_dvd__power,axiom,
! [X_28: complex,N_35: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N_35 )
| ( X_28 = one_one_complex ) )
=> ( dvd_dvd_complex @ X_28 @ ( power_power_complex @ X_28 @ N_35 ) ) ) ).
thf(fact_1494_dvd__power,axiom,
! [X_28: quickcheck_code_int,N_35: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N_35 )
| ( X_28 = one_on1684967323de_int ) )
=> ( dvd_dv1760642554de_int @ X_28 @ ( power_881366806de_int @ X_28 @ N_35 ) ) ) ).
thf(fact_1495_dvd__power,axiom,
! [X_28: rat,N_35: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N_35 )
| ( X_28 = one_one_rat ) )
=> ( dvd_dvd_rat @ X_28 @ ( power_power_rat @ X_28 @ N_35 ) ) ) ).
thf(fact_1496_dvd__power,axiom,
! [X_28: int,N_35: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N_35 )
| ( X_28 = one_one_int ) )
=> ( dvd_dvd_int @ X_28 @ ( power_power_int @ X_28 @ N_35 ) ) ) ).
thf(fact_1497_dvd__power,axiom,
! [X_28: nat,N_35: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N_35 )
| ( X_28 = one_one_nat ) )
=> ( dvd_dvd_nat @ X_28 @ ( power_power_nat @ X_28 @ N_35 ) ) ) ).
thf(fact_1498__096QuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_096,axiom,
quadRes @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( number_number_of_int @ min ) ).
thf(fact_1499__0964_A_K_Am_A_L_A1_Advd_As_A_094_A2_A_N_A_N1_096,axiom,
dvd_dvd_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( minus_minus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of_int @ min ) ) ).
thf(fact_1500_neg__one__power__eq__mod__m,axiom,
! [J: nat,K_1: nat,M: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ M )
=> ( ( zcong @ ( power_power_int @ ( number_number_of_int @ min ) @ J ) @ ( power_power_int @ ( number_number_of_int @ min ) @ K_1 ) @ M )
=> ( ( power_power_int @ ( number_number_of_int @ min ) @ J )
= ( power_power_int @ ( number_number_of_int @ min ) @ K_1 ) ) ) ) ).
thf(fact_1501_one__not__neg__one__mod__m,axiom,
! [M: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ M )
=> ~ ( zcong @ one_one_int @ ( number_number_of_int @ min ) @ M ) ) ).
thf(fact_1502__096s_A_094_A2_A_N_A_N1_A_061_As_A_094_A2_A_L_A1_096,axiom,
( ( minus_minus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number_number_of_int @ min ) )
= ( plus_plus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_int ) ) ).
thf(fact_1503__096_126_AQuadRes_A_I4_A_K_Am_A_L_A1_J_A_N1_A_061_061_062_ALegendre_A_,axiom,
( ~ ( quadRes @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( number_number_of_int @ min ) )
=> ( ( legendre @ ( number_number_of_int @ min ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
!= one_one_int ) ) ).
thf(fact_1504_Legendre__1mod4,axiom,
! [M: int] :
( ( zprime @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ M ) @ one_one_int ) )
=> ( ( legendre @ ( number_number_of_int @ min ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ M ) @ one_one_int ) )
= one_one_int ) ) ).
thf(fact_1505_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_1506_number__of1,axiom,
! [N: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ ( bit0 @ N ) ) )
& ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ ( bit1 @ N ) ) ) ) ) ).
thf(fact_1507_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_1508_zprime__factor__exists,axiom,
! [A: int] :
( ( ord_less_int @ one_one_int @ A )
=> ? [P_4: int] :
( ( zprime @ P_4 )
& ( dvd_dvd_int @ P_4 @ A ) ) ) ).
thf(fact_1509_not__real__square__gt__zero,axiom,
! [X: real] :
( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) )
<=> ( X = zero_zero_real ) ) ).
thf(fact_1510_number__of__diff,axiom,
! [V_6: int,W_3: int] :
( ( number_number_of_int @ ( minus_minus_int @ V_6 @ W_3 ) )
= ( minus_minus_int @ ( number_number_of_int @ V_6 ) @ ( number_number_of_int @ W_3 ) ) ) ).
thf(fact_1511_number__of__diff,axiom,
! [V_6: int,W_3: int] :
( ( number267125858f_real @ ( minus_minus_int @ V_6 @ W_3 ) )
= ( minus_minus_real @ ( number267125858f_real @ V_6 ) @ ( number267125858f_real @ W_3 ) ) ) ).
thf(fact_1512_number__of__diff,axiom,
! [V_6: int,W_3: int] :
( ( number528085621omplex @ ( minus_minus_int @ V_6 @ W_3 ) )
= ( minus_minus_complex @ ( number528085621omplex @ V_6 ) @ ( number528085621omplex @ W_3 ) ) ) ).
thf(fact_1513_number__of__diff,axiom,
! [V_6: int,W_3: int] :
( ( number_number_of_rat @ ( minus_minus_int @ V_6 @ W_3 ) )
= ( minus_minus_rat @ ( number_number_of_rat @ V_6 ) @ ( number_number_of_rat @ W_3 ) ) ) ).
thf(fact_1514_dvd__reduce,axiom,
! [K_1: nat,N: nat] :
( ( dvd_dvd_nat @ K_1 @ ( plus_plus_nat @ N @ K_1 ) )
<=> ( dvd_dvd_nat @ K_1 @ N ) ) ).
thf(fact_1515_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
<=> ( M = one_one_nat ) ) ).
thf(fact_1516_dvd__diff,axiom,
! [Z_7: real,X_27: real,Y_19: real] :
( ( dvd_dvd_real @ X_27 @ Y_19 )
=> ( ( dvd_dvd_real @ X_27 @ Z_7 )
=> ( dvd_dvd_real @ X_27 @ ( minus_minus_real @ Y_19 @ Z_7 ) ) ) ) ).
thf(fact_1517_dvd__diff,axiom,
! [Z_7: complex,X_27: complex,Y_19: complex] :
( ( dvd_dvd_complex @ X_27 @ Y_19 )
=> ( ( dvd_dvd_complex @ X_27 @ Z_7 )
=> ( dvd_dvd_complex @ X_27 @ ( minus_minus_complex @ Y_19 @ Z_7 ) ) ) ) ).
thf(fact_1518_dvd__diff,axiom,
! [Z_7: rat,X_27: rat,Y_19: rat] :
( ( dvd_dvd_rat @ X_27 @ Y_19 )
=> ( ( dvd_dvd_rat @ X_27 @ Z_7 )
=> ( dvd_dvd_rat @ X_27 @ ( minus_minus_rat @ Y_19 @ Z_7 ) ) ) ) ).
thf(fact_1519_dvd__diff,axiom,
! [Z_7: int,X_27: int,Y_19: int] :
( ( dvd_dvd_int @ X_27 @ Y_19 )
=> ( ( dvd_dvd_int @ X_27 @ Z_7 )
=> ( dvd_dvd_int @ X_27 @ ( minus_minus_int @ Y_19 @ Z_7 ) ) ) ) ).
thf(fact_1520_diff__bin__simps_I1_J,axiom,
! [K_1: int] :
( ( minus_minus_int @ K_1 @ pls )
= K_1 ) ).
thf(fact_1521_diff__bin__simps_I7_J,axiom,
! [K_1: int,L: int] :
( ( minus_minus_int @ ( bit0 @ K_1 ) @ ( bit0 @ L ) )
= ( bit0 @ ( minus_minus_int @ K_1 @ L ) ) ) ).
thf(fact_1522_zdiff__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z2 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
thf(fact_1523_zdiff__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z2 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
thf(fact_1524_Int2_Oaux1,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
=> ( A
= ( plus_plus_int @ C @ B ) ) ) ).
thf(fact_1525_zcong__zdiff,axiom,
! [C: int,D: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ C @ D @ M )
=> ( zcong @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) @ M ) ) ) ).
thf(fact_1526_zdvd__zdiffD,axiom,
! [K_1: int,M: int,N: int] :
( ( dvd_dvd_int @ K_1 @ ( minus_minus_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K_1 @ N )
=> ( dvd_dvd_int @ K_1 @ M ) ) ) ).
thf(fact_1527_nat__dvd__not__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
thf(fact_1528_inf__period_I3_J,axiom,
! [T_4: real,D_26: real,D_25: real] :
( ( dvd_dvd_real @ D_26 @ D_25 )
=> ! [X_1: real,K: real] :
( ( dvd_dvd_real @ D_26 @ ( plus_plus_real @ X_1 @ T_4 ) )
<=> ( dvd_dvd_real @ D_26 @ ( plus_plus_real @ ( minus_minus_real @ X_1 @ ( times_times_real @ K @ D_25 ) ) @ T_4 ) ) ) ) ).
thf(fact_1529_inf__period_I3_J,axiom,
! [T_4: complex,D_26: complex,D_25: complex] :
( ( dvd_dvd_complex @ D_26 @ D_25 )
=> ! [X_1: complex,K: complex] :
( ( dvd_dvd_complex @ D_26 @ ( plus_plus_complex @ X_1 @ T_4 ) )
<=> ( dvd_dvd_complex @ D_26 @ ( plus_plus_complex @ ( minus_minus_complex @ X_1 @ ( times_times_complex @ K @ D_25 ) ) @ T_4 ) ) ) ) ).
thf(fact_1530_inf__period_I3_J,axiom,
! [T_4: rat,D_26: rat,D_25: rat] :
( ( dvd_dvd_rat @ D_26 @ D_25 )
=> ! [X_1: rat,K: rat] :
( ( dvd_dvd_rat @ D_26 @ ( plus_plus_rat @ X_1 @ T_4 ) )
<=> ( dvd_dvd_rat @ D_26 @ ( plus_plus_rat @ ( minus_minus_rat @ X_1 @ ( times_times_rat @ K @ D_25 ) ) @ T_4 ) ) ) ) ).
thf(fact_1531_inf__period_I3_J,axiom,
! [T_4: int,D_26: int,D_25: int] :
( ( dvd_dvd_int @ D_26 @ D_25 )
=> ! [X_1: int,K: int] :
( ( dvd_dvd_int @ D_26 @ ( plus_plus_int @ X_1 @ T_4 ) )
<=> ( dvd_dvd_int @ D_26 @ ( plus_plus_int @ ( minus_minus_int @ X_1 @ ( times_times_int @ K @ D_25 ) ) @ T_4 ) ) ) ) ).
thf(fact_1532_inf__period_I4_J,axiom,
! [T_3: real,D_24: real,D_23: real] :
( ( dvd_dvd_real @ D_24 @ D_23 )
=> ! [X_1: real,K: real] :
( ~ ( dvd_dvd_real @ D_24 @ ( plus_plus_real @ X_1 @ T_3 ) )
<=> ~ ( dvd_dvd_real @ D_24 @ ( plus_plus_real @ ( minus_minus_real @ X_1 @ ( times_times_real @ K @ D_23 ) ) @ T_3 ) ) ) ) ).
thf(fact_1533_inf__period_I4_J,axiom,
! [T_3: complex,D_24: complex,D_23: complex] :
( ( dvd_dvd_complex @ D_24 @ D_23 )
=> ! [X_1: complex,K: complex] :
( ~ ( dvd_dvd_complex @ D_24 @ ( plus_plus_complex @ X_1 @ T_3 ) )
<=> ~ ( dvd_dvd_complex @ D_24 @ ( plus_plus_complex @ ( minus_minus_complex @ X_1 @ ( times_times_complex @ K @ D_23 ) ) @ T_3 ) ) ) ) ).
thf(fact_1534_inf__period_I4_J,axiom,
! [T_3: rat,D_24: rat,D_23: rat] :
( ( dvd_dvd_rat @ D_24 @ D_23 )
=> ! [X_1: rat,K: rat] :
( ~ ( dvd_dvd_rat @ D_24 @ ( plus_plus_rat @ X_1 @ T_3 ) )
<=> ~ ( dvd_dvd_rat @ D_24 @ ( plus_plus_rat @ ( minus_minus_rat @ X_1 @ ( times_times_rat @ K @ D_23 ) ) @ T_3 ) ) ) ) ).
thf(fact_1535_inf__period_I4_J,axiom,
! [T_3: int,D_24: int,D_23: int] :
( ( dvd_dvd_int @ D_24 @ D_23 )
=> ! [X_1: int,K: int] :
( ~ ( dvd_dvd_int @ D_24 @ ( plus_plus_int @ X_1 @ T_3 ) )
<=> ~ ( dvd_dvd_int @ D_24 @ ( plus_plus_int @ ( minus_minus_int @ X_1 @ ( times_times_int @ K @ D_23 ) ) @ T_3 ) ) ) ) ).
thf(fact_1536_eq__add__iff2,axiom,
! [A_151: int,E_7: int,C_73: int,B_123: int,D_22: int] :
( ( ( plus_plus_int @ ( times_times_int @ A_151 @ E_7 ) @ C_73 )
= ( plus_plus_int @ ( times_times_int @ B_123 @ E_7 ) @ D_22 ) )
<=> ( C_73
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_123 @ A_151 ) @ E_7 ) @ D_22 ) ) ) ).
thf(fact_1537_eq__add__iff2,axiom,
! [A_151: real,E_7: real,C_73: real,B_123: real,D_22: real] :
( ( ( plus_plus_real @ ( times_times_real @ A_151 @ E_7 ) @ C_73 )
= ( plus_plus_real @ ( times_times_real @ B_123 @ E_7 ) @ D_22 ) )
<=> ( C_73
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B_123 @ A_151 ) @ E_7 ) @ D_22 ) ) ) ).
thf(fact_1538_eq__add__iff2,axiom,
! [A_151: complex,E_7: complex,C_73: complex,B_123: complex,D_22: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ A_151 @ E_7 ) @ C_73 )
= ( plus_plus_complex @ ( times_times_complex @ B_123 @ E_7 ) @ D_22 ) )
<=> ( C_73
= ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B_123 @ A_151 ) @ E_7 ) @ D_22 ) ) ) ).
thf(fact_1539_eq__add__iff2,axiom,
! [A_151: rat,E_7: rat,C_73: rat,B_123: rat,D_22: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A_151 @ E_7 ) @ C_73 )
= ( plus_plus_rat @ ( times_times_rat @ B_123 @ E_7 ) @ D_22 ) )
<=> ( C_73
= ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B_123 @ A_151 ) @ E_7 ) @ D_22 ) ) ) ).
thf(fact_1540_eq__add__iff1,axiom,
! [A_150: int,E_6: int,C_72: int,B_122: int,D_21: int] :
( ( ( plus_plus_int @ ( times_times_int @ A_150 @ E_6 ) @ C_72 )
= ( plus_plus_int @ ( times_times_int @ B_122 @ E_6 ) @ D_21 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_150 @ B_122 ) @ E_6 ) @ C_72 )
= D_21 ) ) ).
thf(fact_1541_eq__add__iff1,axiom,
! [A_150: real,E_6: real,C_72: real,B_122: real,D_21: real] :
( ( ( plus_plus_real @ ( times_times_real @ A_150 @ E_6 ) @ C_72 )
= ( plus_plus_real @ ( times_times_real @ B_122 @ E_6 ) @ D_21 ) )
<=> ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A_150 @ B_122 ) @ E_6 ) @ C_72 )
= D_21 ) ) ).
thf(fact_1542_eq__add__iff1,axiom,
! [A_150: complex,E_6: complex,C_72: complex,B_122: complex,D_21: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ A_150 @ E_6 ) @ C_72 )
= ( plus_plus_complex @ ( times_times_complex @ B_122 @ E_6 ) @ D_21 ) )
<=> ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A_150 @ B_122 ) @ E_6 ) @ C_72 )
= D_21 ) ) ).
thf(fact_1543_eq__add__iff1,axiom,
! [A_150: rat,E_6: rat,C_72: rat,B_122: rat,D_21: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A_150 @ E_6 ) @ C_72 )
= ( plus_plus_rat @ ( times_times_rat @ B_122 @ E_6 ) @ D_21 ) )
<=> ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A_150 @ B_122 ) @ E_6 ) @ C_72 )
= D_21 ) ) ).
thf(fact_1544_right__diff__distrib__number__of,axiom,
! [V_5: int,B_121: int,C_71: int] :
( ( times_times_int @ ( number_number_of_int @ V_5 ) @ ( minus_minus_int @ B_121 @ C_71 ) )
= ( minus_minus_int @ ( times_times_int @ ( number_number_of_int @ V_5 ) @ B_121 ) @ ( times_times_int @ ( number_number_of_int @ V_5 ) @ C_71 ) ) ) ).
thf(fact_1545_right__diff__distrib__number__of,axiom,
! [V_5: int,B_121: real,C_71: real] :
( ( times_times_real @ ( number267125858f_real @ V_5 ) @ ( minus_minus_real @ B_121 @ C_71 ) )
= ( minus_minus_real @ ( times_times_real @ ( number267125858f_real @ V_5 ) @ B_121 ) @ ( times_times_real @ ( number267125858f_real @ V_5 ) @ C_71 ) ) ) ).
thf(fact_1546_right__diff__distrib__number__of,axiom,
! [V_5: int,B_121: complex,C_71: complex] :
( ( times_times_complex @ ( number528085621omplex @ V_5 ) @ ( minus_minus_complex @ B_121 @ C_71 ) )
= ( minus_minus_complex @ ( times_times_complex @ ( number528085621omplex @ V_5 ) @ B_121 ) @ ( times_times_complex @ ( number528085621omplex @ V_5 ) @ C_71 ) ) ) ).
thf(fact_1547_right__diff__distrib__number__of,axiom,
! [V_5: int,B_121: rat,C_71: rat] :
( ( times_times_rat @ ( number_number_of_rat @ V_5 ) @ ( minus_minus_rat @ B_121 @ C_71 ) )
= ( minus_minus_rat @ ( times_times_rat @ ( number_number_of_rat @ V_5 ) @ B_121 ) @ ( times_times_rat @ ( number_number_of_rat @ V_5 ) @ C_71 ) ) ) ).
thf(fact_1548_left__diff__distrib__number__of,axiom,
! [A_149: int,B_120: int,V_4: int] :
( ( times_times_int @ ( minus_minus_int @ A_149 @ B_120 ) @ ( number_number_of_int @ V_4 ) )
= ( minus_minus_int @ ( times_times_int @ A_149 @ ( number_number_of_int @ V_4 ) ) @ ( times_times_int @ B_120 @ ( number_number_of_int @ V_4 ) ) ) ) ).
thf(fact_1549_left__diff__distrib__number__of,axiom,
! [A_149: real,B_120: real,V_4: int] :
( ( times_times_real @ ( minus_minus_real @ A_149 @ B_120 ) @ ( number267125858f_real @ V_4 ) )
= ( minus_minus_real @ ( times_times_real @ A_149 @ ( number267125858f_real @ V_4 ) ) @ ( times_times_real @ B_120 @ ( number267125858f_real @ V_4 ) ) ) ) ).
thf(fact_1550_left__diff__distrib__number__of,axiom,
! [A_149: complex,B_120: complex,V_4: int] :
( ( times_times_complex @ ( minus_minus_complex @ A_149 @ B_120 ) @ ( number528085621omplex @ V_4 ) )
= ( minus_minus_complex @ ( times_times_complex @ A_149 @ ( number528085621omplex @ V_4 ) ) @ ( times_times_complex @ B_120 @ ( number528085621omplex @ V_4 ) ) ) ) ).
thf(fact_1551_left__diff__distrib__number__of,axiom,
! [A_149: rat,B_120: rat,V_4: int] :
( ( times_times_rat @ ( minus_minus_rat @ A_149 @ B_120 ) @ ( number_number_of_rat @ V_4 ) )
= ( minus_minus_rat @ ( times_times_rat @ A_149 @ ( number_number_of_rat @ V_4 ) ) @ ( times_times_rat @ B_120 @ ( number_number_of_rat @ V_4 ) ) ) ) ).
thf(fact_1552_diff__bin__simps_I9_J,axiom,
! [K_1: int,L: int] :
( ( minus_minus_int @ ( bit1 @ K_1 ) @ ( bit0 @ L ) )
= ( bit1 @ ( minus_minus_int @ K_1 @ L ) ) ) ).
thf(fact_1553_diff__bin__simps_I10_J,axiom,
! [K_1: int,L: int] :
( ( minus_minus_int @ ( bit1 @ K_1 ) @ ( bit1 @ L ) )
= ( bit0 @ ( minus_minus_int @ K_1 @ L ) ) ) ).
thf(fact_1554_diff__bin__simps_I3_J,axiom,
! [L: int] :
( ( minus_minus_int @ pls @ ( bit0 @ L ) )
= ( bit0 @ ( minus_minus_int @ pls @ L ) ) ) ).
thf(fact_1555_less__bin__lemma,axiom,
! [K_1: int,L: int] :
( ( ord_less_int @ K_1 @ L )
<=> ( ord_less_int @ ( minus_minus_int @ K_1 @ L ) @ zero_zero_int ) ) ).
thf(fact_1556_dvd__imp__le,axiom,
! [K_1: nat,N: nat] :
( ( dvd_dvd_nat @ K_1 @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K_1 @ N ) ) ) ).
thf(fact_1557_dvd__mult__cancel,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
thf(fact_1558_xzgcda__linear__aux1,axiom,
! [A: int,R_1: int,B: int,M: int,C: int,D: int,N: int] :
( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ ( times_times_int @ R_1 @ B ) ) @ M ) @ ( times_times_int @ ( minus_minus_int @ C @ ( times_times_int @ R_1 @ D ) ) @ N ) )
= ( minus_minus_int @ ( plus_plus_int @ ( times_times_int @ A @ M ) @ ( times_times_int @ C @ N ) ) @ ( times_times_int @ R_1 @ ( plus_plus_int @ ( times_times_int @ B @ M ) @ ( times_times_int @ D @ N ) ) ) ) ) ).
thf(fact_1559_zcong__def,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
<=> ( dvd_dvd_int @ M @ ( minus_minus_int @ A @ B ) ) ) ).
thf(fact_1560_le__add__iff1,axiom,
! [A_148: int,E_5: int,C_70: int,B_119: int,D_20: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A_148 @ E_5 ) @ C_70 ) @ ( plus_plus_int @ ( times_times_int @ B_119 @ E_5 ) @ D_20 ) )
<=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_148 @ B_119 ) @ E_5 ) @ C_70 ) @ D_20 ) ) ).
thf(fact_1561_le__add__iff1,axiom,
! [A_148: real,E_5: real,C_70: real,B_119: real,D_20: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A_148 @ E_5 ) @ C_70 ) @ ( plus_plus_real @ ( times_times_real @ B_119 @ E_5 ) @ D_20 ) )
<=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A_148 @ B_119 ) @ E_5 ) @ C_70 ) @ D_20 ) ) ).
thf(fact_1562_le__add__iff1,axiom,
! [A_148: rat,E_5: rat,C_70: rat,B_119: rat,D_20: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A_148 @ E_5 ) @ C_70 ) @ ( plus_plus_rat @ ( times_times_rat @ B_119 @ E_5 ) @ D_20 ) )
<=> ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A_148 @ B_119 ) @ E_5 ) @ C_70 ) @ D_20 ) ) ).
thf(fact_1563_le__add__iff2,axiom,
! [A_147: int,E_4: int,C_69: int,B_118: int,D_19: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A_147 @ E_4 ) @ C_69 ) @ ( plus_plus_int @ ( times_times_int @ B_118 @ E_4 ) @ D_19 ) )
<=> ( ord_less_eq_int @ C_69 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_118 @ A_147 ) @ E_4 ) @ D_19 ) ) ) ).
thf(fact_1564_le__add__iff2,axiom,
! [A_147: real,E_4: real,C_69: real,B_118: real,D_19: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A_147 @ E_4 ) @ C_69 ) @ ( plus_plus_real @ ( times_times_real @ B_118 @ E_4 ) @ D_19 ) )
<=> ( ord_less_eq_real @ C_69 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B_118 @ A_147 ) @ E_4 ) @ D_19 ) ) ) ).
thf(fact_1565_le__add__iff2,axiom,
! [A_147: rat,E_4: rat,C_69: rat,B_118: rat,D_19: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A_147 @ E_4 ) @ C_69 ) @ ( plus_plus_rat @ ( times_times_rat @ B_118 @ E_4 ) @ D_19 ) )
<=> ( ord_less_eq_rat @ C_69 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B_118 @ A_147 ) @ E_4 ) @ D_19 ) ) ) ).
thf(fact_1566_less__add__iff2,axiom,
! [A_146: int,E_3: int,C_68: int,B_117: int,D_18: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A_146 @ E_3 ) @ C_68 ) @ ( plus_plus_int @ ( times_times_int @ B_117 @ E_3 ) @ D_18 ) )
<=> ( ord_less_int @ C_68 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_117 @ A_146 ) @ E_3 ) @ D_18 ) ) ) ).
thf(fact_1567_less__add__iff2,axiom,
! [A_146: real,E_3: real,C_68: real,B_117: real,D_18: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A_146 @ E_3 ) @ C_68 ) @ ( plus_plus_real @ ( times_times_real @ B_117 @ E_3 ) @ D_18 ) )
<=> ( ord_less_real @ C_68 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B_117 @ A_146 ) @ E_3 ) @ D_18 ) ) ) ).
thf(fact_1568_less__add__iff2,axiom,
! [A_146: rat,E_3: rat,C_68: rat,B_117: rat,D_18: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A_146 @ E_3 ) @ C_68 ) @ ( plus_plus_rat @ ( times_times_rat @ B_117 @ E_3 ) @ D_18 ) )
<=> ( ord_less_rat @ C_68 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B_117 @ A_146 ) @ E_3 ) @ D_18 ) ) ) ).
thf(fact_1569_less__add__iff1,axiom,
! [A_145: int,E_2: int,C_67: int,B_116: int,D_17: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A_145 @ E_2 ) @ C_67 ) @ ( plus_plus_int @ ( times_times_int @ B_116 @ E_2 ) @ D_17 ) )
<=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_145 @ B_116 ) @ E_2 ) @ C_67 ) @ D_17 ) ) ).
thf(fact_1570_less__add__iff1,axiom,
! [A_145: real,E_2: real,C_67: real,B_116: real,D_17: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A_145 @ E_2 ) @ C_67 ) @ ( plus_plus_real @ ( times_times_real @ B_116 @ E_2 ) @ D_17 ) )
<=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A_145 @ B_116 ) @ E_2 ) @ C_67 ) @ D_17 ) ) ).
thf(fact_1571_less__add__iff1,axiom,
! [A_145: rat,E_2: rat,C_67: rat,B_116: rat,D_17: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A_145 @ E_2 ) @ C_67 ) @ ( plus_plus_rat @ ( times_times_rat @ B_116 @ E_2 ) @ D_17 ) )
<=> ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A_145 @ B_116 ) @ E_2 ) @ C_67 ) @ D_17 ) ) ).
thf(fact_1572_dvd__mult__cancel1,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
<=> ( N = one_one_nat ) ) ) ).
thf(fact_1573_dvd__mult__cancel2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
<=> ( N = one_one_nat ) ) ) ).
thf(fact_1574_add__number__of__diff1,axiom,
! [V_3: int,W_2: int,C_66: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_3 ) @ ( minus_minus_int @ ( number_number_of_int @ W_2 ) @ C_66 ) )
= ( minus_minus_int @ ( number_number_of_int @ ( plus_plus_int @ V_3 @ W_2 ) ) @ C_66 ) ) ).
thf(fact_1575_add__number__of__diff1,axiom,
! [V_3: int,W_2: int,C_66: real] :
( ( plus_plus_real @ ( number267125858f_real @ V_3 ) @ ( minus_minus_real @ ( number267125858f_real @ W_2 ) @ C_66 ) )
= ( minus_minus_real @ ( number267125858f_real @ ( plus_plus_int @ V_3 @ W_2 ) ) @ C_66 ) ) ).
thf(fact_1576_add__number__of__diff1,axiom,
! [V_3: int,W_2: int,C_66: complex] :
( ( plus_plus_complex @ ( number528085621omplex @ V_3 ) @ ( minus_minus_complex @ ( number528085621omplex @ W_2 ) @ C_66 ) )
= ( minus_minus_complex @ ( number528085621omplex @ ( plus_plus_int @ V_3 @ W_2 ) ) @ C_66 ) ) ).
thf(fact_1577_add__number__of__diff1,axiom,
! [V_3: int,W_2: int,C_66: rat] :
( ( plus_plus_rat @ ( number_number_of_rat @ V_3 ) @ ( minus_minus_rat @ ( number_number_of_rat @ W_2 ) @ C_66 ) )
= ( minus_minus_rat @ ( number_number_of_rat @ ( plus_plus_int @ V_3 @ W_2 ) ) @ C_66 ) ) ).
thf(fact_1578_Euler_Oaux1,axiom,
! [A: int,X: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ A )
=> ( ( X
!= ( minus_minus_int @ A @ one_one_int ) )
=> ( ord_less_int @ X @ ( minus_minus_int @ A @ one_one_int ) ) ) ) ) ).
thf(fact_1579_zle__diff1__eq,axiom,
! [W: int,Z_1: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z_1 @ one_one_int ) )
<=> ( ord_less_int @ W @ Z_1 ) ) ).
thf(fact_1580_diff__bin__simps_I4_J,axiom,
! [L: int] :
( ( minus_minus_int @ pls @ ( bit1 @ L ) )
= ( bit1 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_1581_diff__bin__simps_I5_J,axiom,
! [L: int] :
( ( minus_minus_int @ min @ ( bit0 @ L ) )
= ( bit1 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_1582_diff__bin__simps_I6_J,axiom,
! [L: int] :
( ( minus_minus_int @ min @ ( bit1 @ L ) )
= ( bit0 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_1583_inv__not__p__minus__1__aux,axiom,
! [A: int,P_3: int] :
( ( zcong @ ( times_times_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) ) @ one_one_int @ P_3 )
<=> ( zcong @ A @ ( minus_minus_int @ P_3 @ one_one_int ) @ P_3 ) ) ).
thf(fact_1584_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_1585_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
<=> ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
thf(fact_1586_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
thf(fact_1587_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_1588_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
thf(fact_1589_less__not__refl3,axiom,
! [S_1: nat,T: nat] :
( ( ord_less_nat @ S_1 @ T )
=> ( S_1 != T ) ) ).
thf(fact_1590_nat__less__cases,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
=> ( P @ N @ M ) )
=> ( ( ( M = N )
=> ( P @ N @ M ) )
=> ( ( ( ord_less_nat @ N @ M )
=> ( P @ N @ M ) )
=> ( P @ N @ M ) ) ) ) ).
thf(fact_1591_nat__add__commute,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ N )
= ( plus_plus_nat @ N @ M ) ) ).
thf(fact_1592_nat__add__left__commute,axiom,
! [X: nat,Y: nat,Z_1: nat] :
( ( plus_plus_nat @ X @ ( plus_plus_nat @ Y @ Z_1 ) )
= ( plus_plus_nat @ Y @ ( plus_plus_nat @ X @ Z_1 ) ) ) ).
thf(fact_1593_nat__add__assoc,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ M @ N ) @ K_1 )
= ( plus_plus_nat @ M @ ( plus_plus_nat @ N @ K_1 ) ) ) ).
thf(fact_1594_nat__add__left__cancel,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ( plus_plus_nat @ K_1 @ M )
= ( plus_plus_nat @ K_1 @ N ) )
<=> ( M = N ) ) ).
thf(fact_1595_nat__add__right__cancel,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ( plus_plus_nat @ M @ K_1 )
= ( plus_plus_nat @ N @ K_1 ) )
<=> ( M = N ) ) ).
thf(fact_1596_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
thf(fact_1597_le__trans,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K_1 )
=> ( ord_less_eq_nat @ I @ K_1 ) ) ) ).
thf(fact_1598_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_1599_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_1600_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
thf(fact_1601_nat__mult__commute,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ N )
= ( times_times_nat @ N @ M ) ) ).
thf(fact_1602_nat__mult__assoc,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( times_times_nat @ ( times_times_nat @ M @ N ) @ K_1 )
= ( times_times_nat @ M @ ( times_times_nat @ N @ K_1 ) ) ) ).
thf(fact_1603_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_1604_zcong__square__zless,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ P_3 )
=> ( ( zcong @ ( times_times_int @ A @ A ) @ one_one_int @ P_3 )
=> ( ( A = one_one_int )
| ( A
= ( minus_minus_int @ P_3 @ one_one_int ) ) ) ) ) ) ) ).
thf(fact_1605_zcong__square,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( zcong @ ( times_times_int @ A @ A ) @ one_one_int @ P_3 )
=> ( ( zcong @ A @ one_one_int @ P_3 )
| ( zcong @ A @ ( minus_minus_int @ P_3 @ one_one_int ) @ P_3 ) ) ) ) ) ).
thf(fact_1606_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_1607_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
<=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_1608_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_1609_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
thf(fact_1610_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_1611_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
thf(fact_1612_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_1613_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
<=> ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
thf(fact_1614_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
thf(fact_1615_conj__le__cong,axiom,
! [P_1: $o,P: $o,X: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( P
<=> P_1 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
<=> ( ( ord_less_eq_int @ zero_zero_int @ X )
& P_1 ) ) ) ).
thf(fact_1616_imp__le__cong,axiom,
! [P_1: $o,P: $o,X: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( P
<=> P_1 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
<=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P_1 ) ) ) ).
thf(fact_1617_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_1618_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_1619_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_1620_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_1621_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
<=> ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
thf(fact_1622_mult__cancel1,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K_1 @ M )
= ( times_times_nat @ K_1 @ N ) )
<=> ( ( M = N )
| ( K_1 = zero_zero_nat ) ) ) ).
thf(fact_1623_mult__cancel2,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ( times_times_nat @ M @ K_1 )
= ( times_times_nat @ N @ K_1 ) )
<=> ( ( M = N )
| ( K_1 = zero_zero_nat ) ) ) ).
thf(fact_1624_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
thf(fact_1625_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
thf(fact_1626_nat__add__left__cancel__less,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K_1 @ M ) @ ( plus_plus_nat @ K_1 @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_1627_trans__less__add1,axiom,
! [M: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
thf(fact_1628_trans__less__add2,axiom,
! [M: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
thf(fact_1629_add__less__mono1,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K_1 ) @ ( plus_plus_nat @ J @ K_1 ) ) ) ).
thf(fact_1630_add__less__mono,axiom,
! [K_1: nat,L: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K_1 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K_1 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
thf(fact_1631_less__add__eq__less,axiom,
! [M: nat,N: nat,K_1: nat,L: nat] :
( ( ord_less_nat @ K_1 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K_1 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_1632_add__lessD1,axiom,
! [I: nat,J: nat,K_1: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K_1 )
=> ( ord_less_nat @ I @ K_1 ) ) ).
thf(fact_1633_nat__less__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
<=> ( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ).
thf(fact_1634_le__eq__less__or__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
<=> ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
thf(fact_1635_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_1636_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_1637_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_1638_zspecial__product,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_1639_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
thf(fact_1640_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
thf(fact_1641_le__iff__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
<=> ? [K: nat] :
( N
= ( plus_plus_nat @ M @ K ) ) ) ).
thf(fact_1642_nat__add__left__cancel__le,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K_1 @ M ) @ ( plus_plus_nat @ K_1 @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_1643_trans__le__add1,axiom,
! [M: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
thf(fact_1644_trans__le__add2,axiom,
! [M: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
thf(fact_1645_add__le__mono1,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K_1 ) @ ( plus_plus_nat @ J @ K_1 ) ) ) ).
thf(fact_1646_add__le__mono,axiom,
! [K_1: nat,L: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K_1 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K_1 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
thf(fact_1647_add__leD2,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K_1 ) @ N )
=> ( ord_less_eq_nat @ K_1 @ N ) ) ).
thf(fact_1648_add__leD1,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K_1 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_1649_add__leE,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K_1 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K_1 @ N ) ) ) ).
thf(fact_1650_add__mult__distrib2,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( times_times_nat @ K_1 @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) ) ) ).
thf(fact_1651_add__mult__distrib,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K_1 )
= ( plus_plus_nat @ ( times_times_nat @ M @ K_1 ) @ ( times_times_nat @ N @ K_1 ) ) ) ).
thf(fact_1652_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
thf(fact_1653_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
thf(fact_1654_mult__le__mono1,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K_1 ) @ ( times_times_nat @ J @ K_1 ) ) ) ).
thf(fact_1655_mult__le__mono2,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K_1 @ I ) @ ( times_times_nat @ K_1 @ J ) ) ) ).
thf(fact_1656_mult__le__mono,axiom,
! [K_1: nat,L: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K_1 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K_1 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
thf(fact_1657_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
thf(fact_1658_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
<=> ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
thf(fact_1659_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
thf(fact_1660_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
<=> ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
thf(fact_1661_power2__diff,axiom,
! [X_26: int,Y_18: int] :
( ( power_power_int @ ( minus_minus_int @ X_26 @ Y_18 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X_26 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_18 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_26 ) @ Y_18 ) ) ) ).
thf(fact_1662_power2__diff,axiom,
! [X_26: real,Y_18: real] :
( ( power_power_real @ ( minus_minus_real @ X_26 @ Y_18 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X_26 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_18 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_26 ) @ Y_18 ) ) ) ).
thf(fact_1663_power2__diff,axiom,
! [X_26: complex,Y_18: complex] :
( ( power_power_complex @ ( minus_minus_complex @ X_26 @ Y_18 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X_26 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_complex @ Y_18 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( number528085621omplex @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_26 ) @ Y_18 ) ) ) ).
thf(fact_1664_power2__diff,axiom,
! [X_26: rat,Y_18: rat] :
( ( power_power_rat @ ( minus_minus_rat @ X_26 @ Y_18 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X_26 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_rat @ Y_18 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( number_number_of_rat @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_26 ) @ Y_18 ) ) ) ).
thf(fact_1665_zdiff__power2,axiom,
! [A: int,B: int] :
( ( power_power_int @ ( minus_minus_int @ A @ B ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ ( minus_minus_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) @ B ) ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_1666_zdiff__power3,axiom,
! [A: int,B: int] :
( ( power_power_int @ ( minus_minus_int @ A @ B ) @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( minus_minus_int @ ( plus_plus_int @ ( minus_minus_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ B ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) @ A ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_1667_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
thf(fact_1668_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
thf(fact_1669_mult__less__cancel1,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
& ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_1670_mult__less__cancel2,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K_1 ) @ ( times_times_nat @ N @ K_1 ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
& ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_1671_mult__less__mono1,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K_1 ) @ ( times_times_nat @ J @ K_1 ) ) ) ) ).
thf(fact_1672_mult__less__mono2,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ord_less_nat @ ( times_times_nat @ K_1 @ I ) @ ( times_times_nat @ K_1 @ J ) ) ) ) ).
thf(fact_1673_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
thf(fact_1674_zdvd__mono,axiom,
! [M: int,T: int,K_1: int] :
( ( K_1 != zero_zero_int )
=> ( ( dvd_dvd_int @ M @ T )
<=> ( dvd_dvd_int @ ( times_times_int @ K_1 @ M ) @ ( times_times_int @ K_1 @ T ) ) ) ) ).
thf(fact_1675_unity__coeff__ex,axiom,
! [P_6: code_code_numeral > $o,L_2: code_code_numeral] :
( ? [X_1: code_code_numeral] : ( P_6 @ ( times_1655362735umeral @ L_2 @ X_1 ) )
<=> ? [X_1: code_code_numeral] :
( ( dvd_dv174992974umeral @ L_2 @ ( plus_p1627245867umeral @ X_1 @ zero_z126310315umeral ) )
& ( P_6 @ X_1 ) ) ) ).
thf(fact_1676_unity__coeff__ex,axiom,
! [P_6: real > $o,L_2: real] :
( ? [X_1: real] : ( P_6 @ ( times_times_real @ L_2 @ X_1 ) )
<=> ? [X_1: real] :
( ( dvd_dvd_real @ L_2 @ ( plus_plus_real @ X_1 @ zero_zero_real ) )
& ( P_6 @ X_1 ) ) ) ).
thf(fact_1677_unity__coeff__ex,axiom,
! [P_6: complex > $o,L_2: complex] :
( ? [X_1: complex] : ( P_6 @ ( times_times_complex @ L_2 @ X_1 ) )
<=> ? [X_1: complex] :
( ( dvd_dvd_complex @ L_2 @ ( plus_plus_complex @ X_1 @ zero_zero_complex ) )
& ( P_6 @ X_1 ) ) ) ).
thf(fact_1678_unity__coeff__ex,axiom,
! [P_6: quickcheck_code_int > $o,L_2: quickcheck_code_int] :
( ? [X_1: quickcheck_code_int] : ( P_6 @ ( times_123202395de_int @ L_2 @ X_1 ) )
<=> ? [X_1: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ L_2 @ ( plus_p1446045655de_int @ X_1 @ zero_z891286103de_int ) )
& ( P_6 @ X_1 ) ) ) ).
thf(fact_1679_unity__coeff__ex,axiom,
! [P_6: rat > $o,L_2: rat] :
( ? [X_1: rat] : ( P_6 @ ( times_times_rat @ L_2 @ X_1 ) )
<=> ? [X_1: rat] :
( ( dvd_dvd_rat @ L_2 @ ( plus_plus_rat @ X_1 @ zero_zero_rat ) )
& ( P_6 @ X_1 ) ) ) ).
thf(fact_1680_unity__coeff__ex,axiom,
! [P_6: int > $o,L_2: int] :
( ? [X_1: int] : ( P_6 @ ( times_times_int @ L_2 @ X_1 ) )
<=> ? [X_1: int] :
( ( dvd_dvd_int @ L_2 @ ( plus_plus_int @ X_1 @ zero_zero_int ) )
& ( P_6 @ X_1 ) ) ) ).
thf(fact_1681_unity__coeff__ex,axiom,
! [P_6: nat > $o,L_2: nat] :
( ? [X_1: nat] : ( P_6 @ ( times_times_nat @ L_2 @ X_1 ) )
<=> ? [X_1: nat] :
( ( dvd_dvd_nat @ L_2 @ ( plus_plus_nat @ X_1 @ zero_zero_nat ) )
& ( P_6 @ X_1 ) ) ) ).
thf(fact_1682_number__of2,axiom,
ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ pls ) ).
thf(fact_1683_mult__le__cancel1,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_1684_mult__le__cancel2,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K_1 ) @ ( times_times_nat @ N @ K_1 ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_1685_neg__one__power,axiom,
! [N: nat] :
( ( ( power_power_int @ ( number_number_of_int @ min ) @ N )
= one_one_int )
| ( ( power_power_int @ ( number_number_of_int @ min ) @ N )
= ( number_number_of_int @ min ) ) ) ).
thf(fact_1686_Legendre__def,axiom,
! [A: int,P_3: int] :
( ( ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ( legendre @ A @ P_3 )
= zero_zero_int ) )
& ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ( ( quadRes @ P_3 @ A )
=> ( ( legendre @ A @ P_3 )
= one_one_int ) )
& ( ~ ( quadRes @ P_3 @ A )
=> ( ( legendre @ A @ P_3 )
= ( number_number_of_int @ min ) ) ) ) ) ) ).
thf(fact_1687_QuadRes__def,axiom,
! [M: int,X: int] :
( ( quadRes @ M @ X )
<=> ? [Y_1: int] : ( zcong @ ( power_power_int @ Y_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ X @ M ) ) ).
thf(fact_1688_real__sum__squared__expand,axiom,
! [X: real,Y: real] :
( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X ) @ Y ) ) ) ).
thf(fact_1689_divides__cases,axiom,
! [N: nat,M: nat] :
( ( dvd_dvd_nat @ N @ M )
=> ( ( M = zero_zero_nat )
| ( M = N )
| ( ord_less_eq_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) @ M ) ) ) ).
thf(fact_1690_decr__mult__lemma,axiom,
! [K_1: int,P: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( minus_minus_int @ X_1 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( minus_minus_int @ X_1 @ ( times_times_int @ K_1 @ D ) ) ) ) ) ) ) ).
thf(fact_1691_ex__least__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ( P @ N )
=> ? [K: nat] :
( ( ord_less_nat @ K @ N )
& ! [I_1: nat] :
( ( ord_less_eq_nat @ I_1 @ K )
=> ~ ( P @ I_1 ) )
& ( P @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ).
thf(fact_1692_incr__mult__lemma,axiom,
! [K_1: int,P: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( plus_plus_int @ X_1 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( plus_plus_int @ X_1 @ ( times_times_int @ K_1 @ D ) ) ) ) ) ) ) ).
thf(fact_1693_two__realpow__ge__one,axiom,
! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) ).
thf(fact_1694_realpow__two__sum__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_real )
<=> ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
thf(fact_1695_zcong__zless__unique,axiom,
! [A: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ? [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
& ( ord_less_int @ X_1 @ M )
& ( zcong @ A @ X_1 @ M )
& ! [Y_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Y_1 )
& ( ord_less_int @ Y_1 @ M )
& ( zcong @ A @ Y_1 @ M ) )
=> ( Y_1 = X_1 ) ) ) ) ).
thf(fact_1696_dvd_Oorder__refl,axiom,
! [X: nat] : ( dvd_dvd_nat @ X @ X ) ).
thf(fact_1697_real__le__eq__diff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
<=> ( ord_less_eq_real @ ( minus_minus_real @ X @ Y ) @ zero_zero_real ) ) ).
thf(fact_1698_dvd_Oless__irrefl,axiom,
! [X: nat] :
~ ( ( dvd_dvd_nat @ X @ X )
& ~ ( dvd_dvd_nat @ X @ X ) ) ).
thf(fact_1699_dvd_Oeq__iff,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
<=> ( ( dvd_dvd_nat @ X @ Y )
& ( dvd_dvd_nat @ Y @ X ) ) ) ).
thf(fact_1700_dvd_Ole__less,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
<=> ( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
| ( X = Y ) ) ) ).
thf(fact_1701_dvd_Oless__le,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
<=> ( ( dvd_dvd_nat @ X @ Y )
& ( X != Y ) ) ) ).
thf(fact_1702_dvd_Oless__le__not__le,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
<=> ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) ) ) ).
thf(fact_1703_divides__antisym,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ( dvd_dvd_nat @ Y @ X ) )
<=> ( X = Y ) ) ).
thf(fact_1704_dvd_Oneq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) ) ) ) ).
thf(fact_1705_dvd_Oeq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( dvd_dvd_nat @ X @ Y ) ) ).
thf(fact_1706_dvd_Oantisym__conv,axiom,
! [Y: nat,X: nat] :
( ( dvd_dvd_nat @ Y @ X )
=> ( ( dvd_dvd_nat @ X @ Y )
<=> ( X = Y ) ) ) ).
thf(fact_1707_dvd_Ole__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
| ( X = Y ) ) ) ).
thf(fact_1708_dvd_Ole__neq__trans,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( A != B )
=> ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) ) ) ) ).
thf(fact_1709_dvd_Oord__eq__le__trans,axiom,
! [C: nat,A: nat,B: nat] :
( ( A = B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
thf(fact_1710_dvd_Oord__le__eq__trans,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( B = C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
thf(fact_1711_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
thf(fact_1712_dvd_Oantisym,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( dvd_dvd_nat @ Y @ X )
=> ( X = Y ) ) ) ).
thf(fact_1713_dvd_Oorder__trans,axiom,
! [Z_1: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( dvd_dvd_nat @ Y @ Z_1 )
=> ( dvd_dvd_nat @ X @ Z_1 ) ) ) ).
thf(fact_1714_dvd__diff__nat,axiom,
! [N: nat,K_1: nat,M: nat] :
( ( dvd_dvd_nat @ K_1 @ M )
=> ( ( dvd_dvd_nat @ K_1 @ N )
=> ( dvd_dvd_nat @ K_1 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_1715_dvd_Oord__eq__less__trans,axiom,
! [C: nat,A: nat,B: nat] :
( ( A = B )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ~ ( dvd_dvd_nat @ C @ B ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ~ ( dvd_dvd_nat @ C @ A ) ) ) ) ).
thf(fact_1716_dvd_Ole__less__trans,axiom,
! [Z_1: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( ( dvd_dvd_nat @ Y @ Z_1 )
& ~ ( dvd_dvd_nat @ Z_1 @ Y ) )
=> ( ( dvd_dvd_nat @ X @ Z_1 )
& ~ ( dvd_dvd_nat @ Z_1 @ X ) ) ) ) ).
thf(fact_1717_dvd_Oless__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( X != Y ) ) ).
thf(fact_1718_dvd_Oless__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ~ ( ( dvd_dvd_nat @ Y @ X )
& ~ ( dvd_dvd_nat @ X @ Y ) ) ) ).
thf(fact_1719_dvd_Oless__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( dvd_dvd_nat @ X @ Y ) ) ).
thf(fact_1720_dvd_Oless__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ~ ( ( dvd_dvd_nat @ Y @ X )
& ~ ( dvd_dvd_nat @ X @ Y ) ) ) ).
thf(fact_1721_dvd_Oless__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( X != Y ) ) ).
thf(fact_1722_dvd_Oless__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( Y != X ) ) ).
thf(fact_1723_dvd_Oless__imp__triv,axiom,
! [P: $o,X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( ( ( dvd_dvd_nat @ Y @ X )
& ~ ( dvd_dvd_nat @ X @ Y ) )
=> P ) ) ).
thf(fact_1724_dvd_Oord__less__eq__trans,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) )
=> ( ( B = C )
=> ( ( dvd_dvd_nat @ A @ C )
& ~ ( dvd_dvd_nat @ C @ A ) ) ) ) ).
thf(fact_1725_dvd_Oless__le__trans,axiom,
! [Z_1: nat,X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( ( dvd_dvd_nat @ Y @ Z_1 )
=> ( ( dvd_dvd_nat @ X @ Z_1 )
& ~ ( dvd_dvd_nat @ Z_1 @ X ) ) ) ) ).
thf(fact_1726_dvd_Oless__asym_H,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) )
=> ~ ( ( dvd_dvd_nat @ B @ A )
& ~ ( dvd_dvd_nat @ A @ B ) ) ) ).
thf(fact_1727_dvd_Oless__trans,axiom,
! [Z_1: nat,X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( ( ( dvd_dvd_nat @ Y @ Z_1 )
& ~ ( dvd_dvd_nat @ Z_1 @ Y ) )
=> ( ( dvd_dvd_nat @ X @ Z_1 )
& ~ ( dvd_dvd_nat @ Z_1 @ X ) ) ) ) ).
thf(fact_1728_dvd_Oless__asym,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ~ ( ( dvd_dvd_nat @ Y @ X )
& ~ ( dvd_dvd_nat @ X @ Y ) ) ) ).
thf(fact_1729_real__zero__not__eq__one,axiom,
zero_zero_real != one_one_real ).
thf(fact_1730_real__le__antisym,axiom,
! [Z_1: real,W: real] :
( ( ord_less_eq_real @ Z_1 @ W )
=> ( ( ord_less_eq_real @ W @ Z_1 )
=> ( Z_1 = W ) ) ) ).
thf(fact_1731_real__le__trans,axiom,
! [K_1: real,I: real,J: real] :
( ( ord_less_eq_real @ I @ J )
=> ( ( ord_less_eq_real @ J @ K_1 )
=> ( ord_less_eq_real @ I @ K_1 ) ) ) ).
thf(fact_1732_real__add__left__mono,axiom,
! [Z_1: real,X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( plus_plus_real @ Z_1 @ X ) @ ( plus_plus_real @ Z_1 @ Y ) ) ) ).
thf(fact_1733_less__eq__real__def,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
<=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
thf(fact_1734_real__less__def,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
<=> ( ( ord_less_eq_real @ X @ Y )
& ( X != Y ) ) ) ).
thf(fact_1735_real__le__linear,axiom,
! [Z_1: real,W: real] :
( ( ord_less_eq_real @ Z_1 @ W )
| ( ord_less_eq_real @ W @ Z_1 ) ) ).
thf(fact_1736_real__le__refl,axiom,
! [W: real] : ( ord_less_eq_real @ W @ W ) ).
thf(fact_1737_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
thf(fact_1738_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
thf(fact_1739_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_1740_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_1741_diff__less__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
thf(fact_1742_less__imp__diff__less,axiom,
! [N: nat,J: nat,K_1: nat] :
( ( ord_less_nat @ J @ K_1 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K_1 ) ) ).
thf(fact_1743_diff__cancel2,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K_1 ) @ ( plus_plus_nat @ N @ K_1 ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_1744_Nat_Odiff__cancel,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K_1 @ M ) @ ( plus_plus_nat @ K_1 @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_1745_diff__diff__left,axiom,
! [I: nat,J: nat,K_1: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K_1 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K_1 ) ) ) ).
thf(fact_1746_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
thf(fact_1747_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
thf(fact_1748_Nat_Odiff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
thf(fact_1749_diff__le__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
thf(fact_1750_diff__le__mono,axiom,
! [L: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
thf(fact_1751_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
thf(fact_1752_eq__diff__iff,axiom,
! [N: nat,K_1: nat,M: nat] :
( ( ord_less_eq_nat @ K_1 @ M )
=> ( ( ord_less_eq_nat @ K_1 @ N )
=> ( ( ( minus_minus_nat @ M @ K_1 )
= ( minus_minus_nat @ N @ K_1 ) )
<=> ( M = N ) ) ) ) ).
thf(fact_1753_Nat_Odiff__diff__eq,axiom,
! [N: nat,K_1: nat,M: nat] :
( ( ord_less_eq_nat @ K_1 @ M )
=> ( ( ord_less_eq_nat @ K_1 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K_1 ) @ ( minus_minus_nat @ N @ K_1 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_1754_le__diff__iff,axiom,
! [N: nat,K_1: nat,M: nat] :
( ( ord_less_eq_nat @ K_1 @ M )
=> ( ( ord_less_eq_nat @ K_1 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K_1 ) @ ( minus_minus_nat @ N @ K_1 ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ) ) ).
thf(fact_1755_dvd__diffD,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K_1 @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K_1 @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K_1 @ M ) ) ) ) ).
thf(fact_1756_dvd__diffD1,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K_1 @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K_1 @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K_1 @ N ) ) ) ) ).
thf(fact_1757_diff__mult__distrib,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K_1 )
= ( minus_minus_nat @ ( times_times_nat @ M @ K_1 ) @ ( times_times_nat @ N @ K_1 ) ) ) ).
thf(fact_1758_diff__mult__distrib2,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( times_times_nat @ K_1 @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) ) ) ).
thf(fact_1759_diff__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
thf(fact_1760_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_1761_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
thf(fact_1762_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
thf(fact_1763_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_1764_add__diff__inverse,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
thf(fact_1765_less__diff__conv,axiom,
! [I: nat,J: nat,K_1: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K_1 ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ I @ K_1 ) @ J ) ) ).
thf(fact_1766_less__diff__iff,axiom,
! [N: nat,K_1: nat,M: nat] :
( ( ord_less_eq_nat @ K_1 @ M )
=> ( ( ord_less_eq_nat @ K_1 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K_1 ) @ ( minus_minus_nat @ N @ K_1 ) )
<=> ( ord_less_nat @ M @ N ) ) ) ) ).
thf(fact_1767_diff__less__mono,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
thf(fact_1768_diff__add__assoc2,axiom,
! [I: nat,K_1: nat,J: nat] :
( ( ord_less_eq_nat @ K_1 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K_1 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K_1 ) @ I ) ) ) ).
thf(fact_1769_add__diff__assoc2,axiom,
! [I: nat,K_1: nat,J: nat] :
( ( ord_less_eq_nat @ K_1 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K_1 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K_1 ) ) ) ).
thf(fact_1770_diff__add__assoc,axiom,
! [I: nat,K_1: nat,J: nat] :
( ( ord_less_eq_nat @ K_1 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K_1 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K_1 ) ) ) ) ).
thf(fact_1771_le__imp__diff__is__add,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K_1 )
<=> ( J
= ( plus_plus_nat @ K_1 @ I ) ) ) ) ).
thf(fact_1772_le__add__diff__inverse2,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ M @ N ) @ N )
= M ) ) ).
thf(fact_1773_le__diff__conv2,axiom,
! [I: nat,K_1: nat,J: nat] :
( ( ord_less_eq_nat @ K_1 @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K_1 ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K_1 ) @ J ) ) ) ).
thf(fact_1774_add__diff__assoc,axiom,
! [I: nat,K_1: nat,J: nat] :
( ( ord_less_eq_nat @ K_1 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K_1 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K_1 ) ) ) ).
thf(fact_1775_le__add__diff__inverse,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
thf(fact_1776_le__add__diff,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ord_less_eq_nat @ K_1 @ N )
=> ( ord_less_eq_nat @ M @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K_1 ) ) ) ).
thf(fact_1777_le__diff__conv,axiom,
! [J: nat,K_1: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K_1 ) @ I )
<=> ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K_1 ) ) ) ).
thf(fact_1778_diff__diff__right,axiom,
! [I: nat,K_1: nat,J: nat] :
( ( ord_less_eq_nat @ K_1 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K_1 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K_1 ) @ J ) ) ) ).
thf(fact_1779_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
<=> ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D_2: nat] :
( ( A
= ( plus_plus_nat @ B @ D_2 ) )
=> ( P @ D_2 ) ) ) ) ).
thf(fact_1780_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
<=> ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D_2: nat] :
( ( A
= ( plus_plus_nat @ B @ D_2 ) )
& ~ ( P @ D_2 ) ) ) ) ).
thf(fact_1781_divides__add__revr,axiom,
! [B: nat,D: nat,A: nat] :
( ( dvd_dvd_nat @ D @ A )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
=> ( dvd_dvd_nat @ D @ B ) ) ) ).
thf(fact_1782_real__mult__1,axiom,
! [Z_1: real] :
( ( times_times_real @ one_one_real @ Z_1 )
= Z_1 ) ).
thf(fact_1783_real__mult__commute,axiom,
! [Z_1: real,W: real] :
( ( times_times_real @ Z_1 @ W )
= ( times_times_real @ W @ Z_1 ) ) ).
thf(fact_1784_real__mult__assoc,axiom,
! [Z1: real,Z2: real,Z3: real] :
( ( times_times_real @ ( times_times_real @ Z1 @ Z2 ) @ Z3 )
= ( times_times_real @ Z1 @ ( times_times_real @ Z2 @ Z3 ) ) ) ).
thf(fact_1785_real__add__mult__distrib,axiom,
! [Z1: real,Z2: real,W: real] :
( ( times_times_real @ ( plus_plus_real @ Z1 @ Z2 ) @ W )
= ( plus_plus_real @ ( times_times_real @ Z1 @ W ) @ ( times_times_real @ Z2 @ W ) ) ) ).
thf(fact_1786_real__two__squares__add__zero__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= zero_zero_real )
<=> ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ).
thf(fact_1787_real__mult__right__cancel,axiom,
! [A: real,B: real,C: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
<=> ( A = B ) ) ) ).
thf(fact_1788_real__mult__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
<=> ( A = B ) ) ) ).
thf(fact_1789_real__mult__less__iff1,axiom,
! [X: real,Y: real,Z_1: real] :
( ( ord_less_real @ zero_zero_real @ Z_1 )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z_1 ) @ ( times_times_real @ Y @ Z_1 ) )
<=> ( ord_less_real @ X @ Y ) ) ) ).
thf(fact_1790_real__mult__order,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ Y ) ) ) ) ).
thf(fact_1791_real__mult__less__mono2,axiom,
! [X: real,Y: real,Z_1: real] :
( ( ord_less_real @ zero_zero_real @ Z_1 )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( times_times_real @ Z_1 @ X ) @ ( times_times_real @ Z_1 @ Y ) ) ) ) ).
thf(fact_1792_real__mult__le__cancel__iff2,axiom,
! [X: real,Y: real,Z_1: real] :
( ( ord_less_real @ zero_zero_real @ Z_1 )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z_1 @ X ) @ ( times_times_real @ Z_1 @ Y ) )
<=> ( ord_less_eq_real @ X @ Y ) ) ) ).
thf(fact_1793_real__mult__le__cancel__iff1,axiom,
! [X: real,Y: real,Z_1: real] :
( ( ord_less_real @ zero_zero_real @ Z_1 )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z_1 ) @ ( times_times_real @ Y @ Z_1 ) )
<=> ( ord_less_eq_real @ X @ Y ) ) ) ).
thf(fact_1794_divides__mul__l,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( dvd_dvd_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
thf(fact_1795_divides__mul__r,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
thf(fact_1796_realpow__minus__mult,axiom,
! [X_25: code_code_numeral,N_34: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_34 )
=> ( ( times_1655362735umeral @ ( power_2100829034umeral @ X_25 @ ( minus_minus_nat @ N_34 @ one_one_nat ) ) @ X_25 )
= ( power_2100829034umeral @ X_25 @ N_34 ) ) ) ).
thf(fact_1797_realpow__minus__mult,axiom,
! [X_25: quickcheck_code_int,N_34: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_34 )
=> ( ( times_123202395de_int @ ( power_881366806de_int @ X_25 @ ( minus_minus_nat @ N_34 @ one_one_nat ) ) @ X_25 )
= ( power_881366806de_int @ X_25 @ N_34 ) ) ) ).
thf(fact_1798_realpow__minus__mult,axiom,
! [X_25: rat,N_34: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_34 )
=> ( ( times_times_rat @ ( power_power_rat @ X_25 @ ( minus_minus_nat @ N_34 @ one_one_nat ) ) @ X_25 )
= ( power_power_rat @ X_25 @ N_34 ) ) ) ).
thf(fact_1799_realpow__minus__mult,axiom,
! [X_25: int,N_34: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_34 )
=> ( ( times_times_int @ ( power_power_int @ X_25 @ ( minus_minus_nat @ N_34 @ one_one_nat ) ) @ X_25 )
= ( power_power_int @ X_25 @ N_34 ) ) ) ).
thf(fact_1800_realpow__minus__mult,axiom,
! [X_25: nat,N_34: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_34 )
=> ( ( times_times_nat @ ( power_power_nat @ X_25 @ ( minus_minus_nat @ N_34 @ one_one_nat ) ) @ X_25 )
= ( power_power_nat @ X_25 @ N_34 ) ) ) ).
thf(fact_1801_realpow__minus__mult,axiom,
! [X_25: real,N_34: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_34 )
=> ( ( times_times_real @ ( power_power_real @ X_25 @ ( minus_minus_nat @ N_34 @ one_one_nat ) ) @ X_25 )
= ( power_power_real @ X_25 @ N_34 ) ) ) ).
thf(fact_1802_realpow__minus__mult,axiom,
! [X_25: complex,N_34: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_34 )
=> ( ( times_times_complex @ ( power_power_complex @ X_25 @ ( minus_minus_nat @ N_34 @ one_one_nat ) ) @ X_25 )
= ( power_power_complex @ X_25 @ N_34 ) ) ) ).
thf(fact_1803_divides__exp,axiom,
! [N: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) ) ) ).
thf(fact_1804_mult__eq__if,axiom,
! [N: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( times_times_nat @ M @ N )
= zero_zero_nat ) )
& ( ( M != zero_zero_nat )
=> ( ( times_times_nat @ M @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
thf(fact_1805_power__eq__if,axiom,
! [P_3: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( power_power_nat @ P_3 @ M )
= one_one_nat ) )
& ( ( M != zero_zero_nat )
=> ( ( power_power_nat @ P_3 @ M )
= ( times_times_nat @ P_3 @ ( power_power_nat @ P_3 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).
thf(fact_1806_diff__square,axiom,
! [X: nat,Y: nat] :
( ( minus_minus_nat @ ( power_power_nat @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_nat @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( times_times_nat @ ( plus_plus_nat @ X @ Y ) @ ( minus_minus_nat @ X @ Y ) ) ) ).
thf(fact_1807_add__diff__add,axiom,
! [A_144: int,C_65: int,B_115: int,D_16: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_144 @ C_65 ) @ ( plus_plus_int @ B_115 @ D_16 ) )
= ( plus_plus_int @ ( minus_minus_int @ A_144 @ B_115 ) @ ( minus_minus_int @ C_65 @ D_16 ) ) ) ).
thf(fact_1808_add__diff__add,axiom,
! [A_144: real,C_65: real,B_115: real,D_16: real] :
( ( minus_minus_real @ ( plus_plus_real @ A_144 @ C_65 ) @ ( plus_plus_real @ B_115 @ D_16 ) )
= ( plus_plus_real @ ( minus_minus_real @ A_144 @ B_115 ) @ ( minus_minus_real @ C_65 @ D_16 ) ) ) ).
thf(fact_1809_add__diff__add,axiom,
! [A_144: complex,C_65: complex,B_115: complex,D_16: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A_144 @ C_65 ) @ ( plus_plus_complex @ B_115 @ D_16 ) )
= ( plus_plus_complex @ ( minus_minus_complex @ A_144 @ B_115 ) @ ( minus_minus_complex @ C_65 @ D_16 ) ) ) ).
thf(fact_1810_add__diff__add,axiom,
! [A_144: rat,C_65: rat,B_115: rat,D_16: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A_144 @ C_65 ) @ ( plus_plus_rat @ B_115 @ D_16 ) )
= ( plus_plus_rat @ ( minus_minus_rat @ A_144 @ B_115 ) @ ( minus_minus_rat @ C_65 @ D_16 ) ) ) ).
thf(fact_1811_divides__ge,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( B = zero_zero_nat )
| ( ord_less_eq_nat @ A @ B ) ) ) ).
thf(fact_1812_nat__mult__dvd__cancel__disj_H,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ M @ K_1 ) @ ( times_times_nat @ N @ K_1 ) )
<=> ( ( K_1 = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
thf(fact_1813_nat__mult__eq__one,axiom,
! [N: nat,M: nat] :
( ( ( times_times_nat @ N @ M )
= one_one_nat )
<=> ( ( N = one_one_nat )
& ( M = one_one_nat ) ) ) ).
thf(fact_1814_nat__power__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( power_power_nat @ M @ N )
= zero_zero_nat )
<=> ( ( N != zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
thf(fact_1815_divides__rev,axiom,
! [A: nat,N: nat,B: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( N != zero_zero_nat )
=> ( dvd_dvd_nat @ A @ B ) ) ) ).
thf(fact_1816_divides__exp2,axiom,
! [X: nat,Y: nat,N: nat] :
( ( N != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ Y )
=> ( dvd_dvd_nat @ X @ Y ) ) ) ).
thf(fact_1817_mult__diff__mult,axiom,
! [X_24: int,Y_17: int,A_143: int,B_114: int] :
( ( minus_minus_int @ ( times_times_int @ X_24 @ Y_17 ) @ ( times_times_int @ A_143 @ B_114 ) )
= ( plus_plus_int @ ( times_times_int @ X_24 @ ( minus_minus_int @ Y_17 @ B_114 ) ) @ ( times_times_int @ ( minus_minus_int @ X_24 @ A_143 ) @ B_114 ) ) ) ).
thf(fact_1818_mult__diff__mult,axiom,
! [X_24: real,Y_17: real,A_143: real,B_114: real] :
( ( minus_minus_real @ ( times_times_real @ X_24 @ Y_17 ) @ ( times_times_real @ A_143 @ B_114 ) )
= ( plus_plus_real @ ( times_times_real @ X_24 @ ( minus_minus_real @ Y_17 @ B_114 ) ) @ ( times_times_real @ ( minus_minus_real @ X_24 @ A_143 ) @ B_114 ) ) ) ).
thf(fact_1819_mult__diff__mult,axiom,
! [X_24: complex,Y_17: complex,A_143: complex,B_114: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X_24 @ Y_17 ) @ ( times_times_complex @ A_143 @ B_114 ) )
= ( plus_plus_complex @ ( times_times_complex @ X_24 @ ( minus_minus_complex @ Y_17 @ B_114 ) ) @ ( times_times_complex @ ( minus_minus_complex @ X_24 @ A_143 ) @ B_114 ) ) ) ).
thf(fact_1820_mult__diff__mult,axiom,
! [X_24: rat,Y_17: rat,A_143: rat,B_114: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X_24 @ Y_17 ) @ ( times_times_rat @ A_143 @ B_114 ) )
= ( plus_plus_rat @ ( times_times_rat @ X_24 @ ( minus_minus_rat @ Y_17 @ B_114 ) ) @ ( times_times_rat @ ( minus_minus_rat @ X_24 @ A_143 ) @ B_114 ) ) ) ).
thf(fact_1821_exp__eq__1,axiom,
! [X: nat,N: nat] :
( ( ( power_power_nat @ X @ N )
= one_one_nat )
<=> ( ( X = one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
thf(fact_1822_real__squared__diff__one__factored,axiom,
! [X_23: int] :
( ( minus_minus_int @ ( times_times_int @ X_23 @ X_23 ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X_23 @ one_one_int ) @ ( minus_minus_int @ X_23 @ one_one_int ) ) ) ).
thf(fact_1823_real__squared__diff__one__factored,axiom,
! [X_23: real] :
( ( minus_minus_real @ ( times_times_real @ X_23 @ X_23 ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X_23 @ one_one_real ) @ ( minus_minus_real @ X_23 @ one_one_real ) ) ) ).
thf(fact_1824_real__squared__diff__one__factored,axiom,
! [X_23: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X_23 @ X_23 ) @ one_one_complex )
= ( times_times_complex @ ( plus_plus_complex @ X_23 @ one_one_complex ) @ ( minus_minus_complex @ X_23 @ one_one_complex ) ) ) ).
thf(fact_1825_real__squared__diff__one__factored,axiom,
! [X_23: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X_23 @ X_23 ) @ one_one_rat )
= ( times_times_rat @ ( plus_plus_rat @ X_23 @ one_one_rat ) @ ( minus_minus_rat @ X_23 @ one_one_rat ) ) ) ).
thf(fact_1826_divides__div__not,axiom,
! [X: nat,Q: nat,N: nat,R_1: nat] :
( ( X
= ( plus_plus_nat @ ( times_times_nat @ Q @ N ) @ R_1 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ R_1 )
=> ( ( ord_less_nat @ R_1 @ N )
=> ~ ( dvd_dvd_nat @ N @ X ) ) ) ) ).
thf(fact_1827_realpow__pos__nth__unique,axiom,
! [A: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X_1: real] :
( ( ord_less_real @ zero_zero_real @ X_1 )
& ( ( power_power_real @ X_1 @ N )
= A )
& ! [Y_1: real] :
( ( ( ord_less_real @ zero_zero_real @ Y_1 )
& ( ( power_power_real @ Y_1 @ N )
= A ) )
=> ( Y_1 = X_1 ) ) ) ) ) ).
thf(fact_1828_realpow__pos__nth,axiom,
! [A: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ N )
= A ) ) ) ) ).
thf(fact_1829_realpow__num__eq__if,axiom,
! [M_22: int,N_33: nat] :
( ( ( N_33 = zero_zero_nat )
=> ( ( power_power_int @ M_22 @ N_33 )
= one_one_int ) )
& ( ( N_33 != zero_zero_nat )
=> ( ( power_power_int @ M_22 @ N_33 )
= ( times_times_int @ M_22 @ ( power_power_int @ M_22 @ ( minus_minus_nat @ N_33 @ one_one_nat ) ) ) ) ) ) ).
thf(fact_1830_realpow__num__eq__if,axiom,
! [M_22: nat,N_33: nat] :
( ( ( N_33 = zero_zero_nat )
=> ( ( power_power_nat @ M_22 @ N_33 )
= one_one_nat ) )
& ( ( N_33 != zero_zero_nat )
=> ( ( power_power_nat @ M_22 @ N_33 )
= ( times_times_nat @ M_22 @ ( power_power_nat @ M_22 @ ( minus_minus_nat @ N_33 @ one_one_nat ) ) ) ) ) ) ).
thf(fact_1831_realpow__num__eq__if,axiom,
! [M_22: real,N_33: nat] :
( ( ( N_33 = zero_zero_nat )
=> ( ( power_power_real @ M_22 @ N_33 )
= one_one_real ) )
& ( ( N_33 != zero_zero_nat )
=> ( ( power_power_real @ M_22 @ N_33 )
= ( times_times_real @ M_22 @ ( power_power_real @ M_22 @ ( minus_minus_nat @ N_33 @ one_one_nat ) ) ) ) ) ) ).
thf(fact_1832_realpow__num__eq__if,axiom,
! [M_22: code_code_numeral,N_33: nat] :
( ( ( N_33 = zero_zero_nat )
=> ( ( power_2100829034umeral @ M_22 @ N_33 )
= one_on1645066479umeral ) )
& ( ( N_33 != zero_zero_nat )
=> ( ( power_2100829034umeral @ M_22 @ N_33 )
= ( times_1655362735umeral @ M_22 @ ( power_2100829034umeral @ M_22 @ ( minus_minus_nat @ N_33 @ one_one_nat ) ) ) ) ) ) ).
thf(fact_1833_realpow__num__eq__if,axiom,
! [M_22: complex,N_33: nat] :
( ( ( N_33 = zero_zero_nat )
=> ( ( power_power_complex @ M_22 @ N_33 )
= one_one_complex ) )
& ( ( N_33 != zero_zero_nat )
=> ( ( power_power_complex @ M_22 @ N_33 )
= ( times_times_complex @ M_22 @ ( power_power_complex @ M_22 @ ( minus_minus_nat @ N_33 @ one_one_nat ) ) ) ) ) ) ).
thf(fact_1834_realpow__num__eq__if,axiom,
! [M_22: quickcheck_code_int,N_33: nat] :
( ( ( N_33 = zero_zero_nat )
=> ( ( power_881366806de_int @ M_22 @ N_33 )
= one_on1684967323de_int ) )
& ( ( N_33 != zero_zero_nat )
=> ( ( power_881366806de_int @ M_22 @ N_33 )
= ( times_123202395de_int @ M_22 @ ( power_881366806de_int @ M_22 @ ( minus_minus_nat @ N_33 @ one_one_nat ) ) ) ) ) ) ).
thf(fact_1835_realpow__num__eq__if,axiom,
! [M_22: rat,N_33: nat] :
( ( ( N_33 = zero_zero_nat )
=> ( ( power_power_rat @ M_22 @ N_33 )
= one_one_rat ) )
& ( ( N_33 != zero_zero_nat )
=> ( ( power_power_rat @ M_22 @ N_33 )
= ( times_times_rat @ M_22 @ ( power_power_rat @ M_22 @ ( minus_minus_nat @ N_33 @ one_one_nat ) ) ) ) ) ) ).
thf(fact_1836__096sum2sq_A_Is_M_A1_J_A_061_A_I4_A_K_Am_A_L_A1_J_A_K_At_096,axiom,
( ( twoSqu2072599593sum2sq @ ( product_Pair_int_int @ s @ one_one_int ) )
= ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ t ) ) ).
thf(fact_1837_norR__mem__unique__aux,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ B @ one_one_int ) )
=> ( ord_less_int @ A @ B ) ) ).
thf(fact_1838_nat__less__add__iff1,axiom,
! [U: nat,M: nat,N: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
thf(fact_1839_nat__less__add__iff2,axiom,
! [U: nat,M: nat,N: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
thf(fact_1840_nat__mult__le__cancel1,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_1841_nat__mult__dvd__cancel1,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
<=> ( dvd_dvd_nat @ M @ N ) ) ) ).
thf(fact_1842_diff__commute,axiom,
! [I: nat,J: nat,K_1: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K_1 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K_1 ) @ J ) ) ).
thf(fact_1843_mult__sum2sq,axiom,
! [A: int,B: int,P_3: int,Q: int] :
( ( times_times_int @ ( twoSqu2072599593sum2sq @ ( product_Pair_int_int @ A @ B ) ) @ ( twoSqu2072599593sum2sq @ ( product_Pair_int_int @ P_3 @ Q ) ) )
= ( twoSqu2072599593sum2sq @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A @ P_3 ) @ ( times_times_int @ B @ Q ) ) @ ( minus_minus_int @ ( times_times_int @ A @ Q ) @ ( times_times_int @ B @ P_3 ) ) ) ) ) ).
thf(fact_1844_nat__mult__eq__cancel__disj,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K_1 @ M )
= ( times_times_nat @ K_1 @ N ) )
<=> ( ( K_1 = zero_zero_nat )
| ( M = N ) ) ) ).
thf(fact_1845_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K_1: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K_1 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K_1 ) ) ).
thf(fact_1846_nat__mult__eq__cancel1,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ( ( times_times_nat @ K_1 @ M )
= ( times_times_nat @ K_1 @ N ) )
<=> ( M = N ) ) ) ).
thf(fact_1847_nat__mult__less__cancel1,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ( ord_less_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_1848_nat__mult__dvd__cancel__disj,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
<=> ( ( K_1 = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
thf(fact_1849_nat__eq__add__iff2,axiom,
! [U: nat,M: nat,N: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
thf(fact_1850_nat__diff__add__eq2,axiom,
! [U: nat,M: nat,N: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
thf(fact_1851_nat__le__add__iff2,axiom,
! [U: nat,M: nat,N: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
thf(fact_1852_nat__eq__add__iff1,axiom,
! [U: nat,M: nat,N: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
thf(fact_1853_nat__diff__add__eq1,axiom,
! [U: nat,M: nat,N: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
thf(fact_1854_nat__le__add__iff1,axiom,
! [U: nat,M: nat,N: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
thf(fact_1855_is__sum2sq__def,axiom,
! [X: int] :
( ( twoSqu1152398899sum2sq @ X )
<=> ? [A_2: int,B_4: int] :
( ( twoSqu2072599593sum2sq @ ( product_Pair_int_int @ A_2 @ B_4 ) )
= X ) ) ).
thf(fact_1856_Wilson__Russ,axiom,
! [P_3: int] :
( ( zprime @ P_3 )
=> ( zcong @ ( zfact @ ( minus_minus_int @ P_3 @ one_one_int ) ) @ ( number_number_of_int @ min ) @ P_3 ) ) ).
thf(fact_1857_add__pos__nonneg,axiom,
! [B_113: code_code_numeral,A_142: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_142 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_113 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ ( plus_p1627245867umeral @ A_142 @ B_113 ) ) ) ) ).
thf(fact_1858_add__pos__nonneg,axiom,
! [B_113: int,A_142: int] :
( ( ord_less_int @ zero_zero_int @ A_142 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_113 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_142 @ B_113 ) ) ) ) ).
thf(fact_1859_add__pos__nonneg,axiom,
! [B_113: nat,A_142: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_142 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_113 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A_142 @ B_113 ) ) ) ) ).
thf(fact_1860_add__pos__nonneg,axiom,
! [B_113: real,A_142: real] :
( ( ord_less_real @ zero_zero_real @ A_142 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_113 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A_142 @ B_113 ) ) ) ) ).
thf(fact_1861_add__pos__nonneg,axiom,
! [B_113: quickcheck_code_int,A_142: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_142 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_113 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ ( plus_p1446045655de_int @ A_142 @ B_113 ) ) ) ) ).
thf(fact_1862_add__pos__nonneg,axiom,
! [B_113: rat,A_142: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_142 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_113 )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A_142 @ B_113 ) ) ) ) ).
thf(fact_1863_add__nonneg__pos,axiom,
! [B_112: code_code_numeral,A_141: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_141 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ B_112 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ ( plus_p1627245867umeral @ A_141 @ B_112 ) ) ) ) ).
thf(fact_1864_add__nonneg__pos,axiom,
! [B_112: int,A_141: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_141 )
=> ( ( ord_less_int @ zero_zero_int @ B_112 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_141 @ B_112 ) ) ) ) ).
thf(fact_1865_add__nonneg__pos,axiom,
! [B_112: nat,A_141: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_141 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_112 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A_141 @ B_112 ) ) ) ) ).
thf(fact_1866_add__nonneg__pos,axiom,
! [B_112: real,A_141: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_141 )
=> ( ( ord_less_real @ zero_zero_real @ B_112 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A_141 @ B_112 ) ) ) ) ).
thf(fact_1867_add__nonneg__pos,axiom,
! [B_112: quickcheck_code_int,A_141: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_141 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ B_112 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ ( plus_p1446045655de_int @ A_141 @ B_112 ) ) ) ) ).
thf(fact_1868_add__nonneg__pos,axiom,
! [B_112: rat,A_141: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_141 )
=> ( ( ord_less_rat @ zero_zero_rat @ B_112 )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A_141 @ B_112 ) ) ) ) ).
thf(fact_1869_add__strict__increasing,axiom,
! [B_111: code_code_numeral,C_64: code_code_numeral,A_140: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_140 )
=> ( ( ord_le565307924umeral @ B_111 @ C_64 )
=> ( ord_le1304079648umeral @ B_111 @ ( plus_p1627245867umeral @ A_140 @ C_64 ) ) ) ) ).
thf(fact_1870_add__strict__increasing,axiom,
! [B_111: int,C_64: int,A_140: int] :
( ( ord_less_int @ zero_zero_int @ A_140 )
=> ( ( ord_less_eq_int @ B_111 @ C_64 )
=> ( ord_less_int @ B_111 @ ( plus_plus_int @ A_140 @ C_64 ) ) ) ) ).
thf(fact_1871_add__strict__increasing,axiom,
! [B_111: nat,C_64: nat,A_140: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_140 )
=> ( ( ord_less_eq_nat @ B_111 @ C_64 )
=> ( ord_less_nat @ B_111 @ ( plus_plus_nat @ A_140 @ C_64 ) ) ) ) ).
thf(fact_1872_add__strict__increasing,axiom,
! [B_111: real,C_64: real,A_140: real] :
( ( ord_less_real @ zero_zero_real @ A_140 )
=> ( ( ord_less_eq_real @ B_111 @ C_64 )
=> ( ord_less_real @ B_111 @ ( plus_plus_real @ A_140 @ C_64 ) ) ) ) ).
thf(fact_1873_add__strict__increasing,axiom,
! [B_111: quickcheck_code_int,C_64: quickcheck_code_int,A_140: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_140 )
=> ( ( ord_le258702272de_int @ B_111 @ C_64 )
=> ( ord_le1860547276de_int @ B_111 @ ( plus_p1446045655de_int @ A_140 @ C_64 ) ) ) ) ).
thf(fact_1874_add__strict__increasing,axiom,
! [B_111: rat,C_64: rat,A_140: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_140 )
=> ( ( ord_less_eq_rat @ B_111 @ C_64 )
=> ( ord_less_rat @ B_111 @ ( plus_plus_rat @ A_140 @ C_64 ) ) ) ) ).
thf(fact_1875_add__strict__increasing2,axiom,
! [B_110: code_code_numeral,C_63: code_code_numeral,A_139: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_139 )
=> ( ( ord_le1304079648umeral @ B_110 @ C_63 )
=> ( ord_le1304079648umeral @ B_110 @ ( plus_p1627245867umeral @ A_139 @ C_63 ) ) ) ) ).
thf(fact_1876_add__strict__increasing2,axiom,
! [B_110: int,C_63: int,A_139: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_139 )
=> ( ( ord_less_int @ B_110 @ C_63 )
=> ( ord_less_int @ B_110 @ ( plus_plus_int @ A_139 @ C_63 ) ) ) ) ).
thf(fact_1877_add__strict__increasing2,axiom,
! [B_110: nat,C_63: nat,A_139: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_139 )
=> ( ( ord_less_nat @ B_110 @ C_63 )
=> ( ord_less_nat @ B_110 @ ( plus_plus_nat @ A_139 @ C_63 ) ) ) ) ).
thf(fact_1878_add__strict__increasing2,axiom,
! [B_110: real,C_63: real,A_139: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_139 )
=> ( ( ord_less_real @ B_110 @ C_63 )
=> ( ord_less_real @ B_110 @ ( plus_plus_real @ A_139 @ C_63 ) ) ) ) ).
thf(fact_1879_add__strict__increasing2,axiom,
! [B_110: quickcheck_code_int,C_63: quickcheck_code_int,A_139: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_139 )
=> ( ( ord_le1860547276de_int @ B_110 @ C_63 )
=> ( ord_le1860547276de_int @ B_110 @ ( plus_p1446045655de_int @ A_139 @ C_63 ) ) ) ) ).
thf(fact_1880_add__strict__increasing2,axiom,
! [B_110: rat,C_63: rat,A_139: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_139 )
=> ( ( ord_less_rat @ B_110 @ C_63 )
=> ( ord_less_rat @ B_110 @ ( plus_plus_rat @ A_139 @ C_63 ) ) ) ) ).
thf(fact_1881_add__neg__nonpos,axiom,
! [B_109: code_code_numeral,A_138: code_code_numeral] :
( ( ord_le1304079648umeral @ A_138 @ zero_z126310315umeral )
=> ( ( ord_le565307924umeral @ B_109 @ zero_z126310315umeral )
=> ( ord_le1304079648umeral @ ( plus_p1627245867umeral @ A_138 @ B_109 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1882_add__neg__nonpos,axiom,
! [B_109: int,A_138: int] :
( ( ord_less_int @ A_138 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B_109 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A_138 @ B_109 ) @ zero_zero_int ) ) ) ).
thf(fact_1883_add__neg__nonpos,axiom,
! [B_109: nat,A_138: nat] :
( ( ord_less_nat @ A_138 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B_109 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A_138 @ B_109 ) @ zero_zero_nat ) ) ) ).
thf(fact_1884_add__neg__nonpos,axiom,
! [B_109: real,A_138: real] :
( ( ord_less_real @ A_138 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B_109 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A_138 @ B_109 ) @ zero_zero_real ) ) ) ).
thf(fact_1885_add__neg__nonpos,axiom,
! [B_109: quickcheck_code_int,A_138: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_138 @ zero_z891286103de_int )
=> ( ( ord_le258702272de_int @ B_109 @ zero_z891286103de_int )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_138 @ B_109 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1886_add__neg__nonpos,axiom,
! [B_109: rat,A_138: rat] :
( ( ord_less_rat @ A_138 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B_109 @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A_138 @ B_109 ) @ zero_zero_rat ) ) ) ).
thf(fact_1887_add__nonpos__neg,axiom,
! [B_108: code_code_numeral,A_137: code_code_numeral] :
( ( ord_le565307924umeral @ A_137 @ zero_z126310315umeral )
=> ( ( ord_le1304079648umeral @ B_108 @ zero_z126310315umeral )
=> ( ord_le1304079648umeral @ ( plus_p1627245867umeral @ A_137 @ B_108 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1888_add__nonpos__neg,axiom,
! [B_108: int,A_137: int] :
( ( ord_less_eq_int @ A_137 @ zero_zero_int )
=> ( ( ord_less_int @ B_108 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A_137 @ B_108 ) @ zero_zero_int ) ) ) ).
thf(fact_1889_add__nonpos__neg,axiom,
! [B_108: nat,A_137: nat] :
( ( ord_less_eq_nat @ A_137 @ zero_zero_nat )
=> ( ( ord_less_nat @ B_108 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A_137 @ B_108 ) @ zero_zero_nat ) ) ) ).
thf(fact_1890_add__nonpos__neg,axiom,
! [B_108: real,A_137: real] :
( ( ord_less_eq_real @ A_137 @ zero_zero_real )
=> ( ( ord_less_real @ B_108 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A_137 @ B_108 ) @ zero_zero_real ) ) ) ).
thf(fact_1891_add__nonpos__neg,axiom,
! [B_108: quickcheck_code_int,A_137: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_137 @ zero_z891286103de_int )
=> ( ( ord_le1860547276de_int @ B_108 @ zero_z891286103de_int )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_137 @ B_108 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1892_add__nonpos__neg,axiom,
! [B_108: rat,A_137: rat] :
( ( ord_less_eq_rat @ A_137 @ zero_zero_rat )
=> ( ( ord_less_rat @ B_108 @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A_137 @ B_108 ) @ zero_zero_rat ) ) ) ).
thf(fact_1893_inv__inv,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_eq_int @ ( number_number_of_int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P_3 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ P_3 )
=> ( ( inv @ P_3 @ ( inv @ P_3 @ A ) )
= A ) ) ) ) ) ).
thf(fact_1894_zero__reorient,axiom,
! [X_22: int] :
( ( zero_zero_int = X_22 )
<=> ( X_22 = zero_zero_int ) ) ).
thf(fact_1895_zero__reorient,axiom,
! [X_22: nat] :
( ( zero_zero_nat = X_22 )
<=> ( X_22 = zero_zero_nat ) ) ).
thf(fact_1896_zero__reorient,axiom,
! [X_22: real] :
( ( zero_zero_real = X_22 )
<=> ( X_22 = zero_zero_real ) ) ).
thf(fact_1897_zero__reorient,axiom,
! [X_22: code_code_numeral] :
( ( zero_z126310315umeral = X_22 )
<=> ( X_22 = zero_z126310315umeral ) ) ).
thf(fact_1898_zero__reorient,axiom,
! [X_22: complex] :
( ( zero_zero_complex = X_22 )
<=> ( X_22 = zero_zero_complex ) ) ).
thf(fact_1899_zero__reorient,axiom,
! [X_22: quickcheck_code_int] :
( ( zero_z891286103de_int = X_22 )
<=> ( X_22 = zero_z891286103de_int ) ) ).
thf(fact_1900_zero__reorient,axiom,
! [X_22: rat] :
( ( zero_zero_rat = X_22 )
<=> ( X_22 = zero_zero_rat ) ) ).
thf(fact_1901_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_136: code_code_numeral,B_107: code_code_numeral,C_62: code_code_numeral] :
( ( times_1655362735umeral @ ( times_1655362735umeral @ A_136 @ B_107 ) @ C_62 )
= ( times_1655362735umeral @ A_136 @ ( times_1655362735umeral @ B_107 @ C_62 ) ) ) ).
thf(fact_1902_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_136: int,B_107: int,C_62: int] :
( ( times_times_int @ ( times_times_int @ A_136 @ B_107 ) @ C_62 )
= ( times_times_int @ A_136 @ ( times_times_int @ B_107 @ C_62 ) ) ) ).
thf(fact_1903_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_136: nat,B_107: nat,C_62: nat] :
( ( times_times_nat @ ( times_times_nat @ A_136 @ B_107 ) @ C_62 )
= ( times_times_nat @ A_136 @ ( times_times_nat @ B_107 @ C_62 ) ) ) ).
thf(fact_1904_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_136: real,B_107: real,C_62: real] :
( ( times_times_real @ ( times_times_real @ A_136 @ B_107 ) @ C_62 )
= ( times_times_real @ A_136 @ ( times_times_real @ B_107 @ C_62 ) ) ) ).
thf(fact_1905_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_136: complex,B_107: complex,C_62: complex] :
( ( times_times_complex @ ( times_times_complex @ A_136 @ B_107 ) @ C_62 )
= ( times_times_complex @ A_136 @ ( times_times_complex @ B_107 @ C_62 ) ) ) ).
thf(fact_1906_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_136: quickcheck_code_int,B_107: quickcheck_code_int,C_62: quickcheck_code_int] :
( ( times_123202395de_int @ ( times_123202395de_int @ A_136 @ B_107 ) @ C_62 )
= ( times_123202395de_int @ A_136 @ ( times_123202395de_int @ B_107 @ C_62 ) ) ) ).
thf(fact_1907_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_136: rat,B_107: rat,C_62: rat] :
( ( times_times_rat @ ( times_times_rat @ A_136 @ B_107 ) @ C_62 )
= ( times_times_rat @ A_136 @ ( times_times_rat @ B_107 @ C_62 ) ) ) ).
thf(fact_1908_add__right__imp__eq,axiom,
! [B_106: code_code_numeral,A_135: code_code_numeral,C_61: code_code_numeral] :
( ( ( plus_p1627245867umeral @ B_106 @ A_135 )
= ( plus_p1627245867umeral @ C_61 @ A_135 ) )
=> ( B_106 = C_61 ) ) ).
thf(fact_1909_add__right__imp__eq,axiom,
! [B_106: int,A_135: int,C_61: int] :
( ( ( plus_plus_int @ B_106 @ A_135 )
= ( plus_plus_int @ C_61 @ A_135 ) )
=> ( B_106 = C_61 ) ) ).
thf(fact_1910_add__right__imp__eq,axiom,
! [B_106: nat,A_135: nat,C_61: nat] :
( ( ( plus_plus_nat @ B_106 @ A_135 )
= ( plus_plus_nat @ C_61 @ A_135 ) )
=> ( B_106 = C_61 ) ) ).
thf(fact_1911_add__right__imp__eq,axiom,
! [B_106: real,A_135: real,C_61: real] :
( ( ( plus_plus_real @ B_106 @ A_135 )
= ( plus_plus_real @ C_61 @ A_135 ) )
=> ( B_106 = C_61 ) ) ).
thf(fact_1912_add__right__imp__eq,axiom,
! [B_106: complex,A_135: complex,C_61: complex] :
( ( ( plus_plus_complex @ B_106 @ A_135 )
= ( plus_plus_complex @ C_61 @ A_135 ) )
=> ( B_106 = C_61 ) ) ).
thf(fact_1913_add__right__imp__eq,axiom,
! [B_106: quickcheck_code_int,A_135: quickcheck_code_int,C_61: quickcheck_code_int] :
( ( ( plus_p1446045655de_int @ B_106 @ A_135 )
= ( plus_p1446045655de_int @ C_61 @ A_135 ) )
=> ( B_106 = C_61 ) ) ).
thf(fact_1914_add__right__imp__eq,axiom,
! [B_106: rat,A_135: rat,C_61: rat] :
( ( ( plus_plus_rat @ B_106 @ A_135 )
= ( plus_plus_rat @ C_61 @ A_135 ) )
=> ( B_106 = C_61 ) ) ).
thf(fact_1915_add__imp__eq,axiom,
! [A_134: code_code_numeral,B_105: code_code_numeral,C_60: code_code_numeral] :
( ( ( plus_p1627245867umeral @ A_134 @ B_105 )
= ( plus_p1627245867umeral @ A_134 @ C_60 ) )
=> ( B_105 = C_60 ) ) ).
thf(fact_1916_add__imp__eq,axiom,
! [A_134: int,B_105: int,C_60: int] :
( ( ( plus_plus_int @ A_134 @ B_105 )
= ( plus_plus_int @ A_134 @ C_60 ) )
=> ( B_105 = C_60 ) ) ).
thf(fact_1917_add__imp__eq,axiom,
! [A_134: nat,B_105: nat,C_60: nat] :
( ( ( plus_plus_nat @ A_134 @ B_105 )
= ( plus_plus_nat @ A_134 @ C_60 ) )
=> ( B_105 = C_60 ) ) ).
thf(fact_1918_add__imp__eq,axiom,
! [A_134: real,B_105: real,C_60: real] :
( ( ( plus_plus_real @ A_134 @ B_105 )
= ( plus_plus_real @ A_134 @ C_60 ) )
=> ( B_105 = C_60 ) ) ).
thf(fact_1919_add__imp__eq,axiom,
! [A_134: complex,B_105: complex,C_60: complex] :
( ( ( plus_plus_complex @ A_134 @ B_105 )
= ( plus_plus_complex @ A_134 @ C_60 ) )
=> ( B_105 = C_60 ) ) ).
thf(fact_1920_add__imp__eq,axiom,
! [A_134: quickcheck_code_int,B_105: quickcheck_code_int,C_60: quickcheck_code_int] :
( ( ( plus_p1446045655de_int @ A_134 @ B_105 )
= ( plus_p1446045655de_int @ A_134 @ C_60 ) )
=> ( B_105 = C_60 ) ) ).
thf(fact_1921_add__imp__eq,axiom,
! [A_134: rat,B_105: rat,C_60: rat] :
( ( ( plus_plus_rat @ A_134 @ B_105 )
= ( plus_plus_rat @ A_134 @ C_60 ) )
=> ( B_105 = C_60 ) ) ).
thf(fact_1922_add__left__imp__eq,axiom,
! [A_133: code_code_numeral,B_104: code_code_numeral,C_59: code_code_numeral] :
( ( ( plus_p1627245867umeral @ A_133 @ B_104 )
= ( plus_p1627245867umeral @ A_133 @ C_59 ) )
=> ( B_104 = C_59 ) ) ).
thf(fact_1923_add__left__imp__eq,axiom,
! [A_133: int,B_104: int,C_59: int] :
( ( ( plus_plus_int @ A_133 @ B_104 )
= ( plus_plus_int @ A_133 @ C_59 ) )
=> ( B_104 = C_59 ) ) ).
thf(fact_1924_add__left__imp__eq,axiom,
! [A_133: nat,B_104: nat,C_59: nat] :
( ( ( plus_plus_nat @ A_133 @ B_104 )
= ( plus_plus_nat @ A_133 @ C_59 ) )
=> ( B_104 = C_59 ) ) ).
thf(fact_1925_add__left__imp__eq,axiom,
! [A_133: real,B_104: real,C_59: real] :
( ( ( plus_plus_real @ A_133 @ B_104 )
= ( plus_plus_real @ A_133 @ C_59 ) )
=> ( B_104 = C_59 ) ) ).
thf(fact_1926_add__left__imp__eq,axiom,
! [A_133: complex,B_104: complex,C_59: complex] :
( ( ( plus_plus_complex @ A_133 @ B_104 )
= ( plus_plus_complex @ A_133 @ C_59 ) )
=> ( B_104 = C_59 ) ) ).
thf(fact_1927_add__left__imp__eq,axiom,
! [A_133: quickcheck_code_int,B_104: quickcheck_code_int,C_59: quickcheck_code_int] :
( ( ( plus_p1446045655de_int @ A_133 @ B_104 )
= ( plus_p1446045655de_int @ A_133 @ C_59 ) )
=> ( B_104 = C_59 ) ) ).
thf(fact_1928_add__left__imp__eq,axiom,
! [A_133: rat,B_104: rat,C_59: rat] :
( ( ( plus_plus_rat @ A_133 @ B_104 )
= ( plus_plus_rat @ A_133 @ C_59 ) )
=> ( B_104 = C_59 ) ) ).
thf(fact_1929_add__right__cancel,axiom,
! [B_103: code_code_numeral,A_132: code_code_numeral,C_58: code_code_numeral] :
( ( ( plus_p1627245867umeral @ B_103 @ A_132 )
= ( plus_p1627245867umeral @ C_58 @ A_132 ) )
<=> ( B_103 = C_58 ) ) ).
thf(fact_1930_add__right__cancel,axiom,
! [B_103: int,A_132: int,C_58: int] :
( ( ( plus_plus_int @ B_103 @ A_132 )
= ( plus_plus_int @ C_58 @ A_132 ) )
<=> ( B_103 = C_58 ) ) ).
thf(fact_1931_add__right__cancel,axiom,
! [B_103: nat,A_132: nat,C_58: nat] :
( ( ( plus_plus_nat @ B_103 @ A_132 )
= ( plus_plus_nat @ C_58 @ A_132 ) )
<=> ( B_103 = C_58 ) ) ).
thf(fact_1932_add__right__cancel,axiom,
! [B_103: real,A_132: real,C_58: real] :
( ( ( plus_plus_real @ B_103 @ A_132 )
= ( plus_plus_real @ C_58 @ A_132 ) )
<=> ( B_103 = C_58 ) ) ).
thf(fact_1933_add__right__cancel,axiom,
! [B_103: complex,A_132: complex,C_58: complex] :
( ( ( plus_plus_complex @ B_103 @ A_132 )
= ( plus_plus_complex @ C_58 @ A_132 ) )
<=> ( B_103 = C_58 ) ) ).
thf(fact_1934_add__right__cancel,axiom,
! [B_103: quickcheck_code_int,A_132: quickcheck_code_int,C_58: quickcheck_code_int] :
( ( ( plus_p1446045655de_int @ B_103 @ A_132 )
= ( plus_p1446045655de_int @ C_58 @ A_132 ) )
<=> ( B_103 = C_58 ) ) ).
thf(fact_1935_add__right__cancel,axiom,
! [B_103: rat,A_132: rat,C_58: rat] :
( ( ( plus_plus_rat @ B_103 @ A_132 )
= ( plus_plus_rat @ C_58 @ A_132 ) )
<=> ( B_103 = C_58 ) ) ).
thf(fact_1936_add__left__cancel,axiom,
! [A_131: code_code_numeral,B_102: code_code_numeral,C_57: code_code_numeral] :
( ( ( plus_p1627245867umeral @ A_131 @ B_102 )
= ( plus_p1627245867umeral @ A_131 @ C_57 ) )
<=> ( B_102 = C_57 ) ) ).
thf(fact_1937_add__left__cancel,axiom,
! [A_131: int,B_102: int,C_57: int] :
( ( ( plus_plus_int @ A_131 @ B_102 )
= ( plus_plus_int @ A_131 @ C_57 ) )
<=> ( B_102 = C_57 ) ) ).
thf(fact_1938_add__left__cancel,axiom,
! [A_131: nat,B_102: nat,C_57: nat] :
( ( ( plus_plus_nat @ A_131 @ B_102 )
= ( plus_plus_nat @ A_131 @ C_57 ) )
<=> ( B_102 = C_57 ) ) ).
thf(fact_1939_add__left__cancel,axiom,
! [A_131: real,B_102: real,C_57: real] :
( ( ( plus_plus_real @ A_131 @ B_102 )
= ( plus_plus_real @ A_131 @ C_57 ) )
<=> ( B_102 = C_57 ) ) ).
thf(fact_1940_add__left__cancel,axiom,
! [A_131: complex,B_102: complex,C_57: complex] :
( ( ( plus_plus_complex @ A_131 @ B_102 )
= ( plus_plus_complex @ A_131 @ C_57 ) )
<=> ( B_102 = C_57 ) ) ).
thf(fact_1941_add__left__cancel,axiom,
! [A_131: quickcheck_code_int,B_102: quickcheck_code_int,C_57: quickcheck_code_int] :
( ( ( plus_p1446045655de_int @ A_131 @ B_102 )
= ( plus_p1446045655de_int @ A_131 @ C_57 ) )
<=> ( B_102 = C_57 ) ) ).
thf(fact_1942_add__left__cancel,axiom,
! [A_131: rat,B_102: rat,C_57: rat] :
( ( ( plus_plus_rat @ A_131 @ B_102 )
= ( plus_plus_rat @ A_131 @ C_57 ) )
<=> ( B_102 = C_57 ) ) ).
thf(fact_1943_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_130: code_code_numeral,B_101: code_code_numeral,C_56: code_code_numeral] :
( ( plus_p1627245867umeral @ ( plus_p1627245867umeral @ A_130 @ B_101 ) @ C_56 )
= ( plus_p1627245867umeral @ A_130 @ ( plus_p1627245867umeral @ B_101 @ C_56 ) ) ) ).
thf(fact_1944_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_130: int,B_101: int,C_56: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_130 @ B_101 ) @ C_56 )
= ( plus_plus_int @ A_130 @ ( plus_plus_int @ B_101 @ C_56 ) ) ) ).
thf(fact_1945_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_130: nat,B_101: nat,C_56: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_130 @ B_101 ) @ C_56 )
= ( plus_plus_nat @ A_130 @ ( plus_plus_nat @ B_101 @ C_56 ) ) ) ).
thf(fact_1946_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_130: real,B_101: real,C_56: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_130 @ B_101 ) @ C_56 )
= ( plus_plus_real @ A_130 @ ( plus_plus_real @ B_101 @ C_56 ) ) ) ).
thf(fact_1947_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_130: complex,B_101: complex,C_56: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A_130 @ B_101 ) @ C_56 )
= ( plus_plus_complex @ A_130 @ ( plus_plus_complex @ B_101 @ C_56 ) ) ) ).
thf(fact_1948_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_130: quickcheck_code_int,B_101: quickcheck_code_int,C_56: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_130 @ B_101 ) @ C_56 )
= ( plus_p1446045655de_int @ A_130 @ ( plus_p1446045655de_int @ B_101 @ C_56 ) ) ) ).
thf(fact_1949_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_130: rat,B_101: rat,C_56: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A_130 @ B_101 ) @ C_56 )
= ( plus_plus_rat @ A_130 @ ( plus_plus_rat @ B_101 @ C_56 ) ) ) ).
thf(fact_1950_one__reorient,axiom,
! [X_21: int] :
( ( one_one_int = X_21 )
<=> ( X_21 = one_one_int ) ) ).
thf(fact_1951_one__reorient,axiom,
! [X_21: nat] :
( ( one_one_nat = X_21 )
<=> ( X_21 = one_one_nat ) ) ).
thf(fact_1952_one__reorient,axiom,
! [X_21: real] :
( ( one_one_real = X_21 )
<=> ( X_21 = one_one_real ) ) ).
thf(fact_1953_one__reorient,axiom,
! [X_21: code_code_numeral] :
( ( one_on1645066479umeral = X_21 )
<=> ( X_21 = one_on1645066479umeral ) ) ).
thf(fact_1954_one__reorient,axiom,
! [X_21: complex] :
( ( one_one_complex = X_21 )
<=> ( X_21 = one_one_complex ) ) ).
thf(fact_1955_one__reorient,axiom,
! [X_21: quickcheck_code_int] :
( ( one_on1684967323de_int = X_21 )
<=> ( X_21 = one_on1684967323de_int ) ) ).
thf(fact_1956_one__reorient,axiom,
! [X_21: rat] :
( ( one_one_rat = X_21 )
<=> ( X_21 = one_one_rat ) ) ).
thf(fact_1957_diff__eq__diff__eq,axiom,
! [A_129: int,B_100: int,C_55: int,D_15: int] :
( ( ( minus_minus_int @ A_129 @ B_100 )
= ( minus_minus_int @ C_55 @ D_15 ) )
=> ( ( A_129 = B_100 )
<=> ( C_55 = D_15 ) ) ) ).
thf(fact_1958_diff__eq__diff__eq,axiom,
! [A_129: real,B_100: real,C_55: real,D_15: real] :
( ( ( minus_minus_real @ A_129 @ B_100 )
= ( minus_minus_real @ C_55 @ D_15 ) )
=> ( ( A_129 = B_100 )
<=> ( C_55 = D_15 ) ) ) ).
thf(fact_1959_diff__eq__diff__eq,axiom,
! [A_129: complex,B_100: complex,C_55: complex,D_15: complex] :
( ( ( minus_minus_complex @ A_129 @ B_100 )
= ( minus_minus_complex @ C_55 @ D_15 ) )
=> ( ( A_129 = B_100 )
<=> ( C_55 = D_15 ) ) ) ).
thf(fact_1960_diff__eq__diff__eq,axiom,
! [A_129: rat,B_100: rat,C_55: rat,D_15: rat] :
( ( ( minus_minus_rat @ A_129 @ B_100 )
= ( minus_minus_rat @ C_55 @ D_15 ) )
=> ( ( A_129 = B_100 )
<=> ( C_55 = D_15 ) ) ) ).
thf(fact_1961_add_Ocomm__neutral,axiom,
! [A_128: code_code_numeral] :
( ( plus_p1627245867umeral @ A_128 @ zero_z126310315umeral )
= A_128 ) ).
thf(fact_1962_add_Ocomm__neutral,axiom,
! [A_128: int] :
( ( plus_plus_int @ A_128 @ zero_zero_int )
= A_128 ) ).
thf(fact_1963_add_Ocomm__neutral,axiom,
! [A_128: nat] :
( ( plus_plus_nat @ A_128 @ zero_zero_nat )
= A_128 ) ).
thf(fact_1964_add_Ocomm__neutral,axiom,
! [A_128: real] :
( ( plus_plus_real @ A_128 @ zero_zero_real )
= A_128 ) ).
thf(fact_1965_add_Ocomm__neutral,axiom,
! [A_128: complex] :
( ( plus_plus_complex @ A_128 @ zero_zero_complex )
= A_128 ) ).
thf(fact_1966_add_Ocomm__neutral,axiom,
! [A_128: quickcheck_code_int] :
( ( plus_p1446045655de_int @ A_128 @ zero_z891286103de_int )
= A_128 ) ).
thf(fact_1967_add_Ocomm__neutral,axiom,
! [A_128: rat] :
( ( plus_plus_rat @ A_128 @ zero_zero_rat )
= A_128 ) ).
thf(fact_1968_add__0__right,axiom,
! [A_127: code_code_numeral] :
( ( plus_p1627245867umeral @ A_127 @ zero_z126310315umeral )
= A_127 ) ).
thf(fact_1969_add__0__right,axiom,
! [A_127: int] :
( ( plus_plus_int @ A_127 @ zero_zero_int )
= A_127 ) ).
thf(fact_1970_add__0__right,axiom,
! [A_127: nat] :
( ( plus_plus_nat @ A_127 @ zero_zero_nat )
= A_127 ) ).
thf(fact_1971_add__0__right,axiom,
! [A_127: real] :
( ( plus_plus_real @ A_127 @ zero_zero_real )
= A_127 ) ).
thf(fact_1972_add__0__right,axiom,
! [A_127: complex] :
( ( plus_plus_complex @ A_127 @ zero_zero_complex )
= A_127 ) ).
thf(fact_1973_add__0__right,axiom,
! [A_127: quickcheck_code_int] :
( ( plus_p1446045655de_int @ A_127 @ zero_z891286103de_int )
= A_127 ) ).
thf(fact_1974_add__0__right,axiom,
! [A_127: rat] :
( ( plus_plus_rat @ A_127 @ zero_zero_rat )
= A_127 ) ).
thf(fact_1975_double__zero__sym,axiom,
! [A_126: int] :
( ( zero_zero_int
= ( plus_plus_int @ A_126 @ A_126 ) )
<=> ( A_126 = zero_zero_int ) ) ).
thf(fact_1976_double__zero__sym,axiom,
! [A_126: real] :
( ( zero_zero_real
= ( plus_plus_real @ A_126 @ A_126 ) )
<=> ( A_126 = zero_zero_real ) ) ).
thf(fact_1977_double__zero__sym,axiom,
! [A_126: rat] :
( ( zero_zero_rat
= ( plus_plus_rat @ A_126 @ A_126 ) )
<=> ( A_126 = zero_zero_rat ) ) ).
thf(fact_1978_add__0,axiom,
! [A_125: code_code_numeral] :
( ( plus_p1627245867umeral @ zero_z126310315umeral @ A_125 )
= A_125 ) ).
thf(fact_1979_add__0,axiom,
! [A_125: int] :
( ( plus_plus_int @ zero_zero_int @ A_125 )
= A_125 ) ).
thf(fact_1980_add__0,axiom,
! [A_125: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_125 )
= A_125 ) ).
thf(fact_1981_add__0,axiom,
! [A_125: real] :
( ( plus_plus_real @ zero_zero_real @ A_125 )
= A_125 ) ).
thf(fact_1982_add__0,axiom,
! [A_125: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A_125 )
= A_125 ) ).
thf(fact_1983_add__0,axiom,
! [A_125: quickcheck_code_int] :
( ( plus_p1446045655de_int @ zero_z891286103de_int @ A_125 )
= A_125 ) ).
thf(fact_1984_add__0,axiom,
! [A_125: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A_125 )
= A_125 ) ).
thf(fact_1985_add__0__left,axiom,
! [A_124: code_code_numeral] :
( ( plus_p1627245867umeral @ zero_z126310315umeral @ A_124 )
= A_124 ) ).
thf(fact_1986_add__0__left,axiom,
! [A_124: int] :
( ( plus_plus_int @ zero_zero_int @ A_124 )
= A_124 ) ).
thf(fact_1987_add__0__left,axiom,
! [A_124: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_124 )
= A_124 ) ).
thf(fact_1988_add__0__left,axiom,
! [A_124: real] :
( ( plus_plus_real @ zero_zero_real @ A_124 )
= A_124 ) ).
thf(fact_1989_add__0__left,axiom,
! [A_124: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A_124 )
= A_124 ) ).
thf(fact_1990_add__0__left,axiom,
! [A_124: quickcheck_code_int] :
( ( plus_p1446045655de_int @ zero_z891286103de_int @ A_124 )
= A_124 ) ).
thf(fact_1991_add__0__left,axiom,
! [A_124: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A_124 )
= A_124 ) ).
thf(fact_1992_add__le__imp__le__left,axiom,
! [C_54: code_code_numeral,A_123: code_code_numeral,B_99: code_code_numeral] :
( ( ord_le565307924umeral @ ( plus_p1627245867umeral @ C_54 @ A_123 ) @ ( plus_p1627245867umeral @ C_54 @ B_99 ) )
=> ( ord_le565307924umeral @ A_123 @ B_99 ) ) ).
thf(fact_1993_add__le__imp__le__left,axiom,
! [C_54: int,A_123: int,B_99: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_54 @ A_123 ) @ ( plus_plus_int @ C_54 @ B_99 ) )
=> ( ord_less_eq_int @ A_123 @ B_99 ) ) ).
thf(fact_1994_add__le__imp__le__left,axiom,
! [C_54: nat,A_123: nat,B_99: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_54 @ A_123 ) @ ( plus_plus_nat @ C_54 @ B_99 ) )
=> ( ord_less_eq_nat @ A_123 @ B_99 ) ) ).
thf(fact_1995_add__le__imp__le__left,axiom,
! [C_54: real,A_123: real,B_99: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C_54 @ A_123 ) @ ( plus_plus_real @ C_54 @ B_99 ) )
=> ( ord_less_eq_real @ A_123 @ B_99 ) ) ).
thf(fact_1996_add__le__imp__le__left,axiom,
! [C_54: quickcheck_code_int,A_123: quickcheck_code_int,B_99: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( plus_p1446045655de_int @ C_54 @ A_123 ) @ ( plus_p1446045655de_int @ C_54 @ B_99 ) )
=> ( ord_le258702272de_int @ A_123 @ B_99 ) ) ).
thf(fact_1997_add__le__imp__le__left,axiom,
! [C_54: rat,A_123: rat,B_99: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C_54 @ A_123 ) @ ( plus_plus_rat @ C_54 @ B_99 ) )
=> ( ord_less_eq_rat @ A_123 @ B_99 ) ) ).
thf(fact_1998_add__le__imp__le__right,axiom,
! [A_122: code_code_numeral,C_53: code_code_numeral,B_98: code_code_numeral] :
( ( ord_le565307924umeral @ ( plus_p1627245867umeral @ A_122 @ C_53 ) @ ( plus_p1627245867umeral @ B_98 @ C_53 ) )
=> ( ord_le565307924umeral @ A_122 @ B_98 ) ) ).
thf(fact_1999_add__le__imp__le__right,axiom,
! [A_122: int,C_53: int,B_98: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_122 @ C_53 ) @ ( plus_plus_int @ B_98 @ C_53 ) )
=> ( ord_less_eq_int @ A_122 @ B_98 ) ) ).
thf(fact_2000_add__le__imp__le__right,axiom,
! [A_122: nat,C_53: nat,B_98: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_122 @ C_53 ) @ ( plus_plus_nat @ B_98 @ C_53 ) )
=> ( ord_less_eq_nat @ A_122 @ B_98 ) ) ).
thf(fact_2001_add__le__imp__le__right,axiom,
! [A_122: real,C_53: real,B_98: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A_122 @ C_53 ) @ ( plus_plus_real @ B_98 @ C_53 ) )
=> ( ord_less_eq_real @ A_122 @ B_98 ) ) ).
thf(fact_2002_add__le__imp__le__right,axiom,
! [A_122: quickcheck_code_int,C_53: quickcheck_code_int,B_98: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( plus_p1446045655de_int @ A_122 @ C_53 ) @ ( plus_p1446045655de_int @ B_98 @ C_53 ) )
=> ( ord_le258702272de_int @ A_122 @ B_98 ) ) ).
thf(fact_2003_add__le__imp__le__right,axiom,
! [A_122: rat,C_53: rat,B_98: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A_122 @ C_53 ) @ ( plus_plus_rat @ B_98 @ C_53 ) )
=> ( ord_less_eq_rat @ A_122 @ B_98 ) ) ).
thf(fact_2004_add__mono,axiom,
! [C_52: code_code_numeral,D_14: code_code_numeral,A_121: code_code_numeral,B_97: code_code_numeral] :
( ( ord_le565307924umeral @ A_121 @ B_97 )
=> ( ( ord_le565307924umeral @ C_52 @ D_14 )
=> ( ord_le565307924umeral @ ( plus_p1627245867umeral @ A_121 @ C_52 ) @ ( plus_p1627245867umeral @ B_97 @ D_14 ) ) ) ) ).
thf(fact_2005_add__mono,axiom,
! [C_52: int,D_14: int,A_121: int,B_97: int] :
( ( ord_less_eq_int @ A_121 @ B_97 )
=> ( ( ord_less_eq_int @ C_52 @ D_14 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_121 @ C_52 ) @ ( plus_plus_int @ B_97 @ D_14 ) ) ) ) ).
thf(fact_2006_add__mono,axiom,
! [C_52: nat,D_14: nat,A_121: nat,B_97: nat] :
( ( ord_less_eq_nat @ A_121 @ B_97 )
=> ( ( ord_less_eq_nat @ C_52 @ D_14 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_121 @ C_52 ) @ ( plus_plus_nat @ B_97 @ D_14 ) ) ) ) ).
thf(fact_2007_add__mono,axiom,
! [C_52: real,D_14: real,A_121: real,B_97: real] :
( ( ord_less_eq_real @ A_121 @ B_97 )
=> ( ( ord_less_eq_real @ C_52 @ D_14 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A_121 @ C_52 ) @ ( plus_plus_real @ B_97 @ D_14 ) ) ) ) ).
thf(fact_2008_add__mono,axiom,
! [C_52: quickcheck_code_int,D_14: quickcheck_code_int,A_121: quickcheck_code_int,B_97: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_121 @ B_97 )
=> ( ( ord_le258702272de_int @ C_52 @ D_14 )
=> ( ord_le258702272de_int @ ( plus_p1446045655de_int @ A_121 @ C_52 ) @ ( plus_p1446045655de_int @ B_97 @ D_14 ) ) ) ) ).
thf(fact_2009_add__mono,axiom,
! [C_52: rat,D_14: rat,A_121: rat,B_97: rat] :
( ( ord_less_eq_rat @ A_121 @ B_97 )
=> ( ( ord_less_eq_rat @ C_52 @ D_14 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A_121 @ C_52 ) @ ( plus_plus_rat @ B_97 @ D_14 ) ) ) ) ).
thf(fact_2010_add__left__mono,axiom,
! [C_51: code_code_numeral,A_120: code_code_numeral,B_96: code_code_numeral] :
( ( ord_le565307924umeral @ A_120 @ B_96 )
=> ( ord_le565307924umeral @ ( plus_p1627245867umeral @ C_51 @ A_120 ) @ ( plus_p1627245867umeral @ C_51 @ B_96 ) ) ) ).
thf(fact_2011_add__left__mono,axiom,
! [C_51: int,A_120: int,B_96: int] :
( ( ord_less_eq_int @ A_120 @ B_96 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C_51 @ A_120 ) @ ( plus_plus_int @ C_51 @ B_96 ) ) ) ).
thf(fact_2012_add__left__mono,axiom,
! [C_51: nat,A_120: nat,B_96: nat] :
( ( ord_less_eq_nat @ A_120 @ B_96 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C_51 @ A_120 ) @ ( plus_plus_nat @ C_51 @ B_96 ) ) ) ).
thf(fact_2013_add__left__mono,axiom,
! [C_51: real,A_120: real,B_96: real] :
( ( ord_less_eq_real @ A_120 @ B_96 )
=> ( ord_less_eq_real @ ( plus_plus_real @ C_51 @ A_120 ) @ ( plus_plus_real @ C_51 @ B_96 ) ) ) ).
thf(fact_2014_add__left__mono,axiom,
! [C_51: quickcheck_code_int,A_120: quickcheck_code_int,B_96: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_120 @ B_96 )
=> ( ord_le258702272de_int @ ( plus_p1446045655de_int @ C_51 @ A_120 ) @ ( plus_p1446045655de_int @ C_51 @ B_96 ) ) ) ).
thf(fact_2015_add__left__mono,axiom,
! [C_51: rat,A_120: rat,B_96: rat] :
( ( ord_less_eq_rat @ A_120 @ B_96 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ C_51 @ A_120 ) @ ( plus_plus_rat @ C_51 @ B_96 ) ) ) ).
thf(fact_2016_add__right__mono,axiom,
! [C_50: code_code_numeral,A_119: code_code_numeral,B_95: code_code_numeral] :
( ( ord_le565307924umeral @ A_119 @ B_95 )
=> ( ord_le565307924umeral @ ( plus_p1627245867umeral @ A_119 @ C_50 ) @ ( plus_p1627245867umeral @ B_95 @ C_50 ) ) ) ).
thf(fact_2017_add__right__mono,axiom,
! [C_50: int,A_119: int,B_95: int] :
( ( ord_less_eq_int @ A_119 @ B_95 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_119 @ C_50 ) @ ( plus_plus_int @ B_95 @ C_50 ) ) ) ).
thf(fact_2018_add__right__mono,axiom,
! [C_50: nat,A_119: nat,B_95: nat] :
( ( ord_less_eq_nat @ A_119 @ B_95 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_119 @ C_50 ) @ ( plus_plus_nat @ B_95 @ C_50 ) ) ) ).
thf(fact_2019_add__right__mono,axiom,
! [C_50: real,A_119: real,B_95: real] :
( ( ord_less_eq_real @ A_119 @ B_95 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A_119 @ C_50 ) @ ( plus_plus_real @ B_95 @ C_50 ) ) ) ).
thf(fact_2020_add__right__mono,axiom,
! [C_50: quickcheck_code_int,A_119: quickcheck_code_int,B_95: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_119 @ B_95 )
=> ( ord_le258702272de_int @ ( plus_p1446045655de_int @ A_119 @ C_50 ) @ ( plus_p1446045655de_int @ B_95 @ C_50 ) ) ) ).
thf(fact_2021_add__right__mono,axiom,
! [C_50: rat,A_119: rat,B_95: rat] :
( ( ord_less_eq_rat @ A_119 @ B_95 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A_119 @ C_50 ) @ ( plus_plus_rat @ B_95 @ C_50 ) ) ) ).
thf(fact_2022_add__le__cancel__left,axiom,
! [C_49: code_code_numeral,A_118: code_code_numeral,B_94: code_code_numeral] :
( ( ord_le565307924umeral @ ( plus_p1627245867umeral @ C_49 @ A_118 ) @ ( plus_p1627245867umeral @ C_49 @ B_94 ) )
<=> ( ord_le565307924umeral @ A_118 @ B_94 ) ) ).
thf(fact_2023_add__le__cancel__left,axiom,
! [C_49: int,A_118: int,B_94: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_49 @ A_118 ) @ ( plus_plus_int @ C_49 @ B_94 ) )
<=> ( ord_less_eq_int @ A_118 @ B_94 ) ) ).
thf(fact_2024_add__le__cancel__left,axiom,
! [C_49: nat,A_118: nat,B_94: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_49 @ A_118 ) @ ( plus_plus_nat @ C_49 @ B_94 ) )
<=> ( ord_less_eq_nat @ A_118 @ B_94 ) ) ).
thf(fact_2025_add__le__cancel__left,axiom,
! [C_49: real,A_118: real,B_94: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C_49 @ A_118 ) @ ( plus_plus_real @ C_49 @ B_94 ) )
<=> ( ord_less_eq_real @ A_118 @ B_94 ) ) ).
thf(fact_2026_add__le__cancel__left,axiom,
! [C_49: quickcheck_code_int,A_118: quickcheck_code_int,B_94: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( plus_p1446045655de_int @ C_49 @ A_118 ) @ ( plus_p1446045655de_int @ C_49 @ B_94 ) )
<=> ( ord_le258702272de_int @ A_118 @ B_94 ) ) ).
thf(fact_2027_add__le__cancel__left,axiom,
! [C_49: rat,A_118: rat,B_94: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C_49 @ A_118 ) @ ( plus_plus_rat @ C_49 @ B_94 ) )
<=> ( ord_less_eq_rat @ A_118 @ B_94 ) ) ).
thf(fact_2028_add__le__cancel__right,axiom,
! [A_117: code_code_numeral,C_48: code_code_numeral,B_93: code_code_numeral] :
( ( ord_le565307924umeral @ ( plus_p1627245867umeral @ A_117 @ C_48 ) @ ( plus_p1627245867umeral @ B_93 @ C_48 ) )
<=> ( ord_le565307924umeral @ A_117 @ B_93 ) ) ).
thf(fact_2029_add__le__cancel__right,axiom,
! [A_117: int,C_48: int,B_93: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_117 @ C_48 ) @ ( plus_plus_int @ B_93 @ C_48 ) )
<=> ( ord_less_eq_int @ A_117 @ B_93 ) ) ).
thf(fact_2030_add__le__cancel__right,axiom,
! [A_117: nat,C_48: nat,B_93: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_117 @ C_48 ) @ ( plus_plus_nat @ B_93 @ C_48 ) )
<=> ( ord_less_eq_nat @ A_117 @ B_93 ) ) ).
thf(fact_2031_add__le__cancel__right,axiom,
! [A_117: real,C_48: real,B_93: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A_117 @ C_48 ) @ ( plus_plus_real @ B_93 @ C_48 ) )
<=> ( ord_less_eq_real @ A_117 @ B_93 ) ) ).
thf(fact_2032_add__le__cancel__right,axiom,
! [A_117: quickcheck_code_int,C_48: quickcheck_code_int,B_93: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( plus_p1446045655de_int @ A_117 @ C_48 ) @ ( plus_p1446045655de_int @ B_93 @ C_48 ) )
<=> ( ord_le258702272de_int @ A_117 @ B_93 ) ) ).
thf(fact_2033_add__le__cancel__right,axiom,
! [A_117: rat,C_48: rat,B_93: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A_117 @ C_48 ) @ ( plus_plus_rat @ B_93 @ C_48 ) )
<=> ( ord_less_eq_rat @ A_117 @ B_93 ) ) ).
thf(fact_2034_add__less__imp__less__left,axiom,
! [C_47: code_code_numeral,A_116: code_code_numeral,B_92: code_code_numeral] :
( ( ord_le1304079648umeral @ ( plus_p1627245867umeral @ C_47 @ A_116 ) @ ( plus_p1627245867umeral @ C_47 @ B_92 ) )
=> ( ord_le1304079648umeral @ A_116 @ B_92 ) ) ).
thf(fact_2035_add__less__imp__less__left,axiom,
! [C_47: int,A_116: int,B_92: int] :
( ( ord_less_int @ ( plus_plus_int @ C_47 @ A_116 ) @ ( plus_plus_int @ C_47 @ B_92 ) )
=> ( ord_less_int @ A_116 @ B_92 ) ) ).
thf(fact_2036_add__less__imp__less__left,axiom,
! [C_47: nat,A_116: nat,B_92: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_47 @ A_116 ) @ ( plus_plus_nat @ C_47 @ B_92 ) )
=> ( ord_less_nat @ A_116 @ B_92 ) ) ).
thf(fact_2037_add__less__imp__less__left,axiom,
! [C_47: real,A_116: real,B_92: real] :
( ( ord_less_real @ ( plus_plus_real @ C_47 @ A_116 ) @ ( plus_plus_real @ C_47 @ B_92 ) )
=> ( ord_less_real @ A_116 @ B_92 ) ) ).
thf(fact_2038_add__less__imp__less__left,axiom,
! [C_47: quickcheck_code_int,A_116: quickcheck_code_int,B_92: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ C_47 @ A_116 ) @ ( plus_p1446045655de_int @ C_47 @ B_92 ) )
=> ( ord_le1860547276de_int @ A_116 @ B_92 ) ) ).
thf(fact_2039_add__less__imp__less__left,axiom,
! [C_47: rat,A_116: rat,B_92: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C_47 @ A_116 ) @ ( plus_plus_rat @ C_47 @ B_92 ) )
=> ( ord_less_rat @ A_116 @ B_92 ) ) ).
thf(fact_2040_add__less__imp__less__right,axiom,
! [A_115: code_code_numeral,C_46: code_code_numeral,B_91: code_code_numeral] :
( ( ord_le1304079648umeral @ ( plus_p1627245867umeral @ A_115 @ C_46 ) @ ( plus_p1627245867umeral @ B_91 @ C_46 ) )
=> ( ord_le1304079648umeral @ A_115 @ B_91 ) ) ).
thf(fact_2041_add__less__imp__less__right,axiom,
! [A_115: int,C_46: int,B_91: int] :
( ( ord_less_int @ ( plus_plus_int @ A_115 @ C_46 ) @ ( plus_plus_int @ B_91 @ C_46 ) )
=> ( ord_less_int @ A_115 @ B_91 ) ) ).
thf(fact_2042_add__less__imp__less__right,axiom,
! [A_115: nat,C_46: nat,B_91: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_115 @ C_46 ) @ ( plus_plus_nat @ B_91 @ C_46 ) )
=> ( ord_less_nat @ A_115 @ B_91 ) ) ).
thf(fact_2043_add__less__imp__less__right,axiom,
! [A_115: real,C_46: real,B_91: real] :
( ( ord_less_real @ ( plus_plus_real @ A_115 @ C_46 ) @ ( plus_plus_real @ B_91 @ C_46 ) )
=> ( ord_less_real @ A_115 @ B_91 ) ) ).
thf(fact_2044_add__less__imp__less__right,axiom,
! [A_115: quickcheck_code_int,C_46: quickcheck_code_int,B_91: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_115 @ C_46 ) @ ( plus_p1446045655de_int @ B_91 @ C_46 ) )
=> ( ord_le1860547276de_int @ A_115 @ B_91 ) ) ).
thf(fact_2045_add__less__imp__less__right,axiom,
! [A_115: rat,C_46: rat,B_91: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A_115 @ C_46 ) @ ( plus_plus_rat @ B_91 @ C_46 ) )
=> ( ord_less_rat @ A_115 @ B_91 ) ) ).
thf(fact_2046_add__strict__mono,axiom,
! [C_45: code_code_numeral,D_13: code_code_numeral,A_114: code_code_numeral,B_90: code_code_numeral] :
( ( ord_le1304079648umeral @ A_114 @ B_90 )
=> ( ( ord_le1304079648umeral @ C_45 @ D_13 )
=> ( ord_le1304079648umeral @ ( plus_p1627245867umeral @ A_114 @ C_45 ) @ ( plus_p1627245867umeral @ B_90 @ D_13 ) ) ) ) ).
thf(fact_2047_add__strict__mono,axiom,
! [C_45: int,D_13: int,A_114: int,B_90: int] :
( ( ord_less_int @ A_114 @ B_90 )
=> ( ( ord_less_int @ C_45 @ D_13 )
=> ( ord_less_int @ ( plus_plus_int @ A_114 @ C_45 ) @ ( plus_plus_int @ B_90 @ D_13 ) ) ) ) ).
thf(fact_2048_add__strict__mono,axiom,
! [C_45: nat,D_13: nat,A_114: nat,B_90: nat] :
( ( ord_less_nat @ A_114 @ B_90 )
=> ( ( ord_less_nat @ C_45 @ D_13 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_114 @ C_45 ) @ ( plus_plus_nat @ B_90 @ D_13 ) ) ) ) ).
thf(fact_2049_add__strict__mono,axiom,
! [C_45: real,D_13: real,A_114: real,B_90: real] :
( ( ord_less_real @ A_114 @ B_90 )
=> ( ( ord_less_real @ C_45 @ D_13 )
=> ( ord_less_real @ ( plus_plus_real @ A_114 @ C_45 ) @ ( plus_plus_real @ B_90 @ D_13 ) ) ) ) ).
thf(fact_2050_add__strict__mono,axiom,
! [C_45: quickcheck_code_int,D_13: quickcheck_code_int,A_114: quickcheck_code_int,B_90: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_114 @ B_90 )
=> ( ( ord_le1860547276de_int @ C_45 @ D_13 )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_114 @ C_45 ) @ ( plus_p1446045655de_int @ B_90 @ D_13 ) ) ) ) ).
thf(fact_2051_add__strict__mono,axiom,
! [C_45: rat,D_13: rat,A_114: rat,B_90: rat] :
( ( ord_less_rat @ A_114 @ B_90 )
=> ( ( ord_less_rat @ C_45 @ D_13 )
=> ( ord_less_rat @ ( plus_plus_rat @ A_114 @ C_45 ) @ ( plus_plus_rat @ B_90 @ D_13 ) ) ) ) ).
thf(fact_2052_add__strict__left__mono,axiom,
! [C_44: code_code_numeral,A_113: code_code_numeral,B_89: code_code_numeral] :
( ( ord_le1304079648umeral @ A_113 @ B_89 )
=> ( ord_le1304079648umeral @ ( plus_p1627245867umeral @ C_44 @ A_113 ) @ ( plus_p1627245867umeral @ C_44 @ B_89 ) ) ) ).
thf(fact_2053_add__strict__left__mono,axiom,
! [C_44: int,A_113: int,B_89: int] :
( ( ord_less_int @ A_113 @ B_89 )
=> ( ord_less_int @ ( plus_plus_int @ C_44 @ A_113 ) @ ( plus_plus_int @ C_44 @ B_89 ) ) ) ).
thf(fact_2054_add__strict__left__mono,axiom,
! [C_44: nat,A_113: nat,B_89: nat] :
( ( ord_less_nat @ A_113 @ B_89 )
=> ( ord_less_nat @ ( plus_plus_nat @ C_44 @ A_113 ) @ ( plus_plus_nat @ C_44 @ B_89 ) ) ) ).
thf(fact_2055_add__strict__left__mono,axiom,
! [C_44: real,A_113: real,B_89: real] :
( ( ord_less_real @ A_113 @ B_89 )
=> ( ord_less_real @ ( plus_plus_real @ C_44 @ A_113 ) @ ( plus_plus_real @ C_44 @ B_89 ) ) ) ).
thf(fact_2056_add__strict__left__mono,axiom,
! [C_44: quickcheck_code_int,A_113: quickcheck_code_int,B_89: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_113 @ B_89 )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ C_44 @ A_113 ) @ ( plus_p1446045655de_int @ C_44 @ B_89 ) ) ) ).
thf(fact_2057_add__strict__left__mono,axiom,
! [C_44: rat,A_113: rat,B_89: rat] :
( ( ord_less_rat @ A_113 @ B_89 )
=> ( ord_less_rat @ ( plus_plus_rat @ C_44 @ A_113 ) @ ( plus_plus_rat @ C_44 @ B_89 ) ) ) ).
thf(fact_2058_add__strict__right__mono,axiom,
! [C_43: code_code_numeral,A_112: code_code_numeral,B_88: code_code_numeral] :
( ( ord_le1304079648umeral @ A_112 @ B_88 )
=> ( ord_le1304079648umeral @ ( plus_p1627245867umeral @ A_112 @ C_43 ) @ ( plus_p1627245867umeral @ B_88 @ C_43 ) ) ) ).
thf(fact_2059_add__strict__right__mono,axiom,
! [C_43: int,A_112: int,B_88: int] :
( ( ord_less_int @ A_112 @ B_88 )
=> ( ord_less_int @ ( plus_plus_int @ A_112 @ C_43 ) @ ( plus_plus_int @ B_88 @ C_43 ) ) ) ).
thf(fact_2060_add__strict__right__mono,axiom,
! [C_43: nat,A_112: nat,B_88: nat] :
( ( ord_less_nat @ A_112 @ B_88 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_112 @ C_43 ) @ ( plus_plus_nat @ B_88 @ C_43 ) ) ) ).
thf(fact_2061_add__strict__right__mono,axiom,
! [C_43: real,A_112: real,B_88: real] :
( ( ord_less_real @ A_112 @ B_88 )
=> ( ord_less_real @ ( plus_plus_real @ A_112 @ C_43 ) @ ( plus_plus_real @ B_88 @ C_43 ) ) ) ).
thf(fact_2062_add__strict__right__mono,axiom,
! [C_43: quickcheck_code_int,A_112: quickcheck_code_int,B_88: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_112 @ B_88 )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_112 @ C_43 ) @ ( plus_p1446045655de_int @ B_88 @ C_43 ) ) ) ).
thf(fact_2063_add__strict__right__mono,axiom,
! [C_43: rat,A_112: rat,B_88: rat] :
( ( ord_less_rat @ A_112 @ B_88 )
=> ( ord_less_rat @ ( plus_plus_rat @ A_112 @ C_43 ) @ ( plus_plus_rat @ B_88 @ C_43 ) ) ) ).
thf(fact_2064_add__less__cancel__left,axiom,
! [C_42: code_code_numeral,A_111: code_code_numeral,B_87: code_code_numeral] :
( ( ord_le1304079648umeral @ ( plus_p1627245867umeral @ C_42 @ A_111 ) @ ( plus_p1627245867umeral @ C_42 @ B_87 ) )
<=> ( ord_le1304079648umeral @ A_111 @ B_87 ) ) ).
thf(fact_2065_add__less__cancel__left,axiom,
! [C_42: int,A_111: int,B_87: int] :
( ( ord_less_int @ ( plus_plus_int @ C_42 @ A_111 ) @ ( plus_plus_int @ C_42 @ B_87 ) )
<=> ( ord_less_int @ A_111 @ B_87 ) ) ).
thf(fact_2066_add__less__cancel__left,axiom,
! [C_42: nat,A_111: nat,B_87: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_42 @ A_111 ) @ ( plus_plus_nat @ C_42 @ B_87 ) )
<=> ( ord_less_nat @ A_111 @ B_87 ) ) ).
thf(fact_2067_add__less__cancel__left,axiom,
! [C_42: real,A_111: real,B_87: real] :
( ( ord_less_real @ ( plus_plus_real @ C_42 @ A_111 ) @ ( plus_plus_real @ C_42 @ B_87 ) )
<=> ( ord_less_real @ A_111 @ B_87 ) ) ).
thf(fact_2068_add__less__cancel__left,axiom,
! [C_42: quickcheck_code_int,A_111: quickcheck_code_int,B_87: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ C_42 @ A_111 ) @ ( plus_p1446045655de_int @ C_42 @ B_87 ) )
<=> ( ord_le1860547276de_int @ A_111 @ B_87 ) ) ).
thf(fact_2069_add__less__cancel__left,axiom,
! [C_42: rat,A_111: rat,B_87: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C_42 @ A_111 ) @ ( plus_plus_rat @ C_42 @ B_87 ) )
<=> ( ord_less_rat @ A_111 @ B_87 ) ) ).
thf(fact_2070_add__less__cancel__right,axiom,
! [A_110: code_code_numeral,C_41: code_code_numeral,B_86: code_code_numeral] :
( ( ord_le1304079648umeral @ ( plus_p1627245867umeral @ A_110 @ C_41 ) @ ( plus_p1627245867umeral @ B_86 @ C_41 ) )
<=> ( ord_le1304079648umeral @ A_110 @ B_86 ) ) ).
thf(fact_2071_add__less__cancel__right,axiom,
! [A_110: int,C_41: int,B_86: int] :
( ( ord_less_int @ ( plus_plus_int @ A_110 @ C_41 ) @ ( plus_plus_int @ B_86 @ C_41 ) )
<=> ( ord_less_int @ A_110 @ B_86 ) ) ).
thf(fact_2072_add__less__cancel__right,axiom,
! [A_110: nat,C_41: nat,B_86: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_110 @ C_41 ) @ ( plus_plus_nat @ B_86 @ C_41 ) )
<=> ( ord_less_nat @ A_110 @ B_86 ) ) ).
thf(fact_2073_add__less__cancel__right,axiom,
! [A_110: real,C_41: real,B_86: real] :
( ( ord_less_real @ ( plus_plus_real @ A_110 @ C_41 ) @ ( plus_plus_real @ B_86 @ C_41 ) )
<=> ( ord_less_real @ A_110 @ B_86 ) ) ).
thf(fact_2074_add__less__cancel__right,axiom,
! [A_110: quickcheck_code_int,C_41: quickcheck_code_int,B_86: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_110 @ C_41 ) @ ( plus_p1446045655de_int @ B_86 @ C_41 ) )
<=> ( ord_le1860547276de_int @ A_110 @ B_86 ) ) ).
thf(fact_2075_add__less__cancel__right,axiom,
! [A_110: rat,C_41: rat,B_86: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A_110 @ C_41 ) @ ( plus_plus_rat @ B_86 @ C_41 ) )
<=> ( ord_less_rat @ A_110 @ B_86 ) ) ).
thf(fact_2076_right__minus__eq,axiom,
! [A_109: int,B_85: int] :
( ( ( minus_minus_int @ A_109 @ B_85 )
= zero_zero_int )
<=> ( A_109 = B_85 ) ) ).
thf(fact_2077_right__minus__eq,axiom,
! [A_109: real,B_85: real] :
( ( ( minus_minus_real @ A_109 @ B_85 )
= zero_zero_real )
<=> ( A_109 = B_85 ) ) ).
thf(fact_2078_right__minus__eq,axiom,
! [A_109: complex,B_85: complex] :
( ( ( minus_minus_complex @ A_109 @ B_85 )
= zero_zero_complex )
<=> ( A_109 = B_85 ) ) ).
thf(fact_2079_right__minus__eq,axiom,
! [A_109: rat,B_85: rat] :
( ( ( minus_minus_rat @ A_109 @ B_85 )
= zero_zero_rat )
<=> ( A_109 = B_85 ) ) ).
thf(fact_2080_eq__iff__diff__eq__0,axiom,
! [A_108: int,B_84: int] :
( ( A_108 = B_84 )
<=> ( ( minus_minus_int @ A_108 @ B_84 )
= zero_zero_int ) ) ).
thf(fact_2081_eq__iff__diff__eq__0,axiom,
! [A_108: real,B_84: real] :
( ( A_108 = B_84 )
<=> ( ( minus_minus_real @ A_108 @ B_84 )
= zero_zero_real ) ) ).
thf(fact_2082_eq__iff__diff__eq__0,axiom,
! [A_108: complex,B_84: complex] :
( ( A_108 = B_84 )
<=> ( ( minus_minus_complex @ A_108 @ B_84 )
= zero_zero_complex ) ) ).
thf(fact_2083_eq__iff__diff__eq__0,axiom,
! [A_108: rat,B_84: rat] :
( ( A_108 = B_84 )
<=> ( ( minus_minus_rat @ A_108 @ B_84 )
= zero_zero_rat ) ) ).
thf(fact_2084_diff__self,axiom,
! [A_107: int] :
( ( minus_minus_int @ A_107 @ A_107 )
= zero_zero_int ) ).
thf(fact_2085_diff__self,axiom,
! [A_107: real] :
( ( minus_minus_real @ A_107 @ A_107 )
= zero_zero_real ) ).
thf(fact_2086_diff__self,axiom,
! [A_107: complex] :
( ( minus_minus_complex @ A_107 @ A_107 )
= zero_zero_complex ) ).
thf(fact_2087_diff__self,axiom,
! [A_107: rat] :
( ( minus_minus_rat @ A_107 @ A_107 )
= zero_zero_rat ) ).
thf(fact_2088_diff__0__right,axiom,
! [A_106: int] :
( ( minus_minus_int @ A_106 @ zero_zero_int )
= A_106 ) ).
thf(fact_2089_diff__0__right,axiom,
! [A_106: real] :
( ( minus_minus_real @ A_106 @ zero_zero_real )
= A_106 ) ).
thf(fact_2090_diff__0__right,axiom,
! [A_106: complex] :
( ( minus_minus_complex @ A_106 @ zero_zero_complex )
= A_106 ) ).
thf(fact_2091_diff__0__right,axiom,
! [A_106: rat] :
( ( minus_minus_rat @ A_106 @ zero_zero_rat )
= A_106 ) ).
thf(fact_2092_diff__eq__diff__less__eq,axiom,
! [A_105: int,B_83: int,C_40: int,D_12: int] :
( ( ( minus_minus_int @ A_105 @ B_83 )
= ( minus_minus_int @ C_40 @ D_12 ) )
=> ( ( ord_less_eq_int @ A_105 @ B_83 )
<=> ( ord_less_eq_int @ C_40 @ D_12 ) ) ) ).
thf(fact_2093_diff__eq__diff__less__eq,axiom,
! [A_105: real,B_83: real,C_40: real,D_12: real] :
( ( ( minus_minus_real @ A_105 @ B_83 )
= ( minus_minus_real @ C_40 @ D_12 ) )
=> ( ( ord_less_eq_real @ A_105 @ B_83 )
<=> ( ord_less_eq_real @ C_40 @ D_12 ) ) ) ).
thf(fact_2094_diff__eq__diff__less__eq,axiom,
! [A_105: rat,B_83: rat,C_40: rat,D_12: rat] :
( ( ( minus_minus_rat @ A_105 @ B_83 )
= ( minus_minus_rat @ C_40 @ D_12 ) )
=> ( ( ord_less_eq_rat @ A_105 @ B_83 )
<=> ( ord_less_eq_rat @ C_40 @ D_12 ) ) ) ).
thf(fact_2095_diff__eq__diff__less,axiom,
! [A_104: int,B_82: int,C_39: int,D_11: int] :
( ( ( minus_minus_int @ A_104 @ B_82 )
= ( minus_minus_int @ C_39 @ D_11 ) )
=> ( ( ord_less_int @ A_104 @ B_82 )
<=> ( ord_less_int @ C_39 @ D_11 ) ) ) ).
thf(fact_2096_diff__eq__diff__less,axiom,
! [A_104: real,B_82: real,C_39: real,D_11: real] :
( ( ( minus_minus_real @ A_104 @ B_82 )
= ( minus_minus_real @ C_39 @ D_11 ) )
=> ( ( ord_less_real @ A_104 @ B_82 )
<=> ( ord_less_real @ C_39 @ D_11 ) ) ) ).
thf(fact_2097_diff__eq__diff__less,axiom,
! [A_104: rat,B_82: rat,C_39: rat,D_11: rat] :
( ( ( minus_minus_rat @ A_104 @ B_82 )
= ( minus_minus_rat @ C_39 @ D_11 ) )
=> ( ( ord_less_rat @ A_104 @ B_82 )
<=> ( ord_less_rat @ C_39 @ D_11 ) ) ) ).
thf(fact_2098_mult_Ocomm__neutral,axiom,
! [A_103: int] :
( ( times_times_int @ A_103 @ one_one_int )
= A_103 ) ).
thf(fact_2099_mult_Ocomm__neutral,axiom,
! [A_103: nat] :
( ( times_times_nat @ A_103 @ one_one_nat )
= A_103 ) ).
thf(fact_2100_mult_Ocomm__neutral,axiom,
! [A_103: real] :
( ( times_times_real @ A_103 @ one_one_real )
= A_103 ) ).
thf(fact_2101_mult_Ocomm__neutral,axiom,
! [A_103: code_code_numeral] :
( ( times_1655362735umeral @ A_103 @ one_on1645066479umeral )
= A_103 ) ).
thf(fact_2102_mult_Ocomm__neutral,axiom,
! [A_103: complex] :
( ( times_times_complex @ A_103 @ one_one_complex )
= A_103 ) ).
thf(fact_2103_mult_Ocomm__neutral,axiom,
! [A_103: quickcheck_code_int] :
( ( times_123202395de_int @ A_103 @ one_on1684967323de_int )
= A_103 ) ).
thf(fact_2104_mult_Ocomm__neutral,axiom,
! [A_103: rat] :
( ( times_times_rat @ A_103 @ one_one_rat )
= A_103 ) ).
thf(fact_2105_mult__1__right,axiom,
! [A_102: int] :
( ( times_times_int @ A_102 @ one_one_int )
= A_102 ) ).
thf(fact_2106_mult__1__right,axiom,
! [A_102: nat] :
( ( times_times_nat @ A_102 @ one_one_nat )
= A_102 ) ).
thf(fact_2107_mult__1__right,axiom,
! [A_102: real] :
( ( times_times_real @ A_102 @ one_one_real )
= A_102 ) ).
thf(fact_2108_mult__1__right,axiom,
! [A_102: code_code_numeral] :
( ( times_1655362735umeral @ A_102 @ one_on1645066479umeral )
= A_102 ) ).
thf(fact_2109_mult__1__right,axiom,
! [A_102: complex] :
( ( times_times_complex @ A_102 @ one_one_complex )
= A_102 ) ).
thf(fact_2110_mult__1__right,axiom,
! [A_102: quickcheck_code_int] :
( ( times_123202395de_int @ A_102 @ one_on1684967323de_int )
= A_102 ) ).
thf(fact_2111_mult__1__right,axiom,
! [A_102: rat] :
( ( times_times_rat @ A_102 @ one_one_rat )
= A_102 ) ).
thf(fact_2112_mult__1,axiom,
! [A_101: int] :
( ( times_times_int @ one_one_int @ A_101 )
= A_101 ) ).
thf(fact_2113_mult__1,axiom,
! [A_101: nat] :
( ( times_times_nat @ one_one_nat @ A_101 )
= A_101 ) ).
thf(fact_2114_mult__1,axiom,
! [A_101: real] :
( ( times_times_real @ one_one_real @ A_101 )
= A_101 ) ).
thf(fact_2115_mult__1,axiom,
! [A_101: code_code_numeral] :
( ( times_1655362735umeral @ one_on1645066479umeral @ A_101 )
= A_101 ) ).
thf(fact_2116_mult__1,axiom,
! [A_101: complex] :
( ( times_times_complex @ one_one_complex @ A_101 )
= A_101 ) ).
thf(fact_2117_mult__1,axiom,
! [A_101: quickcheck_code_int] :
( ( times_123202395de_int @ one_on1684967323de_int @ A_101 )
= A_101 ) ).
thf(fact_2118_mult__1,axiom,
! [A_101: rat] :
( ( times_times_rat @ one_one_rat @ A_101 )
= A_101 ) ).
thf(fact_2119_mult__1__left,axiom,
! [A_100: int] :
( ( times_times_int @ one_one_int @ A_100 )
= A_100 ) ).
thf(fact_2120_mult__1__left,axiom,
! [A_100: nat] :
( ( times_times_nat @ one_one_nat @ A_100 )
= A_100 ) ).
thf(fact_2121_mult__1__left,axiom,
! [A_100: real] :
( ( times_times_real @ one_one_real @ A_100 )
= A_100 ) ).
thf(fact_2122_mult__1__left,axiom,
! [A_100: code_code_numeral] :
( ( times_1655362735umeral @ one_on1645066479umeral @ A_100 )
= A_100 ) ).
thf(fact_2123_mult__1__left,axiom,
! [A_100: complex] :
( ( times_times_complex @ one_one_complex @ A_100 )
= A_100 ) ).
thf(fact_2124_mult__1__left,axiom,
! [A_100: quickcheck_code_int] :
( ( times_123202395de_int @ one_on1684967323de_int @ A_100 )
= A_100 ) ).
thf(fact_2125_mult__1__left,axiom,
! [A_100: rat] :
( ( times_times_rat @ one_one_rat @ A_100 )
= A_100 ) ).
thf(fact_2126_inv__less__p__minus__1,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ord_less_int @ ( inv @ P_3 @ A ) @ ( minus_minus_int @ P_3 @ one_one_int ) ) ) ) ) ).
thf(fact_2127_inv__g__1,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ord_less_int @ one_one_int @ ( inv @ P_3 @ A ) ) ) ) ) ).
thf(fact_2128_inv__not__p__minus__1,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( inv @ P_3 @ A )
!= ( minus_minus_int @ P_3 @ one_one_int ) ) ) ) ) ).
thf(fact_2129_inv__not__1,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( inv @ P_3 @ A )
!= one_one_int ) ) ) ) ).
thf(fact_2130_inv__distinct,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( A
!= ( inv @ P_3 @ A ) ) ) ) ) ).
thf(fact_2131_add__diff__cancel,axiom,
! [A_99: int,B_81: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_99 @ B_81 ) @ B_81 )
= A_99 ) ).
thf(fact_2132_add__diff__cancel,axiom,
! [A_99: real,B_81: real] :
( ( minus_minus_real @ ( plus_plus_real @ A_99 @ B_81 ) @ B_81 )
= A_99 ) ).
thf(fact_2133_add__diff__cancel,axiom,
! [A_99: complex,B_81: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A_99 @ B_81 ) @ B_81 )
= A_99 ) ).
thf(fact_2134_add__diff__cancel,axiom,
! [A_99: rat,B_81: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A_99 @ B_81 ) @ B_81 )
= A_99 ) ).
thf(fact_2135_diff__add__cancel,axiom,
! [A_98: int,B_80: int] :
( ( plus_plus_int @ ( minus_minus_int @ A_98 @ B_80 ) @ B_80 )
= A_98 ) ).
thf(fact_2136_diff__add__cancel,axiom,
! [A_98: real,B_80: real] :
( ( plus_plus_real @ ( minus_minus_real @ A_98 @ B_80 ) @ B_80 )
= A_98 ) ).
thf(fact_2137_diff__add__cancel,axiom,
! [A_98: complex,B_80: complex] :
( ( plus_plus_complex @ ( minus_minus_complex @ A_98 @ B_80 ) @ B_80 )
= A_98 ) ).
thf(fact_2138_diff__add__cancel,axiom,
! [A_98: rat,B_80: rat] :
( ( plus_plus_rat @ ( minus_minus_rat @ A_98 @ B_80 ) @ B_80 )
= A_98 ) ).
thf(fact_2139_inv__not__0,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( inv @ P_3 @ A )
!= zero_zero_int ) ) ) ) ).
thf(fact_2140_inv__is__inv,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ P_3 )
=> ( zcong @ ( times_times_int @ A @ ( inv @ P_3 @ A ) ) @ one_one_int @ P_3 ) ) ) ) ).
thf(fact_2141_add__nonpos__nonpos,axiom,
! [B_79: code_code_numeral,A_97: code_code_numeral] :
( ( ord_le565307924umeral @ A_97 @ zero_z126310315umeral )
=> ( ( ord_le565307924umeral @ B_79 @ zero_z126310315umeral )
=> ( ord_le565307924umeral @ ( plus_p1627245867umeral @ A_97 @ B_79 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_2142_add__nonpos__nonpos,axiom,
! [B_79: int,A_97: int] :
( ( ord_less_eq_int @ A_97 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B_79 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_97 @ B_79 ) @ zero_zero_int ) ) ) ).
thf(fact_2143_add__nonpos__nonpos,axiom,
! [B_79: nat,A_97: nat] :
( ( ord_less_eq_nat @ A_97 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B_79 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_97 @ B_79 ) @ zero_zero_nat ) ) ) ).
thf(fact_2144_add__nonpos__nonpos,axiom,
! [B_79: real,A_97: real] :
( ( ord_less_eq_real @ A_97 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B_79 @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A_97 @ B_79 ) @ zero_zero_real ) ) ) ).
thf(fact_2145_add__nonpos__nonpos,axiom,
! [B_79: quickcheck_code_int,A_97: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_97 @ zero_z891286103de_int )
=> ( ( ord_le258702272de_int @ B_79 @ zero_z891286103de_int )
=> ( ord_le258702272de_int @ ( plus_p1446045655de_int @ A_97 @ B_79 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_2146_add__nonpos__nonpos,axiom,
! [B_79: rat,A_97: rat] :
( ( ord_less_eq_rat @ A_97 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B_79 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A_97 @ B_79 ) @ zero_zero_rat ) ) ) ).
thf(fact_2147_add__increasing2,axiom,
! [B_78: code_code_numeral,A_96: code_code_numeral,C_38: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ C_38 )
=> ( ( ord_le565307924umeral @ B_78 @ A_96 )
=> ( ord_le565307924umeral @ B_78 @ ( plus_p1627245867umeral @ A_96 @ C_38 ) ) ) ) ).
thf(fact_2148_add__increasing2,axiom,
! [B_78: int,A_96: int,C_38: int] :
( ( ord_less_eq_int @ zero_zero_int @ C_38 )
=> ( ( ord_less_eq_int @ B_78 @ A_96 )
=> ( ord_less_eq_int @ B_78 @ ( plus_plus_int @ A_96 @ C_38 ) ) ) ) ).
thf(fact_2149_add__increasing2,axiom,
! [B_78: nat,A_96: nat,C_38: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C_38 )
=> ( ( ord_less_eq_nat @ B_78 @ A_96 )
=> ( ord_less_eq_nat @ B_78 @ ( plus_plus_nat @ A_96 @ C_38 ) ) ) ) ).
thf(fact_2150_add__increasing2,axiom,
! [B_78: real,A_96: real,C_38: real] :
( ( ord_less_eq_real @ zero_zero_real @ C_38 )
=> ( ( ord_less_eq_real @ B_78 @ A_96 )
=> ( ord_less_eq_real @ B_78 @ ( plus_plus_real @ A_96 @ C_38 ) ) ) ) ).
thf(fact_2151_add__increasing2,axiom,
! [B_78: quickcheck_code_int,A_96: quickcheck_code_int,C_38: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ C_38 )
=> ( ( ord_le258702272de_int @ B_78 @ A_96 )
=> ( ord_le258702272de_int @ B_78 @ ( plus_p1446045655de_int @ A_96 @ C_38 ) ) ) ) ).
thf(fact_2152_add__increasing2,axiom,
! [B_78: rat,A_96: rat,C_38: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ C_38 )
=> ( ( ord_less_eq_rat @ B_78 @ A_96 )
=> ( ord_less_eq_rat @ B_78 @ ( plus_plus_rat @ A_96 @ C_38 ) ) ) ) ).
thf(fact_2153_add__increasing,axiom,
! [B_77: code_code_numeral,C_37: code_code_numeral,A_95: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_95 )
=> ( ( ord_le565307924umeral @ B_77 @ C_37 )
=> ( ord_le565307924umeral @ B_77 @ ( plus_p1627245867umeral @ A_95 @ C_37 ) ) ) ) ).
thf(fact_2154_add__increasing,axiom,
! [B_77: int,C_37: int,A_95: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_95 )
=> ( ( ord_less_eq_int @ B_77 @ C_37 )
=> ( ord_less_eq_int @ B_77 @ ( plus_plus_int @ A_95 @ C_37 ) ) ) ) ).
thf(fact_2155_add__increasing,axiom,
! [B_77: nat,C_37: nat,A_95: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_95 )
=> ( ( ord_less_eq_nat @ B_77 @ C_37 )
=> ( ord_less_eq_nat @ B_77 @ ( plus_plus_nat @ A_95 @ C_37 ) ) ) ) ).
thf(fact_2156_add__increasing,axiom,
! [B_77: real,C_37: real,A_95: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_95 )
=> ( ( ord_less_eq_real @ B_77 @ C_37 )
=> ( ord_less_eq_real @ B_77 @ ( plus_plus_real @ A_95 @ C_37 ) ) ) ) ).
thf(fact_2157_add__increasing,axiom,
! [B_77: quickcheck_code_int,C_37: quickcheck_code_int,A_95: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_95 )
=> ( ( ord_le258702272de_int @ B_77 @ C_37 )
=> ( ord_le258702272de_int @ B_77 @ ( plus_p1446045655de_int @ A_95 @ C_37 ) ) ) ) ).
thf(fact_2158_add__increasing,axiom,
! [B_77: rat,C_37: rat,A_95: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_95 )
=> ( ( ord_less_eq_rat @ B_77 @ C_37 )
=> ( ord_less_eq_rat @ B_77 @ ( plus_plus_rat @ A_95 @ C_37 ) ) ) ) ).
thf(fact_2159_add__nonneg__eq__0__iff,axiom,
! [Y_16: code_code_numeral,X_20: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ X_20 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ Y_16 )
=> ( ( ( plus_p1627245867umeral @ X_20 @ Y_16 )
= zero_z126310315umeral )
<=> ( ( X_20 = zero_z126310315umeral )
& ( Y_16 = zero_z126310315umeral ) ) ) ) ) ).
thf(fact_2160_add__nonneg__eq__0__iff,axiom,
! [Y_16: int,X_20: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_20 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_16 )
=> ( ( ( plus_plus_int @ X_20 @ Y_16 )
= zero_zero_int )
<=> ( ( X_20 = zero_zero_int )
& ( Y_16 = zero_zero_int ) ) ) ) ) ).
thf(fact_2161_add__nonneg__eq__0__iff,axiom,
! [Y_16: nat,X_20: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X_20 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_16 )
=> ( ( ( plus_plus_nat @ X_20 @ Y_16 )
= zero_zero_nat )
<=> ( ( X_20 = zero_zero_nat )
& ( Y_16 = zero_zero_nat ) ) ) ) ) ).
thf(fact_2162_add__nonneg__eq__0__iff,axiom,
! [Y_16: real,X_20: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_20 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_16 )
=> ( ( ( plus_plus_real @ X_20 @ Y_16 )
= zero_zero_real )
<=> ( ( X_20 = zero_zero_real )
& ( Y_16 = zero_zero_real ) ) ) ) ) ).
thf(fact_2163_add__nonneg__eq__0__iff,axiom,
! [Y_16: quickcheck_code_int,X_20: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ X_20 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ Y_16 )
=> ( ( ( plus_p1446045655de_int @ X_20 @ Y_16 )
= zero_z891286103de_int )
<=> ( ( X_20 = zero_z891286103de_int )
& ( Y_16 = zero_z891286103de_int ) ) ) ) ) ).
thf(fact_2164_add__nonneg__eq__0__iff,axiom,
! [Y_16: rat,X_20: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X_20 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y_16 )
=> ( ( ( plus_plus_rat @ X_20 @ Y_16 )
= zero_zero_rat )
<=> ( ( X_20 = zero_zero_rat )
& ( Y_16 = zero_zero_rat ) ) ) ) ) ).
thf(fact_2165_add__nonneg__nonneg,axiom,
! [B_76: code_code_numeral,A_94: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_94 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_76 )
=> ( ord_le565307924umeral @ zero_z126310315umeral @ ( plus_p1627245867umeral @ A_94 @ B_76 ) ) ) ) ).
thf(fact_2166_add__nonneg__nonneg,axiom,
! [B_76: int,A_94: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_94 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_76 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A_94 @ B_76 ) ) ) ) ).
thf(fact_2167_add__nonneg__nonneg,axiom,
! [B_76: nat,A_94: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_94 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_76 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A_94 @ B_76 ) ) ) ) ).
thf(fact_2168_add__nonneg__nonneg,axiom,
! [B_76: real,A_94: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_94 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_76 )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A_94 @ B_76 ) ) ) ) ).
thf(fact_2169_add__nonneg__nonneg,axiom,
! [B_76: quickcheck_code_int,A_94: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_94 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_76 )
=> ( ord_le258702272de_int @ zero_z891286103de_int @ ( plus_p1446045655de_int @ A_94 @ B_76 ) ) ) ) ).
thf(fact_2170_add__nonneg__nonneg,axiom,
! [B_76: rat,A_94: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_94 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_76 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A_94 @ B_76 ) ) ) ) ).
thf(fact_2171_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A_93: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_93 @ A_93 ) @ zero_zero_int )
<=> ( ord_less_eq_int @ A_93 @ zero_zero_int ) ) ).
thf(fact_2172_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A_93: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A_93 @ A_93 ) @ zero_zero_real )
<=> ( ord_less_eq_real @ A_93 @ zero_zero_real ) ) ).
thf(fact_2173_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A_93: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A_93 @ A_93 ) @ zero_zero_rat )
<=> ( ord_less_eq_rat @ A_93 @ zero_zero_rat ) ) ).
thf(fact_2174_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A_92: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A_92 @ A_92 ) )
<=> ( ord_less_eq_int @ zero_zero_int @ A_92 ) ) ).
thf(fact_2175_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A_92: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A_92 @ A_92 ) )
<=> ( ord_less_eq_real @ zero_zero_real @ A_92 ) ) ).
thf(fact_2176_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A_92: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A_92 @ A_92 ) )
<=> ( ord_less_eq_rat @ zero_zero_rat @ A_92 ) ) ).
thf(fact_2177_add__neg__neg,axiom,
! [B_75: code_code_numeral,A_91: code_code_numeral] :
( ( ord_le1304079648umeral @ A_91 @ zero_z126310315umeral )
=> ( ( ord_le1304079648umeral @ B_75 @ zero_z126310315umeral )
=> ( ord_le1304079648umeral @ ( plus_p1627245867umeral @ A_91 @ B_75 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_2178_add__neg__neg,axiom,
! [B_75: int,A_91: int] :
( ( ord_less_int @ A_91 @ zero_zero_int )
=> ( ( ord_less_int @ B_75 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A_91 @ B_75 ) @ zero_zero_int ) ) ) ).
thf(fact_2179_add__neg__neg,axiom,
! [B_75: nat,A_91: nat] :
( ( ord_less_nat @ A_91 @ zero_zero_nat )
=> ( ( ord_less_nat @ B_75 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A_91 @ B_75 ) @ zero_zero_nat ) ) ) ).
thf(fact_2180_add__neg__neg,axiom,
! [B_75: real,A_91: real] :
( ( ord_less_real @ A_91 @ zero_zero_real )
=> ( ( ord_less_real @ B_75 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A_91 @ B_75 ) @ zero_zero_real ) ) ) ).
thf(fact_2181_add__neg__neg,axiom,
! [B_75: quickcheck_code_int,A_91: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_91 @ zero_z891286103de_int )
=> ( ( ord_le1860547276de_int @ B_75 @ zero_z891286103de_int )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_91 @ B_75 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_2182_add__neg__neg,axiom,
! [B_75: rat,A_91: rat] :
( ( ord_less_rat @ A_91 @ zero_zero_rat )
=> ( ( ord_less_rat @ B_75 @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A_91 @ B_75 ) @ zero_zero_rat ) ) ) ).
thf(fact_2183_add__pos__pos,axiom,
! [B_74: code_code_numeral,A_90: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_90 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ B_74 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ ( plus_p1627245867umeral @ A_90 @ B_74 ) ) ) ) ).
thf(fact_2184_add__pos__pos,axiom,
! [B_74: int,A_90: int] :
( ( ord_less_int @ zero_zero_int @ A_90 )
=> ( ( ord_less_int @ zero_zero_int @ B_74 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_90 @ B_74 ) ) ) ) ).
thf(fact_2185_add__pos__pos,axiom,
! [B_74: nat,A_90: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_90 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_74 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A_90 @ B_74 ) ) ) ) ).
thf(fact_2186_add__pos__pos,axiom,
! [B_74: real,A_90: real] :
( ( ord_less_real @ zero_zero_real @ A_90 )
=> ( ( ord_less_real @ zero_zero_real @ B_74 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A_90 @ B_74 ) ) ) ) ).
thf(fact_2187_add__pos__pos,axiom,
! [B_74: quickcheck_code_int,A_90: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_90 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ B_74 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ ( plus_p1446045655de_int @ A_90 @ B_74 ) ) ) ) ).
thf(fact_2188_add__pos__pos,axiom,
! [B_74: rat,A_90: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_90 )
=> ( ( ord_less_rat @ zero_zero_rat @ B_74 )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A_90 @ B_74 ) ) ) ) ).
thf(fact_2189_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_89: int] :
( ( ord_less_int @ ( plus_plus_int @ A_89 @ A_89 ) @ zero_zero_int )
<=> ( ord_less_int @ A_89 @ zero_zero_int ) ) ).
thf(fact_2190_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_89: real] :
( ( ord_less_real @ ( plus_plus_real @ A_89 @ A_89 ) @ zero_zero_real )
<=> ( ord_less_real @ A_89 @ zero_zero_real ) ) ).
thf(fact_2191_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_89: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A_89 @ A_89 ) @ zero_zero_rat )
<=> ( ord_less_rat @ A_89 @ zero_zero_rat ) ) ).
thf(fact_2192_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_88: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_88 @ A_88 ) )
<=> ( ord_less_int @ zero_zero_int @ A_88 ) ) ).
thf(fact_2193_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_88: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A_88 @ A_88 ) )
<=> ( ord_less_real @ zero_zero_real @ A_88 ) ) ).
thf(fact_2194_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_88: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A_88 @ A_88 ) )
<=> ( ord_less_rat @ zero_zero_rat @ A_88 ) ) ).
thf(fact_2195_add__le__less__mono,axiom,
! [C_36: code_code_numeral,D_10: code_code_numeral,A_87: code_code_numeral,B_73: code_code_numeral] :
( ( ord_le565307924umeral @ A_87 @ B_73 )
=> ( ( ord_le1304079648umeral @ C_36 @ D_10 )
=> ( ord_le1304079648umeral @ ( plus_p1627245867umeral @ A_87 @ C_36 ) @ ( plus_p1627245867umeral @ B_73 @ D_10 ) ) ) ) ).
thf(fact_2196_add__le__less__mono,axiom,
! [C_36: int,D_10: int,A_87: int,B_73: int] :
( ( ord_less_eq_int @ A_87 @ B_73 )
=> ( ( ord_less_int @ C_36 @ D_10 )
=> ( ord_less_int @ ( plus_plus_int @ A_87 @ C_36 ) @ ( plus_plus_int @ B_73 @ D_10 ) ) ) ) ).
thf(fact_2197_add__le__less__mono,axiom,
! [C_36: nat,D_10: nat,A_87: nat,B_73: nat] :
( ( ord_less_eq_nat @ A_87 @ B_73 )
=> ( ( ord_less_nat @ C_36 @ D_10 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_87 @ C_36 ) @ ( plus_plus_nat @ B_73 @ D_10 ) ) ) ) ).
thf(fact_2198_add__le__less__mono,axiom,
! [C_36: real,D_10: real,A_87: real,B_73: real] :
( ( ord_less_eq_real @ A_87 @ B_73 )
=> ( ( ord_less_real @ C_36 @ D_10 )
=> ( ord_less_real @ ( plus_plus_real @ A_87 @ C_36 ) @ ( plus_plus_real @ B_73 @ D_10 ) ) ) ) ).
thf(fact_2199_add__le__less__mono,axiom,
! [C_36: quickcheck_code_int,D_10: quickcheck_code_int,A_87: quickcheck_code_int,B_73: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_87 @ B_73 )
=> ( ( ord_le1860547276de_int @ C_36 @ D_10 )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_87 @ C_36 ) @ ( plus_p1446045655de_int @ B_73 @ D_10 ) ) ) ) ).
thf(fact_2200_add__le__less__mono,axiom,
! [C_36: rat,D_10: rat,A_87: rat,B_73: rat] :
( ( ord_less_eq_rat @ A_87 @ B_73 )
=> ( ( ord_less_rat @ C_36 @ D_10 )
=> ( ord_less_rat @ ( plus_plus_rat @ A_87 @ C_36 ) @ ( plus_plus_rat @ B_73 @ D_10 ) ) ) ) ).
thf(fact_2201_add__less__le__mono,axiom,
! [C_35: code_code_numeral,D_9: code_code_numeral,A_86: code_code_numeral,B_72: code_code_numeral] :
( ( ord_le1304079648umeral @ A_86 @ B_72 )
=> ( ( ord_le565307924umeral @ C_35 @ D_9 )
=> ( ord_le1304079648umeral @ ( plus_p1627245867umeral @ A_86 @ C_35 ) @ ( plus_p1627245867umeral @ B_72 @ D_9 ) ) ) ) ).
thf(fact_2202_add__less__le__mono,axiom,
! [C_35: int,D_9: int,A_86: int,B_72: int] :
( ( ord_less_int @ A_86 @ B_72 )
=> ( ( ord_less_eq_int @ C_35 @ D_9 )
=> ( ord_less_int @ ( plus_plus_int @ A_86 @ C_35 ) @ ( plus_plus_int @ B_72 @ D_9 ) ) ) ) ).
thf(fact_2203_add__less__le__mono,axiom,
! [C_35: nat,D_9: nat,A_86: nat,B_72: nat] :
( ( ord_less_nat @ A_86 @ B_72 )
=> ( ( ord_less_eq_nat @ C_35 @ D_9 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_86 @ C_35 ) @ ( plus_plus_nat @ B_72 @ D_9 ) ) ) ) ).
thf(fact_2204_add__less__le__mono,axiom,
! [C_35: real,D_9: real,A_86: real,B_72: real] :
( ( ord_less_real @ A_86 @ B_72 )
=> ( ( ord_less_eq_real @ C_35 @ D_9 )
=> ( ord_less_real @ ( plus_plus_real @ A_86 @ C_35 ) @ ( plus_plus_real @ B_72 @ D_9 ) ) ) ) ).
thf(fact_2205_add__less__le__mono,axiom,
! [C_35: quickcheck_code_int,D_9: quickcheck_code_int,A_86: quickcheck_code_int,B_72: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_86 @ B_72 )
=> ( ( ord_le258702272de_int @ C_35 @ D_9 )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_86 @ C_35 ) @ ( plus_p1446045655de_int @ B_72 @ D_9 ) ) ) ) ).
thf(fact_2206_add__less__le__mono,axiom,
! [C_35: rat,D_9: rat,A_86: rat,B_72: rat] :
( ( ord_less_rat @ A_86 @ B_72 )
=> ( ( ord_less_eq_rat @ C_35 @ D_9 )
=> ( ord_less_rat @ ( plus_plus_rat @ A_86 @ C_35 ) @ ( plus_plus_rat @ B_72 @ D_9 ) ) ) ) ).
thf(fact_2207_le__iff__diff__le__0,axiom,
! [A_85: int,B_71: int] :
( ( ord_less_eq_int @ A_85 @ B_71 )
<=> ( ord_less_eq_int @ ( minus_minus_int @ A_85 @ B_71 ) @ zero_zero_int ) ) ).
thf(fact_2208_le__iff__diff__le__0,axiom,
! [A_85: real,B_71: real] :
( ( ord_less_eq_real @ A_85 @ B_71 )
<=> ( ord_less_eq_real @ ( minus_minus_real @ A_85 @ B_71 ) @ zero_zero_real ) ) ).
thf(fact_2209_le__iff__diff__le__0,axiom,
! [A_85: rat,B_71: rat] :
( ( ord_less_eq_rat @ A_85 @ B_71 )
<=> ( ord_less_eq_rat @ ( minus_minus_rat @ A_85 @ B_71 ) @ zero_zero_rat ) ) ).
thf(fact_2210_less__iff__diff__less__0,axiom,
! [A_84: int,B_70: int] :
( ( ord_less_int @ A_84 @ B_70 )
<=> ( ord_less_int @ ( minus_minus_int @ A_84 @ B_70 ) @ zero_zero_int ) ) ).
thf(fact_2211_less__iff__diff__less__0,axiom,
! [A_84: real,B_70: real] :
( ( ord_less_real @ A_84 @ B_70 )
<=> ( ord_less_real @ ( minus_minus_real @ A_84 @ B_70 ) @ zero_zero_real ) ) ).
thf(fact_2212_less__iff__diff__less__0,axiom,
! [A_84: rat,B_70: rat] :
( ( ord_less_rat @ A_84 @ B_70 )
<=> ( ord_less_rat @ ( minus_minus_rat @ A_84 @ B_70 ) @ zero_zero_rat ) ) ).
thf(fact_2213_zfact_Osimps,axiom,
! [N: int] :
( ( ( ord_less_eq_int @ N @ zero_zero_int )
=> ( ( zfact @ N )
= one_one_int ) )
& ( ~ ( ord_less_eq_int @ N @ zero_zero_int )
=> ( ( zfact @ N )
= ( times_times_int @ N @ ( zfact @ ( minus_minus_int @ N @ one_one_int ) ) ) ) ) ) ).
thf(fact_2214_wset__mem__inv__mem,axiom,
! [B: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_eq_int @ ( number_number_of_int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P_3 )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( member_int @ B @ ( wset @ A @ P_3 ) )
=> ( member_int @ ( inv @ P_3 @ B ) @ ( wset @ A @ P_3 ) ) ) ) ) ) ).
thf(fact_2215_wset__inv__mem__mem,axiom,
! [B: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_eq_int @ ( number_number_of_int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P_3 )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ B @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( member_int @ ( inv @ P_3 @ B ) @ ( wset @ A @ P_3 ) )
=> ( member_int @ B @ ( wset @ A @ P_3 ) ) ) ) ) ) ) ) ).
thf(fact_2216_MultInvPair__distinct,axiom,
! [J: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ~ ( zcong @ J @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A )
=> ~ ( zcong @ J @ ( times_times_int @ A @ ( multInv @ P_3 @ J ) ) @ P_3 ) ) ) ) ) ) ).
thf(fact_2217_int__le__induct,axiom,
! [P: int > $o,I: int,K_1: int] :
( ( ord_less_eq_int @ I @ K_1 )
=> ( ( P @ K_1 )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ I_1 @ K_1 )
=> ( ( P @ I_1 )
=> ( P @ ( minus_minus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_2218_d22set__induct__old,axiom,
! [X: int,P: int > $o] :
( ! [A_2: int] :
( ( ( ord_less_int @ one_one_int @ A_2 )
=> ( P @ ( minus_minus_int @ A_2 @ one_one_int ) ) )
=> ( P @ A_2 ) )
=> ( P @ X ) ) ).
thf(fact_2219_int__less__induct,axiom,
! [P: int > $o,I: int,K_1: int] :
( ( ord_less_int @ I @ K_1 )
=> ( ( P @ ( minus_minus_int @ K_1 @ one_one_int ) )
=> ( ! [I_1: int] :
( ( ord_less_int @ I_1 @ K_1 )
=> ( ( P @ I_1 )
=> ( P @ ( minus_minus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_2220_int__ge__induct,axiom,
! [P: int > $o,K_1: int,I: int] :
( ( ord_less_eq_int @ K_1 @ I )
=> ( ( P @ K_1 )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ K_1 @ I_1 )
=> ( ( P @ I_1 )
=> ( P @ ( plus_plus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_2221_aux______1,axiom,
! [J: int,A: int,P_3: int,K_1: int] :
( ( zcong @ J @ ( times_times_int @ A @ ( multInv @ P_3 @ K_1 ) ) @ P_3 )
=> ( zcong @ ( times_times_int @ J @ K_1 ) @ ( times_times_int @ ( times_times_int @ A @ ( multInv @ P_3 @ K_1 ) ) @ K_1 ) @ P_3 ) ) ).
thf(fact_2222_aux______3,axiom,
! [J: int,K_1: int,A: int,P_3: int] :
( ( zcong @ ( times_times_int @ J @ K_1 ) @ A @ P_3 )
=> ( zcong @ ( times_times_int @ ( times_times_int @ ( multInv @ P_3 @ J ) @ J ) @ K_1 ) @ ( times_times_int @ ( multInv @ P_3 @ J ) @ A ) @ P_3 ) ) ).
thf(fact_2223_wset__mem__mem,axiom,
! [P_3: int,A: int] :
( ( ord_less_int @ one_one_int @ A )
=> ( member_int @ A @ ( wset @ A @ P_3 ) ) ) ).
thf(fact_2224_wset__subset,axiom,
! [B: int,P_3: int,A: int] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( member_int @ B @ ( wset @ ( minus_minus_int @ A @ one_one_int ) @ P_3 ) )
=> ( member_int @ B @ ( wset @ A @ P_3 ) ) ) ) ).
thf(fact_2225_wset__g__1,axiom,
! [B: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( member_int @ B @ ( wset @ A @ P_3 ) )
=> ( ord_less_int @ one_one_int @ B ) ) ) ) ).
thf(fact_2226_wset__less,axiom,
! [B: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( member_int @ B @ ( wset @ A @ P_3 ) )
=> ( ord_less_int @ B @ ( minus_minus_int @ P_3 @ one_one_int ) ) ) ) ) ).
thf(fact_2227_wset__mem__imp__or,axiom,
! [B: int,P_3: int,A: int] :
( ( ord_less_int @ one_one_int @ A )
=> ( ~ ( member_int @ B @ ( wset @ ( minus_minus_int @ A @ one_one_int ) @ P_3 ) )
=> ( ( member_int @ B @ ( wset @ A @ P_3 ) )
=> ( ( B = A )
| ( B
= ( inv @ P_3 @ A ) ) ) ) ) ) ).
thf(fact_2228_MultInv__prop1,axiom,
! [X: int,Y: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zcong @ X @ Y @ P_3 )
=> ( zcong @ ( multInv @ P_3 @ X ) @ ( multInv @ P_3 @ Y ) @ P_3 ) ) ) ).
thf(fact_2229_wset__mem,axiom,
! [B: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( member_int @ B @ ( wset @ A @ P_3 ) ) ) ) ) ) ).
thf(fact_2230_MultInv__zcong__prop1,axiom,
! [A: int,J: int,K_1: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zcong @ J @ K_1 @ P_3 )
=> ( zcong @ ( times_times_int @ A @ ( multInv @ P_3 @ J ) ) @ ( times_times_int @ A @ ( multInv @ P_3 @ K_1 ) ) @ P_3 ) ) ) ).
thf(fact_2231_MultInv__prop3,axiom,
! [X: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ X @ zero_zero_int @ P_3 )
=> ~ ( zcong @ ( multInv @ P_3 @ X ) @ zero_zero_int @ P_3 ) ) ) ) ).
thf(fact_2232_MultInv__prop4,axiom,
! [X: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ X @ zero_zero_int @ P_3 )
=> ( zcong @ ( multInv @ P_3 @ ( multInv @ P_3 @ X ) ) @ X @ P_3 ) ) ) ) ).
thf(fact_2233_MultInv__prop5,axiom,
! [Y: int,X: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ X @ zero_zero_int @ P_3 )
=> ( ~ ( zcong @ Y @ zero_zero_int @ P_3 )
=> ( ( zcong @ ( multInv @ P_3 @ X ) @ ( multInv @ P_3 @ Y ) @ P_3 )
=> ( zcong @ X @ Y @ P_3 ) ) ) ) ) ) ).
thf(fact_2234_Int2_Oaux____1,axiom,
! [X: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ X @ zero_zero_int @ P_3 )
=> ( zcong @ ( multInv @ P_3 @ ( multInv @ P_3 @ X ) ) @ ( times_times_int @ ( times_times_int @ X @ ( multInv @ P_3 @ X ) ) @ ( multInv @ P_3 @ ( multInv @ P_3 @ X ) ) ) @ P_3 ) ) ) ) ).
thf(fact_2235_Int2_Oaux____2,axiom,
! [X: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ X @ zero_zero_int @ P_3 )
=> ( zcong @ ( times_times_int @ ( times_times_int @ X @ ( multInv @ P_3 @ X ) ) @ ( multInv @ P_3 @ ( multInv @ P_3 @ X ) ) ) @ X @ P_3 ) ) ) ) ).
thf(fact_2236_MultInv__zcong__prop3,axiom,
! [J: int,K_1: int,A: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ~ ( zcong @ K_1 @ zero_zero_int @ P_3 )
=> ( ~ ( zcong @ J @ zero_zero_int @ P_3 )
=> ( ( zcong @ ( times_times_int @ A @ ( multInv @ P_3 @ J ) ) @ ( times_times_int @ A @ ( multInv @ P_3 @ K_1 ) ) @ P_3 )
=> ( zcong @ J @ K_1 @ P_3 ) ) ) ) ) ) ) ).
thf(fact_2237_MultInv__zcong__prop2,axiom,
! [A: int,J: int,K_1: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ K_1 @ zero_zero_int @ P_3 )
=> ( ~ ( zcong @ J @ zero_zero_int @ P_3 )
=> ( ( zcong @ J @ ( times_times_int @ A @ ( multInv @ P_3 @ K_1 ) ) @ P_3 )
=> ( zcong @ K_1 @ ( times_times_int @ A @ ( multInv @ P_3 @ J ) ) @ P_3 ) ) ) ) ) ) ).
thf(fact_2238_aux______2,axiom,
! [J: int,A: int,K_1: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ K_1 @ zero_zero_int @ P_3 )
=> ( ( zcong @ ( times_times_int @ J @ K_1 ) @ ( times_times_int @ ( times_times_int @ A @ ( multInv @ P_3 @ K_1 ) ) @ K_1 ) @ P_3 )
=> ( zcong @ ( times_times_int @ J @ K_1 ) @ A @ P_3 ) ) ) ) ) ).
thf(fact_2239_aux______4,axiom,
! [K_1: int,A: int,J: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ J @ zero_zero_int @ P_3 )
=> ( ( zcong @ ( times_times_int @ ( times_times_int @ ( multInv @ P_3 @ J ) @ J ) @ K_1 ) @ ( times_times_int @ ( multInv @ P_3 @ J ) @ A ) @ P_3 )
=> ( zcong @ K_1 @ ( times_times_int @ A @ ( multInv @ P_3 @ J ) ) @ P_3 ) ) ) ) ) ).
thf(fact_2240_MultInv__prop2,axiom,
! [X: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ X @ zero_zero_int @ P_3 )
=> ( zcong @ ( times_times_int @ X @ ( multInv @ P_3 @ X ) ) @ one_one_int @ P_3 ) ) ) ) ).
thf(fact_2241_MultInv__prop2a,axiom,
! [X: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ X @ zero_zero_int @ P_3 )
=> ( zcong @ ( times_times_int @ ( multInv @ P_3 @ X ) @ X ) @ one_one_int @ P_3 ) ) ) ) ).
thf(fact_2242_int__gr__induct,axiom,
! [P: int > $o,K_1: int,I: int] :
( ( ord_less_int @ K_1 @ I )
=> ( ( P @ ( plus_plus_int @ K_1 @ one_one_int ) )
=> ( ! [I_1: int] :
( ( ord_less_int @ K_1 @ I_1 )
=> ( ( P @ I_1 )
=> ( P @ ( plus_plus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_2243_mono__nat__linear__lb,axiom,
! [M: nat,K_1: nat,F: nat > nat] :
( ! [M_2: nat,N_1: nat] :
( ( ord_less_nat @ M_2 @ N_1 )
=> ( ord_less_nat @ ( F @ M_2 ) @ ( F @ N_1 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K_1 ) @ ( F @ ( plus_plus_nat @ M @ K_1 ) ) ) ) ).
thf(fact_2244_d22set__eq__wset,axiom,
! [P_3: int] :
( ( zprime @ P_3 )
=> ( ( d22set @ ( minus_minus_int @ P_3 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( wset @ ( minus_minus_int @ P_3 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P_3 ) ) ) ).
thf(fact_2245_ex__least__nat__le,axiom,
! [N: nat,P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ( P @ N )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N )
& ! [I_1: nat] :
( ( ord_less_nat @ I_1 @ K )
=> ~ ( P @ I_1 ) )
& ( P @ K ) ) ) ) ).
thf(fact_2246_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
& ( ( plus_plus_nat @ I @ K )
= J ) ) ) ).
thf(fact_2247_minf_I10_J,axiom,
! [D_8: code_code_numeral,S_7: code_code_numeral] :
? [Z: code_code_numeral] :
! [X_1: code_code_numeral] :
( ( ord_le1304079648umeral @ X_1 @ Z )
=> ( ~ ( dvd_dv174992974umeral @ D_8 @ ( plus_p1627245867umeral @ X_1 @ S_7 ) )
<=> ~ ( dvd_dv174992974umeral @ D_8 @ ( plus_p1627245867umeral @ X_1 @ S_7 ) ) ) ) ).
thf(fact_2248_minf_I10_J,axiom,
! [D_8: real,S_7: real] :
? [Z: real] :
! [X_1: real] :
( ( ord_less_real @ X_1 @ Z )
=> ( ~ ( dvd_dvd_real @ D_8 @ ( plus_plus_real @ X_1 @ S_7 ) )
<=> ~ ( dvd_dvd_real @ D_8 @ ( plus_plus_real @ X_1 @ S_7 ) ) ) ) ).
thf(fact_2249_minf_I10_J,axiom,
! [D_8: quickcheck_code_int,S_7: quickcheck_code_int] :
? [Z: quickcheck_code_int] :
! [X_1: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_1 @ Z )
=> ( ~ ( dvd_dv1760642554de_int @ D_8 @ ( plus_p1446045655de_int @ X_1 @ S_7 ) )
<=> ~ ( dvd_dv1760642554de_int @ D_8 @ ( plus_p1446045655de_int @ X_1 @ S_7 ) ) ) ) ).
thf(fact_2250_minf_I10_J,axiom,
! [D_8: rat,S_7: rat] :
? [Z: rat] :
! [X_1: rat] :
( ( ord_less_rat @ X_1 @ Z )
=> ( ~ ( dvd_dvd_rat @ D_8 @ ( plus_plus_rat @ X_1 @ S_7 ) )
<=> ~ ( dvd_dvd_rat @ D_8 @ ( plus_plus_rat @ X_1 @ S_7 ) ) ) ) ).
thf(fact_2251_minf_I10_J,axiom,
! [D_8: int,S_7: int] :
? [Z: int] :
! [X_1: int] :
( ( ord_less_int @ X_1 @ Z )
=> ( ~ ( dvd_dvd_int @ D_8 @ ( plus_plus_int @ X_1 @ S_7 ) )
<=> ~ ( dvd_dvd_int @ D_8 @ ( plus_plus_int @ X_1 @ S_7 ) ) ) ) ).
thf(fact_2252_minf_I10_J,axiom,
! [D_8: nat,S_7: nat] :
? [Z: nat] :
! [X_1: nat] :
( ( ord_less_nat @ X_1 @ Z )
=> ( ~ ( dvd_dvd_nat @ D_8 @ ( plus_plus_nat @ X_1 @ S_7 ) )
<=> ~ ( dvd_dvd_nat @ D_8 @ ( plus_plus_nat @ X_1 @ S_7 ) ) ) ) ).
thf(fact_2253_pinf_I10_J,axiom,
! [D_7: code_code_numeral,S_6: code_code_numeral] :
? [Z: code_code_numeral] :
! [X_1: code_code_numeral] :
( ( ord_le1304079648umeral @ Z @ X_1 )
=> ( ~ ( dvd_dv174992974umeral @ D_7 @ ( plus_p1627245867umeral @ X_1 @ S_6 ) )
<=> ~ ( dvd_dv174992974umeral @ D_7 @ ( plus_p1627245867umeral @ X_1 @ S_6 ) ) ) ) ).
thf(fact_2254_pinf_I10_J,axiom,
! [D_7: real,S_6: real] :
? [Z: real] :
! [X_1: real] :
( ( ord_less_real @ Z @ X_1 )
=> ( ~ ( dvd_dvd_real @ D_7 @ ( plus_plus_real @ X_1 @ S_6 ) )
<=> ~ ( dvd_dvd_real @ D_7 @ ( plus_plus_real @ X_1 @ S_6 ) ) ) ) ).
thf(fact_2255_pinf_I10_J,axiom,
! [D_7: quickcheck_code_int,S_6: quickcheck_code_int] :
? [Z: quickcheck_code_int] :
! [X_1: quickcheck_code_int] :
( ( ord_le1860547276de_int @ Z @ X_1 )
=> ( ~ ( dvd_dv1760642554de_int @ D_7 @ ( plus_p1446045655de_int @ X_1 @ S_6 ) )
<=> ~ ( dvd_dv1760642554de_int @ D_7 @ ( plus_p1446045655de_int @ X_1 @ S_6 ) ) ) ) ).
thf(fact_2256_pinf_I10_J,axiom,
! [D_7: rat,S_6: rat] :
? [Z: rat] :
! [X_1: rat] :
( ( ord_less_rat @ Z @ X_1 )
=> ( ~ ( dvd_dvd_rat @ D_7 @ ( plus_plus_rat @ X_1 @ S_6 ) )
<=> ~ ( dvd_dvd_rat @ D_7 @ ( plus_plus_rat @ X_1 @ S_6 ) ) ) ) ).
thf(fact_2257_pinf_I10_J,axiom,
! [D_7: int,S_6: int] :
? [Z: int] :
! [X_1: int] :
( ( ord_less_int @ Z @ X_1 )
=> ( ~ ( dvd_dvd_int @ D_7 @ ( plus_plus_int @ X_1 @ S_6 ) )
<=> ~ ( dvd_dvd_int @ D_7 @ ( plus_plus_int @ X_1 @ S_6 ) ) ) ) ).
thf(fact_2258_pinf_I10_J,axiom,
! [D_7: nat,S_6: nat] :
? [Z: nat] :
! [X_1: nat] :
( ( ord_less_nat @ Z @ X_1 )
=> ( ~ ( dvd_dvd_nat @ D_7 @ ( plus_plus_nat @ X_1 @ S_6 ) )
<=> ~ ( dvd_dvd_nat @ D_7 @ ( plus_plus_nat @ X_1 @ S_6 ) ) ) ) ).
thf(fact_2259_minf_I9_J,axiom,
! [D_6: code_code_numeral,S_5: code_code_numeral] :
? [Z: code_code_numeral] :
! [X_1: code_code_numeral] :
( ( ord_le1304079648umeral @ X_1 @ Z )
=> ( ( dvd_dv174992974umeral @ D_6 @ ( plus_p1627245867umeral @ X_1 @ S_5 ) )
<=> ( dvd_dv174992974umeral @ D_6 @ ( plus_p1627245867umeral @ X_1 @ S_5 ) ) ) ) ).
thf(fact_2260_minf_I9_J,axiom,
! [D_6: real,S_5: real] :
? [Z: real] :
! [X_1: real] :
( ( ord_less_real @ X_1 @ Z )
=> ( ( dvd_dvd_real @ D_6 @ ( plus_plus_real @ X_1 @ S_5 ) )
<=> ( dvd_dvd_real @ D_6 @ ( plus_plus_real @ X_1 @ S_5 ) ) ) ) ).
thf(fact_2261_minf_I9_J,axiom,
! [D_6: quickcheck_code_int,S_5: quickcheck_code_int] :
? [Z: quickcheck_code_int] :
! [X_1: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_1 @ Z )
=> ( ( dvd_dv1760642554de_int @ D_6 @ ( plus_p1446045655de_int @ X_1 @ S_5 ) )
<=> ( dvd_dv1760642554de_int @ D_6 @ ( plus_p1446045655de_int @ X_1 @ S_5 ) ) ) ) ).
thf(fact_2262_minf_I9_J,axiom,
! [D_6: rat,S_5: rat] :
? [Z: rat] :
! [X_1: rat] :
( ( ord_less_rat @ X_1 @ Z )
=> ( ( dvd_dvd_rat @ D_6 @ ( plus_plus_rat @ X_1 @ S_5 ) )
<=> ( dvd_dvd_rat @ D_6 @ ( plus_plus_rat @ X_1 @ S_5 ) ) ) ) ).
thf(fact_2263_minf_I9_J,axiom,
! [D_6: int,S_5: int] :
? [Z: int] :
! [X_1: int] :
( ( ord_less_int @ X_1 @ Z )
=> ( ( dvd_dvd_int @ D_6 @ ( plus_plus_int @ X_1 @ S_5 ) )
<=> ( dvd_dvd_int @ D_6 @ ( plus_plus_int @ X_1 @ S_5 ) ) ) ) ).
thf(fact_2264_minf_I9_J,axiom,
! [D_6: nat,S_5: nat] :
? [Z: nat] :
! [X_1: nat] :
( ( ord_less_nat @ X_1 @ Z )
=> ( ( dvd_dvd_nat @ D_6 @ ( plus_plus_nat @ X_1 @ S_5 ) )
<=> ( dvd_dvd_nat @ D_6 @ ( plus_plus_nat @ X_1 @ S_5 ) ) ) ) ).
thf(fact_2265_d22set__le__swap,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( member_int @ B @ ( d22set @ A ) ) ) ).
thf(fact_2266_d22set__le,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( d22set @ A ) )
=> ( ord_less_eq_int @ B @ A ) ) ).
thf(fact_2267_d22set__g__1,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( d22set @ A ) )
=> ( ord_less_int @ one_one_int @ B ) ) ).
thf(fact_2268_d22set__mem,axiom,
! [A: int,B: int] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( member_int @ B @ ( d22set @ A ) ) ) ) ).
thf(fact_2269_pinf_I9_J,axiom,
! [D_5: code_code_numeral,S_4: code_code_numeral] :
? [Z: code_code_numeral] :
! [X_1: code_code_numeral] :
( ( ord_le1304079648umeral @ Z @ X_1 )
=> ( ( dvd_dv174992974umeral @ D_5 @ ( plus_p1627245867umeral @ X_1 @ S_4 ) )
<=> ( dvd_dv174992974umeral @ D_5 @ ( plus_p1627245867umeral @ X_1 @ S_4 ) ) ) ) ).
thf(fact_2270_pinf_I9_J,axiom,
! [D_5: real,S_4: real] :
? [Z: real] :
! [X_1: real] :
( ( ord_less_real @ Z @ X_1 )
=> ( ( dvd_dvd_real @ D_5 @ ( plus_plus_real @ X_1 @ S_4 ) )
<=> ( dvd_dvd_real @ D_5 @ ( plus_plus_real @ X_1 @ S_4 ) ) ) ) ).
thf(fact_2271_pinf_I9_J,axiom,
! [D_5: quickcheck_code_int,S_4: quickcheck_code_int] :
? [Z: quickcheck_code_int] :
! [X_1: quickcheck_code_int] :
( ( ord_le1860547276de_int @ Z @ X_1 )
=> ( ( dvd_dv1760642554de_int @ D_5 @ ( plus_p1446045655de_int @ X_1 @ S_4 ) )
<=> ( dvd_dv1760642554de_int @ D_5 @ ( plus_p1446045655de_int @ X_1 @ S_4 ) ) ) ) ).
thf(fact_2272_pinf_I9_J,axiom,
! [D_5: rat,S_4: rat] :
? [Z: rat] :
! [X_1: rat] :
( ( ord_less_rat @ Z @ X_1 )
=> ( ( dvd_dvd_rat @ D_5 @ ( plus_plus_rat @ X_1 @ S_4 ) )
<=> ( dvd_dvd_rat @ D_5 @ ( plus_plus_rat @ X_1 @ S_4 ) ) ) ) ).
thf(fact_2273_pinf_I9_J,axiom,
! [D_5: int,S_4: int] :
? [Z: int] :
! [X_1: int] :
( ( ord_less_int @ Z @ X_1 )
=> ( ( dvd_dvd_int @ D_5 @ ( plus_plus_int @ X_1 @ S_4 ) )
<=> ( dvd_dvd_int @ D_5 @ ( plus_plus_int @ X_1 @ S_4 ) ) ) ) ).
thf(fact_2274_pinf_I9_J,axiom,
! [D_5: nat,S_4: nat] :
? [Z: nat] :
! [X_1: nat] :
( ( ord_less_nat @ Z @ X_1 )
=> ( ( dvd_dvd_nat @ D_5 @ ( plus_plus_nat @ X_1 @ S_4 ) )
<=> ( dvd_dvd_nat @ D_5 @ ( plus_plus_nat @ X_1 @ S_4 ) ) ) ) ).
thf(fact_2275_pow__divides__eq__int,axiom,
! [A: int,B: int,N: nat] :
( ( N != zero_zero_nat )
=> ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
<=> ( dvd_dvd_int @ A @ B ) ) ) ).
thf(fact_2276_pow__divides__pow__int,axiom,
! [A: int,N: nat,B: int] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( N != zero_zero_nat )
=> ( dvd_dvd_int @ A @ B ) ) ) ).
thf(fact_2277_pow__divides__eq__nat,axiom,
! [A: nat,B: nat,N: nat] :
( ( N != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
<=> ( dvd_dvd_nat @ A @ B ) ) ) ).
thf(fact_2278_divides__le,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( ord_less_eq_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
thf(fact_2279_mult__left__cancel,axiom,
! [N: nat,M: nat,K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ( ( times_times_nat @ K_1 @ N )
= ( times_times_nat @ K_1 @ M ) )
=> ( N = M ) ) ) ).
thf(fact_2280_dvd__pos__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
thf(fact_2281_mult_Oprod__diff__prod,axiom,
! [X_19: real,Y_15: real,A_83: real,B_69: real] :
( ( minus_minus_real @ ( times_times_real @ X_19 @ Y_15 ) @ ( times_times_real @ A_83 @ B_69 ) )
= ( plus_plus_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ X_19 @ A_83 ) @ ( minus_minus_real @ Y_15 @ B_69 ) ) @ ( times_times_real @ ( minus_minus_real @ X_19 @ A_83 ) @ B_69 ) ) @ ( times_times_real @ A_83 @ ( minus_minus_real @ Y_15 @ B_69 ) ) ) ) ).
thf(fact_2282_mult_Oprod__diff__prod,axiom,
! [X_19: complex,Y_15: complex,A_83: complex,B_69: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X_19 @ Y_15 ) @ ( times_times_complex @ A_83 @ B_69 ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ X_19 @ A_83 ) @ ( minus_minus_complex @ Y_15 @ B_69 ) ) @ ( times_times_complex @ ( minus_minus_complex @ X_19 @ A_83 ) @ B_69 ) ) @ ( times_times_complex @ A_83 @ ( minus_minus_complex @ Y_15 @ B_69 ) ) ) ) ).
thf(fact_2283_mult_Ozero__left,axiom,
! [B_68: real] :
( ( times_times_real @ zero_zero_real @ B_68 )
= zero_zero_real ) ).
thf(fact_2284_mult_Ozero__left,axiom,
! [B_68: complex] :
( ( times_times_complex @ zero_zero_complex @ B_68 )
= zero_zero_complex ) ).
thf(fact_2285_mult__left_Ozero,axiom,
! [Y_14: real] :
( ( times_times_real @ zero_zero_real @ Y_14 )
= zero_zero_real ) ).
thf(fact_2286_mult__left_Ozero,axiom,
! [Y_14: complex] :
( ( times_times_complex @ zero_zero_complex @ Y_14 )
= zero_zero_complex ) ).
thf(fact_2287_mult_Ozero__right,axiom,
! [A_82: real] :
( ( times_times_real @ A_82 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_2288_mult_Ozero__right,axiom,
! [A_82: complex] :
( ( times_times_complex @ A_82 @ zero_zero_complex )
= zero_zero_complex ) ).
thf(fact_2289_mult__right_Ozero,axiom,
! [X_18: real] :
( ( times_times_real @ X_18 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_2290_mult__right_Ozero,axiom,
! [X_18: complex] :
( ( times_times_complex @ X_18 @ zero_zero_complex )
= zero_zero_complex ) ).
thf(fact_2291_mult_Oadd__right,axiom,
! [A_81: real,B_67: real,B_66: real] :
( ( times_times_real @ A_81 @ ( plus_plus_real @ B_67 @ B_66 ) )
= ( plus_plus_real @ ( times_times_real @ A_81 @ B_67 ) @ ( times_times_real @ A_81 @ B_66 ) ) ) ).
thf(fact_2292_mult_Oadd__right,axiom,
! [A_81: complex,B_67: complex,B_66: complex] :
( ( times_times_complex @ A_81 @ ( plus_plus_complex @ B_67 @ B_66 ) )
= ( plus_plus_complex @ ( times_times_complex @ A_81 @ B_67 ) @ ( times_times_complex @ A_81 @ B_66 ) ) ) ).
thf(fact_2293_mult__right_Oadd,axiom,
! [Xa_3: real,X_17: real,Y_13: real] :
( ( times_times_real @ Xa_3 @ ( plus_plus_real @ X_17 @ Y_13 ) )
= ( plus_plus_real @ ( times_times_real @ Xa_3 @ X_17 ) @ ( times_times_real @ Xa_3 @ Y_13 ) ) ) ).
thf(fact_2294_mult__right_Oadd,axiom,
! [Xa_3: complex,X_17: complex,Y_13: complex] :
( ( times_times_complex @ Xa_3 @ ( plus_plus_complex @ X_17 @ Y_13 ) )
= ( plus_plus_complex @ ( times_times_complex @ Xa_3 @ X_17 ) @ ( times_times_complex @ Xa_3 @ Y_13 ) ) ) ).
thf(fact_2295_mult_Oadd__left,axiom,
! [A_80: real,A_79: real,B_65: real] :
( ( times_times_real @ ( plus_plus_real @ A_80 @ A_79 ) @ B_65 )
= ( plus_plus_real @ ( times_times_real @ A_80 @ B_65 ) @ ( times_times_real @ A_79 @ B_65 ) ) ) ).
thf(fact_2296_mult_Oadd__left,axiom,
! [A_80: complex,A_79: complex,B_65: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A_80 @ A_79 ) @ B_65 )
= ( plus_plus_complex @ ( times_times_complex @ A_80 @ B_65 ) @ ( times_times_complex @ A_79 @ B_65 ) ) ) ).
thf(fact_2297_mult__left_Oadd,axiom,
! [X_16: real,Y_12: real,Ya_2: real] :
( ( times_times_real @ ( plus_plus_real @ X_16 @ Y_12 ) @ Ya_2 )
= ( plus_plus_real @ ( times_times_real @ X_16 @ Ya_2 ) @ ( times_times_real @ Y_12 @ Ya_2 ) ) ) ).
thf(fact_2298_mult__left_Oadd,axiom,
! [X_16: complex,Y_12: complex,Ya_2: complex] :
( ( times_times_complex @ ( plus_plus_complex @ X_16 @ Y_12 ) @ Ya_2 )
= ( plus_plus_complex @ ( times_times_complex @ X_16 @ Ya_2 ) @ ( times_times_complex @ Y_12 @ Ya_2 ) ) ) ).
thf(fact_2299_mult__left_Odiff,axiom,
! [X_15: real,Y_11: real,Ya_1: real] :
( ( times_times_real @ ( minus_minus_real @ X_15 @ Y_11 ) @ Ya_1 )
= ( minus_minus_real @ ( times_times_real @ X_15 @ Ya_1 ) @ ( times_times_real @ Y_11 @ Ya_1 ) ) ) ).
thf(fact_2300_mult__left_Odiff,axiom,
! [X_15: complex,Y_11: complex,Ya_1: complex] :
( ( times_times_complex @ ( minus_minus_complex @ X_15 @ Y_11 ) @ Ya_1 )
= ( minus_minus_complex @ ( times_times_complex @ X_15 @ Ya_1 ) @ ( times_times_complex @ Y_11 @ Ya_1 ) ) ) ).
thf(fact_2301_mult_Odiff__left,axiom,
! [A_78: real,A_77: real,B_64: real] :
( ( times_times_real @ ( minus_minus_real @ A_78 @ A_77 ) @ B_64 )
= ( minus_minus_real @ ( times_times_real @ A_78 @ B_64 ) @ ( times_times_real @ A_77 @ B_64 ) ) ) ).
thf(fact_2302_mult_Odiff__left,axiom,
! [A_78: complex,A_77: complex,B_64: complex] :
( ( times_times_complex @ ( minus_minus_complex @ A_78 @ A_77 ) @ B_64 )
= ( minus_minus_complex @ ( times_times_complex @ A_78 @ B_64 ) @ ( times_times_complex @ A_77 @ B_64 ) ) ) ).
thf(fact_2303_mult__right_Odiff,axiom,
! [Xa_2: real,X_14: real,Y_10: real] :
( ( times_times_real @ Xa_2 @ ( minus_minus_real @ X_14 @ Y_10 ) )
= ( minus_minus_real @ ( times_times_real @ Xa_2 @ X_14 ) @ ( times_times_real @ Xa_2 @ Y_10 ) ) ) ).
thf(fact_2304_mult__right_Odiff,axiom,
! [Xa_2: complex,X_14: complex,Y_10: complex] :
( ( times_times_complex @ Xa_2 @ ( minus_minus_complex @ X_14 @ Y_10 ) )
= ( minus_minus_complex @ ( times_times_complex @ Xa_2 @ X_14 ) @ ( times_times_complex @ Xa_2 @ Y_10 ) ) ) ).
thf(fact_2305_mult_Odiff__right,axiom,
! [A_76: real,B_63: real,B_62: real] :
( ( times_times_real @ A_76 @ ( minus_minus_real @ B_63 @ B_62 ) )
= ( minus_minus_real @ ( times_times_real @ A_76 @ B_63 ) @ ( times_times_real @ A_76 @ B_62 ) ) ) ).
thf(fact_2306_mult_Odiff__right,axiom,
! [A_76: complex,B_63: complex,B_62: complex] :
( ( times_times_complex @ A_76 @ ( minus_minus_complex @ B_63 @ B_62 ) )
= ( minus_minus_complex @ ( times_times_complex @ A_76 @ B_63 ) @ ( times_times_complex @ A_76 @ B_62 ) ) ) ).
thf(fact_2307_gcd__lcm__complete__lattice__nat_Onot__top__less,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ~ ( dvd_dvd_nat @ A @ zero_zero_nat ) ) ).
thf(fact_2308_gcd__lcm__complete__lattice__nat_Otop__greatest,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
thf(fact_2309_gcd__lcm__complete__lattice__nat_Oless__top,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
<=> ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ~ ( dvd_dvd_nat @ zero_zero_nat @ A ) ) ) ).
thf(fact_2310_gcd__lcm__complete__lattice__nat_Otop__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
<=> ( A = zero_zero_nat ) ) ).
thf(fact_2311_gcd__lcm__complete__lattice__nat_Otop__le,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
thf(fact_2312_gcd__lcm__complete__lattice__nat_Onot__less__bot,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ A @ one_one_nat )
& ~ ( dvd_dvd_nat @ one_one_nat @ A ) ) ).
thf(fact_2313_gcd__lcm__complete__lattice__nat_Obot__least,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
thf(fact_2314_gcd__lcm__complete__lattice__nat_Obot__less,axiom,
! [A: nat] :
( ( A != one_one_nat )
<=> ( ( dvd_dvd_nat @ one_one_nat @ A )
& ~ ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
thf(fact_2315_gcd__lcm__complete__lattice__nat_Obot__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
<=> ( A = one_one_nat ) ) ).
thf(fact_2316_gcd__lcm__complete__lattice__nat_Ole__bot,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( A = one_one_nat ) ) ).
thf(fact_2317_SR__def,axiom,
! [P_3: int] :
( ( sr @ P_3 )
= ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( ord_less_int @ X_1 @ P_3 ) ) ) ) ).
thf(fact_2318_zmod__number__of__Bit1,axiom,
! [V: int,W: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ W ) )
=> ( ( div_mod_int @ ( number_number_of_int @ ( bit1 @ V ) ) @ ( number_number_of_int @ ( bit0 @ W ) ) )
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( div_mod_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) ) ) @ one_one_int ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ W ) )
=> ( ( div_mod_int @ ( number_number_of_int @ ( bit1 @ V ) ) @ ( number_number_of_int @ ( bit0 @ W ) ) )
= ( minus_minus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( div_mod_int @ ( plus_plus_int @ ( number_number_of_int @ V ) @ one_one_int ) @ ( number_number_of_int @ W ) ) ) @ one_one_int ) ) ) ) ).
thf(fact_2319_neg__zmod__mult__2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( div_mod_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) )
= ( minus_minus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( div_mod_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).
thf(fact_2320_StandardRes__prop4,axiom,
! [X: int,Y: int,M: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ M )
=> ( zcong @ ( times_times_int @ ( standardRes @ M @ X ) @ ( standardRes @ M @ Y ) ) @ ( times_times_int @ X @ Y ) @ M ) ) ).
thf(fact_2321_negDivAlg__eqn__1__number__of,axiom,
! [W: int] :
( ( ord_less_int @ zero_zero_int @ ( number_number_of_int @ W ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( number_number_of_int @ W ) ) )
=> ( ( negDivAlg @ one_one_int @ ( number_number_of_int @ W ) )
= ( product_Pair_int_int @ ( number_number_of_int @ min ) @ ( plus_plus_int @ one_one_int @ ( number_number_of_int @ W ) ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( number_number_of_int @ W ) ) )
=> ( ( negDivAlg @ one_one_int @ ( number_number_of_int @ W ) )
= ( adjust @ ( number_number_of_int @ W ) @ ( negDivAlg @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( number_number_of_int @ W ) ) ) ) ) ) ) ) ).
thf(fact_2322_zOddI,axiom,
! [X: int,K_1: int] :
( ( X
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_1 ) @ one_one_int ) )
=> ( member_int @ X @ zOdd ) ) ).
thf(fact_2323_pos__zmod__mult__2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( div_mod_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( div_mod_int @ B @ A ) ) ) ) ) ).
thf(fact_2324_pos__zdiv__mult__2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( div_div_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) )
= ( div_div_int @ B @ A ) ) ) ).
thf(fact_2325_mod__mod__trivial,axiom,
! [A_75: int,B_61: int] :
( ( div_mod_int @ ( div_mod_int @ A_75 @ B_61 ) @ B_61 )
= ( div_mod_int @ A_75 @ B_61 ) ) ).
thf(fact_2326_mod__mod__trivial,axiom,
! [A_75: nat,B_61: nat] :
( ( div_mod_nat @ ( div_mod_nat @ A_75 @ B_61 ) @ B_61 )
= ( div_mod_nat @ A_75 @ B_61 ) ) ).
thf(fact_2327_mod__mod__trivial,axiom,
! [A_75: code_code_numeral,B_61: code_code_numeral] :
( ( div_mo1740067990umeral @ ( div_mo1740067990umeral @ A_75 @ B_61 ) @ B_61 )
= ( div_mo1740067990umeral @ A_75 @ B_61 ) ) ).
thf(fact_2328_mod__mod__trivial,axiom,
! [A_75: quickcheck_code_int,B_61: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( div_mo231679042de_int @ A_75 @ B_61 ) @ B_61 )
= ( div_mo231679042de_int @ A_75 @ B_61 ) ) ).
thf(fact_2329_zmod__zdiv__trivial,axiom,
! [A: int,B: int] :
( ( div_div_int @ ( div_mod_int @ A @ B ) @ B )
= zero_zero_int ) ).
thf(fact_2330_DIVISION__BY__ZERO,axiom,
! [A: int] :
( ( ( div_div_int @ A @ zero_zero_int )
= zero_zero_int )
& ( ( div_mod_int @ A @ zero_zero_int )
= A ) ) ).
thf(fact_2331_mod__div__trivial,axiom,
! [A_74: int,B_60: int] :
( ( div_div_int @ ( div_mod_int @ A_74 @ B_60 ) @ B_60 )
= zero_zero_int ) ).
thf(fact_2332_mod__div__trivial,axiom,
! [A_74: nat,B_60: nat] :
( ( div_div_nat @ ( div_mod_nat @ A_74 @ B_60 ) @ B_60 )
= zero_zero_nat ) ).
thf(fact_2333_mod__div__trivial,axiom,
! [A_74: code_code_numeral,B_60: code_code_numeral] :
( ( div_di1218280263umeral @ ( div_mo1740067990umeral @ A_74 @ B_60 ) @ B_60 )
= zero_z126310315umeral ) ).
thf(fact_2334_mod__div__trivial,axiom,
! [A_74: quickcheck_code_int,B_60: quickcheck_code_int] :
( ( div_di1430059507de_int @ ( div_mo231679042de_int @ A_74 @ B_60 ) @ B_60 )
= zero_z891286103de_int ) ).
thf(fact_2335_StandardRes__SR__prop,axiom,
! [X: int,P_3: int] :
( ( member_int @ X @ ( sr @ P_3 ) )
=> ( ( standardRes @ P_3 @ X )
= X ) ) ).
thf(fact_2336_StandardRes__def,axiom,
! [M: int,X: int] :
( ( standardRes @ M @ X )
= ( div_mod_int @ X @ M ) ) ).
thf(fact_2337_zdiv__zadd1__eq,axiom,
! [A: int,B: int,C: int] :
( ( div_div_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( plus_plus_int @ ( div_div_int @ A @ C ) @ ( div_div_int @ B @ C ) ) @ ( div_div_int @ ( plus_plus_int @ ( div_mod_int @ A @ C ) @ ( div_mod_int @ B @ C ) ) @ C ) ) ) ).
thf(fact_2338_semiring__div__class_Omod__div__equality_H,axiom,
! [A_73: int,B_59: int] :
( ( plus_plus_int @ ( div_mod_int @ A_73 @ B_59 ) @ ( times_times_int @ ( div_div_int @ A_73 @ B_59 ) @ B_59 ) )
= A_73 ) ).
thf(fact_2339_semiring__div__class_Omod__div__equality_H,axiom,
! [A_73: nat,B_59: nat] :
( ( plus_plus_nat @ ( div_mod_nat @ A_73 @ B_59 ) @ ( times_times_nat @ ( div_div_nat @ A_73 @ B_59 ) @ B_59 ) )
= A_73 ) ).
thf(fact_2340_semiring__div__class_Omod__div__equality_H,axiom,
! [A_73: code_code_numeral,B_59: code_code_numeral] :
( ( plus_p1627245867umeral @ ( div_mo1740067990umeral @ A_73 @ B_59 ) @ ( times_1655362735umeral @ ( div_di1218280263umeral @ A_73 @ B_59 ) @ B_59 ) )
= A_73 ) ).
thf(fact_2341_semiring__div__class_Omod__div__equality_H,axiom,
! [A_73: quickcheck_code_int,B_59: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( div_mo231679042de_int @ A_73 @ B_59 ) @ ( times_123202395de_int @ ( div_di1430059507de_int @ A_73 @ B_59 ) @ B_59 ) )
= A_73 ) ).
thf(fact_2342_mod__div__equality2,axiom,
! [B_58: int,A_72: int] :
( ( plus_plus_int @ ( times_times_int @ B_58 @ ( div_div_int @ A_72 @ B_58 ) ) @ ( div_mod_int @ A_72 @ B_58 ) )
= A_72 ) ).
thf(fact_2343_mod__div__equality2,axiom,
! [B_58: nat,A_72: nat] :
( ( plus_plus_nat @ ( times_times_nat @ B_58 @ ( div_div_nat @ A_72 @ B_58 ) ) @ ( div_mod_nat @ A_72 @ B_58 ) )
= A_72 ) ).
thf(fact_2344_mod__div__equality2,axiom,
! [B_58: code_code_numeral,A_72: code_code_numeral] :
( ( plus_p1627245867umeral @ ( times_1655362735umeral @ B_58 @ ( div_di1218280263umeral @ A_72 @ B_58 ) ) @ ( div_mo1740067990umeral @ A_72 @ B_58 ) )
= A_72 ) ).
thf(fact_2345_mod__div__equality2,axiom,
! [B_58: quickcheck_code_int,A_72: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( times_123202395de_int @ B_58 @ ( div_di1430059507de_int @ A_72 @ B_58 ) ) @ ( div_mo231679042de_int @ A_72 @ B_58 ) )
= A_72 ) ).
thf(fact_2346_mod__div__equality,axiom,
! [A_71: int,B_57: int] :
( ( plus_plus_int @ ( times_times_int @ ( div_div_int @ A_71 @ B_57 ) @ B_57 ) @ ( div_mod_int @ A_71 @ B_57 ) )
= A_71 ) ).
thf(fact_2347_mod__div__equality,axiom,
! [A_71: nat,B_57: nat] :
( ( plus_plus_nat @ ( times_times_nat @ ( div_div_nat @ A_71 @ B_57 ) @ B_57 ) @ ( div_mod_nat @ A_71 @ B_57 ) )
= A_71 ) ).
thf(fact_2348_mod__div__equality,axiom,
! [A_71: code_code_numeral,B_57: code_code_numeral] :
( ( plus_p1627245867umeral @ ( times_1655362735umeral @ ( div_di1218280263umeral @ A_71 @ B_57 ) @ B_57 ) @ ( div_mo1740067990umeral @ A_71 @ B_57 ) )
= A_71 ) ).
thf(fact_2349_mod__div__equality,axiom,
! [A_71: quickcheck_code_int,B_57: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( times_123202395de_int @ ( div_di1430059507de_int @ A_71 @ B_57 ) @ B_57 ) @ ( div_mo231679042de_int @ A_71 @ B_57 ) )
= A_71 ) ).
thf(fact_2350_div__mod__equality2,axiom,
! [B_56: int,A_70: int,C_34: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B_56 @ ( div_div_int @ A_70 @ B_56 ) ) @ ( div_mod_int @ A_70 @ B_56 ) ) @ C_34 )
= ( plus_plus_int @ A_70 @ C_34 ) ) ).
thf(fact_2351_div__mod__equality2,axiom,
! [B_56: nat,A_70: nat,C_34: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B_56 @ ( div_div_nat @ A_70 @ B_56 ) ) @ ( div_mod_nat @ A_70 @ B_56 ) ) @ C_34 )
= ( plus_plus_nat @ A_70 @ C_34 ) ) ).
thf(fact_2352_div__mod__equality2,axiom,
! [B_56: code_code_numeral,A_70: code_code_numeral,C_34: code_code_numeral] :
( ( plus_p1627245867umeral @ ( plus_p1627245867umeral @ ( times_1655362735umeral @ B_56 @ ( div_di1218280263umeral @ A_70 @ B_56 ) ) @ ( div_mo1740067990umeral @ A_70 @ B_56 ) ) @ C_34 )
= ( plus_p1627245867umeral @ A_70 @ C_34 ) ) ).
thf(fact_2353_div__mod__equality2,axiom,
! [B_56: quickcheck_code_int,A_70: quickcheck_code_int,C_34: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ ( times_123202395de_int @ B_56 @ ( div_di1430059507de_int @ A_70 @ B_56 ) ) @ ( div_mo231679042de_int @ A_70 @ B_56 ) ) @ C_34 )
= ( plus_p1446045655de_int @ A_70 @ C_34 ) ) ).
thf(fact_2354_div__mod__equality,axiom,
! [A_69: int,B_55: int,C_33: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( div_div_int @ A_69 @ B_55 ) @ B_55 ) @ ( div_mod_int @ A_69 @ B_55 ) ) @ C_33 )
= ( plus_plus_int @ A_69 @ C_33 ) ) ).
thf(fact_2355_div__mod__equality,axiom,
! [A_69: nat,B_55: nat,C_33: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( div_div_nat @ A_69 @ B_55 ) @ B_55 ) @ ( div_mod_nat @ A_69 @ B_55 ) ) @ C_33 )
= ( plus_plus_nat @ A_69 @ C_33 ) ) ).
thf(fact_2356_div__mod__equality,axiom,
! [A_69: code_code_numeral,B_55: code_code_numeral,C_33: code_code_numeral] :
( ( plus_p1627245867umeral @ ( plus_p1627245867umeral @ ( times_1655362735umeral @ ( div_di1218280263umeral @ A_69 @ B_55 ) @ B_55 ) @ ( div_mo1740067990umeral @ A_69 @ B_55 ) ) @ C_33 )
= ( plus_p1627245867umeral @ A_69 @ C_33 ) ) ).
thf(fact_2357_div__mod__equality,axiom,
! [A_69: quickcheck_code_int,B_55: quickcheck_code_int,C_33: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ ( times_123202395de_int @ ( div_di1430059507de_int @ A_69 @ B_55 ) @ B_55 ) @ ( div_mo231679042de_int @ A_69 @ B_55 ) ) @ C_33 )
= ( plus_p1446045655de_int @ A_69 @ C_33 ) ) ).
thf(fact_2358_zmod__zdiv__equality,axiom,
! [A: int,B: int] :
( A
= ( plus_plus_int @ ( times_times_int @ B @ ( div_div_int @ A @ B ) ) @ ( div_mod_int @ A @ B ) ) ) ).
thf(fact_2359_zdiv__zmult1__eq,axiom,
! [A: int,B: int,C: int] :
( ( div_div_int @ ( times_times_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ ( div_div_int @ B @ C ) ) @ ( div_div_int @ ( times_times_int @ A @ ( div_mod_int @ B @ C ) ) @ C ) ) ) ).
thf(fact_2360_zdiv__zmod__equality,axiom,
! [B: int,A: int,K_1: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( div_div_int @ A @ B ) ) @ ( div_mod_int @ A @ B ) ) @ K_1 )
= ( plus_plus_int @ A @ K_1 ) ) ).
thf(fact_2361_zdiv__zmod__equality2,axiom,
! [A: int,B: int,K_1: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( div_div_int @ A @ B ) @ B ) @ ( div_mod_int @ A @ B ) ) @ K_1 )
= ( plus_plus_int @ A @ K_1 ) ) ).
thf(fact_2362_zmod__zdiv__equality_H,axiom,
! [M: int,N: int] :
( ( div_mod_int @ M @ N )
= ( minus_minus_int @ M @ ( times_times_int @ ( div_div_int @ M @ N ) @ N ) ) ) ).
thf(fact_2363_zmult__div__cancel,axiom,
! [N: int,M: int] :
( ( times_times_int @ N @ ( div_div_int @ M @ N ) )
= ( minus_minus_int @ M @ ( div_mod_int @ M @ N ) ) ) ).
thf(fact_2364_mod__self,axiom,
! [A_68: int] :
( ( div_mod_int @ A_68 @ A_68 )
= zero_zero_int ) ).
thf(fact_2365_mod__self,axiom,
! [A_68: nat] :
( ( div_mod_nat @ A_68 @ A_68 )
= zero_zero_nat ) ).
thf(fact_2366_mod__self,axiom,
! [A_68: code_code_numeral] :
( ( div_mo1740067990umeral @ A_68 @ A_68 )
= zero_z126310315umeral ) ).
thf(fact_2367_mod__self,axiom,
! [A_68: quickcheck_code_int] :
( ( div_mo231679042de_int @ A_68 @ A_68 )
= zero_z891286103de_int ) ).
thf(fact_2368_mod__by__0,axiom,
! [A_67: int] :
( ( div_mod_int @ A_67 @ zero_zero_int )
= A_67 ) ).
thf(fact_2369_mod__by__0,axiom,
! [A_67: nat] :
( ( div_mod_nat @ A_67 @ zero_zero_nat )
= A_67 ) ).
thf(fact_2370_mod__by__0,axiom,
! [A_67: code_code_numeral] :
( ( div_mo1740067990umeral @ A_67 @ zero_z126310315umeral )
= A_67 ) ).
thf(fact_2371_mod__by__0,axiom,
! [A_67: quickcheck_code_int] :
( ( div_mo231679042de_int @ A_67 @ zero_z891286103de_int )
= A_67 ) ).
thf(fact_2372_mod__0,axiom,
! [A_66: int] :
( ( div_mod_int @ zero_zero_int @ A_66 )
= zero_zero_int ) ).
thf(fact_2373_mod__0,axiom,
! [A_66: nat] :
( ( div_mod_nat @ zero_zero_nat @ A_66 )
= zero_zero_nat ) ).
thf(fact_2374_mod__0,axiom,
! [A_66: code_code_numeral] :
( ( div_mo1740067990umeral @ zero_z126310315umeral @ A_66 )
= zero_z126310315umeral ) ).
thf(fact_2375_mod__0,axiom,
! [A_66: quickcheck_code_int] :
( ( div_mo231679042de_int @ zero_z891286103de_int @ A_66 )
= zero_z891286103de_int ) ).
thf(fact_2376_negDivAlg__div__mod,axiom,
! [L: int,K_1: int] :
( ( ord_less_int @ K_1 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ L )
=> ( ( negDivAlg @ K_1 @ L )
= ( product_Pair_int_int @ ( div_div_int @ K_1 @ L ) @ ( div_mod_int @ K_1 @ L ) ) ) ) ) ).
thf(fact_2377_mod__mult__right__eq,axiom,
! [A_65: int,B_54: int,C_32: int] :
( ( div_mod_int @ ( times_times_int @ A_65 @ B_54 ) @ C_32 )
= ( div_mod_int @ ( times_times_int @ A_65 @ ( div_mod_int @ B_54 @ C_32 ) ) @ C_32 ) ) ).
thf(fact_2378_mod__mult__right__eq,axiom,
! [A_65: nat,B_54: nat,C_32: nat] :
( ( div_mod_nat @ ( times_times_nat @ A_65 @ B_54 ) @ C_32 )
= ( div_mod_nat @ ( times_times_nat @ A_65 @ ( div_mod_nat @ B_54 @ C_32 ) ) @ C_32 ) ) ).
thf(fact_2379_mod__mult__right__eq,axiom,
! [A_65: code_code_numeral,B_54: code_code_numeral,C_32: code_code_numeral] :
( ( div_mo1740067990umeral @ ( times_1655362735umeral @ A_65 @ B_54 ) @ C_32 )
= ( div_mo1740067990umeral @ ( times_1655362735umeral @ A_65 @ ( div_mo1740067990umeral @ B_54 @ C_32 ) ) @ C_32 ) ) ).
thf(fact_2380_mod__mult__right__eq,axiom,
! [A_65: quickcheck_code_int,B_54: quickcheck_code_int,C_32: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( times_123202395de_int @ A_65 @ B_54 ) @ C_32 )
= ( div_mo231679042de_int @ ( times_123202395de_int @ A_65 @ ( div_mo231679042de_int @ B_54 @ C_32 ) ) @ C_32 ) ) ).
thf(fact_2381_mod__mult__left__eq,axiom,
! [A_64: int,B_53: int,C_31: int] :
( ( div_mod_int @ ( times_times_int @ A_64 @ B_53 ) @ C_31 )
= ( div_mod_int @ ( times_times_int @ ( div_mod_int @ A_64 @ C_31 ) @ B_53 ) @ C_31 ) ) ).
thf(fact_2382_mod__mult__left__eq,axiom,
! [A_64: nat,B_53: nat,C_31: nat] :
( ( div_mod_nat @ ( times_times_nat @ A_64 @ B_53 ) @ C_31 )
= ( div_mod_nat @ ( times_times_nat @ ( div_mod_nat @ A_64 @ C_31 ) @ B_53 ) @ C_31 ) ) ).
thf(fact_2383_mod__mult__left__eq,axiom,
! [A_64: code_code_numeral,B_53: code_code_numeral,C_31: code_code_numeral] :
( ( div_mo1740067990umeral @ ( times_1655362735umeral @ A_64 @ B_53 ) @ C_31 )
= ( div_mo1740067990umeral @ ( times_1655362735umeral @ ( div_mo1740067990umeral @ A_64 @ C_31 ) @ B_53 ) @ C_31 ) ) ).
thf(fact_2384_mod__mult__left__eq,axiom,
! [A_64: quickcheck_code_int,B_53: quickcheck_code_int,C_31: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( times_123202395de_int @ A_64 @ B_53 ) @ C_31 )
= ( div_mo231679042de_int @ ( times_123202395de_int @ ( div_mo231679042de_int @ A_64 @ C_31 ) @ B_53 ) @ C_31 ) ) ).
thf(fact_2385_mod__mult__eq,axiom,
! [A_63: int,B_52: int,C_30: int] :
( ( div_mod_int @ ( times_times_int @ A_63 @ B_52 ) @ C_30 )
= ( div_mod_int @ ( times_times_int @ ( div_mod_int @ A_63 @ C_30 ) @ ( div_mod_int @ B_52 @ C_30 ) ) @ C_30 ) ) ).
thf(fact_2386_mod__mult__eq,axiom,
! [A_63: nat,B_52: nat,C_30: nat] :
( ( div_mod_nat @ ( times_times_nat @ A_63 @ B_52 ) @ C_30 )
= ( div_mod_nat @ ( times_times_nat @ ( div_mod_nat @ A_63 @ C_30 ) @ ( div_mod_nat @ B_52 @ C_30 ) ) @ C_30 ) ) ).
thf(fact_2387_mod__mult__eq,axiom,
! [A_63: code_code_numeral,B_52: code_code_numeral,C_30: code_code_numeral] :
( ( div_mo1740067990umeral @ ( times_1655362735umeral @ A_63 @ B_52 ) @ C_30 )
= ( div_mo1740067990umeral @ ( times_1655362735umeral @ ( div_mo1740067990umeral @ A_63 @ C_30 ) @ ( div_mo1740067990umeral @ B_52 @ C_30 ) ) @ C_30 ) ) ).
thf(fact_2388_mod__mult__eq,axiom,
! [A_63: quickcheck_code_int,B_52: quickcheck_code_int,C_30: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( times_123202395de_int @ A_63 @ B_52 ) @ C_30 )
= ( div_mo231679042de_int @ ( times_123202395de_int @ ( div_mo231679042de_int @ A_63 @ C_30 ) @ ( div_mo231679042de_int @ B_52 @ C_30 ) ) @ C_30 ) ) ).
thf(fact_2389_mod__mult__mult1,axiom,
! [C_29: int,A_62: int,B_51: int] :
( ( div_mod_int @ ( times_times_int @ C_29 @ A_62 ) @ ( times_times_int @ C_29 @ B_51 ) )
= ( times_times_int @ C_29 @ ( div_mod_int @ A_62 @ B_51 ) ) ) ).
thf(fact_2390_mod__mult__mult1,axiom,
! [C_29: nat,A_62: nat,B_51: nat] :
( ( div_mod_nat @ ( times_times_nat @ C_29 @ A_62 ) @ ( times_times_nat @ C_29 @ B_51 ) )
= ( times_times_nat @ C_29 @ ( div_mod_nat @ A_62 @ B_51 ) ) ) ).
thf(fact_2391_mod__mult__mult1,axiom,
! [C_29: code_code_numeral,A_62: code_code_numeral,B_51: code_code_numeral] :
( ( div_mo1740067990umeral @ ( times_1655362735umeral @ C_29 @ A_62 ) @ ( times_1655362735umeral @ C_29 @ B_51 ) )
= ( times_1655362735umeral @ C_29 @ ( div_mo1740067990umeral @ A_62 @ B_51 ) ) ) ).
thf(fact_2392_mod__mult__mult1,axiom,
! [C_29: quickcheck_code_int,A_62: quickcheck_code_int,B_51: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( times_123202395de_int @ C_29 @ A_62 ) @ ( times_123202395de_int @ C_29 @ B_51 ) )
= ( times_123202395de_int @ C_29 @ ( div_mo231679042de_int @ A_62 @ B_51 ) ) ) ).
thf(fact_2393_mod__mult__mult2,axiom,
! [A_61: int,C_28: int,B_50: int] :
( ( div_mod_int @ ( times_times_int @ A_61 @ C_28 ) @ ( times_times_int @ B_50 @ C_28 ) )
= ( times_times_int @ ( div_mod_int @ A_61 @ B_50 ) @ C_28 ) ) ).
thf(fact_2394_mod__mult__mult2,axiom,
! [A_61: nat,C_28: nat,B_50: nat] :
( ( div_mod_nat @ ( times_times_nat @ A_61 @ C_28 ) @ ( times_times_nat @ B_50 @ C_28 ) )
= ( times_times_nat @ ( div_mod_nat @ A_61 @ B_50 ) @ C_28 ) ) ).
thf(fact_2395_mod__mult__mult2,axiom,
! [A_61: code_code_numeral,C_28: code_code_numeral,B_50: code_code_numeral] :
( ( div_mo1740067990umeral @ ( times_1655362735umeral @ A_61 @ C_28 ) @ ( times_1655362735umeral @ B_50 @ C_28 ) )
= ( times_1655362735umeral @ ( div_mo1740067990umeral @ A_61 @ B_50 ) @ C_28 ) ) ).
thf(fact_2396_mod__mult__mult2,axiom,
! [A_61: quickcheck_code_int,C_28: quickcheck_code_int,B_50: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( times_123202395de_int @ A_61 @ C_28 ) @ ( times_123202395de_int @ B_50 @ C_28 ) )
= ( times_123202395de_int @ ( div_mo231679042de_int @ A_61 @ B_50 ) @ C_28 ) ) ).
thf(fact_2397_zmod__simps_I4_J,axiom,
! [A_60: int,C_27: int,B_49: int] :
( ( div_mod_int @ ( times_times_int @ ( div_mod_int @ A_60 @ C_27 ) @ B_49 ) @ C_27 )
= ( div_mod_int @ ( times_times_int @ A_60 @ B_49 ) @ C_27 ) ) ).
thf(fact_2398_zmod__simps_I4_J,axiom,
! [A_60: nat,C_27: nat,B_49: nat] :
( ( div_mod_nat @ ( times_times_nat @ ( div_mod_nat @ A_60 @ C_27 ) @ B_49 ) @ C_27 )
= ( div_mod_nat @ ( times_times_nat @ A_60 @ B_49 ) @ C_27 ) ) ).
thf(fact_2399_zmod__simps_I4_J,axiom,
! [A_60: code_code_numeral,C_27: code_code_numeral,B_49: code_code_numeral] :
( ( div_mo1740067990umeral @ ( times_1655362735umeral @ ( div_mo1740067990umeral @ A_60 @ C_27 ) @ B_49 ) @ C_27 )
= ( div_mo1740067990umeral @ ( times_1655362735umeral @ A_60 @ B_49 ) @ C_27 ) ) ).
thf(fact_2400_zmod__simps_I4_J,axiom,
! [A_60: quickcheck_code_int,C_27: quickcheck_code_int,B_49: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( times_123202395de_int @ ( div_mo231679042de_int @ A_60 @ C_27 ) @ B_49 ) @ C_27 )
= ( div_mo231679042de_int @ ( times_123202395de_int @ A_60 @ B_49 ) @ C_27 ) ) ).
thf(fact_2401_mod__mult__cong,axiom,
! [B_48: int,B_47: int,A_59: int,C_26: int,A_58: int] :
( ( ( div_mod_int @ A_59 @ C_26 )
= ( div_mod_int @ A_58 @ C_26 ) )
=> ( ( ( div_mod_int @ B_48 @ C_26 )
= ( div_mod_int @ B_47 @ C_26 ) )
=> ( ( div_mod_int @ ( times_times_int @ A_59 @ B_48 ) @ C_26 )
= ( div_mod_int @ ( times_times_int @ A_58 @ B_47 ) @ C_26 ) ) ) ) ).
thf(fact_2402_mod__mult__cong,axiom,
! [B_48: nat,B_47: nat,A_59: nat,C_26: nat,A_58: nat] :
( ( ( div_mod_nat @ A_59 @ C_26 )
= ( div_mod_nat @ A_58 @ C_26 ) )
=> ( ( ( div_mod_nat @ B_48 @ C_26 )
= ( div_mod_nat @ B_47 @ C_26 ) )
=> ( ( div_mod_nat @ ( times_times_nat @ A_59 @ B_48 ) @ C_26 )
= ( div_mod_nat @ ( times_times_nat @ A_58 @ B_47 ) @ C_26 ) ) ) ) ).
thf(fact_2403_mod__mult__cong,axiom,
! [B_48: code_code_numeral,B_47: code_code_numeral,A_59: code_code_numeral,C_26: code_code_numeral,A_58: code_code_numeral] :
( ( ( div_mo1740067990umeral @ A_59 @ C_26 )
= ( div_mo1740067990umeral @ A_58 @ C_26 ) )
=> ( ( ( div_mo1740067990umeral @ B_48 @ C_26 )
= ( div_mo1740067990umeral @ B_47 @ C_26 ) )
=> ( ( div_mo1740067990umeral @ ( times_1655362735umeral @ A_59 @ B_48 ) @ C_26 )
= ( div_mo1740067990umeral @ ( times_1655362735umeral @ A_58 @ B_47 ) @ C_26 ) ) ) ) ).
thf(fact_2404_mod__mult__cong,axiom,
! [B_48: quickcheck_code_int,B_47: quickcheck_code_int,A_59: quickcheck_code_int,C_26: quickcheck_code_int,A_58: quickcheck_code_int] :
( ( ( div_mo231679042de_int @ A_59 @ C_26 )
= ( div_mo231679042de_int @ A_58 @ C_26 ) )
=> ( ( ( div_mo231679042de_int @ B_48 @ C_26 )
= ( div_mo231679042de_int @ B_47 @ C_26 ) )
=> ( ( div_mo231679042de_int @ ( times_123202395de_int @ A_59 @ B_48 ) @ C_26 )
= ( div_mo231679042de_int @ ( times_123202395de_int @ A_58 @ B_47 ) @ C_26 ) ) ) ) ).
thf(fact_2405_mod__add__cong,axiom,
! [B_46: int,B_45: int,A_57: int,C_25: int,A_56: int] :
( ( ( div_mod_int @ A_57 @ C_25 )
= ( div_mod_int @ A_56 @ C_25 ) )
=> ( ( ( div_mod_int @ B_46 @ C_25 )
= ( div_mod_int @ B_45 @ C_25 ) )
=> ( ( div_mod_int @ ( plus_plus_int @ A_57 @ B_46 ) @ C_25 )
= ( div_mod_int @ ( plus_plus_int @ A_56 @ B_45 ) @ C_25 ) ) ) ) ).
thf(fact_2406_mod__add__cong,axiom,
! [B_46: nat,B_45: nat,A_57: nat,C_25: nat,A_56: nat] :
( ( ( div_mod_nat @ A_57 @ C_25 )
= ( div_mod_nat @ A_56 @ C_25 ) )
=> ( ( ( div_mod_nat @ B_46 @ C_25 )
= ( div_mod_nat @ B_45 @ C_25 ) )
=> ( ( div_mod_nat @ ( plus_plus_nat @ A_57 @ B_46 ) @ C_25 )
= ( div_mod_nat @ ( plus_plus_nat @ A_56 @ B_45 ) @ C_25 ) ) ) ) ).
thf(fact_2407_mod__add__cong,axiom,
! [B_46: code_code_numeral,B_45: code_code_numeral,A_57: code_code_numeral,C_25: code_code_numeral,A_56: code_code_numeral] :
( ( ( div_mo1740067990umeral @ A_57 @ C_25 )
= ( div_mo1740067990umeral @ A_56 @ C_25 ) )
=> ( ( ( div_mo1740067990umeral @ B_46 @ C_25 )
= ( div_mo1740067990umeral @ B_45 @ C_25 ) )
=> ( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_57 @ B_46 ) @ C_25 )
= ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_56 @ B_45 ) @ C_25 ) ) ) ) ).
thf(fact_2408_mod__add__cong,axiom,
! [B_46: quickcheck_code_int,B_45: quickcheck_code_int,A_57: quickcheck_code_int,C_25: quickcheck_code_int,A_56: quickcheck_code_int] :
( ( ( div_mo231679042de_int @ A_57 @ C_25 )
= ( div_mo231679042de_int @ A_56 @ C_25 ) )
=> ( ( ( div_mo231679042de_int @ B_46 @ C_25 )
= ( div_mo231679042de_int @ B_45 @ C_25 ) )
=> ( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_57 @ B_46 ) @ C_25 )
= ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_56 @ B_45 ) @ C_25 ) ) ) ) ).
thf(fact_2409_zmod__simps_I1_J,axiom,
! [A_55: int,C_24: int,B_44: int] :
( ( div_mod_int @ ( plus_plus_int @ ( div_mod_int @ A_55 @ C_24 ) @ B_44 ) @ C_24 )
= ( div_mod_int @ ( plus_plus_int @ A_55 @ B_44 ) @ C_24 ) ) ).
thf(fact_2410_zmod__simps_I1_J,axiom,
! [A_55: nat,C_24: nat,B_44: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ ( div_mod_nat @ A_55 @ C_24 ) @ B_44 ) @ C_24 )
= ( div_mod_nat @ ( plus_plus_nat @ A_55 @ B_44 ) @ C_24 ) ) ).
thf(fact_2411_zmod__simps_I1_J,axiom,
! [A_55: code_code_numeral,C_24: code_code_numeral,B_44: code_code_numeral] :
( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ ( div_mo1740067990umeral @ A_55 @ C_24 ) @ B_44 ) @ C_24 )
= ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_55 @ B_44 ) @ C_24 ) ) ).
thf(fact_2412_zmod__simps_I1_J,axiom,
! [A_55: quickcheck_code_int,C_24: quickcheck_code_int,B_44: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ ( div_mo231679042de_int @ A_55 @ C_24 ) @ B_44 ) @ C_24 )
= ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_55 @ B_44 ) @ C_24 ) ) ).
thf(fact_2413_zmod__simps_I2_J,axiom,
! [A_54: int,B_43: int,C_23: int] :
( ( div_mod_int @ ( plus_plus_int @ A_54 @ ( div_mod_int @ B_43 @ C_23 ) ) @ C_23 )
= ( div_mod_int @ ( plus_plus_int @ A_54 @ B_43 ) @ C_23 ) ) ).
thf(fact_2414_zmod__simps_I2_J,axiom,
! [A_54: nat,B_43: nat,C_23: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ A_54 @ ( div_mod_nat @ B_43 @ C_23 ) ) @ C_23 )
= ( div_mod_nat @ ( plus_plus_nat @ A_54 @ B_43 ) @ C_23 ) ) ).
thf(fact_2415_zmod__simps_I2_J,axiom,
! [A_54: code_code_numeral,B_43: code_code_numeral,C_23: code_code_numeral] :
( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_54 @ ( div_mo1740067990umeral @ B_43 @ C_23 ) ) @ C_23 )
= ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_54 @ B_43 ) @ C_23 ) ) ).
thf(fact_2416_zmod__simps_I2_J,axiom,
! [A_54: quickcheck_code_int,B_43: quickcheck_code_int,C_23: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_54 @ ( div_mo231679042de_int @ B_43 @ C_23 ) ) @ C_23 )
= ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_54 @ B_43 ) @ C_23 ) ) ).
thf(fact_2417_mod__add__eq,axiom,
! [A_53: int,B_42: int,C_22: int] :
( ( div_mod_int @ ( plus_plus_int @ A_53 @ B_42 ) @ C_22 )
= ( div_mod_int @ ( plus_plus_int @ ( div_mod_int @ A_53 @ C_22 ) @ ( div_mod_int @ B_42 @ C_22 ) ) @ C_22 ) ) ).
thf(fact_2418_mod__add__eq,axiom,
! [A_53: nat,B_42: nat,C_22: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ A_53 @ B_42 ) @ C_22 )
= ( div_mod_nat @ ( plus_plus_nat @ ( div_mod_nat @ A_53 @ C_22 ) @ ( div_mod_nat @ B_42 @ C_22 ) ) @ C_22 ) ) ).
thf(fact_2419_mod__add__eq,axiom,
! [A_53: code_code_numeral,B_42: code_code_numeral,C_22: code_code_numeral] :
( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_53 @ B_42 ) @ C_22 )
= ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ ( div_mo1740067990umeral @ A_53 @ C_22 ) @ ( div_mo1740067990umeral @ B_42 @ C_22 ) ) @ C_22 ) ) ).
thf(fact_2420_mod__add__eq,axiom,
! [A_53: quickcheck_code_int,B_42: quickcheck_code_int,C_22: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_53 @ B_42 ) @ C_22 )
= ( div_mo231679042de_int @ ( plus_p1446045655de_int @ ( div_mo231679042de_int @ A_53 @ C_22 ) @ ( div_mo231679042de_int @ B_42 @ C_22 ) ) @ C_22 ) ) ).
thf(fact_2421_mod__add__left__eq,axiom,
! [A_52: int,B_41: int,C_21: int] :
( ( div_mod_int @ ( plus_plus_int @ A_52 @ B_41 ) @ C_21 )
= ( div_mod_int @ ( plus_plus_int @ ( div_mod_int @ A_52 @ C_21 ) @ B_41 ) @ C_21 ) ) ).
thf(fact_2422_mod__add__left__eq,axiom,
! [A_52: nat,B_41: nat,C_21: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ A_52 @ B_41 ) @ C_21 )
= ( div_mod_nat @ ( plus_plus_nat @ ( div_mod_nat @ A_52 @ C_21 ) @ B_41 ) @ C_21 ) ) ).
thf(fact_2423_mod__add__left__eq,axiom,
! [A_52: code_code_numeral,B_41: code_code_numeral,C_21: code_code_numeral] :
( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_52 @ B_41 ) @ C_21 )
= ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ ( div_mo1740067990umeral @ A_52 @ C_21 ) @ B_41 ) @ C_21 ) ) ).
thf(fact_2424_mod__add__left__eq,axiom,
! [A_52: quickcheck_code_int,B_41: quickcheck_code_int,C_21: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_52 @ B_41 ) @ C_21 )
= ( div_mo231679042de_int @ ( plus_p1446045655de_int @ ( div_mo231679042de_int @ A_52 @ C_21 ) @ B_41 ) @ C_21 ) ) ).
thf(fact_2425_mod__add__right__eq,axiom,
! [A_51: int,B_40: int,C_20: int] :
( ( div_mod_int @ ( plus_plus_int @ A_51 @ B_40 ) @ C_20 )
= ( div_mod_int @ ( plus_plus_int @ A_51 @ ( div_mod_int @ B_40 @ C_20 ) ) @ C_20 ) ) ).
thf(fact_2426_mod__add__right__eq,axiom,
! [A_51: nat,B_40: nat,C_20: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ A_51 @ B_40 ) @ C_20 )
= ( div_mod_nat @ ( plus_plus_nat @ A_51 @ ( div_mod_nat @ B_40 @ C_20 ) ) @ C_20 ) ) ).
thf(fact_2427_mod__add__right__eq,axiom,
! [A_51: code_code_numeral,B_40: code_code_numeral,C_20: code_code_numeral] :
( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_51 @ B_40 ) @ C_20 )
= ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_51 @ ( div_mo1740067990umeral @ B_40 @ C_20 ) ) @ C_20 ) ) ).
thf(fact_2428_mod__add__right__eq,axiom,
! [A_51: quickcheck_code_int,B_40: quickcheck_code_int,C_20: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_51 @ B_40 ) @ C_20 )
= ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_51 @ ( div_mo231679042de_int @ B_40 @ C_20 ) ) @ C_20 ) ) ).
thf(fact_2429_mod__add__self1,axiom,
! [B_39: int,A_50: int] :
( ( div_mod_int @ ( plus_plus_int @ B_39 @ A_50 ) @ B_39 )
= ( div_mod_int @ A_50 @ B_39 ) ) ).
thf(fact_2430_mod__add__self1,axiom,
! [B_39: nat,A_50: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ B_39 @ A_50 ) @ B_39 )
= ( div_mod_nat @ A_50 @ B_39 ) ) ).
thf(fact_2431_mod__add__self1,axiom,
! [B_39: code_code_numeral,A_50: code_code_numeral] :
( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ B_39 @ A_50 ) @ B_39 )
= ( div_mo1740067990umeral @ A_50 @ B_39 ) ) ).
thf(fact_2432_mod__add__self1,axiom,
! [B_39: quickcheck_code_int,A_50: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ B_39 @ A_50 ) @ B_39 )
= ( div_mo231679042de_int @ A_50 @ B_39 ) ) ).
thf(fact_2433_mod__add__self2,axiom,
! [A_49: int,B_38: int] :
( ( div_mod_int @ ( plus_plus_int @ A_49 @ B_38 ) @ B_38 )
= ( div_mod_int @ A_49 @ B_38 ) ) ).
thf(fact_2434_mod__add__self2,axiom,
! [A_49: nat,B_38: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ A_49 @ B_38 ) @ B_38 )
= ( div_mod_nat @ A_49 @ B_38 ) ) ).
thf(fact_2435_mod__add__self2,axiom,
! [A_49: code_code_numeral,B_38: code_code_numeral] :
( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_49 @ B_38 ) @ B_38 )
= ( div_mo1740067990umeral @ A_49 @ B_38 ) ) ).
thf(fact_2436_mod__add__self2,axiom,
! [A_49: quickcheck_code_int,B_38: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_49 @ B_38 ) @ B_38 )
= ( div_mo231679042de_int @ A_49 @ B_38 ) ) ).
thf(fact_2437_mod__diff__cong,axiom,
! [B_37: int,B_36: int,A_48: int,C_19: int,A_47: int] :
( ( ( div_mod_int @ A_48 @ C_19 )
= ( div_mod_int @ A_47 @ C_19 ) )
=> ( ( ( div_mod_int @ B_37 @ C_19 )
= ( div_mod_int @ B_36 @ C_19 ) )
=> ( ( div_mod_int @ ( minus_minus_int @ A_48 @ B_37 ) @ C_19 )
= ( div_mod_int @ ( minus_minus_int @ A_47 @ B_36 ) @ C_19 ) ) ) ) ).
thf(fact_2438_mod__diff__eq,axiom,
! [A_46: int,B_35: int,C_18: int] :
( ( div_mod_int @ ( minus_minus_int @ A_46 @ B_35 ) @ C_18 )
= ( div_mod_int @ ( minus_minus_int @ ( div_mod_int @ A_46 @ C_18 ) @ ( div_mod_int @ B_35 @ C_18 ) ) @ C_18 ) ) ).
thf(fact_2439_mod__diff__left__eq,axiom,
! [A_45: int,B_34: int,C_17: int] :
( ( div_mod_int @ ( minus_minus_int @ A_45 @ B_34 ) @ C_17 )
= ( div_mod_int @ ( minus_minus_int @ ( div_mod_int @ A_45 @ C_17 ) @ B_34 ) @ C_17 ) ) ).
thf(fact_2440_mod__diff__right__eq,axiom,
! [A_44: int,B_33: int,C_16: int] :
( ( div_mod_int @ ( minus_minus_int @ A_44 @ B_33 ) @ C_16 )
= ( div_mod_int @ ( minus_minus_int @ A_44 @ ( div_mod_int @ B_33 @ C_16 ) ) @ C_16 ) ) ).
thf(fact_2441_dvd__mod__iff,axiom,
! [M_21: code_code_numeral,K_6: code_code_numeral,N_32: code_code_numeral] :
( ( dvd_dv174992974umeral @ K_6 @ N_32 )
=> ( ( dvd_dv174992974umeral @ K_6 @ ( div_mo1740067990umeral @ M_21 @ N_32 ) )
<=> ( dvd_dv174992974umeral @ K_6 @ M_21 ) ) ) ).
thf(fact_2442_dvd__mod__iff,axiom,
! [M_21: quickcheck_code_int,K_6: quickcheck_code_int,N_32: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ K_6 @ N_32 )
=> ( ( dvd_dv1760642554de_int @ K_6 @ ( div_mo231679042de_int @ M_21 @ N_32 ) )
<=> ( dvd_dv1760642554de_int @ K_6 @ M_21 ) ) ) ).
thf(fact_2443_dvd__mod__iff,axiom,
! [M_21: int,K_6: int,N_32: int] :
( ( dvd_dvd_int @ K_6 @ N_32 )
=> ( ( dvd_dvd_int @ K_6 @ ( div_mod_int @ M_21 @ N_32 ) )
<=> ( dvd_dvd_int @ K_6 @ M_21 ) ) ) ).
thf(fact_2444_dvd__mod__iff,axiom,
! [M_21: nat,K_6: nat,N_32: nat] :
( ( dvd_dvd_nat @ K_6 @ N_32 )
=> ( ( dvd_dvd_nat @ K_6 @ ( div_mod_nat @ M_21 @ N_32 ) )
<=> ( dvd_dvd_nat @ K_6 @ M_21 ) ) ) ).
thf(fact_2445_mod__mod__cancel,axiom,
! [A_43: code_code_numeral,C_15: code_code_numeral,B_32: code_code_numeral] :
( ( dvd_dv174992974umeral @ C_15 @ B_32 )
=> ( ( div_mo1740067990umeral @ ( div_mo1740067990umeral @ A_43 @ B_32 ) @ C_15 )
= ( div_mo1740067990umeral @ A_43 @ C_15 ) ) ) ).
thf(fact_2446_mod__mod__cancel,axiom,
! [A_43: quickcheck_code_int,C_15: quickcheck_code_int,B_32: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ C_15 @ B_32 )
=> ( ( div_mo231679042de_int @ ( div_mo231679042de_int @ A_43 @ B_32 ) @ C_15 )
= ( div_mo231679042de_int @ A_43 @ C_15 ) ) ) ).
thf(fact_2447_mod__mod__cancel,axiom,
! [A_43: int,C_15: int,B_32: int] :
( ( dvd_dvd_int @ C_15 @ B_32 )
=> ( ( div_mod_int @ ( div_mod_int @ A_43 @ B_32 ) @ C_15 )
= ( div_mod_int @ A_43 @ C_15 ) ) ) ).
thf(fact_2448_mod__mod__cancel,axiom,
! [A_43: nat,C_15: nat,B_32: nat] :
( ( dvd_dvd_nat @ C_15 @ B_32 )
=> ( ( div_mod_nat @ ( div_mod_nat @ A_43 @ B_32 ) @ C_15 )
= ( div_mod_nat @ A_43 @ C_15 ) ) ) ).
thf(fact_2449_dvd__mod,axiom,
! [N_31: code_code_numeral,K_5: code_code_numeral,M_20: code_code_numeral] :
( ( dvd_dv174992974umeral @ K_5 @ M_20 )
=> ( ( dvd_dv174992974umeral @ K_5 @ N_31 )
=> ( dvd_dv174992974umeral @ K_5 @ ( div_mo1740067990umeral @ M_20 @ N_31 ) ) ) ) ).
thf(fact_2450_dvd__mod,axiom,
! [N_31: quickcheck_code_int,K_5: quickcheck_code_int,M_20: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ K_5 @ M_20 )
=> ( ( dvd_dv1760642554de_int @ K_5 @ N_31 )
=> ( dvd_dv1760642554de_int @ K_5 @ ( div_mo231679042de_int @ M_20 @ N_31 ) ) ) ) ).
thf(fact_2451_dvd__mod,axiom,
! [N_31: int,K_5: int,M_20: int] :
( ( dvd_dvd_int @ K_5 @ M_20 )
=> ( ( dvd_dvd_int @ K_5 @ N_31 )
=> ( dvd_dvd_int @ K_5 @ ( div_mod_int @ M_20 @ N_31 ) ) ) ) ).
thf(fact_2452_dvd__mod,axiom,
! [N_31: nat,K_5: nat,M_20: nat] :
( ( dvd_dvd_nat @ K_5 @ M_20 )
=> ( ( dvd_dvd_nat @ K_5 @ N_31 )
=> ( dvd_dvd_nat @ K_5 @ ( div_mod_nat @ M_20 @ N_31 ) ) ) ) ).
thf(fact_2453_dvd__mod__imp__dvd,axiom,
! [K_4: code_code_numeral,M_19: code_code_numeral,N_30: code_code_numeral] :
( ( dvd_dv174992974umeral @ K_4 @ ( div_mo1740067990umeral @ M_19 @ N_30 ) )
=> ( ( dvd_dv174992974umeral @ K_4 @ N_30 )
=> ( dvd_dv174992974umeral @ K_4 @ M_19 ) ) ) ).
thf(fact_2454_dvd__mod__imp__dvd,axiom,
! [K_4: quickcheck_code_int,M_19: quickcheck_code_int,N_30: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ K_4 @ ( div_mo231679042de_int @ M_19 @ N_30 ) )
=> ( ( dvd_dv1760642554de_int @ K_4 @ N_30 )
=> ( dvd_dv1760642554de_int @ K_4 @ M_19 ) ) ) ).
thf(fact_2455_dvd__mod__imp__dvd,axiom,
! [K_4: int,M_19: int,N_30: int] :
( ( dvd_dvd_int @ K_4 @ ( div_mod_int @ M_19 @ N_30 ) )
=> ( ( dvd_dvd_int @ K_4 @ N_30 )
=> ( dvd_dvd_int @ K_4 @ M_19 ) ) ) ).
thf(fact_2456_dvd__mod__imp__dvd,axiom,
! [K_4: nat,M_19: nat,N_30: nat] :
( ( dvd_dvd_nat @ K_4 @ ( div_mod_nat @ M_19 @ N_30 ) )
=> ( ( dvd_dvd_nat @ K_4 @ N_30 )
=> ( dvd_dvd_nat @ K_4 @ M_19 ) ) ) ).
thf(fact_2457_zdiv__zero,axiom,
! [B: int] :
( ( div_div_int @ zero_zero_int @ B )
= zero_zero_int ) ).
thf(fact_2458_zmod__self,axiom,
! [A: int] :
( ( div_mod_int @ A @ A )
= zero_zero_int ) ).
thf(fact_2459_zmod__zero,axiom,
! [B: int] :
( ( div_mod_int @ zero_zero_int @ B )
= zero_zero_int ) ).
thf(fact_2460_zmod__zmult1__eq,axiom,
! [A: int,B: int,C: int] :
( ( div_mod_int @ ( times_times_int @ A @ B ) @ C )
= ( div_mod_int @ ( times_times_int @ A @ ( div_mod_int @ B @ C ) ) @ C ) ) ).
thf(fact_2461_zmod__simps_I3_J,axiom,
! [A: int,B: int,C: int] :
( ( div_mod_int @ ( times_times_int @ A @ ( div_mod_int @ B @ C ) ) @ C )
= ( div_mod_int @ ( times_times_int @ A @ B ) @ C ) ) ).
thf(fact_2462_zdiff__zmod__left,axiom,
! [X: int,M: int,Y: int] :
( ( div_mod_int @ ( minus_minus_int @ ( div_mod_int @ X @ M ) @ Y ) @ M )
= ( div_mod_int @ ( minus_minus_int @ X @ Y ) @ M ) ) ).
thf(fact_2463_zdiff__zmod__right,axiom,
! [X: int,Y: int,M: int] :
( ( div_mod_int @ ( minus_minus_int @ X @ ( div_mod_int @ Y @ M ) ) @ M )
= ( div_mod_int @ ( minus_minus_int @ X @ Y ) @ M ) ) ).
thf(fact_2464_mod__mod__is__mod,axiom,
! [X: int,M: int] : ( zcong @ X @ ( div_mod_int @ X @ M ) @ M ) ).
thf(fact_2465_Residues_Oaux,axiom,
! [X: int,M: int,Y: int] :
( ( ( div_mod_int @ X @ M )
= ( div_mod_int @ Y @ M ) )
=> ( zcong @ X @ Y @ M ) ) ).
thf(fact_2466_zcong__zmod,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
<=> ( zcong @ ( div_mod_int @ A @ M ) @ ( div_mod_int @ B @ M ) @ M ) ) ).
thf(fact_2467_zdvd__zmod,axiom,
! [N: int,F: int,M: int] :
( ( dvd_dvd_int @ F @ M )
=> ( ( dvd_dvd_int @ F @ N )
=> ( dvd_dvd_int @ F @ ( div_mod_int @ M @ N ) ) ) ) ).
thf(fact_2468_zdvd__zmod__imp__zdvd,axiom,
! [K_1: int,M: int,N: int] :
( ( dvd_dvd_int @ K_1 @ ( div_mod_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K_1 @ N )
=> ( dvd_dvd_int @ K_1 @ M ) ) ) ).
thf(fact_2469_zpower__zmod,axiom,
! [X: int,M: int,Y: nat] :
( ( div_mod_int @ ( power_power_int @ ( div_mod_int @ X @ M ) @ Y ) @ M )
= ( div_mod_int @ ( power_power_int @ X @ Y ) @ M ) ) ).
thf(fact_2470_div__by__0,axiom,
! [A_42: int] :
( ( div_div_int @ A_42 @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_2471_div__by__0,axiom,
! [A_42: nat] :
( ( div_div_nat @ A_42 @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_2472_div__by__0,axiom,
! [A_42: code_code_numeral] :
( ( div_di1218280263umeral @ A_42 @ zero_z126310315umeral )
= zero_z126310315umeral ) ).
thf(fact_2473_div__by__0,axiom,
! [A_42: quickcheck_code_int] :
( ( div_di1430059507de_int @ A_42 @ zero_z891286103de_int )
= zero_z891286103de_int ) ).
thf(fact_2474_div__0,axiom,
! [A_41: int] :
( ( div_div_int @ zero_zero_int @ A_41 )
= zero_zero_int ) ).
thf(fact_2475_div__0,axiom,
! [A_41: nat] :
( ( div_div_nat @ zero_zero_nat @ A_41 )
= zero_zero_nat ) ).
thf(fact_2476_div__0,axiom,
! [A_41: code_code_numeral] :
( ( div_di1218280263umeral @ zero_z126310315umeral @ A_41 )
= zero_z126310315umeral ) ).
thf(fact_2477_div__0,axiom,
! [A_41: quickcheck_code_int] :
( ( div_di1430059507de_int @ zero_z891286103de_int @ A_41 )
= zero_z891286103de_int ) ).
thf(fact_2478_div__by__1,axiom,
! [A_40: int] :
( ( div_div_int @ A_40 @ one_one_int )
= A_40 ) ).
thf(fact_2479_div__by__1,axiom,
! [A_40: nat] :
( ( div_div_nat @ A_40 @ one_one_nat )
= A_40 ) ).
thf(fact_2480_div__by__1,axiom,
! [A_40: code_code_numeral] :
( ( div_di1218280263umeral @ A_40 @ one_on1645066479umeral )
= A_40 ) ).
thf(fact_2481_div__by__1,axiom,
! [A_40: quickcheck_code_int] :
( ( div_di1430059507de_int @ A_40 @ one_on1684967323de_int )
= A_40 ) ).
thf(fact_2482_div__dvd__div,axiom,
! [C_14: code_code_numeral,A_39: code_code_numeral,B_31: code_code_numeral] :
( ( dvd_dv174992974umeral @ A_39 @ B_31 )
=> ( ( dvd_dv174992974umeral @ A_39 @ C_14 )
=> ( ( dvd_dv174992974umeral @ ( div_di1218280263umeral @ B_31 @ A_39 ) @ ( div_di1218280263umeral @ C_14 @ A_39 ) )
<=> ( dvd_dv174992974umeral @ B_31 @ C_14 ) ) ) ) ).
thf(fact_2483_div__dvd__div,axiom,
! [C_14: quickcheck_code_int,A_39: quickcheck_code_int,B_31: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ A_39 @ B_31 )
=> ( ( dvd_dv1760642554de_int @ A_39 @ C_14 )
=> ( ( dvd_dv1760642554de_int @ ( div_di1430059507de_int @ B_31 @ A_39 ) @ ( div_di1430059507de_int @ C_14 @ A_39 ) )
<=> ( dvd_dv1760642554de_int @ B_31 @ C_14 ) ) ) ) ).
thf(fact_2484_div__dvd__div,axiom,
! [C_14: int,A_39: int,B_31: int] :
( ( dvd_dvd_int @ A_39 @ B_31 )
=> ( ( dvd_dvd_int @ A_39 @ C_14 )
=> ( ( dvd_dvd_int @ ( div_div_int @ B_31 @ A_39 ) @ ( div_div_int @ C_14 @ A_39 ) )
<=> ( dvd_dvd_int @ B_31 @ C_14 ) ) ) ) ).
thf(fact_2485_div__dvd__div,axiom,
! [C_14: nat,A_39: nat,B_31: nat] :
( ( dvd_dvd_nat @ A_39 @ B_31 )
=> ( ( dvd_dvd_nat @ A_39 @ C_14 )
=> ( ( dvd_dvd_nat @ ( div_div_nat @ B_31 @ A_39 ) @ ( div_div_nat @ C_14 @ A_39 ) )
<=> ( dvd_dvd_nat @ B_31 @ C_14 ) ) ) ) ).
thf(fact_2486_xzgcda__linear__aux2,axiom,
! [R_1: int,S_1: int,T: int,R_3: int,S_3: int,M: int,T_2: int,N: int] :
( ( R_3
= ( plus_plus_int @ ( times_times_int @ S_3 @ M ) @ ( times_times_int @ T_2 @ N ) ) )
=> ( ( R_1
= ( plus_plus_int @ ( times_times_int @ S_1 @ M ) @ ( times_times_int @ T @ N ) ) )
=> ( ( div_mod_int @ R_3 @ R_1 )
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ S_3 @ ( times_times_int @ ( div_div_int @ R_3 @ R_1 ) @ S_1 ) ) @ M ) @ ( times_times_int @ ( minus_minus_int @ T_2 @ ( times_times_int @ ( div_div_int @ R_3 @ R_1 ) @ T ) ) @ N ) ) ) ) ) ).
thf(fact_2487_zcong__zmod__aux,axiom,
! [A: int,B: int,M: int] :
( ( minus_minus_int @ A @ B )
= ( plus_plus_int @ ( times_times_int @ M @ ( minus_minus_int @ ( div_div_int @ A @ M ) @ ( div_div_int @ B @ M ) ) ) @ ( minus_minus_int @ ( div_mod_int @ A @ M ) @ ( div_mod_int @ B @ M ) ) ) ) ).
thf(fact_2488_EvenOdd_Oodd__times__odd,axiom,
! [Y: int,X: int] :
( ( member_int @ X @ zOdd )
=> ( ( member_int @ Y @ zOdd )
=> ( member_int @ ( times_times_int @ X @ Y ) @ zOdd ) ) ) ).
thf(fact_2489_odd__mult__odd__prop,axiom,
! [X: int,Y: int] :
( ( member_int @ ( times_times_int @ X @ Y ) @ zOdd )
=> ( member_int @ X @ zOdd ) ) ).
thf(fact_2490_StandardRes__prop1,axiom,
! [X: int,M: int] : ( zcong @ X @ ( standardRes @ M @ X ) @ M ) ).
thf(fact_2491_mod__mult__self1__is__0,axiom,
! [B_30: int,A_38: int] :
( ( div_mod_int @ ( times_times_int @ B_30 @ A_38 ) @ B_30 )
= zero_zero_int ) ).
thf(fact_2492_mod__mult__self1__is__0,axiom,
! [B_30: nat,A_38: nat] :
( ( div_mod_nat @ ( times_times_nat @ B_30 @ A_38 ) @ B_30 )
= zero_zero_nat ) ).
thf(fact_2493_mod__mult__self1__is__0,axiom,
! [B_30: code_code_numeral,A_38: code_code_numeral] :
( ( div_mo1740067990umeral @ ( times_1655362735umeral @ B_30 @ A_38 ) @ B_30 )
= zero_z126310315umeral ) ).
thf(fact_2494_mod__mult__self1__is__0,axiom,
! [B_30: quickcheck_code_int,A_38: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( times_123202395de_int @ B_30 @ A_38 ) @ B_30 )
= zero_z891286103de_int ) ).
thf(fact_2495_mod__mult__self2__is__0,axiom,
! [A_37: int,B_29: int] :
( ( div_mod_int @ ( times_times_int @ A_37 @ B_29 ) @ B_29 )
= zero_zero_int ) ).
thf(fact_2496_mod__mult__self2__is__0,axiom,
! [A_37: nat,B_29: nat] :
( ( div_mod_nat @ ( times_times_nat @ A_37 @ B_29 ) @ B_29 )
= zero_zero_nat ) ).
thf(fact_2497_mod__mult__self2__is__0,axiom,
! [A_37: code_code_numeral,B_29: code_code_numeral] :
( ( div_mo1740067990umeral @ ( times_1655362735umeral @ A_37 @ B_29 ) @ B_29 )
= zero_z126310315umeral ) ).
thf(fact_2498_mod__mult__self2__is__0,axiom,
! [A_37: quickcheck_code_int,B_29: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( times_123202395de_int @ A_37 @ B_29 ) @ B_29 )
= zero_z891286103de_int ) ).
thf(fact_2499_mod__by__1,axiom,
! [A_36: int] :
( ( div_mod_int @ A_36 @ one_one_int )
= zero_zero_int ) ).
thf(fact_2500_mod__by__1,axiom,
! [A_36: nat] :
( ( div_mod_nat @ A_36 @ one_one_nat )
= zero_zero_nat ) ).
thf(fact_2501_mod__by__1,axiom,
! [A_36: code_code_numeral] :
( ( div_mo1740067990umeral @ A_36 @ one_on1645066479umeral )
= zero_z126310315umeral ) ).
thf(fact_2502_mod__by__1,axiom,
! [A_36: quickcheck_code_int] :
( ( div_mo231679042de_int @ A_36 @ one_on1684967323de_int )
= zero_z891286103de_int ) ).
thf(fact_2503_zmod__zmult2__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( div_mod_int @ A @ ( times_times_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ B @ ( div_mod_int @ ( div_div_int @ A @ B ) @ C ) ) @ ( div_mod_int @ A @ B ) ) ) ) ).
thf(fact_2504_mod__mult__self1,axiom,
! [A_35: int,C_13: int,B_28: int] :
( ( div_mod_int @ ( plus_plus_int @ A_35 @ ( times_times_int @ C_13 @ B_28 ) ) @ B_28 )
= ( div_mod_int @ A_35 @ B_28 ) ) ).
thf(fact_2505_mod__mult__self1,axiom,
! [A_35: nat,C_13: nat,B_28: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ A_35 @ ( times_times_nat @ C_13 @ B_28 ) ) @ B_28 )
= ( div_mod_nat @ A_35 @ B_28 ) ) ).
thf(fact_2506_mod__mult__self1,axiom,
! [A_35: code_code_numeral,C_13: code_code_numeral,B_28: code_code_numeral] :
( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_35 @ ( times_1655362735umeral @ C_13 @ B_28 ) ) @ B_28 )
= ( div_mo1740067990umeral @ A_35 @ B_28 ) ) ).
thf(fact_2507_mod__mult__self1,axiom,
! [A_35: quickcheck_code_int,C_13: quickcheck_code_int,B_28: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_35 @ ( times_123202395de_int @ C_13 @ B_28 ) ) @ B_28 )
= ( div_mo231679042de_int @ A_35 @ B_28 ) ) ).
thf(fact_2508_mod__mult__self2,axiom,
! [A_34: int,B_27: int,C_12: int] :
( ( div_mod_int @ ( plus_plus_int @ A_34 @ ( times_times_int @ B_27 @ C_12 ) ) @ B_27 )
= ( div_mod_int @ A_34 @ B_27 ) ) ).
thf(fact_2509_mod__mult__self2,axiom,
! [A_34: nat,B_27: nat,C_12: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ A_34 @ ( times_times_nat @ B_27 @ C_12 ) ) @ B_27 )
= ( div_mod_nat @ A_34 @ B_27 ) ) ).
thf(fact_2510_mod__mult__self2,axiom,
! [A_34: code_code_numeral,B_27: code_code_numeral,C_12: code_code_numeral] :
( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_34 @ ( times_1655362735umeral @ B_27 @ C_12 ) ) @ B_27 )
= ( div_mo1740067990umeral @ A_34 @ B_27 ) ) ).
thf(fact_2511_mod__mult__self2,axiom,
! [A_34: quickcheck_code_int,B_27: quickcheck_code_int,C_12: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_34 @ ( times_123202395de_int @ B_27 @ C_12 ) ) @ B_27 )
= ( div_mo231679042de_int @ A_34 @ B_27 ) ) ).
thf(fact_2512_dvd__imp__mod__0,axiom,
! [A_33: code_code_numeral,B_26: code_code_numeral] :
( ( dvd_dv174992974umeral @ A_33 @ B_26 )
=> ( ( div_mo1740067990umeral @ B_26 @ A_33 )
= zero_z126310315umeral ) ) ).
thf(fact_2513_dvd__imp__mod__0,axiom,
! [A_33: quickcheck_code_int,B_26: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ A_33 @ B_26 )
=> ( ( div_mo231679042de_int @ B_26 @ A_33 )
= zero_z891286103de_int ) ) ).
thf(fact_2514_dvd__imp__mod__0,axiom,
! [A_33: int,B_26: int] :
( ( dvd_dvd_int @ A_33 @ B_26 )
=> ( ( div_mod_int @ B_26 @ A_33 )
= zero_zero_int ) ) ).
thf(fact_2515_dvd__imp__mod__0,axiom,
! [A_33: nat,B_26: nat] :
( ( dvd_dvd_nat @ A_33 @ B_26 )
=> ( ( div_mod_nat @ B_26 @ A_33 )
= zero_zero_nat ) ) ).
thf(fact_2516_dvd__eq__mod__eq__0,axiom,
! [A_32: code_code_numeral,B_25: code_code_numeral] :
( ( dvd_dv174992974umeral @ A_32 @ B_25 )
<=> ( ( div_mo1740067990umeral @ B_25 @ A_32 )
= zero_z126310315umeral ) ) ).
thf(fact_2517_dvd__eq__mod__eq__0,axiom,
! [A_32: quickcheck_code_int,B_25: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ A_32 @ B_25 )
<=> ( ( div_mo231679042de_int @ B_25 @ A_32 )
= zero_z891286103de_int ) ) ).
thf(fact_2518_dvd__eq__mod__eq__0,axiom,
! [A_32: int,B_25: int] :
( ( dvd_dvd_int @ A_32 @ B_25 )
<=> ( ( div_mod_int @ B_25 @ A_32 )
= zero_zero_int ) ) ).
thf(fact_2519_dvd__eq__mod__eq__0,axiom,
! [A_32: nat,B_25: nat] :
( ( dvd_dvd_nat @ A_32 @ B_25 )
<=> ( ( div_mod_nat @ B_25 @ A_32 )
= zero_zero_nat ) ) ).
thf(fact_2520_div__neg__pos__less0,axiom,
! [B: int,A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( div_div_int @ A @ B ) @ zero_zero_int ) ) ) ).
thf(fact_2521_neg__imp__zdiv__neg__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( div_div_int @ A @ B ) @ zero_zero_int )
<=> ( ord_less_int @ zero_zero_int @ A ) ) ) ).
thf(fact_2522_pos__imp__zdiv__neg__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( div_div_int @ A @ B ) @ zero_zero_int )
<=> ( ord_less_int @ A @ zero_zero_int ) ) ) ).
thf(fact_2523_zdiv__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( div_div_int @ A @ A )
= one_one_int ) ) ).
thf(fact_2524_Divides_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( div_div_int @ X @ Y ) ) ) ) ).
thf(fact_2525_zdiv__number__of__Bit0,axiom,
! [V: int,W: int] :
( ( div_div_int @ ( number_number_of_int @ ( bit0 @ V ) ) @ ( number_number_of_int @ ( bit0 @ W ) ) )
= ( div_div_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) ) ) ).
thf(fact_2526_neg__mod__bound,axiom,
! [A: int,B: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ B @ ( div_mod_int @ A @ B ) ) ) ).
thf(fact_2527_pos__mod__bound,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( div_mod_int @ A @ B ) @ B ) ) ).
thf(fact_2528_zmod__le__nonneg__dividend,axiom,
! [K_1: int,M: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ord_less_eq_int @ ( div_mod_int @ M @ K_1 ) @ M ) ) ).
thf(fact_2529_Divides_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( div_mod_int @ X @ Y ) ) ) ) ).
thf(fact_2530_zmod__eq__0__iff,axiom,
! [M: int,D: int] :
( ( ( div_mod_int @ M @ D )
= zero_zero_int )
<=> ? [Q_2: int] :
( M
= ( times_times_int @ D @ Q_2 ) ) ) ).
thf(fact_2531_zdvd__mult__div__cancel,axiom,
! [N: int,M: int] :
( ( dvd_dvd_int @ N @ M )
=> ( ( times_times_int @ N @ ( div_div_int @ M @ N ) )
= M ) ) ).
thf(fact_2532_zmod__eq__dvd__iff,axiom,
! [X: int,N: int,Y: int] :
( ( ( div_mod_int @ X @ N )
= ( div_mod_int @ Y @ N ) )
<=> ( dvd_dvd_int @ N @ ( minus_minus_int @ X @ Y ) ) ) ).
thf(fact_2533_div__mult__mult1,axiom,
! [A_31: int,B_24: int,C_11: int] :
( ( C_11 != zero_zero_int )
=> ( ( div_div_int @ ( times_times_int @ C_11 @ A_31 ) @ ( times_times_int @ C_11 @ B_24 ) )
= ( div_div_int @ A_31 @ B_24 ) ) ) ).
thf(fact_2534_div__mult__mult1,axiom,
! [A_31: nat,B_24: nat,C_11: nat] :
( ( C_11 != zero_zero_nat )
=> ( ( div_div_nat @ ( times_times_nat @ C_11 @ A_31 ) @ ( times_times_nat @ C_11 @ B_24 ) )
= ( div_div_nat @ A_31 @ B_24 ) ) ) ).
thf(fact_2535_div__mult__mult1,axiom,
! [A_31: code_code_numeral,B_24: code_code_numeral,C_11: code_code_numeral] :
( ( C_11 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( times_1655362735umeral @ C_11 @ A_31 ) @ ( times_1655362735umeral @ C_11 @ B_24 ) )
= ( div_di1218280263umeral @ A_31 @ B_24 ) ) ) ).
thf(fact_2536_div__mult__mult1,axiom,
! [A_31: quickcheck_code_int,B_24: quickcheck_code_int,C_11: quickcheck_code_int] :
( ( C_11 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( times_123202395de_int @ C_11 @ A_31 ) @ ( times_123202395de_int @ C_11 @ B_24 ) )
= ( div_di1430059507de_int @ A_31 @ B_24 ) ) ) ).
thf(fact_2537_div__mult__mult2,axiom,
! [A_30: int,B_23: int,C_10: int] :
( ( C_10 != zero_zero_int )
=> ( ( div_div_int @ ( times_times_int @ A_30 @ C_10 ) @ ( times_times_int @ B_23 @ C_10 ) )
= ( div_div_int @ A_30 @ B_23 ) ) ) ).
thf(fact_2538_div__mult__mult2,axiom,
! [A_30: nat,B_23: nat,C_10: nat] :
( ( C_10 != zero_zero_nat )
=> ( ( div_div_nat @ ( times_times_nat @ A_30 @ C_10 ) @ ( times_times_nat @ B_23 @ C_10 ) )
= ( div_div_nat @ A_30 @ B_23 ) ) ) ).
thf(fact_2539_div__mult__mult2,axiom,
! [A_30: code_code_numeral,B_23: code_code_numeral,C_10: code_code_numeral] :
( ( C_10 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( times_1655362735umeral @ A_30 @ C_10 ) @ ( times_1655362735umeral @ B_23 @ C_10 ) )
= ( div_di1218280263umeral @ A_30 @ B_23 ) ) ) ).
thf(fact_2540_div__mult__mult2,axiom,
! [A_30: quickcheck_code_int,B_23: quickcheck_code_int,C_10: quickcheck_code_int] :
( ( C_10 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( times_123202395de_int @ A_30 @ C_10 ) @ ( times_123202395de_int @ B_23 @ C_10 ) )
= ( div_di1430059507de_int @ A_30 @ B_23 ) ) ) ).
thf(fact_2541_div__mult__self1__is__id,axiom,
! [A_29: int,B_22: int] :
( ( B_22 != zero_zero_int )
=> ( ( div_div_int @ ( times_times_int @ B_22 @ A_29 ) @ B_22 )
= A_29 ) ) ).
thf(fact_2542_div__mult__self1__is__id,axiom,
! [A_29: nat,B_22: nat] :
( ( B_22 != zero_zero_nat )
=> ( ( div_div_nat @ ( times_times_nat @ B_22 @ A_29 ) @ B_22 )
= A_29 ) ) ).
thf(fact_2543_div__mult__self1__is__id,axiom,
! [A_29: code_code_numeral,B_22: code_code_numeral] :
( ( B_22 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( times_1655362735umeral @ B_22 @ A_29 ) @ B_22 )
= A_29 ) ) ).
thf(fact_2544_div__mult__self1__is__id,axiom,
! [A_29: quickcheck_code_int,B_22: quickcheck_code_int] :
( ( B_22 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( times_123202395de_int @ B_22 @ A_29 ) @ B_22 )
= A_29 ) ) ).
thf(fact_2545_div__mult__self2__is__id,axiom,
! [A_28: int,B_21: int] :
( ( B_21 != zero_zero_int )
=> ( ( div_div_int @ ( times_times_int @ A_28 @ B_21 ) @ B_21 )
= A_28 ) ) ).
thf(fact_2546_div__mult__self2__is__id,axiom,
! [A_28: nat,B_21: nat] :
( ( B_21 != zero_zero_nat )
=> ( ( div_div_nat @ ( times_times_nat @ A_28 @ B_21 ) @ B_21 )
= A_28 ) ) ).
thf(fact_2547_div__mult__self2__is__id,axiom,
! [A_28: code_code_numeral,B_21: code_code_numeral] :
( ( B_21 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( times_1655362735umeral @ A_28 @ B_21 ) @ B_21 )
= A_28 ) ) ).
thf(fact_2548_div__mult__self2__is__id,axiom,
! [A_28: quickcheck_code_int,B_21: quickcheck_code_int] :
( ( B_21 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( times_123202395de_int @ A_28 @ B_21 ) @ B_21 )
= A_28 ) ) ).
thf(fact_2549_div__mult__mult1__if,axiom,
! [A_27: int,B_20: int,C_9: int] :
( ( ( C_9 = zero_zero_int )
=> ( ( div_div_int @ ( times_times_int @ C_9 @ A_27 ) @ ( times_times_int @ C_9 @ B_20 ) )
= zero_zero_int ) )
& ( ( C_9 != zero_zero_int )
=> ( ( div_div_int @ ( times_times_int @ C_9 @ A_27 ) @ ( times_times_int @ C_9 @ B_20 ) )
= ( div_div_int @ A_27 @ B_20 ) ) ) ) ).
thf(fact_2550_div__mult__mult1__if,axiom,
! [A_27: nat,B_20: nat,C_9: nat] :
( ( ( C_9 = zero_zero_nat )
=> ( ( div_div_nat @ ( times_times_nat @ C_9 @ A_27 ) @ ( times_times_nat @ C_9 @ B_20 ) )
= zero_zero_nat ) )
& ( ( C_9 != zero_zero_nat )
=> ( ( div_div_nat @ ( times_times_nat @ C_9 @ A_27 ) @ ( times_times_nat @ C_9 @ B_20 ) )
= ( div_div_nat @ A_27 @ B_20 ) ) ) ) ).
thf(fact_2551_div__mult__mult1__if,axiom,
! [A_27: code_code_numeral,B_20: code_code_numeral,C_9: code_code_numeral] :
( ( ( C_9 = zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( times_1655362735umeral @ C_9 @ A_27 ) @ ( times_1655362735umeral @ C_9 @ B_20 ) )
= zero_z126310315umeral ) )
& ( ( C_9 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( times_1655362735umeral @ C_9 @ A_27 ) @ ( times_1655362735umeral @ C_9 @ B_20 ) )
= ( div_di1218280263umeral @ A_27 @ B_20 ) ) ) ) ).
thf(fact_2552_div__mult__mult1__if,axiom,
! [A_27: quickcheck_code_int,B_20: quickcheck_code_int,C_9: quickcheck_code_int] :
( ( ( C_9 = zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( times_123202395de_int @ C_9 @ A_27 ) @ ( times_123202395de_int @ C_9 @ B_20 ) )
= zero_z891286103de_int ) )
& ( ( C_9 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( times_123202395de_int @ C_9 @ A_27 ) @ ( times_123202395de_int @ C_9 @ B_20 ) )
= ( div_di1430059507de_int @ A_27 @ B_20 ) ) ) ) ).
thf(fact_2553_div__self,axiom,
! [A_26: int] :
( ( A_26 != zero_zero_int )
=> ( ( div_div_int @ A_26 @ A_26 )
= one_one_int ) ) ).
thf(fact_2554_div__self,axiom,
! [A_26: nat] :
( ( A_26 != zero_zero_nat )
=> ( ( div_div_nat @ A_26 @ A_26 )
= one_one_nat ) ) ).
thf(fact_2555_div__self,axiom,
! [A_26: code_code_numeral] :
( ( A_26 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ A_26 @ A_26 )
= one_on1645066479umeral ) ) ).
thf(fact_2556_div__self,axiom,
! [A_26: quickcheck_code_int] :
( ( A_26 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ A_26 @ A_26 )
= one_on1684967323de_int ) ) ).
thf(fact_2557_div__mult__div__if__dvd,axiom,
! [Z_6: code_code_numeral,W_1: code_code_numeral,Y_9: code_code_numeral,X_13: code_code_numeral] :
( ( dvd_dv174992974umeral @ Y_9 @ X_13 )
=> ( ( dvd_dv174992974umeral @ Z_6 @ W_1 )
=> ( ( times_1655362735umeral @ ( div_di1218280263umeral @ X_13 @ Y_9 ) @ ( div_di1218280263umeral @ W_1 @ Z_6 ) )
= ( div_di1218280263umeral @ ( times_1655362735umeral @ X_13 @ W_1 ) @ ( times_1655362735umeral @ Y_9 @ Z_6 ) ) ) ) ) ).
thf(fact_2558_div__mult__div__if__dvd,axiom,
! [Z_6: quickcheck_code_int,W_1: quickcheck_code_int,Y_9: quickcheck_code_int,X_13: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ Y_9 @ X_13 )
=> ( ( dvd_dv1760642554de_int @ Z_6 @ W_1 )
=> ( ( times_123202395de_int @ ( div_di1430059507de_int @ X_13 @ Y_9 ) @ ( div_di1430059507de_int @ W_1 @ Z_6 ) )
= ( div_di1430059507de_int @ ( times_123202395de_int @ X_13 @ W_1 ) @ ( times_123202395de_int @ Y_9 @ Z_6 ) ) ) ) ) ).
thf(fact_2559_div__mult__div__if__dvd,axiom,
! [Z_6: int,W_1: int,Y_9: int,X_13: int] :
( ( dvd_dvd_int @ Y_9 @ X_13 )
=> ( ( dvd_dvd_int @ Z_6 @ W_1 )
=> ( ( times_times_int @ ( div_div_int @ X_13 @ Y_9 ) @ ( div_div_int @ W_1 @ Z_6 ) )
= ( div_div_int @ ( times_times_int @ X_13 @ W_1 ) @ ( times_times_int @ Y_9 @ Z_6 ) ) ) ) ) ).
thf(fact_2560_div__mult__div__if__dvd,axiom,
! [Z_6: nat,W_1: nat,Y_9: nat,X_13: nat] :
( ( dvd_dvd_nat @ Y_9 @ X_13 )
=> ( ( dvd_dvd_nat @ Z_6 @ W_1 )
=> ( ( times_times_nat @ ( div_div_nat @ X_13 @ Y_9 ) @ ( div_div_nat @ W_1 @ Z_6 ) )
= ( div_div_nat @ ( times_times_nat @ X_13 @ W_1 ) @ ( times_times_nat @ Y_9 @ Z_6 ) ) ) ) ) ).
thf(fact_2561_dvd__div__mult,axiom,
! [C_8: code_code_numeral,A_25: code_code_numeral,B_19: code_code_numeral] :
( ( dvd_dv174992974umeral @ A_25 @ B_19 )
=> ( ( times_1655362735umeral @ ( div_di1218280263umeral @ B_19 @ A_25 ) @ C_8 )
= ( div_di1218280263umeral @ ( times_1655362735umeral @ B_19 @ C_8 ) @ A_25 ) ) ) ).
thf(fact_2562_dvd__div__mult,axiom,
! [C_8: quickcheck_code_int,A_25: quickcheck_code_int,B_19: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ A_25 @ B_19 )
=> ( ( times_123202395de_int @ ( div_di1430059507de_int @ B_19 @ A_25 ) @ C_8 )
= ( div_di1430059507de_int @ ( times_123202395de_int @ B_19 @ C_8 ) @ A_25 ) ) ) ).
thf(fact_2563_dvd__div__mult,axiom,
! [C_8: int,A_25: int,B_19: int] :
( ( dvd_dvd_int @ A_25 @ B_19 )
=> ( ( times_times_int @ ( div_div_int @ B_19 @ A_25 ) @ C_8 )
= ( div_div_int @ ( times_times_int @ B_19 @ C_8 ) @ A_25 ) ) ) ).
thf(fact_2564_dvd__div__mult,axiom,
! [C_8: nat,A_25: nat,B_19: nat] :
( ( dvd_dvd_nat @ A_25 @ B_19 )
=> ( ( times_times_nat @ ( div_div_nat @ B_19 @ A_25 ) @ C_8 )
= ( div_div_nat @ ( times_times_nat @ B_19 @ C_8 ) @ A_25 ) ) ) ).
thf(fact_2565_dvd__div__mult__self,axiom,
! [A_24: code_code_numeral,B_18: code_code_numeral] :
( ( dvd_dv174992974umeral @ A_24 @ B_18 )
=> ( ( times_1655362735umeral @ ( div_di1218280263umeral @ B_18 @ A_24 ) @ A_24 )
= B_18 ) ) ).
thf(fact_2566_dvd__div__mult__self,axiom,
! [A_24: quickcheck_code_int,B_18: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ A_24 @ B_18 )
=> ( ( times_123202395de_int @ ( div_di1430059507de_int @ B_18 @ A_24 ) @ A_24 )
= B_18 ) ) ).
thf(fact_2567_dvd__div__mult__self,axiom,
! [A_24: int,B_18: int] :
( ( dvd_dvd_int @ A_24 @ B_18 )
=> ( ( times_times_int @ ( div_div_int @ B_18 @ A_24 ) @ A_24 )
= B_18 ) ) ).
thf(fact_2568_dvd__div__mult__self,axiom,
! [A_24: nat,B_18: nat] :
( ( dvd_dvd_nat @ A_24 @ B_18 )
=> ( ( times_times_nat @ ( div_div_nat @ B_18 @ A_24 ) @ A_24 )
= B_18 ) ) ).
thf(fact_2569_div__mult__swap,axiom,
! [A_23: code_code_numeral,C_7: code_code_numeral,B_17: code_code_numeral] :
( ( dvd_dv174992974umeral @ C_7 @ B_17 )
=> ( ( times_1655362735umeral @ A_23 @ ( div_di1218280263umeral @ B_17 @ C_7 ) )
= ( div_di1218280263umeral @ ( times_1655362735umeral @ A_23 @ B_17 ) @ C_7 ) ) ) ).
thf(fact_2570_div__mult__swap,axiom,
! [A_23: quickcheck_code_int,C_7: quickcheck_code_int,B_17: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ C_7 @ B_17 )
=> ( ( times_123202395de_int @ A_23 @ ( div_di1430059507de_int @ B_17 @ C_7 ) )
= ( div_di1430059507de_int @ ( times_123202395de_int @ A_23 @ B_17 ) @ C_7 ) ) ) ).
thf(fact_2571_div__mult__swap,axiom,
! [A_23: int,C_7: int,B_17: int] :
( ( dvd_dvd_int @ C_7 @ B_17 )
=> ( ( times_times_int @ A_23 @ ( div_div_int @ B_17 @ C_7 ) )
= ( div_div_int @ ( times_times_int @ A_23 @ B_17 ) @ C_7 ) ) ) ).
thf(fact_2572_div__mult__swap,axiom,
! [A_23: nat,C_7: nat,B_17: nat] :
( ( dvd_dvd_nat @ C_7 @ B_17 )
=> ( ( times_times_nat @ A_23 @ ( div_div_nat @ B_17 @ C_7 ) )
= ( div_div_nat @ ( times_times_nat @ A_23 @ B_17 ) @ C_7 ) ) ) ).
thf(fact_2573_dvd__mult__div__cancel,axiom,
! [A_22: code_code_numeral,B_16: code_code_numeral] :
( ( dvd_dv174992974umeral @ A_22 @ B_16 )
=> ( ( times_1655362735umeral @ A_22 @ ( div_di1218280263umeral @ B_16 @ A_22 ) )
= B_16 ) ) ).
thf(fact_2574_dvd__mult__div__cancel,axiom,
! [A_22: quickcheck_code_int,B_16: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ A_22 @ B_16 )
=> ( ( times_123202395de_int @ A_22 @ ( div_di1430059507de_int @ B_16 @ A_22 ) )
= B_16 ) ) ).
thf(fact_2575_dvd__mult__div__cancel,axiom,
! [A_22: int,B_16: int] :
( ( dvd_dvd_int @ A_22 @ B_16 )
=> ( ( times_times_int @ A_22 @ ( div_div_int @ B_16 @ A_22 ) )
= B_16 ) ) ).
thf(fact_2576_dvd__mult__div__cancel,axiom,
! [A_22: nat,B_16: nat] :
( ( dvd_dvd_nat @ A_22 @ B_16 )
=> ( ( times_times_nat @ A_22 @ ( div_div_nat @ B_16 @ A_22 ) )
= B_16 ) ) ).
thf(fact_2577_div__add,axiom,
! [Y_8: code_code_numeral,Z_5: code_code_numeral,X_12: code_code_numeral] :
( ( dvd_dv174992974umeral @ Z_5 @ X_12 )
=> ( ( dvd_dv174992974umeral @ Z_5 @ Y_8 )
=> ( ( div_di1218280263umeral @ ( plus_p1627245867umeral @ X_12 @ Y_8 ) @ Z_5 )
= ( plus_p1627245867umeral @ ( div_di1218280263umeral @ X_12 @ Z_5 ) @ ( div_di1218280263umeral @ Y_8 @ Z_5 ) ) ) ) ) ).
thf(fact_2578_div__add,axiom,
! [Y_8: quickcheck_code_int,Z_5: quickcheck_code_int,X_12: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ Z_5 @ X_12 )
=> ( ( dvd_dv1760642554de_int @ Z_5 @ Y_8 )
=> ( ( div_di1430059507de_int @ ( plus_p1446045655de_int @ X_12 @ Y_8 ) @ Z_5 )
= ( plus_p1446045655de_int @ ( div_di1430059507de_int @ X_12 @ Z_5 ) @ ( div_di1430059507de_int @ Y_8 @ Z_5 ) ) ) ) ) ).
thf(fact_2579_div__add,axiom,
! [Y_8: int,Z_5: int,X_12: int] :
( ( dvd_dvd_int @ Z_5 @ X_12 )
=> ( ( dvd_dvd_int @ Z_5 @ Y_8 )
=> ( ( div_div_int @ ( plus_plus_int @ X_12 @ Y_8 ) @ Z_5 )
= ( plus_plus_int @ ( div_div_int @ X_12 @ Z_5 ) @ ( div_div_int @ Y_8 @ Z_5 ) ) ) ) ) ).
thf(fact_2580_div__add,axiom,
! [Y_8: nat,Z_5: nat,X_12: nat] :
( ( dvd_dvd_nat @ Z_5 @ X_12 )
=> ( ( dvd_dvd_nat @ Z_5 @ Y_8 )
=> ( ( div_div_nat @ ( plus_plus_nat @ X_12 @ Y_8 ) @ Z_5 )
= ( plus_plus_nat @ ( div_div_nat @ X_12 @ Z_5 ) @ ( div_div_nat @ Y_8 @ Z_5 ) ) ) ) ) ).
thf(fact_2581_div__power,axiom,
! [N_29: nat,Y_7: code_code_numeral,X_11: code_code_numeral] :
( ( dvd_dv174992974umeral @ Y_7 @ X_11 )
=> ( ( power_2100829034umeral @ ( div_di1218280263umeral @ X_11 @ Y_7 ) @ N_29 )
= ( div_di1218280263umeral @ ( power_2100829034umeral @ X_11 @ N_29 ) @ ( power_2100829034umeral @ Y_7 @ N_29 ) ) ) ) ).
thf(fact_2582_div__power,axiom,
! [N_29: nat,Y_7: quickcheck_code_int,X_11: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ Y_7 @ X_11 )
=> ( ( power_881366806de_int @ ( div_di1430059507de_int @ X_11 @ Y_7 ) @ N_29 )
= ( div_di1430059507de_int @ ( power_881366806de_int @ X_11 @ N_29 ) @ ( power_881366806de_int @ Y_7 @ N_29 ) ) ) ) ).
thf(fact_2583_div__power,axiom,
! [N_29: nat,Y_7: int,X_11: int] :
( ( dvd_dvd_int @ Y_7 @ X_11 )
=> ( ( power_power_int @ ( div_div_int @ X_11 @ Y_7 ) @ N_29 )
= ( div_div_int @ ( power_power_int @ X_11 @ N_29 ) @ ( power_power_int @ Y_7 @ N_29 ) ) ) ) ).
thf(fact_2584_div__power,axiom,
! [N_29: nat,Y_7: nat,X_11: nat] :
( ( dvd_dvd_nat @ Y_7 @ X_11 )
=> ( ( power_power_nat @ ( div_div_nat @ X_11 @ Y_7 ) @ N_29 )
= ( div_div_nat @ ( power_power_nat @ X_11 @ N_29 ) @ ( power_power_nat @ Y_7 @ N_29 ) ) ) ) ).
thf(fact_2585_split__neg__lemma,axiom,
! [P: int > int > $o,N: int,K_1: int] :
( ( ord_less_int @ K_1 @ zero_zero_int )
=> ( ( P @ ( div_div_int @ N @ K_1 ) @ ( div_mod_int @ N @ K_1 ) )
<=> ! [I_1: int,J_1: int] :
( ( ( ord_less_int @ K_1 @ J_1 )
& ( ord_less_eq_int @ J_1 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K_1 @ I_1 ) @ J_1 ) ) )
=> ( P @ I_1 @ J_1 ) ) ) ) ).
thf(fact_2586_split__pos__lemma,axiom,
! [P: int > int > $o,N: int,K_1: int] :
( ( ord_less_int @ zero_zero_int @ K_1 )
=> ( ( P @ ( div_div_int @ N @ K_1 ) @ ( div_mod_int @ N @ K_1 ) )
<=> ! [I_1: int,J_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J_1 )
& ( ord_less_int @ J_1 @ K_1 )
& ( N
= ( plus_plus_int @ ( times_times_int @ K_1 @ I_1 ) @ J_1 ) ) )
=> ( P @ I_1 @ J_1 ) ) ) ) ).
thf(fact_2587_dvd__eq__mod__eq__0__number__of,axiom,
! [X_10: int,Y_6: int] :
( ( dvd_dv174992974umeral @ ( number1443263063umeral @ X_10 ) @ ( number1443263063umeral @ Y_6 ) )
<=> ( ( div_mo1740067990umeral @ ( number1443263063umeral @ Y_6 ) @ ( number1443263063umeral @ X_10 ) )
= zero_z126310315umeral ) ) ).
thf(fact_2588_dvd__eq__mod__eq__0__number__of,axiom,
! [X_10: int,Y_6: int] :
( ( dvd_dv1760642554de_int @ ( number1226105091de_int @ X_10 ) @ ( number1226105091de_int @ Y_6 ) )
<=> ( ( div_mo231679042de_int @ ( number1226105091de_int @ Y_6 ) @ ( number1226105091de_int @ X_10 ) )
= zero_z891286103de_int ) ) ).
thf(fact_2589_dvd__eq__mod__eq__0__number__of,axiom,
! [X_10: int,Y_6: int] :
( ( dvd_dvd_int @ ( number_number_of_int @ X_10 ) @ ( number_number_of_int @ Y_6 ) )
<=> ( ( div_mod_int @ ( number_number_of_int @ Y_6 ) @ ( number_number_of_int @ X_10 ) )
= zero_zero_int ) ) ).
thf(fact_2590_dvd__eq__mod__eq__0__number__of,axiom,
! [X_10: int,Y_6: int] :
( ( dvd_dvd_nat @ ( number_number_of_nat @ X_10 ) @ ( number_number_of_nat @ Y_6 ) )
<=> ( ( div_mod_nat @ ( number_number_of_nat @ Y_6 ) @ ( number_number_of_nat @ X_10 ) )
= zero_zero_nat ) ) ).
thf(fact_2591_StandardRes__ubound,axiom,
! [X: int,P_3: int] :
( ( ord_less_int @ zero_zero_int @ P_3 )
=> ( ord_less_int @ ( standardRes @ P_3 @ X ) @ P_3 ) ) ).
thf(fact_2592_StandardRes__eq__zcong,axiom,
! [M: int,X: int] :
( ( ( standardRes @ M @ X )
= zero_zero_int )
<=> ( zcong @ X @ zero_zero_int @ M ) ) ).
thf(fact_2593_StandardRes__prop3,axiom,
! [X: int,P_3: int] :
( ~ ( zcong @ X @ zero_zero_int @ P_3 )
<=> ( ( standardRes @ P_3 @ X )
!= zero_zero_int ) ) ).
thf(fact_2594_int__div__less__self,axiom,
! [K_1: int,X: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K_1 )
=> ( ord_less_int @ ( div_div_int @ X @ K_1 ) @ X ) ) ) ).
thf(fact_2595_zdiv__mono1__neg,axiom,
! [B: int,A: int,A_5: int] :
( ( ord_less_eq_int @ A @ A_5 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( div_div_int @ A_5 @ B ) @ ( div_div_int @ A @ B ) ) ) ) ).
thf(fact_2596_zdiv__mono1,axiom,
! [B: int,A: int,A_5: int] :
( ( ord_less_eq_int @ A @ A_5 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( div_div_int @ A @ B ) @ ( div_div_int @ A_5 @ B ) ) ) ) ).
thf(fact_2597_div__neg__neg__trivial,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ A )
=> ( ( div_div_int @ A @ B )
= zero_zero_int ) ) ) ).
thf(fact_2598_zdiv__mono2__neg,axiom,
! [B: int,B_5: int,A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B_5 )
=> ( ( ord_less_eq_int @ B_5 @ B )
=> ( ord_less_eq_int @ ( div_div_int @ A @ B_5 ) @ ( div_div_int @ A @ B ) ) ) ) ) ).
thf(fact_2599_div__nonpos__pos__le0,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( div_div_int @ A @ B ) @ zero_zero_int ) ) ) ).
thf(fact_2600_neg__imp__zdiv__nonneg__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( div_div_int @ A @ B ) )
<=> ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
thf(fact_2601_div__pos__pos__trivial,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ B )
=> ( ( div_div_int @ A @ B )
= zero_zero_int ) ) ) ).
thf(fact_2602_div__nonneg__neg__le0,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( div_div_int @ A @ B ) @ zero_zero_int ) ) ) ).
thf(fact_2603_zdiv__mono2,axiom,
! [B: int,B_5: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B_5 )
=> ( ( ord_less_eq_int @ B_5 @ B )
=> ( ord_less_eq_int @ ( div_div_int @ A @ B ) @ ( div_div_int @ A @ B_5 ) ) ) ) ) ).
thf(fact_2604_nonneg1__imp__zdiv__pos__iff,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( div_div_int @ A @ B ) )
<=> ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
thf(fact_2605_pos__imp__zdiv__pos__iff,axiom,
! [I: int,K_1: int] :
( ( ord_less_int @ zero_zero_int @ K_1 )
=> ( ( ord_less_int @ zero_zero_int @ ( div_div_int @ I @ K_1 ) )
<=> ( ord_less_eq_int @ K_1 @ I ) ) ) ).
thf(fact_2606_pos__imp__zdiv__nonneg__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( div_div_int @ A @ B ) )
<=> ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
thf(fact_2607_zdiv__eq__0__iff,axiom,
! [I: int,K_1: int] :
( ( ( div_div_int @ I @ K_1 )
= zero_zero_int )
<=> ( ( K_1 = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K_1 ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K_1 @ I ) ) ) ) ).
thf(fact_2608_div__prop1,axiom,
! [X: int,Y: int,Z_1: int] :
( ( ord_less_int @ zero_zero_int @ Z_1 )
=> ( ( ord_less_int @ X @ ( times_times_int @ Y @ Z_1 ) )
=> ( ord_less_int @ ( div_div_int @ X @ Z_1 ) @ Y ) ) ) ).
thf(fact_2609_zdiv__zmult2__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( div_div_int @ A @ ( times_times_int @ B @ C ) )
= ( div_div_int @ ( div_div_int @ A @ B ) @ C ) ) ) ).
thf(fact_2610_mod__neg__neg__trivial,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ A )
=> ( ( div_mod_int @ A @ B )
= A ) ) ) ).
thf(fact_2611_neg__mod__conj,axiom,
! [A: int,B: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( div_mod_int @ A @ B ) @ zero_zero_int )
& ( ord_less_int @ B @ ( div_mod_int @ A @ B ) ) ) ) ).
thf(fact_2612_neg__mod__sign,axiom,
! [A: int,B: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( div_mod_int @ A @ B ) @ zero_zero_int ) ) ).
thf(fact_2613_mod__pos__pos__trivial,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ B )
=> ( ( div_mod_int @ A @ B )
= A ) ) ) ).
thf(fact_2614_pos__mod__conj,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( div_mod_int @ A @ B ) )
& ( ord_less_int @ ( div_mod_int @ A @ B ) @ B ) ) ) ).
thf(fact_2615_pos__mod__sign,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( div_mod_int @ A @ B ) ) ) ).
thf(fact_2616_zcong__zmod__eq,axiom,
! [A: int,B: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( zcong @ A @ B @ M )
<=> ( ( div_mod_int @ A @ M )
= ( div_mod_int @ B @ M ) ) ) ) ).
thf(fact_2617_zmod__zdvd__zmod,axiom,
! [A: int,B: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( dvd_dvd_int @ M @ B )
=> ( ( div_mod_int @ ( div_mod_int @ A @ B ) @ M )
= ( div_mod_int @ A @ M ) ) ) ) ).
thf(fact_2618_zmod__minus1__right,axiom,
! [A: int] :
( ( div_mod_int @ A @ ( number_number_of_int @ min ) )
= zero_zero_int ) ).
thf(fact_2619_zdvd__iff__zmod__eq__0__number__of,axiom,
! [X: int,Y: int] :
( ( dvd_dvd_int @ ( number_number_of_int @ X ) @ ( number_number_of_int @ Y ) )
<=> ( ( div_mod_int @ ( number_number_of_int @ Y ) @ ( number_number_of_int @ X ) )
= zero_zero_int ) ) ).
thf(fact_2620_div__mult__self1,axiom,
! [A_21: int,C_6: int,B_15: int] :
( ( B_15 != zero_zero_int )
=> ( ( div_div_int @ ( plus_plus_int @ A_21 @ ( times_times_int @ C_6 @ B_15 ) ) @ B_15 )
= ( plus_plus_int @ C_6 @ ( div_div_int @ A_21 @ B_15 ) ) ) ) ).
thf(fact_2621_div__mult__self1,axiom,
! [A_21: nat,C_6: nat,B_15: nat] :
( ( B_15 != zero_zero_nat )
=> ( ( div_div_nat @ ( plus_plus_nat @ A_21 @ ( times_times_nat @ C_6 @ B_15 ) ) @ B_15 )
= ( plus_plus_nat @ C_6 @ ( div_div_nat @ A_21 @ B_15 ) ) ) ) ).
thf(fact_2622_div__mult__self1,axiom,
! [A_21: code_code_numeral,C_6: code_code_numeral,B_15: code_code_numeral] :
( ( B_15 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( plus_p1627245867umeral @ A_21 @ ( times_1655362735umeral @ C_6 @ B_15 ) ) @ B_15 )
= ( plus_p1627245867umeral @ C_6 @ ( div_di1218280263umeral @ A_21 @ B_15 ) ) ) ) ).
thf(fact_2623_div__mult__self1,axiom,
! [A_21: quickcheck_code_int,C_6: quickcheck_code_int,B_15: quickcheck_code_int] :
( ( B_15 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( plus_p1446045655de_int @ A_21 @ ( times_123202395de_int @ C_6 @ B_15 ) ) @ B_15 )
= ( plus_p1446045655de_int @ C_6 @ ( div_di1430059507de_int @ A_21 @ B_15 ) ) ) ) ).
thf(fact_2624_div__mult__self2,axiom,
! [A_20: int,C_5: int,B_14: int] :
( ( B_14 != zero_zero_int )
=> ( ( div_div_int @ ( plus_plus_int @ A_20 @ ( times_times_int @ B_14 @ C_5 ) ) @ B_14 )
= ( plus_plus_int @ C_5 @ ( div_div_int @ A_20 @ B_14 ) ) ) ) ).
thf(fact_2625_div__mult__self2,axiom,
! [A_20: nat,C_5: nat,B_14: nat] :
( ( B_14 != zero_zero_nat )
=> ( ( div_div_nat @ ( plus_plus_nat @ A_20 @ ( times_times_nat @ B_14 @ C_5 ) ) @ B_14 )
= ( plus_plus_nat @ C_5 @ ( div_div_nat @ A_20 @ B_14 ) ) ) ) ).
thf(fact_2626_div__mult__self2,axiom,
! [A_20: code_code_numeral,C_5: code_code_numeral,B_14: code_code_numeral] :
( ( B_14 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( plus_p1627245867umeral @ A_20 @ ( times_1655362735umeral @ B_14 @ C_5 ) ) @ B_14 )
= ( plus_p1627245867umeral @ C_5 @ ( div_di1218280263umeral @ A_20 @ B_14 ) ) ) ) ).
thf(fact_2627_div__mult__self2,axiom,
! [A_20: quickcheck_code_int,C_5: quickcheck_code_int,B_14: quickcheck_code_int] :
( ( B_14 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( plus_p1446045655de_int @ A_20 @ ( times_123202395de_int @ B_14 @ C_5 ) ) @ B_14 )
= ( plus_p1446045655de_int @ C_5 @ ( div_di1430059507de_int @ A_20 @ B_14 ) ) ) ) ).
thf(fact_2628_div__add__self1,axiom,
! [A_19: int,B_13: int] :
( ( B_13 != zero_zero_int )
=> ( ( div_div_int @ ( plus_plus_int @ B_13 @ A_19 ) @ B_13 )
= ( plus_plus_int @ ( div_div_int @ A_19 @ B_13 ) @ one_one_int ) ) ) ).
thf(fact_2629_div__add__self1,axiom,
! [A_19: nat,B_13: nat] :
( ( B_13 != zero_zero_nat )
=> ( ( div_div_nat @ ( plus_plus_nat @ B_13 @ A_19 ) @ B_13 )
= ( plus_plus_nat @ ( div_div_nat @ A_19 @ B_13 ) @ one_one_nat ) ) ) ).
thf(fact_2630_div__add__self1,axiom,
! [A_19: code_code_numeral,B_13: code_code_numeral] :
( ( B_13 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( plus_p1627245867umeral @ B_13 @ A_19 ) @ B_13 )
= ( plus_p1627245867umeral @ ( div_di1218280263umeral @ A_19 @ B_13 ) @ one_on1645066479umeral ) ) ) ).
thf(fact_2631_div__add__self1,axiom,
! [A_19: quickcheck_code_int,B_13: quickcheck_code_int] :
( ( B_13 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( plus_p1446045655de_int @ B_13 @ A_19 ) @ B_13 )
= ( plus_p1446045655de_int @ ( div_di1430059507de_int @ A_19 @ B_13 ) @ one_on1684967323de_int ) ) ) ).
thf(fact_2632_div__add__self2,axiom,
! [A_18: int,B_12: int] :
( ( B_12 != zero_zero_int )
=> ( ( div_div_int @ ( plus_plus_int @ A_18 @ B_12 ) @ B_12 )
= ( plus_plus_int @ ( div_div_int @ A_18 @ B_12 ) @ one_one_int ) ) ) ).
thf(fact_2633_div__add__self2,axiom,
! [A_18: nat,B_12: nat] :
( ( B_12 != zero_zero_nat )
=> ( ( div_div_nat @ ( plus_plus_nat @ A_18 @ B_12 ) @ B_12 )
= ( plus_plus_nat @ ( div_div_nat @ A_18 @ B_12 ) @ one_one_nat ) ) ) ).
thf(fact_2634_div__add__self2,axiom,
! [A_18: code_code_numeral,B_12: code_code_numeral] :
( ( B_12 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( plus_p1627245867umeral @ A_18 @ B_12 ) @ B_12 )
= ( plus_p1627245867umeral @ ( div_di1218280263umeral @ A_18 @ B_12 ) @ one_on1645066479umeral ) ) ) ).
thf(fact_2635_div__add__self2,axiom,
! [A_18: quickcheck_code_int,B_12: quickcheck_code_int] :
( ( B_12 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( plus_p1446045655de_int @ A_18 @ B_12 ) @ B_12 )
= ( plus_p1446045655de_int @ ( div_di1430059507de_int @ A_18 @ B_12 ) @ one_on1684967323de_int ) ) ) ).
thf(fact_2636_dvd__div__div__eq__mult,axiom,
! [D_4: code_code_numeral,B_11: code_code_numeral,C_4: code_code_numeral,A_17: code_code_numeral] :
( ( A_17 != zero_z126310315umeral )
=> ( ( C_4 != zero_z126310315umeral )
=> ( ( dvd_dv174992974umeral @ A_17 @ B_11 )
=> ( ( dvd_dv174992974umeral @ C_4 @ D_4 )
=> ( ( ( div_di1218280263umeral @ B_11 @ A_17 )
= ( div_di1218280263umeral @ D_4 @ C_4 ) )
<=> ( ( times_1655362735umeral @ B_11 @ C_4 )
= ( times_1655362735umeral @ A_17 @ D_4 ) ) ) ) ) ) ) ).
thf(fact_2637_dvd__div__div__eq__mult,axiom,
! [D_4: quickcheck_code_int,B_11: quickcheck_code_int,C_4: quickcheck_code_int,A_17: quickcheck_code_int] :
( ( A_17 != zero_z891286103de_int )
=> ( ( C_4 != zero_z891286103de_int )
=> ( ( dvd_dv1760642554de_int @ A_17 @ B_11 )
=> ( ( dvd_dv1760642554de_int @ C_4 @ D_4 )
=> ( ( ( div_di1430059507de_int @ B_11 @ A_17 )
= ( div_di1430059507de_int @ D_4 @ C_4 ) )
<=> ( ( times_123202395de_int @ B_11 @ C_4 )
= ( times_123202395de_int @ A_17 @ D_4 ) ) ) ) ) ) ) ).
thf(fact_2638_dvd__div__div__eq__mult,axiom,
! [D_4: int,B_11: int,C_4: int,A_17: int] :
( ( A_17 != zero_zero_int )
=> ( ( C_4 != zero_zero_int )
=> ( ( dvd_dvd_int @ A_17 @ B_11 )
=> ( ( dvd_dvd_int @ C_4 @ D_4 )
=> ( ( ( div_div_int @ B_11 @ A_17 )
= ( div_div_int @ D_4 @ C_4 ) )
<=> ( ( times_times_int @ B_11 @ C_4 )
= ( times_times_int @ A_17 @ D_4 ) ) ) ) ) ) ) ).
thf(fact_2639_dvd__div__div__eq__mult,axiom,
! [D_4: nat,B_11: nat,C_4: nat,A_17: nat] :
( ( A_17 != zero_zero_nat )
=> ( ( C_4 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A_17 @ B_11 )
=> ( ( dvd_dvd_nat @ C_4 @ D_4 )
=> ( ( ( div_div_nat @ B_11 @ A_17 )
= ( div_div_nat @ D_4 @ C_4 ) )
<=> ( ( times_times_nat @ B_11 @ C_4 )
= ( times_times_nat @ A_17 @ D_4 ) ) ) ) ) ) ) ).
thf(fact_2640_dvd__div__eq__mult,axiom,
! [C_3: code_code_numeral,B_10: code_code_numeral,A_16: code_code_numeral] :
( ( A_16 != zero_z126310315umeral )
=> ( ( dvd_dv174992974umeral @ A_16 @ B_10 )
=> ( ( ( div_di1218280263umeral @ B_10 @ A_16 )
= C_3 )
<=> ( B_10
= ( times_1655362735umeral @ C_3 @ A_16 ) ) ) ) ) ).
thf(fact_2641_dvd__div__eq__mult,axiom,
! [C_3: quickcheck_code_int,B_10: quickcheck_code_int,A_16: quickcheck_code_int] :
( ( A_16 != zero_z891286103de_int )
=> ( ( dvd_dv1760642554de_int @ A_16 @ B_10 )
=> ( ( ( div_di1430059507de_int @ B_10 @ A_16 )
= C_3 )
<=> ( B_10
= ( times_123202395de_int @ C_3 @ A_16 ) ) ) ) ) ).
thf(fact_2642_dvd__div__eq__mult,axiom,
! [C_3: int,B_10: int,A_16: int] :
( ( A_16 != zero_zero_int )
=> ( ( dvd_dvd_int @ A_16 @ B_10 )
=> ( ( ( div_div_int @ B_10 @ A_16 )
= C_3 )
<=> ( B_10
= ( times_times_int @ C_3 @ A_16 ) ) ) ) ) ).
thf(fact_2643_dvd__div__eq__mult,axiom,
! [C_3: nat,B_10: nat,A_16: nat] :
( ( A_16 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A_16 @ B_10 )
=> ( ( ( div_div_nat @ B_10 @ A_16 )
= C_3 )
<=> ( B_10
= ( times_times_nat @ C_3 @ A_16 ) ) ) ) ) ).
thf(fact_2644_zdiv__leq__prop,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ ( div_div_int @ X @ Y ) ) @ X ) ) ).
thf(fact_2645_power__preserves__odd,axiom,
! [X: int,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_int @ ( power_power_int @ X @ N ) @ zOdd )
<=> ( member_int @ X @ zOdd ) ) ) ).
thf(fact_2646_div__eq__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( div_div_int @ ( number_number_of_int @ min ) @ B )
= ( number_number_of_int @ min ) ) ) ).
thf(fact_2647_mod__pos__neg__trivial,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int )
=> ( ( div_mod_int @ A @ B )
= ( plus_plus_int @ A @ B ) ) ) ) ).
thf(fact_2648_StandardRes__lbound,axiom,
! [X: int,P_3: int] :
( ( ord_less_int @ zero_zero_int @ P_3 )
=> ( ord_less_eq_int @ zero_zero_int @ ( standardRes @ P_3 @ X ) ) ) ).
thf(fact_2649_Euler_Oaux__2,axiom,
! [P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( member_int @ P_3 @ zOdd )
=> ( ord_less_int @ zero_zero_int @ ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_2650_StandardRes__prop2,axiom,
! [X1: int,X2: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( standardRes @ M @ X1 )
= ( standardRes @ M @ X2 ) )
<=> ( zcong @ X1 @ X2 @ M ) ) ) ).
thf(fact_2651_zmod__number__of__Bit0,axiom,
! [V: int,W: int] :
( ( div_mod_int @ ( number_number_of_int @ ( bit0 @ V ) ) @ ( number_number_of_int @ ( bit0 @ W ) ) )
= ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( div_mod_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) ) ) ) ).
thf(fact_2652_div__prop2,axiom,
! [X: int,Y: int,Z_1: int] :
( ( ord_less_int @ zero_zero_int @ Z_1 )
=> ( ( ord_less_int @ X @ ( plus_plus_int @ ( times_times_int @ Y @ Z_1 ) @ Z_1 ) )
=> ( ord_less_eq_int @ ( div_div_int @ X @ Z_1 ) @ Y ) ) ) ).
thf(fact_2653_divmod__int__rel__div__eq,axiom,
! [A_4: int,B_3: int,Y: int,R_5: int] :
( ( A_4
= ( plus_plus_int @ ( times_times_int @ B_3 @ Y ) @ R_5 ) )
=> ( ( ( ( ord_less_int @ zero_zero_int @ B_3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_5 )
& ( ord_less_int @ R_5 @ B_3 ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ B_3 )
=> ( ( ord_less_int @ B_3 @ R_5 )
& ( ord_less_eq_int @ R_5 @ zero_zero_int ) ) ) )
=> ( ( B_3 != zero_zero_int )
=> ( ( div_div_int @ A_4 @ B_3 )
= Y ) ) ) ) ).
thf(fact_2654_split__zdiv,axiom,
! [P: int > $o,N: int,K_1: int] :
( ( P @ ( div_div_int @ N @ K_1 ) )
<=> ( ( ( K_1 = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K_1 )
=> ! [I_1: int,J_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J_1 )
& ( ord_less_int @ J_1 @ K_1 )
& ( N
= ( plus_plus_int @ ( times_times_int @ K_1 @ I_1 ) @ J_1 ) ) )
=> ( P @ I_1 ) ) )
& ( ( ord_less_int @ K_1 @ zero_zero_int )
=> ! [I_1: int,J_1: int] :
( ( ( ord_less_int @ K_1 @ J_1 )
& ( ord_less_eq_int @ J_1 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K_1 @ I_1 ) @ J_1 ) ) )
=> ( P @ I_1 ) ) ) ) ) ).
thf(fact_2655_divmod__int__rel__mod__eq,axiom,
! [A_4: int,B_3: int,Q_7: int,Y: int] :
( ( A_4
= ( plus_plus_int @ ( times_times_int @ B_3 @ Q_7 ) @ Y ) )
=> ( ( ( ( ord_less_int @ zero_zero_int @ B_3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
& ( ord_less_int @ Y @ B_3 ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ B_3 )
=> ( ( ord_less_int @ B_3 @ Y )
& ( ord_less_eq_int @ Y @ zero_zero_int ) ) ) )
=> ( ( B_3 != zero_zero_int )
=> ( ( div_mod_int @ A_4 @ B_3 )
= Y ) ) ) ) ).
thf(fact_2656_zmult2__lemma__aux2,axiom,
! [Q: int,B: int,R_1: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ B @ R_1 )
=> ( ( ord_less_eq_int @ R_1 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ ( div_mod_int @ Q @ C ) ) @ R_1 ) @ zero_zero_int ) ) ) ) ).
thf(fact_2657_zmult2__lemma__aux1,axiom,
! [Q: int,B: int,R_1: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ B @ R_1 )
=> ( ( ord_less_eq_int @ R_1 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ ( div_mod_int @ Q @ C ) ) @ R_1 ) ) ) ) ) ).
thf(fact_2658_zmult2__lemma__aux4,axiom,
! [Q: int,B: int,R_1: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ( ord_less_int @ R_1 @ B )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B @ ( div_mod_int @ Q @ C ) ) @ R_1 ) @ ( times_times_int @ B @ C ) ) ) ) ) ).
thf(fact_2659_zmult2__lemma__aux3,axiom,
! [Q: int,B: int,R_1: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ( ord_less_int @ R_1 @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B @ ( div_mod_int @ Q @ C ) ) @ R_1 ) ) ) ) ) ).
thf(fact_2660_split__zmod,axiom,
! [P: int > $o,N: int,K_1: int] :
( ( P @ ( div_mod_int @ N @ K_1 ) )
<=> ( ( ( K_1 = zero_zero_int )
=> ( P @ N ) )
& ( ( ord_less_int @ zero_zero_int @ K_1 )
=> ! [I_1: int,J_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J_1 )
& ( ord_less_int @ J_1 @ K_1 )
& ( N
= ( plus_plus_int @ ( times_times_int @ K_1 @ I_1 ) @ J_1 ) ) )
=> ( P @ J_1 ) ) )
& ( ( ord_less_int @ K_1 @ zero_zero_int )
=> ! [I_1: int,J_1: int] :
( ( ( ord_less_int @ K_1 @ J_1 )
& ( ord_less_eq_int @ J_1 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K_1 @ I_1 ) @ J_1 ) ) )
=> ( P @ J_1 ) ) ) ) ) ).
thf(fact_2661_neq__one__mod__two,axiom,
! [X: int] :
( ( ( div_mod_int @ X @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= zero_zero_int )
<=> ( ( div_mod_int @ X @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_int ) ) ).
thf(fact_2662_div__pos__neg__trivial,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int )
=> ( ( div_div_int @ A @ B )
= ( number_number_of_int @ min ) ) ) ) ).
thf(fact_2663_zmod__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( div_mod_int @ ( number_number_of_int @ min ) @ B )
= ( minus_minus_int @ B @ one_one_int ) ) ) ).
thf(fact_2664_zdiv__number__of__Bit1,axiom,
! [V: int,W: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ W ) )
=> ( ( div_div_int @ ( number_number_of_int @ ( bit1 @ V ) ) @ ( number_number_of_int @ ( bit0 @ W ) ) )
= ( div_div_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ W ) )
=> ( ( div_div_int @ ( number_number_of_int @ ( bit1 @ V ) ) @ ( number_number_of_int @ ( bit0 @ W ) ) )
= ( div_div_int @ ( plus_plus_int @ ( number_number_of_int @ V ) @ one_one_int ) @ ( number_number_of_int @ W ) ) ) ) ) ).
thf(fact_2665_negDivAlg__minus1,axiom,
! [B: int] :
( ( negDivAlg @ ( number_number_of_int @ min ) @ B )
= ( product_Pair_int_int @ ( number_number_of_int @ min ) @ ( minus_minus_int @ B @ one_one_int ) ) ) ).
thf(fact_2666_negDivAlg_Osimps,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) )
| ( ord_less_eq_int @ B @ zero_zero_int ) )
=> ( ( negDivAlg @ A @ B )
= ( product_Pair_int_int @ ( number_number_of_int @ min ) @ ( plus_plus_int @ A @ B ) ) ) )
& ( ~ ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) )
| ( ord_less_eq_int @ B @ zero_zero_int ) )
=> ( ( negDivAlg @ A @ B )
= ( adjust @ B @ ( negDivAlg @ A @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) ) ) ) ) ).
thf(fact_2667_zprime__zOdd__eq__grt__2,axiom,
! [P_3: int] :
( ( zprime @ P_3 )
=> ( ( member_int @ P_3 @ zOdd )
<=> ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 ) ) ) ).
thf(fact_2668_negDivAlg__eqn__number__of,axiom,
! [W: int,V: int] :
( ( ord_less_int @ zero_zero_int @ ( number_number_of_int @ V ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( number_number_of_int @ W ) @ ( number_number_of_int @ V ) ) )
=> ( ( negDivAlg @ ( number_number_of_int @ W ) @ ( number_number_of_int @ V ) )
= ( product_Pair_int_int @ ( number_number_of_int @ min ) @ ( plus_plus_int @ ( number_number_of_int @ W ) @ ( number_number_of_int @ V ) ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( number_number_of_int @ W ) @ ( number_number_of_int @ V ) ) )
=> ( ( negDivAlg @ ( number_number_of_int @ W ) @ ( number_number_of_int @ V ) )
= ( adjust @ ( number_number_of_int @ V ) @ ( negDivAlg @ ( number_number_of_int @ W ) @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( number_number_of_int @ V ) ) ) ) ) ) ) ) ).
thf(fact_2669_negDivAlg__eqn,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) )
=> ( ( negDivAlg @ A @ B )
= ( product_Pair_int_int @ ( number_number_of_int @ min ) @ ( plus_plus_int @ A @ B ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) )
=> ( ( negDivAlg @ A @ B )
= ( adjust @ B @ ( negDivAlg @ A @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) ) ) ) ) ) ).
thf(fact_2670_neg__zdiv__mult__2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( div_div_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) )
= ( div_div_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
thf(fact_2671_zOddE,axiom,
! [X: int] :
( ( member_int @ X @ zOdd )
=> ~ ! [K: int] :
( X
!= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ K ) @ one_one_int ) ) ) ).
thf(fact_2672_SRStar__mult__prop2,axiom,
! [X: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ( member_int @ X @ ( sRStar @ P_3 ) )
=> ( member_int @ ( standardRes @ P_3 @ ( times_times_int @ A @ ( multInv @ P_3 @ X ) ) ) @ ( sRStar @ P_3 ) ) ) ) ) ) ).
thf(fact_2673_mod__div__decomp,axiom,
! [A_15: int,B_9: int] :
( A_15
= ( plus_plus_int @ ( times_times_int @ ( div_div_int @ A_15 @ B_9 ) @ B_9 ) @ ( div_mod_int @ A_15 @ B_9 ) ) ) ).
thf(fact_2674_mod__div__decomp,axiom,
! [A_15: nat,B_9: nat] :
( A_15
= ( plus_plus_nat @ ( times_times_nat @ ( div_div_nat @ A_15 @ B_9 ) @ B_9 ) @ ( div_mod_nat @ A_15 @ B_9 ) ) ) ).
thf(fact_2675_mod__div__decomp,axiom,
! [A_15: code_code_numeral,B_9: code_code_numeral] :
( A_15
= ( plus_p1627245867umeral @ ( times_1655362735umeral @ ( div_di1218280263umeral @ A_15 @ B_9 ) @ B_9 ) @ ( div_mo1740067990umeral @ A_15 @ B_9 ) ) ) ).
thf(fact_2676_mod__div__decomp,axiom,
! [A_15: quickcheck_code_int,B_9: quickcheck_code_int] :
( A_15
= ( plus_p1446045655de_int @ ( times_123202395de_int @ ( div_di1430059507de_int @ A_15 @ B_9 ) @ B_9 ) @ ( div_mo231679042de_int @ A_15 @ B_9 ) ) ) ).
thf(fact_2677_zfact__prop,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A )
=> ( zcong @ ( zfact @ ( minus_minus_int @ P_3 @ one_one_int ) ) @ ( power_power_int @ A @ ( nat_1 @ ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ P_3 ) ) ) ) ) ).
thf(fact_2678_Euler__part1,axiom,
! [X: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ X @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ X )
=> ( zcong @ ( power_power_int @ X @ ( nat_1 @ ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( number_number_of_int @ min ) @ P_3 ) ) ) ) ) ).
thf(fact_2679_posDivAlg_Osimps,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_int @ A @ B )
| ( ord_less_eq_int @ B @ zero_zero_int ) )
=> ( ( posDivAlg @ A @ B )
= ( product_Pair_int_int @ zero_zero_int @ A ) ) )
& ( ~ ( ( ord_less_int @ A @ B )
| ( ord_less_eq_int @ B @ zero_zero_int ) )
=> ( ( posDivAlg @ A @ B )
= ( adjust @ B @ ( posDivAlg @ A @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) ) ) ) ) ).
thf(fact_2680_posDivAlg__eqn__1__number__of,axiom,
! [W: int] :
( ( ord_less_int @ zero_zero_int @ ( number_number_of_int @ W ) )
=> ( ( ( ord_less_int @ one_one_int @ ( number_number_of_int @ W ) )
=> ( ( posDivAlg @ one_one_int @ ( number_number_of_int @ W ) )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) )
& ( ~ ( ord_less_int @ one_one_int @ ( number_number_of_int @ W ) )
=> ( ( posDivAlg @ one_one_int @ ( number_number_of_int @ W ) )
= ( adjust @ ( number_number_of_int @ W ) @ ( posDivAlg @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( number_number_of_int @ W ) ) ) ) ) ) ) ) ).
thf(fact_2681_Euler__part3,axiom,
! [X: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ X @ zero_zero_int @ P_3 )
=> ( ( quadRes @ P_3 @ X )
=> ( zcong @ ( power_power_int @ X @ ( nat_1 @ ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ one_one_int @ P_3 ) ) ) ) ) ).
thf(fact_2682_div__add1__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( div_div_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( plus_plus_nat @ ( div_div_nat @ A @ C ) @ ( div_div_nat @ B @ C ) ) @ ( div_div_nat @ ( plus_plus_nat @ ( div_mod_nat @ A @ C ) @ ( div_mod_nat @ B @ C ) ) @ C ) ) ) ).
thf(fact_2683_nat__if__cong,axiom,
! [X: int,Y: int,P: $o] :
( ( P
=> ( ( nat_1 @ X )
= ( nat_1 @ ( if_int @ P @ X @ Y ) ) ) )
& ( ~ P
=> ( ( nat_1 @ Y )
= ( nat_1 @ ( if_int @ P @ X @ Y ) ) ) ) ) ).
thf(fact_2684_mod__mult2__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( div_mod_nat @ A @ ( times_times_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ B @ ( div_mod_nat @ ( div_div_nat @ A @ B ) @ C ) ) @ ( div_mod_nat @ A @ B ) ) ) ).
thf(fact_2685_div__mult1__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( div_div_nat @ ( times_times_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( div_div_nat @ B @ C ) ) @ ( div_div_nat @ ( times_times_nat @ A @ ( div_mod_nat @ B @ C ) ) @ C ) ) ) ).
thf(fact_2686_div__mod__equality_H,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( div_div_nat @ M @ N ) @ N )
= ( minus_minus_nat @ M @ ( div_mod_nat @ M @ N ) ) ) ).
thf(fact_2687_mult__div__cancel,axiom,
! [N: nat,M: nat] :
( ( times_times_nat @ N @ ( div_div_nat @ M @ N ) )
= ( minus_minus_nat @ M @ ( div_mod_nat @ M @ N ) ) ) ).
thf(fact_2688_Divides_Omod__div__equality_H,axiom,
! [M: nat,N: nat] :
( ( div_mod_nat @ M @ N )
= ( minus_minus_nat @ M @ ( times_times_nat @ ( div_div_nat @ M @ N ) @ N ) ) ) ).
thf(fact_2689_div__le__mono,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( div_div_nat @ M @ K_1 ) @ ( div_div_nat @ N @ K_1 ) ) ) ).
thf(fact_2690_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( div_div_nat @ M @ N ) @ M ) ).
thf(fact_2691_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( div_mod_nat @ M @ N )
= M ) ) ).
thf(fact_2692_div__mult2__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( div_div_nat @ A @ ( times_times_nat @ B @ C ) )
= ( div_div_nat @ ( div_div_nat @ A @ B ) @ C ) ) ).
thf(fact_2693_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( div_mod_nat @ M @ N ) @ M ) ).
thf(fact_2694_mod__mult__distrib2,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( times_times_nat @ K_1 @ ( div_mod_nat @ M @ N ) )
= ( div_mod_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) ) ) ).
thf(fact_2695_mod__mult__distrib,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( times_times_nat @ ( div_mod_nat @ M @ N ) @ K_1 )
= ( div_mod_nat @ ( times_times_nat @ M @ K_1 ) @ ( times_times_nat @ N @ K_1 ) ) ) ).
thf(fact_2696_nat__div__distrib,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( nat_1 @ ( div_div_int @ X @ Y ) )
= ( div_div_nat @ ( nat_1 @ X ) @ ( nat_1 @ Y ) ) ) ) ).
thf(fact_2697_Divides_Otransfer__nat__int__functions_I1_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( div_div_nat @ ( nat_1 @ X ) @ ( nat_1 @ Y ) )
= ( nat_1 @ ( div_div_int @ X @ Y ) ) ) ) ) ).
thf(fact_2698_nat__mod__distrib,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat_1 @ ( div_mod_int @ X @ Y ) )
= ( div_mod_nat @ ( nat_1 @ X ) @ ( nat_1 @ Y ) ) ) ) ) ).
thf(fact_2699_Divides_Otransfer__nat__int__functions_I2_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( div_mod_nat @ ( nat_1 @ X ) @ ( nat_1 @ Y ) )
= ( nat_1 @ ( div_mod_int @ X @ Y ) ) ) ) ) ).
thf(fact_2700_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( div_div_nat @ M @ N )
= zero_zero_nat ) ) ).
thf(fact_2701_nat__mult__div__cancel__disj,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( ( K_1 = zero_zero_nat )
=> ( ( div_div_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
= zero_zero_nat ) )
& ( ( K_1 != zero_zero_nat )
=> ( ( div_div_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
= ( div_div_nat @ M @ N ) ) ) ) ).
thf(fact_2702_StandardRes__SRStar__prop3,axiom,
! [X: int,P_3: int] :
( ( member_int @ X @ ( sRStar @ P_3 ) )
=> ( ( standardRes @ P_3 @ X )
= X ) ) ).
thf(fact_2703_mod__less__divisor,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( div_mod_nat @ M @ N ) @ N ) ) ).
thf(fact_2704_transfer__nat__int__numerals_I1_J,axiom,
( zero_zero_nat
= ( nat_1 @ zero_zero_int ) ) ).
thf(fact_2705_nat__0,axiom,
( ( nat_1 @ zero_zero_int )
= zero_zero_nat ) ).
thf(fact_2706_mod__eq__0__iff,axiom,
! [M: nat,D: nat] :
( ( ( div_mod_nat @ M @ D )
= zero_zero_nat )
<=> ? [Q_2: nat] :
( M
= ( times_times_nat @ D @ Q_2 ) ) ) ).
thf(fact_2707_transfer__nat__int__relations_I1_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( nat_1 @ X )
= ( nat_1 @ Y ) )
<=> ( X = Y ) ) ) ) ).
thf(fact_2708_all__nat,axiom,
! [P: nat > $o] :
( ( all @ P )
<=> ! [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( P @ ( nat_1 @ X_1 ) ) ) ) ).
thf(fact_2709_ex__nat,axiom,
! [P: nat > $o] :
( ( ?? @ nat @ P )
<=> ? [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
& ( P @ ( nat_1 @ X_1 ) ) ) ) ).
thf(fact_2710_eq__nat__nat__iff,axiom,
! [Z_3: int,Z_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z_3 )
=> ( ( ( nat_1 @ Z_1 )
= ( nat_1 @ Z_3 ) )
<=> ( Z_1 = Z_3 ) ) ) ) ).
thf(fact_2711_mod__geq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( div_mod_nat @ M @ N )
= ( div_mod_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
thf(fact_2712_mod__if,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
=> ( ( div_mod_nat @ M @ N )
= M ) )
& ( ~ ( ord_less_nat @ M @ N )
=> ( ( div_mod_nat @ M @ N )
= ( div_mod_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ).
thf(fact_2713_mod__mult__self3,axiom,
! [K_1: nat,N: nat,M: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ ( times_times_nat @ K_1 @ N ) @ M ) @ N )
= ( div_mod_nat @ M @ N ) ) ).
thf(fact_2714_nat__number__of__def,axiom,
! [V: int] :
( ( number_number_of_nat @ V )
= ( nat_1 @ ( number_number_of_int @ V ) ) ) ).
thf(fact_2715_nat__number__of,axiom,
! [W: int] :
( ( nat_1 @ ( number_number_of_int @ W ) )
= ( number_number_of_nat @ W ) ) ).
thf(fact_2716_le__mod__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( div_mod_nat @ M @ N )
= ( div_mod_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
thf(fact_2717_transfer__nat__int__numerals_I2_J,axiom,
( one_one_nat
= ( nat_1 @ one_one_int ) ) ).
thf(fact_2718_SRStar__SR__prop,axiom,
! [X: int,P_3: int] :
( ( member_int @ X @ ( sRStar @ P_3 ) )
=> ( member_int @ X @ ( sr @ P_3 ) ) ) ).
thf(fact_2719_div__le__mono2,axiom,
! [K_1: nat,N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( div_div_nat @ K_1 @ N ) @ ( div_div_nat @ K_1 @ M ) ) ) ) ).
thf(fact_2720_nat__mult__div__cancel1,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ( div_div_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
= ( div_div_nat @ M @ N ) ) ) ).
thf(fact_2721_div__mult__self__is__m,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( div_div_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
thf(fact_2722_div__mult__self1__is__m,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( div_div_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
thf(fact_2723_div__less__dividend,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( div_div_nat @ M @ N ) @ M ) ) ) ).
thf(fact_2724_mod__le__divisor,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( div_mod_nat @ M @ N ) @ N ) ) ).
thf(fact_2725_nat__le__0,axiom,
! [Z_1: int] :
( ( ord_less_eq_int @ Z_1 @ zero_zero_int )
=> ( ( nat_1 @ Z_1 )
= zero_zero_nat ) ) ).
thf(fact_2726_nat__0__iff,axiom,
! [I: int] :
( ( ( nat_1 @ I )
= zero_zero_nat )
<=> ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
thf(fact_2727_zless__nat__conj,axiom,
! [W: int,Z_1: int] :
( ( ord_less_nat @ ( nat_1 @ W ) @ ( nat_1 @ Z_1 ) )
<=> ( ( ord_less_int @ zero_zero_int @ Z_1 )
& ( ord_less_int @ W @ Z_1 ) ) ) ).
thf(fact_2728_nat__mono__iff,axiom,
! [W: int,Z_1: int] :
( ( ord_less_int @ zero_zero_int @ Z_1 )
=> ( ( ord_less_nat @ ( nat_1 @ W ) @ ( nat_1 @ Z_1 ) )
<=> ( ord_less_int @ W @ Z_1 ) ) ) ).
thf(fact_2729_transfer__nat__int__relations_I3_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_nat @ ( nat_1 @ X ) @ ( nat_1 @ Y ) )
<=> ( ord_less_eq_int @ X @ Y ) ) ) ) ).
thf(fact_2730_posDivAlg__0,axiom,
! [B: int] :
( ( posDivAlg @ zero_zero_int @ B )
= ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) ).
thf(fact_2731_SRStar__def,axiom,
! [P_3: int] :
( ( sRStar @ P_3 )
= ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_int @ zero_zero_int @ X_1 ) @ ( ord_less_int @ X_1 @ P_3 ) ) ) ) ).
thf(fact_2732_StandardRes__SRStar__prop1a,axiom,
! [X: int,P_3: int] :
( ( member_int @ X @ ( sRStar @ P_3 ) )
=> ~ ( zcong @ X @ zero_zero_int @ P_3 ) ) ).
thf(fact_2733_split__div,axiom,
! [P: nat > $o,N: nat,K_1: nat] :
( ( P @ ( div_div_nat @ N @ K_1 ) )
<=> ( ( ( K_1 = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( K_1 != zero_zero_nat )
=> ! [I_1: nat,J_1: nat] :
( ( ord_less_nat @ J_1 @ K_1 )
=> ( ( N
= ( plus_plus_nat @ ( times_times_nat @ K_1 @ I_1 ) @ J_1 ) )
=> ( P @ I_1 ) ) ) ) ) ) ).
thf(fact_2734_zero__less__nat__eq,axiom,
! [Z_1: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat_1 @ Z_1 ) )
<=> ( ord_less_int @ zero_zero_int @ Z_1 ) ) ).
thf(fact_2735_split__mod,axiom,
! [P: nat > $o,N: nat,K_1: nat] :
( ( P @ ( div_mod_nat @ N @ K_1 ) )
<=> ( ( ( K_1 = zero_zero_nat )
=> ( P @ N ) )
& ( ( K_1 != zero_zero_nat )
=> ! [I_1: nat,J_1: nat] :
( ( ord_less_nat @ J_1 @ K_1 )
=> ( ( N
= ( plus_plus_nat @ ( times_times_nat @ K_1 @ I_1 ) @ J_1 ) )
=> ( P @ J_1 ) ) ) ) ) ) ).
thf(fact_2736_mod__lemma,axiom,
! [Q: nat,R_1: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ( ord_less_nat @ R_1 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( div_mod_nat @ Q @ C ) ) @ R_1 ) @ ( times_times_nat @ B @ C ) ) ) ) ).
thf(fact_2737_transfer__nat__int__numerals_I4_J,axiom,
( ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) )
= ( nat_1 @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_2738_transfer__nat__int__relations_I2_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_nat @ ( nat_1 @ X ) @ ( nat_1 @ Y ) )
<=> ( ord_less_int @ X @ Y ) ) ) ) ).
thf(fact_2739_nat__less__eq__zless,axiom,
! [Z_1: int,W: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat_1 @ W ) @ ( nat_1 @ Z_1 ) )
<=> ( ord_less_int @ W @ Z_1 ) ) ) ).
thf(fact_2740_nat__le__eq__zle,axiom,
! [Z_1: int,W: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z_1 ) )
=> ( ( ord_less_eq_nat @ ( nat_1 @ W ) @ ( nat_1 @ Z_1 ) )
<=> ( ord_less_eq_int @ W @ Z_1 ) ) ) ).
thf(fact_2741_nat__mult__distrib,axiom,
! [Z_3: int,Z_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z_1 )
=> ( ( nat_1 @ ( times_times_int @ Z_1 @ Z_3 ) )
= ( times_times_nat @ ( nat_1 @ Z_1 ) @ ( nat_1 @ Z_3 ) ) ) ) ).
thf(fact_2742_Nat__Transfer_Otransfer__nat__int__functions_I2_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( times_times_nat @ ( nat_1 @ X ) @ ( nat_1 @ Y ) )
= ( nat_1 @ ( times_times_int @ X @ Y ) ) ) ) ) ).
thf(fact_2743_Nat__Transfer_Otransfer__nat__int__functions_I1_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( plus_plus_nat @ ( nat_1 @ X ) @ ( nat_1 @ Y ) )
= ( nat_1 @ ( plus_plus_int @ X @ Y ) ) ) ) ) ).
thf(fact_2744_nat__add__distrib,axiom,
! [Z_3: int,Z_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z_3 )
=> ( ( nat_1 @ ( plus_plus_int @ Z_1 @ Z_3 ) )
= ( plus_plus_nat @ ( nat_1 @ Z_1 ) @ ( nat_1 @ Z_3 ) ) ) ) ) ).
thf(fact_2745_transfer__nat__int__relations_I4_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( dvd_dvd_nat @ ( nat_1 @ X ) @ ( nat_1 @ Y ) )
<=> ( dvd_dvd_int @ X @ Y ) ) ) ) ).
thf(fact_2746_nat__diff__distrib,axiom,
! [Z_1: int,Z_3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z_3 )
=> ( ( ord_less_eq_int @ Z_3 @ Z_1 )
=> ( ( nat_1 @ ( minus_minus_int @ Z_1 @ Z_3 ) )
= ( minus_minus_nat @ ( nat_1 @ Z_1 ) @ ( nat_1 @ Z_3 ) ) ) ) ) ).
thf(fact_2747_Nat__Transfer_Otransfer__nat__int__functions_I4_J,axiom,
! [N: nat,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( power_power_nat @ ( nat_1 @ X ) @ N )
= ( nat_1 @ ( power_power_int @ X @ N ) ) ) ) ).
thf(fact_2748_nat__power__eq,axiom,
! [N: nat,Z_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z_1 )
=> ( ( nat_1 @ ( power_power_int @ Z_1 @ N ) )
= ( power_power_nat @ ( nat_1 @ Z_1 ) @ N ) ) ) ).
thf(fact_2749_add__self__div__2,axiom,
! [M: nat] :
( ( div_div_nat @ ( plus_plus_nat @ M @ M ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= M ) ).
thf(fact_2750_transfer__nat__int__numerals_I3_J,axiom,
( ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) )
= ( nat_1 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_2751_Euler_Oaux__1,axiom,
! [A: int,P_3: int] :
( ( ord_less_int @ zero_zero_int @ P_3 )
=> ( ( power_power_int @ A @ ( nat_1 @ P_3 ) )
= ( times_times_int @ A @ ( power_power_int @ A @ ( minus_minus_nat @ ( nat_1 @ P_3 ) @ one_one_nat ) ) ) ) ) ).
thf(fact_2752_EvenOdd_Oneg__one__odd__power,axiom,
! [X: int] :
( ( member_int @ X @ zOdd )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( power_power_int @ ( number_number_of_int @ min ) @ ( nat_1 @ X ) )
= ( number_number_of_int @ min ) ) ) ) ).
thf(fact_2753_Little__Fermat,axiom,
! [X: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ~ ( dvd_dvd_int @ P_3 @ X )
=> ( zcong @ ( power_power_int @ X @ ( nat_1 @ ( minus_minus_int @ P_3 @ one_one_int ) ) ) @ one_one_int @ P_3 ) ) ) ).
thf(fact_2754_div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( div_div_nat @ N @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_2755_mod2__gr__0,axiom,
! [M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( div_mod_nat @ M @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( ( div_mod_nat @ M @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_nat ) ) ).
thf(fact_2756_MultInv__def,axiom,
! [P_3: int,X: int] :
( ( multInv @ P_3 @ X )
= ( power_power_int @ X @ ( nat_1 @ ( minus_minus_int @ P_3 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_2757_posDivAlg__div__mod,axiom,
! [L: int,K_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( posDivAlg @ K_1 @ L )
= ( product_Pair_int_int @ ( div_div_int @ K_1 @ L ) @ ( div_mod_int @ K_1 @ L ) ) ) ) ) ).
thf(fact_2758_Suc__n__div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( div_div_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_2759_Int2_Oaux__2,axiom,
! [P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ord_less_nat @ zero_zero_nat @ ( nat_1 @ ( minus_minus_int @ P_3 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_2760_Int2_Oaux__1,axiom,
! [P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( minus_minus_nat @ ( nat_1 @ P_3 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( nat_1 @ ( minus_minus_int @ P_3 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_2761_WilsonRuss_Oinv__def,axiom,
! [P_3: int,A: int] :
( ( inv @ P_3 @ A )
= ( div_mod_int @ ( power_power_int @ A @ ( nat_1 @ ( minus_minus_int @ P_3 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ P_3 ) ) ).
thf(fact_2762_Euler_Oaux____2,axiom,
! [P_3: int] :
( ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( nat_1 @ ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
= ( nat_1 @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_2763_StandardRes__SRStar__prop4,axiom,
! [X: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( member_int @ X @ ( sRStar @ P_3 ) )
=> ( member_int @ ( standardRes @ P_3 @ X ) @ ( sRStar @ P_3 ) ) ) ) ) ).
thf(fact_2764_StandardRes__SRStar__prop1,axiom,
! [X: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( member_int @ ( standardRes @ P_3 @ X ) @ ( sRStar @ P_3 ) )
<=> ~ ( zcong @ X @ zero_zero_int @ P_3 ) ) ) ).
thf(fact_2765_SRStar__mult__prop1,axiom,
! [Y: int,X: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( member_int @ X @ ( sRStar @ P_3 ) )
=> ( ( member_int @ Y @ ( sRStar @ P_3 ) )
=> ( member_int @ ( standardRes @ P_3 @ ( times_times_int @ X @ Y ) ) @ ( sRStar @ P_3 ) ) ) ) ) ) ).
thf(fact_2766_Euler__part2,axiom,
! [A: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ( zcong @ A @ zero_zero_int @ P_3 )
=> ( zcong @ zero_zero_int @ ( power_power_int @ A @ ( nat_1 @ ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ P_3 ) ) ) ) ).
thf(fact_2767_StandardRes__SRStar__prop2,axiom,
! [X: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ( member_int @ X @ ( sRStar @ P_3 ) )
=> ( member_int @ ( standardRes @ P_3 @ ( multInv @ P_3 @ X ) ) @ ( sRStar @ P_3 ) ) ) ) ) ).
thf(fact_2768_Euler__Criterion,axiom,
! [A: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( zcong @ ( legendre @ A @ P_3 ) @ ( power_power_int @ A @ ( nat_1 @ ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ P_3 ) ) ) ).
thf(fact_2769_posDivAlg__eqn,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ( ord_less_int @ A @ B )
=> ( ( posDivAlg @ A @ B )
= ( product_Pair_int_int @ zero_zero_int @ A ) ) )
& ( ~ ( ord_less_int @ A @ B )
=> ( ( posDivAlg @ A @ B )
= ( adjust @ B @ ( posDivAlg @ A @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) ) ) ) ) ) ).
thf(fact_2770_posDivAlg__eqn__number__of,axiom,
! [W: int,V: int] :
( ( ord_less_int @ zero_zero_int @ ( number_number_of_int @ V ) )
=> ( ( ( ord_less_int @ ( number_number_of_int @ W ) @ ( number_number_of_int @ V ) )
=> ( ( posDivAlg @ ( number_number_of_int @ W ) @ ( number_number_of_int @ V ) )
= ( product_Pair_int_int @ zero_zero_int @ ( number_number_of_int @ W ) ) ) )
& ( ~ ( ord_less_int @ ( number_number_of_int @ W ) @ ( number_number_of_int @ V ) )
=> ( ( posDivAlg @ ( number_number_of_int @ W ) @ ( number_number_of_int @ V ) )
= ( adjust @ ( number_number_of_int @ V ) @ ( posDivAlg @ ( number_number_of_int @ W ) @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( number_number_of_int @ V ) ) ) ) ) ) ) ) ).
thf(fact_2771_mod__exhaust__less__4,axiom,
! [M: nat] :
( ( ( div_mod_nat @ M @ ( number_number_of_nat @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_nat )
| ( ( div_mod_nat @ M @ ( number_number_of_nat @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= one_one_nat )
| ( ( div_mod_nat @ M @ ( number_number_of_nat @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
| ( ( div_mod_nat @ M @ ( number_number_of_nat @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_2772_nat__aux__def,axiom,
! [I: int,N: nat] :
( ( nat_aux @ I @ N )
= ( plus_plus_nat @ ( nat_1 @ I ) @ N ) ) ).
thf(fact_2773_zmod__eq__0D,axiom,
! [M_18: int,D_3: int] :
( ( ( div_mod_int @ M_18 @ D_3 )
= zero_zero_int )
=> ? [Q_2: int] :
( M_18
= ( times_times_int @ D_3 @ Q_2 ) ) ) ).
thf(fact_2774_nat__mod__eq__lemma,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( div_mod_nat @ X @ N )
= ( div_mod_nat @ Y @ N ) )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ? [Q_2: nat] :
( X
= ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q_2 ) ) ) ) ) ).
thf(fact_2775_phi__prime,axiom,
! [P_3: int] :
( ( zprime @ P_3 )
=> ( ( phi @ P_3 )
= ( nat_1 @ ( minus_minus_int @ P_3 @ one_one_int ) ) ) ) ).
thf(fact_2776_negDivAlg_Opsimps,axiom,
! [A: int,B: int] :
( ( accp_P2006205492nt_int @ negDivAlg_rel @ ( product_Pair_int_int @ A @ B ) )
=> ( ( ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) )
| ( ord_less_eq_int @ B @ zero_zero_int ) )
=> ( ( negDivAlg @ A @ B )
= ( product_Pair_int_int @ ( number_number_of_int @ min ) @ ( plus_plus_int @ A @ B ) ) ) )
& ( ~ ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) )
| ( ord_less_eq_int @ B @ zero_zero_int ) )
=> ( ( negDivAlg @ A @ B )
= ( adjust @ B @ ( negDivAlg @ A @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) ) ) ) ) ) ).
thf(fact_2777_transfer__morphism__nat__int,axiom,
nat_tr876908586nt_nat @ nat_1 @ ( ord_less_eq_int @ zero_zero_int ) ).
thf(fact_2778_transfer__morphismI,axiom,
! [F_4: int > nat,A_14: int > $o] : ( nat_tr876908586nt_nat @ F_4 @ A_14 ) ).
thf(fact_2779_transfer__morphismI,axiom,
! [F_4: nat > int,A_14: nat > $o] : ( nat_tr160667106at_int @ F_4 @ A_14 ) ).
thf(fact_2780_transfer__morphism__def,axiom,
! [F_3: int > nat,A_13: int > $o] : ( nat_tr876908586nt_nat @ F_3 @ A_13 ) ).
thf(fact_2781_transfer__morphism__def,axiom,
! [F_3: nat > int,A_13: nat > $o] : ( nat_tr160667106at_int @ F_3 @ A_13 ) ).
thf(fact_2782_eq__diff__eq_H,axiom,
! [X: real,Y: real,Z_1: real] :
( ( X
= ( minus_minus_real @ Y @ Z_1 ) )
<=> ( Y
= ( plus_plus_real @ X @ Z_1 ) ) ) ).
thf(fact_2783_negDivAlg_Opinduct,axiom,
! [P: int > int > $o,A0: int,A1: int] :
( ( accp_P2006205492nt_int @ negDivAlg_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [A_2: int,B_4: int] :
( ( accp_P2006205492nt_int @ negDivAlg_rel @ ( product_Pair_int_int @ A_2 @ B_4 ) )
=> ( ( ~ ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A_2 @ B_4 ) )
| ( ord_less_eq_int @ B_4 @ zero_zero_int ) )
=> ( P @ A_2 @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B_4 ) ) )
=> ( P @ A_2 @ B_4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
thf(fact_2784_posDivAlg_Opsimps,axiom,
! [A: int,B: int] :
( ( accp_P2006205492nt_int @ posDivAlg_rel @ ( product_Pair_int_int @ A @ B ) )
=> ( ( ( ( ord_less_int @ A @ B )
| ( ord_less_eq_int @ B @ zero_zero_int ) )
=> ( ( posDivAlg @ A @ B )
= ( product_Pair_int_int @ zero_zero_int @ A ) ) )
& ( ~ ( ( ord_less_int @ A @ B )
| ( ord_less_eq_int @ B @ zero_zero_int ) )
=> ( ( posDivAlg @ A @ B )
= ( adjust @ B @ ( posDivAlg @ A @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) ) ) ) ) ) ).
thf(fact_2785_posDivAlg_Opinduct,axiom,
! [P: int > int > $o,A0: int,A1: int] :
( ( accp_P2006205492nt_int @ posDivAlg_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [A_2: int,B_4: int] :
( ( accp_P2006205492nt_int @ posDivAlg_rel @ ( product_Pair_int_int @ A_2 @ B_4 ) )
=> ( ( ~ ( ( ord_less_int @ A_2 @ B_4 )
| ( ord_less_eq_int @ B_4 @ zero_zero_int ) )
=> ( P @ A_2 @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B_4 ) ) )
=> ( P @ A_2 @ B_4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
thf(fact_2786_mod__eqD,axiom,
! [M: nat,D: nat,R_1: nat] :
( ( ( div_mod_nat @ M @ D )
= R_1 )
=> ? [Q_2: nat] :
( M
= ( plus_plus_nat @ R_1 @ ( times_times_nat @ Q_2 @ D ) ) ) ) ).
thf(fact_2787_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( div_mod_nat @ X @ N )
= ( div_mod_nat @ Y @ N ) )
<=> ? [Q1: nat,Q2: nat] :
( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
= ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
thf(fact_2788_mod__eq__0D,axiom,
! [M_18: nat,D_3: nat] :
( ( ( div_mod_nat @ M_18 @ D_3 )
= zero_zero_nat )
=> ? [Q_2: nat] :
( M_18
= ( times_times_nat @ D_3 @ Q_2 ) ) ) ).
thf(fact_2789_zmult2__lemma,axiom,
! [C: int,A: int,B: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( B != zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( divmod_int_rel @ A @ ( times_times_int @ B @ C ) @ ( product_Pair_int_int @ ( div_div_int @ Q @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ ( div_mod_int @ Q @ C ) ) @ R_1 ) ) ) ) ) ) ).
thf(fact_2790_unique__remainder,axiom,
! [Q_6: int,R_3: int,A: int,B: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q_6 @ R_3 ) )
=> ( ( B != zero_zero_int )
=> ( R_1 = R_3 ) ) ) ) ).
thf(fact_2791_unique__quotient,axiom,
! [Q_6: int,R_3: int,A: int,B: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q_6 @ R_3 ) )
=> ( ( B != zero_zero_int )
=> ( Q = Q_6 ) ) ) ) ).
thf(fact_2792_self__remainder,axiom,
! [A: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A @ A @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( A != zero_zero_int )
=> ( R_1 = zero_zero_int ) ) ) ).
thf(fact_2793_divmod__int__rel__0,axiom,
! [B: int] :
( ( B != zero_zero_int )
=> ( divmod_int_rel @ zero_zero_int @ B @ ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) ) ).
thf(fact_2794_self__quotient,axiom,
! [A: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A @ A @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( A != zero_zero_int )
=> ( Q = one_one_int ) ) ) ).
thf(fact_2795_divmod__int__rel__div,axiom,
! [A: int,B: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( B != zero_zero_int )
=> ( ( div_div_int @ A @ B )
= Q ) ) ) ).
thf(fact_2796_divmod__int__rel__mod,axiom,
! [A: int,B: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( B != zero_zero_int )
=> ( ( div_mod_int @ A @ B )
= R_1 ) ) ) ).
thf(fact_2797_negDivAlg__correct,axiom,
! [B: int,A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( divmod_int_rel @ A @ B @ ( negDivAlg @ A @ B ) ) ) ) ).
thf(fact_2798_divmod__int__rel__div__mod,axiom,
! [A: int,B: int] :
( ( B != zero_zero_int )
=> ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ ( div_div_int @ A @ B ) @ ( div_mod_int @ A @ B ) ) ) ) ).
thf(fact_2799_posDivAlg__correct,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( divmod_int_rel @ A @ B @ ( posDivAlg @ A @ B ) ) ) ) ).
thf(fact_2800_zadd1__lemma,axiom,
! [B: int,Bq: int,Br: int,A: int,C: int,Aq: int,Ar: int] :
( ( divmod_int_rel @ A @ C @ ( product_Pair_int_int @ Aq @ Ar ) )
=> ( ( divmod_int_rel @ B @ C @ ( product_Pair_int_int @ Bq @ Br ) )
=> ( ( C != zero_zero_int )
=> ( divmod_int_rel @ ( plus_plus_int @ A @ B ) @ C @ ( product_Pair_int_int @ ( plus_plus_int @ ( plus_plus_int @ Aq @ Bq ) @ ( div_div_int @ ( plus_plus_int @ Ar @ Br ) @ C ) ) @ ( div_mod_int @ ( plus_plus_int @ Ar @ Br ) @ C ) ) ) ) ) ) ).
thf(fact_2801_divmod__int__relI,axiom,
! [A: int,B: int,Q: int,R_1: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R_1 ) )
=> ( ( ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
& ( ord_less_int @ R_1 @ B ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ R_1 )
& ( ord_less_eq_int @ R_1 @ zero_zero_int ) ) ) )
=> ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q @ R_1 ) ) ) ) ).
thf(fact_2802_zmult1__lemma,axiom,
! [A: int,B: int,C: int,Q: int,R_1: int] :
( ( divmod_int_rel @ B @ C @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( C != zero_zero_int )
=> ( divmod_int_rel @ ( times_times_int @ A @ B ) @ C @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A @ Q ) @ ( div_div_int @ ( times_times_int @ A @ R_1 ) @ C ) ) @ ( div_mod_int @ ( times_times_int @ A @ R_1 ) @ C ) ) ) ) ) ).
thf(fact_2803_Nat__Transfer_Otransfer__nat__int__functions_I3_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( minus_minus_nat @ ( nat_1 @ X ) @ ( nat_1 @ Y ) )
= ( nat_1 @ ( nat_tsub @ X @ Y ) ) ) ) ) ).
thf(fact_2804_inv__inv__aux,axiom,
! [P_3: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P_3 )
=> ( ( times_times_nat @ ( nat_1 @ ( minus_minus_int @ P_3 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( nat_1 @ ( minus_minus_int @ P_3 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
= ( suc @ ( times_times_nat @ ( nat_1 @ ( minus_minus_int @ P_3 @ one_one_int ) ) @ ( nat_1 @ ( minus_minus_int @ P_3 @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_2805_two__times__odd__div__two__plus__one,axiom,
! [X: int] :
( ~ ( even_odd_even_int @ X )
=> ( ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( div_div_int @ X @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ one_one_int )
= X ) ) ).
thf(fact_2806_Code__Numeral_Oint__of__code,axiom,
! [K_1: code_code_numeral] :
( ( ( K_1 = zero_z126310315umeral )
=> ( ( code_int_of @ K_1 )
= zero_zero_int ) )
& ( ( K_1 != zero_z126310315umeral )
=> ( ( ( ( div_mo1740067990umeral @ K_1 @ ( number1443263063umeral @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_z126310315umeral )
=> ( ( code_int_of @ K_1 )
= ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( code_int_of @ ( div_di1218280263umeral @ K_1 @ ( number1443263063umeral @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) )
& ( ( ( div_mo1740067990umeral @ K_1 @ ( number1443263063umeral @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= zero_z126310315umeral )
=> ( ( code_int_of @ K_1 )
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( code_int_of @ ( div_di1218280263umeral @ K_1 @ ( number1443263063umeral @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
thf(fact_2807_EvenOdd_Oneg__one__even__power,axiom,
! [X: int] :
( ( member_int @ X @ zEven )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( power_power_int @ ( number_number_of_int @ min ) @ ( nat_1 @ X ) )
= one_one_int ) ) ) ).
thf(fact_2808_of__nat__double,axiom,
! [X_9: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_9 )
=> ( ( semiri1619134803umeral @ ( nat_1 @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_9 ) ) )
= ( plus_p1627245867umeral @ ( semiri1619134803umeral @ ( nat_1 @ X_9 ) ) @ ( semiri1619134803umeral @ ( nat_1 @ X_9 ) ) ) ) ) ).
thf(fact_2809_of__nat__double,axiom,
! [X_9: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_9 )
=> ( ( semiri1621563631at_int @ ( nat_1 @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_9 ) ) )
= ( plus_plus_int @ ( semiri1621563631at_int @ ( nat_1 @ X_9 ) ) @ ( semiri1621563631at_int @ ( nat_1 @ X_9 ) ) ) ) ) ).
thf(fact_2810_of__nat__double,axiom,
! [X_9: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_9 )
=> ( ( semiri984289939at_nat @ ( nat_1 @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_9 ) ) )
= ( plus_plus_nat @ ( semiri984289939at_nat @ ( nat_1 @ X_9 ) ) @ ( semiri984289939at_nat @ ( nat_1 @ X_9 ) ) ) ) ) ).
thf(fact_2811_of__nat__double,axiom,
! [X_9: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_9 )
=> ( ( semiri132038758t_real @ ( nat_1 @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_9 ) ) )
= ( plus_plus_real @ ( semiri132038758t_real @ ( nat_1 @ X_9 ) ) @ ( semiri132038758t_real @ ( nat_1 @ X_9 ) ) ) ) ) ).
thf(fact_2812_of__nat__double,axiom,
! [X_9: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_9 )
=> ( ( semiri2020571505omplex @ ( nat_1 @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_9 ) ) )
= ( plus_plus_complex @ ( semiri2020571505omplex @ ( nat_1 @ X_9 ) ) @ ( semiri2020571505omplex @ ( nat_1 @ X_9 ) ) ) ) ) ).
thf(fact_2813_of__nat__double,axiom,
! [X_9: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_9 )
=> ( ( semiri1424489471de_int @ ( nat_1 @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_9 ) ) )
= ( plus_p1446045655de_int @ ( semiri1424489471de_int @ ( nat_1 @ X_9 ) ) @ ( semiri1424489471de_int @ ( nat_1 @ X_9 ) ) ) ) ) ).
thf(fact_2814_of__nat__double,axiom,
! [X_9: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_9 )
=> ( ( semiri151668891at_rat @ ( nat_1 @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_9 ) ) )
= ( plus_plus_rat @ ( semiri151668891at_rat @ ( nat_1 @ X_9 ) ) @ ( semiri151668891at_rat @ ( nat_1 @ X_9 ) ) ) ) ) ).
thf(fact_2815_less__mono__imp__le__mono,axiom,
! [I: nat,J: nat,F: nat > nat] :
( ! [I_1: nat,J_1: nat] :
( ( ord_less_nat @ I_1 @ J_1 )
=> ( ord_less_nat @ ( F @ I_1 ) @ ( F @ J_1 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
thf(fact_2816_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
thf(fact_2817_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
thf(fact_2818_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
thf(fact_2819_dvd__1__left,axiom,
! [K_1: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K_1 ) ).
thf(fact_2820_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
! [X: nat,Y: nat] :
( ( nat_tsub @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
= ( semiri1621563631at_int @ ( minus_minus_nat @ X @ Y ) ) ) ).
thf(fact_2821_code__numeral__zero__minus__one,axiom,
( ( minus_1690775515umeral @ zero_z126310315umeral @ one_on1645066479umeral )
= zero_z126310315umeral ) ).
thf(fact_2822_number__of__int,axiom,
! [N_28: nat] :
( ( number_number_of_nat @ ( semiri1621563631at_int @ N_28 ) )
= ( semiri984289939at_nat @ N_28 ) ) ).
thf(fact_2823_number__of__int,axiom,
! [N_28: nat] :
( ( number_number_of_int @ ( semiri1621563631at_int @ N_28 ) )
= ( semiri1621563631at_int @ N_28 ) ) ).
thf(fact_2824_number__of__int,axiom,
! [N_28: nat] :
( ( number267125858f_real @ ( semiri1621563631at_int @ N_28 ) )
= ( semiri132038758t_real @ N_28 ) ) ).
thf(fact_2825_number__of__int,axiom,
! [N_28: nat] :
( ( number528085621omplex @ ( semiri1621563631at_int @ N_28 ) )
= ( semiri2020571505omplex @ N_28 ) ) ).
thf(fact_2826_number__of__int,axiom,
! [N_28: nat] :
( ( number_number_of_rat @ ( semiri1621563631at_int @ N_28 ) )
= ( semiri151668891at_rat @ N_28 ) ) ).
thf(fact_2827_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
thf(fact_2828_of__nat__eq__iff,axiom,
! [M_17: nat,N_27: nat] :
( ( ( semiri1424489471de_int @ M_17 )
= ( semiri1424489471de_int @ N_27 ) )
<=> ( M_17 = N_27 ) ) ).
thf(fact_2829_of__nat__eq__iff,axiom,
! [M_17: nat,N_27: nat] :
( ( ( semiri1619134803umeral @ M_17 )
= ( semiri1619134803umeral @ N_27 ) )
<=> ( M_17 = N_27 ) ) ).
thf(fact_2830_of__nat__eq__iff,axiom,
! [M_17: nat,N_27: nat] :
( ( ( semiri984289939at_nat @ M_17 )
= ( semiri984289939at_nat @ N_27 ) )
<=> ( M_17 = N_27 ) ) ).
thf(fact_2831_of__nat__eq__iff,axiom,
! [M_17: nat,N_27: nat] :
( ( ( semiri1621563631at_int @ M_17 )
= ( semiri1621563631at_int @ N_27 ) )
<=> ( M_17 = N_27 ) ) ).
thf(fact_2832_of__nat__eq__iff,axiom,
! [M_17: nat,N_27: nat] :
( ( ( semiri132038758t_real @ M_17 )
= ( semiri132038758t_real @ N_27 ) )
<=> ( M_17 = N_27 ) ) ).
thf(fact_2833_of__nat__eq__iff,axiom,
! [M_17: nat,N_27: nat] :
( ( ( semiri2020571505omplex @ M_17 )
= ( semiri2020571505omplex @ N_27 ) )
<=> ( M_17 = N_27 ) ) ).
thf(fact_2834_of__nat__eq__iff,axiom,
! [M_17: nat,N_27: nat] :
( ( ( semiri151668891at_rat @ M_17 )
= ( semiri151668891at_rat @ N_27 ) )
<=> ( M_17 = N_27 ) ) ).
thf(fact_2835_nat_Oinject,axiom,
! [Nat: nat,Nat_2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat_2 ) )
<=> ( Nat = Nat_2 ) ) ).
thf(fact_2836_Suc__n__not__n,axiom,
! [N: nat] :
( ( suc @ N )
!= N ) ).
thf(fact_2837_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
thf(fact_2838_of__nat__Suc,axiom,
! [M_16: nat] :
( ( semiri1621563631at_int @ ( suc @ M_16 ) )
= ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ M_16 ) ) ) ).
thf(fact_2839_of__nat__Suc,axiom,
! [M_16: nat] :
( ( semiri984289939at_nat @ ( suc @ M_16 ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri984289939at_nat @ M_16 ) ) ) ).
thf(fact_2840_of__nat__Suc,axiom,
! [M_16: nat] :
( ( semiri132038758t_real @ ( suc @ M_16 ) )
= ( plus_plus_real @ one_one_real @ ( semiri132038758t_real @ M_16 ) ) ) ).
thf(fact_2841_of__nat__Suc,axiom,
! [M_16: nat] :
( ( semiri1619134803umeral @ ( suc @ M_16 ) )
= ( plus_p1627245867umeral @ one_on1645066479umeral @ ( semiri1619134803umeral @ M_16 ) ) ) ).
thf(fact_2842_of__nat__Suc,axiom,
! [M_16: nat] :
( ( semiri2020571505omplex @ ( suc @ M_16 ) )
= ( plus_plus_complex @ one_one_complex @ ( semiri2020571505omplex @ M_16 ) ) ) ).
thf(fact_2843_of__nat__Suc,axiom,
! [M_16: nat] :
( ( semiri1424489471de_int @ ( suc @ M_16 ) )
= ( plus_p1446045655de_int @ one_on1684967323de_int @ ( semiri1424489471de_int @ M_16 ) ) ) ).
thf(fact_2844_of__nat__Suc,axiom,
! [M_16: nat] :
( ( semiri151668891at_rat @ ( suc @ M_16 ) )
= ( plus_plus_rat @ one_one_rat @ ( semiri151668891at_rat @ M_16 ) ) ) ).
thf(fact_2845_int__Suc0__eq__1,axiom,
( ( semiri1621563631at_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
thf(fact_2846_zless__iff__Suc__zadd,axiom,
! [W: int,Z_1: int] :
( ( ord_less_int @ W @ Z_1 )
<=> ? [N_1: nat] :
( Z_1
= ( plus_plus_int @ W @ ( semiri1621563631at_int @ ( suc @ N_1 ) ) ) ) ) ).
thf(fact_2847_int__Suc,axiom,
! [M: nat] :
( ( semiri1621563631at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ M ) ) ) ).
thf(fact_2848_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_2849_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_2850_nat_Osimps_I3_J,axiom,
! [Nat_3: nat] :
( ( suc @ Nat_3 )
!= zero_zero_nat ) ).
thf(fact_2851_Suc__not__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_2852_nat_Osimps_I2_J,axiom,
! [Nat_2: nat] :
( zero_zero_nat
!= ( suc @ Nat_2 ) ) ).
thf(fact_2853_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_2854_nat__int,axiom,
! [N: nat] :
( ( nat_1 @ ( semiri1621563631at_int @ N ) )
= N ) ).
thf(fact_2855_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_2856_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_2857_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
thf(fact_2858_less__trans__Suc,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K_1 )
=> ( ord_less_nat @ ( suc @ I ) @ K_1 ) ) ) ).
thf(fact_2859_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
thf(fact_2860_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_2861_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
thf(fact_2862_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
<=> ( N = M ) ) ) ).
thf(fact_2863_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_2864_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
<=> ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
thf(fact_2865_not__less__eq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
<=> ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
thf(fact_2866_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_2867_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
thf(fact_2868_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
thf(fact_2869_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
thf(fact_2870_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_eq_nat @ M @ N )
<=> ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
thf(fact_2871_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
<=> ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
thf(fact_2872_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
<=> ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_2873_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_2874_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
thf(fact_2875_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_2876_Suc__mult__cancel1,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K_1 ) @ M )
= ( times_times_nat @ ( suc @ K_1 ) @ N ) )
<=> ( M = N ) ) ).
thf(fact_2877_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_2878_Suc__diff__diff,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K_1 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K_1 ) ) ).
thf(fact_2879_mod__Suc__eq__Suc__mod,axiom,
! [M: nat,N: nat] :
( ( div_mod_nat @ ( suc @ M ) @ N )
= ( div_mod_nat @ ( suc @ ( div_mod_nat @ M @ N ) ) @ N ) ) ).
thf(fact_2880_exp__mono__eq,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( power_power_nat @ X @ ( suc @ N ) )
= ( power_power_nat @ Y @ ( suc @ N ) ) )
<=> ( X = Y ) ) ).
thf(fact_2881_even__zero__int,axiom,
even_odd_even_int @ zero_zero_int ).
thf(fact_2882_odd__one__int,axiom,
~ ( even_odd_even_int @ one_one_int ) ).
thf(fact_2883_Parity_Oeven__product,axiom,
! [X: int,Y: int] :
( ( even_odd_even_int @ ( times_times_int @ X @ Y ) )
<=> ( ( even_odd_even_int @ X )
| ( even_odd_even_int @ Y ) ) ) ).
thf(fact_2884_even__times__anything,axiom,
! [Y: int,X: int] :
( ( even_odd_even_int @ X )
=> ( even_odd_even_int @ ( times_times_int @ X @ Y ) ) ) ).
thf(fact_2885_anything__times__even,axiom,
! [X: int,Y: int] :
( ( even_odd_even_int @ Y )
=> ( even_odd_even_int @ ( times_times_int @ X @ Y ) ) ) ).
thf(fact_2886_Parity_Oodd__times__odd,axiom,
! [Y: int,X: int] :
( ~ ( even_odd_even_int @ X )
=> ( ~ ( even_odd_even_int @ Y )
=> ~ ( even_odd_even_int @ ( times_times_int @ X @ Y ) ) ) ) ).
thf(fact_2887_Parity_Oodd__plus__odd,axiom,
! [Y: int,X: int] :
( ~ ( even_odd_even_int @ X )
=> ( ~ ( even_odd_even_int @ Y )
=> ( even_odd_even_int @ ( plus_plus_int @ X @ Y ) ) ) ) ).
thf(fact_2888_odd__plus__even,axiom,
! [Y: int,X: int] :
( ~ ( even_odd_even_int @ X )
=> ( ( even_odd_even_int @ Y )
=> ~ ( even_odd_even_int @ ( plus_plus_int @ X @ Y ) ) ) ) ).
thf(fact_2889_Parity_Oeven__plus__odd,axiom,
! [Y: int,X: int] :
( ( even_odd_even_int @ X )
=> ( ~ ( even_odd_even_int @ Y )
=> ~ ( even_odd_even_int @ ( plus_plus_int @ X @ Y ) ) ) ) ).
thf(fact_2890_Parity_Oeven__plus__even,axiom,
! [Y: int,X: int] :
( ( even_odd_even_int @ X )
=> ( ( even_odd_even_int @ Y )
=> ( even_odd_even_int @ ( plus_plus_int @ X @ Y ) ) ) ) ).
thf(fact_2891_even__sum,axiom,
! [X: int,Y: int] :
( ( even_odd_even_int @ ( plus_plus_int @ X @ Y ) )
<=> ( ( ( even_odd_even_int @ X )
& ( even_odd_even_int @ Y ) )
| ( ~ ( even_odd_even_int @ X )
& ~ ( even_odd_even_int @ Y ) ) ) ) ).
thf(fact_2892_even__difference,axiom,
! [X: int,Y: int] :
( ( even_odd_even_int @ ( minus_minus_int @ X @ Y ) )
<=> ( ( ( even_odd_even_int @ X )
& ( even_odd_even_int @ Y ) )
| ( ~ ( even_odd_even_int @ X )
& ~ ( even_odd_even_int @ Y ) ) ) ) ).
thf(fact_2893_odd__pow,axiom,
! [N: nat,X: int] :
( ~ ( even_odd_even_int @ X )
=> ~ ( even_odd_even_int @ ( power_power_int @ X @ N ) ) ) ).
thf(fact_2894_one__not__even,axiom,
~ ( member_int @ one_one_int @ zEven ) ).
thf(fact_2895_even__times__either,axiom,
! [Y: int,X: int] :
( ( member_int @ X @ zEven )
=> ( member_int @ ( times_times_int @ X @ Y ) @ zEven ) ) ).
thf(fact_2896_EvenOdd_Oeven__product,axiom,
! [X: int,Y: int] :
( ( member_int @ ( times_times_int @ X @ Y ) @ zEven )
=> ( ( member_int @ X @ zEven )
| ( member_int @ Y @ zEven ) ) ) ).
thf(fact_2897_EvenOdd_Oeven__plus__even,axiom,
! [Y: int,X: int] :
( ( member_int @ X @ zEven )
=> ( ( member_int @ Y @ zEven )
=> ( member_int @ ( plus_plus_int @ X @ Y ) @ zEven ) ) ) ).
thf(fact_2898_even__diff,axiom,
! [X: int,Y: int] :
( ( member_int @ ( minus_minus_int @ X @ Y ) @ zEven )
<=> ( ( member_int @ X @ zEven )
<=> ( member_int @ Y @ zEven ) ) ) ).
thf(fact_2899_even__minus__even,axiom,
! [Y: int,X: int] :
( ( member_int @ X @ zEven )
=> ( ( member_int @ Y @ zEven )
=> ( member_int @ ( minus_minus_int @ X @ Y ) @ zEven ) ) ) ).
thf(fact_2900_not__odd__impl__even,axiom,
! [X: int] :
( ~ ( member_int @ X @ zOdd )
=> ( member_int @ X @ zEven ) ) ).
thf(fact_2901_even__odd__disj,axiom,
! [X: int] :
( ( member_int @ X @ zOdd )
| ( member_int @ X @ zEven ) ) ).
thf(fact_2902_odd__iff__not__even,axiom,
! [X: int] :
( ( member_int @ X @ zOdd )
<=> ~ ( member_int @ X @ zEven ) ) ).
thf(fact_2903_even__odd__conj,axiom,
! [X: int] :
~ ( ( member_int @ X @ zOdd )
& ( member_int @ X @ zEven ) ) ).
thf(fact_2904_zero__le__imp__of__nat,axiom,
! [M_15: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri151668891at_rat @ M_15 ) ) ).
thf(fact_2905_zero__le__imp__of__nat,axiom,
! [M_15: nat] : ( ord_le258702272de_int @ zero_z891286103de_int @ ( semiri1424489471de_int @ M_15 ) ) ).
thf(fact_2906_zero__le__imp__of__nat,axiom,
! [M_15: nat] : ( ord_le565307924umeral @ zero_z126310315umeral @ ( semiri1619134803umeral @ M_15 ) ) ).
thf(fact_2907_zero__le__imp__of__nat,axiom,
! [M_15: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri132038758t_real @ M_15 ) ) ).
thf(fact_2908_zero__le__imp__of__nat,axiom,
! [M_15: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri984289939at_nat @ M_15 ) ) ).
thf(fact_2909_zero__le__imp__of__nat,axiom,
! [M_15: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ M_15 ) ) ).
thf(fact_2910_of__nat__0__le__iff,axiom,
! [N_26: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri151668891at_rat @ N_26 ) ) ).
thf(fact_2911_of__nat__0__le__iff,axiom,
! [N_26: nat] : ( ord_le258702272de_int @ zero_z891286103de_int @ ( semiri1424489471de_int @ N_26 ) ) ).
thf(fact_2912_of__nat__0__le__iff,axiom,
! [N_26: nat] : ( ord_le565307924umeral @ zero_z126310315umeral @ ( semiri1619134803umeral @ N_26 ) ) ).
thf(fact_2913_of__nat__0__le__iff,axiom,
! [N_26: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri132038758t_real @ N_26 ) ) ).
thf(fact_2914_of__nat__0__le__iff,axiom,
! [N_26: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri984289939at_nat @ N_26 ) ) ).
thf(fact_2915_of__nat__0__le__iff,axiom,
! [N_26: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ N_26 ) ) ).
thf(fact_2916_of__nat__less__0__iff,axiom,
! [M_14: nat] :
~ ( ord_less_rat @ ( semiri151668891at_rat @ M_14 ) @ zero_zero_rat ) ).
thf(fact_2917_of__nat__less__0__iff,axiom,
! [M_14: nat] :
~ ( ord_le1860547276de_int @ ( semiri1424489471de_int @ M_14 ) @ zero_z891286103de_int ) ).
thf(fact_2918_of__nat__less__0__iff,axiom,
! [M_14: nat] :
~ ( ord_le1304079648umeral @ ( semiri1619134803umeral @ M_14 ) @ zero_z126310315umeral ) ).
thf(fact_2919_of__nat__less__0__iff,axiom,
! [M_14: nat] :
~ ( ord_less_real @ ( semiri132038758t_real @ M_14 ) @ zero_zero_real ) ).
thf(fact_2920_of__nat__less__0__iff,axiom,
! [M_14: nat] :
~ ( ord_less_nat @ ( semiri984289939at_nat @ M_14 ) @ zero_zero_nat ) ).
thf(fact_2921_of__nat__less__0__iff,axiom,
! [M_14: nat] :
~ ( ord_less_int @ ( semiri1621563631at_int @ M_14 ) @ zero_zero_int ) ).
thf(fact_2922_of__nat__0,axiom,
( ( semiri151668891at_rat @ zero_zero_nat )
= zero_zero_rat ) ).
thf(fact_2923_of__nat__0,axiom,
( ( semiri1424489471de_int @ zero_zero_nat )
= zero_z891286103de_int ) ).
thf(fact_2924_of__nat__0,axiom,
( ( semiri2020571505omplex @ zero_zero_nat )
= zero_zero_complex ) ).
thf(fact_2925_of__nat__0,axiom,
( ( semiri1619134803umeral @ zero_zero_nat )
= zero_z126310315umeral ) ).
thf(fact_2926_of__nat__0,axiom,
( ( semiri132038758t_real @ zero_zero_nat )
= zero_zero_real ) ).
thf(fact_2927_of__nat__0,axiom,
( ( semiri984289939at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_2928_of__nat__0,axiom,
( ( semiri1621563631at_int @ zero_zero_nat )
= zero_zero_int ) ).
thf(fact_2929_of__nat__less__imp__less,axiom,
! [M_13: nat,N_25: nat] :
( ( ord_le1304079648umeral @ ( semiri1619134803umeral @ M_13 ) @ ( semiri1619134803umeral @ N_25 ) )
=> ( ord_less_nat @ M_13 @ N_25 ) ) ).
thf(fact_2930_of__nat__less__imp__less,axiom,
! [M_13: nat,N_25: nat] :
( ( ord_less_rat @ ( semiri151668891at_rat @ M_13 ) @ ( semiri151668891at_rat @ N_25 ) )
=> ( ord_less_nat @ M_13 @ N_25 ) ) ).
thf(fact_2931_of__nat__less__imp__less,axiom,
! [M_13: nat,N_25: nat] :
( ( ord_le1860547276de_int @ ( semiri1424489471de_int @ M_13 ) @ ( semiri1424489471de_int @ N_25 ) )
=> ( ord_less_nat @ M_13 @ N_25 ) ) ).
thf(fact_2932_of__nat__less__imp__less,axiom,
! [M_13: nat,N_25: nat] :
( ( ord_less_real @ ( semiri132038758t_real @ M_13 ) @ ( semiri132038758t_real @ N_25 ) )
=> ( ord_less_nat @ M_13 @ N_25 ) ) ).
thf(fact_2933_of__nat__less__imp__less,axiom,
! [M_13: nat,N_25: nat] :
( ( ord_less_nat @ ( semiri984289939at_nat @ M_13 ) @ ( semiri984289939at_nat @ N_25 ) )
=> ( ord_less_nat @ M_13 @ N_25 ) ) ).
thf(fact_2934_of__nat__less__imp__less,axiom,
! [M_13: nat,N_25: nat] :
( ( ord_less_int @ ( semiri1621563631at_int @ M_13 ) @ ( semiri1621563631at_int @ N_25 ) )
=> ( ord_less_nat @ M_13 @ N_25 ) ) ).
thf(fact_2935_less__imp__of__nat__less,axiom,
! [M_12: nat,N_24: nat] :
( ( ord_less_nat @ M_12 @ N_24 )
=> ( ord_le1304079648umeral @ ( semiri1619134803umeral @ M_12 ) @ ( semiri1619134803umeral @ N_24 ) ) ) ).
thf(fact_2936_less__imp__of__nat__less,axiom,
! [M_12: nat,N_24: nat] :
( ( ord_less_nat @ M_12 @ N_24 )
=> ( ord_less_rat @ ( semiri151668891at_rat @ M_12 ) @ ( semiri151668891at_rat @ N_24 ) ) ) ).
thf(fact_2937_less__imp__of__nat__less,axiom,
! [M_12: nat,N_24: nat] :
( ( ord_less_nat @ M_12 @ N_24 )
=> ( ord_le1860547276de_int @ ( semiri1424489471de_int @ M_12 ) @ ( semiri1424489471de_int @ N_24 ) ) ) ).
thf(fact_2938_less__imp__of__nat__less,axiom,
! [M_12: nat,N_24: nat] :
( ( ord_less_nat @ M_12 @ N_24 )
=> ( ord_less_real @ ( semiri132038758t_real @ M_12 ) @ ( semiri132038758t_real @ N_24 ) ) ) ).
thf(fact_2939_less__imp__of__nat__less,axiom,
! [M_12: nat,N_24: nat] :
( ( ord_less_nat @ M_12 @ N_24 )
=> ( ord_less_nat @ ( semiri984289939at_nat @ M_12 ) @ ( semiri984289939at_nat @ N_24 ) ) ) ).
thf(fact_2940_less__imp__of__nat__less,axiom,
! [M_12: nat,N_24: nat] :
( ( ord_less_nat @ M_12 @ N_24 )
=> ( ord_less_int @ ( semiri1621563631at_int @ M_12 ) @ ( semiri1621563631at_int @ N_24 ) ) ) ).
thf(fact_2941_of__nat__less__iff,axiom,
! [M_11: nat,N_23: nat] :
( ( ord_le1304079648umeral @ ( semiri1619134803umeral @ M_11 ) @ ( semiri1619134803umeral @ N_23 ) )
<=> ( ord_less_nat @ M_11 @ N_23 ) ) ).
thf(fact_2942_of__nat__less__iff,axiom,
! [M_11: nat,N_23: nat] :
( ( ord_less_rat @ ( semiri151668891at_rat @ M_11 ) @ ( semiri151668891at_rat @ N_23 ) )
<=> ( ord_less_nat @ M_11 @ N_23 ) ) ).
thf(fact_2943_of__nat__less__iff,axiom,
! [M_11: nat,N_23: nat] :
( ( ord_le1860547276de_int @ ( semiri1424489471de_int @ M_11 ) @ ( semiri1424489471de_int @ N_23 ) )
<=> ( ord_less_nat @ M_11 @ N_23 ) ) ).
thf(fact_2944_of__nat__less__iff,axiom,
! [M_11: nat,N_23: nat] :
( ( ord_less_real @ ( semiri132038758t_real @ M_11 ) @ ( semiri132038758t_real @ N_23 ) )
<=> ( ord_less_nat @ M_11 @ N_23 ) ) ).
thf(fact_2945_of__nat__less__iff,axiom,
! [M_11: nat,N_23: nat] :
( ( ord_less_nat @ ( semiri984289939at_nat @ M_11 ) @ ( semiri984289939at_nat @ N_23 ) )
<=> ( ord_less_nat @ M_11 @ N_23 ) ) ).
thf(fact_2946_of__nat__less__iff,axiom,
! [M_11: nat,N_23: nat] :
( ( ord_less_int @ ( semiri1621563631at_int @ M_11 ) @ ( semiri1621563631at_int @ N_23 ) )
<=> ( ord_less_nat @ M_11 @ N_23 ) ) ).
thf(fact_2947_of__nat__le__iff,axiom,
! [M_10: nat,N_22: nat] :
( ( ord_le565307924umeral @ ( semiri1619134803umeral @ M_10 ) @ ( semiri1619134803umeral @ N_22 ) )
<=> ( ord_less_eq_nat @ M_10 @ N_22 ) ) ).
thf(fact_2948_of__nat__le__iff,axiom,
! [M_10: nat,N_22: nat] :
( ( ord_le258702272de_int @ ( semiri1424489471de_int @ M_10 ) @ ( semiri1424489471de_int @ N_22 ) )
<=> ( ord_less_eq_nat @ M_10 @ N_22 ) ) ).
thf(fact_2949_of__nat__le__iff,axiom,
! [M_10: nat,N_22: nat] :
( ( ord_less_eq_nat @ ( semiri984289939at_nat @ M_10 ) @ ( semiri984289939at_nat @ N_22 ) )
<=> ( ord_less_eq_nat @ M_10 @ N_22 ) ) ).
thf(fact_2950_of__nat__le__iff,axiom,
! [M_10: nat,N_22: nat] :
( ( ord_less_eq_rat @ ( semiri151668891at_rat @ M_10 ) @ ( semiri151668891at_rat @ N_22 ) )
<=> ( ord_less_eq_nat @ M_10 @ N_22 ) ) ).
thf(fact_2951_of__nat__le__iff,axiom,
! [M_10: nat,N_22: nat] :
( ( ord_less_eq_real @ ( semiri132038758t_real @ M_10 ) @ ( semiri132038758t_real @ N_22 ) )
<=> ( ord_less_eq_nat @ M_10 @ N_22 ) ) ).
thf(fact_2952_of__nat__le__iff,axiom,
! [M_10: nat,N_22: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ M_10 ) @ ( semiri1621563631at_int @ N_22 ) )
<=> ( ord_less_eq_nat @ M_10 @ N_22 ) ) ).
thf(fact_2953_of__nat__add,axiom,
! [M_9: nat,N_21: nat] :
( ( semiri1619134803umeral @ ( plus_plus_nat @ M_9 @ N_21 ) )
= ( plus_p1627245867umeral @ ( semiri1619134803umeral @ M_9 ) @ ( semiri1619134803umeral @ N_21 ) ) ) ).
thf(fact_2954_of__nat__add,axiom,
! [M_9: nat,N_21: nat] :
( ( semiri151668891at_rat @ ( plus_plus_nat @ M_9 @ N_21 ) )
= ( plus_plus_rat @ ( semiri151668891at_rat @ M_9 ) @ ( semiri151668891at_rat @ N_21 ) ) ) ).
thf(fact_2955_of__nat__add,axiom,
! [M_9: nat,N_21: nat] :
( ( semiri1424489471de_int @ ( plus_plus_nat @ M_9 @ N_21 ) )
= ( plus_p1446045655de_int @ ( semiri1424489471de_int @ M_9 ) @ ( semiri1424489471de_int @ N_21 ) ) ) ).
thf(fact_2956_of__nat__add,axiom,
! [M_9: nat,N_21: nat] :
( ( semiri2020571505omplex @ ( plus_plus_nat @ M_9 @ N_21 ) )
= ( plus_plus_complex @ ( semiri2020571505omplex @ M_9 ) @ ( semiri2020571505omplex @ N_21 ) ) ) ).
thf(fact_2957_of__nat__add,axiom,
! [M_9: nat,N_21: nat] :
( ( semiri132038758t_real @ ( plus_plus_nat @ M_9 @ N_21 ) )
= ( plus_plus_real @ ( semiri132038758t_real @ M_9 ) @ ( semiri132038758t_real @ N_21 ) ) ) ).
thf(fact_2958_of__nat__add,axiom,
! [M_9: nat,N_21: nat] :
( ( semiri984289939at_nat @ ( plus_plus_nat @ M_9 @ N_21 ) )
= ( plus_plus_nat @ ( semiri984289939at_nat @ M_9 ) @ ( semiri984289939at_nat @ N_21 ) ) ) ).
thf(fact_2959_of__nat__add,axiom,
! [M_9: nat,N_21: nat] :
( ( semiri1621563631at_int @ ( plus_plus_nat @ M_9 @ N_21 ) )
= ( plus_plus_int @ ( semiri1621563631at_int @ M_9 ) @ ( semiri1621563631at_int @ N_21 ) ) ) ).
thf(fact_2960_of__nat__mult,axiom,
! [M_8: nat,N_20: nat] :
( ( semiri1619134803umeral @ ( times_times_nat @ M_8 @ N_20 ) )
= ( times_1655362735umeral @ ( semiri1619134803umeral @ M_8 ) @ ( semiri1619134803umeral @ N_20 ) ) ) ).
thf(fact_2961_of__nat__mult,axiom,
! [M_8: nat,N_20: nat] :
( ( semiri151668891at_rat @ ( times_times_nat @ M_8 @ N_20 ) )
= ( times_times_rat @ ( semiri151668891at_rat @ M_8 ) @ ( semiri151668891at_rat @ N_20 ) ) ) ).
thf(fact_2962_of__nat__mult,axiom,
! [M_8: nat,N_20: nat] :
( ( semiri1424489471de_int @ ( times_times_nat @ M_8 @ N_20 ) )
= ( times_123202395de_int @ ( semiri1424489471de_int @ M_8 ) @ ( semiri1424489471de_int @ N_20 ) ) ) ).
thf(fact_2963_of__nat__mult,axiom,
! [M_8: nat,N_20: nat] :
( ( semiri2020571505omplex @ ( times_times_nat @ M_8 @ N_20 ) )
= ( times_times_complex @ ( semiri2020571505omplex @ M_8 ) @ ( semiri2020571505omplex @ N_20 ) ) ) ).
thf(fact_2964_of__nat__mult,axiom,
! [M_8: nat,N_20: nat] :
( ( semiri132038758t_real @ ( times_times_nat @ M_8 @ N_20 ) )
= ( times_times_real @ ( semiri132038758t_real @ M_8 ) @ ( semiri132038758t_real @ N_20 ) ) ) ).
thf(fact_2965_of__nat__mult,axiom,
! [M_8: nat,N_20: nat] :
( ( semiri984289939at_nat @ ( times_times_nat @ M_8 @ N_20 ) )
= ( times_times_nat @ ( semiri984289939at_nat @ M_8 ) @ ( semiri984289939at_nat @ N_20 ) ) ) ).
thf(fact_2966_of__nat__mult,axiom,
! [M_8: nat,N_20: nat] :
( ( semiri1621563631at_int @ ( times_times_nat @ M_8 @ N_20 ) )
= ( times_times_int @ ( semiri1621563631at_int @ M_8 ) @ ( semiri1621563631at_int @ N_20 ) ) ) ).
thf(fact_2967_of__nat__1,axiom,
( ( semiri151668891at_rat @ one_one_nat )
= one_one_rat ) ).
thf(fact_2968_of__nat__1,axiom,
( ( semiri1424489471de_int @ one_one_nat )
= one_on1684967323de_int ) ).
thf(fact_2969_of__nat__1,axiom,
( ( semiri2020571505omplex @ one_one_nat )
= one_one_complex ) ).
thf(fact_2970_of__nat__1,axiom,
( ( semiri1619134803umeral @ one_one_nat )
= one_on1645066479umeral ) ).
thf(fact_2971_of__nat__1,axiom,
( ( semiri132038758t_real @ one_one_nat )
= one_one_real ) ).
thf(fact_2972_of__nat__1,axiom,
( ( semiri984289939at_nat @ one_one_nat )
= one_one_nat ) ).
thf(fact_2973_of__nat__1,axiom,
( ( semiri1621563631at_int @ one_one_nat )
= one_one_int ) ).
thf(fact_2974_int__0,axiom,
( ( semiri1621563631at_int @ zero_zero_nat )
= zero_zero_int ) ).
thf(fact_2975_int__eq__0__conv,axiom,
! [N: nat] :
( ( ( semiri1621563631at_int @ N )
= zero_zero_int )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_2976_transfer__int__nat__numerals_I1_J,axiom,
( zero_zero_int
= ( semiri1621563631at_int @ zero_zero_nat ) ) ).
thf(fact_2977_of__nat__power,axiom,
! [M_7: nat,N_19: nat] :
( ( semiri1424489471de_int @ ( power_power_nat @ M_7 @ N_19 ) )
= ( power_881366806de_int @ ( semiri1424489471de_int @ M_7 ) @ N_19 ) ) ).
thf(fact_2978_of__nat__power,axiom,
! [M_7: nat,N_19: nat] :
( ( semiri1619134803umeral @ ( power_power_nat @ M_7 @ N_19 ) )
= ( power_2100829034umeral @ ( semiri1619134803umeral @ M_7 ) @ N_19 ) ) ).
thf(fact_2979_of__nat__power,axiom,
! [M_7: nat,N_19: nat] :
( ( semiri151668891at_rat @ ( power_power_nat @ M_7 @ N_19 ) )
= ( power_power_rat @ ( semiri151668891at_rat @ M_7 ) @ N_19 ) ) ).
thf(fact_2980_of__nat__power,axiom,
! [M_7: nat,N_19: nat] :
( ( semiri2020571505omplex @ ( power_power_nat @ M_7 @ N_19 ) )
= ( power_power_complex @ ( semiri2020571505omplex @ M_7 ) @ N_19 ) ) ).
thf(fact_2981_of__nat__power,axiom,
! [M_7: nat,N_19: nat] :
( ( semiri132038758t_real @ ( power_power_nat @ M_7 @ N_19 ) )
= ( power_power_real @ ( semiri132038758t_real @ M_7 ) @ N_19 ) ) ).
thf(fact_2982_of__nat__power,axiom,
! [M_7: nat,N_19: nat] :
( ( semiri984289939at_nat @ ( power_power_nat @ M_7 @ N_19 ) )
= ( power_power_nat @ ( semiri984289939at_nat @ M_7 ) @ N_19 ) ) ).
thf(fact_2983_of__nat__power,axiom,
! [M_7: nat,N_19: nat] :
( ( semiri1621563631at_int @ ( power_power_nat @ M_7 @ N_19 ) )
= ( power_power_int @ ( semiri1621563631at_int @ M_7 ) @ N_19 ) ) ).
thf(fact_2984_int__less__0__conv,axiom,
! [K_1: nat] :
~ ( ord_less_int @ ( semiri1621563631at_int @ K_1 ) @ zero_zero_int ) ).
thf(fact_2985_zero__zle__int,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ N ) ) ).
thf(fact_2986_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z_1: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ Z_1 ) ) ).
thf(fact_2987_transfer__int__nat__quantifiers_I2_J,axiom,
! [P: int > $o] :
( ? [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
& ( P @ X_1 ) )
<=> ? [X_1: nat] : ( P @ ( semiri1621563631at_int @ X_1 ) ) ) ).
thf(fact_2988_transfer__int__nat__quantifiers_I1_J,axiom,
! [P: int > $o] :
( ! [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( P @ X_1 ) )
<=> ! [X_1: nat] : ( P @ ( semiri1621563631at_int @ X_1 ) ) ) ).
thf(fact_2989_zless__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_2990_Nat__Transfer_Otransfer__int__nat__relations_I2_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
<=> ( ord_less_nat @ X @ Y ) ) ).
thf(fact_2991_power__0__Suc,axiom,
! [N_18: nat] :
( ( power_power_rat @ zero_zero_rat @ ( suc @ N_18 ) )
= zero_zero_rat ) ).
thf(fact_2992_power__0__Suc,axiom,
! [N_18: nat] :
( ( power_881366806de_int @ zero_z891286103de_int @ ( suc @ N_18 ) )
= zero_z891286103de_int ) ).
thf(fact_2993_power__0__Suc,axiom,
! [N_18: nat] :
( ( power_power_complex @ zero_zero_complex @ ( suc @ N_18 ) )
= zero_zero_complex ) ).
thf(fact_2994_power__0__Suc,axiom,
! [N_18: nat] :
( ( power_2100829034umeral @ zero_z126310315umeral @ ( suc @ N_18 ) )
= zero_z126310315umeral ) ).
thf(fact_2995_power__0__Suc,axiom,
! [N_18: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N_18 ) )
= zero_zero_real ) ).
thf(fact_2996_power__0__Suc,axiom,
! [N_18: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N_18 ) )
= zero_zero_nat ) ).
thf(fact_2997_power__0__Suc,axiom,
! [N_18: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N_18 ) )
= zero_zero_int ) ).
thf(fact_2998_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_2999_Nat__Transfer_Otransfer__int__nat__relations_I3_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
<=> ( ord_less_eq_nat @ X @ Y ) ) ).
thf(fact_3000_int__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1621563631at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) ) ) ).
thf(fact_3001_zmult__int,axiom,
! [M: nat,N: nat] :
( ( times_times_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
= ( semiri1621563631at_int @ ( times_times_nat @ M @ N ) ) ) ).
thf(fact_3002_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
! [X: nat,Y: nat] :
( ( times_times_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
= ( semiri1621563631at_int @ ( times_times_nat @ X @ Y ) ) ) ).
thf(fact_3003_zle__iff__zadd,axiom,
! [W: int,Z_1: int] :
( ( ord_less_eq_int @ W @ Z_1 )
<=> ? [N_1: nat] :
( Z_1
= ( plus_plus_int @ W @ ( semiri1621563631at_int @ N_1 ) ) ) ) ).
thf(fact_3004_power__Suc2,axiom,
! [A_12: quickcheck_code_int,N_17: nat] :
( ( power_881366806de_int @ A_12 @ ( suc @ N_17 ) )
= ( times_123202395de_int @ ( power_881366806de_int @ A_12 @ N_17 ) @ A_12 ) ) ).
thf(fact_3005_power__Suc2,axiom,
! [A_12: code_code_numeral,N_17: nat] :
( ( power_2100829034umeral @ A_12 @ ( suc @ N_17 ) )
= ( times_1655362735umeral @ ( power_2100829034umeral @ A_12 @ N_17 ) @ A_12 ) ) ).
thf(fact_3006_power__Suc2,axiom,
! [A_12: rat,N_17: nat] :
( ( power_power_rat @ A_12 @ ( suc @ N_17 ) )
= ( times_times_rat @ ( power_power_rat @ A_12 @ N_17 ) @ A_12 ) ) ).
thf(fact_3007_power__Suc2,axiom,
! [A_12: complex,N_17: nat] :
( ( power_power_complex @ A_12 @ ( suc @ N_17 ) )
= ( times_times_complex @ ( power_power_complex @ A_12 @ N_17 ) @ A_12 ) ) ).
thf(fact_3008_power__Suc2,axiom,
! [A_12: real,N_17: nat] :
( ( power_power_real @ A_12 @ ( suc @ N_17 ) )
= ( times_times_real @ ( power_power_real @ A_12 @ N_17 ) @ A_12 ) ) ).
thf(fact_3009_power__Suc2,axiom,
! [A_12: nat,N_17: nat] :
( ( power_power_nat @ A_12 @ ( suc @ N_17 ) )
= ( times_times_nat @ ( power_power_nat @ A_12 @ N_17 ) @ A_12 ) ) ).
thf(fact_3010_power__Suc2,axiom,
! [A_12: int,N_17: nat] :
( ( power_power_int @ A_12 @ ( suc @ N_17 ) )
= ( times_times_int @ ( power_power_int @ A_12 @ N_17 ) @ A_12 ) ) ).
thf(fact_3011_power__Suc,axiom,
! [A_11: quickcheck_code_int,N_16: nat] :
( ( power_881366806de_int @ A_11 @ ( suc @ N_16 ) )
= ( times_123202395de_int @ A_11 @ ( power_881366806de_int @ A_11 @ N_16 ) ) ) ).
thf(fact_3012_power__Suc,axiom,
! [A_11: code_code_numeral,N_16: nat] :
( ( power_2100829034umeral @ A_11 @ ( suc @ N_16 ) )
= ( times_1655362735umeral @ A_11 @ ( power_2100829034umeral @ A_11 @ N_16 ) ) ) ).
thf(fact_3013_power__Suc,axiom,
! [A_11: rat,N_16: nat] :
( ( power_power_rat @ A_11 @ ( suc @ N_16 ) )
= ( times_times_rat @ A_11 @ ( power_power_rat @ A_11 @ N_16 ) ) ) ).
thf(fact_3014_power__Suc,axiom,
! [A_11: complex,N_16: nat] :
( ( power_power_complex @ A_11 @ ( suc @ N_16 ) )
= ( times_times_complex @ A_11 @ ( power_power_complex @ A_11 @ N_16 ) ) ) ).
thf(fact_3015_power__Suc,axiom,
! [A_11: real,N_16: nat] :
( ( power_power_real @ A_11 @ ( suc @ N_16 ) )
= ( times_times_real @ A_11 @ ( power_power_real @ A_11 @ N_16 ) ) ) ).
thf(fact_3016_power__Suc,axiom,
! [A_11: nat,N_16: nat] :
( ( power_power_nat @ A_11 @ ( suc @ N_16 ) )
= ( times_times_nat @ A_11 @ ( power_power_nat @ A_11 @ N_16 ) ) ) ).
thf(fact_3017_power__Suc,axiom,
! [A_11: int,N_16: nat] :
( ( power_power_int @ A_11 @ ( suc @ N_16 ) )
= ( times_times_int @ A_11 @ ( power_power_int @ A_11 @ N_16 ) ) ) ).
thf(fact_3018_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
! [X_8: quickcheck_code_int,Q_5: nat] :
( ( times_123202395de_int @ ( power_881366806de_int @ X_8 @ Q_5 ) @ X_8 )
= ( power_881366806de_int @ X_8 @ ( suc @ Q_5 ) ) ) ).
thf(fact_3019_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
! [X_8: code_code_numeral,Q_5: nat] :
( ( times_1655362735umeral @ ( power_2100829034umeral @ X_8 @ Q_5 ) @ X_8 )
= ( power_2100829034umeral @ X_8 @ ( suc @ Q_5 ) ) ) ).
thf(fact_3020_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
! [X_8: rat,Q_5: nat] :
( ( times_times_rat @ ( power_power_rat @ X_8 @ Q_5 ) @ X_8 )
= ( power_power_rat @ X_8 @ ( suc @ Q_5 ) ) ) ).
thf(fact_3021_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
! [X_8: complex,Q_5: nat] :
( ( times_times_complex @ ( power_power_complex @ X_8 @ Q_5 ) @ X_8 )
= ( power_power_complex @ X_8 @ ( suc @ Q_5 ) ) ) ).
thf(fact_3022_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
! [X_8: real,Q_5: nat] :
( ( times_times_real @ ( power_power_real @ X_8 @ Q_5 ) @ X_8 )
= ( power_power_real @ X_8 @ ( suc @ Q_5 ) ) ) ).
thf(fact_3023_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
! [X_8: nat,Q_5: nat] :
( ( times_times_nat @ ( power_power_nat @ X_8 @ Q_5 ) @ X_8 )
= ( power_power_nat @ X_8 @ ( suc @ Q_5 ) ) ) ).
thf(fact_3024_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
! [X_8: int,Q_5: nat] :
( ( times_times_int @ ( power_power_int @ X_8 @ Q_5 ) @ X_8 )
= ( power_power_int @ X_8 @ ( suc @ Q_5 ) ) ) ).
thf(fact_3025_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
! [X_7: quickcheck_code_int,Q_4: nat] :
( ( times_123202395de_int @ X_7 @ ( power_881366806de_int @ X_7 @ Q_4 ) )
= ( power_881366806de_int @ X_7 @ ( suc @ Q_4 ) ) ) ).
thf(fact_3026_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
! [X_7: code_code_numeral,Q_4: nat] :
( ( times_1655362735umeral @ X_7 @ ( power_2100829034umeral @ X_7 @ Q_4 ) )
= ( power_2100829034umeral @ X_7 @ ( suc @ Q_4 ) ) ) ).
thf(fact_3027_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
! [X_7: rat,Q_4: nat] :
( ( times_times_rat @ X_7 @ ( power_power_rat @ X_7 @ Q_4 ) )
= ( power_power_rat @ X_7 @ ( suc @ Q_4 ) ) ) ).
thf(fact_3028_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
! [X_7: complex,Q_4: nat] :
( ( times_times_complex @ X_7 @ ( power_power_complex @ X_7 @ Q_4 ) )
= ( power_power_complex @ X_7 @ ( suc @ Q_4 ) ) ) ).
thf(fact_3029_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
! [X_7: real,Q_4: nat] :
( ( times_times_real @ X_7 @ ( power_power_real @ X_7 @ Q_4 ) )
= ( power_power_real @ X_7 @ ( suc @ Q_4 ) ) ) ).
thf(fact_3030_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
! [X_7: nat,Q_4: nat] :
( ( times_times_nat @ X_7 @ ( power_power_nat @ X_7 @ Q_4 ) )
= ( power_power_nat @ X_7 @ ( suc @ Q_4 ) ) ) ).
thf(fact_3031_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
! [X_7: int,Q_4: nat] :
( ( times_times_int @ X_7 @ ( power_power_int @ X_7 @ Q_4 ) )
= ( power_power_int @ X_7 @ ( suc @ Q_4 ) ) ) ).
thf(fact_3032_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
! [X_6: quickcheck_code_int,Q_3: nat] :
( ( power_881366806de_int @ X_6 @ ( suc @ Q_3 ) )
= ( times_123202395de_int @ X_6 @ ( power_881366806de_int @ X_6 @ Q_3 ) ) ) ).
thf(fact_3033_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
! [X_6: code_code_numeral,Q_3: nat] :
( ( power_2100829034umeral @ X_6 @ ( suc @ Q_3 ) )
= ( times_1655362735umeral @ X_6 @ ( power_2100829034umeral @ X_6 @ Q_3 ) ) ) ).
thf(fact_3034_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
! [X_6: rat,Q_3: nat] :
( ( power_power_rat @ X_6 @ ( suc @ Q_3 ) )
= ( times_times_rat @ X_6 @ ( power_power_rat @ X_6 @ Q_3 ) ) ) ).
thf(fact_3035_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
! [X_6: complex,Q_3: nat] :
( ( power_power_complex @ X_6 @ ( suc @ Q_3 ) )
= ( times_times_complex @ X_6 @ ( power_power_complex @ X_6 @ Q_3 ) ) ) ).
thf(fact_3036_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
! [X_6: real,Q_3: nat] :
( ( power_power_real @ X_6 @ ( suc @ Q_3 ) )
= ( times_times_real @ X_6 @ ( power_power_real @ X_6 @ Q_3 ) ) ) ).
thf(fact_3037_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
! [X_6: nat,Q_3: nat] :
( ( power_power_nat @ X_6 @ ( suc @ Q_3 ) )
= ( times_times_nat @ X_6 @ ( power_power_nat @ X_6 @ Q_3 ) ) ) ).
thf(fact_3038_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
! [X_6: int,Q_3: nat] :
( ( power_power_int @ X_6 @ ( suc @ Q_3 ) )
= ( times_times_int @ X_6 @ ( power_power_int @ X_6 @ Q_3 ) ) ) ).
thf(fact_3039_zadd__int,axiom,
! [M: nat,N: nat] :
( ( plus_plus_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
= ( semiri1621563631at_int @ ( plus_plus_nat @ M @ N ) ) ) ).
thf(fact_3040_zadd__int__left,axiom,
! [M: nat,N: nat,Z_1: int] :
( ( plus_plus_int @ ( semiri1621563631at_int @ M ) @ ( plus_plus_int @ ( semiri1621563631at_int @ N ) @ Z_1 ) )
= ( plus_plus_int @ ( semiri1621563631at_int @ ( plus_plus_nat @ M @ N ) ) @ Z_1 ) ) ).
thf(fact_3041_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
! [X: nat,Y: nat] :
( ( plus_plus_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
= ( semiri1621563631at_int @ ( plus_plus_nat @ X @ Y ) ) ) ).
thf(fact_3042_int__1,axiom,
( ( semiri1621563631at_int @ one_one_nat )
= one_one_int ) ).
thf(fact_3043_transfer__int__nat__numerals_I2_J,axiom,
( one_one_int
= ( semiri1621563631at_int @ one_one_nat ) ) ).
thf(fact_3044_Nat__Transfer_Otransfer__int__nat__relations_I4_J,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
<=> ( dvd_dvd_nat @ X @ Y ) ) ).
thf(fact_3045_zdvd__int,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
<=> ( dvd_dvd_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) ) ) ).
thf(fact_3046_nat__lt__two__imp__zero__or__one,axiom,
! [X: nat] :
( ( ord_less_nat @ X @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( ( X = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
thf(fact_3047_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
<=> ( ( M = zero_zero_nat )
| ? [J_1: nat] :
( ( M
= ( suc @ J_1 ) )
& ( ord_less_nat @ J_1 @ N ) ) ) ) ).
thf(fact_3048_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_3049_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
<=> ? [M_2: nat] :
( N
= ( suc @ M_2 ) ) ) ).
thf(fact_3050_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
<=> ( M
= ( suc @ zero_zero_nat ) ) ) ).
thf(fact_3051_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
<=> ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_3052_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
<=> ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_3053_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
<=> ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
thf(fact_3054_less__iff__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
<=> ? [K: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K ) ) ) ) ).
thf(fact_3055_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
thf(fact_3056_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
thf(fact_3057_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
thf(fact_3058_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_3059_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
<=> ( N = M ) ) ) ).
thf(fact_3060_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
thf(fact_3061_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_3062_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_3063_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_3064_less__eq__Suc__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
<=> ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
thf(fact_3065_Suc__mult__less__cancel1,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K_1 ) @ M ) @ ( times_times_nat @ ( suc @ K_1 ) @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_3066_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
thf(fact_3067_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
thf(fact_3068_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
thf(fact_3069_Suc__mult__le__cancel1,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K_1 ) @ M ) @ ( times_times_nat @ ( suc @ K_1 ) @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_3070_Suc__eq__plus1,axiom,
! [N: nat] :
( ( suc @ N )
= ( plus_plus_nat @ N @ one_one_nat ) ) ).
thf(fact_3071_Suc__eq__plus1__left,axiom,
! [N: nat] :
( ( suc @ N )
= ( plus_plus_nat @ one_one_nat @ N ) ) ).
thf(fact_3072_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_3073_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
thf(fact_3074_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
thf(fact_3075_zdiv__int,axiom,
! [A: nat,B: nat] :
( ( semiri1621563631at_int @ ( div_div_nat @ A @ B ) )
= ( div_div_int @ ( semiri1621563631at_int @ A ) @ ( semiri1621563631at_int @ B ) ) ) ).
thf(fact_3076_Divides_Otransfer__int__nat__functions_I1_J,axiom,
! [X: nat,Y: nat] :
( ( div_div_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
= ( semiri1621563631at_int @ ( div_div_nat @ X @ Y ) ) ) ).
thf(fact_3077_div__1,axiom,
! [M: nat] :
( ( div_div_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
thf(fact_3078_mod__1,axiom,
! [M: nat] :
( ( div_mod_nat @ M @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
thf(fact_3079_mod__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( suc @ ( div_mod_nat @ M @ N ) )
= N )
=> ( ( div_mod_nat @ ( suc @ M ) @ N )
= zero_zero_nat ) )
& ( ( ( suc @ ( div_mod_nat @ M @ N ) )
!= N )
=> ( ( div_mod_nat @ ( suc @ M ) @ N )
= ( suc @ ( div_mod_nat @ M @ N ) ) ) ) ) ).
thf(fact_3080_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
<=> ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
thf(fact_3081_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
thf(fact_3082_int__power,axiom,
! [M: nat,N: nat] :
( ( semiri1621563631at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri1621563631at_int @ M ) @ N ) ) ).
thf(fact_3083_zpower__int,axiom,
! [M: nat,N: nat] :
( ( power_power_int @ ( semiri1621563631at_int @ M ) @ N )
= ( semiri1621563631at_int @ ( power_power_nat @ M @ N ) ) ) ).
thf(fact_3084_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
! [X: nat,N: nat] :
( ( power_power_int @ ( semiri1621563631at_int @ X ) @ N )
= ( semiri1621563631at_int @ ( power_power_nat @ X @ N ) ) ) ).
thf(fact_3085_zmod__int,axiom,
! [A: nat,B: nat] :
( ( semiri1621563631at_int @ ( div_mod_nat @ A @ B ) )
= ( div_mod_int @ ( semiri1621563631at_int @ A ) @ ( semiri1621563631at_int @ B ) ) ) ).
thf(fact_3086_Divides_Otransfer__int__nat__functions_I2_J,axiom,
! [X: nat,Y: nat] :
( ( div_mod_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
= ( semiri1621563631at_int @ ( div_mod_nat @ X @ Y ) ) ) ).
thf(fact_3087_exp__mono__lt,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ord_less_nat @ ( power_power_nat @ X @ ( suc @ N ) ) @ ( power_power_nat @ Y @ ( suc @ N ) ) )
<=> ( ord_less_nat @ X @ Y ) ) ).
thf(fact_3088_divides__rexp,axiom,
! [N: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( dvd_dvd_nat @ X @ ( power_power_nat @ Y @ ( suc @ N ) ) ) ) ).
thf(fact_3089_exp__mono__le,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( suc @ N ) ) @ ( power_power_nat @ Y @ ( suc @ N ) ) )
<=> ( ord_less_eq_nat @ X @ Y ) ) ).
thf(fact_3090_even__power,axiom,
! [X: int,N: nat] :
( ( even_odd_even_int @ ( power_power_int @ X @ N ) )
<=> ( ( even_odd_even_int @ X )
& ( N != zero_zero_nat ) ) ) ).
thf(fact_3091_IntNatAux_Oodd__plus__odd,axiom,
! [B: int,A: int] :
( ( member_int @ A @ zOdd )
=> ( ( member_int @ B @ zOdd )
=> ( member_int @ ( plus_plus_int @ A @ B ) @ zEven ) ) ) ).
thf(fact_3092_IntNatAux_Oeven__plus__odd,axiom,
! [B: int,A: int] :
( ( member_int @ A @ zEven )
=> ( ( member_int @ B @ zOdd )
=> ( member_int @ ( plus_plus_int @ A @ B ) @ zOdd ) ) ) ).
thf(fact_3093_even__plus__odd__prop1,axiom,
! [A: int,B: int] :
( ( member_int @ ( plus_plus_int @ A @ B ) @ zOdd )
=> ( ( member_int @ A @ zOdd )
=> ( member_int @ B @ zEven ) ) ) ).
thf(fact_3094_even__plus__odd__prop2,axiom,
! [A: int,B: int] :
( ( member_int @ ( plus_plus_int @ A @ B ) @ zOdd )
=> ( ( member_int @ A @ zEven )
=> ( member_int @ B @ zOdd ) ) ) ).
thf(fact_3095_even__minus__odd,axiom,
! [Y: int,X: int] :
( ( member_int @ X @ zEven )
=> ( ( member_int @ Y @ zOdd )
=> ( member_int @ ( minus_minus_int @ X @ Y ) @ zOdd ) ) ) ).
thf(fact_3096_odd__minus__even,axiom,
! [Y: int,X: int] :
( ( member_int @ X @ zOdd )
=> ( ( member_int @ Y @ zEven )
=> ( member_int @ ( minus_minus_int @ X @ Y ) @ zOdd ) ) ) ).
thf(fact_3097_odd__minus__odd,axiom,
! [Y: int,X: int] :
( ( member_int @ X @ zOdd )
=> ( ( member_int @ Y @ zOdd )
=> ( member_int @ ( minus_minus_int @ X @ Y ) @ zEven ) ) ) ).
thf(fact_3098_of__nat__diff,axiom,
! [N_15: nat,M_6: nat] :
( ( ord_less_eq_nat @ N_15 @ M_6 )
=> ( ( semiri151668891at_rat @ ( minus_minus_nat @ M_6 @ N_15 ) )
= ( minus_minus_rat @ ( semiri151668891at_rat @ M_6 ) @ ( semiri151668891at_rat @ N_15 ) ) ) ) ).
thf(fact_3099_of__nat__diff,axiom,
! [N_15: nat,M_6: nat] :
( ( ord_less_eq_nat @ N_15 @ M_6 )
=> ( ( semiri2020571505omplex @ ( minus_minus_nat @ M_6 @ N_15 ) )
= ( minus_minus_complex @ ( semiri2020571505omplex @ M_6 ) @ ( semiri2020571505omplex @ N_15 ) ) ) ) ).
thf(fact_3100_of__nat__diff,axiom,
! [N_15: nat,M_6: nat] :
( ( ord_less_eq_nat @ N_15 @ M_6 )
=> ( ( semiri132038758t_real @ ( minus_minus_nat @ M_6 @ N_15 ) )
= ( minus_minus_real @ ( semiri132038758t_real @ M_6 ) @ ( semiri132038758t_real @ N_15 ) ) ) ) ).
thf(fact_3101_of__nat__diff,axiom,
! [N_15: nat,M_6: nat] :
( ( ord_less_eq_nat @ N_15 @ M_6 )
=> ( ( semiri1621563631at_int @ ( minus_minus_nat @ M_6 @ N_15 ) )
= ( minus_minus_int @ ( semiri1621563631at_int @ M_6 ) @ ( semiri1621563631at_int @ N_15 ) ) ) ) ).
thf(fact_3102_int__le__0__conv,axiom,
! [N: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ N ) @ zero_zero_int )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_3103_power__inject__base,axiom,
! [A_10: rat,N_14: nat,B_8: rat] :
( ( ( power_power_rat @ A_10 @ ( suc @ N_14 ) )
= ( power_power_rat @ B_8 @ ( suc @ N_14 ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A_10 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_8 )
=> ( A_10 = B_8 ) ) ) ) ).
thf(fact_3104_power__inject__base,axiom,
! [A_10: quickcheck_code_int,N_14: nat,B_8: quickcheck_code_int] :
( ( ( power_881366806de_int @ A_10 @ ( suc @ N_14 ) )
= ( power_881366806de_int @ B_8 @ ( suc @ N_14 ) ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_10 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_8 )
=> ( A_10 = B_8 ) ) ) ) ).
thf(fact_3105_power__inject__base,axiom,
! [A_10: code_code_numeral,N_14: nat,B_8: code_code_numeral] :
( ( ( power_2100829034umeral @ A_10 @ ( suc @ N_14 ) )
= ( power_2100829034umeral @ B_8 @ ( suc @ N_14 ) ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_10 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_8 )
=> ( A_10 = B_8 ) ) ) ) ).
thf(fact_3106_power__inject__base,axiom,
! [A_10: real,N_14: nat,B_8: real] :
( ( ( power_power_real @ A_10 @ ( suc @ N_14 ) )
= ( power_power_real @ B_8 @ ( suc @ N_14 ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_10 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_8 )
=> ( A_10 = B_8 ) ) ) ) ).
thf(fact_3107_power__inject__base,axiom,
! [A_10: nat,N_14: nat,B_8: nat] :
( ( ( power_power_nat @ A_10 @ ( suc @ N_14 ) )
= ( power_power_nat @ B_8 @ ( suc @ N_14 ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_10 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_8 )
=> ( A_10 = B_8 ) ) ) ) ).
thf(fact_3108_power__inject__base,axiom,
! [A_10: int,N_14: nat,B_8: int] :
( ( ( power_power_int @ A_10 @ ( suc @ N_14 ) )
= ( power_power_int @ B_8 @ ( suc @ N_14 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_10 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_8 )
=> ( A_10 = B_8 ) ) ) ) ).
thf(fact_3109_power__le__imp__le__base,axiom,
! [A_9: rat,N_13: nat,B_7: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A_9 @ ( suc @ N_13 ) ) @ ( power_power_rat @ B_7 @ ( suc @ N_13 ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_7 )
=> ( ord_less_eq_rat @ A_9 @ B_7 ) ) ) ).
thf(fact_3110_power__le__imp__le__base,axiom,
! [A_9: quickcheck_code_int,N_13: nat,B_7: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( power_881366806de_int @ A_9 @ ( suc @ N_13 ) ) @ ( power_881366806de_int @ B_7 @ ( suc @ N_13 ) ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_7 )
=> ( ord_le258702272de_int @ A_9 @ B_7 ) ) ) ).
thf(fact_3111_power__le__imp__le__base,axiom,
! [A_9: code_code_numeral,N_13: nat,B_7: code_code_numeral] :
( ( ord_le565307924umeral @ ( power_2100829034umeral @ A_9 @ ( suc @ N_13 ) ) @ ( power_2100829034umeral @ B_7 @ ( suc @ N_13 ) ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_7 )
=> ( ord_le565307924umeral @ A_9 @ B_7 ) ) ) ).
thf(fact_3112_power__le__imp__le__base,axiom,
! [A_9: real,N_13: nat,B_7: real] :
( ( ord_less_eq_real @ ( power_power_real @ A_9 @ ( suc @ N_13 ) ) @ ( power_power_real @ B_7 @ ( suc @ N_13 ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_7 )
=> ( ord_less_eq_real @ A_9 @ B_7 ) ) ) ).
thf(fact_3113_power__le__imp__le__base,axiom,
! [A_9: nat,N_13: nat,B_7: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A_9 @ ( suc @ N_13 ) ) @ ( power_power_nat @ B_7 @ ( suc @ N_13 ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_7 )
=> ( ord_less_eq_nat @ A_9 @ B_7 ) ) ) ).
thf(fact_3114_power__le__imp__le__base,axiom,
! [A_9: int,N_13: nat,B_7: int] :
( ( ord_less_eq_int @ ( power_power_int @ A_9 @ ( suc @ N_13 ) ) @ ( power_power_int @ B_7 @ ( suc @ N_13 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_7 )
=> ( ord_less_eq_int @ A_9 @ B_7 ) ) ) ).
thf(fact_3115_power__gt1,axiom,
! [N_12: nat,A_8: rat] :
( ( ord_less_rat @ one_one_rat @ A_8 )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A_8 @ ( suc @ N_12 ) ) ) ) ).
thf(fact_3116_power__gt1,axiom,
! [N_12: nat,A_8: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ A_8 )
=> ( ord_le1860547276de_int @ one_on1684967323de_int @ ( power_881366806de_int @ A_8 @ ( suc @ N_12 ) ) ) ) ).
thf(fact_3117_power__gt1,axiom,
! [N_12: nat,A_8: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ A_8 )
=> ( ord_le1304079648umeral @ one_on1645066479umeral @ ( power_2100829034umeral @ A_8 @ ( suc @ N_12 ) ) ) ) ).
thf(fact_3118_power__gt1,axiom,
! [N_12: nat,A_8: real] :
( ( ord_less_real @ one_one_real @ A_8 )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A_8 @ ( suc @ N_12 ) ) ) ) ).
thf(fact_3119_power__gt1,axiom,
! [N_12: nat,A_8: nat] :
( ( ord_less_nat @ one_one_nat @ A_8 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A_8 @ ( suc @ N_12 ) ) ) ) ).
thf(fact_3120_power__gt1,axiom,
! [N_12: nat,A_8: int] :
( ( ord_less_int @ one_one_int @ A_8 )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A_8 @ ( suc @ N_12 ) ) ) ) ).
thf(fact_3121_int__nat__eq,axiom,
! [Z_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z_1 )
=> ( ( semiri1621563631at_int @ ( nat_1 @ Z_1 ) )
= Z_1 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z_1 )
=> ( ( semiri1621563631at_int @ ( nat_1 @ Z_1 ) )
= zero_zero_int ) ) ) ).
thf(fact_3122_int__eq__iff,axiom,
! [M: nat,Z_1: int] :
( ( ( semiri1621563631at_int @ M )
= Z_1 )
<=> ( ( M
= ( nat_1 @ Z_1 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z_1 ) ) ) ).
thf(fact_3123_nat__0__le,axiom,
! [Z_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z_1 )
=> ( ( semiri1621563631at_int @ ( nat_1 @ Z_1 ) )
= Z_1 ) ) ).
thf(fact_3124_zless__nat__eq__int__zless,axiom,
! [M: nat,Z_1: int] :
( ( ord_less_nat @ M @ ( nat_1 @ Z_1 ) )
<=> ( ord_less_int @ ( semiri1621563631at_int @ M ) @ Z_1 ) ) ).
thf(fact_3125_zdiff__int,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
= ( semiri1621563631at_int @ ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_3126_one__less__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
thf(fact_3127_n__less__n__mult__m,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
thf(fact_3128_n__less__m__mult__n,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
thf(fact_3129_mn__eq__m__one,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( times_times_nat @ M @ N )
= M )
=> ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
thf(fact_3130_prod__mn__less__k,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ( ( times_times_nat @ M @ N )
= K_1 )
=> ( ord_less_nat @ N @ K_1 ) ) ) ) ) ).
thf(fact_3131_one__less__k,axiom,
! [M: nat,K_1: nat] :
( ( M
!= ( times_times_nat @ M @ K_1 ) )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ K_1 ) )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ K_1 ) ) ) ).
thf(fact_3132_one__less__m,axiom,
! [M: nat,K_1: nat] :
( ( M
!= ( times_times_nat @ M @ K_1 ) )
=> ( ( M
!= ( suc @ zero_zero_nat ) )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M ) ) ) ).
thf(fact_3133_diff__Suc__less,axiom,
! [I: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
thf(fact_3134_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
thf(fact_3135_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
<=> ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
thf(fact_3136_nat__1,axiom,
( ( nat_1 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
thf(fact_3137_diff__Suc__diff__eq1,axiom,
! [M: nat,K_1: nat,J: nat] :
( ( ord_less_eq_nat @ K_1 @ J )
=> ( ( minus_minus_nat @ M @ ( suc @ ( minus_minus_nat @ J @ K_1 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ M @ K_1 ) @ ( suc @ J ) ) ) ) ).
thf(fact_3138_diff__Suc__diff__eq2,axiom,
! [M: nat,K_1: nat,J: nat] :
( ( ord_less_eq_nat @ K_1 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K_1 ) ) @ M )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K_1 @ M ) ) ) ) ).
thf(fact_3139_nat__one__le__power,axiom,
! [N: nat,I: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
thf(fact_3140_mod__mult__self4,axiom,
! [K_1: nat,N: nat,M: nat] :
( ( div_mod_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K_1 @ N ) @ M ) ) @ N )
= ( div_mod_nat @ ( suc @ M ) @ N ) ) ).
thf(fact_3141_zero__code__numeral__code,axiom,
( zero_z126310315umeral
= ( number1443263063umeral @ pls ) ) ).
thf(fact_3142_of__nat__0__less__iff,axiom,
! [N_11: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( semiri151668891at_rat @ N_11 ) )
<=> ( ord_less_nat @ zero_zero_nat @ N_11 ) ) ).
thf(fact_3143_of__nat__0__less__iff,axiom,
! [N_11: nat] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ ( semiri1424489471de_int @ N_11 ) )
<=> ( ord_less_nat @ zero_zero_nat @ N_11 ) ) ).
thf(fact_3144_of__nat__0__less__iff,axiom,
! [N_11: nat] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ ( semiri1619134803umeral @ N_11 ) )
<=> ( ord_less_nat @ zero_zero_nat @ N_11 ) ) ).
thf(fact_3145_of__nat__0__less__iff,axiom,
! [N_11: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri132038758t_real @ N_11 ) )
<=> ( ord_less_nat @ zero_zero_nat @ N_11 ) ) ).
thf(fact_3146_of__nat__0__less__iff,axiom,
! [N_11: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri984289939at_nat @ N_11 ) )
<=> ( ord_less_nat @ zero_zero_nat @ N_11 ) ) ).
thf(fact_3147_of__nat__0__less__iff,axiom,
! [N_11: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1621563631at_int @ N_11 ) )
<=> ( ord_less_nat @ zero_zero_nat @ N_11 ) ) ).
thf(fact_3148_power__preserves__even,axiom,
! [X: int,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_int @ ( power_power_int @ X @ N ) @ zEven )
<=> ( member_int @ X @ zEven ) ) ) ).
thf(fact_3149_zero__less__int__conv,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1621563631at_int @ N ) )
<=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_3150_odd__minus__one__even,axiom,
! [X: int] :
( ( member_int @ X @ zOdd )
=> ( member_int @ ( minus_minus_int @ X @ one_one_int ) @ zEven ) ) ).
thf(fact_3151_transfer__int__nat__numerals_I4_J,axiom,
( ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) )
= ( semiri1621563631at_int @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3152_zmult__zless__mono2__lemma,axiom,
! [K_1: nat,I: int,J: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ord_less_int @ ( times_times_int @ ( semiri1621563631at_int @ K_1 ) @ I ) @ ( times_times_int @ ( semiri1621563631at_int @ K_1 ) @ J ) ) ) ) ).
thf(fact_3153_realpow__Suc__le__self,axiom,
! [N_10: nat,R_4: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ R_4 )
=> ( ( ord_less_eq_rat @ R_4 @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ R_4 @ ( suc @ N_10 ) ) @ R_4 ) ) ) ).
thf(fact_3154_realpow__Suc__le__self,axiom,
! [N_10: nat,R_4: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ R_4 )
=> ( ( ord_le258702272de_int @ R_4 @ one_on1684967323de_int )
=> ( ord_le258702272de_int @ ( power_881366806de_int @ R_4 @ ( suc @ N_10 ) ) @ R_4 ) ) ) ).
thf(fact_3155_realpow__Suc__le__self,axiom,
! [N_10: nat,R_4: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ R_4 )
=> ( ( ord_le565307924umeral @ R_4 @ one_on1645066479umeral )
=> ( ord_le565307924umeral @ ( power_2100829034umeral @ R_4 @ ( suc @ N_10 ) ) @ R_4 ) ) ) ).
thf(fact_3156_realpow__Suc__le__self,axiom,
! [N_10: nat,R_4: real] :
( ( ord_less_eq_real @ zero_zero_real @ R_4 )
=> ( ( ord_less_eq_real @ R_4 @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ R_4 @ ( suc @ N_10 ) ) @ R_4 ) ) ) ).
thf(fact_3157_realpow__Suc__le__self,axiom,
! [N_10: nat,R_4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ R_4 )
=> ( ( ord_less_eq_nat @ R_4 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ R_4 @ ( suc @ N_10 ) ) @ R_4 ) ) ) ).
thf(fact_3158_realpow__Suc__le__self,axiom,
! [N_10: nat,R_4: int] :
( ( ord_less_eq_int @ zero_zero_int @ R_4 )
=> ( ( ord_less_eq_int @ R_4 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ R_4 @ ( suc @ N_10 ) ) @ R_4 ) ) ) ).
thf(fact_3159_power__Suc__less__one,axiom,
! [N_9: nat,A_7: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_7 )
=> ( ( ord_less_rat @ A_7 @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A_7 @ ( suc @ N_9 ) ) @ one_one_rat ) ) ) ).
thf(fact_3160_power__Suc__less__one,axiom,
! [N_9: nat,A_7: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_7 )
=> ( ( ord_le1860547276de_int @ A_7 @ one_on1684967323de_int )
=> ( ord_le1860547276de_int @ ( power_881366806de_int @ A_7 @ ( suc @ N_9 ) ) @ one_on1684967323de_int ) ) ) ).
thf(fact_3161_power__Suc__less__one,axiom,
! [N_9: nat,A_7: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_7 )
=> ( ( ord_le1304079648umeral @ A_7 @ one_on1645066479umeral )
=> ( ord_le1304079648umeral @ ( power_2100829034umeral @ A_7 @ ( suc @ N_9 ) ) @ one_on1645066479umeral ) ) ) ).
thf(fact_3162_power__Suc__less__one,axiom,
! [N_9: nat,A_7: real] :
( ( ord_less_real @ zero_zero_real @ A_7 )
=> ( ( ord_less_real @ A_7 @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A_7 @ ( suc @ N_9 ) ) @ one_one_real ) ) ) ).
thf(fact_3163_power__Suc__less__one,axiom,
! [N_9: nat,A_7: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_7 )
=> ( ( ord_less_nat @ A_7 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A_7 @ ( suc @ N_9 ) ) @ one_one_nat ) ) ) ).
thf(fact_3164_power__Suc__less__one,axiom,
! [N_9: nat,A_7: int] :
( ( ord_less_int @ zero_zero_int @ A_7 )
=> ( ( ord_less_int @ A_7 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A_7 @ ( suc @ N_9 ) ) @ one_one_int ) ) ) ).
thf(fact_3165_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat_1 @ I ) )
<=> ( ! [N_1: nat] :
( ( I
= ( semiri1621563631at_int @ N_1 ) )
=> ( P @ N_1 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
thf(fact_3166_nat__eq__iff2,axiom,
! [M: nat,W: int] :
( ( M
= ( nat_1 @ W ) )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1621563631at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
thf(fact_3167_nat__eq__iff,axiom,
! [W: int,M: nat] :
( ( ( nat_1 @ W )
= M )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1621563631at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
thf(fact_3168_int__eq__iff__number__of,axiom,
! [M: nat,V: int] :
( ( ( semiri1621563631at_int @ M )
= ( number_number_of_int @ V ) )
<=> ( ( M
= ( nat_1 @ ( number_number_of_int @ V ) ) )
& ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ V ) ) ) ) ).
thf(fact_3169_numeral__3__eq__3,axiom,
( ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) )
= ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
thf(fact_3170_numeral__1__eq__Suc__0,axiom,
( ( number_number_of_nat @ ( bit1 @ pls ) )
= ( suc @ zero_zero_nat ) ) ).
thf(fact_3171_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) @ N ) ) ).
thf(fact_3172_lemma__realpow__diff,axiom,
! [Y_5: quickcheck_code_int,P_5: nat,N_8: nat] :
( ( ord_less_eq_nat @ P_5 @ N_8 )
=> ( ( power_881366806de_int @ Y_5 @ ( minus_minus_nat @ ( suc @ N_8 ) @ P_5 ) )
= ( times_123202395de_int @ ( power_881366806de_int @ Y_5 @ ( minus_minus_nat @ N_8 @ P_5 ) ) @ Y_5 ) ) ) ).
thf(fact_3173_lemma__realpow__diff,axiom,
! [Y_5: code_code_numeral,P_5: nat,N_8: nat] :
( ( ord_less_eq_nat @ P_5 @ N_8 )
=> ( ( power_2100829034umeral @ Y_5 @ ( minus_minus_nat @ ( suc @ N_8 ) @ P_5 ) )
= ( times_1655362735umeral @ ( power_2100829034umeral @ Y_5 @ ( minus_minus_nat @ N_8 @ P_5 ) ) @ Y_5 ) ) ) ).
thf(fact_3174_lemma__realpow__diff,axiom,
! [Y_5: rat,P_5: nat,N_8: nat] :
( ( ord_less_eq_nat @ P_5 @ N_8 )
=> ( ( power_power_rat @ Y_5 @ ( minus_minus_nat @ ( suc @ N_8 ) @ P_5 ) )
= ( times_times_rat @ ( power_power_rat @ Y_5 @ ( minus_minus_nat @ N_8 @ P_5 ) ) @ Y_5 ) ) ) ).
thf(fact_3175_lemma__realpow__diff,axiom,
! [Y_5: complex,P_5: nat,N_8: nat] :
( ( ord_less_eq_nat @ P_5 @ N_8 )
=> ( ( power_power_complex @ Y_5 @ ( minus_minus_nat @ ( suc @ N_8 ) @ P_5 ) )
= ( times_times_complex @ ( power_power_complex @ Y_5 @ ( minus_minus_nat @ N_8 @ P_5 ) ) @ Y_5 ) ) ) ).
thf(fact_3176_lemma__realpow__diff,axiom,
! [Y_5: real,P_5: nat,N_8: nat] :
( ( ord_less_eq_nat @ P_5 @ N_8 )
=> ( ( power_power_real @ Y_5 @ ( minus_minus_nat @ ( suc @ N_8 ) @ P_5 ) )
= ( times_times_real @ ( power_power_real @ Y_5 @ ( minus_minus_nat @ N_8 @ P_5 ) ) @ Y_5 ) ) ) ).
thf(fact_3177_lemma__realpow__diff,axiom,
! [Y_5: nat,P_5: nat,N_8: nat] :
( ( ord_less_eq_nat @ P_5 @ N_8 )
=> ( ( power_power_nat @ Y_5 @ ( minus_minus_nat @ ( suc @ N_8 ) @ P_5 ) )
= ( times_times_nat @ ( power_power_nat @ Y_5 @ ( minus_minus_nat @ N_8 @ P_5 ) ) @ Y_5 ) ) ) ).
thf(fact_3178_lemma__realpow__diff,axiom,
! [Y_5: int,P_5: nat,N_8: nat] :
( ( ord_less_eq_nat @ P_5 @ N_8 )
=> ( ( power_power_int @ Y_5 @ ( minus_minus_nat @ ( suc @ N_8 ) @ P_5 ) )
= ( times_times_int @ ( power_power_int @ Y_5 @ ( minus_minus_nat @ N_8 @ P_5 ) ) @ Y_5 ) ) ) ).
thf(fact_3179_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
thf(fact_3180_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
thf(fact_3181_add__eq__if,axiom,
! [N: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( plus_plus_nat @ M @ N )
= N ) )
& ( ( M != zero_zero_nat )
=> ( ( plus_plus_nat @ M @ N )
= ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
thf(fact_3182_div__geq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( ( div_div_nat @ M @ N )
= ( suc @ ( div_div_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
thf(fact_3183_div__if,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( ord_less_nat @ M @ N )
=> ( ( div_div_nat @ M @ N )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ M @ N )
=> ( ( div_div_nat @ M @ N )
= ( suc @ ( div_div_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ) ).
thf(fact_3184_Suc__times__mod__eq,axiom,
! [M: nat,K_1: nat] :
( ( ord_less_nat @ one_one_nat @ K_1 )
=> ( ( div_mod_nat @ ( suc @ ( times_times_nat @ K_1 @ M ) ) @ K_1 )
= one_one_nat ) ) ).
thf(fact_3185_div__Suc,axiom,
! [A: nat,C: nat] :
( ( div_div_nat @ ( suc @ A ) @ C )
= ( plus_plus_nat @ ( plus_plus_nat @ ( div_div_nat @ A @ C ) @ ( div_div_nat @ ( suc @ zero_zero_nat ) @ C ) ) @ ( div_div_nat @ ( plus_plus_nat @ ( div_mod_nat @ A @ C ) @ ( div_mod_nat @ ( suc @ zero_zero_nat ) @ C ) ) @ C ) ) ) ).
thf(fact_3186_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( nat_tsub @ X @ Y ) ) ) ) ).
thf(fact_3187_transfer__int__nat__numerals_I3_J,axiom,
( ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) )
= ( semiri1621563631at_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3188_tsub__eq,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( nat_tsub @ X @ Y )
= ( minus_minus_int @ X @ Y ) ) ) ).
thf(fact_3189_nat__less__iff,axiom,
! [M: nat,W: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat_1 @ W ) @ M )
<=> ( ord_less_int @ W @ ( semiri1621563631at_int @ M ) ) ) ) ).
thf(fact_3190_realpow__two__diff,axiom,
! [X_5: rat,Y_4: rat] :
( ( minus_minus_rat @ ( power_power_rat @ X_5 @ ( suc @ ( suc @ zero_zero_nat ) ) ) @ ( power_power_rat @ Y_4 @ ( suc @ ( suc @ zero_zero_nat ) ) ) )
= ( times_times_rat @ ( minus_minus_rat @ X_5 @ Y_4 ) @ ( plus_plus_rat @ X_5 @ Y_4 ) ) ) ).
thf(fact_3191_realpow__two__diff,axiom,
! [X_5: complex,Y_4: complex] :
( ( minus_minus_complex @ ( power_power_complex @ X_5 @ ( suc @ ( suc @ zero_zero_nat ) ) ) @ ( power_power_complex @ Y_4 @ ( suc @ ( suc @ zero_zero_nat ) ) ) )
= ( times_times_complex @ ( minus_minus_complex @ X_5 @ Y_4 ) @ ( plus_plus_complex @ X_5 @ Y_4 ) ) ) ).
thf(fact_3192_realpow__two__diff,axiom,
! [X_5: real,Y_4: real] :
( ( minus_minus_real @ ( power_power_real @ X_5 @ ( suc @ ( suc @ zero_zero_nat ) ) ) @ ( power_power_real @ Y_4 @ ( suc @ ( suc @ zero_zero_nat ) ) ) )
= ( times_times_real @ ( minus_minus_real @ X_5 @ Y_4 ) @ ( plus_plus_real @ X_5 @ Y_4 ) ) ) ).
thf(fact_3193_realpow__two__diff,axiom,
! [X_5: int,Y_4: int] :
( ( minus_minus_int @ ( power_power_int @ X_5 @ ( suc @ ( suc @ zero_zero_nat ) ) ) @ ( power_power_int @ Y_4 @ ( suc @ ( suc @ zero_zero_nat ) ) ) )
= ( times_times_int @ ( minus_minus_int @ X_5 @ Y_4 ) @ ( plus_plus_int @ X_5 @ Y_4 ) ) ) ).
thf(fact_3194_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1621563631at_int @ ( minus_minus_nat @ X @ Y ) ) )
<=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
thf(fact_3195_numeral__2__eq__2,axiom,
( ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_3196_semiring__norm_I115_J,axiom,
( ( suc @ ( suc @ zero_zero_nat ) )
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3197_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
thf(fact_3198_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( suc @ ( suc @ N ) ) ) ).
thf(fact_3199_Suc__diff__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( number_number_of_nat @ pls ) @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ ( number_number_of_nat @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3200_one__less__nat__eq,axiom,
! [Z_1: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat_1 @ Z_1 ) )
<=> ( ord_less_int @ one_one_int @ Z_1 ) ) ).
thf(fact_3201_expand__Suc,axiom,
! [V: int] :
( ( ord_less_nat @ zero_zero_nat @ ( number_number_of_nat @ V ) )
=> ( ( number_number_of_nat @ V )
= ( suc @ ( minus_minus_nat @ ( number_number_of_nat @ V ) @ one_one_nat ) ) ) ) ).
thf(fact_3202_div2__Suc__Suc,axiom,
! [M: nat] :
( ( div_div_nat @ ( suc @ ( suc @ M ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( suc @ ( div_div_nat @ M @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3203_mod2__Suc__Suc,axiom,
! [M: nat] :
( ( div_mod_nat @ ( suc @ ( suc @ M ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( div_mod_nat @ M @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3204_Suc__nat__eq__nat__zadd1,axiom,
! [Z_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z_1 )
=> ( ( suc @ ( nat_1 @ Z_1 ) )
= ( nat_1 @ ( plus_plus_int @ one_one_int @ Z_1 ) ) ) ) ).
thf(fact_3205_Suc__div__eq__add3__div__number__of,axiom,
! [M: nat,V: int] :
( ( div_div_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( number_number_of_nat @ V ) )
= ( div_div_nat @ ( plus_plus_nat @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) @ M ) @ ( number_number_of_nat @ V ) ) ) ).
thf(fact_3206_Suc__div__eq__add3__div,axiom,
! [M: nat,N: nat] :
( ( div_div_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
= ( div_div_nat @ ( plus_plus_nat @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) @ M ) @ N ) ) ).
thf(fact_3207_div__Suc__eq__div__add3,axiom,
! [M: nat,N: nat] :
( ( div_div_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( div_div_nat @ M @ ( plus_plus_nat @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) @ N ) ) ) ).
thf(fact_3208_Suc__mod__eq__add3__mod__number__of,axiom,
! [M: nat,V: int] :
( ( div_mod_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( number_number_of_nat @ V ) )
= ( div_mod_nat @ ( plus_plus_nat @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) @ M ) @ ( number_number_of_nat @ V ) ) ) ).
thf(fact_3209_Suc__mod__eq__add3__mod,axiom,
! [M: nat,N: nat] :
( ( div_mod_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
= ( div_mod_nat @ ( plus_plus_nat @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) @ M ) @ N ) ) ).
thf(fact_3210_mod__Suc__eq__mod__add3,axiom,
! [M: nat,N: nat] :
( ( div_mod_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( div_mod_nat @ M @ ( plus_plus_nat @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) @ N ) ) ) ).
thf(fact_3211_split__div__lemma,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q ) ) ) )
<=> ( Q
= ( div_div_nat @ M @ N ) ) ) ) ).
thf(fact_3212_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( div_div_nat @ M @ N ) )
<=> ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q_2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q_2 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q_2 ) ) )
& ( P @ Q_2 ) ) ) ) ).
thf(fact_3213_le__div__geq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( div_div_nat @ M @ N )
= ( suc @ ( div_div_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
thf(fact_3214_of__nat__less__two__power,axiom,
! [N_7: nat] : ( ord_less_rat @ ( semiri151668891at_rat @ N_7 ) @ ( power_power_rat @ ( number_number_of_rat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_7 ) ) ).
thf(fact_3215_of__nat__less__two__power,axiom,
! [N_7: nat] : ( ord_less_real @ ( semiri132038758t_real @ N_7 ) @ ( power_power_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_7 ) ) ).
thf(fact_3216_of__nat__less__two__power,axiom,
! [N_7: nat] : ( ord_less_int @ ( semiri1621563631at_int @ N_7 ) @ ( power_power_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_7 ) ) ).
thf(fact_3217_of__nat__number__of__lemma,axiom,
! [V_2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ V_2 ) )
=> ( ( semiri151668891at_rat @ ( number_number_of_nat @ V_2 ) )
= ( number_number_of_rat @ V_2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ V_2 ) )
=> ( ( semiri151668891at_rat @ ( number_number_of_nat @ V_2 ) )
= zero_zero_rat ) ) ) ).
thf(fact_3218_of__nat__number__of__lemma,axiom,
! [V_2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ V_2 ) )
=> ( ( semiri2020571505omplex @ ( number_number_of_nat @ V_2 ) )
= ( number528085621omplex @ V_2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ V_2 ) )
=> ( ( semiri2020571505omplex @ ( number_number_of_nat @ V_2 ) )
= zero_zero_complex ) ) ) ).
thf(fact_3219_of__nat__number__of__lemma,axiom,
! [V_2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ V_2 ) )
=> ( ( semiri132038758t_real @ ( number_number_of_nat @ V_2 ) )
= ( number267125858f_real @ V_2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ V_2 ) )
=> ( ( semiri132038758t_real @ ( number_number_of_nat @ V_2 ) )
= zero_zero_real ) ) ) ).
thf(fact_3220_of__nat__number__of__lemma,axiom,
! [V_2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ V_2 ) )
=> ( ( semiri1621563631at_int @ ( number_number_of_nat @ V_2 ) )
= ( number_number_of_int @ V_2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ V_2 ) )
=> ( ( semiri1621563631at_int @ ( number_number_of_nat @ V_2 ) )
= zero_zero_int ) ) ) ).
thf(fact_3221_even__equiv__def,axiom,
! [X: int] :
( ( even_odd_even_int @ X )
<=> ? [Y_1: int] :
( X
= ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Y_1 ) ) ) ).
thf(fact_3222_int__even__iff__2__dvd,axiom,
! [X: int] :
( ( even_odd_even_int @ X )
<=> ( dvd_dvd_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ X ) ) ).
thf(fact_3223_nat__dvd__iff,axiom,
! [Z_1: int,M: nat] :
( ( dvd_dvd_nat @ ( nat_1 @ Z_1 ) @ M )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ Z_1 )
=> ( dvd_dvd_int @ Z_1 @ ( semiri1621563631at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z_1 )
=> ( M = zero_zero_nat ) ) ) ) ).
thf(fact_3224_zEvenI,axiom,
! [X: int,K_1: int] :
( ( X
= ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_1 ) )
=> ( member_int @ X @ zEven ) ) ).
thf(fact_3225_less__2__cases,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
=> ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
thf(fact_3226_nat__2,axiom,
( ( nat_1 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_3227_even__def,axiom,
! [X: int] :
( ( even_odd_even_int @ X )
<=> ( ( div_mod_int @ X @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_int ) ) ).
thf(fact_3228_two__times__even__div__two,axiom,
! [X: int] :
( ( even_odd_even_int @ X )
=> ( ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( div_div_int @ X @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= X ) ) ).
thf(fact_3229_one__code__numeral__code,axiom,
( one_on1645066479umeral
= ( number1443263063umeral @ ( bit1 @ pls ) ) ) ).
thf(fact_3230_power__odd__eq,axiom,
! [A_6: nat,N_6: nat] :
( ( power_power_nat @ A_6 @ ( suc @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_6 ) ) )
= ( times_times_nat @ A_6 @ ( power_power_nat @ ( power_power_nat @ A_6 @ N_6 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3231_power__odd__eq,axiom,
! [A_6: int,N_6: nat] :
( ( power_power_int @ A_6 @ ( suc @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_6 ) ) )
= ( times_times_int @ A_6 @ ( power_power_int @ ( power_power_int @ A_6 @ N_6 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3232_tsub__def,axiom,
! [Y: int,X: int] :
( ( ( ord_less_eq_int @ Y @ X )
=> ( ( nat_tsub @ X @ Y )
= ( minus_minus_int @ X @ Y ) ) )
& ( ~ ( ord_less_eq_int @ Y @ X )
=> ( ( nat_tsub @ X @ Y )
= zero_zero_int ) ) ) ).
thf(fact_3233_lemma__Suc__Suc__4n__diff__2,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( suc @ ( suc @ ( minus_minus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ N ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
= ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ N ) ) ) ).
thf(fact_3234_Suc__Suc__mult__two__diff__two,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( suc @ ( suc @ ( minus_minus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
= ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) ) ).
thf(fact_3235_even__div__2__prop1,axiom,
! [X: int] :
( ( member_int @ X @ zEven )
=> ( ( div_mod_int @ X @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_int ) ) ).
thf(fact_3236_even__div__2__l,axiom,
! [X: int,Y: int] :
( ( member_int @ Y @ zEven )
=> ( ( ord_less_int @ X @ Y )
=> ( ord_less_int @ ( div_div_int @ X @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( div_div_int @ Y @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3237_int__power__div__base,axiom,
! [K_1: int,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_int @ zero_zero_int @ K_1 )
=> ( ( div_div_int @ ( power_power_int @ K_1 @ M ) @ K_1 )
= ( power_power_int @ K_1 @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
thf(fact_3238_even__prod__div__2,axiom,
! [Y: int,X: int] :
( ( member_int @ X @ zEven )
=> ( ( div_div_int @ ( times_times_int @ X @ Y ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_int @ ( div_div_int @ X @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ Y ) ) ) ).
thf(fact_3239_even__div__2__prop2,axiom,
! [X: int] :
( ( member_int @ X @ zEven )
=> ( ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( div_div_int @ X @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= X ) ) ).
thf(fact_3240_even__sum__div__2,axiom,
! [Y: int,X: int] :
( ( member_int @ X @ zEven )
=> ( ( member_int @ Y @ zEven )
=> ( ( div_div_int @ ( plus_plus_int @ X @ Y ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ ( div_div_int @ X @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( div_div_int @ Y @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ).
thf(fact_3241_one__div__nat__number__of,axiom,
! [V_1: int] :
( ( div_div_nat @ ( suc @ zero_zero_nat ) @ ( number_number_of_nat @ V_1 ) )
= ( nat_1 @ ( div_div_int @ one_one_int @ ( number_number_of_int @ V_1 ) ) ) ) ).
thf(fact_3242_neg__one__power__parity,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( member_int @ X @ zEven )
<=> ( member_int @ Y @ zEven ) )
=> ( ( power_power_int @ ( number_number_of_int @ min ) @ ( nat_1 @ X ) )
= ( power_power_int @ ( number_number_of_int @ min ) @ ( nat_1 @ Y ) ) ) ) ) ) ).
thf(fact_3243_odd__equiv__def,axiom,
! [X: int] :
( ~ ( even_odd_even_int @ X )
<=> ? [Y_1: int] :
( X
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Y_1 ) @ one_one_int ) ) ) ).
thf(fact_3244_odd__plus__one__div__two,axiom,
! [X: int] :
( ~ ( even_odd_even_int @ X )
=> ( ( div_div_int @ ( plus_plus_int @ X @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ ( div_div_int @ X @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_int ) ) ) ).
thf(fact_3245_even__plus__one__div__two,axiom,
! [X: int] :
( ( even_odd_even_int @ X )
=> ( ( div_div_int @ ( plus_plus_int @ X @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( div_div_int @ X @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3246_Suc__mult__two__diff__one,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( suc @ ( minus_minus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) @ one_one_nat ) )
= ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) ) ).
thf(fact_3247_inv__is__inv__aux,axiom,
! [M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( suc @ ( nat_1 @ ( minus_minus_int @ M @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
= ( nat_1 @ ( minus_minus_int @ M @ one_one_int ) ) ) ) ).
thf(fact_3248_zEvenE,axiom,
! [X: int] :
( ( member_int @ X @ zEven )
=> ~ ! [K: int] :
( X
!= ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ K ) ) ) ).
thf(fact_3249_mod__induct__0,axiom,
! [I: nat,P: nat > $o,P_3: nat] :
( ! [I_1: nat] :
( ( ord_less_nat @ I_1 @ P_3 )
=> ( ( P @ I_1 )
=> ( P @ ( div_mod_nat @ ( suc @ I_1 ) @ P_3 ) ) ) )
=> ( ( P @ I )
=> ( ( ord_less_nat @ I @ P_3 )
=> ( P @ zero_zero_nat ) ) ) ) ).
thf(fact_3250_realpow__pos__nth2,axiom,
! [N: nat,A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ ( suc @ N ) )
= A ) ) ) ).
thf(fact_3251_zero__less__imp__eq__int,axiom,
! [K_1: int] :
( ( ord_less_int @ zero_zero_int @ K_1 )
=> ? [N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_1 )
& ( K_1
= ( semiri1621563631at_int @ N_1 ) ) ) ) ).
thf(fact_3252_mod__induct,axiom,
! [J: nat,I: nat,P: nat > $o,P_3: nat] :
( ! [I_1: nat] :
( ( ord_less_nat @ I_1 @ P_3 )
=> ( ( P @ I_1 )
=> ( P @ ( div_mod_nat @ ( suc @ I_1 ) @ P_3 ) ) ) )
=> ( ( P @ I )
=> ( ( ord_less_nat @ I @ P_3 )
=> ( ( ord_less_nat @ J @ P_3 )
=> ( P @ J ) ) ) ) ) ).
thf(fact_3253_dvd_Olift__Suc__mono__le,axiom,
! [N: nat,N_5: nat,F: nat > nat] :
( ! [N_1: nat] : ( dvd_dvd_nat @ ( F @ N_1 ) @ ( F @ ( suc @ N_1 ) ) )
=> ( ( ord_less_eq_nat @ N @ N_5 )
=> ( dvd_dvd_nat @ ( F @ N ) @ ( F @ N_5 ) ) ) ) ).
thf(fact_3254_inc__induct,axiom,
! [P: nat > $o,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [I_1: nat] :
( ( ord_less_nat @ I_1 @ J )
=> ( ( P @ ( suc @ I_1 ) )
=> ( P @ I_1 ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_3255_int__if__cong,axiom,
! [X: nat,Y: nat,P: $o] :
( ( P
=> ( ( semiri1621563631at_int @ X )
= ( semiri1621563631at_int @ ( if_nat @ P @ X @ Y ) ) ) )
& ( ~ P
=> ( ( semiri1621563631at_int @ Y )
= ( semiri1621563631at_int @ ( if_nat @ P @ X @ Y ) ) ) ) ) ).
thf(fact_3256_Nat__Transfer_Otransfer__int__nat__relations_I1_J,axiom,
! [X: nat,Y: nat] :
( ( ( semiri1621563631at_int @ X )
= ( semiri1621563631at_int @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_3257_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1621563631at_int @ M )
= ( semiri1621563631at_int @ N ) )
<=> ( M = N ) ) ).
thf(fact_3258_nat__of__aux__code,axiom,
! [N: nat,I: code_code_numeral] :
( ( ( I = zero_z126310315umeral )
=> ( ( code_nat_of_aux @ I @ N )
= N ) )
& ( ( I != zero_z126310315umeral )
=> ( ( code_nat_of_aux @ I @ N )
= ( code_nat_of_aux @ ( minus_1690775515umeral @ I @ one_on1645066479umeral ) @ ( suc @ N ) ) ) ) ) ).
thf(fact_3259_nonneg__eq__int,axiom,
! [Z_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z_1 )
=> ~ ! [M_2: nat] :
( Z_1
!= ( semiri1621563631at_int @ M_2 ) ) ) ).
thf(fact_3260_nonneg__int__cases,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ~ ! [N_1: nat] :
( K_1
!= ( semiri1621563631at_int @ N_1 ) ) ) ).
thf(fact_3261_zero__le__imp__eq__int,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ? [N_1: nat] :
( K_1
= ( semiri1621563631at_int @ N_1 ) ) ) ).
thf(fact_3262_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K ) ) ) ) ).
thf(fact_3263_dvd_Olift__Suc__mono__less,axiom,
! [N: nat,N_5: nat,F: nat > nat] :
( ! [N_1: nat] :
( ( dvd_dvd_nat @ ( F @ N_1 ) @ ( F @ ( suc @ N_1 ) ) )
& ~ ( dvd_dvd_nat @ ( F @ ( suc @ N_1 ) ) @ ( F @ N_1 ) ) )
=> ( ( ord_less_nat @ N @ N_5 )
=> ( ( dvd_dvd_nat @ ( F @ N ) @ ( F @ N_5 ) )
& ~ ( dvd_dvd_nat @ ( F @ N_5 ) @ ( F @ N ) ) ) ) ) ).
thf(fact_3264_dvd_Olift__Suc__mono__less__iff,axiom,
! [N: nat,M: nat,F: nat > nat] :
( ! [N_1: nat] :
( ( dvd_dvd_nat @ ( F @ N_1 ) @ ( F @ ( suc @ N_1 ) ) )
& ~ ( dvd_dvd_nat @ ( F @ ( suc @ N_1 ) ) @ ( F @ N_1 ) ) )
=> ( ( ( dvd_dvd_nat @ ( F @ N ) @ ( F @ M ) )
& ~ ( dvd_dvd_nat @ ( F @ M ) @ ( F @ N ) ) )
<=> ( ord_less_nat @ N @ M ) ) ) ).
thf(fact_3265_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M_2: nat] :
( N
= ( suc @ M_2 ) ) ) ).
thf(fact_3266_one__mod__nat__number__of,axiom,
! [V_1: int] :
( ( ( nat_neg @ ( number_number_of_int @ V_1 ) )
=> ( ( div_mod_nat @ ( suc @ zero_zero_nat ) @ ( number_number_of_nat @ V_1 ) )
= ( suc @ zero_zero_nat ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V_1 ) )
=> ( ( div_mod_nat @ ( suc @ zero_zero_nat ) @ ( number_number_of_nat @ V_1 ) )
= ( nat_1 @ ( div_mod_int @ one_one_int @ ( number_number_of_int @ V_1 ) ) ) ) ) ) ).
thf(fact_3267_not__neg__0,axiom,
~ ( nat_neg @ zero_zero_int ) ).
thf(fact_3268_not__neg__1,axiom,
~ ( nat_neg @ one_one_int ) ).
thf(fact_3269_not__neg__int,axiom,
! [N: nat] :
~ ( nat_neg @ ( semiri1621563631at_int @ N ) ) ).
thf(fact_3270_neg__def,axiom,
! [Z_4: int] :
( ( nat_neg @ Z_4 )
<=> ( ord_less_int @ Z_4 @ zero_zero_int ) ) ).
thf(fact_3271_not__neg__eq__ge__0,axiom,
! [X: int] :
( ~ ( nat_neg @ X )
<=> ( ord_less_eq_int @ zero_zero_int @ X ) ) ).
thf(fact_3272_neg__number__of__Bit1,axiom,
! [W: int] :
( ( nat_neg @ ( number_number_of_int @ ( bit1 @ W ) ) )
<=> ( nat_neg @ ( number_number_of_int @ W ) ) ) ).
thf(fact_3273_not__neg__number__of__Pls,axiom,
~ ( nat_neg @ ( number_number_of_int @ pls ) ) ).
thf(fact_3274_neg__number__of__Bit0,axiom,
! [W: int] :
( ( nat_neg @ ( number_number_of_int @ ( bit0 @ W ) ) )
<=> ( nat_neg @ ( number_number_of_int @ W ) ) ) ).
thf(fact_3275_neg__nat,axiom,
! [Z_1: int] :
( ( nat_neg @ Z_1 )
=> ( ( nat_1 @ Z_1 )
= zero_zero_nat ) ) ).
thf(fact_3276_neg__number__of__Min,axiom,
nat_neg @ ( number_number_of_int @ min ) ).
thf(fact_3277_not__neg__nat,axiom,
! [Z_1: int] :
( ~ ( nat_neg @ Z_1 )
=> ( ( semiri1621563631at_int @ ( nat_1 @ Z_1 ) )
= Z_1 ) ) ).
thf(fact_3278_neg__imp__number__of__eq__0,axiom,
! [V: int] :
( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( number_number_of_nat @ V )
= zero_zero_nat ) ) ).
thf(fact_3279_eq__nat__number__of,axiom,
! [V: int,V_1: int] :
( ( ( number_number_of_nat @ V )
= ( number_number_of_nat @ V_1 ) )
<=> ( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ord_less_eq_int @ ( number_number_of_int @ V_1 ) @ zero_zero_int ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( ( nat_neg @ ( number_number_of_int @ V_1 ) )
=> ( ( number_number_of_int @ V )
= zero_zero_int ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V_1 ) )
=> ( V = V_1 ) ) ) ) ) ) ).
thf(fact_3280_nat__number__of__add__left,axiom,
! [V_1: int,K_1: nat,V: int] :
( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K_1 ) )
= ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K_1 ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( ( nat_neg @ ( number_number_of_int @ V_1 ) )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K_1 ) )
= ( plus_plus_nat @ ( number_number_of_nat @ V ) @ K_1 ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V_1 ) )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K_1 ) )
= ( plus_plus_nat @ ( number_number_of_nat @ ( plus_plus_int @ V @ V_1 ) ) @ K_1 ) ) ) ) ) ) ).
thf(fact_3281_int__nat__number__of,axiom,
! [V: int] :
( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( semiri1621563631at_int @ ( number_number_of_nat @ V ) )
= zero_zero_int ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( semiri1621563631at_int @ ( number_number_of_nat @ V ) )
= ( number_number_of_int @ V ) ) ) ) ).
thf(fact_3282_div__nat__number__of,axiom,
! [V_1: int,V: int] :
( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( div_div_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
= zero_zero_nat ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( div_div_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
= ( nat_1 @ ( div_div_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ V_1 ) ) ) ) ) ) ).
thf(fact_3283_power__nat__number__of__number__of,axiom,
! [W: int,V: int] :
( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( power_power_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ W ) )
= ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ W ) ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( power_power_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ W ) )
= ( nat_1 @ ( power_power_int @ ( number_number_of_int @ V ) @ ( number_number_of_nat @ W ) ) ) ) ) ) ).
thf(fact_3284_power__nat__number__of,axiom,
! [N: nat,V: int] :
( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( power_power_nat @ ( number_number_of_nat @ V ) @ N )
= ( power_power_nat @ zero_zero_nat @ N ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( power_power_nat @ ( number_number_of_nat @ V ) @ N )
= ( nat_1 @ ( power_power_int @ ( number_number_of_int @ V ) @ N ) ) ) ) ) ).
thf(fact_3285_mod__nat__number__of,axiom,
! [V_1: int,V: int] :
( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( div_mod_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
= zero_zero_nat ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( ( nat_neg @ ( number_number_of_int @ V_1 ) )
=> ( ( div_mod_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
= ( number_number_of_nat @ V ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V_1 ) )
=> ( ( div_mod_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
= ( nat_1 @ ( div_mod_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ V_1 ) ) ) ) ) ) ) ) ).
thf(fact_3286_Suc__nat__number__of__add,axiom,
! [N: nat,V: int] :
( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( suc @ ( plus_plus_nat @ ( number_number_of_nat @ V ) @ N ) )
= ( plus_plus_nat @ one_one_nat @ N ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( suc @ ( plus_plus_nat @ ( number_number_of_nat @ V ) @ N ) )
= ( plus_plus_nat @ ( number_number_of_nat @ ( succ @ V ) ) @ N ) ) ) ) ).
thf(fact_3287_SRStar__card,axiom,
! [P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( semiri1621563631at_int @ ( finite_card_int @ ( sRStar @ P_3 ) ) )
= ( minus_minus_int @ P_3 @ one_one_int ) ) ) ).
thf(fact_3288_int__diff__cases,axiom,
! [Z_1: int] :
~ ! [M_2: nat,N_1: nat] :
( Z_1
!= ( minus_minus_int @ ( semiri1621563631at_int @ M_2 ) @ ( semiri1621563631at_int @ N_1 ) ) ) ).
thf(fact_3289_zero__induct__lemma,axiom,
! [I: nat,P: nat > $o,K_1: nat] :
( ( P @ K_1 )
=> ( ! [N_1: nat] :
( ( P @ ( suc @ N_1 ) )
=> ( P @ N_1 ) )
=> ( P @ ( minus_minus_nat @ K_1 @ I ) ) ) ) ).
thf(fact_3290_Suc__le__D,axiom,
! [N: nat,M_5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M_5 )
=> ? [M_2: nat] :
( M_5
= ( suc @ M_2 ) ) ) ).
thf(fact_3291_succ__Pls,axiom,
( ( succ @ pls )
= ( bit1 @ pls ) ) ).
thf(fact_3292_succ__Bit1,axiom,
! [K_1: int] :
( ( succ @ ( bit1 @ K_1 ) )
= ( bit0 @ ( succ @ K_1 ) ) ) ).
thf(fact_3293_succ__Bit0,axiom,
! [K_1: int] :
( ( succ @ ( bit0 @ K_1 ) )
= ( bit1 @ K_1 ) ) ).
thf(fact_3294_succ__def,axiom,
! [K_1: int] :
( ( succ @ K_1 )
= ( plus_plus_int @ K_1 @ one_one_int ) ) ).
thf(fact_3295_succ__Min,axiom,
( ( succ @ min )
= pls ) ).
thf(fact_3296_diff__bin__simps_I2_J,axiom,
! [K_1: int] :
( ( minus_minus_int @ K_1 @ min )
= ( succ @ K_1 ) ) ).
thf(fact_3297_add__Bit1__Bit1,axiom,
! [K_1: int,L: int] :
( ( plus_plus_int @ ( bit1 @ K_1 ) @ ( bit1 @ L ) )
= ( bit0 @ ( plus_plus_int @ K_1 @ ( succ @ L ) ) ) ) ).
thf(fact_3298_nat__number__of__add__1,axiom,
! [V: int] :
( ( ( ord_less_int @ V @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ one_one_nat )
= one_one_nat ) )
& ( ~ ( ord_less_int @ V @ pls )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ one_one_nat )
= ( number_number_of_nat @ ( succ @ V ) ) ) ) ) ).
thf(fact_3299_nat__1__add__number__of,axiom,
! [V: int] :
( ( ( ord_less_int @ V @ pls )
=> ( ( plus_plus_nat @ one_one_nat @ ( number_number_of_nat @ V ) )
= one_one_nat ) )
& ( ~ ( ord_less_int @ V @ pls )
=> ( ( plus_plus_nat @ one_one_nat @ ( number_number_of_nat @ V ) )
= ( number_number_of_nat @ ( succ @ V ) ) ) ) ) ).
thf(fact_3300_Suc__nat__number__of,axiom,
! [V: int] :
( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( suc @ ( number_number_of_nat @ V ) )
= one_one_nat ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( suc @ ( number_number_of_nat @ V ) )
= ( number_number_of_nat @ ( succ @ V ) ) ) ) ) ).
thf(fact_3301_MultInvPair__card__two,axiom,
! [J: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A )
=> ( ~ ( zcong @ J @ zero_zero_int @ P_3 )
=> ( ( finite_card_int @ ( multInvPair @ A @ P_3 @ J ) )
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_3302_SetS__elems__card,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A )
=> ! [X_1: int > $o] :
( ( member_int_o @ X_1 @ ( setS @ A @ P_3 ) )
=> ( ( finite_card_int @ X_1 )
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_3303_lessE,axiom,
! [I: nat,K_1: nat] :
( ( ord_less_nat @ I @ K_1 )
=> ( ( K_1
!= ( suc @ I ) )
=> ~ ! [J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( K_1
!= ( suc @ J_1 ) ) ) ) ) ).
thf(fact_3304_Suc__lessE,axiom,
! [I: nat,K_1: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K_1 )
=> ~ ! [J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( K_1
!= ( suc @ J_1 ) ) ) ) ).
thf(fact_3305_SetS__card,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A )
=> ( ( semiri1621563631at_int @ ( finite_card_int_o @ ( setS @ A @ P_3 ) ) )
= ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_3306_MultInvPair__prop2,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ( comple1092985777_int_o @ ( setS @ A @ P_3 ) )
= ( sRStar @ P_3 ) ) ) ) ) ).
thf(fact_3307_real__lbound__gt__zero,axiom,
! [D2: real,D1: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
thf(fact_3308_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X: real,Y: real,Xa_1: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Ya @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ).
thf(fact_3309_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat_1: nat] :
( Y
!= ( suc @ Nat_1 ) ) ) ).
thf(fact_3310_zero__induct,axiom,
! [P: nat > $o,K_1: nat] :
( ( P @ K_1 )
=> ( ! [N_1: nat] :
( ( P @ ( suc @ N_1 ) )
=> ( P @ N_1 ) )
=> ( P @ zero_zero_nat ) ) ) ).
thf(fact_3311_real__sqrt__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( sqrt @ X )
= ( sqrt @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_3312_real__sqrt__le__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
thf(fact_3313_real__sqrt__le__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
<=> ( ord_less_eq_real @ X @ Y ) ) ).
thf(fact_3314_real__sqrt__eq__0__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= zero_zero_real )
<=> ( X = zero_zero_real ) ) ).
thf(fact_3315_real__sqrt__zero,axiom,
( ( sqrt @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_3316_real__sqrt__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
thf(fact_3317_real__sqrt__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
<=> ( ord_less_real @ X @ Y ) ) ).
thf(fact_3318_real__sqrt__mult,axiom,
! [X: real,Y: real] :
( ( sqrt @ ( times_times_real @ X @ Y ) )
= ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
thf(fact_3319_real__sqrt__power,axiom,
! [X: real,K_1: nat] :
( ( sqrt @ ( power_power_real @ X @ K_1 ) )
= ( power_power_real @ ( sqrt @ X ) @ K_1 ) ) ).
thf(fact_3320_real__sqrt__eq__1__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= one_one_real )
<=> ( X = one_one_real ) ) ).
thf(fact_3321_real__sqrt__one,axiom,
( ( sqrt @ one_one_real )
= one_one_real ) ).
thf(fact_3322_real__sqrt__eq__zero__cancel,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ( sqrt @ X )
= zero_zero_real )
=> ( X = zero_zero_real ) ) ) ).
thf(fact_3323_real__sqrt__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
thf(fact_3324_real__sqrt__le__0__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
<=> ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
thf(fact_3325_real__sqrt__ge__0__iff,axiom,
! [Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y ) )
<=> ( ord_less_eq_real @ zero_zero_real @ Y ) ) ).
thf(fact_3326_real__sqrt__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
thf(fact_3327_real__sqrt__not__eq__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( sqrt @ X )
!= zero_zero_real ) ) ).
thf(fact_3328_real__sqrt__lt__0__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
<=> ( ord_less_real @ X @ zero_zero_real ) ) ).
thf(fact_3329_real__sqrt__gt__0__iff,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
<=> ( ord_less_real @ zero_zero_real @ Y ) ) ).
thf(fact_3330_real__sqrt__ge__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).
thf(fact_3331_real__sqrt__le__1__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
<=> ( ord_less_eq_real @ X @ one_one_real ) ) ).
thf(fact_3332_real__sqrt__ge__1__iff,axiom,
! [Y: real] :
( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y ) )
<=> ( ord_less_eq_real @ one_one_real @ Y ) ) ).
thf(fact_3333_real__sqrt__lt__1__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
<=> ( ord_less_real @ X @ one_one_real ) ) ).
thf(fact_3334_real__sqrt__gt__1__iff,axiom,
! [Y: real] :
( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
<=> ( ord_less_real @ one_one_real @ Y ) ) ).
thf(fact_3335_le__real__sqrt__sumsq,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ) ).
thf(fact_3336_real__sqrt__mult__self__sum__ge__zero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ) ).
thf(fact_3337_real__sqrt__two__ge__zero,axiom,
ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3338_real__sqrt__two__gt__zero,axiom,
ord_less_real @ zero_zero_real @ ( sqrt @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3339_real__sqrt__two__gt__one,axiom,
ord_less_real @ one_one_real @ ( sqrt @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3340_real__sqrt__unique,axiom,
! [Y: real,X: real] :
( ( ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( sqrt @ X )
= Y ) ) ) ).
thf(fact_3341_real__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( sqrt @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= X ) ) ).
thf(fact_3342_real__sqrt__pow2__iff,axiom,
! [X: real] :
( ( ( power_power_real @ ( sqrt @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= X )
<=> ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
thf(fact_3343_real__sqrt__pow2__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( sqrt @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3344_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( plus_plus_real @ B @ D ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ D @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ).
thf(fact_3345_real__sqrt__sum__squares__ge2,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3346_real__sqrt__sum__squares__ge1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3347_real__sqrt__sum__squares__eq__cancel2,axiom,
! [X: real,Y: real] :
( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
= Y )
=> ( X = zero_zero_real ) ) ).
thf(fact_3348_real__sqrt__sum__squares__eq__cancel,axiom,
! [X: real,Y: real] :
( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
= X )
=> ( Y = zero_zero_real ) ) ).
thf(fact_3349_real__sqrt__sum__squares__mult__squared__eq,axiom,
! [X: real,Y: real,Xa_1: real,Ya: real] :
( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Ya @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Ya @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3350_real__sqrt__sum__squares__ge__zero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3351_lemma__sqrt__hcomplex__capprox,axiom,
! [Y: real,X: real,U: real] :
( ( ord_less_real @ zero_zero_real @ U )
=> ( ( ord_less_real @ X @ ( inverse_divide_real @ U @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_real @ Y @ ( inverse_divide_real @ U @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ U ) ) ) ) ) ) ).
thf(fact_3352_Union__SetS__finite,axiom,
! [A: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( finite_finite_int @ ( comple1092985777_int_o @ ( setS @ A @ P_3 ) ) ) ) ).
thf(fact_3353_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M_2: nat] :
( N
= ( suc @ M_2 ) ) ) ).
thf(fact_3354_nat__induct,axiom,
! [N: nat,P: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ! [N_1: nat] :
( ( P @ N_1 )
=> ( P @ ( suc @ N_1 ) ) )
=> ( P @ N ) ) ) ).
thf(fact_3355_real__divide__square__eq,axiom,
! [R_1: real,A: real] :
( ( inverse_divide_real @ ( times_times_real @ R_1 @ A ) @ ( times_times_real @ R_1 @ R_1 ) )
= ( inverse_divide_real @ A @ R_1 ) ) ).
thf(fact_3356_real__sqrt__divide,axiom,
! [X: real,Y: real] :
( ( sqrt @ ( inverse_divide_real @ X @ Y ) )
= ( inverse_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
thf(fact_3357_wset__fin,axiom,
! [A: int,P_3: int] : ( finite_finite_int @ ( wset @ A @ P_3 ) ) ).
thf(fact_3358_SetS__elems__finite,axiom,
! [A: int,P_3: int,X_1: int > $o] :
( ( member_int_o @ X_1 @ ( setS @ A @ P_3 ) )
=> ( finite_finite_int @ X_1 ) ) ).
thf(fact_3359_d22set__fin,axiom,
! [A: int] : ( finite_finite_int @ ( d22set @ A ) ) ).
thf(fact_3360_real__0__le__divide__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( inverse_divide_real @ X @ Y ) )
<=> ( ( ( ord_less_eq_real @ X @ zero_zero_real )
| ( ord_less_eq_real @ zero_zero_real @ Y ) )
& ( ( ord_less_eq_real @ zero_zero_real @ X )
| ( ord_less_eq_real @ Y @ zero_zero_real ) ) ) ) ).
thf(fact_3361_real__less__half__sum,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ X @ ( inverse_divide_real @ ( plus_plus_real @ X @ Y ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3362_real__gt__half__sum,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( inverse_divide_real @ ( plus_plus_real @ X @ Y ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ Y ) ) ).
thf(fact_3363_SRStar__finite,axiom,
! [P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( finite_finite_int @ ( sRStar @ P_3 ) ) ) ).
thf(fact_3364_lemma__real__divide__sqrt__less,axiom,
! [U: real] :
( ( ord_less_real @ zero_zero_real @ U )
=> ( ord_less_real @ ( inverse_divide_real @ U @ ( sqrt @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ U ) ) ).
thf(fact_3365_real__average__minus__first,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( inverse_divide_real @ ( plus_plus_real @ A @ B ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A )
= ( inverse_divide_real @ ( minus_minus_real @ B @ A ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3366_real__average__minus__second,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( inverse_divide_real @ ( plus_plus_real @ B @ A ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A )
= ( inverse_divide_real @ ( minus_minus_real @ B @ A ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3367_real__sum__of__halves,axiom,
! [X: real] :
( ( plus_plus_real @ ( inverse_divide_real @ X @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( inverse_divide_real @ X @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= X ) ).
thf(fact_3368_eq__divide__2__times__iff,axiom,
! [X: real,Y: real,Z_1: real] :
( ( X
= ( inverse_divide_real @ Y @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_1 ) ) )
<=> ( ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X )
= ( inverse_divide_real @ Y @ Z_1 ) ) ) ).
thf(fact_3369_complex__mod__triangle__ineq2,axiom,
! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( norm_norm_complex @ ( plus_plus_complex @ B @ A ) ) @ ( norm_norm_complex @ B ) ) @ ( norm_norm_complex @ A ) ) ).
thf(fact_3370_SetS__finite,axiom,
! [A: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( finite_finite_int_o @ ( setS @ A @ P_3 ) ) ) ).
thf(fact_3371_le__Suc__ex__iff,axiom,
! [K_1: nat,L: nat] :
( ( ord_less_eq_nat @ K_1 @ L )
<=> ? [N_1: nat] :
( L
= ( plus_plus_nat @ K_1 @ N_1 ) ) ) ).
thf(fact_3372_lemma__MVT,axiom,
! [F: real > real,A: real,B: real] :
( ( minus_minus_real @ ( F @ A ) @ ( times_times_real @ ( inverse_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) ) @ A ) )
= ( minus_minus_real @ ( F @ B ) @ ( times_times_real @ ( inverse_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) ) @ B ) ) ) ).
thf(fact_3373_arctan__half,axiom,
! [X: real] :
( ( arctan @ X )
= ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( arctan @ ( inverse_divide_real @ X @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ) ) ).
thf(fact_3374_lemma__f__mono__add,axiom,
! [M: nat,No: nat,F: nat > real] :
( ! [N_1: nat] : ( ord_less_eq_real @ ( F @ N_1 ) @ ( F @ ( suc @ N_1 ) ) )
=> ( ord_less_eq_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ M @ No ) ) ) ) ).
thf(fact_3375_arctan__monotone_H,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
thf(fact_3376_arctan__zero__zero,axiom,
( ( arctan @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_3377_arctan__monotone,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
thf(fact_3378_natceiling__add__number__of,axiom,
! [X: real,N: int] :
( ~ ( nat_neg @ ( number_number_of_int @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( natceiling @ ( plus_plus_real @ X @ ( number267125858f_real @ N ) ) )
= ( plus_plus_nat @ ( natceiling @ X ) @ ( number_number_of_nat @ N ) ) ) ) ) ).
thf(fact_3379_real__sqrt__sum__squares__less,axiom,
! [Y: real,X: real,U: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ ( inverse_divide_real @ U @ ( sqrt @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
=> ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( inverse_divide_real @ U @ ( sqrt @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
=> ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ U ) ) ) ).
thf(fact_3380_cos__x__y__le__one,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( inverse_divide_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) @ one_one_real ) ).
thf(fact_3381_natfloor__add__number__of,axiom,
! [X: real,N: int] :
( ~ ( nat_neg @ ( number_number_of_int @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( natfloor @ ( plus_plus_real @ X @ ( number267125858f_real @ N ) ) )
= ( plus_plus_nat @ ( natfloor @ X ) @ ( number_number_of_nat @ N ) ) ) ) ) ).
thf(fact_3382_real__norm__def,axiom,
! [R_1: real] :
( ( norm_norm_real @ R_1 )
= ( abs_abs_real @ R_1 ) ) ).
thf(fact_3383_real__sqrt__abs2,axiom,
! [X: real] :
( ( sqrt @ ( times_times_real @ X @ X ) )
= ( abs_abs_real @ X ) ) ).
thf(fact_3384_natfloor__zero,axiom,
( ( natfloor @ zero_zero_real )
= zero_zero_nat ) ).
thf(fact_3385_zero__le__natfloor,axiom,
! [X: real] : ( ord_less_eq_nat @ zero_zero_nat @ ( natfloor @ X ) ) ).
thf(fact_3386_natfloor__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( natfloor @ X ) @ ( natfloor @ Y ) ) ) ).
thf(fact_3387_natfloor__number__of__eq,axiom,
! [N: int] :
( ( natfloor @ ( number267125858f_real @ N ) )
= ( number_number_of_nat @ N ) ) ).
thf(fact_3388_natfloor__one,axiom,
( ( natfloor @ one_one_real )
= one_one_nat ) ).
thf(fact_3389_natceiling__zero,axiom,
( ( natceiling @ zero_zero_real )
= zero_zero_nat ) ).
thf(fact_3390_zero__le__natceiling,axiom,
! [X: real] : ( ord_less_eq_nat @ zero_zero_nat @ ( natceiling @ X ) ) ).
thf(fact_3391_natceiling__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( natceiling @ X ) @ ( natceiling @ Y ) ) ) ).
thf(fact_3392_natceiling__number__of__eq,axiom,
! [N: int] :
( ( natceiling @ ( number267125858f_real @ N ) )
= ( number_number_of_nat @ N ) ) ).
thf(fact_3393_natceiling__one,axiom,
( ( natceiling @ one_one_real )
= one_one_nat ) ).
thf(fact_3394_rabs__ratiotest__lemma,axiom,
! [X: real,Y: real,C: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( times_times_real @ C @ ( abs_abs_real @ Y ) ) )
=> ( X = zero_zero_real ) ) ) ).
thf(fact_3395_abs__add__one__not__less__self,axiom,
! [X: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( abs_abs_real @ X ) @ one_one_real ) @ X ) ).
thf(fact_3396_sin__bound__lemma,axiom,
! [U: real,V: real,X: real,Y: real] :
( ( X = Y )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).
thf(fact_3397_natfloor__neg,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( natfloor @ X )
= zero_zero_nat ) ) ).
thf(fact_3398_abs__add__one__gt__zero,axiom,
! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).
thf(fact_3399_natceiling__neg,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( natceiling @ X )
= zero_zero_nat ) ) ).
thf(fact_3400_le__natfloor__eq__one,axiom,
! [X: real] :
( ( ord_less_eq_nat @ one_one_nat @ ( natfloor @ X ) )
<=> ( ord_less_eq_real @ one_one_real @ X ) ) ).
thf(fact_3401_natceiling__le__eq__one,axiom,
! [X: real] :
( ( ord_less_eq_nat @ ( natceiling @ X ) @ one_one_nat )
<=> ( ord_less_eq_real @ X @ one_one_real ) ) ).
thf(fact_3402_real__sqrt__abs,axiom,
! [X: real] :
( ( sqrt @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( abs_abs_real @ X ) ) ).
thf(fact_3403_le__mult__natfloor,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( natfloor @ A ) @ ( natfloor @ B ) ) @ ( natfloor @ ( times_times_real @ A @ B ) ) ) ) ) ).
thf(fact_3404_less__one__imp__sqr__less__one,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_real ) ) ).
thf(fact_3405_natfloor__add__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( natfloor @ ( plus_plus_real @ X @ one_one_real ) )
= ( plus_plus_nat @ ( natfloor @ X ) @ one_one_nat ) ) ) ).
thf(fact_3406_real__sqrt__ge__abs1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3407_real__sqrt__ge__abs2,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3408_arctan__add,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( arctan @ ( inverse_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).
thf(fact_3409_natceiling__add__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( natceiling @ ( plus_plus_real @ X @ one_one_real ) )
= ( plus_plus_nat @ ( natceiling @ X ) @ one_one_nat ) ) ) ).
thf(fact_3410_le__natfloor__eq__number__of,axiom,
! [X: real,N: int] :
( ~ ( nat_neg @ ( number_number_of_int @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_nat @ ( number_number_of_nat @ N ) @ ( natfloor @ X ) )
<=> ( ord_less_eq_real @ ( number267125858f_real @ N ) @ X ) ) ) ) ).
thf(fact_3411_natceiling__le__eq__number__of,axiom,
! [X: real,N: int] :
( ~ ( nat_neg @ ( number_number_of_int @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_nat @ ( natceiling @ X ) @ ( number_number_of_nat @ N ) )
<=> ( ord_less_eq_real @ X @ ( number267125858f_real @ N ) ) ) ) ) ).
thf(fact_3412_lemma__interval,axiom,
! [B: real,A: real,X: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D_2: real] :
( ( ord_less_real @ zero_zero_real @ D_2 )
& ! [Y_1: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y_1 ) ) @ D_2 )
=> ( ( ord_less_eq_real @ A @ Y_1 )
& ( ord_less_eq_real @ Y_1 @ B ) ) ) ) ) ) ).
thf(fact_3413_lemma__interval__lt,axiom,
! [B: real,A: real,X: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D_2: real] :
( ( ord_less_real @ zero_zero_real @ D_2 )
& ! [Y_1: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y_1 ) ) @ D_2 )
=> ( ( ord_less_real @ A @ Y_1 )
& ( ord_less_real @ Y_1 @ B ) ) ) ) ) ) ).
thf(fact_3414_abs__ln__one__plus__x__minus__x__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( inverse_divide_real @ one_one_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3415_machin,axiom,
( ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( minus_minus_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( arctan @ ( inverse_divide_real @ one_one_real @ ( number267125858f_real @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) @ ( arctan @ ( inverse_divide_real @ one_one_real @ ( number267125858f_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(fact_3416_pi__neq__zero,axiom,
pi != zero_zero_real ).
thf(fact_3417_abs__int__eq,axiom,
! [M: nat] :
( ( abs_abs_int @ ( semiri1621563631at_int @ M ) )
= ( semiri1621563631at_int @ M ) ) ).
thf(fact_3418_zdvd__antisym__abs,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( abs_abs_int @ A )
= ( abs_abs_int @ B ) ) ) ) ).
thf(fact_3419_zdvd__self__abs2,axiom,
! [D: int] : ( dvd_dvd_int @ ( abs_abs_int @ D ) @ D ) ).
thf(fact_3420_zdvd__self__abs1,axiom,
! [D: int] : ( dvd_dvd_int @ D @ ( abs_abs_int @ D ) ) ).
thf(fact_3421_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
thf(fact_3422_pi__ge__zero,axiom,
ord_less_eq_real @ zero_zero_real @ pi ).
thf(fact_3423_pi__not__less__zero,axiom,
~ ( ord_less_real @ pi @ zero_zero_real ) ).
thf(fact_3424_pi__gt__zero,axiom,
ord_less_real @ zero_zero_real @ pi ).
thf(fact_3425_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
<=> ( ( abs_abs_int @ X )
= one_one_int ) ) ).
thf(fact_3426_int__nat__abs,axiom,
! [X: int] :
( ( semiri1621563631at_int @ ( nat_1 @ ( abs_abs_int @ X ) ) )
= ( abs_abs_int @ X ) ) ).
thf(fact_3427_ln__less__self,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ ( ln @ X ) @ X ) ) ).
thf(fact_3428_ln__inj__iff,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ( ln @ X )
= ( ln @ Y ) )
<=> ( X = Y ) ) ) ) ).
thf(fact_3429_ln__less__cancel__iff,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( ln @ X ) @ ( ln @ Y ) )
<=> ( ord_less_real @ X @ Y ) ) ) ) ).
thf(fact_3430_abs__div,axiom,
! [Y: int,X: int] :
( ( dvd_dvd_int @ Y @ X )
=> ( ( abs_abs_int @ ( div_div_int @ X @ Y ) )
= ( div_div_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) ) ) ) ).
thf(fact_3431_ln__one,axiom,
( ( ln @ one_one_real )
= zero_zero_real ) ).
thf(fact_3432_ln__eq__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln @ X )
= ( minus_minus_real @ X @ one_one_real ) )
=> ( X = one_one_real ) ) ) ).
thf(fact_3433_zabs__less__one__iff,axiom,
! [Z_1: int] :
( ( ord_less_int @ ( abs_abs_int @ Z_1 ) @ one_one_int )
<=> ( Z_1 = zero_zero_int ) ) ).
thf(fact_3434_dvd__imp__le__int,axiom,
! [D: int,I: int] :
( ( I != zero_zero_int )
=> ( ( dvd_dvd_int @ D @ I )
=> ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
thf(fact_3435_zero__le__zpower__abs,axiom,
! [X: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ X ) @ N ) ) ).
thf(fact_3436_nat__abs__mult__distrib,axiom,
! [W: int,Z_1: int] :
( ( nat_1 @ ( abs_abs_int @ ( times_times_int @ W @ Z_1 ) ) )
= ( times_times_nat @ ( nat_1 @ ( abs_abs_int @ W ) ) @ ( nat_1 @ ( abs_abs_int @ Z_1 ) ) ) ) ).
thf(fact_3437_abs__eq__1__iff,axiom,
! [Z_1: int] :
( ( ( abs_abs_int @ Z_1 )
= one_one_int )
<=> ( ( Z_1 = one_one_int )
| ( Z_1
= ( number_number_of_int @ min ) ) ) ) ).
thf(fact_3438_ln__le__cancel__iff,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( ln @ X ) @ ( ln @ Y ) )
<=> ( ord_less_eq_real @ X @ Y ) ) ) ) ).
thf(fact_3439_ln__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln @ X ) ) ) ).
thf(fact_3440_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( ln @ X ) ) ) ).
thf(fact_3441_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( ln @ X ) )
<=> ( ord_less_real @ one_one_real @ X ) ) ) ).
thf(fact_3442_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln @ X )
= zero_zero_real )
<=> ( X = one_one_real ) ) ) ).
thf(fact_3443_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( ln @ X ) @ zero_zero_real )
<=> ( ord_less_real @ X @ one_one_real ) ) ) ).
thf(fact_3444_ln__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( ln @ X ) @ zero_zero_real ) ) ) ).
thf(fact_3445_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( ln @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ X ) ) ) ).
thf(fact_3446_ln__le__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
thf(fact_3447_zero__less__zpower__abs__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ X ) @ N ) )
<=> ( ( X != zero_zero_int )
| ( N = zero_zero_nat ) ) ) ).
thf(fact_3448_abs__power3__distrib,axiom,
! [X: int] :
( ( abs_abs_int @ ( power_power_int @ X @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) )
= ( power_power_int @ ( abs_abs_int @ X ) @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3449_zdvd__mult__cancel1,axiom,
! [N: int,M: int] :
( ( M != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
<=> ( ( abs_abs_int @ N )
= one_one_int ) ) ) ).
thf(fact_3450_dvd__int__iff,axiom,
! [Z_1: int,M: nat] :
( ( dvd_dvd_int @ Z_1 @ ( semiri1621563631at_int @ M ) )
<=> ( dvd_dvd_nat @ ( nat_1 @ ( abs_abs_int @ Z_1 ) ) @ M ) ) ).
thf(fact_3451_int__dvd__iff,axiom,
! [M: nat,Z_1: int] :
( ( dvd_dvd_int @ ( semiri1621563631at_int @ M ) @ Z_1 )
<=> ( dvd_dvd_nat @ M @ ( nat_1 @ ( abs_abs_int @ Z_1 ) ) ) ) ).
thf(fact_3452_ln__ge__zero__imp__ge__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
thf(fact_3453_ln__ge__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln @ X ) )
<=> ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
thf(fact_3454_ln__mult,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln @ ( times_times_real @ X @ Y ) )
= ( plus_plus_real @ ( ln @ X ) @ ( ln @ Y ) ) ) ) ) ).
thf(fact_3455_ln__add__one__self__le__self,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
thf(fact_3456_ln__div,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln @ ( inverse_divide_real @ X @ Y ) )
= ( minus_minus_real @ ( ln @ X ) @ ( ln @ Y ) ) ) ) ) ).
thf(fact_3457_ln__add__one__self__le__self2,axiom,
! [X: real] :
( ( ord_less_real @ ( number267125858f_real @ min ) @ X )
=> ( ord_less_eq_real @ ( ln @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
thf(fact_3458_power2__eq__iff__abs__eq,axiom,
! [A: int,B: int] :
( ( ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_int @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( ( abs_abs_int @ A )
= ( abs_abs_int @ B ) ) ) ).
thf(fact_3459_abs__power2__distrib,axiom,
! [A: int] :
( ( abs_abs_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( power_power_int @ ( abs_abs_int @ A ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3460_pi__ge__two,axiom,
ord_less_eq_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ).
thf(fact_3461_pi__less__4,axiom,
ord_less_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3462_pi__half__neq__two,axiom,
( ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3463_power2__eq1__iff,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_int )
=> ( ( abs_abs_int @ A )
= one_one_int ) ) ).
thf(fact_3464_decr__lemma,axiom,
! [X: int,Z_1: int,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z_1 ) ) @ one_one_int ) @ D ) ) @ Z_1 ) ) ).
thf(fact_3465_incr__lemma,axiom,
! [Z_1: int,X: int,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ Z_1 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z_1 ) ) @ one_one_int ) @ D ) ) ) ) ).
thf(fact_3466_pi__half__le__two,axiom,
ord_less_eq_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_3467_pi__half__neq__zero,axiom,
( ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= zero_zero_real ) ).
thf(fact_3468_pi__half__less__two,axiom,
ord_less_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_3469_pi__half__ge__zero,axiom,
ord_less_eq_real @ zero_zero_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3470_pi__half__gt__zero,axiom,
ord_less_real @ zero_zero_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3471_arctan__ubound,axiom,
! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3472_arctan1__eq__pi4,axiom,
( ( arctan @ one_one_real )
= ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3473_ln__one__plus__pos__lower__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( ln @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ) ).
thf(fact_3474_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3475_best__odd__division__abs,axiom,
! [Y: int,X: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( member_int @ X @ zOdd )
=> ? [N_1: int] : ( ord_less_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ Y @ ( times_times_int @ N_1 @ X ) ) ) ) @ X ) ) ) ).
thf(fact_3476_best__division__abs,axiom,
! [Y: int,X: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ? [N_1: int] : ( ord_less_eq_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ Y @ ( times_times_int @ N_1 @ X ) ) ) ) @ X ) ) ).
thf(fact_3477_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( inverse_divide_real @ one_one_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ).
thf(fact_3478_int__val__lemma,axiom,
! [K_1: int,F: nat > int,N: nat] :
( ! [I_1: nat] :
( ( ord_less_nat @ I_1 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I_1 @ one_one_nat ) ) @ ( F @ I_1 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K_1 )
=> ( ( ord_less_eq_int @ K_1 @ ( F @ N ) )
=> ? [I_1: nat] :
( ( ord_less_eq_nat @ I_1 @ N )
& ( ( F @ I_1 )
= K_1 ) ) ) ) ) ).
thf(fact_3479_nat0__intermed__int__val,axiom,
! [K_1: int,F: nat > int,N: nat] :
( ! [I_1: nat] :
( ( ord_less_nat @ I_1 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I_1 @ one_one_nat ) ) @ ( F @ I_1 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K_1 )
=> ( ( ord_less_eq_int @ K_1 @ ( F @ N ) )
=> ? [I_1: nat] :
( ( ord_less_eq_nat @ I_1 @ N )
& ( ( F @ I_1 )
= K_1 ) ) ) ) ) ).
thf(fact_3480_real__sqrt__minus,axiom,
! [X: real] :
( ( sqrt @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).
thf(fact_3481_arctan__minus,axiom,
! [X: real] :
( ( arctan @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( arctan @ X ) ) ) ).
thf(fact_3482_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
thf(fact_3483_real__add__minus__iff,axiom,
! [X: real,A: real] :
( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
<=> ( X = A ) ) ).
thf(fact_3484_real__add__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
<=> ( Y
= ( uminus_uminus_real @ X ) ) ) ).
thf(fact_3485_abs__le__interval__iff,axiom,
! [X: real,R_1: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ R_1 )
<=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ R_1 ) @ X )
& ( ord_less_eq_real @ X @ R_1 ) ) ) ).
thf(fact_3486_real__diff__def,axiom,
! [R_1: real,S_1: real] :
( ( minus_minus_real @ R_1 @ S_1 )
= ( plus_plus_real @ R_1 @ ( uminus_uminus_real @ S_1 ) ) ) ).
thf(fact_3487_minus__real__def,axiom,
! [X: real,Y: real] :
( ( minus_minus_real @ X @ Y )
= ( plus_plus_real @ X @ ( uminus_uminus_real @ Y ) ) ) ).
thf(fact_3488_abs__minus__add__cancel,axiom,
! [X: real,Y: real] :
( ( abs_abs_real @ ( plus_plus_real @ X @ ( uminus_uminus_real @ Y ) ) )
= ( abs_abs_real @ ( plus_plus_real @ Y @ ( uminus_uminus_real @ X ) ) ) ) ).
thf(fact_3489_complex__mod__minus__le__complex__mod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( norm_norm_complex @ X ) ) @ ( norm_norm_complex @ X ) ) ).
thf(fact_3490_real__0__le__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
<=> ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
thf(fact_3491_real__add__le__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
<=> ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
thf(fact_3492_real__add__less__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
<=> ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
thf(fact_3493_real__0__less__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
<=> ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
thf(fact_3494_abs__real__def,axiom,
! [A: real] :
( ( ( ord_less_real @ A @ zero_zero_real )
=> ( ( abs_abs_real @ A )
= ( uminus_uminus_real @ A ) ) )
& ( ~ ( ord_less_real @ A @ zero_zero_real )
=> ( ( abs_abs_real @ A )
= A ) ) ) ).
thf(fact_3495_real__abs__def,axiom,
! [R_1: real] :
( ( ( ord_less_real @ R_1 @ zero_zero_real )
=> ( ( abs_abs_real @ R_1 )
= ( uminus_uminus_real @ R_1 ) ) )
& ( ~ ( ord_less_real @ R_1 @ zero_zero_real )
=> ( ( abs_abs_real @ R_1 )
= R_1 ) ) ) ).
thf(fact_3496_abs__sum__triangle__ineq,axiom,
! [X: real,Y: real,L: real,M: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ ( uminus_uminus_real @ L ) @ ( uminus_uminus_real @ M ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( plus_plus_real @ X @ ( uminus_uminus_real @ L ) ) ) @ ( abs_abs_real @ ( plus_plus_real @ Y @ ( uminus_uminus_real @ M ) ) ) ) ) ).
thf(fact_3497_realpow__square__minus__le,axiom,
! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3498_ln__one__minus__pos__upper__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( ln @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
thf(fact_3499_aux5,axiom,
! [X: real] :
( ( ord_less_real @ X @ one_one_real )
=> ( ( ln @ ( minus_minus_real @ one_one_real @ X ) )
= ( uminus_uminus_real @ ( ln @ ( plus_plus_real @ one_one_real @ ( inverse_divide_real @ X @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).
thf(fact_3500_m2pi__less__pi,axiom,
ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ) ) @ pi ).
thf(fact_3501_minus__pi__half__less__zero,axiom,
ord_less_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_real ).
thf(fact_3502_arctan__bounded,axiom,
! [Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( arctan @ Y ) )
& ( ord_less_real @ ( arctan @ Y ) @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3503_arctan__lbound,axiom,
! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( arctan @ Y ) ) ).
thf(fact_3504_ln__one__minus__pos__lower__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ ( inverse_divide_real @ one_one_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( ln @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
thf(fact_3505_nat__intermed__int__val,axiom,
! [K_1: int,F: nat > int,N: nat,M: nat] :
( ! [I_1: nat] :
( ( ( ord_less_eq_nat @ M @ I_1 )
& ( ord_less_nat @ I_1 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I_1 @ one_one_nat ) ) @ ( F @ I_1 ) ) ) @ one_one_int ) )
=> ( ( ord_less_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K_1 )
=> ( ( ord_less_eq_int @ K_1 @ ( F @ N ) )
=> ? [I_1: nat] :
( ( ord_less_eq_nat @ M @ I_1 )
& ( ord_less_eq_nat @ I_1 @ N )
& ( ( F @ I_1 )
= K_1 ) ) ) ) ) ) ).
thf(fact_3506_arcsin__lt__bounded,axiom,
! [Y: real] :
( ( ord_less_real @ ( number267125858f_real @ min ) @ Y )
=> ( ( ord_less_real @ Y @ one_one_real )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_real @ ( arcsin @ Y ) @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ).
thf(fact_3507_arcsin__bounded,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( number267125858f_real @ min ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq_real @ ( arcsin @ Y ) @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ).
thf(fact_3508_arcsin__lbound,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( number267125858f_real @ min ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).
thf(fact_3509_arcsin__ubound,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( number267125858f_real @ min ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( arcsin @ Y ) @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3510_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1621563631at_int @ N ) ) @ ( semiri1621563631at_int @ M ) ) ).
thf(fact_3511_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1621563631at_int @ ( suc @ N ) ) ) @ ( semiri1621563631at_int @ M ) ) ).
thf(fact_3512_minus__Pls,axiom,
( ( uminus_uminus_int @ pls )
= pls ) ).
thf(fact_3513_minus__Bit0,axiom,
! [K_1: int] :
( ( uminus_uminus_int @ ( bit0 @ K_1 ) )
= ( bit0 @ ( uminus_uminus_int @ K_1 ) ) ) ).
thf(fact_3514_zminus__0,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_3515_zmult__zminus,axiom,
! [Z_1: int,W: int] :
( ( times_times_int @ ( uminus_uminus_int @ Z_1 ) @ W )
= ( uminus_uminus_int @ ( times_times_int @ Z_1 @ W ) ) ) ).
thf(fact_3516_minus__numeral__code_I5_J,axiom,
! [W: int] :
( ( uminus_uminus_int @ ( number_number_of_int @ W ) )
= ( number_number_of_int @ ( uminus_uminus_int @ W ) ) ) ).
thf(fact_3517_zminus__zadd__distrib,axiom,
! [Z_1: int,W: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ Z_1 @ W ) )
= ( plus_plus_int @ ( uminus_uminus_int @ Z_1 ) @ ( uminus_uminus_int @ W ) ) ) ).
thf(fact_3518_zminus__zminus,axiom,
! [Z_1: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ Z_1 ) )
= Z_1 ) ).
thf(fact_3519_zcong__zminus,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ ( uminus_uminus_int @ M ) )
<=> ( zcong @ A @ B @ M ) ) ).
thf(fact_3520_uminus__dvd__conv_I1_J,axiom,
! [D: int,T: int] :
( ( dvd_dvd_int @ D @ T )
<=> ( dvd_dvd_int @ ( uminus_uminus_int @ D ) @ T ) ) ).
thf(fact_3521_uminus__dvd__conv_I2_J,axiom,
! [D: int,T: int] :
( ( dvd_dvd_int @ D @ T )
<=> ( dvd_dvd_int @ D @ ( uminus_uminus_int @ T ) ) ) ).
thf(fact_3522_zdiv__zminus2,axiom,
! [A: int,B: int] :
( ( div_div_int @ A @ ( uminus_uminus_int @ B ) )
= ( div_div_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
thf(fact_3523_zdiv__zminus__zminus,axiom,
! [A: int,B: int] :
( ( div_div_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( div_div_int @ A @ B ) ) ).
thf(fact_3524_zmod__zminus2,axiom,
! [A: int,B: int] :
( ( div_mod_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( div_mod_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
thf(fact_3525_zmod__zminus__zminus,axiom,
! [A: int,B: int] :
( ( div_mod_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( div_mod_int @ A @ B ) ) ) ).
thf(fact_3526_zminus__zmod,axiom,
! [X: int,M: int] :
( ( div_mod_int @ ( uminus_uminus_int @ ( div_mod_int @ X @ M ) ) @ M )
= ( div_mod_int @ ( uminus_uminus_int @ X ) @ M ) ) ).
thf(fact_3527_even__neg,axiom,
! [X: int] :
( ( even_odd_even_int @ ( uminus_uminus_int @ X ) )
<=> ( even_odd_even_int @ X ) ) ).
thf(fact_3528_complex__diff__def,axiom,
! [X: complex,Y: complex] :
( ( minus_minus_complex @ X @ Y )
= ( plus_plus_complex @ X @ ( uminus473333897omplex @ Y ) ) ) ).
thf(fact_3529_zadd__zminus__inverse2,axiom,
! [Z_1: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ Z_1 ) @ Z_1 )
= zero_zero_int ) ).
thf(fact_3530_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1621563631at_int @ N ) )
= ( semiri1621563631at_int @ M ) )
<=> ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
thf(fact_3531_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1621563631at_int @ N ) @ ( uminus_uminus_int @ ( semiri1621563631at_int @ M ) ) ) ).
thf(fact_3532_Int_OMin__def,axiom,
( min
= ( uminus_uminus_int @ one_one_int ) ) ).
thf(fact_3533_zmod__zminus1__not__zero,axiom,
! [K_1: int,L: int] :
( ( ( div_mod_int @ ( uminus_uminus_int @ K_1 ) @ L )
!= zero_zero_int )
=> ( ( div_mod_int @ K_1 @ L )
!= zero_zero_int ) ) ).
thf(fact_3534_zmod__zminus2__not__zero,axiom,
! [K_1: int,L: int] :
( ( ( div_mod_int @ K_1 @ ( uminus_uminus_int @ L ) )
!= zero_zero_int )
=> ( ( div_mod_int @ K_1 @ L )
!= zero_zero_int ) ) ).
thf(fact_3535_mult__Min,axiom,
! [K_1: int] :
( ( times_times_int @ min @ K_1 )
= ( uminus_uminus_int @ K_1 ) ) ).
thf(fact_3536_diff__int__def,axiom,
! [Z_1: int,W: int] :
( ( minus_minus_int @ Z_1 @ W )
= ( plus_plus_int @ Z_1 @ ( uminus_uminus_int @ W ) ) ) ).
thf(fact_3537_diff__int__def__symmetric,axiom,
! [Z_1: int,W: int] :
( ( plus_plus_int @ Z_1 @ ( uminus_uminus_int @ W ) )
= ( minus_minus_int @ Z_1 @ W ) ) ).
thf(fact_3538_minus__Min,axiom,
( ( uminus_uminus_int @ min )
= ( bit1 @ pls ) ) ).
thf(fact_3539_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ N ) @ ( uminus_uminus_int @ ( semiri1621563631at_int @ M ) ) )
<=> ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
thf(fact_3540_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1621563631at_int @ N ) ) @ zero_zero_int ) ).
thf(fact_3541_nat__zminus__int,axiom,
! [N: nat] :
( ( nat_1 @ ( uminus_uminus_int @ ( semiri1621563631at_int @ N ) ) )
= zero_zero_nat ) ).
thf(fact_3542_zabs__def,axiom,
! [I: int] :
( ( ( ord_less_int @ I @ zero_zero_int )
=> ( ( abs_abs_int @ I )
= ( uminus_uminus_int @ I ) ) )
& ( ~ ( ord_less_int @ I @ zero_zero_int )
=> ( ( abs_abs_int @ I )
= I ) ) ) ).
thf(fact_3543_minus__numeral__code_I6_J,axiom,
! [V: int,W: int] :
( ( minus_minus_int @ ( number_number_of_int @ V ) @ ( number_number_of_int @ W ) )
= ( number_number_of_int @ ( plus_plus_int @ V @ ( uminus_uminus_int @ W ) ) ) ) ).
thf(fact_3544_zmod__zminus2__eq__if,axiom,
! [A: int,B: int] :
( ( ( ( div_mod_int @ A @ B )
= zero_zero_int )
=> ( ( div_mod_int @ A @ ( uminus_uminus_int @ B ) )
= zero_zero_int ) )
& ( ( ( div_mod_int @ A @ B )
!= zero_zero_int )
=> ( ( div_mod_int @ A @ ( uminus_uminus_int @ B ) )
= ( minus_minus_int @ ( div_mod_int @ A @ B ) @ B ) ) ) ) ).
thf(fact_3545_zmod__zminus1__eq__if,axiom,
! [A: int,B: int] :
( ( ( ( div_mod_int @ A @ B )
= zero_zero_int )
=> ( ( div_mod_int @ ( uminus_uminus_int @ A ) @ B )
= zero_zero_int ) )
& ( ( ( div_mod_int @ A @ B )
!= zero_zero_int )
=> ( ( div_mod_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ ( div_mod_int @ A @ B ) ) ) ) ) ).
thf(fact_3546_zdiv__minus1__right,axiom,
! [A: int] :
( ( div_div_int @ A @ ( number_number_of_int @ min ) )
= ( uminus_uminus_int @ A ) ) ).
thf(fact_3547_neg__zminus__int,axiom,
! [N: nat] : ( nat_neg @ ( uminus_uminus_int @ ( semiri1621563631at_int @ ( suc @ N ) ) ) ) ).
thf(fact_3548_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1621563631at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
thf(fact_3549_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1621563631at_int @ ( suc @ N ) ) ) ) ).
thf(fact_3550_power3__minus,axiom,
! [A: int] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( uminus_uminus_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3551_nat__mult__distrib__neg,axiom,
! [Z_3: int,Z_1: int] :
( ( ord_less_eq_int @ Z_1 @ zero_zero_int )
=> ( ( nat_1 @ ( times_times_int @ Z_1 @ Z_3 ) )
= ( times_times_nat @ ( nat_1 @ ( uminus_uminus_int @ Z_1 ) ) @ ( nat_1 @ ( uminus_uminus_int @ Z_3 ) ) ) ) ) ).
thf(fact_3552_zdiv__zminus2__eq__if,axiom,
! [A: int,B: int] :
( ( B != zero_zero_int )
=> ( ( ( ( div_mod_int @ A @ B )
= zero_zero_int )
=> ( ( div_div_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( div_div_int @ A @ B ) ) ) )
& ( ( ( div_mod_int @ A @ B )
!= zero_zero_int )
=> ( ( div_div_int @ A @ ( uminus_uminus_int @ B ) )
= ( minus_minus_int @ ( uminus_uminus_int @ ( div_div_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
thf(fact_3553_zdiv__zminus1__eq__if,axiom,
! [A: int,B: int] :
( ( B != zero_zero_int )
=> ( ( ( ( div_mod_int @ A @ B )
= zero_zero_int )
=> ( ( div_div_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( div_div_int @ A @ B ) ) ) )
& ( ( ( div_mod_int @ A @ B )
!= zero_zero_int )
=> ( ( div_div_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ ( uminus_uminus_int @ ( div_div_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
thf(fact_3554_neg__even__power,axiom,
! [A: int,X: int] :
( ( member_int @ X @ zEven )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( nat_1 @ X ) )
= ( power_power_int @ A @ ( nat_1 @ X ) ) ) ) ) ).
thf(fact_3555_neg__odd__power,axiom,
! [A: int,X: int] :
( ( member_int @ X @ zOdd )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( nat_1 @ X ) )
= ( uminus_uminus_int @ ( power_power_int @ A @ ( nat_1 @ X ) ) ) ) ) ) ).
thf(fact_3556_zminus1__lemma,axiom,
! [A: int,B: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( divmod_int_rel @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R_1 = zero_zero_int ) @ ( uminus_uminus_int @ Q ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q ) @ one_one_int ) ) @ ( if_int @ ( R_1 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R_1 ) ) ) ) ) ).
thf(fact_3557_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N_1: nat] :
( X
= ( uminus_uminus_int @ ( semiri1621563631at_int @ ( suc @ N_1 ) ) ) ) ) ).
thf(fact_3558_arctan__inverse,axiom,
! [X: real] :
( ( X != zero_zero_real )
=> ( ( arctan @ ( inverse_divide_real @ one_one_real @ X ) )
= ( minus_minus_real @ ( inverse_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( arctan @ X ) ) ) ) ).
thf(fact_3559_cos__arcsin,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( number267125858f_real @ min ) @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ( cos @ ( arcsin @ X ) )
= ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_3560_ln__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( ln @ X ) @ X ) ) ).
thf(fact_3561_cos__minus,axiom,
! [X: real] :
( ( cos @ ( uminus_uminus_real @ X ) )
= ( cos @ X ) ) ).
thf(fact_3562_cos__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( cos @ X ) @ one_one_real ) ).
thf(fact_3563_cos__zero,axiom,
( ( cos @ zero_zero_real )
= one_one_real ) ).
thf(fact_3564_cos__arctan__not__zero,axiom,
! [X: real] :
( ( cos @ ( arctan @ X ) )
!= zero_zero_real ) ).
thf(fact_3565_real__sgn__eq,axiom,
! [X: real] :
( ( sgn_sgn_real @ X )
= ( inverse_divide_real @ X @ ( abs_abs_real @ X ) ) ) ).
thf(fact_3566_cos__monotone__0__pi_H,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ord_less_eq_real @ ( cos @ X ) @ ( cos @ Y ) ) ) ) ) ).
thf(fact_3567_cos__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( number267125858f_real @ min ) @ ( cos @ X ) ) ).
thf(fact_3568_abs__cos__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos @ X ) ) @ one_one_real ) ).
thf(fact_3569_cos__pi,axiom,
( ( cos @ pi )
= ( number267125858f_real @ min ) ) ).
thf(fact_3570_cos__periodic__pi,axiom,
! [X: real] :
( ( cos @ ( plus_plus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( cos @ X ) ) ) ).
thf(fact_3571_real__sgn__pos,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( sgn_sgn_real @ X )
= one_one_real ) ) ).
thf(fact_3572_cos__monotone__0__pi,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ord_less_real @ ( cos @ X ) @ ( cos @ Y ) ) ) ) ) ).
thf(fact_3573_cos__monotone__minus__pi__0_H,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_eq_real @ ( cos @ Y ) @ ( cos @ X ) ) ) ) ) ).
thf(fact_3574_real__sgn__neg,axiom,
! [X: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( sgn_sgn_real @ X )
= ( number267125858f_real @ min ) ) ) ).
thf(fact_3575_real__sgn__def,axiom,
! [X: real] :
( ( ( X = zero_zero_real )
=> ( ( sgn_sgn_real @ X )
= zero_zero_real ) )
& ( ( X != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( sgn_sgn_real @ X )
= one_one_real ) )
& ( ~ ( ord_less_real @ zero_zero_real @ X )
=> ( ( sgn_sgn_real @ X )
= ( uminus_uminus_real @ one_one_real ) ) ) ) ) ) ).
thf(fact_3576_sgn__real__def,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( sgn_sgn_real @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( sgn_sgn_real @ A )
= one_one_real ) )
& ( ~ ( ord_less_real @ zero_zero_real @ A )
=> ( ( sgn_sgn_real @ A )
= ( uminus_uminus_real @ one_one_real ) ) ) ) ) ) ).
thf(fact_3577_cos__two__neq__zero,axiom,
( ( cos @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= zero_zero_real ) ).
thf(fact_3578_cos__monotone__minus__pi__0,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
=> ( ( ord_less_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_real @ ( cos @ Y ) @ ( cos @ X ) ) ) ) ) ).
thf(fact_3579_cos__two__le__zero,axiom,
ord_less_eq_real @ ( cos @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_real ).
thf(fact_3580_cos__two__less__zero,axiom,
ord_less_real @ ( cos @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_real ).
thf(fact_3581_cos__pi__half,axiom,
( ( cos @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_real ) ).
thf(fact_3582_cos__two__pi,axiom,
( ( cos @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ) )
= one_one_real ) ).
thf(fact_3583_cos__periodic,axiom,
! [X: real] :
( ( cos @ ( plus_plus_real @ X @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ) ) )
= ( cos @ X ) ) ).
thf(fact_3584_cos__60,axiom,
( ( cos @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit1 @ ( bit1 @ pls ) ) ) ) )
= ( inverse_divide_real @ one_one_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3585_cos__30,axiom,
( ( cos @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) )
= ( inverse_divide_real @ ( sqrt @ ( number267125858f_real @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3586_cos__45,axiom,
( ( cos @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
= ( inverse_divide_real @ ( sqrt @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3587_cos__double__less__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
=> ( ord_less_real @ ( cos @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X ) ) @ one_one_real ) ) ) ).
thf(fact_3588_cos__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( cos @ X ) ) ) ) ).
thf(fact_3589_cos__3over2__pi,axiom,
( ( cos @ ( times_times_real @ ( inverse_divide_real @ ( number267125858f_real @ ( bit1 @ ( bit1 @ pls ) ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ pi ) )
= zero_zero_real ) ).
thf(fact_3590_cos__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( cos @ X ) ) ) ) ).
thf(fact_3591_cos__gt__zero__pi,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( cos @ X ) ) ) ) ).
thf(fact_3592_cos__total,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( number267125858f_real @ min ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ? [X_1: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_1 )
& ( ord_less_eq_real @ X_1 @ pi )
& ( ( cos @ X_1 )
= Y )
& ! [Y_1: real] :
( ( ( ord_less_eq_real @ zero_zero_real @ Y_1 )
& ( ord_less_eq_real @ Y_1 @ pi )
& ( ( cos @ Y_1 )
= Y ) )
=> ( Y_1 = X_1 ) ) ) ) ) ).
thf(fact_3593_cos__is__zero,axiom,
? [X_1: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_1 )
& ( ord_less_eq_real @ X_1 @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
& ( ( cos @ X_1 )
= zero_zero_real )
& ! [Y_1: real] :
( ( ( ord_less_eq_real @ zero_zero_real @ Y_1 )
& ( ord_less_eq_real @ Y_1 @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
& ( ( cos @ Y_1 )
= zero_zero_real ) )
=> ( Y_1 = X_1 ) ) ) ).
thf(fact_3594_tan__double,axiom,
! [X: real] :
( ( ( cos @ X )
!= zero_zero_real )
=> ( ( ( cos @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X ) )
!= zero_zero_real )
=> ( ( tan @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X ) )
= ( inverse_divide_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( tan @ X ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_3595_cos__arccos__lemma1,axiom,
! [X_4: real,Y_3: real] :
( ( cos @ ( arccos @ ( inverse_divide_real @ X_4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X_4 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_3 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) )
= ( inverse_divide_real @ X_4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X_4 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_3 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ).
thf(fact_3596_tan__minus,axiom,
! [X: real] :
( ( tan @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( tan @ X ) ) ) ).
thf(fact_3597_tan__zero,axiom,
( ( tan @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_3598_tan__arctan,axiom,
! [Y: real] :
( ( tan @ ( arctan @ Y ) )
= Y ) ).
thf(fact_3599_tan__pi,axiom,
( ( tan @ pi )
= zero_zero_real ) ).
thf(fact_3600_tan__periodic__pi,axiom,
! [X: real] :
( ( tan @ ( plus_plus_real @ X @ pi ) )
= ( tan @ X ) ) ).
thf(fact_3601_tan__periodic__n,axiom,
! [X: real,N: int] :
( ( tan @ ( plus_plus_real @ X @ ( times_times_real @ ( number267125858f_real @ N ) @ pi ) ) )
= ( tan @ X ) ) ).
thf(fact_3602_zsgn__def,axiom,
! [I: int] :
( ( ( I = zero_zero_int )
=> ( ( sgn_sgn_int @ I )
= zero_zero_int ) )
& ( ( I != zero_zero_int )
=> ( ( ( ord_less_int @ zero_zero_int @ I )
=> ( ( sgn_sgn_int @ I )
= one_one_int ) )
& ( ~ ( ord_less_int @ zero_zero_int @ I )
=> ( ( sgn_sgn_int @ I )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ) ).
thf(fact_3603_arccos__cos,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ( arccos @ ( cos @ X ) )
= X ) ) ) ).
thf(fact_3604_cos__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( cos @ ( arccos @ Y ) )
= Y ) ) ).
thf(fact_3605_tan__60,axiom,
( ( tan @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit1 @ ( bit1 @ pls ) ) ) ) )
= ( sqrt @ ( number267125858f_real @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3606_arccos__lbound,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( number267125858f_real @ min ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) ) ) ) ).
thf(fact_3607_cos__arccos,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( number267125858f_real @ min ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( cos @ ( arccos @ Y ) )
= Y ) ) ) ).
thf(fact_3608_arccos__ubound,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( number267125858f_real @ min ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ).
thf(fact_3609_arccos__cos2,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X )
=> ( ( arccos @ ( cos @ X ) )
= ( uminus_uminus_real @ X ) ) ) ) ).
thf(fact_3610_tan__periodic,axiom,
! [X: real] :
( ( tan @ ( plus_plus_real @ X @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ) ) )
= ( tan @ X ) ) ).
thf(fact_3611_tan__45,axiom,
( ( tan @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
= one_one_real ) ).
thf(fact_3612_tan__add,axiom,
! [Y: real,X: real] :
( ( ( cos @ X )
!= zero_zero_real )
=> ( ( ( cos @ Y )
!= zero_zero_real )
=> ( ( ( cos @ ( plus_plus_real @ X @ Y ) )
!= zero_zero_real )
=> ( ( tan @ ( plus_plus_real @ X @ Y ) )
= ( inverse_divide_real @ ( plus_plus_real @ ( tan @ X ) @ ( tan @ Y ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan @ X ) @ ( tan @ Y ) ) ) ) ) ) ) ) ).
thf(fact_3613_lemma__tan__add1,axiom,
! [Y: real,X: real] :
( ( ( cos @ X )
!= zero_zero_real )
=> ( ( ( cos @ Y )
!= zero_zero_real )
=> ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan @ X ) @ ( tan @ Y ) ) )
= ( inverse_divide_real @ ( cos @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos @ X ) @ ( cos @ Y ) ) ) ) ) ) ).
thf(fact_3614_tan__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( tan @ X ) ) ) ) ).
thf(fact_3615_arccos__bounded,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( number267125858f_real @ min ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
& ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
thf(fact_3616_tan__monotone,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ Y )
=> ( ( ord_less_real @ Y @ X )
=> ( ( ord_less_real @ X @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ord_less_real @ ( tan @ Y ) @ ( tan @ X ) ) ) ) ) ).
thf(fact_3617_tan__monotone_H,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ Y )
=> ( ( ord_less_real @ Y @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_real @ Y @ X )
<=> ( ord_less_real @ ( tan @ Y ) @ ( tan @ X ) ) ) ) ) ) ) ).
thf(fact_3618_arccos__lt__bounded,axiom,
! [Y: real] :
( ( ord_less_real @ ( number267125858f_real @ min ) @ Y )
=> ( ( ord_less_real @ Y @ one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y ) )
& ( ord_less_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
thf(fact_3619_tan__inverse,axiom,
! [Y: real] :
( ( inverse_divide_real @ one_one_real @ ( tan @ Y ) )
= ( tan @ ( minus_minus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ Y ) ) ) ).
thf(fact_3620_tan__30,axiom,
( ( tan @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) )
= ( inverse_divide_real @ one_one_real @ ( sqrt @ ( number267125858f_real @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3621_tan__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ ( inverse_divide_real @ ( uminus_uminus_real @ pi ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( ord_less_real @ ( tan @ X ) @ zero_zero_real ) ) ) ).
thf(fact_3622_arctan,axiom,
! [Y: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( arctan @ Y ) )
& ( ord_less_real @ ( arctan @ Y ) @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
& ( ( tan @ ( arctan @ Y ) )
= Y ) ) ).
thf(fact_3623_arctan__tan,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( arctan @ ( tan @ X ) )
= X ) ) ) ).
thf(fact_3624_arccos,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( number267125858f_real @ min ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
& ( ord_less_eq_real @ ( arccos @ Y ) @ pi )
& ( ( cos @ ( arccos @ Y ) )
= Y ) ) ) ) ).
thf(fact_3625_tan__total__pi4,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ? [Z: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ Z )
& ( ord_less_real @ Z @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
& ( ( tan @ Z )
= X ) ) ) ).
thf(fact_3626_tan__total__pos,axiom,
! [Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ? [X_1: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_1 )
& ( ord_less_real @ X_1 @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
& ( ( tan @ X_1 )
= Y ) ) ) ).
thf(fact_3627_tan__total,axiom,
! [Y: real] :
? [X_1: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ X_1 )
& ( ord_less_real @ X_1 @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
& ( ( tan @ X_1 )
= Y )
& ! [Y_1: real] :
( ( ( ord_less_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ Y_1 )
& ( ord_less_real @ Y_1 @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
& ( ( tan @ Y_1 )
= Y ) )
=> ( Y_1 = X_1 ) ) ) ).
thf(fact_3628_lemma__tan__total1,axiom,
! [Y: real] :
? [X_1: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ X_1 )
& ( ord_less_real @ X_1 @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
& ( ( tan @ X_1 )
= Y ) ) ).
thf(fact_3629_lemma__tan__total,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ? [X_1: real] :
( ( ord_less_real @ zero_zero_real @ X_1 )
& ( ord_less_real @ X_1 @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
& ( ord_less_real @ Y @ ( tan @ X_1 ) ) ) ) ).
thf(fact_3630_tan__half,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( tan @ X )
= ( inverse_divide_real @ ( sin @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X ) ) @ ( plus_plus_real @ ( cos @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X ) ) @ one_one_real ) ) ) ) ) ).
thf(fact_3631_cos__pi__eq__zero,axiom,
! [M: nat] :
( ( cos @ ( inverse_divide_real @ ( times_times_real @ pi @ ( real_nat @ ( suc @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ M ) ) ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_real ) ).
thf(fact_3632_sin__arccos,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( number267125858f_real @ min ) @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ( sin @ ( arccos @ X ) )
= ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_3633_real__of__nat__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( real_nat @ N ) ) ).
thf(fact_3634_sin__minus,axiom,
! [X: real] :
( ( sin @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( sin @ X ) ) ) ).
thf(fact_3635_abs__real__of__nat__cancel,axiom,
! [X: nat] :
( ( abs_abs_real @ ( real_nat @ X ) )
= ( real_nat @ X ) ) ).
thf(fact_3636_real__of__nat__inject,axiom,
! [N: nat,M: nat] :
( ( ( real_nat @ N )
= ( real_nat @ M ) )
<=> ( N = M ) ) ).
thf(fact_3637_sin__zero,axiom,
( ( sin @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_3638_natfloor__real__of__nat,axiom,
! [N: nat] :
( ( natfloor @ ( real_nat @ N ) )
= N ) ).
thf(fact_3639_natceiling__real__of__nat,axiom,
! [N: nat] :
( ( natceiling @ ( real_nat @ N ) )
= N ) ).
thf(fact_3640_sin__npi2,axiom,
! [N: nat] :
( ( sin @ ( times_times_real @ pi @ ( real_nat @ N ) ) )
= zero_zero_real ) ).
thf(fact_3641_sin__npi,axiom,
! [N: nat] :
( ( sin @ ( times_times_real @ ( real_nat @ N ) @ pi ) )
= zero_zero_real ) ).
thf(fact_3642_real__of__nat__def,axiom,
real_nat = semiri132038758t_real ).
thf(fact_3643_real__eq__of__nat,axiom,
real_nat = semiri132038758t_real ).
thf(fact_3644_sin__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( sin @ X ) @ one_one_real ) ).
thf(fact_3645_real__of__nat__zero__iff,axiom,
! [N: nat] :
( ( ( real_nat @ N )
= zero_zero_real )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_3646_real__of__nat__zero,axiom,
( ( real_nat @ zero_zero_nat )
= zero_zero_real ) ).
thf(fact_3647_not__real__of__nat__less__zero,axiom,
! [N: nat] :
~ ( ord_less_real @ ( real_nat @ N ) @ zero_zero_real ) ).
thf(fact_3648_real__of__nat__less__iff,axiom,
! [N: nat,M: nat] :
( ( ord_less_real @ ( real_nat @ N ) @ ( real_nat @ M ) )
<=> ( ord_less_nat @ N @ M ) ) ).
thf(fact_3649_sin__pi,axiom,
( ( sin @ pi )
= zero_zero_real ) ).
thf(fact_3650_real__of__nat__le__iff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_real @ ( real_nat @ N ) @ ( real_nat @ M ) )
<=> ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_3651_real__of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( real_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( real_nat @ M ) @ ( real_nat @ N ) ) ) ).
thf(fact_3652_real__of__nat__add,axiom,
! [M: nat,N: nat] :
( ( real_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_real @ ( real_nat @ M ) @ ( real_nat @ N ) ) ) ).
thf(fact_3653_real__of__nat__1,axiom,
( ( real_nat @ one_one_nat )
= one_one_real ) ).
thf(fact_3654_power__real__of__nat,axiom,
! [M: nat,N: nat] :
( ( power_power_real @ ( real_nat @ M ) @ N )
= ( real_nat @ ( power_power_nat @ M @ N ) ) ) ).
thf(fact_3655_real__of__nat__power,axiom,
! [M: nat,N: nat] :
( ( real_nat @ ( power_power_nat @ M @ N ) )
= ( power_power_real @ ( real_nat @ M ) @ N ) ) ).
thf(fact_3656_real__natceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( real_nat @ ( natceiling @ X ) ) ) ).
thf(fact_3657_sin__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ pi )
=> ( ord_less_eq_real @ zero_zero_real @ ( sin @ X ) ) ) ) ).
thf(fact_3658_sin__gt__zero__pi,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ pi )
=> ( ord_less_real @ zero_zero_real @ ( sin @ X ) ) ) ) ).
thf(fact_3659_real__of__nat__le__zero__cancel__iff,axiom,
! [N: nat] :
( ( ord_less_eq_real @ ( real_nat @ N ) @ zero_zero_real )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_3660_sin__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( number267125858f_real @ min ) @ ( sin @ X ) ) ).
thf(fact_3661_real__of__nat__Suc__gt__zero,axiom,
! [N: nat] : ( ord_less_real @ zero_zero_real @ ( real_nat @ ( suc @ N ) ) ) ).
thf(fact_3662_cos__one__sin__zero,axiom,
! [X: real] :
( ( ( cos @ X )
= one_one_real )
=> ( ( sin @ X )
= zero_zero_real ) ) ).
thf(fact_3663_abs__sin__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin @ X ) ) @ one_one_real ) ).
thf(fact_3664_real__of__nat__one,axiom,
( ( real_nat @ ( suc @ zero_zero_nat ) )
= one_one_real ) ).
thf(fact_3665_sin__add,axiom,
! [X: real,Y: real] :
( ( sin @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ ( sin @ X ) @ ( cos @ Y ) ) @ ( times_times_real @ ( cos @ X ) @ ( sin @ Y ) ) ) ) ).
thf(fact_3666_sin__diff2,axiom,
! [X: real,Y: real] :
( ( sin @ ( minus_minus_real @ X @ Y ) )
= ( minus_minus_real @ ( times_times_real @ ( cos @ Y ) @ ( sin @ X ) ) @ ( times_times_real @ ( sin @ Y ) @ ( cos @ X ) ) ) ) ).
thf(fact_3667_sin__diff,axiom,
! [X: real,Y: real] :
( ( sin @ ( minus_minus_real @ X @ Y ) )
= ( minus_minus_real @ ( times_times_real @ ( sin @ X ) @ ( cos @ Y ) ) @ ( times_times_real @ ( cos @ X ) @ ( sin @ Y ) ) ) ) ).
thf(fact_3668_real__of__nat__Suc,axiom,
! [N: nat] :
( ( real_nat @ ( suc @ N ) )
= ( plus_plus_real @ ( real_nat @ N ) @ one_one_real ) ) ).
thf(fact_3669_sin__periodic__pi2,axiom,
! [X: real] :
( ( sin @ ( plus_plus_real @ pi @ X ) )
= ( uminus_uminus_real @ ( sin @ X ) ) ) ).
thf(fact_3670_sin__periodic__pi,axiom,
! [X: real] :
( ( sin @ ( plus_plus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( sin @ X ) ) ) ).
thf(fact_3671_real__of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( real_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_real @ ( real_nat @ M ) @ ( real_nat @ N ) ) ) ) ).
thf(fact_3672_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( real_nat @ ( div_div_nat @ N @ X ) ) @ ( inverse_divide_real @ ( real_nat @ N ) @ ( real_nat @ X ) ) ) ).
thf(fact_3673_tan__def,axiom,
! [X: real] :
( ( tan @ X )
= ( inverse_divide_real @ ( sin @ X ) @ ( cos @ X ) ) ) ).
thf(fact_3674_real__natfloor__le,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_nat @ ( natfloor @ X ) ) @ X ) ) ).
thf(fact_3675_le__natfloor,axiom,
! [X: nat,A: real] :
( ( ord_less_eq_real @ ( real_nat @ X ) @ A )
=> ( ord_less_eq_nat @ X @ ( natfloor @ A ) ) ) ).
thf(fact_3676_natceiling__le,axiom,
! [X: real,A: nat] :
( ( ord_less_eq_real @ X @ ( real_nat @ A ) )
=> ( ord_less_eq_nat @ ( natceiling @ X ) @ A ) ) ).
thf(fact_3677_sin__2npi,axiom,
! [N: nat] :
( ( sin @ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( real_nat @ N ) ) @ pi ) )
= zero_zero_real ) ).
thf(fact_3678_natfloor__power,axiom,
! [N: nat,X: real] :
( ( X
= ( real_nat @ ( natfloor @ X ) ) )
=> ( ( natfloor @ ( power_power_real @ X @ N ) )
= ( power_power_nat @ ( natfloor @ X ) @ N ) ) ) ).
thf(fact_3679_real__of__nat__gt__zero__cancel__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( real_nat @ N ) )
<=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_3680_sin__cos__squared__add3,axiom,
! [X: real] :
( ( plus_plus_real @ ( times_times_real @ ( cos @ X ) @ ( cos @ X ) ) @ ( times_times_real @ ( sin @ X ) @ ( sin @ X ) ) )
= one_one_real ) ).
thf(fact_3681_sin__zero__abs__cos__one,axiom,
! [X: real] :
( ( ( sin @ X )
= zero_zero_real )
=> ( ( abs_abs_real @ ( cos @ X ) )
= one_one_real ) ) ).
thf(fact_3682_cos__diff2,axiom,
! [X: real,Y: real] :
( ( cos @ ( minus_minus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ ( cos @ Y ) @ ( cos @ X ) ) @ ( times_times_real @ ( sin @ Y ) @ ( sin @ X ) ) ) ) ).
thf(fact_3683_cos__diff,axiom,
! [X: real,Y: real] :
( ( cos @ ( minus_minus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ ( cos @ X ) @ ( cos @ Y ) ) @ ( times_times_real @ ( sin @ X ) @ ( sin @ Y ) ) ) ) ).
thf(fact_3684_cos__add,axiom,
! [X: real,Y: real] :
( ( cos @ ( plus_plus_real @ X @ Y ) )
= ( minus_minus_real @ ( times_times_real @ ( cos @ X ) @ ( cos @ Y ) ) @ ( times_times_real @ ( sin @ X ) @ ( sin @ Y ) ) ) ) ).
thf(fact_3685_nat__less__real__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
<=> ( ord_less_eq_real @ ( plus_plus_real @ ( real_nat @ N ) @ one_one_real ) @ ( real_nat @ M ) ) ) ).
thf(fact_3686_nat__le__real__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
<=> ( ord_less_real @ ( real_nat @ N ) @ ( plus_plus_real @ ( real_nat @ M ) @ one_one_real ) ) ) ).
thf(fact_3687_le__natfloor__eq,axiom,
! [A: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_nat @ A @ ( natfloor @ X ) )
<=> ( ord_less_eq_real @ ( real_nat @ A ) @ X ) ) ) ).
thf(fact_3688_tan__npi,axiom,
! [N: nat] :
( ( tan @ ( times_times_real @ ( real_nat @ N ) @ pi ) )
= zero_zero_real ) ).
thf(fact_3689_natceiling__le__eq,axiom,
! [A: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_nat @ ( natceiling @ X ) @ A )
<=> ( ord_less_eq_real @ X @ ( real_nat @ A ) ) ) ) ).
thf(fact_3690_real__natfloor__add__one__gt,axiom,
! [X: real] : ( ord_less_real @ X @ ( plus_plus_real @ ( real_nat @ ( natfloor @ X ) ) @ one_one_real ) ) ).
thf(fact_3691_tan__periodic__nat,axiom,
! [X: real,N: nat] :
( ( tan @ ( plus_plus_real @ X @ ( times_times_real @ ( real_nat @ N ) @ pi ) ) )
= ( tan @ X ) ) ).
thf(fact_3692_natfloor__subtract,axiom,
! [A: nat,X: real] :
( ( ord_less_eq_real @ ( real_nat @ A ) @ X )
=> ( ( natfloor @ ( minus_minus_real @ X @ ( real_nat @ A ) ) )
= ( minus_minus_nat @ ( natfloor @ X ) @ A ) ) ) ).
thf(fact_3693_real__natfloor__gt__diff__one,axiom,
! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ one_one_real ) @ ( real_nat @ ( natfloor @ X ) ) ) ).
thf(fact_3694_natceiling__subtract,axiom,
! [A: nat,X: real] :
( ( ord_less_eq_real @ ( real_nat @ A ) @ X )
=> ( ( natceiling @ ( minus_minus_real @ X @ ( real_nat @ A ) ) )
= ( minus_minus_nat @ ( natceiling @ X ) @ A ) ) ) ).
thf(fact_3695_sin__expansion__lemma,axiom,
! [X: real,M: nat] :
( ( sin @ ( plus_plus_real @ X @ ( inverse_divide_real @ ( times_times_real @ ( real_nat @ ( suc @ M ) ) @ pi ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
= ( cos @ ( plus_plus_real @ X @ ( inverse_divide_real @ ( times_times_real @ ( real_nat @ M ) @ pi ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3696_real__of__nat__div,axiom,
! [N: nat,D: nat] :
( ( ord_less_nat @ zero_zero_nat @ D )
=> ( ( dvd_dvd_nat @ D @ N )
=> ( ( real_nat @ ( div_div_nat @ N @ D ) )
= ( inverse_divide_real @ ( real_nat @ N ) @ ( real_nat @ D ) ) ) ) ) ).
thf(fact_3697_ln__realpow,axiom,
! [N: nat,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( real_nat @ N ) @ ( ln @ X ) ) ) ) ).
thf(fact_3698_less__natfloor,axiom,
! [N: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( real_nat @ N ) )
=> ( ord_less_nat @ ( natfloor @ X ) @ N ) ) ) ).
thf(fact_3699_real__of__nat__div2,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( inverse_divide_real @ ( real_nat @ N ) @ ( real_nat @ X ) ) @ ( real_nat @ ( div_div_nat @ N @ X ) ) ) ) ).
thf(fact_3700_real__of__nat__number__of,axiom,
! [V: int] :
( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( real_nat @ ( number_number_of_nat @ V ) )
= zero_zero_real ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( real_nat @ ( number_number_of_nat @ V ) )
= ( number267125858f_real @ V ) ) ) ) ).
thf(fact_3701_real__of__nat__div3,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( inverse_divide_real @ ( real_nat @ N ) @ ( real_nat @ X ) ) @ ( real_nat @ ( div_div_nat @ N @ X ) ) ) @ one_one_real ) ).
thf(fact_3702_natfloor__add,axiom,
! [A: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( natfloor @ ( plus_plus_real @ X @ ( real_nat @ A ) ) )
= ( plus_plus_nat @ ( natfloor @ X ) @ A ) ) ) ).
thf(fact_3703_sin__arcsin,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( number267125858f_real @ min ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( sin @ ( arcsin @ Y ) )
= Y ) ) ) ).
thf(fact_3704_ge__natfloor__plus__one__imp__gt,axiom,
! [Z_1: real,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( natfloor @ Z_1 ) @ one_one_nat ) @ N )
=> ( ord_less_real @ Z_1 @ ( real_nat @ N ) ) ) ).
thf(fact_3705_natfloor__eq,axiom,
! [N: nat,X: real] :
( ( ord_less_eq_real @ ( real_nat @ N ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( real_nat @ N ) @ one_one_real ) )
=> ( ( natfloor @ X )
= N ) ) ) ).
thf(fact_3706_natceiling__add,axiom,
! [A: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( natceiling @ ( plus_plus_real @ X @ ( real_nat @ A ) ) )
= ( plus_plus_nat @ ( natceiling @ X ) @ A ) ) ) ).
thf(fact_3707_cos__expansion__lemma,axiom,
! [X: real,M: nat] :
( ( cos @ ( plus_plus_real @ X @ ( inverse_divide_real @ ( times_times_real @ ( real_nat @ ( suc @ M ) ) @ pi ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
= ( uminus_uminus_real @ ( sin @ ( plus_plus_real @ X @ ( inverse_divide_real @ ( times_times_real @ ( real_nat @ M ) @ pi ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ).
thf(fact_3708_sin__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( sin @ X ) ) ) ) ).
thf(fact_3709_sin__cos__npi,axiom,
! [N: nat] :
( ( sin @ ( inverse_divide_real @ ( times_times_real @ ( real_nat @ ( suc @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) ) @ pi ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( power_power_real @ ( number267125858f_real @ min ) @ N ) ) ).
thf(fact_3710_sin__double,axiom,
! [X: real] :
( ( sin @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X ) )
= ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( sin @ X ) ) @ ( cos @ X ) ) ) ).
thf(fact_3711_two__realpow__gt,axiom,
! [N: nat] : ( ord_less_real @ ( real_nat @ N ) @ ( power_power_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) ).
thf(fact_3712_LIMSEQ__inverse__realpow__zero__lemma,axiom,
! [N: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( real_nat @ N ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N ) ) ) ).
thf(fact_3713_add__tan__eq,axiom,
! [Y: real,X: real] :
( ( ( cos @ X )
!= zero_zero_real )
=> ( ( ( cos @ Y )
!= zero_zero_real )
=> ( ( plus_plus_real @ ( tan @ X ) @ ( tan @ Y ) )
= ( inverse_divide_real @ ( sin @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos @ X ) @ ( cos @ Y ) ) ) ) ) ) ).
thf(fact_3714_real__of__nat__div__aux,axiom,
! [X: nat,D: nat] :
( ( ord_less_nat @ zero_zero_nat @ D )
=> ( ( inverse_divide_real @ ( real_nat @ X ) @ ( real_nat @ D ) )
= ( plus_plus_real @ ( real_nat @ ( div_div_nat @ X @ D ) ) @ ( inverse_divide_real @ ( real_nat @ ( div_mod_nat @ X @ D ) ) @ ( real_nat @ D ) ) ) ) ) ).
thf(fact_3715_cos__npi,axiom,
! [N: nat] :
( ( cos @ ( times_times_real @ ( real_nat @ N ) @ pi ) )
= ( power_power_real @ ( number267125858f_real @ min ) @ N ) ) ).
thf(fact_3716_cos__npi2,axiom,
! [N: nat] :
( ( cos @ ( times_times_real @ pi @ ( real_nat @ N ) ) )
= ( power_power_real @ ( number267125858f_real @ min ) @ N ) ) ).
thf(fact_3717_sin__two__pi,axiom,
( ( sin @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ) )
= zero_zero_real ) ).
thf(fact_3718_sin__periodic,axiom,
! [X: real] :
( ( sin @ ( plus_plus_real @ X @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ) ) )
= ( sin @ X ) ) ).
thf(fact_3719_sin__pi__half,axiom,
( ( sin @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= one_one_real ) ).
thf(fact_3720_sin__30,axiom,
( ( sin @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) )
= ( inverse_divide_real @ one_one_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3721_sin__60,axiom,
( ( sin @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit1 @ ( bit1 @ pls ) ) ) ) )
= ( inverse_divide_real @ ( sqrt @ ( number267125858f_real @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3722_sin__45,axiom,
( ( sin @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
= ( inverse_divide_real @ ( sqrt @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3723_fact__lemma,axiom,
! [N: nat] :
( ( times_times_real @ ( real_nat @ N ) @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( real_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ N ) ) ) ).
thf(fact_3724_natfloor__div__nat,axiom,
! [Y: nat,X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_nat @ zero_zero_nat @ Y )
=> ( ( natfloor @ ( inverse_divide_real @ X @ ( real_nat @ Y ) ) )
= ( div_div_nat @ ( natfloor @ X ) @ Y ) ) ) ) ).
thf(fact_3725_natceiling__eq,axiom,
! [N: nat,X: real] :
( ( ord_less_real @ ( real_nat @ N ) @ X )
=> ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( real_nat @ N ) @ one_one_real ) )
=> ( ( natceiling @ X )
= ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ).
thf(fact_3726_sin__gt__zero2,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( sin @ X ) ) ) ) ).
thf(fact_3727_sin__cos__squared__add2__mult,axiom,
! [R_1: real,A: real] :
( ( plus_plus_real @ ( power_power_real @ ( times_times_real @ R_1 @ ( cos @ A ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( times_times_real @ R_1 @ ( sin @ A ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( power_power_real @ R_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3728_sin__cos__squared__add,axiom,
! [X: real] :
( ( plus_plus_real @ ( power_power_real @ ( sin @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( cos @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= one_one_real ) ).
thf(fact_3729_sin__cos__squared__add2,axiom,
! [X: real] :
( ( plus_plus_real @ ( power_power_real @ ( cos @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( sin @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= one_one_real ) ).
thf(fact_3730_sin__squared__eq,axiom,
! [X: real] :
( ( power_power_real @ ( sin @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3731_cos__squared__eq,axiom,
! [X: real] :
( ( power_power_real @ ( cos @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3732_sin__monotone__2pi_H,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ord_less_eq_real @ ( sin @ Y ) @ ( sin @ X ) ) ) ) ) ).
thf(fact_3733_cos__sin__eq,axiom,
! [X: real] :
( ( cos @ X )
= ( sin @ ( minus_minus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ X ) ) ) ).
thf(fact_3734_sin__cos__eq,axiom,
! [X: real] :
( ( sin @ X )
= ( cos @ ( minus_minus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ X ) ) ) ).
thf(fact_3735_cos__2npi,axiom,
! [N: nat] :
( ( cos @ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( real_nat @ N ) ) @ pi ) )
= one_one_real ) ).
thf(fact_3736_sin__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ ( inverse_divide_real @ ( uminus_uminus_real @ pi ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( ord_less_real @ ( sin @ X ) @ zero_zero_real ) ) ) ).
thf(fact_3737_cos__double,axiom,
! [X: real] :
( ( cos @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X ) )
= ( minus_minus_real @ ( power_power_real @ ( cos @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( sin @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3738_sin__3over2__pi,axiom,
( ( sin @ ( times_times_real @ ( inverse_divide_real @ ( number267125858f_real @ ( bit1 @ ( bit1 @ pls ) ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ pi ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
thf(fact_3739_minus__sin__cos__eq,axiom,
! [X: real] :
( ( uminus_uminus_real @ ( sin @ X ) )
= ( cos @ ( plus_plus_real @ X @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3740_arcsin__sin,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( arcsin @ ( sin @ X ) )
= X ) ) ) ).
thf(fact_3741_sin__cos__add,axiom,
! [X: real,Y: real] :
( ( plus_plus_real @ ( power_power_real @ ( minus_minus_real @ ( sin @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( times_times_real @ ( sin @ X ) @ ( cos @ Y ) ) @ ( times_times_real @ ( cos @ X ) @ ( sin @ Y ) ) ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ ( cos @ ( plus_plus_real @ X @ Y ) ) @ ( minus_minus_real @ ( times_times_real @ ( cos @ X ) @ ( cos @ Y ) ) @ ( times_times_real @ ( sin @ X ) @ ( sin @ Y ) ) ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_real ) ).
thf(fact_3742_sin__cos__minus,axiom,
! [X: real] :
( ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ ( sin @ ( uminus_uminus_real @ X ) ) @ ( sin @ X ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ ( cos @ ( uminus_uminus_real @ X ) ) @ ( cos @ X ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_real ) ).
thf(fact_3743_sin__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( sin @ ( arccos @ Y ) )
= ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ).
thf(fact_3744_arcsin__pi,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( number267125858f_real @ min ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq_real @ ( arcsin @ Y ) @ pi )
& ( ( sin @ ( arcsin @ Y ) )
= Y ) ) ) ) ).
thf(fact_3745_arcsin,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( number267125858f_real @ min ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less_eq_real @ ( arcsin @ Y ) @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
& ( ( sin @ ( arcsin @ Y ) )
= Y ) ) ) ) ).
thf(fact_3746_sin__arccos__lemma1,axiom,
! [X_4: real,Y_3: real] :
( ( sin @ ( arccos @ ( inverse_divide_real @ X_4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X_4 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_3 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) )
= ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( inverse_divide_real @ X_4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X_4 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_3 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3747_sin__total,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( number267125858f_real @ min ) @ Y )
=> ( ( ord_less_eq_real @ Y @ one_one_real )
=> ? [X_1: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ X_1 )
& ( ord_less_eq_real @ X_1 @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
& ( ( sin @ X_1 )
= Y )
& ! [Y_1: real] :
( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ Y_1 )
& ( ord_less_eq_real @ Y_1 @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
& ( ( sin @ Y_1 )
= Y ) )
=> ( Y_1 = X_1 ) ) ) ) ) ).
thf(fact_3748_reals__Archimedean6,axiom,
! [R_1: real] :
( ( ord_less_eq_real @ zero_zero_real @ R_1 )
=> ? [N_1: nat] :
( ( ord_less_eq_real @ ( real_nat @ ( minus_minus_nat @ N_1 @ one_one_nat ) ) @ R_1 )
& ( ord_less_real @ R_1 @ ( real_nat @ N_1 ) ) ) ) ).
thf(fact_3749_reals__Archimedean4,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ? [N_1: nat] :
( ( ord_less_eq_real @ ( times_times_real @ ( real_nat @ N_1 ) @ Y ) @ X )
& ( ord_less_real @ X @ ( times_times_real @ ( real_nat @ ( suc @ N_1 ) ) @ Y ) ) ) ) ) ).
thf(fact_3750_reals__Archimedean6a,axiom,
! [R_1: real] :
( ( ord_less_eq_real @ zero_zero_real @ R_1 )
=> ? [N_1: nat] :
( ( ord_less_eq_real @ ( real_nat @ N_1 ) @ R_1 )
& ( ord_less_real @ R_1 @ ( real_nat @ ( suc @ N_1 ) ) ) ) ) ).
thf(fact_3751_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y_1: real] :
? [N_1: nat] : ( ord_less_real @ Y_1 @ ( times_times_real @ ( real_nat @ N_1 ) @ X ) ) ) ).
thf(fact_3752_lemma__STAR__cos,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ ( div_div_nat @ N @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ zero_zero_real @ N ) )
= zero_zero_real ) ) ).
thf(fact_3753_sin__zero__iff,axiom,
! [X: real] :
( ( ( sin @ X )
= zero_zero_real )
<=> ( ? [N_1: nat] :
( ( even_odd_even_nat @ N_1 )
& ( X
= ( times_times_real @ ( real_nat @ N_1 ) @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) )
| ? [N_1: nat] :
( ( even_odd_even_nat @ N_1 )
& ( X
= ( uminus_uminus_real @ ( times_times_real @ ( real_nat @ N_1 ) @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ) ).
thf(fact_3754_cos__zero__iff,axiom,
! [X: real] :
( ( ( cos @ X )
= zero_zero_real )
<=> ( ? [N_1: nat] :
( ~ ( even_odd_even_nat @ N_1 )
& ( X
= ( times_times_real @ ( real_nat @ N_1 ) @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) )
| ? [N_1: nat] :
( ~ ( even_odd_even_nat @ N_1 )
& ( X
= ( uminus_uminus_real @ ( times_times_real @ ( real_nat @ N_1 ) @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ) ).
thf(fact_3755_odd__add,axiom,
! [M: nat,N: nat] :
( ~ ( even_odd_even_nat @ ( plus_plus_nat @ M @ N ) )
<=> ~ ( ~ ( even_odd_even_nat @ M )
<=> ~ ( even_odd_even_nat @ N ) ) ) ).
thf(fact_3756_even__add,axiom,
! [M: nat,N: nat] :
( ( even_odd_even_nat @ ( plus_plus_nat @ M @ N ) )
<=> ( ( even_odd_even_nat @ M )
<=> ( even_odd_even_nat @ N ) ) ) ).
thf(fact_3757_even__sum__nat,axiom,
! [X: nat,Y: nat] :
( ( even_odd_even_nat @ ( plus_plus_nat @ X @ Y ) )
<=> ( ( ( even_odd_even_nat @ X )
& ( even_odd_even_nat @ Y ) )
| ( ~ ( even_odd_even_nat @ X )
& ~ ( even_odd_even_nat @ Y ) ) ) ) ).
thf(fact_3758_even__zero__nat,axiom,
even_odd_even_nat @ zero_zero_nat ).
thf(fact_3759_even__Suc,axiom,
! [X: nat] :
( ( even_odd_even_nat @ ( suc @ X ) )
<=> ~ ( even_odd_even_nat @ X ) ) ).
thf(fact_3760_odd__1__nat,axiom,
~ ( even_odd_even_nat @ one_one_nat ) ).
thf(fact_3761_even__product__nat,axiom,
! [X: nat,Y: nat] :
( ( even_odd_even_nat @ ( times_times_nat @ X @ Y ) )
<=> ( ( even_odd_even_nat @ X )
| ( even_odd_even_nat @ Y ) ) ) ).
thf(fact_3762_real__of__nat__fact__not__zero,axiom,
! [N: nat] :
( ( real_nat @ ( fact_fact_nat @ N ) )
!= zero_zero_real ) ).
thf(fact_3763_odd__pos,axiom,
! [N: nat] :
( ~ ( even_odd_even_nat @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_3764_even__difference__nat,axiom,
! [X: nat,Y: nat] :
( ( even_odd_even_nat @ ( minus_minus_nat @ X @ Y ) )
<=> ( ( ord_less_nat @ X @ Y )
| ( ( even_odd_even_nat @ X )
& ( even_odd_even_nat @ Y ) )
| ( ~ ( even_odd_even_nat @ X )
& ~ ( even_odd_even_nat @ Y ) ) ) ) ).
thf(fact_3765_Parity_Otransfer__int__nat__relations,axiom,
! [X: nat] :
( ( even_odd_even_int @ ( semiri1621563631at_int @ X ) )
<=> ( even_odd_even_nat @ X ) ) ).
thf(fact_3766_even__nat__def,axiom,
! [X: nat] :
( ( even_odd_even_nat @ X )
<=> ( even_odd_even_int @ ( semiri1621563631at_int @ X ) ) ) ).
thf(fact_3767_real__of__nat__fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( real_nat @ ( fact_fact_nat @ N ) ) ) ).
thf(fact_3768_real__of__nat__fact__gt__zero,axiom,
! [N: nat] : ( ord_less_real @ zero_zero_real @ ( real_nat @ ( fact_fact_nat @ N ) ) ) ).
thf(fact_3769_odd__nat__equiv__def2,axiom,
! [X: nat] :
( ~ ( even_odd_even_nat @ X )
<=> ? [Y_1: nat] :
( X
= ( suc @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ Y_1 ) ) ) ) ).
thf(fact_3770_even__nat__equiv__def2,axiom,
! [X: nat] :
( ( even_odd_even_nat @ X )
<=> ? [Y_1: nat] :
( X
= ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ Y_1 ) ) ) ).
thf(fact_3771_odd__nat__plus__one__div__two,axiom,
! [X: nat] :
( ~ ( even_odd_even_nat @ X )
=> ( ( div_div_nat @ ( suc @ X ) @ ( suc @ ( suc @ zero_zero_nat ) ) )
= ( suc @ ( div_div_nat @ X @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
thf(fact_3772_even__nat__plus__one__div__two,axiom,
! [X: nat] :
( ( even_odd_even_nat @ X )
=> ( ( div_div_nat @ ( suc @ X ) @ ( suc @ ( suc @ zero_zero_nat ) ) )
= ( div_div_nat @ X @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_3773_odd__nat__mod__two__eq__one,axiom,
! [X: nat] :
( ~ ( even_odd_even_nat @ X )
=> ( ( div_mod_nat @ X @ ( suc @ ( suc @ zero_zero_nat ) ) )
= ( suc @ zero_zero_nat ) ) ) ).
thf(fact_3774_even__nat__mod__two__eq__zero,axiom,
! [X: nat] :
( ( even_odd_even_nat @ X )
=> ( ( div_mod_nat @ X @ ( suc @ ( suc @ zero_zero_nat ) ) )
= zero_zero_nat ) ) ).
thf(fact_3775_odd__nat__equiv__def,axiom,
! [X: nat] :
( ~ ( even_odd_even_nat @ X )
<=> ( ( div_mod_nat @ X @ ( suc @ ( suc @ zero_zero_nat ) ) )
= ( suc @ zero_zero_nat ) ) ) ).
thf(fact_3776_even__nat__equiv__def,axiom,
! [X: nat] :
( ( even_odd_even_nat @ X )
<=> ( ( div_mod_nat @ X @ ( suc @ ( suc @ zero_zero_nat ) ) )
= zero_zero_nat ) ) ).
thf(fact_3777_even__power__nat,axiom,
! [X: nat,Y: nat] :
( ( even_odd_even_nat @ ( power_power_nat @ X @ Y ) )
<=> ( ( even_odd_even_nat @ X )
& ( ord_less_nat @ zero_zero_nat @ Y ) ) ) ).
thf(fact_3778_fact__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( ( fact_fact_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
= ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( fact_fact_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
thf(fact_3779_even__num__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( even_odd_even_nat @ N )
<=> ~ ( even_odd_even_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
thf(fact_3780_odd__nat__div__two__times__two__plus__one,axiom,
! [X: nat] :
( ~ ( even_odd_even_nat @ X )
=> ( ( suc @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ ( div_div_nat @ X @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) )
= X ) ) ).
thf(fact_3781_even__nat__div__two__times__two,axiom,
! [X: nat] :
( ( even_odd_even_nat @ X )
=> ( ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ ( div_div_nat @ X @ ( suc @ ( suc @ zero_zero_nat ) ) ) )
= X ) ) ).
thf(fact_3782_pos__int__even__equiv__nat__even,axiom,
! [X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( even_odd_even_int @ X )
<=> ( even_odd_even_nat @ ( nat_1 @ X ) ) ) ) ).
thf(fact_3783_even__dvd,axiom,
! [N: nat] :
( ( even_odd_even_nat @ N )
<=> ( dvd_dvd_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) ).
thf(fact_3784_nat__even__iff__2__dvd,axiom,
! [X: nat] :
( ( even_odd_even_nat @ X )
<=> ( dvd_dvd_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ X ) ) ).
thf(fact_3785_even__mult__two__ex,axiom,
! [N: nat] :
( ( even_odd_even_nat @ N )
<=> ? [M_2: nat] :
( N
= ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ M_2 ) ) ) ).
thf(fact_3786_even__even__mod__4__iff,axiom,
! [N: nat] :
( ( even_odd_even_nat @ N )
<=> ( even_odd_even_nat @ ( div_mod_nat @ N @ ( number_number_of_nat @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3787_odd__Suc__mult__two__ex,axiom,
! [N: nat] :
( ~ ( even_odd_even_nat @ N )
<=> ? [M_2: nat] :
( N
= ( suc @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ M_2 ) ) ) ) ).
thf(fact_3788_lemma__even__div2,axiom,
! [N: nat] :
( ( even_odd_even_nat @ N )
=> ( ( div_div_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( div_div_nat @ N @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3789_lemma__STAR__sin,axiom,
! [N: nat] :
( ( times_times_real @ ( if_real @ ( even_odd_even_nat @ N ) @ zero_zero_real @ ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ ( div_div_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) ) @ ( power_power_real @ zero_zero_real @ N ) )
= zero_zero_real ) ).
thf(fact_3790_lemma__not__even__div2,axiom,
! [N: nat] :
( ~ ( even_odd_even_nat @ N )
=> ( ( div_div_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( suc @ ( div_div_nat @ N @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3791_lemma__odd__mod__4__div__2,axiom,
! [N: nat] :
( ( ( div_mod_nat @ N @ ( number_number_of_nat @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
=> ~ ( even_odd_even_nat @ ( div_div_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3792_lemma__even__mod__4__div__2,axiom,
! [N: nat] :
( ( ( div_mod_nat @ N @ ( number_number_of_nat @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= one_one_nat )
=> ( even_odd_even_nat @ ( div_div_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3793_sin__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ( sin @ X )
= zero_zero_real )
=> ? [N_1: nat] :
( ( even_odd_even_nat @ N_1 )
& ( X
= ( times_times_real @ ( real_nat @ N_1 ) @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_3794_cos__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ( cos @ X )
= zero_zero_real )
=> ? [N_1: nat] :
( ~ ( even_odd_even_nat @ N_1 )
& ( X
= ( times_times_real @ ( real_nat @ N_1 ) @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_3795_odd__square,axiom,
! [N: nat] :
( ~ ( even_odd_even_nat @ N )
=> ? [X_1: nat] :
( ( power_power_nat @ N @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ X_1 ) @ one_one_nat ) ) ) ).
thf(fact_3796_sin__coeff__def,axiom,
! [X_1: nat] :
( ( ( even_odd_even_nat @ X_1 )
=> ( ( sin_coeff @ X_1 )
= zero_zero_real ) )
& ( ~ ( even_odd_even_nat @ X_1 )
=> ( ( sin_coeff @ X_1 )
= ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ ( div_div_nat @ ( minus_minus_nat @ X_1 @ ( suc @ zero_zero_nat ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( real_nat @ ( fact_fact_nat @ X_1 ) ) ) ) ) ) ).
thf(fact_3797_even__square,axiom,
! [N: nat] :
( ( even_odd_even_nat @ N )
=> ? [X_1: nat] :
( ( power_power_nat @ N @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ X_1 ) ) ) ).
thf(fact_3798_cos__coeff__def,axiom,
! [X_1: nat] :
( ( ( even_odd_even_nat @ X_1 )
=> ( ( cos_coeff @ X_1 )
= ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ ( div_div_nat @ X_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( real_nat @ ( fact_fact_nat @ X_1 ) ) ) ) )
& ( ~ ( even_odd_even_nat @ X_1 )
=> ( ( cos_coeff @ X_1 )
= zero_zero_real ) ) ) ).
thf(fact_3799_fact__add__num__eq__if2__nat,axiom,
! [N: nat,M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( fact_fact_nat @ ( plus_plus_nat @ M @ N ) )
= ( fact_fact_nat @ N ) ) )
& ( ( M != zero_zero_nat )
=> ( ( fact_fact_nat @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ ( fact_fact_nat @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ) ).
thf(fact_3800_fact__add__num__eq__if__nat,axiom,
! [M: nat,N: nat] :
( ( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
=> ( ( fact_fact_nat @ ( plus_plus_nat @ M @ N ) )
= one_one_nat ) )
& ( ( ( plus_plus_nat @ M @ N )
!= zero_zero_nat )
=> ( ( fact_fact_nat @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ ( fact_fact_nat @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ one_one_nat ) ) ) ) ) ) ).
thf(fact_3801_fact__nonzero__nat,axiom,
! [N: nat] :
( ( fact_fact_nat @ N )
!= zero_zero_nat ) ).
thf(fact_3802_fact__mono__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( fact_fact_nat @ M ) @ ( fact_fact_nat @ N ) ) ) ).
thf(fact_3803_fact__1__nat,axiom,
( ( fact_fact_nat @ one_one_nat )
= one_one_nat ) ).
thf(fact_3804_Fact_Ofact__0__nat,axiom,
( ( fact_fact_nat @ zero_zero_nat )
= ( suc @ zero_zero_nat ) ) ).
thf(fact_3805_fact__Suc__0__nat,axiom,
( ( fact_fact_nat @ ( suc @ zero_zero_nat ) )
= ( suc @ zero_zero_nat ) ) ).
thf(fact_3806_fact__gt__zero__nat,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( fact_fact_nat @ N ) ) ).
thf(fact_3807_fact__less__mono__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( fact_fact_nat @ M ) @ ( fact_fact_nat @ N ) ) ) ) ).
thf(fact_3808_Fact_Ofact__Suc,axiom,
! [X: nat] :
( ( fact_fact_nat @ ( suc @ X ) )
= ( times_times_nat @ ( suc @ X ) @ ( fact_fact_nat @ X ) ) ) ).
thf(fact_3809_fact__dvd,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ ( fact_fact_nat @ N ) @ ( fact_fact_nat @ M ) ) ) ).
thf(fact_3810_fact__ge__one__nat,axiom,
! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( fact_fact_nat @ N ) ) ).
thf(fact_3811_fact__ge__Suc__0__nat,axiom,
! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( fact_fact_nat @ N ) ) ).
thf(fact_3812_dvd__fact__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ M @ ( fact_fact_nat @ N ) ) ) ) ).
thf(fact_3813_fact__plus__one__nat,axiom,
! [N: nat] :
( ( fact_fact_nat @ ( plus_plus_nat @ N @ one_one_nat ) )
= ( times_times_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( fact_fact_nat @ N ) ) ) ).
thf(fact_3814_fact__mod,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( div_mod_nat @ ( fact_fact_nat @ N ) @ ( fact_fact_nat @ M ) )
= zero_zero_nat ) ) ).
thf(fact_3815_fact__num__eq__if__nat,axiom,
! [M: nat] :
( ( ( M = zero_zero_nat )
=> ( ( fact_fact_nat @ M )
= one_one_nat ) )
& ( ( M != zero_zero_nat )
=> ( ( fact_fact_nat @ M )
= ( times_times_nat @ M @ ( fact_fact_nat @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ) ).
thf(fact_3816_fact__reduce__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( fact_fact_nat @ N )
= ( times_times_nat @ N @ ( fact_fact_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
thf(fact_3817_Ln_Oaux1,axiom,
! [N: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ ( inverse_inverse_real @ ( real_nat @ ( fact_fact_nat @ ( plus_plus_nat @ N @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ N @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( times_times_real @ ( inverse_divide_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( inverse_divide_real @ one_one_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ N ) ) ) ) ) ).
thf(fact_3818_aux4,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( exp_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ).
thf(fact_3819_exp__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
thf(fact_3820_ln__unique,axiom,
! [Y: real,X: real] :
( ( ( exp_real @ Y )
= X )
=> ( ( ln @ X )
= Y ) ) ).
thf(fact_3821_exp__ln__eq,axiom,
! [U: real,X: real] :
( ( ( exp_real @ U )
= X )
=> ( ( ln @ X )
= U ) ) ).
thf(fact_3822_ln__exp,axiom,
! [X: real] :
( ( ln @ ( exp_real @ X ) )
= X ) ).
thf(fact_3823_exp__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
<=> ( ord_less_eq_real @ X @ Y ) ) ).
thf(fact_3824_exp__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
thf(fact_3825_exp__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
<=> ( ord_less_real @ X @ Y ) ) ).
thf(fact_3826_INVERSE__ZERO,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_3827_exp__inj__iff,axiom,
! [X: real,Y: real] :
( ( ( exp_real @ X )
= ( exp_real @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_3828_abs__exp__cancel,axiom,
! [X: real] :
( ( abs_abs_real @ ( exp_real @ X ) )
= ( exp_real @ X ) ) ).
thf(fact_3829_real__sqrt__inverse,axiom,
! [X: real] :
( ( sqrt @ ( inverse_inverse_real @ X ) )
= ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).
thf(fact_3830_divide__real__def,axiom,
! [X: real,Y: real] :
( ( inverse_divide_real @ X @ Y )
= ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ).
thf(fact_3831_real__divide__def,axiom,
! [R_2: real,S: real] :
( ( inverse_divide_real @ R_2 @ S )
= ( times_times_real @ R_2 @ ( inverse_inverse_real @ S ) ) ) ).
thf(fact_3832_not__exp__le__zero,axiom,
! [X: real] :
~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).
thf(fact_3833_exp__ge__zero,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).
thf(fact_3834_exp__gt__zero,axiom,
! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).
thf(fact_3835_not__exp__less__zero,axiom,
! [X: real] :
~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).
thf(fact_3836_exp__eq__one__iff,axiom,
! [X: real] :
( ( ( exp_real @ X )
= one_one_real )
<=> ( X = zero_zero_real ) ) ).
thf(fact_3837_real__mult__inverse__cancel,axiom,
! [Y: real,U: real,X1: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ X1 )
=> ( ( ord_less_real @ ( times_times_real @ X1 @ Y ) @ ( times_times_real @ X @ U ) )
=> ( ord_less_real @ ( times_times_real @ ( inverse_inverse_real @ X ) @ Y ) @ ( times_times_real @ ( inverse_inverse_real @ X1 ) @ U ) ) ) ) ) ).
thf(fact_3838_real__mult__inverse__cancel2,axiom,
! [Y: real,U: real,X1: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ X1 )
=> ( ( ord_less_real @ ( times_times_real @ X1 @ Y ) @ ( times_times_real @ X @ U ) )
=> ( ord_less_real @ ( times_times_real @ Y @ ( inverse_inverse_real @ X ) ) @ ( times_times_real @ U @ ( inverse_inverse_real @ X1 ) ) ) ) ) ) ).
thf(fact_3839_real__mult__inverse__left,axiom,
! [X: real] :
( ( X != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ X ) @ X )
= one_one_real ) ) ).
thf(fact_3840_one__le__exp__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
<=> ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
thf(fact_3841_exp__le__one__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
<=> ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
thf(fact_3842_exp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
thf(fact_3843_exp__less__one__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
<=> ( ord_less_real @ X @ zero_zero_real ) ) ).
thf(fact_3844_one__less__exp__iff,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
<=> ( ord_less_real @ zero_zero_real @ X ) ) ).
thf(fact_3845_exp__ge__add__one__self,axiom,
! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).
thf(fact_3846_exp__real__of__nat__mult,axiom,
! [N: nat,X: real] :
( ( exp_real @ ( times_times_real @ ( real_nat @ N ) @ X ) )
= ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
thf(fact_3847_exp__ln__iff,axiom,
! [X: real] :
( ( ( exp_real @ ( ln @ X ) )
= X )
<=> ( ord_less_real @ zero_zero_real @ X ) ) ).
thf(fact_3848_exp__ln,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( exp_real @ ( ln @ X ) )
= X ) ) ).
thf(fact_3849_inv__real__of__nat__fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ ( real_nat @ ( fact_fact_nat @ N ) ) ) ) ).
thf(fact_3850_sqrt__divide__self__eq,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( inverse_divide_real @ ( sqrt @ X ) @ X )
= ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).
thf(fact_3851_inv__real__of__nat__fact__gt__zero,axiom,
! [N: nat] : ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( real_nat @ ( fact_fact_nat @ N ) ) ) ) ).
thf(fact_3852_ln__inverse,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln @ ( inverse_inverse_real @ X ) )
= ( uminus_uminus_real @ ( ln @ X ) ) ) ) ).
thf(fact_3853_exp__ge__add__one__self__aux,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).
thf(fact_3854_ln__x__over__x__mono,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( inverse_divide_real @ ( ln @ Y ) @ Y ) @ ( inverse_divide_real @ ( ln @ X ) @ X ) ) ) ) ).
thf(fact_3855_real__inv__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( inverse_inverse_real @ X ) ) ) ).
thf(fact_3856_exp__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ).
thf(fact_3857_tan__sec,axiom,
! [X: real] :
( ( ( cos @ X )
!= zero_zero_real )
=> ( ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( power_power_real @ ( inverse_inverse_real @ ( cos @ X ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3858_lemma__exp__total,axiom,
! [Y: real] :
( ( ord_less_eq_real @ one_one_real @ Y )
=> ? [X_1: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_1 )
& ( ord_less_eq_real @ X_1 @ ( minus_minus_real @ Y @ one_one_real ) )
& ( ( exp_real @ X_1 )
= Y ) ) ) ).
thf(fact_3859_reals__Archimedean,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N_1: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( real_nat @ ( suc @ N_1 ) ) ) @ X ) ) ).
thf(fact_3860_log__base__10__eq1,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ ( number267125858f_real @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ X )
= ( times_times_real @ ( inverse_divide_real @ ( ln @ ( exp_real @ one_one_real ) ) @ ( ln @ ( number267125858f_real @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) @ ( ln @ X ) ) ) ) ).
thf(fact_3861_DERIV__arccos,axiom,
! [X: real] :
( ( ord_less_real @ ( number267125858f_real @ min ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( deriv_real @ arccos @ X @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ) ) ).
thf(fact_3862_DERIV__sin,axiom,
! [X: real] : ( deriv_real @ sin @ X @ ( cos @ X ) ) ).
thf(fact_3863_log__one,axiom,
! [A: real] :
( ( log @ A @ one_one_real )
= zero_zero_real ) ).
thf(fact_3864_Log_Olog__def,axiom,
! [A: real,X: real] :
( ( log @ A @ X )
= ( inverse_divide_real @ ( ln @ X ) @ ( ln @ A ) ) ) ).
thf(fact_3865_DERIV__cos,axiom,
! [X: real] : ( deriv_real @ cos @ X @ ( uminus_uminus_real @ ( sin @ X ) ) ) ).
thf(fact_3866_DERIV__ln,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( deriv_real @ ln @ X @ ( inverse_inverse_real @ X ) ) ) ).
thf(fact_3867_log__eq__one,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ A )
= one_one_real ) ) ) ).
thf(fact_3868_log__less__cancel__iff,axiom,
! [Y: real,X: real,A: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
<=> ( ord_less_real @ X @ Y ) ) ) ) ) ).
thf(fact_3869_log__ln,axiom,
! [X: real] :
( ( ln @ X )
= ( log @ ( exp_real @ one_one_real ) @ X ) ) ).
thf(fact_3870_DERIV__ln__divide,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( deriv_real @ ln @ X @ ( inverse_divide_real @ one_one_real @ X ) ) ) ).
thf(fact_3871_log__le__cancel__iff,axiom,
! [Y: real,X: real,A: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
<=> ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).
thf(fact_3872_log__nat__power,axiom,
! [B: real,N: nat,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ B @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( real_nat @ N ) @ ( log @ B @ X ) ) ) ) ).
thf(fact_3873_log__mult,axiom,
! [Y: real,X: real,A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( log @ A @ ( times_times_real @ X @ Y ) )
= ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
thf(fact_3874_log__divide,axiom,
! [Y: real,X: real,A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( log @ A @ ( inverse_divide_real @ X @ Y ) )
= ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
thf(fact_3875_log__inverse,axiom,
! [X: real,A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ A @ ( inverse_inverse_real @ X ) )
= ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).
thf(fact_3876_log__eq__div__ln__mult__log,axiom,
! [X: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ A @ X )
= ( times_times_real @ ( inverse_divide_real @ ( ln @ B ) @ ( ln @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).
thf(fact_3877_DERIV__real__sqrt,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( deriv_real @ sqrt @ X @ ( inverse_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3878_DERIV__tan,axiom,
! [X: real] :
( ( ( cos @ X )
!= zero_zero_real )
=> ( deriv_real @ tan @ X @ ( inverse_inverse_real @ ( power_power_real @ ( cos @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3879_DERIV__arctan,axiom,
! [X: real] : ( deriv_real @ arctan @ X @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3880_DERIV__real__sqrt__generic,axiom,
! [D_1: real,X: real] :
( ( X != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ X )
=> ( D_1
= ( inverse_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
=> ( ( ( ord_less_real @ X @ zero_zero_real )
=> ( D_1
= ( inverse_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
=> ( deriv_real @ sqrt @ X @ D_1 ) ) ) ) ).
thf(fact_3881_DERIV__arcsin,axiom,
! [X: real] :
( ( ord_less_real @ ( number267125858f_real @ min ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( deriv_real @ arcsin @ X @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ) ).
thf(fact_3882_log__base__10__eq2,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ ( number267125858f_real @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ X )
= ( times_times_real @ ( log @ ( number267125858f_real @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln @ X ) ) ) ) ).
thf(fact_3883_DERIV__const__average,axiom,
! [V: real > real,K_1: real,A: real,B: real] :
( ( A != B )
=> ( ! [X_1: real] : ( deriv_real @ V @ X_1 @ K_1 )
=> ( ( V @ ( inverse_divide_real @ ( plus_plus_real @ A @ B ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( inverse_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3884_DERIV__local__min,axiom,
! [D: real,F: real > real,X: real,L: real] :
( ( deriv_real @ F @ X @ L )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y_1: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y_1 ) ) @ D )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y_1 ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
thf(fact_3885_DERIV__local__max,axiom,
! [D: real,F: real > real,X: real,L: real] :
( ( deriv_real @ F @ X @ L )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y_1: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y_1 ) ) @ D )
=> ( ord_less_eq_real @ ( F @ Y_1 ) @ ( F @ X ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
thf(fact_3886_DERIV__local__const,axiom,
! [D: real,F: real > real,X: real,L: real] :
( ( deriv_real @ F @ X @ L )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y_1: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y_1 ) ) @ D )
=> ( ( F @ X )
= ( F @ Y_1 ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
thf(fact_3887_DERIV__pos__inc__left,axiom,
! [F: real > real,X: real,L: real] :
( ( deriv_real @ F @ X @ L )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D_2: real] :
( ( ord_less_real @ zero_zero_real @ D_2 )
& ! [H_1: real] :
( ( ord_less_real @ zero_zero_real @ H_1 )
=> ( ( ord_less_real @ H_1 @ D_2 )
=> ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H_1 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
thf(fact_3888_DERIV__neg__dec__left,axiom,
! [F: real > real,X: real,L: real] :
( ( deriv_real @ F @ X @ L )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D_2: real] :
( ( ord_less_real @ zero_zero_real @ D_2 )
& ! [H_1: real] :
( ( ord_less_real @ zero_zero_real @ H_1 )
=> ( ( ord_less_real @ H_1 @ D_2 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H_1 ) ) ) ) ) ) ) ) ).
thf(fact_3889_DERIV__neg__dec__right,axiom,
! [F: real > real,X: real,L: real] :
( ( deriv_real @ F @ X @ L )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [D_2: real] :
( ( ord_less_real @ zero_zero_real @ D_2 )
& ! [H_1: real] :
( ( ord_less_real @ zero_zero_real @ H_1 )
=> ( ( ord_less_real @ H_1 @ D_2 )
=> ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H_1 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
thf(fact_3890_DERIV__pos__inc__right,axiom,
! [F: real > real,X: real,L: real] :
( ( deriv_real @ F @ X @ L )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [D_2: real] :
( ( ord_less_real @ zero_zero_real @ D_2 )
& ! [H_1: real] :
( ( ord_less_real @ zero_zero_real @ H_1 )
=> ( ( ord_less_real @ H_1 @ D_2 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H_1 ) ) ) ) ) ) ) ) ).
thf(fact_3891_DERIV__const__ratio__const2,axiom,
! [F: real > real,K_1: real,A: real,B: real] :
( ( A != B )
=> ( ! [X_1: real] : ( deriv_real @ F @ X_1 @ K_1 )
=> ( ( inverse_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
= K_1 ) ) ) ).
thf(fact_3892_DERIV__const__ratio__const,axiom,
! [F: real > real,K_1: real,A: real,B: real] :
( ( A != B )
=> ( ! [X_1: real] : ( deriv_real @ F @ X_1 @ K_1 )
=> ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
= ( times_times_real @ ( minus_minus_real @ B @ A ) @ K_1 ) ) ) ) ).
thf(fact_3893_DERIV__even__real__root,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( even_odd_even_nat @ N )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( deriv_real @ ( root @ N ) @ X @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( real_nat @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) ) ).
thf(fact_3894_DERIV__real__root__generic,axiom,
! [D_1: real,X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( X != zero_zero_real )
=> ( ( ( even_odd_even_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( D_1
= ( inverse_inverse_real @ ( times_times_real @ ( real_nat @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
=> ( ( ( even_odd_even_nat @ N )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( D_1
= ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( real_nat @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
=> ( ( ~ ( even_odd_even_nat @ N )
=> ( D_1
= ( inverse_inverse_real @ ( times_times_real @ ( real_nat @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
=> ( deriv_real @ ( root @ N ) @ X @ D_1 ) ) ) ) ) ) ).
thf(fact_3895_real__root__zero,axiom,
! [N: nat] :
( ( root @ N @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_3896_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ zero_zero_nat ) @ X )
= X ) ).
thf(fact_3897_real__root__eq__iff,axiom,
! [X: real,Y: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= ( root @ N @ Y ) )
<=> ( X = Y ) ) ) ).
thf(fact_3898_real__root__commute,axiom,
! [X: real,N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ M @ ( root @ N @ X ) )
= ( root @ N @ ( root @ M @ X ) ) ) ) ) ).
thf(fact_3899_odd__real__root__unique,axiom,
! [Y: real,X: real,N: nat] :
( ~ ( even_odd_even_nat @ N )
=> ( ( ( power_power_real @ Y @ N )
= X )
=> ( ( root @ N @ X )
= Y ) ) ) ).
thf(fact_3900_odd__real__root__pow,axiom,
! [X: real,N: nat] :
( ~ ( even_odd_even_nat @ N )
=> ( ( power_power_real @ ( root @ N @ X ) @ N )
= X ) ) ).
thf(fact_3901_odd__real__root__power__cancel,axiom,
! [X: real,N: nat] :
( ~ ( even_odd_even_nat @ N )
=> ( ( root @ N @ ( power_power_real @ X @ N ) )
= X ) ) ).
thf(fact_3902_real__root__le__iff,axiom,
! [X: real,Y: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
<=> ( ord_less_eq_real @ X @ Y ) ) ) ).
thf(fact_3903_real__root__le__mono,axiom,
! [X: real,Y: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
thf(fact_3904_real__root__eq__0__iff,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= zero_zero_real )
<=> ( X = zero_zero_real ) ) ) ).
thf(fact_3905_real__root__less__iff,axiom,
! [X: real,Y: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
<=> ( ord_less_real @ X @ Y ) ) ) ).
thf(fact_3906_real__root__less__mono,axiom,
! [X: real,Y: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
thf(fact_3907_real__root__mult,axiom,
! [X: real,Y: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( times_times_real @ X @ Y ) )
= ( times_times_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
thf(fact_3908_real__root__mult__exp,axiom,
! [X: real,N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ ( times_times_nat @ M @ N ) @ X )
= ( root @ M @ ( root @ N @ X ) ) ) ) ) ).
thf(fact_3909_real__root__power,axiom,
! [X: real,K_1: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( power_power_real @ X @ K_1 ) )
= ( power_power_real @ ( root @ N @ X ) @ K_1 ) ) ) ).
thf(fact_3910_real__root__one,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ one_one_real )
= one_one_real ) ) ).
thf(fact_3911_real__root__eq__1__iff,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= one_one_real )
<=> ( X = one_one_real ) ) ) ).
thf(fact_3912_real__root__divide,axiom,
! [X: real,Y: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( inverse_divide_real @ X @ Y ) )
= ( inverse_divide_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
thf(fact_3913_real__root__minus,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( root @ N @ X ) ) ) ) ).
thf(fact_3914_real__root__abs,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( abs_abs_real @ X ) )
= ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).
thf(fact_3915_real__root__inverse,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( inverse_inverse_real @ X ) )
= ( inverse_inverse_real @ ( root @ N @ X ) ) ) ) ).
thf(fact_3916_real__root__ge__0__iff,axiom,
! [Y: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y ) )
<=> ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).
thf(fact_3917_real__root__le__0__iff,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
<=> ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
thf(fact_3918_real__root__ge__zero,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
thf(fact_3919_real__root__gt__0__iff,axiom,
! [Y: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y ) )
<=> ( ord_less_real @ zero_zero_real @ Y ) ) ) ).
thf(fact_3920_real__root__lt__0__iff,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
<=> ( ord_less_real @ X @ zero_zero_real ) ) ) ).
thf(fact_3921_real__root__gt__zero,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
thf(fact_3922_real__root__ge__1__iff,axiom,
! [Y: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
<=> ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).
thf(fact_3923_real__root__le__1__iff,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
<=> ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
thf(fact_3924_real__root__decreasing,axiom,
! [X: real,N_3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N_3 )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( root @ N_3 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
thf(fact_3925_real__root__gt__1__iff,axiom,
! [Y: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
<=> ( ord_less_real @ one_one_real @ Y ) ) ) ).
thf(fact_3926_real__root__lt__1__iff,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
<=> ( ord_less_real @ X @ one_one_real ) ) ) ).
thf(fact_3927_real__root__strict__decreasing,axiom,
! [X: real,N_3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N_3 )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( root @ N_3 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
thf(fact_3928_real__root__pos,axiom,
! [N: nat,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( root @ ( suc @ N ) @ ( power_power_real @ X @ ( suc @ N ) ) )
= X ) ) ).
thf(fact_3929_real__root__pos__pos,axiom,
! [X_4: real,N_4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_4 )
=> ( ( ord_less_real @ zero_zero_real @ X_4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N_4 @ X_4 ) ) ) ) ).
thf(fact_3930_real__root__less__mono__lemma,axiom,
! [Y: real,X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ) ).
thf(fact_3931_real__root__mult__lemma,axiom,
! [Y: real,X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( root @ N @ ( times_times_real @ X @ Y ) )
= ( times_times_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ) ) ).
thf(fact_3932_real__root__pos__mult__exp,axiom,
! [X: real,N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( root @ ( times_times_nat @ M @ N ) @ X )
= ( root @ M @ ( root @ N @ X ) ) ) ) ) ) ).
thf(fact_3933_real__root__pos2,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( root @ N @ ( power_power_real @ X @ N ) )
= X ) ) ) ).
thf(fact_3934_real__root__pow__pos2,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
thf(fact_3935_real__root__pos__unique,axiom,
! [X: real,Y: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( power_power_real @ Y @ N )
= X )
=> ( ( root @ N @ X )
= Y ) ) ) ) ).
thf(fact_3936_real__root__pow__pos,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
thf(fact_3937_real__root__increasing,axiom,
! [X: real,N_3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N_3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N_3 @ X ) ) ) ) ) ) ).
thf(fact_3938_real__root__strict__increasing,axiom,
! [X: real,N_3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N_3 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( root @ N @ X ) @ ( root @ N_3 @ X ) ) ) ) ) ) ).
thf(fact_3939_sqrt__def,axiom,
( sqrt
= ( root @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3940_real__root__inverse__lemma,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( root @ N @ ( inverse_inverse_real @ X ) )
= ( inverse_inverse_real @ ( root @ N @ X ) ) ) ) ) ).
thf(fact_3941_DERIV__odd__real__root,axiom,
! [X: real,N: nat] :
( ~ ( even_odd_even_nat @ N )
=> ( ( X != zero_zero_real )
=> ( deriv_real @ ( root @ N ) @ X @ ( inverse_inverse_real @ ( times_times_real @ ( real_nat @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) ).
thf(fact_3942_DERIV__real__root,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( deriv_real @ ( root @ N ) @ X @ ( inverse_inverse_real @ ( times_times_real @ ( real_nat @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) ).
thf(fact_3943_exp__total,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ? [X_1: real] :
( ( exp_real @ X_1 )
= Y ) ) ).
thf(fact_3944_pdivmod__posDivAlg,axiom,
! [K_1: int,L: int] :
( ( ( L = zero_zero_int )
=> ( ( pdivmod @ K_1 @ L )
= ( product_Pair_int_int @ zero_zero_int @ ( abs_abs_int @ K_1 ) ) ) )
& ( ( L != zero_zero_int )
=> ( ( pdivmod @ K_1 @ L )
= ( posDivAlg @ ( abs_abs_int @ K_1 ) @ ( abs_abs_int @ L ) ) ) ) ) ).
thf(fact_3945_MVT2,axiom,
! [F: real > real,F_1: real > real,A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A @ X_1 )
& ( ord_less_eq_real @ X_1 @ B ) )
=> ( deriv_real @ F @ X_1 @ ( F_1 @ X_1 ) ) )
=> ? [Z: real] :
( ( ord_less_real @ A @ Z )
& ( ord_less_real @ Z @ B )
& ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
= ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F_1 @ Z ) ) ) ) ) ) ).
thf(fact_3946_pdivmod__def,axiom,
! [K_1: int,L: int] :
( ( pdivmod @ K_1 @ L )
= ( product_Pair_int_int @ ( div_div_int @ ( abs_abs_int @ K_1 ) @ ( abs_abs_int @ L ) ) @ ( div_mod_int @ ( abs_abs_int @ K_1 ) @ ( abs_abs_int @ L ) ) ) ) ).
thf(fact_3947_polar__ex2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ Y @ zero_zero_real )
=> ? [R: real,A_2: real] :
( ( X
= ( times_times_real @ R @ ( cos @ A_2 ) ) )
& ( Y
= ( times_times_real @ R @ ( sin @ A_2 ) ) ) ) ) ).
thf(fact_3948_polar__ex1,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ? [R: real,A_2: real] :
( ( X
= ( times_times_real @ R @ ( cos @ A_2 ) ) )
& ( Y
= ( times_times_real @ R @ ( sin @ A_2 ) ) ) ) ) ).
thf(fact_3949_z3mod__def,axiom,
! [K_1: int,L: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( z3mod @ K_1 @ L )
= ( div_mod_int @ K_1 @ L ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( z3mod @ K_1 @ L )
= ( div_mod_int @ K_1 @ ( uminus_uminus_int @ L ) ) ) ) ) ).
thf(fact_3950_z3div__def,axiom,
! [K_1: int,L: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( z3div @ K_1 @ L )
= ( div_div_int @ K_1 @ L ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( z3div @ K_1 @ L )
= ( uminus_uminus_int @ ( div_div_int @ K_1 @ ( uminus_uminus_int @ L ) ) ) ) ) ) ).
thf(fact_3951_DERIV__isconst__all,axiom,
! [X: real,Y: real,F: real > real] :
( ! [X_1: real] : ( deriv_real @ F @ X_1 @ zero_zero_real )
=> ( ( F @ X )
= ( F @ Y ) ) ) ).
thf(fact_3952_reals__Archimedean2,axiom,
! [X: real] :
? [N_1: nat] : ( ord_less_real @ X @ ( real_nat @ N_1 ) ) ).
thf(fact_3953_le__Suc__ex,axiom,
! [K_1: nat,L: nat] :
( ( ord_less_eq_nat @ K_1 @ L )
=> ? [N_1: nat] :
( L
= ( plus_plus_nat @ K_1 @ N_1 ) ) ) ).
thf(fact_3954_complex__norm,axiom,
! [X: real,Y: real] :
( ( norm_norm_complex @ ( complex_1 @ X @ Y ) )
= ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3955_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ( ?? @ nat @ P )
=> ? [N_1: nat] :
( ~ ( P @ N_1 )
& ( P @ ( suc @ N_1 ) ) ) ) ) ).
thf(fact_3956_complex_Oinject,axiom,
! [Real1: real,Real2: real,Real1_1: real,Real2_1: real] :
( ( ( complex_1 @ Real1 @ Real2 )
= ( complex_1 @ Real1_1 @ Real2_1 ) )
<=> ( ( Real1 = Real1_1 )
& ( Real2 = Real2_1 ) ) ) ).
thf(fact_3957_Complex__eq__0,axiom,
! [A: real,B: real] :
( ( ( complex_1 @ A @ B )
= zero_zero_complex )
<=> ( ( A = zero_zero_real )
& ( B = zero_zero_real ) ) ) ).
thf(fact_3958_complex__zero__def,axiom,
( zero_zero_complex
= ( complex_1 @ zero_zero_real @ zero_zero_real ) ) ).
thf(fact_3959_Complex__eq__number__of,axiom,
! [A: real,B: real,W: int] :
( ( ( complex_1 @ A @ B )
= ( number528085621omplex @ W ) )
<=> ( ( A
= ( number267125858f_real @ W ) )
& ( B = zero_zero_real ) ) ) ).
thf(fact_3960_complex__one__def,axiom,
( one_one_complex
= ( complex_1 @ one_one_real @ zero_zero_real ) ) ).
thf(fact_3961_Complex__eq__1,axiom,
! [A: real,B: real] :
( ( ( complex_1 @ A @ B )
= one_one_complex )
<=> ( ( A = one_one_real )
& ( B = zero_zero_real ) ) ) ).
thf(fact_3962_complex__add,axiom,
! [A: real,B: real,C: real,D: real] :
( ( plus_plus_complex @ ( complex_1 @ A @ B ) @ ( complex_1 @ C @ D ) )
= ( complex_1 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
thf(fact_3963_complex__minus,axiom,
! [A: real,B: real] :
( ( uminus473333897omplex @ ( complex_1 @ A @ B ) )
= ( complex_1 @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
thf(fact_3964_complex__diff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( minus_minus_complex @ ( complex_1 @ A @ B ) @ ( complex_1 @ C @ D ) )
= ( complex_1 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).
thf(fact_3965_complex__mult,axiom,
! [A: real,B: real,C: real,D: real] :
( ( times_times_complex @ ( complex_1 @ A @ B ) @ ( complex_1 @ C @ D ) )
= ( complex_1 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
thf(fact_3966_cmod__unit__one,axiom,
! [A: real] :
( ( norm_norm_complex @ ( complex_1 @ ( cos @ A ) @ ( sin @ A ) ) )
= one_one_real ) ).
thf(fact_3967_complex__inverse,axiom,
! [A: real,B: real] :
( ( invers1449016382omplex @ ( complex_1 @ A @ B ) )
= ( complex_1 @ ( inverse_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( inverse_divide_real @ ( uminus_uminus_real @ B ) @ ( plus_plus_real @ ( power_power_real @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ).
thf(fact_3968_complex_Osize_I1_J,axiom,
! [Real1: real,Real2: real] :
( ( complex_size @ ( complex_1 @ Real1 @ Real2 ) )
= zero_zero_nat ) ).
thf(fact_3969_cos__arg__i__mult__zero__pos,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( cos @ ( arg @ ( complex_1 @ zero_zero_real @ Y ) ) )
= zero_zero_real ) ) ).
thf(fact_3970_cos__arg__i__mult__zero__neg,axiom,
! [Y: real] :
( ( ord_less_real @ Y @ zero_zero_real )
=> ( ( cos @ ( arg @ ( complex_1 @ zero_zero_real @ Y ) ) )
= zero_zero_real ) ) ).
thf(fact_3971_complex_Osize_I2_J,axiom,
! [Real1: real,Real2: real] :
( ( size_size_complex @ ( complex_1 @ Real1 @ Real2 ) )
= zero_zero_nat ) ).
thf(fact_3972_complex__divide__def,axiom,
! [X: complex,Y: complex] :
( ( invers1025623611omplex @ X @ Y )
= ( times_times_complex @ X @ ( invers1449016382omplex @ Y ) ) ) ).
thf(fact_3973_cos__arg__i__mult__zero,axiom,
! [Y: real] :
( ( Y != zero_zero_real )
=> ( ( cos @ ( arg @ ( complex_1 @ zero_zero_real @ Y ) ) )
= zero_zero_real ) ) ).
thf(fact_3974_power2__i,axiom,
( ( power_power_complex @ ii @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( number528085621omplex @ min ) ) ).
thf(fact_3975_complex__i__not__zero,axiom,
ii != zero_zero_complex ).
thf(fact_3976_complex__i__not__one,axiom,
ii != one_one_complex ).
thf(fact_3977_complex__i__not__number__of,axiom,
! [W: int] :
( ii
!= ( number528085621omplex @ W ) ) ).
thf(fact_3978_complex__i__mult__minus,axiom,
! [X: complex] :
( ( times_times_complex @ ii @ ( times_times_complex @ ii @ X ) )
= ( uminus473333897omplex @ X ) ) ).
thf(fact_3979_inverse__i,axiom,
( ( invers1449016382omplex @ ii )
= ( uminus473333897omplex @ ii ) ) ).
thf(fact_3980_i__def,axiom,
( ii
= ( complex_1 @ zero_zero_real @ one_one_real ) ) ).
thf(fact_3981_Complex__eq__i,axiom,
! [X: real,Y: real] :
( ( ( complex_1 @ X @ Y )
= ii )
<=> ( ( X = zero_zero_real )
& ( Y = one_one_real ) ) ) ).
thf(fact_3982_Complex__mult__i,axiom,
! [A: real,B: real] :
( ( times_times_complex @ ( complex_1 @ A @ B ) @ ii )
= ( complex_1 @ ( uminus_uminus_real @ B ) @ A ) ) ).
thf(fact_3983_i__mult__Complex,axiom,
! [A: real,B: real] :
( ( times_times_complex @ ii @ ( complex_1 @ A @ B ) )
= ( complex_1 @ ( uminus_uminus_real @ B ) @ A ) ) ).
thf(fact_3984_i__squared,axiom,
( ( times_times_complex @ ii @ ii )
= ( number528085621omplex @ min ) ) ).
thf(fact_3985_i__mult__eq2,axiom,
( ( times_times_complex @ ii @ ii )
= ( uminus473333897omplex @ one_one_complex ) ) ).
thf(fact_3986_complex__inverse__complex__split,axiom,
! [X: real,Y: real] :
( ( invers1449016382omplex @ ( plus_plus_complex @ ( of_real_complex @ X ) @ ( times_times_complex @ ii @ ( of_real_complex @ Y ) ) ) )
= ( minus_minus_complex @ ( of_real_complex @ ( inverse_divide_real @ X @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) @ ( times_times_complex @ ii @ ( of_real_complex @ ( inverse_divide_real @ Y @ ( plus_plus_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ) ).
thf(fact_3987_complex__eq__cancel__iff2,axiom,
! [X: real,Y: real,Xa_1: real] :
( ( ( complex_1 @ X @ Y )
= ( of_real_complex @ Xa_1 ) )
<=> ( ( X = Xa_1 )
& ( Y = zero_zero_real ) ) ) ).
thf(fact_3988_complex__of__real__def,axiom,
! [R_1: real] :
( ( of_real_complex @ R_1 )
= ( complex_1 @ R_1 @ zero_zero_real ) ) ).
thf(fact_3989_sgn__eq,axiom,
! [Z_1: complex] :
( ( sgn_sgn_complex @ Z_1 )
= ( invers1025623611omplex @ Z_1 @ ( of_real_complex @ ( norm_norm_complex @ Z_1 ) ) ) ) ).
thf(fact_3990_Complex__mult__complex__of__real,axiom,
! [X: real,Y: real,R_1: real] :
( ( times_times_complex @ ( complex_1 @ X @ Y ) @ ( of_real_complex @ R_1 ) )
= ( complex_1 @ ( times_times_real @ X @ R_1 ) @ ( times_times_real @ Y @ R_1 ) ) ) ).
thf(fact_3991_complex__of__real__mult__Complex,axiom,
! [R_1: real,X: real,Y: real] :
( ( times_times_complex @ ( of_real_complex @ R_1 ) @ ( complex_1 @ X @ Y ) )
= ( complex_1 @ ( times_times_real @ R_1 @ X ) @ ( times_times_real @ R_1 @ Y ) ) ) ).
thf(fact_3992_Complex__add__complex__of__real,axiom,
! [X: real,Y: real,R_1: real] :
( ( plus_plus_complex @ ( complex_1 @ X @ Y ) @ ( of_real_complex @ R_1 ) )
= ( complex_1 @ ( plus_plus_real @ X @ R_1 ) @ Y ) ) ).
thf(fact_3993_complex__of__real__add__Complex,axiom,
! [R_1: real,X: real,Y: real] :
( ( plus_plus_complex @ ( of_real_complex @ R_1 ) @ ( complex_1 @ X @ Y ) )
= ( complex_1 @ ( plus_plus_real @ R_1 @ X ) @ Y ) ) ).
thf(fact_3994_i__mult__eq,axiom,
( ( times_times_complex @ ii @ ii )
= ( of_real_complex @ ( number267125858f_real @ min ) ) ) ).
thf(fact_3995_complex__of__real__i,axiom,
! [R_1: real] :
( ( times_times_complex @ ( of_real_complex @ R_1 ) @ ii )
= ( complex_1 @ zero_zero_real @ R_1 ) ) ).
thf(fact_3996_i__complex__of__real,axiom,
! [R_1: real] :
( ( times_times_complex @ ii @ ( of_real_complex @ R_1 ) )
= ( complex_1 @ zero_zero_real @ R_1 ) ) ).
thf(fact_3997_complex__of__real__minus__one,axiom,
( ( of_real_complex @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus473333897omplex @ one_one_complex ) ) ).
thf(fact_3998_cmod__complex__polar,axiom,
! [R_1: real,A: real] :
( ( norm_norm_complex @ ( times_times_complex @ ( of_real_complex @ R_1 ) @ ( complex_1 @ ( cos @ A ) @ ( sin @ A ) ) ) )
= ( abs_abs_real @ R_1 ) ) ).
thf(fact_3999_expi__two__pi__i,axiom,
( ( expi @ ( times_times_complex @ ( times_times_complex @ ( number528085621omplex @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( of_real_complex @ pi ) ) @ ii ) )
= one_one_complex ) ).
thf(fact_4000_expi__add,axiom,
! [A: complex,B: complex] :
( ( expi @ ( plus_plus_complex @ A @ B ) )
= ( times_times_complex @ ( expi @ A ) @ ( expi @ B ) ) ) ).
thf(fact_4001_expi__zero,axiom,
( ( expi @ zero_zero_complex )
= one_one_complex ) ).
thf(fact_4002_complex__split__polar,axiom,
! [Z_1: complex] :
? [R: real,A_2: real] :
( Z_1
= ( times_times_complex @ ( of_real_complex @ R ) @ ( complex_1 @ ( cos @ A_2 ) @ ( sin @ A_2 ) ) ) ) ).
thf(fact_4003_DERIV__pos__imp__increasing,axiom,
! [F: real > real,A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A @ X_1 )
& ( ord_less_eq_real @ X_1 @ B ) )
=> ? [Y_1: real] :
( ( deriv_real @ F @ X_1 @ Y_1 )
& ( ord_less_real @ zero_zero_real @ Y_1 ) ) )
=> ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
thf(fact_4004_DERIV__neg__imp__decreasing,axiom,
! [F: real > real,A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A @ X_1 )
& ( ord_less_eq_real @ X_1 @ B ) )
=> ? [Y_1: real] :
( ( deriv_real @ F @ X_1 @ Y_1 )
& ( ord_less_real @ Y_1 @ zero_zero_real ) ) )
=> ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
thf(fact_4005_int__induct,axiom,
! [I: int,P: int > $o,K_1: int] :
( ( P @ K_1 )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ K_1 @ I_1 )
=> ( ( P @ I_1 )
=> ( P @ ( plus_plus_int @ I_1 @ one_one_int ) ) ) )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ I_1 @ K_1 )
=> ( ( P @ I_1 )
=> ( P @ ( minus_minus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_4006_minusinfinity,axiom,
! [P: int > $o,P1: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int,K: int] :
( ( P1 @ X_1 )
<=> ( P1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K @ D ) ) ) )
=> ( ? [Z: int] :
! [X_1: int] :
( ( ord_less_int @ X_1 @ Z )
=> ( ( P @ X_1 )
<=> ( P1 @ X_1 ) ) )
=> ( ( ?? @ int @ P1 )
=> ( ?? @ int @ P ) ) ) ) ) ).
thf(fact_4007_plusinfinity,axiom,
! [P: int > $o,P_1: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int,K: int] :
( ( P_1 @ X_1 )
<=> ( P_1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K @ D ) ) ) )
=> ( ? [Z: int] :
! [X_1: int] :
( ( ord_less_int @ Z @ X_1 )
=> ( ( P @ X_1 )
<=> ( P_1 @ X_1 ) ) )
=> ( ( ?? @ int @ P_1 )
=> ( ?? @ int @ P ) ) ) ) ) ).
thf(fact_4008_coprime__sos,axiom,
! [X: nat,Y: nat] :
( ( coprime @ X @ Y )
=> ( coprime @ ( times_times_nat @ X @ Y ) @ ( plus_plus_nat @ ( power_power_nat @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_nat @ Y @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_4009_coprime__lmul2,axiom,
! [D: nat,A: nat,B: nat] :
( ( coprime @ D @ ( times_times_nat @ A @ B ) )
=> ( coprime @ D @ B ) ) ).
thf(fact_4010_coprime__rmul2,axiom,
! [D: nat,A: nat,B: nat] :
( ( coprime @ D @ ( times_times_nat @ A @ B ) )
=> ( coprime @ D @ A ) ) ).
thf(fact_4011_coprime__mul,axiom,
! [B: nat,D: nat,A: nat] :
( ( coprime @ D @ A )
=> ( ( coprime @ D @ B )
=> ( coprime @ D @ ( times_times_nat @ A @ B ) ) ) ) ).
thf(fact_4012_coprime__mul__eq,axiom,
! [D: nat,A: nat,B: nat] :
( ( coprime @ D @ ( times_times_nat @ A @ B ) )
<=> ( ( coprime @ D @ A )
& ( coprime @ D @ B ) ) ) ).
thf(fact_4013_coprime__refl,axiom,
! [N: nat] :
( ( coprime @ N @ N )
<=> ( N = one_one_nat ) ) ).
thf(fact_4014_coprime__1,axiom,
! [A: nat] : ( coprime @ A @ one_one_nat ) ).
thf(fact_4015_coprime__1_H,axiom,
! [A: nat] : ( coprime @ one_one_nat @ A ) ).
thf(fact_4016_coprime__commute,axiom,
! [A: nat,B: nat] :
( ( coprime @ A @ B )
<=> ( coprime @ B @ A ) ) ).
thf(fact_4017_coprime__divisors,axiom,
! [E_1: nat,B: nat,D: nat,A: nat] :
( ( dvd_dvd_nat @ D @ A )
=> ( ( dvd_dvd_nat @ E_1 @ B )
=> ( ( coprime @ A @ B )
=> ( coprime @ D @ E_1 ) ) ) ) ).
thf(fact_4018_coprime__exp__imp,axiom,
! [N: nat,A: nat,B: nat] :
( ( coprime @ A @ B )
=> ( coprime @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
thf(fact_4019_coprime__exp,axiom,
! [N: nat,D: nat,A: nat] :
( ( coprime @ D @ A )
=> ( coprime @ D @ ( power_power_nat @ A @ N ) ) ) ).
thf(fact_4020_floor__real__of__nat,axiom,
! [N: nat] :
( ( archim1246769320r_real @ ( real_nat @ N ) )
= ( semiri1621563631at_int @ N ) ) ).
thf(fact_4021_natfloor__def,axiom,
! [X: real] :
( ( natfloor @ X )
= ( nat_1 @ ( archim1246769320r_real @ X ) ) ) ).
thf(fact_4022_coprime__Suc0_H,axiom,
! [A: nat] : ( coprime @ ( suc @ zero_zero_nat ) @ A ) ).
thf(fact_4023_coprime__Suc0,axiom,
! [A: nat] : ( coprime @ A @ ( suc @ zero_zero_nat ) ) ).
thf(fact_4024_coprime__0,axiom,
! [D: nat] :
( ( coprime @ D @ zero_zero_nat )
<=> ( D = one_one_nat ) ) ).
thf(fact_4025_coprime__0_H,axiom,
! [D: nat] :
( ( coprime @ zero_zero_nat @ D )
<=> ( D = one_one_nat ) ) ).
thf(fact_4026_divides__mul,axiom,
! [N: nat,M: nat,R_1: nat] :
( ( dvd_dvd_nat @ M @ R_1 )
=> ( ( dvd_dvd_nat @ N @ R_1 )
=> ( ( coprime @ M @ N )
=> ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ R_1 ) ) ) ) ).
thf(fact_4027_coprime__divprod,axiom,
! [D: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ D @ ( times_times_nat @ A @ B ) )
=> ( ( coprime @ D @ A )
=> ( dvd_dvd_nat @ D @ B ) ) ) ).
thf(fact_4028_coprime,axiom,
! [A: nat,B: nat] :
( ( coprime @ A @ B )
<=> ! [D_2: nat] :
( ( ( dvd_dvd_nat @ D_2 @ A )
& ( dvd_dvd_nat @ D_2 @ B ) )
<=> ( D_2 = one_one_nat ) ) ) ).
thf(fact_4029_coprime__plus1,axiom,
! [N: nat] : ( coprime @ ( plus_plus_nat @ N @ one_one_nat ) @ N ) ).
thf(fact_4030_coprime__exp2,axiom,
! [A: nat,N: nat,B: nat] :
( ( coprime @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
<=> ( coprime @ A @ B ) ) ).
thf(fact_4031_coprime__minus1,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( coprime @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
thf(fact_4032_floor__eq3,axiom,
! [N: nat,X: real] :
( ( ord_less_real @ ( real_nat @ N ) @ X )
=> ( ( ord_less_real @ X @ ( real_nat @ ( suc @ N ) ) )
=> ( ( nat_1 @ ( archim1246769320r_real @ X ) )
= N ) ) ) ).
thf(fact_4033_floor__minus__real__of__nat,axiom,
! [N: nat] :
( ( archim1246769320r_real @ ( uminus_uminus_real @ ( real_nat @ N ) ) )
= ( uminus_uminus_int @ ( semiri1621563631at_int @ N ) ) ) ).
thf(fact_4034_le__mult__floor,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_int @ ( times_times_int @ ( archim1246769320r_real @ A ) @ ( archim1246769320r_real @ B ) ) @ ( archim1246769320r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
thf(fact_4035_floor__eq4,axiom,
! [N: nat,X: real] :
( ( ord_less_eq_real @ ( real_nat @ N ) @ X )
=> ( ( ord_less_real @ X @ ( real_nat @ ( suc @ N ) ) )
=> ( ( nat_1 @ ( archim1246769320r_real @ X ) )
= N ) ) ) ).
thf(fact_4036_coprime__bezout,axiom,
! [A: nat,B: nat] :
( ( coprime @ A @ B )
<=> ? [X_1: nat,Y_1: nat] :
( ( ( minus_minus_nat @ ( times_times_nat @ A @ X_1 ) @ ( times_times_nat @ B @ Y_1 ) )
= one_one_nat )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X_1 ) @ ( times_times_nat @ A @ Y_1 ) )
= one_one_nat ) ) ) ).
thf(fact_4037_coprime__bezout__strong,axiom,
! [A: nat,B: nat] :
( ( coprime @ A @ B )
=> ( ( B != one_one_nat )
=> ? [X_1: nat,Y_1: nat] :
( ( times_times_nat @ A @ X_1 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y_1 ) @ one_one_nat ) ) ) ) ).
thf(fact_4038_nat__number__of__diff__1,axiom,
! [V: int] :
( ( ( ord_less_eq_int @ V @ pls )
=> ( ( minus_minus_nat @ ( number_number_of_nat @ V ) @ one_one_nat )
= zero_zero_nat ) )
& ( ~ ( ord_less_eq_int @ V @ pls )
=> ( ( minus_minus_nat @ ( number_number_of_nat @ V ) @ one_one_nat )
= ( number_number_of_nat @ ( pred @ V ) ) ) ) ) ).
thf(fact_4039_succ__pred,axiom,
! [X: int] :
( ( succ @ ( pred @ X ) )
= X ) ).
thf(fact_4040_pred__Bit1,axiom,
! [K_1: int] :
( ( pred @ ( bit1 @ K_1 ) )
= ( bit0 @ K_1 ) ) ).
thf(fact_4041_pred__Bit0,axiom,
! [K_1: int] :
( ( pred @ ( bit0 @ K_1 ) )
= ( bit1 @ ( pred @ K_1 ) ) ) ).
thf(fact_4042_le__iff__pred__less,axiom,
! [K_1: int,L: int] :
( ( ord_less_eq_int @ K_1 @ L )
<=> ( ord_less_int @ ( pred @ K_1 ) @ L ) ) ).
thf(fact_4043_pred__Pls,axiom,
( ( pred @ pls )
= min ) ).
thf(fact_4044_pred__Min,axiom,
( ( pred @ min )
= ( bit0 @ min ) ) ).
thf(fact_4045_pred__def,axiom,
! [K_1: int] :
( ( pred @ K_1 )
= ( minus_minus_int @ K_1 @ one_one_int ) ) ).
thf(fact_4046_minus__Bit1,axiom,
! [K_1: int] :
( ( uminus_uminus_int @ ( bit1 @ K_1 ) )
= ( bit1 @ ( pred @ ( uminus_uminus_int @ K_1 ) ) ) ) ).
thf(fact_4047_add__Min__right,axiom,
! [K_1: int] :
( ( plus_plus_int @ K_1 @ min )
= ( pred @ K_1 ) ) ).
thf(fact_4048_add__Min,axiom,
! [K_1: int] :
( ( plus_plus_int @ min @ K_1 )
= ( pred @ K_1 ) ) ).
thf(fact_4049_diff__bin__simps_I8_J,axiom,
! [K_1: int,L: int] :
( ( minus_minus_int @ ( bit0 @ K_1 ) @ ( bit1 @ L ) )
= ( bit1 @ ( minus_minus_int @ ( pred @ K_1 ) @ L ) ) ) ).
thf(fact_4050_neg__number__of__pred__iff__0,axiom,
! [V: int] :
( ( nat_neg @ ( number_number_of_int @ ( pred @ V ) ) )
<=> ( ( number_number_of_nat @ V )
= zero_zero_nat ) ) ).
thf(fact_4051_Suc__diff__number__of,axiom,
! [M: nat,V: int] :
( ( ord_less_int @ pls @ V )
=> ( ( minus_minus_nat @ ( suc @ M ) @ ( number_number_of_nat @ V ) )
= ( minus_minus_nat @ M @ ( number_number_of_nat @ ( pred @ V ) ) ) ) ) ).
thf(fact_4052_complex__expi__Ex,axiom,
! [Z_1: complex] :
? [A_2: complex,R: real] :
( Z_1
= ( times_times_complex @ ( of_real_complex @ R ) @ ( expi @ A_2 ) ) ) ).
thf(fact_4053_ResSet__finite,axiom,
! [X_2: int > $o,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( resSet @ M @ X_2 )
=> ( finite_finite_int @ X_2 ) ) ) ).
thf(fact_4054_ResSet__def,axiom,
! [M: int,X_2: int > $o] :
( ( resSet @ M @ X_2 )
<=> ! [Y1: int,Y2: int] :
( ( ( member_int @ Y1 @ X_2 )
& ( member_int @ Y2 @ X_2 )
& ( zcong @ Y1 @ Y2 @ M ) )
=> ( Y1 = Y2 ) ) ) ).
thf(fact_4055_ResSet__SRStar__prop,axiom,
! [P_3: int] : ( resSet @ P_3 @ ( sRStar @ P_3 ) ) ).
thf(fact_4056_nat_Osize_I2_J,axiom,
! [Nat: nat] :
( ( nat_size @ ( suc @ Nat ) )
= ( plus_plus_nat @ ( nat_size @ Nat ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_4057_nat_Osize_I4_J,axiom,
! [Nat: nat] :
( ( size_size_nat @ ( suc @ Nat ) )
= ( plus_plus_nat @ ( size_size_nat @ Nat ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_4058_code__numeral_Osize_I1_J,axiom,
( ( code_c271388182l_size @ zero_z126310315umeral )
= zero_zero_nat ) ).
thf(fact_4059_nat__size,axiom,
! [N: nat] :
( ( size_size_nat @ N )
= N ) ).
thf(fact_4060_nat_Osize_I3_J,axiom,
( ( size_size_nat @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_4061_nat_Osize_I1_J,axiom,
( ( nat_size @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_4062_code__numeral_Osize_I2_J,axiom,
! [Code_numeral_1: code_code_numeral] :
( ( code_c271388182l_size @ ( code_S1047413653umeral @ Code_numeral_1 ) )
= ( plus_plus_nat @ ( code_c271388182l_size @ Code_numeral_1 ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_4063_complex__mod__mult__cnj,axiom,
! [Z_1: complex] :
( ( norm_norm_complex @ ( times_times_complex @ Z_1 @ ( cnj @ Z_1 ) ) )
= ( power_power_real @ ( norm_norm_complex @ Z_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_4064_complex__cnj__inverse,axiom,
! [X: complex] :
( ( cnj @ ( invers1449016382omplex @ X ) )
= ( invers1449016382omplex @ ( cnj @ X ) ) ) ).
thf(fact_4065_complex__cnj__zero__iff,axiom,
! [Z_1: complex] :
( ( ( cnj @ Z_1 )
= zero_zero_complex )
<=> ( Z_1 = zero_zero_complex ) ) ).
thf(fact_4066_cnj_Ozero,axiom,
( ( cnj @ zero_zero_complex )
= zero_zero_complex ) ).
thf(fact_4067_complex__cnj__number__of,axiom,
! [W: int] :
( ( cnj @ ( number528085621omplex @ W ) )
= ( number528085621omplex @ W ) ) ).
thf(fact_4068_complex__cnj__mult,axiom,
! [X: complex,Y: complex] :
( ( cnj @ ( times_times_complex @ X @ Y ) )
= ( times_times_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
thf(fact_4069_complex__cnj__one,axiom,
( ( cnj @ one_one_complex )
= one_one_complex ) ).
thf(fact_4070_cnj_Ominus,axiom,
! [X: complex] :
( ( cnj @ ( uminus473333897omplex @ X ) )
= ( uminus473333897omplex @ ( cnj @ X ) ) ) ).
thf(fact_4071_cnj_Oadd,axiom,
! [X: complex,Y: complex] :
( ( cnj @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
thf(fact_4072_complex__cnj__cancel__iff,axiom,
! [X: complex,Y: complex] :
( ( ( cnj @ X )
= ( cnj @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_4073_complex__cnj__cnj,axiom,
! [Z_1: complex] :
( ( cnj @ ( cnj @ Z_1 ) )
= Z_1 ) ).
thf(fact_4074_code__numeral_Oinject,axiom,
! [Code_numeral_1: code_code_numeral,Code_numeral_2: code_code_numeral] :
( ( ( code_S1047413653umeral @ Code_numeral_1 )
= ( code_S1047413653umeral @ Code_numeral_2 ) )
<=> ( Code_numeral_1 = Code_numeral_2 ) ) ).
thf(fact_4075_complex__cnj__of__nat,axiom,
! [N: nat] :
( ( cnj @ ( semiri2020571505omplex @ N ) )
= ( semiri2020571505omplex @ N ) ) ).
thf(fact_4076_complex__mod__cnj,axiom,
! [Z_1: complex] :
( ( norm_norm_complex @ ( cnj @ Z_1 ) )
= ( norm_norm_complex @ Z_1 ) ) ).
thf(fact_4077_cnj_Odiff,axiom,
! [X: complex,Y: complex] :
( ( cnj @ ( minus_minus_complex @ X @ Y ) )
= ( minus_minus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
thf(fact_4078_complex__cnj__power,axiom,
! [X: complex,N: nat] :
( ( cnj @ ( power_power_complex @ X @ N ) )
= ( power_power_complex @ ( cnj @ X ) @ N ) ) ).
thf(fact_4079_complex__cnj__divide,axiom,
! [X: complex,Y: complex] :
( ( cnj @ ( invers1025623611omplex @ X @ Y ) )
= ( invers1025623611omplex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
thf(fact_4080_complex__cnj__complex__of__real,axiom,
! [X: real] :
( ( cnj @ ( of_real_complex @ X ) )
= ( of_real_complex @ X ) ) ).
thf(fact_4081_code__numeral_Osimps_I3_J,axiom,
! [Code_numeral_3: code_code_numeral] :
( ( code_S1047413653umeral @ Code_numeral_3 )
!= zero_z126310315umeral ) ).
thf(fact_4082_code__numeral_Osimps_I2_J,axiom,
! [Code_numeral_2: code_code_numeral] :
( zero_z126310315umeral
!= ( code_S1047413653umeral @ Code_numeral_2 ) ) ).
thf(fact_4083_complex__cnj,axiom,
! [A: real,B: real] :
( ( cnj @ ( complex_1 @ A @ B ) )
= ( complex_1 @ A @ ( uminus_uminus_real @ B ) ) ) ).
thf(fact_4084_complex__cnj__i,axiom,
( ( cnj @ ii )
= ( uminus473333897omplex @ ii ) ) ).
thf(fact_4085_Suc__code__numeral__minus__one,axiom,
! [N: code_code_numeral] :
( ( minus_1690775515umeral @ ( code_S1047413653umeral @ N ) @ one_on1645066479umeral )
= N ) ).
thf(fact_4086_cnj_Opos__bounded,axiom,
? [K_3: real] :
( ( ord_less_real @ zero_zero_real @ K_3 )
& ! [X_1: complex] : ( ord_less_eq_real @ ( norm_norm_complex @ ( cnj @ X_1 ) ) @ ( times_times_real @ ( norm_norm_complex @ X_1 ) @ K_3 ) ) ) ).
thf(fact_4087_cnj_Ononneg__bounded,axiom,
? [K_3: real] :
( ( ord_less_eq_real @ zero_zero_real @ K_3 )
& ! [X_1: complex] : ( ord_less_eq_real @ ( norm_norm_complex @ ( cnj @ X_1 ) ) @ ( times_times_real @ ( norm_norm_complex @ X_1 ) @ K_3 ) ) ) ).
thf(fact_4088_cnj_Obounded,axiom,
? [K_3: real] :
! [X_1: complex] : ( ord_less_eq_real @ ( norm_norm_complex @ ( cnj @ X_1 ) ) @ ( times_times_real @ ( norm_norm_complex @ X_1 ) @ K_3 ) ) ).
thf(fact_4089_code__numeral_Osize_I4_J,axiom,
! [Code_numeral_1: code_code_numeral] :
( ( size_s945831648umeral @ ( code_S1047413653umeral @ Code_numeral_1 ) )
= ( plus_plus_nat @ ( size_s945831648umeral @ Code_numeral_1 ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_4090_complex__diff__cnj,axiom,
! [Z_1: complex] :
( ( minus_minus_complex @ Z_1 @ ( cnj @ Z_1 ) )
= ( times_times_complex @ ( of_real_complex @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( im @ Z_1 ) ) ) @ ii ) ) ).
thf(fact_4091_code__numeral_Oexhaust,axiom,
! [Y: code_code_numeral] :
( ( Y != zero_z126310315umeral )
=> ~ ! [Code_numeral: code_code_numeral] :
( Y
!= ( code_S1047413653umeral @ Code_numeral ) ) ) ).
thf(fact_4092_Im,axiom,
! [X: real,Y: real] :
( ( im @ ( complex_1 @ X @ Y ) )
= Y ) ).
thf(fact_4093_complex__Im__cnj,axiom,
! [X: complex] :
( ( im @ ( cnj @ X ) )
= ( uminus_uminus_real @ ( im @ X ) ) ) ).
thf(fact_4094_Im__complex__of__real,axiom,
! [Z_1: real] :
( ( im @ ( of_real_complex @ Z_1 ) )
= zero_zero_real ) ).
thf(fact_4095_complex__Im__i,axiom,
( ( im @ ii )
= one_one_real ) ).
thf(fact_4096_Im_Ozero,axiom,
( ( im @ zero_zero_complex )
= zero_zero_real ) ).
thf(fact_4097_complex__Im__one,axiom,
( ( im @ one_one_complex )
= zero_zero_real ) ).
thf(fact_4098_Im_Oadd,axiom,
! [X: complex,Y: complex] :
( ( im @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).
thf(fact_4099_complex__Im__number__of,axiom,
! [V: int] :
( ( im @ ( number528085621omplex @ V ) )
= zero_zero_real ) ).
thf(fact_4100_Im_Ominus,axiom,
! [X: complex] :
( ( im @ ( uminus473333897omplex @ X ) )
= ( uminus_uminus_real @ ( im @ X ) ) ) ).
thf(fact_4101_Im_Odiff,axiom,
! [X: complex,Y: complex] :
( ( im @ ( minus_minus_complex @ X @ Y ) )
= ( minus_minus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).
thf(fact_4102_complex__Im__of__nat,axiom,
! [N: nat] :
( ( im @ ( semiri2020571505omplex @ N ) )
= zero_zero_real ) ).
thf(fact_4103_abs__Im__le__cmod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( norm_norm_complex @ X ) ) ).
thf(fact_4104_complex__In__mult__cnj__zero,axiom,
! [Z_1: complex] :
( ( im @ ( times_times_complex @ Z_1 @ ( cnj @ Z_1 ) ) )
= zero_zero_real ) ).
thf(fact_4105_Im__sgn,axiom,
! [Z_1: complex] :
( ( im @ ( sgn_sgn_complex @ Z_1 ) )
= ( inverse_divide_real @ ( im @ Z_1 ) @ ( norm_norm_complex @ Z_1 ) ) ) ).
thf(fact_4106_code__numeral_Osize_I3_J,axiom,
( ( size_s945831648umeral @ zero_z126310315umeral )
= zero_zero_nat ) ).
thf(fact_4107_Im_Opos__bounded,axiom,
? [K_3: real] :
( ( ord_less_real @ zero_zero_real @ K_3 )
& ! [X_1: complex] : ( ord_less_eq_real @ ( norm_norm_real @ ( im @ X_1 ) ) @ ( times_times_real @ ( norm_norm_complex @ X_1 ) @ K_3 ) ) ) ).
thf(fact_4108_Im_Ononneg__bounded,axiom,
? [K_3: real] :
( ( ord_less_eq_real @ zero_zero_real @ K_3 )
& ! [X_1: complex] : ( ord_less_eq_real @ ( norm_norm_real @ ( im @ X_1 ) ) @ ( times_times_real @ ( norm_norm_complex @ X_1 ) @ K_3 ) ) ) ).
thf(fact_4109_Im_Obounded,axiom,
? [K_3: real] :
! [X_1: complex] : ( ord_less_eq_real @ ( norm_norm_real @ ( im @ X_1 ) ) @ ( times_times_real @ ( norm_norm_complex @ X_1 ) @ K_3 ) ) ).
thf(fact_4110_complex__inverse__def,axiom,
! [X: complex] :
( ( invers1449016382omplex @ X )
= ( complex_1 @ ( inverse_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( im @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( inverse_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( im @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ).
thf(fact_4111_complex__mult__cnj,axiom,
! [Z_1: complex] :
( ( times_times_complex @ Z_1 @ ( cnj @ Z_1 ) )
= ( of_real_complex @ ( plus_plus_real @ ( power_power_real @ ( re @ Z_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( im @ Z_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_4112_complex__Im__inverse,axiom,
! [X: complex] :
( ( im @ ( invers1449016382omplex @ X ) )
= ( inverse_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( im @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_4113_Re,axiom,
! [X: real,Y: real] :
( ( re @ ( complex_1 @ X @ Y ) )
= X ) ).
thf(fact_4114_Re__complex__of__real,axiom,
! [Z_1: real] :
( ( re @ ( of_real_complex @ Z_1 ) )
= Z_1 ) ).
thf(fact_4115_complex__Re__cnj,axiom,
! [X: complex] :
( ( re @ ( cnj @ X ) )
= ( re @ X ) ) ).
thf(fact_4116_complex__eq__iff,axiom,
! [X: complex,Y: complex] :
( ( X = Y )
<=> ( ( ( re @ X )
= ( re @ Y ) )
& ( ( im @ X )
= ( im @ Y ) ) ) ) ).
thf(fact_4117_complex__eqI,axiom,
! [X: complex,Y: complex] :
( ( ( re @ X )
= ( re @ Y ) )
=> ( ( ( im @ X )
= ( im @ Y ) )
=> ( X = Y ) ) ) ).
thf(fact_4118_complex__Re__le__cmod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( re @ X ) @ ( norm_norm_complex @ X ) ) ).
thf(fact_4119_complex__Re__i,axiom,
( ( re @ ii )
= zero_zero_real ) ).
thf(fact_4120_complex__surj,axiom,
! [Z_1: complex] :
( ( complex_1 @ ( re @ Z_1 ) @ ( im @ Z_1 ) )
= Z_1 ) ).
thf(fact_4121_Re_Ozero,axiom,
( ( re @ zero_zero_complex )
= zero_zero_real ) ).
thf(fact_4122_Re_Oadd,axiom,
! [X: complex,Y: complex] :
( ( re @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).
thf(fact_4123_complex__Re__one,axiom,
( ( re @ one_one_complex )
= one_one_real ) ).
thf(fact_4124_Re_Ominus,axiom,
! [X: complex] :
( ( re @ ( uminus473333897omplex @ X ) )
= ( uminus_uminus_real @ ( re @ X ) ) ) ).
thf(fact_4125_complex__Re__number__of,axiom,
! [V: int] :
( ( re @ ( number528085621omplex @ V ) )
= ( number267125858f_real @ V ) ) ).
thf(fact_4126_Re_Odiff,axiom,
! [X: complex,Y: complex] :
( ( re @ ( minus_minus_complex @ X @ Y ) )
= ( minus_minus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).
thf(fact_4127_complex__Re__of__nat,axiom,
! [N: nat] :
( ( re @ ( semiri2020571505omplex @ N ) )
= ( semiri132038758t_real @ N ) ) ).
thf(fact_4128_abs__Re__le__cmod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( norm_norm_complex @ X ) ) ).
thf(fact_4129_Re__sgn,axiom,
! [Z_1: complex] :
( ( re @ ( sgn_sgn_complex @ Z_1 ) )
= ( inverse_divide_real @ ( re @ Z_1 ) @ ( norm_norm_complex @ Z_1 ) ) ) ).
thf(fact_4130_complex__Im__mult,axiom,
! [X: complex,Y: complex] :
( ( im @ ( times_times_complex @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ).
thf(fact_4131_complex__Re__mult,axiom,
! [X: complex,Y: complex] :
( ( re @ ( times_times_complex @ X @ Y ) )
= ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ).
thf(fact_4132_cnj__def,axiom,
! [Z_1: complex] :
( ( cnj @ Z_1 )
= ( complex_1 @ ( re @ Z_1 ) @ ( uminus_uminus_real @ ( im @ Z_1 ) ) ) ) ).
thf(fact_4133_complex__mod__sqrt__Re__mult__cnj,axiom,
! [Z_1: complex] :
( ( norm_norm_complex @ Z_1 )
= ( sqrt @ ( re @ ( times_times_complex @ Z_1 @ ( cnj @ Z_1 ) ) ) ) ) ).
thf(fact_4134_complex__add__def,axiom,
! [X: complex,Y: complex] :
( ( plus_plus_complex @ X @ Y )
= ( complex_1 @ ( plus_plus_real @ ( re @ X ) @ ( re @ Y ) ) @ ( plus_plus_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ).
thf(fact_4135_complex__minus__def,axiom,
! [X: complex] :
( ( uminus473333897omplex @ X )
= ( complex_1 @ ( uminus_uminus_real @ ( re @ X ) ) @ ( uminus_uminus_real @ ( im @ X ) ) ) ) ).
thf(fact_4136_complex__mult__def,axiom,
! [X: complex,Y: complex] :
( ( times_times_complex @ X @ Y )
= ( complex_1 @ ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ) ).
thf(fact_4137_complex__add__cnj,axiom,
! [Z_1: complex] :
( ( plus_plus_complex @ Z_1 @ ( cnj @ Z_1 ) )
= ( of_real_complex @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( re @ Z_1 ) ) ) ) ).
thf(fact_4138_cmod__def,axiom,
! [Z_1: complex] :
( ( norm_norm_complex @ Z_1 )
= ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( im @ Z_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_4139_complex__Re__inverse,axiom,
! [X: complex] :
( ( re @ ( invers1449016382omplex @ X ) )
= ( inverse_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( im @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_4140_Re_Opos__bounded,axiom,
? [K_3: real] :
( ( ord_less_real @ zero_zero_real @ K_3 )
& ! [X_1: complex] : ( ord_less_eq_real @ ( norm_norm_real @ ( re @ X_1 ) ) @ ( times_times_real @ ( norm_norm_complex @ X_1 ) @ K_3 ) ) ) ).
thf(fact_4141_Re_Ononneg__bounded,axiom,
? [K_3: real] :
( ( ord_less_eq_real @ zero_zero_real @ K_3 )
& ! [X_1: complex] : ( ord_less_eq_real @ ( norm_norm_real @ ( re @ X_1 ) ) @ ( times_times_real @ ( norm_norm_complex @ X_1 ) @ K_3 ) ) ) ).
thf(fact_4142_Re_Obounded,axiom,
? [K_3: real] :
! [X_1: complex] : ( ord_less_eq_real @ ( norm_norm_real @ ( re @ X_1 ) ) @ ( times_times_real @ ( norm_norm_complex @ X_1 ) @ K_3 ) ) ).
thf(fact_4143_expi__def,axiom,
! [Z_1: complex] :
( ( expi @ Z_1 )
= ( times_times_complex @ ( of_real_complex @ ( exp_real @ ( re @ Z_1 ) ) ) @ ( cis @ ( im @ Z_1 ) ) ) ) ).
thf(fact_4144_sin__n__Im__cis__pow__n,axiom,
! [N: nat,A: real] :
( ( sin @ ( times_times_real @ ( real_nat @ N ) @ A ) )
= ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
thf(fact_4145_cos__n__Re__cis__pow__n,axiom,
! [N: nat,A: real] :
( ( cos @ ( times_times_real @ ( real_nat @ N ) @ A ) )
= ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
thf(fact_4146_Re__cis,axiom,
! [A: real] :
( ( re @ ( cis @ A ) )
= ( cos @ A ) ) ).
thf(fact_4147_Im__cis,axiom,
! [A: real] :
( ( im @ ( cis @ A ) )
= ( sin @ A ) ) ).
thf(fact_4148_cis__mult,axiom,
! [A: real,B: real] :
( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
= ( cis @ ( plus_plus_real @ A @ B ) ) ) ).
thf(fact_4149_cis__zero,axiom,
( ( cis @ zero_zero_real )
= one_one_complex ) ).
thf(fact_4150_cis__inverse,axiom,
! [A: real] :
( ( invers1449016382omplex @ ( cis @ A ) )
= ( cis @ ( uminus_uminus_real @ A ) ) ) ).
thf(fact_4151_cis__divide,axiom,
! [A: real,B: real] :
( ( invers1025623611omplex @ ( cis @ A ) @ ( cis @ B ) )
= ( cis @ ( minus_minus_real @ A @ B ) ) ) ).
thf(fact_4152_cis__def,axiom,
! [A: real] :
( ( cis @ A )
= ( complex_1 @ ( cos @ A ) @ ( sin @ A ) ) ) ).
thf(fact_4153_DeMoivre,axiom,
! [A: real,N: nat] :
( ( power_power_complex @ ( cis @ A ) @ N )
= ( cis @ ( times_times_real @ ( real_nat @ N ) @ A ) ) ) ).
thf(fact_4154_cis__real__of__nat__Suc__mult,axiom,
! [N: nat,A: real] :
( ( cis @ ( times_times_real @ ( real_nat @ ( suc @ N ) ) @ A ) )
= ( times_times_complex @ ( cis @ A ) @ ( cis @ ( times_times_real @ ( real_nat @ N ) @ A ) ) ) ) ).
thf(fact_4155_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
thf(fact_4156_termination__basic__simps_I4_J,axiom,
! [Y: nat,X: nat,Z_1: nat] :
( ( ord_less_eq_nat @ X @ Z_1 )
=> ( ord_less_eq_nat @ X @ ( plus_plus_nat @ Y @ Z_1 ) ) ) ).
thf(fact_4157_termination__basic__simps_I1_J,axiom,
! [Z_1: nat,X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z_1 ) ) ) ).
thf(fact_4158_termination__basic__simps_I2_J,axiom,
! [Y: nat,X: nat,Z_1: nat] :
( ( ord_less_nat @ X @ Z_1 )
=> ( ord_less_nat @ X @ ( plus_plus_nat @ Y @ Z_1 ) ) ) ).
thf(fact_4159_termination__basic__simps_I5_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
thf(fact_4160_termination__basic__simps_I3_J,axiom,
! [Z_1: nat,X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ ( plus_plus_nat @ Y @ Z_1 ) ) ) ).
thf(fact_4161_prime__dvd__power__two,axiom,
! [M: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( dvd_dvd_nat @ P_3 @ ( power_power_nat @ M @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( dvd_dvd_nat @ P_3 @ M ) ) ) ).
thf(fact_4162_divmod__int__def,axiom,
! [B: int,A: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( divmod_int @ A @ B )
= ( posDivAlg @ A @ B ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( A = zero_zero_int )
=> ( ( divmod_int @ A @ B )
= ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) )
& ( ( A != zero_zero_int )
=> ( ( divmod_int @ A @ B )
= ( negateSnd @ ( negDivAlg @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( divmod_int @ A @ B )
= ( negDivAlg @ A @ B ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ B )
=> ( ( divmod_int @ A @ B )
= ( negateSnd @ ( posDivAlg @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ) ) ) ) ) ).
thf(fact_4163_prime__0,axiom,
~ ( prime @ zero_zero_nat ) ).
thf(fact_4164_prime__1,axiom,
~ ( prime @ one_one_nat ) ).
thf(fact_4165_primes__eq,axiom,
! [Q: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( prime @ Q )
=> ( ( dvd_dvd_nat @ P_3 @ Q )
=> ( P_3 = Q ) ) ) ) ).
thf(fact_4166_distinct__prime__coprime,axiom,
! [Q: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( prime @ Q )
=> ( ( P_3 != Q )
=> ( coprime @ P_3 @ Q ) ) ) ) ).
thf(fact_4167_prime__Suc0,axiom,
~ ( prime @ ( suc @ zero_zero_nat ) ) ).
thf(fact_4168_prime__g__zero,axiom,
! [P_3: nat] :
( ( prime @ P_3 )
=> ( ord_less_nat @ zero_zero_nat @ P_3 ) ) ).
thf(fact_4169_prime__divprod__eq,axiom,
! [A: nat,B: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( dvd_dvd_nat @ P_3 @ ( times_times_nat @ A @ B ) )
<=> ( ( dvd_dvd_nat @ P_3 @ A )
| ( dvd_dvd_nat @ P_3 @ B ) ) ) ) ).
thf(fact_4170_prime__divprod,axiom,
! [A: nat,B: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( dvd_dvd_nat @ P_3 @ ( times_times_nat @ A @ B ) )
=> ( ( dvd_dvd_nat @ P_3 @ A )
| ( dvd_dvd_nat @ P_3 @ B ) ) ) ) ).
thf(fact_4171_prime__dvd__mult,axiom,
! [M: nat,N: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( dvd_dvd_nat @ P_3 @ ( times_times_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ P_3 @ M )
| ( dvd_dvd_nat @ P_3 @ N ) ) ) ) ).
thf(fact_4172_prime__product,axiom,
! [P_3: nat,Q: nat] :
( ( prime @ ( times_times_nat @ P_3 @ Q ) )
=> ( ( P_3 = one_one_nat )
| ( Q = one_one_nat ) ) ) ).
thf(fact_4173_prime__divexp__n,axiom,
! [X: nat,N: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( dvd_dvd_nat @ P_3 @ ( power_power_nat @ X @ N ) )
=> ( dvd_dvd_nat @ ( power_power_nat @ P_3 @ N ) @ ( power_power_nat @ X @ N ) ) ) ) ).
thf(fact_4174_prime__divexp,axiom,
! [X: nat,N: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( dvd_dvd_nat @ P_3 @ ( power_power_nat @ X @ N ) )
=> ( dvd_dvd_nat @ P_3 @ X ) ) ) ).
thf(fact_4175_prime__dvd__power,axiom,
! [A: nat,N: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( dvd_dvd_nat @ P_3 @ ( power_power_nat @ A @ N ) )
=> ( dvd_dvd_nat @ P_3 @ A ) ) ) ).
thf(fact_4176_prime__exp,axiom,
! [P_3: nat,N: nat] :
( ( prime @ ( power_power_nat @ P_3 @ N ) )
<=> ( ( prime @ P_3 )
& ( N = one_one_nat ) ) ) ).
thf(fact_4177_coprime__prime,axiom,
! [P_3: nat,A: nat,B: nat] :
( ( coprime @ A @ B )
=> ~ ( ( prime @ P_3 )
& ( dvd_dvd_nat @ P_3 @ A )
& ( dvd_dvd_nat @ P_3 @ B ) ) ) ).
thf(fact_4178_prime__coprime__strong,axiom,
! [N: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( dvd_dvd_nat @ P_3 @ N )
| ( coprime @ P_3 @ N ) ) ) ).
thf(fact_4179_coprime__prime__eq,axiom,
! [A: nat,B: nat] :
( ( coprime @ A @ B )
<=> ! [P_4: nat] :
~ ( ( prime @ P_4 )
& ( dvd_dvd_nat @ P_4 @ A )
& ( dvd_dvd_nat @ P_4 @ B ) ) ) ).
thf(fact_4180_prime__impl__zprime__int,axiom,
! [A: nat] :
( ( prime @ A )
=> ( zprime @ ( semiri1621563631at_int @ A ) ) ) ).
thf(fact_4181_prime__g__one,axiom,
! [P_3: nat] :
( ( prime @ P_3 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ P_3 ) ) ).
thf(fact_4182_prime__nd__one,axiom,
! [P_3: nat] :
( ( prime @ P_3 )
=> ~ ( dvd_dvd_nat @ P_3 @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_4183_prime__factor__lt,axiom,
! [M: nat,N: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( N != zero_zero_nat )
=> ( ( N
= ( times_times_nat @ P_3 @ M ) )
=> ( ord_less_nat @ M @ N ) ) ) ) ).
thf(fact_4184_prime__def,axiom,
! [P_3: nat] :
( ( prime @ P_3 )
<=> ( ( ord_less_nat @ one_one_nat @ P_3 )
& ! [M_2: nat] :
( ( dvd_dvd_nat @ M_2 @ P_3 )
=> ( ( M_2 = one_one_nat )
| ( M_2 = P_3 ) ) ) ) ) ).
thf(fact_4185_divides__primepow,axiom,
! [D: nat,K_1: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( dvd_dvd_nat @ D @ ( power_power_nat @ P_3 @ K_1 ) )
<=> ? [I_1: nat] :
( ( ord_less_eq_nat @ I_1 @ K_1 )
& ( D
= ( power_power_nat @ P_3 @ I_1 ) ) ) ) ) ).
thf(fact_4186_prime__power__dvd__cancel__right,axiom,
! [N: nat,A: nat,B: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ~ ( dvd_dvd_nat @ P_3 @ B )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ P_3 @ N ) @ ( times_times_nat @ A @ B ) )
=> ( dvd_dvd_nat @ ( power_power_nat @ P_3 @ N ) @ A ) ) ) ) ).
thf(fact_4187_prime__coprime__lt,axiom,
! [X: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ord_less_nat @ X @ P_3 )
=> ( coprime @ X @ P_3 ) ) ) ) ).
thf(fact_4188_prime__coprime,axiom,
! [N: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( N = one_one_nat )
| ( dvd_dvd_nat @ P_3 @ N )
| ( coprime @ P_3 @ N ) ) ) ).
thf(fact_4189_negateSnd__eq,axiom,
! [Q: int,R_1: int] :
( ( negateSnd @ ( product_Pair_int_int @ Q @ R_1 ) )
= ( product_Pair_int_int @ Q @ ( uminus_uminus_int @ R_1 ) ) ) ).
thf(fact_4190_divmod__int__rel__neg,axiom,
! [A: int,B: int,Qr: product_prod_int_int] :
( ( divmod_int_rel @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) @ Qr )
=> ( divmod_int_rel @ A @ B @ ( negateSnd @ Qr ) ) ) ).
thf(fact_4191_two__is__prime,axiom,
prime @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_4192_divmod__int__correct,axiom,
! [A: int,B: int] :
( ( B != zero_zero_int )
=> ( divmod_int_rel @ A @ B @ ( divmod_int @ A @ B ) ) ) ).
thf(fact_4193_prime__dvd__square,axiom,
! [M: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( dvd_dvd_nat @ P_3 @ ( power_power_nat @ M @ ( suc @ ( suc @ zero_zero_nat ) ) ) )
=> ( dvd_dvd_nat @ P_3 @ M ) ) ) ).
thf(fact_4194_prime__divprod__pow,axiom,
! [N: nat,A: nat,B: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( coprime @ A @ B )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ P_3 @ N ) @ ( times_times_nat @ A @ B ) )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ P_3 @ N ) @ A )
| ( dvd_dvd_nat @ ( power_power_nat @ P_3 @ N ) @ B ) ) ) ) ) ).
thf(fact_4195_prime__ge__2,axiom,
! [P_3: nat] :
( ( prime @ P_3 )
=> ( ord_less_eq_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 ) ) ).
thf(fact_4196_prime__odd,axiom,
! [P_3: nat] :
( ( prime @ P_3 )
=> ( ( P_3
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
| ~ ( even_odd_even_nat @ P_3 ) ) ) ).
thf(fact_4197_divmod__int__mod__div,axiom,
! [P_3: int,Q: int] :
( ( divmod_int @ P_3 @ Q )
= ( product_Pair_int_int @ ( div_div_int @ P_3 @ Q ) @ ( div_mod_int @ P_3 @ Q ) ) ) ).
thf(fact_4198_div__mod__code__numeral__def,axiom,
! [N: code_code_numeral,M: code_code_numeral] :
( ( code_d418564891umeral @ N @ M )
= ( produc2136830103umeral @ ( div_di1218280263umeral @ N @ M ) @ ( div_mo1740067990umeral @ N @ M ) ) ) ).
thf(fact_4199_xzgcd__linear,axiom,
! [M: int,R_1: int,S_1: int,T: int,N: int] :
( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( xzgcd @ M @ N )
= ( produc282740534nt_int @ R_1 @ ( product_Pair_int_int @ S_1 @ T ) ) )
=> ( R_1
= ( plus_plus_int @ ( times_times_int @ S_1 @ M ) @ ( times_times_int @ T @ N ) ) ) ) ) ).
thf(fact_4200_xzgcda_Osimps,axiom,
! [M: int,N: int,R_3: int,S_3: int,S_1: int,T_2: int,T: int,R_1: int] :
( ( ( ord_less_eq_int @ R_1 @ zero_zero_int )
=> ( ( xzgcda @ M @ N @ R_3 @ R_1 @ S_3 @ S_1 @ T_2 @ T )
= ( produc282740534nt_int @ R_3 @ ( product_Pair_int_int @ S_3 @ T_2 ) ) ) )
& ( ~ ( ord_less_eq_int @ R_1 @ zero_zero_int )
=> ( ( xzgcda @ M @ N @ R_3 @ R_1 @ S_3 @ S_1 @ T_2 @ T )
= ( xzgcda @ M @ N @ R_1 @ ( div_mod_int @ R_3 @ R_1 ) @ S_1 @ ( minus_minus_int @ S_3 @ ( times_times_int @ ( div_div_int @ R_3 @ R_1 ) @ S_1 ) ) @ T @ ( minus_minus_int @ T_2 @ ( times_times_int @ ( div_div_int @ R_3 @ R_1 ) @ T ) ) ) ) ) ) ).
thf(fact_4201_xzgcd__def,axiom,
! [M: int,N: int] :
( ( xzgcd @ M @ N )
= ( xzgcda @ M @ N @ M @ N @ one_one_int @ zero_zero_int @ zero_zero_int @ one_one_int ) ) ).
thf(fact_4202_xzgcda__linear,axiom,
! [M: int,N: int,R_3: int,S_3: int,S_1: int,T_2: int,T: int,Rn: int,Sn_1: int,Tn_1: int,R_1: int] :
( ( ord_less_int @ zero_zero_int @ R_1 )
=> ( ( ( xzgcda @ M @ N @ R_3 @ R_1 @ S_3 @ S_1 @ T_2 @ T )
= ( produc282740534nt_int @ Rn @ ( product_Pair_int_int @ Sn_1 @ Tn_1 ) ) )
=> ( ( R_3
= ( plus_plus_int @ ( times_times_int @ S_3 @ M ) @ ( times_times_int @ T_2 @ N ) ) )
=> ( ( R_1
= ( plus_plus_int @ ( times_times_int @ S_1 @ M ) @ ( times_times_int @ T @ N ) ) )
=> ( Rn
= ( plus_plus_int @ ( times_times_int @ Sn_1 @ M ) @ ( times_times_int @ Tn_1 @ N ) ) ) ) ) ) ) ).
thf(fact_4203_coprime__prime__dvd__ex,axiom,
! [X: nat,Y: nat] :
( ~ ( coprime @ X @ Y )
=> ? [P_4: nat] :
( ( prime @ P_4 )
& ( dvd_dvd_nat @ P_4 @ X )
& ( dvd_dvd_nat @ P_4 @ Y ) ) ) ).
thf(fact_4204_prime__power__exp,axiom,
! [X: nat,K_1: nat,N: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( N != zero_zero_nat )
=> ( ( ( power_power_nat @ X @ N )
= ( power_power_nat @ P_3 @ K_1 ) )
=> ? [I_1: nat] :
( X
= ( power_power_nat @ P_3 @ I_1 ) ) ) ) ) ).
thf(fact_4205_prime__factor,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ? [P_4: nat] :
( ( prime @ P_4 )
& ( dvd_dvd_nat @ P_4 @ N ) ) ) ).
thf(fact_4206_divmod__nat__step,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( divmod_nat @ M @ N )
= ( product_Pair_nat_nat @ ( suc @ ( div_div_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) @ ( div_mod_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
thf(fact_4207_bezout__prime,axiom,
! [A: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ~ ( dvd_dvd_nat @ P_3 @ A )
=> ? [X_1: nat,Y_1: nat] :
( ( times_times_nat @ A @ X_1 )
= ( plus_plus_nat @ ( times_times_nat @ P_3 @ Y_1 ) @ one_one_nat ) ) ) ) ).
thf(fact_4208_not__prime__ex__mk,axiom,
! [N: nat] :
( ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
& ~ ( prime @ N ) )
=> ? [M_2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M_2 )
& ( ord_less_nat @ ( suc @ zero_zero_nat ) @ K )
& ( ord_less_nat @ M_2 @ N )
& ( ord_less_nat @ K @ N )
& ( N
= ( times_times_nat @ M_2 @ K ) ) ) ) ).
thf(fact_4209_divmod__nat__zero,axiom,
! [M: nat] :
( ( divmod_nat @ M @ zero_zero_nat )
= ( product_Pair_nat_nat @ zero_zero_nat @ M ) ) ).
thf(fact_4210_divmod__nat__base,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divmod_nat @ M @ N )
= ( product_Pair_nat_nat @ zero_zero_nat @ M ) ) ) ).
thf(fact_4211_divmod__nat__div__mod,axiom,
! [M: nat,N: nat] :
( ( divmod_nat @ M @ N )
= ( product_Pair_nat_nat @ ( div_div_nat @ M @ N ) @ ( div_mod_nat @ M @ N ) ) ) ).
thf(fact_4212_divmod__nat__rel__mult1__eq,axiom,
! [A: nat,B: nat,C: nat,Q: nat,R_1: nat] :
( ( divmod_nat_rel @ B @ C @ ( product_Pair_nat_nat @ Q @ R_1 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( divmod_nat_rel @ ( times_times_nat @ A @ B ) @ C @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ Q ) @ ( div_div_nat @ ( times_times_nat @ A @ R_1 ) @ C ) ) @ ( div_mod_nat @ ( times_times_nat @ A @ R_1 ) @ C ) ) ) ) ) ).
thf(fact_4213_divmod__nat__rel__mult2__eq,axiom,
! [C: nat,A: nat,B: nat,Q: nat,R_1: nat] :
( ( divmod_nat_rel @ A @ B @ ( product_Pair_nat_nat @ Q @ R_1 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( divmod_nat_rel @ A @ ( times_times_nat @ B @ C ) @ ( product_Pair_nat_nat @ ( div_div_nat @ Q @ C ) @ ( plus_plus_nat @ ( times_times_nat @ B @ ( div_mod_nat @ Q @ C ) ) @ R_1 ) ) ) ) ) ) ).
thf(fact_4214_divmod__nat__rel__add1__eq,axiom,
! [B: nat,Bq: nat,Br: nat,A: nat,C: nat,Aq: nat,Ar: nat] :
( ( divmod_nat_rel @ A @ C @ ( product_Pair_nat_nat @ Aq @ Ar ) )
=> ( ( divmod_nat_rel @ B @ C @ ( product_Pair_nat_nat @ Bq @ Br ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( divmod_nat_rel @ ( plus_plus_nat @ A @ B ) @ C @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( plus_plus_nat @ Aq @ Bq ) @ ( div_div_nat @ ( plus_plus_nat @ Ar @ Br ) @ C ) ) @ ( div_mod_nat @ ( plus_plus_nat @ Ar @ Br ) @ C ) ) ) ) ) ) ).
thf(fact_4215_divmod__nat__rel__unique,axiom,
! [Qr_1: product_prod_nat_nat,M: nat,N: nat,Qr: product_prod_nat_nat] :
( ( divmod_nat_rel @ M @ N @ Qr )
=> ( ( divmod_nat_rel @ M @ N @ Qr_1 )
=> ( Qr = Qr_1 ) ) ) ).
thf(fact_4216_divmod__nat__eq,axiom,
! [M: nat,N: nat,Qr: product_prod_nat_nat] :
( ( divmod_nat_rel @ M @ N @ Qr )
=> ( ( divmod_nat @ M @ N )
= Qr ) ) ).
thf(fact_4217_divmod__nat__rel__divmod__nat,axiom,
! [M: nat,N: nat] : ( divmod_nat_rel @ M @ N @ ( divmod_nat @ M @ N ) ) ).
thf(fact_4218_div__eq,axiom,
! [M: nat,N: nat,Q: nat,R_1: nat] :
( ( divmod_nat_rel @ M @ N @ ( product_Pair_nat_nat @ Q @ R_1 ) )
=> ( ( div_div_nat @ M @ N )
= Q ) ) ).
thf(fact_4219_mod__eq,axiom,
! [M: nat,N: nat,Q: nat,R_1: nat] :
( ( divmod_nat_rel @ M @ N @ ( product_Pair_nat_nat @ Q @ R_1 ) )
=> ( ( div_mod_nat @ M @ N )
= R_1 ) ) ).
thf(fact_4220_divmod__nat__rel,axiom,
! [M: nat,N: nat] : ( divmod_nat_rel @ M @ N @ ( product_Pair_nat_nat @ ( div_div_nat @ M @ N ) @ ( div_mod_nat @ M @ N ) ) ) ).
thf(fact_4221_divmod__nat__rel__ex,axiom,
! [M: nat,N: nat] :
~ ! [Q_2: nat,R: nat] :
~ ( divmod_nat_rel @ M @ N @ ( product_Pair_nat_nat @ Q_2 @ R ) ) ).
thf(fact_4222_euclid,axiom,
! [N: nat] :
? [P_4: nat] :
( ( prime @ P_4 )
& ( ord_less_nat @ N @ P_4 ) ) ).
thf(fact_4223_div__pos__neg__1__number__of,axiom,
! [W: int] :
( ( ord_less_int @ ( number_number_of_int @ W ) @ zero_zero_int )
=> ( ( div_div_int @ one_one_int @ ( number_number_of_int @ W ) )
= ( product_fst_int_int @ ( negateSnd @ ( negDivAlg @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( number_number_of_int @ W ) ) ) ) ) ) ) ).
thf(fact_4224_div__int__def,axiom,
! [A: int,B: int] :
( ( div_div_int @ A @ B )
= ( product_fst_int_int @ ( divmod_int @ A @ B ) ) ) ).
thf(fact_4225_div__neg__pos,axiom,
! [B: int,A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( div_div_int @ A @ B )
= ( product_fst_int_int @ ( negDivAlg @ A @ B ) ) ) ) ) ).
thf(fact_4226_div__pos__pos,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( div_div_int @ A @ B )
= ( product_fst_int_int @ ( posDivAlg @ A @ B ) ) ) ) ) ).
thf(fact_4227_div__pos__pos__1__number__of,axiom,
! [W: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ W ) )
=> ( ( div_div_int @ one_one_int @ ( number_number_of_int @ W ) )
= ( product_fst_int_int @ ( posDivAlg @ one_one_int @ ( number_number_of_int @ W ) ) ) ) ) ).
thf(fact_4228_div__pos__neg,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ( div_div_int @ A @ B )
= ( product_fst_int_int @ ( negateSnd @ ( negDivAlg @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ) ) ) ) ).
thf(fact_4229_div__neg__neg,axiom,
! [B: int,A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ( div_div_int @ A @ B )
= ( product_fst_int_int @ ( negateSnd @ ( posDivAlg @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ) ) ) ) ).
thf(fact_4230_bezw_Osimps,axiom,
! [X: nat,Y: nat] :
( ( ( Y = zero_zero_nat )
=> ( ( bezw @ X @ Y )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
& ( ( Y != zero_zero_nat )
=> ( ( bezw @ X @ Y )
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( div_mod_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( div_mod_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( div_mod_nat @ X @ Y ) ) ) @ ( semiri1621563631at_int @ ( div_div_nat @ X @ Y ) ) ) ) ) ) ) ) ).
thf(fact_4231_bezw__non__0,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ zero_zero_nat @ Y )
=> ( ( bezw @ X @ Y )
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( div_mod_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( div_mod_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( div_mod_nat @ X @ Y ) ) ) @ ( semiri1621563631at_int @ ( div_div_nat @ X @ Y ) ) ) ) ) ) ) ).
thf(fact_4232_mod__pos__neg__1__number__of,axiom,
! [W: int] :
( ( ord_less_int @ ( number_number_of_int @ W ) @ zero_zero_int )
=> ( ( div_mod_int @ one_one_int @ ( number_number_of_int @ W ) )
= ( product_snd_int_int @ ( negateSnd @ ( negDivAlg @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( number_number_of_int @ W ) ) ) ) ) ) ) ).
thf(fact_4233_mod__int__def,axiom,
! [A: int,B: int] :
( ( div_mod_int @ A @ B )
= ( product_snd_int_int @ ( divmod_int @ A @ B ) ) ) ).
thf(fact_4234_mod__neg__pos,axiom,
! [B: int,A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( div_mod_int @ A @ B )
= ( product_snd_int_int @ ( negDivAlg @ A @ B ) ) ) ) ) ).
thf(fact_4235_mod__pos__pos,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( div_mod_int @ A @ B )
= ( product_snd_int_int @ ( posDivAlg @ A @ B ) ) ) ) ) ).
thf(fact_4236_mod__pos__pos__1__number__of,axiom,
! [W: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ W ) )
=> ( ( div_mod_int @ one_one_int @ ( number_number_of_int @ W ) )
= ( product_snd_int_int @ ( posDivAlg @ one_one_int @ ( number_number_of_int @ W ) ) ) ) ) ).
thf(fact_4237_mod__pos__neg,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ( div_mod_int @ A @ B )
= ( product_snd_int_int @ ( negateSnd @ ( negDivAlg @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ) ) ) ) ).
thf(fact_4238_mod__neg__neg,axiom,
! [B: int,A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ( div_mod_int @ A @ B )
= ( product_snd_int_int @ ( negateSnd @ ( posDivAlg @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ) ) ) ) ).
thf(fact_4239_div__nat__def,axiom,
! [M: nat,N: nat] :
( ( div_div_nat @ M @ N )
= ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) ) ) ).
thf(fact_4240_mod__nat__def,axiom,
! [M: nat,N: nat] :
( ( div_mod_nat @ M @ N )
= ( product_snd_nat_nat @ ( divmod_nat @ M @ N ) ) ) ).
thf(fact_4241_divmod__nat__rel__def,axiom,
! [M: nat,N: nat,Qr: product_prod_nat_nat] :
( ( divmod_nat_rel @ M @ N @ Qr )
<=> ( ( M
= ( plus_plus_nat @ ( times_times_nat @ ( product_fst_nat_nat @ Qr ) @ N ) @ ( product_snd_nat_nat @ Qr ) ) )
& ( ( N = zero_zero_nat )
=> ( ( product_fst_nat_nat @ Qr )
= zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( product_snd_nat_nat @ Qr ) )
& ( ord_less_nat @ ( product_snd_nat_nat @ Qr ) @ N ) ) )
& ( ~ ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ ( product_snd_nat_nat @ Qr ) )
& ( ord_less_eq_nat @ ( product_snd_nat_nat @ Qr ) @ zero_zero_nat ) ) ) ) ) ) ) ).
thf(fact_4242_Bolzano__bisect__diff,axiom,
! [P: produc914805421l_real > $o,N: nat,A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( minus_minus_real @ ( produc556554744l_real @ ( bolzano_bisect @ P @ A @ B @ N ) ) @ ( produc1935615926l_real @ ( bolzano_bisect @ P @ A @ B @ N ) ) )
= ( inverse_divide_real @ ( minus_minus_real @ B @ A ) @ ( power_power_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) ) ) ).
thf(fact_4243_Bolzano__bisect__le,axiom,
! [P: produc914805421l_real > $o,A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ! [N_1: nat] : ( ord_less_eq_real @ ( produc1935615926l_real @ ( bolzano_bisect @ P @ A @ B @ N_1 ) ) @ ( produc556554744l_real @ ( bolzano_bisect @ P @ A @ B @ N_1 ) ) ) ) ).
thf(fact_4244_Bolzano__bisect_Osimps_I1_J,axiom,
! [P: produc914805421l_real > $o,A: real,B: real] :
( ( bolzano_bisect @ P @ A @ B @ zero_zero_nat )
= ( produc865579683l_real @ A @ B ) ) ).
thf(fact_4245_Bolzano__bisect__fst__le__Suc,axiom,
! [P: produc914805421l_real > $o,A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ! [N_1: nat] : ( ord_less_eq_real @ ( produc1935615926l_real @ ( bolzano_bisect @ P @ A @ B @ N_1 ) ) @ ( produc1935615926l_real @ ( bolzano_bisect @ P @ A @ B @ ( suc @ N_1 ) ) ) ) ) ).
thf(fact_4246_Bolzano__bisect__Suc__le__snd,axiom,
! [P: produc914805421l_real > $o,A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ! [N_1: nat] : ( ord_less_eq_real @ ( produc556554744l_real @ ( bolzano_bisect @ P @ A @ B @ ( suc @ N_1 ) ) ) @ ( produc556554744l_real @ ( bolzano_bisect @ P @ A @ B @ N_1 ) ) ) ) ).
thf(fact_4247_not__P__Bolzano__bisect,axiom,
! [N: nat,A: real,B: real,P: produc914805421l_real > $o] :
( ! [A_2: real,B_4: real,C_2: real] :
( ( P @ ( produc865579683l_real @ A_2 @ B_4 ) )
=> ( ( P @ ( produc865579683l_real @ B_4 @ C_2 ) )
=> ( ( ord_less_eq_real @ A_2 @ B_4 )
=> ( ( ord_less_eq_real @ B_4 @ C_2 )
=> ( P @ ( produc865579683l_real @ A_2 @ C_2 ) ) ) ) ) )
=> ( ~ ( P @ ( produc865579683l_real @ A @ B ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ~ ( P @ ( produc865579683l_real @ ( produc1935615926l_real @ ( bolzano_bisect @ P @ A @ B @ N ) ) @ ( produc556554744l_real @ ( bolzano_bisect @ P @ A @ B @ N ) ) ) ) ) ) ) ).
thf(fact_4248_not__P__Bolzano__bisect_H,axiom,
! [A: real,B: real,P: produc914805421l_real > $o] :
( ! [A_2: real,B_4: real,C_2: real] :
( ( ( P @ ( produc865579683l_real @ A_2 @ B_4 ) )
& ( P @ ( produc865579683l_real @ B_4 @ C_2 ) )
& ( ord_less_eq_real @ A_2 @ B_4 )
& ( ord_less_eq_real @ B_4 @ C_2 ) )
=> ( P @ ( produc865579683l_real @ A_2 @ C_2 ) ) )
=> ( ~ ( P @ ( produc865579683l_real @ A @ B ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ! [N_1: nat] :
~ ( P @ ( produc865579683l_real @ ( produc1935615926l_real @ ( bolzano_bisect @ P @ A @ B @ N_1 ) ) @ ( produc556554744l_real @ ( bolzano_bisect @ P @ A @ B @ N_1 ) ) ) ) ) ) ) ).
thf(fact_4249_one__code__int__code,axiom,
( one_on1684967323de_int
= ( number1226105091de_int @ ( bit1 @ pls ) ) ) ).
thf(fact_4250_Quickcheck__Narrowing_Oint__of__code,axiom,
! [K_1: quickcheck_code_int] :
( ( ( K_1 = zero_z891286103de_int )
=> ( ( quickcheck_int_of @ K_1 )
= zero_zero_int ) )
& ( ( K_1 != zero_z891286103de_int )
=> ( ( ( ( div_mo231679042de_int @ K_1 @ ( number1226105091de_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_z891286103de_int )
=> ( ( quickcheck_int_of @ K_1 )
= ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( quickcheck_int_of @ ( div_di1430059507de_int @ K_1 @ ( number1226105091de_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) )
& ( ( ( div_mo231679042de_int @ K_1 @ ( number1226105091de_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= zero_z891286103de_int )
=> ( ( quickcheck_int_of @ K_1 )
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( quickcheck_int_of @ ( div_di1430059507de_int @ K_1 @ ( number1226105091de_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
thf(fact_4251_DeMoivre2,axiom,
! [R_1: real,A: real,N: nat] :
( ( power_power_complex @ ( rcis @ R_1 @ A ) @ N )
= ( rcis @ ( power_power_real @ R_1 @ N ) @ ( times_times_real @ ( real_nat @ N ) @ A ) ) ) ).
thf(fact_4252_less__eq__code__int__def,axiom,
! [N: quickcheck_code_int,M: quickcheck_code_int] :
( ( ord_le258702272de_int @ N @ M )
<=> ( ord_less_eq_int @ ( quickcheck_int_of @ N ) @ ( quickcheck_int_of @ M ) ) ) ).
thf(fact_4253_less__code__int__def,axiom,
! [N: quickcheck_code_int,M: quickcheck_code_int] :
( ( ord_le1860547276de_int @ N @ M )
<=> ( ord_less_int @ ( quickcheck_int_of @ N ) @ ( quickcheck_int_of @ M ) ) ) ).
thf(fact_4254_code__int_Oint__of__inject,axiom,
! [X: quickcheck_code_int,Y: quickcheck_code_int] :
( ( ( quickcheck_int_of @ X )
= ( quickcheck_int_of @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_4255_Quickcheck__Narrowing_Oint__of__inject,axiom,
! [K_1: quickcheck_code_int,L: quickcheck_code_int] :
( ( ( quickcheck_int_of @ K_1 )
= ( quickcheck_int_of @ L ) )
<=> ( K_1 = L ) ) ).
thf(fact_4256_int__of__number,axiom,
! [K_1: int] :
( ( quickcheck_int_of @ ( number1226105091de_int @ K_1 ) )
= ( number_number_of_int @ K_1 ) ) ).
thf(fact_4257_zero__code__int__code,axiom,
( zero_z891286103de_int
= ( number1226105091de_int @ pls ) ) ).
thf(fact_4258_rcis__zero__arg,axiom,
! [R_1: real] :
( ( rcis @ R_1 @ zero_zero_real )
= ( of_real_complex @ R_1 ) ) ).
thf(fact_4259_complex__mod__rcis,axiom,
! [R_1: real,A: real] :
( ( norm_norm_complex @ ( rcis @ R_1 @ A ) )
= ( abs_abs_real @ R_1 ) ) ).
thf(fact_4260_cis__rcis__eq,axiom,
! [A: real] :
( ( cis @ A )
= ( rcis @ one_one_real @ A ) ) ).
thf(fact_4261_rcis__zero__mod,axiom,
! [A: real] :
( ( rcis @ zero_zero_real @ A )
= zero_zero_complex ) ).
thf(fact_4262_Re__rcis,axiom,
! [R_1: real,A: real] :
( ( re @ ( rcis @ R_1 @ A ) )
= ( times_times_real @ R_1 @ ( cos @ A ) ) ) ).
thf(fact_4263_Im__rcis,axiom,
! [R_1: real,A: real] :
( ( im @ ( rcis @ R_1 @ A ) )
= ( times_times_real @ R_1 @ ( sin @ A ) ) ) ).
thf(fact_4264_rcis__mult,axiom,
! [R1: real,A: real,R2: real,B: real] :
( ( times_times_complex @ ( rcis @ R1 @ A ) @ ( rcis @ R2 @ B ) )
= ( rcis @ ( times_times_real @ R1 @ R2 ) @ ( plus_plus_real @ A @ B ) ) ) ).
thf(fact_4265_rcis__divide,axiom,
! [R1: real,A: real,R2: real,B: real] :
( ( invers1025623611omplex @ ( rcis @ R1 @ A ) @ ( rcis @ R2 @ B ) )
= ( rcis @ ( inverse_divide_real @ R1 @ R2 ) @ ( minus_minus_real @ A @ B ) ) ) ).
thf(fact_4266_rcis__def,axiom,
! [R_1: real,A: real] :
( ( rcis @ R_1 @ A )
= ( times_times_complex @ ( of_real_complex @ R_1 ) @ ( cis @ A ) ) ) ).
thf(fact_4267_rcis__inverse,axiom,
! [R_1: real,A: real] :
( ( invers1449016382omplex @ ( rcis @ R_1 @ A ) )
= ( rcis @ ( inverse_divide_real @ one_one_real @ R_1 ) @ ( uminus_uminus_real @ A ) ) ) ).
thf(fact_4268_div__mod__code__int__def,axiom,
! [N: quickcheck_code_int,M: quickcheck_code_int] :
( ( quickc495462417de_int @ N @ M )
= ( produc1318306967de_int @ ( div_di1430059507de_int @ N @ M ) @ ( div_mo231679042de_int @ N @ M ) ) ) ).
thf(fact_4269_nat__of__def,axiom,
! [I: quickcheck_code_int] :
( ( quickcheck_nat_of @ I )
= ( nat_1 @ ( quickcheck_int_of @ I ) ) ) ).
thf(fact_4270_around__zero_Opinduct,axiom,
! [P: int > $o,A0: int] :
( ( accp_int @ quickc1265749348ro_rel @ A0 )
=> ( ! [I_1: int] :
( ( accp_int @ quickc1265749348ro_rel @ I_1 )
=> ( ( ~ ( ord_less_int @ I_1 @ zero_zero_int )
=> ( ( I_1 != zero_zero_int )
=> ( P @ ( minus_minus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I_1 ) ) )
=> ( P @ A0 ) ) ) ).
thf(fact_4271_fact__int__def,axiom,
! [X: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( fact_fact_int @ X )
= ( semiri1621563631at_int @ ( fact_fact_nat @ ( nat_1 @ X ) ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( fact_fact_int @ X )
= zero_zero_int ) ) ) ).
thf(fact_4272_fact__mono__int,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ M @ N )
=> ( ord_less_eq_int @ ( fact_fact_int @ M ) @ ( fact_fact_int @ N ) ) ) ).
thf(fact_4273_fact__1__int,axiom,
( ( fact_fact_int @ one_one_int )
= one_one_int ) ).
thf(fact_4274_fact__less__mono__int,axiom,
! [N: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_int @ M @ N )
=> ( ord_less_int @ ( fact_fact_int @ M ) @ ( fact_fact_int @ N ) ) ) ) ).
thf(fact_4275_fact__neg__int,axiom,
! [M: int] :
( ( ord_less_int @ M @ zero_zero_int )
=> ( ( fact_fact_int @ M )
= zero_zero_int ) ) ).
thf(fact_4276_fact__0__int,axiom,
( ( fact_fact_int @ zero_zero_int )
= one_one_int ) ).
thf(fact_4277_transfer__nat__int__factorial__closure,axiom,
! [X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ord_less_eq_int @ zero_zero_int @ ( fact_fact_int @ X ) ) ) ).
thf(fact_4278_fact__nonzero__int,axiom,
! [N: int] :
( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( fact_fact_int @ N )
!= zero_zero_int ) ) ).
thf(fact_4279_fact__ge__zero__int,axiom,
! [M: int] : ( ord_less_eq_int @ zero_zero_int @ ( fact_fact_int @ M ) ) ).
thf(fact_4280_transfer__int__nat__factorial,axiom,
! [X: nat] :
( ( fact_fact_int @ ( semiri1621563631at_int @ X ) )
= ( semiri1621563631at_int @ ( fact_fact_nat @ X ) ) ) ).
thf(fact_4281_fact__gt__zero__int,axiom,
! [N: int] :
( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ord_less_int @ zero_zero_int @ ( fact_fact_int @ N ) ) ) ).
thf(fact_4282_fact__ge__one__int,axiom,
! [N: int] :
( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ one_one_int @ ( fact_fact_int @ N ) ) ) ).
thf(fact_4283_fact__mono__int__aux,axiom,
! [M: int,K_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ( ord_less_eq_int @ ( fact_fact_int @ M ) @ ( fact_fact_int @ ( plus_plus_int @ M @ K_1 ) ) ) ) ).
thf(fact_4284_dvd__fact__int,axiom,
! [N: int,M: int] :
( ( ord_less_eq_int @ one_one_int @ M )
=> ( ( ord_less_eq_int @ M @ N )
=> ( dvd_dvd_int @ M @ ( fact_fact_int @ N ) ) ) ) ).
thf(fact_4285_transfer__nat__int__factorial,axiom,
! [X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( fact_fact_nat @ ( nat_1 @ X ) )
= ( nat_1 @ ( fact_fact_int @ X ) ) ) ) ).
thf(fact_4286_fact__less__mono__int__aux,axiom,
! [M: int,K_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ( ( ord_less_int @ zero_zero_int @ M )
=> ( ord_less_int @ ( fact_fact_int @ M ) @ ( fact_fact_int @ ( plus_plus_int @ ( plus_plus_int @ M @ one_one_int ) @ K_1 ) ) ) ) ) ).
thf(fact_4287_fact__plus__one__int,axiom,
! [N: int] :
( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( fact_fact_int @ ( plus_plus_int @ N @ one_one_int ) )
= ( times_times_int @ ( plus_plus_int @ N @ one_one_int ) @ ( fact_fact_int @ N ) ) ) ) ).
thf(fact_4288_fact__reduce__int,axiom,
! [N: int] :
( ( ord_less_int @ zero_zero_int @ N )
=> ( ( fact_fact_int @ N )
= ( times_times_int @ N @ ( fact_fact_int @ ( minus_minus_int @ N @ one_one_int ) ) ) ) ) ).
thf(fact_4289_length__around__zero,axiom,
! [I: int] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( size_size_list_int @ ( quickc666637781d_zero @ I ) )
= ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( nat_1 @ I ) ) @ one_one_nat ) ) ) ).
thf(fact_4290_Bnor__prime,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ A @ P_3 )
=> ( ( finite_card_int @ ( bnorRset @ A @ P_3 ) )
= ( nat_1 @ A ) ) ) ) ).
thf(fact_4291_lemma__BOLZANO,axiom,
! [A: real,B: real,P: produc914805421l_real > $o] :
( ! [A_2: real,B_4: real,C_2: real] :
( ( ( P @ ( produc865579683l_real @ A_2 @ B_4 ) )
& ( P @ ( produc865579683l_real @ B_4 @ C_2 ) )
& ( ord_less_eq_real @ A_2 @ B_4 )
& ( ord_less_eq_real @ B_4 @ C_2 ) )
=> ( P @ ( produc865579683l_real @ A_2 @ C_2 ) ) )
=> ( ! [X_1: real] :
? [D_2: real] :
( ( ord_less_real @ zero_zero_real @ D_2 )
& ! [A_2: real,B_4: real] :
( ( ( ord_less_eq_real @ A_2 @ X_1 )
& ( ord_less_eq_real @ X_1 @ B_4 )
& ( ord_less_real @ ( minus_minus_real @ B_4 @ A_2 ) @ D_2 ) )
=> ( P @ ( produc865579683l_real @ A_2 @ B_4 ) ) ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( P @ ( produc865579683l_real @ A @ B ) ) ) ) ) ).
thf(fact_4292_Bnor__mem__zle__swap,axiom,
! [M: int,A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( member_int @ B @ ( bnorRset @ A @ M ) ) ) ).
thf(fact_4293_Bnor__mem__zle,axiom,
! [B: int,A: int,M: int] :
( ( member_int @ B @ ( bnorRset @ A @ M ) )
=> ( ord_less_eq_int @ B @ A ) ) ).
thf(fact_4294_Bnor__fin,axiom,
! [A: int,M: int] : ( finite_finite_int @ ( bnorRset @ A @ M ) ) ).
thf(fact_4295_Bnor__mem__zg,axiom,
! [B: int,A: int,M: int] :
( ( member_int @ B @ ( bnorRset @ A @ M ) )
=> ( ord_less_int @ zero_zero_int @ B ) ) ).
thf(fact_4296_lemma__BOLZANO2,axiom,
! [P: produc914805421l_real > $o] :
( ( ! [A_2: real,B_4: real,C_2: real] :
( ( ( ord_less_eq_real @ A_2 @ B_4 )
& ( ord_less_eq_real @ B_4 @ C_2 )
& ( P @ ( produc865579683l_real @ A_2 @ B_4 ) )
& ( P @ ( produc865579683l_real @ B_4 @ C_2 ) ) )
=> ( P @ ( produc865579683l_real @ A_2 @ C_2 ) ) )
& ! [X_1: real] :
? [D_2: real] :
( ( ord_less_real @ zero_zero_real @ D_2 )
& ! [A_2: real,B_4: real] :
( ( ( ord_less_eq_real @ A_2 @ X_1 )
& ( ord_less_eq_real @ X_1 @ B_4 )
& ( ord_less_real @ ( minus_minus_real @ B_4 @ A_2 ) @ D_2 ) )
=> ( P @ ( produc865579683l_real @ A_2 @ B_4 ) ) ) ) )
=> ! [A_2: real,B_4: real] :
( ( ord_less_eq_real @ A_2 @ B_4 )
=> ( P @ ( produc865579683l_real @ A_2 @ B_4 ) ) ) ) ).
thf(fact_4297_bezw__aux,axiom,
! [X: nat,Y: nat] :
( ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1621563631at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1621563631at_int @ Y ) ) )
= ( semiri1621563631at_int @ ( gcd_gcd_nat @ X @ Y ) ) ) ).
thf(fact_4298_gcd__dvd2__nat,axiom,
! [M: nat,N: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ M @ N ) @ N ) ).
thf(fact_4299_gcd__dvd1__nat,axiom,
! [M: nat,N: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ M @ N ) @ M ) ).
thf(fact_4300_gcd__dvd__prod__nat,axiom,
! [M: nat,N: nat,K_1: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ M @ N ) @ ( times_times_nat @ K_1 @ N ) ) ).
thf(fact_4301_coprime__exp2__nat,axiom,
! [N: nat,M: nat,A: nat,B: nat] :
( ( ( gcd_gcd_nat @ A @ B )
= one_one_nat )
=> ( ( gcd_gcd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ M ) )
= one_one_nat ) ) ).
thf(fact_4302_gcd__exp__nat,axiom,
! [A: nat,N: nat,B: nat] :
( ( gcd_gcd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( power_power_nat @ ( gcd_gcd_nat @ A @ B ) @ N ) ) ).
thf(fact_4303_gcd__semilattice__nat_Oinf_Oidem,axiom,
! [A: nat] :
( ( gcd_gcd_nat @ A @ A )
= A ) ).
thf(fact_4304_gcd__idem__nat,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ X @ X )
= X ) ).
thf(fact_4305_gcd__commute__nat,axiom,
! [A: nat,B: nat] :
( ( gcd_gcd_nat @ A @ B )
= ( gcd_gcd_nat @ B @ A ) ) ).
thf(fact_4306_gcd__semilattice__nat_Oinf__commute,axiom,
! [X: nat,Y: nat] :
( ( gcd_gcd_nat @ X @ Y )
= ( gcd_gcd_nat @ Y @ X ) ) ).
thf(fact_4307_gcd__semilattice__nat_Oinf_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( gcd_gcd_nat @ A @ ( gcd_gcd_nat @ A @ B ) )
= ( gcd_gcd_nat @ A @ B ) ) ).
thf(fact_4308_gcd__semilattice__nat_Oinf__left__idem,axiom,
! [X: nat,Y: nat] :
( ( gcd_gcd_nat @ X @ ( gcd_gcd_nat @ X @ Y ) )
= ( gcd_gcd_nat @ X @ Y ) ) ).
thf(fact_4309_gcd__left__commute__nat,axiom,
! [B: nat,A: nat,C: nat] :
( ( gcd_gcd_nat @ B @ ( gcd_gcd_nat @ A @ C ) )
= ( gcd_gcd_nat @ A @ ( gcd_gcd_nat @ B @ C ) ) ) ).
thf(fact_4310_gcd__semilattice__nat_Oinf__left__commute,axiom,
! [X: nat,Y: nat,Z_1: nat] :
( ( gcd_gcd_nat @ X @ ( gcd_gcd_nat @ Y @ Z_1 ) )
= ( gcd_gcd_nat @ Y @ ( gcd_gcd_nat @ X @ Z_1 ) ) ) ).
thf(fact_4311_gcd__assoc__nat,axiom,
! [A: nat,B: nat,C: nat] :
( ( gcd_gcd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
= ( gcd_gcd_nat @ A @ ( gcd_gcd_nat @ B @ C ) ) ) ).
thf(fact_4312_gcd__semilattice__nat_Oinf__assoc,axiom,
! [X: nat,Y: nat,Z_1: nat] :
( ( gcd_gcd_nat @ ( gcd_gcd_nat @ X @ Y ) @ Z_1 )
= ( gcd_gcd_nat @ X @ ( gcd_gcd_nat @ Y @ Z_1 ) ) ) ).
thf(fact_4313_gcd__semilattice__nat_Oless__infI2,axiom,
! [A: nat,B: nat,X: nat] :
( ( ( dvd_dvd_nat @ B @ X )
& ~ ( dvd_dvd_nat @ X @ B ) )
=> ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ X )
& ~ ( dvd_dvd_nat @ X @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
thf(fact_4314_gcd__semilattice__nat_Oless__infI1,axiom,
! [B: nat,A: nat,X: nat] :
( ( ( dvd_dvd_nat @ A @ X )
& ~ ( dvd_dvd_nat @ X @ A ) )
=> ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ X )
& ~ ( dvd_dvd_nat @ X @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
thf(fact_4315_gcd__semilattice__nat_Ole__infE,axiom,
! [X: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ X @ ( gcd_gcd_nat @ A @ B ) )
=> ~ ( ( dvd_dvd_nat @ X @ A )
=> ~ ( dvd_dvd_nat @ X @ B ) ) ) ).
thf(fact_4316_dvd__gcd__D2__nat,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K_1 @ ( gcd_gcd_nat @ M @ N ) )
=> ( dvd_dvd_nat @ K_1 @ N ) ) ).
thf(fact_4317_dvd__gcd__D1__nat,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K_1 @ ( gcd_gcd_nat @ M @ N ) )
=> ( dvd_dvd_nat @ K_1 @ M ) ) ).
thf(fact_4318_gcd__semilattice__nat_Oinf__mono,axiom,
! [B: nat,D: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ B @ D )
=> ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ ( gcd_gcd_nat @ C @ D ) ) ) ) ).
thf(fact_4319_gcd__semilattice__nat_Oinf__greatest,axiom,
! [Z_1: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( dvd_dvd_nat @ X @ Z_1 )
=> ( dvd_dvd_nat @ X @ ( gcd_gcd_nat @ Y @ Z_1 ) ) ) ) ).
thf(fact_4320_gcd__semilattice__nat_Ole__infI,axiom,
! [B: nat,X: nat,A: nat] :
( ( dvd_dvd_nat @ X @ A )
=> ( ( dvd_dvd_nat @ X @ B )
=> ( dvd_dvd_nat @ X @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
thf(fact_4321_gcd__greatest__nat,axiom,
! [N: nat,K_1: nat,M: nat] :
( ( dvd_dvd_nat @ K_1 @ M )
=> ( ( dvd_dvd_nat @ K_1 @ N )
=> ( dvd_dvd_nat @ K_1 @ ( gcd_gcd_nat @ M @ N ) ) ) ) ).
thf(fact_4322_gcd__proj2__if__dvd__nat,axiom,
! [Y: nat,X: nat] :
( ( dvd_dvd_nat @ Y @ X )
=> ( ( gcd_gcd_nat @ X @ Y )
= Y ) ) ).
thf(fact_4323_gcd__proj1__if__dvd__nat,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( gcd_gcd_nat @ X @ Y )
= X ) ) ).
thf(fact_4324_gcd__semilattice__nat_Ole__infI2,axiom,
! [A: nat,B: nat,X: nat] :
( ( dvd_dvd_nat @ B @ X )
=> ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ X ) ) ).
thf(fact_4325_gcd__semilattice__nat_Ole__infI1,axiom,
! [B: nat,A: nat,X: nat] :
( ( dvd_dvd_nat @ A @ X )
=> ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ X ) ) ).
thf(fact_4326_gcd__unique__nat,axiom,
! [B: nat,D: nat,A: nat] :
( ( ( dvd_dvd_nat @ D @ A )
& ( dvd_dvd_nat @ D @ B )
& ! [E: nat] :
( ( ( dvd_dvd_nat @ E @ A )
& ( dvd_dvd_nat @ E @ B ) )
=> ( dvd_dvd_nat @ E @ D ) ) )
<=> ( D
= ( gcd_gcd_nat @ A @ B ) ) ) ).
thf(fact_4327_gcd__semilattice__nat_Ole__inf__iff,axiom,
! [X: nat,Y: nat,Z_1: nat] :
( ( dvd_dvd_nat @ X @ ( gcd_gcd_nat @ Y @ Z_1 ) )
<=> ( ( dvd_dvd_nat @ X @ Y )
& ( dvd_dvd_nat @ X @ Z_1 ) ) ) ).
thf(fact_4328_gcd__greatest__iff__nat,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K_1 @ ( gcd_gcd_nat @ M @ N ) )
<=> ( ( dvd_dvd_nat @ K_1 @ M )
& ( dvd_dvd_nat @ K_1 @ N ) ) ) ).
thf(fact_4329_gcd__semilattice__nat_Ole__iff__inf,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
<=> ( ( gcd_gcd_nat @ X @ Y )
= X ) ) ).
thf(fact_4330_gcd__semilattice__nat_Oinf__le2,axiom,
! [X: nat,Y: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ X @ Y ) @ Y ) ).
thf(fact_4331_gcd__semilattice__nat_Oinf__le1,axiom,
! [X: nat,Y: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ X @ Y ) @ X ) ).
thf(fact_4332_gcd__add2__nat,axiom,
! [M: nat,N: nat] :
( ( gcd_gcd_nat @ M @ ( plus_plus_nat @ M @ N ) )
= ( gcd_gcd_nat @ M @ N ) ) ).
thf(fact_4333_gcd__add1__nat,axiom,
! [M: nat,N: nat] :
( ( gcd_gcd_nat @ ( plus_plus_nat @ M @ N ) @ N )
= ( gcd_gcd_nat @ M @ N ) ) ).
thf(fact_4334_gcd__mult__distrib__nat,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( times_times_nat @ K_1 @ ( gcd_gcd_nat @ M @ N ) )
= ( gcd_gcd_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) ) ) ).
thf(fact_4335_gcd__lcm__complete__lattice__nat_Oinf__bot__left,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ one_one_nat @ X )
= one_one_nat ) ).
thf(fact_4336_gcd__1__nat,axiom,
! [M: nat] :
( ( gcd_gcd_nat @ M @ one_one_nat )
= one_one_nat ) ).
thf(fact_4337_gcd__lcm__complete__lattice__nat_Oinf__bot__right,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ X @ one_one_nat )
= one_one_nat ) ).
thf(fact_4338_gcd__0__left__nat,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ zero_zero_nat @ X )
= X ) ).
thf(fact_4339_gcd__0__nat,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ X @ zero_zero_nat )
= X ) ).
thf(fact_4340_gcd__zero__nat,axiom,
! [M: nat,N: nat] :
( ( ( gcd_gcd_nat @ M @ N )
= zero_zero_nat )
<=> ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
thf(fact_4341_gcd__lcm__complete__lattice__nat_Oinf__eq__top__iff,axiom,
! [X: nat,Y: nat] :
( ( ( gcd_gcd_nat @ X @ Y )
= zero_zero_nat )
<=> ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
thf(fact_4342_gcd__red__nat,axiom,
! [X: nat,Y: nat] :
( ( gcd_gcd_nat @ X @ Y )
= ( gcd_gcd_nat @ Y @ ( div_mod_nat @ X @ Y ) ) ) ).
thf(fact_4343_gcd__Suc__0,axiom,
! [M: nat] :
( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( suc @ zero_zero_nat ) ) ).
thf(fact_4344_gcd__pos__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
<=> ( ( M != zero_zero_nat )
| ( N != zero_zero_nat ) ) ) ).
thf(fact_4345_gcd__le1__nat,axiom,
! [B: nat,A: nat] :
( ( A != zero_zero_nat )
=> ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
thf(fact_4346_gcd__le2__nat,axiom,
! [A: nat,B: nat] :
( ( B != zero_zero_nat )
=> ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
thf(fact_4347_coprime__Suc__nat,axiom,
! [N: nat] :
( ( gcd_gcd_nat @ ( suc @ N ) @ N )
= one_one_nat ) ).
thf(fact_4348_gcd__add__mult__nat,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( gcd_gcd_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ K_1 @ M ) @ N ) )
= ( gcd_gcd_nat @ M @ N ) ) ).
thf(fact_4349_coprime__common__divisor__nat,axiom,
! [X: nat,A: nat,B: nat] :
( ( ( gcd_gcd_nat @ A @ B )
= one_one_nat )
=> ( ( dvd_dvd_nat @ X @ A )
=> ( ( dvd_dvd_nat @ X @ B )
=> ( X = one_one_nat ) ) ) ) ).
thf(fact_4350_coprime__nat,axiom,
! [A: nat,B: nat] :
( ( ( gcd_gcd_nat @ A @ B )
= one_one_nat )
<=> ! [D_2: nat] :
( ( ( dvd_dvd_nat @ D_2 @ A )
& ( dvd_dvd_nat @ D_2 @ B ) )
<=> ( D_2 = one_one_nat ) ) ) ).
thf(fact_4351_coprime__plus__one__nat,axiom,
! [N: nat] :
( ( gcd_gcd_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ N )
= one_one_nat ) ).
thf(fact_4352_coprime__mul__eq__nat,axiom,
! [D: nat,A: nat,B: nat] :
( ( ( gcd_gcd_nat @ D @ ( times_times_nat @ A @ B ) )
= one_one_nat )
<=> ( ( ( gcd_gcd_nat @ D @ A )
= one_one_nat )
& ( ( gcd_gcd_nat @ D @ B )
= one_one_nat ) ) ) ).
thf(fact_4353_gcd__mult__cancel__nat,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ( gcd_gcd_nat @ K_1 @ N )
= one_one_nat )
=> ( ( gcd_gcd_nat @ ( times_times_nat @ K_1 @ M ) @ N )
= ( gcd_gcd_nat @ M @ N ) ) ) ).
thf(fact_4354_coprime__mult__nat,axiom,
! [B: nat,D: nat,A: nat] :
( ( ( gcd_gcd_nat @ D @ A )
= one_one_nat )
=> ( ( ( gcd_gcd_nat @ D @ B )
= one_one_nat )
=> ( ( gcd_gcd_nat @ D @ ( times_times_nat @ A @ B ) )
= one_one_nat ) ) ) ).
thf(fact_4355_coprime__crossproduct__nat,axiom,
! [B: nat,C: nat,A: nat,D: nat] :
( ( ( gcd_gcd_nat @ A @ D )
= one_one_nat )
=> ( ( ( gcd_gcd_nat @ B @ C )
= one_one_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ D ) )
<=> ( ( A = B )
& ( C = D ) ) ) ) ) ).
thf(fact_4356_coprime__lmult__nat,axiom,
! [D: nat,A: nat,B: nat] :
( ( ( gcd_gcd_nat @ D @ ( times_times_nat @ A @ B ) )
= one_one_nat )
=> ( ( gcd_gcd_nat @ D @ A )
= one_one_nat ) ) ).
thf(fact_4357_coprime__rmult__nat,axiom,
! [D: nat,A: nat,B: nat] :
( ( ( gcd_gcd_nat @ D @ ( times_times_nat @ A @ B ) )
= one_one_nat )
=> ( ( gcd_gcd_nat @ D @ B )
= one_one_nat ) ) ).
thf(fact_4358_gcd__diff2__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
= ( gcd_gcd_nat @ M @ N ) ) ) ).
thf(fact_4359_gcd__diff1__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
= ( gcd_gcd_nat @ M @ N ) ) ) ).
thf(fact_4360_gcd__nat_Osimps,axiom,
! [X: nat,Y: nat] :
( ( ( Y = zero_zero_nat )
=> ( ( gcd_gcd_nat @ X @ Y )
= X ) )
& ( ( Y != zero_zero_nat )
=> ( ( gcd_gcd_nat @ X @ Y )
= ( gcd_gcd_nat @ Y @ ( div_mod_nat @ X @ Y ) ) ) ) ) ).
thf(fact_4361_gcd__non__0__nat,axiom,
! [X: nat,Y: nat] :
( ( Y != zero_zero_nat )
=> ( ( gcd_gcd_nat @ X @ Y )
= ( gcd_gcd_nat @ Y @ ( div_mod_nat @ X @ Y ) ) ) ) ).
thf(fact_4362_coprime__exp__nat,axiom,
! [N: nat,D: nat,A: nat] :
( ( ( gcd_gcd_nat @ D @ A )
= one_one_nat )
=> ( ( gcd_gcd_nat @ D @ ( power_power_nat @ A @ N ) )
= one_one_nat ) ) ).
thf(fact_4363_ceiling__real__of__nat,axiom,
! [N: nat] :
( ( archim856651990g_real @ ( real_nat @ N ) )
= ( semiri1621563631at_int @ N ) ) ).
thf(fact_4364_natceiling__def,axiom,
! [X: real] :
( ( natceiling @ X )
= ( nat_1 @ ( archim856651990g_real @ X ) ) ) ).
thf(fact_4365_gcd__coprime__nat,axiom,
! [B_5: nat,A_5: nat,A: nat,B: nat] :
( ( ( gcd_gcd_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A
= ( times_times_nat @ A_5 @ ( gcd_gcd_nat @ A @ B ) ) )
=> ( ( B
= ( times_times_nat @ B_5 @ ( gcd_gcd_nat @ A @ B ) ) )
=> ( ( gcd_gcd_nat @ A_5 @ B_5 )
= one_one_nat ) ) ) ) ).
thf(fact_4366_coprime__minus__one__nat,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N )
= one_one_nat ) ) ).
thf(fact_4367_divides__mult__nat,axiom,
! [N: nat,M: nat,R_1: nat] :
( ( dvd_dvd_nat @ M @ R_1 )
=> ( ( dvd_dvd_nat @ N @ R_1 )
=> ( ( ( gcd_gcd_nat @ M @ N )
= one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ R_1 ) ) ) ) ).
thf(fact_4368_coprime__dvd__mult__iff__nat,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ( gcd_gcd_nat @ K_1 @ N )
= one_one_nat )
=> ( ( dvd_dvd_nat @ K_1 @ ( times_times_nat @ M @ N ) )
<=> ( dvd_dvd_nat @ K_1 @ M ) ) ) ).
thf(fact_4369_coprime__dvd__mult__nat,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ( gcd_gcd_nat @ K_1 @ N )
= one_one_nat )
=> ( ( dvd_dvd_nat @ K_1 @ ( times_times_nat @ M @ N ) )
=> ( dvd_dvd_nat @ K_1 @ M ) ) ) ).
thf(fact_4370_div__gcd__coprime__nat,axiom,
! [B: nat,A: nat] :
( ( ( A != zero_zero_nat )
| ( B != zero_zero_nat ) )
=> ( ( gcd_gcd_nat @ ( div_div_nat @ A @ ( gcd_gcd_nat @ A @ B ) ) @ ( div_div_nat @ B @ ( gcd_gcd_nat @ A @ B ) ) )
= one_one_nat ) ) ).
thf(fact_4371_invertible__coprime__nat,axiom,
! [X: nat,Y: nat,M: nat] :
( ( ( div_mod_nat @ ( times_times_nat @ X @ Y ) @ M )
= one_one_nat )
=> ( ( gcd_gcd_nat @ X @ M )
= one_one_nat ) ) ).
thf(fact_4372_coprime__Suc__0__nat,axiom,
! [A: nat,B: nat] :
( ( ( gcd_gcd_nat @ A @ B )
= one_one_nat )
<=> ! [D_2: nat] :
( ( ( dvd_dvd_nat @ D_2 @ A )
& ( dvd_dvd_nat @ D_2 @ B ) )
<=> ( D_2
= ( suc @ zero_zero_nat ) ) ) ) ).
thf(fact_4373_gcd__coprime__exists__nat,axiom,
! [A: nat,B: nat] :
( ( ( gcd_gcd_nat @ A @ B )
!= zero_zero_nat )
=> ? [A_3: nat,B_2: nat] :
( ( A
= ( times_times_nat @ A_3 @ ( gcd_gcd_nat @ A @ B ) ) )
& ( B
= ( times_times_nat @ B_2 @ ( gcd_gcd_nat @ A @ B ) ) )
& ( ( gcd_gcd_nat @ A_3 @ B_2 )
= one_one_nat ) ) ) ).
thf(fact_4374_bezout__nat,axiom,
! [B: nat,A: nat] :
( ( A != zero_zero_nat )
=> ? [X_1: nat,Y_1: nat] :
( ( times_times_nat @ A @ X_1 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y_1 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
thf(fact_4375_Cauchy__iff2,axiom,
! [X_2: nat > real] :
( ( cauchy_real @ X_2 )
<=> ! [J_1: nat] :
? [M_4: nat] :
! [M_2: nat] :
( ( ord_less_eq_nat @ M_4 @ M_2 )
=> ! [N_1: nat] :
( ( ord_less_eq_nat @ M_4 @ N_1 )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X_2 @ M_2 ) @ ( X_2 @ N_1 ) ) ) @ ( inverse_inverse_real @ ( real_nat @ ( suc @ J_1 ) ) ) ) ) ) ) ).
thf(fact_4376_ceiling__eq2,axiom,
! [N: int,X: real] :
( ( ord_less_real @ ( real_int @ N ) @ X )
=> ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( real_int @ N ) @ one_one_real ) )
=> ( ( archim856651990g_real @ X )
= ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
thf(fact_4377_ln__powr__bound,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( ln @ X ) @ ( inverse_divide_real @ ( powr @ X @ A ) @ A ) ) ) ) ).
thf(fact_4378_real__of__int__less__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_real @ ( real_int @ X ) @ ( real_int @ Y ) )
<=> ( ord_less_int @ X @ Y ) ) ).
thf(fact_4379_powr__not__zero,axiom,
! [X: real,A: real] :
( ( powr @ X @ A )
!= zero_zero_real ) ).
thf(fact_4380_powr__one__eq__one,axiom,
! [A: real] :
( ( powr @ one_one_real @ A )
= one_one_real ) ).
thf(fact_4381_floor__real__of__int,axiom,
! [N: int] :
( ( archim1246769320r_real @ ( real_int @ N ) )
= N ) ).
thf(fact_4382_real__of__int__floor__cancel,axiom,
! [X: real] :
( ( ( real_int @ ( archim1246769320r_real @ X ) )
= X )
<=> ? [N_1: int] :
( X
= ( real_int @ N_1 ) ) ) ).
thf(fact_4383_powr__powr,axiom,
! [X: real,A: real,B: real] :
( ( powr @ ( powr @ X @ A ) @ B )
= ( powr @ X @ ( times_times_real @ A @ B ) ) ) ).
thf(fact_4384_powr__powr__swap,axiom,
! [X: real,A: real,B: real] :
( ( powr @ ( powr @ X @ A ) @ B )
= ( powr @ ( powr @ X @ B ) @ A ) ) ).
thf(fact_4385_real__of__int__inject,axiom,
! [X: int,Y: int] :
( ( ( real_int @ X )
= ( real_int @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_4386_real__of__int__zero,axiom,
( ( real_int @ zero_zero_int )
= zero_zero_real ) ).
thf(fact_4387_real__of__int__zero__cancel,axiom,
! [X: int] :
( ( ( real_int @ X )
= zero_zero_real )
<=> ( X = zero_zero_int ) ) ).
thf(fact_4388_real__of__int__le__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_real @ ( real_int @ X ) @ ( real_int @ Y ) )
<=> ( ord_less_eq_int @ X @ Y ) ) ).
thf(fact_4389_real__of__int__diff,axiom,
! [X: int,Y: int] :
( ( real_int @ ( minus_minus_int @ X @ Y ) )
= ( minus_minus_real @ ( real_int @ X ) @ ( real_int @ Y ) ) ) ).
thf(fact_4390_power__real__of__int,axiom,
! [X: int,N: nat] :
( ( power_power_real @ ( real_int @ X ) @ N )
= ( real_int @ ( power_power_int @ X @ N ) ) ) ).
thf(fact_4391_real__of__int__power,axiom,
! [X: int,N: nat] :
( ( real_int @ ( power_power_int @ X @ N ) )
= ( power_power_real @ ( real_int @ X ) @ N ) ) ).
thf(fact_4392_real__of__int__of__nat__eq,axiom,
! [N: nat] :
( ( real_int @ ( semiri1621563631at_int @ N ) )
= ( real_nat @ N ) ) ).
thf(fact_4393_real__of__int__add,axiom,
! [X: int,Y: int] :
( ( real_int @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_real @ ( real_int @ X ) @ ( real_int @ Y ) ) ) ).
thf(fact_4394_real__number__of,axiom,
! [V: int] :
( ( real_int @ ( number_number_of_int @ V ) )
= ( number267125858f_real @ V ) ) ).
thf(fact_4395_real__of__one,axiom,
( ( real_int @ one_one_int )
= one_one_real ) ).
thf(fact_4396_real__of__int__mult,axiom,
! [X: int,Y: int] :
( ( real_int @ ( times_times_int @ X @ Y ) )
= ( times_times_real @ ( real_int @ X ) @ ( real_int @ Y ) ) ) ).
thf(fact_4397_ceiling__real__of__int,axiom,
! [N: int] :
( ( archim856651990g_real @ ( real_int @ N ) )
= N ) ).
thf(fact_4398_real__of__int__ceiling__cancel,axiom,
! [X: real] :
( ( ( real_int @ ( archim856651990g_real @ X ) )
= X )
<=> ? [N_1: int] :
( X
= ( real_int @ N_1 ) ) ) ).
thf(fact_4399_real__of__int__minus,axiom,
! [X: int] :
( ( real_int @ ( uminus_uminus_int @ X ) )
= ( uminus_uminus_real @ ( real_int @ X ) ) ) ).
thf(fact_4400_real__of__int__abs,axiom,
! [X: int] :
( ( real_int @ ( abs_abs_int @ X ) )
= ( abs_abs_real @ ( real_int @ X ) ) ) ).
thf(fact_4401_real__of__int__floor__le,axiom,
! [R_1: real] : ( ord_less_eq_real @ ( real_int @ ( archim1246769320r_real @ R_1 ) ) @ R_1 ) ).
thf(fact_4402_real__of__int__ceiling__ge,axiom,
! [R_1: real] : ( ord_less_eq_real @ R_1 @ ( real_int @ ( archim856651990g_real @ R_1 ) ) ) ).
thf(fact_4403_powr__ge__pzero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr @ X @ Y ) ) ).
thf(fact_4404_powr__gt__zero,axiom,
! [X: real,A: real] : ( ord_less_real @ zero_zero_real @ ( powr @ X @ A ) ) ).
thf(fact_4405_powr__less__mono2,axiom,
! [Y: real,X: real,A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( powr @ X @ A ) @ ( powr @ Y @ A ) ) ) ) ) ).
thf(fact_4406_powr__less__mono2__neg,axiom,
! [Y: real,X: real,A: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( powr @ Y @ A ) @ ( powr @ X @ A ) ) ) ) ) ).
thf(fact_4407_powr__mono,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( powr @ X @ A ) @ ( powr @ X @ B ) ) ) ) ).
thf(fact_4408_powr__zero__eq__one,axiom,
! [X: real] :
( ( powr @ X @ zero_zero_real )
= one_one_real ) ).
thf(fact_4409_powr__less__cancel__iff,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ ( powr @ X @ A ) @ ( powr @ X @ B ) )
<=> ( ord_less_real @ A @ B ) ) ) ).
thf(fact_4410_powr__less__mono,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( powr @ X @ A ) @ ( powr @ X @ B ) ) ) ) ).
thf(fact_4411_powr__less__cancel,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_real @ ( powr @ X @ A ) @ ( powr @ X @ B ) )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ A @ B ) ) ) ).
thf(fact_4412_powr__add,axiom,
! [X: real,A: real,B: real] :
( ( powr @ X @ ( plus_plus_real @ A @ B ) )
= ( times_times_real @ ( powr @ X @ A ) @ ( powr @ X @ B ) ) ) ).
thf(fact_4413_powr__divide2,axiom,
! [X: real,A: real,B: real] :
( ( inverse_divide_real @ ( powr @ X @ A ) @ ( powr @ X @ B ) )
= ( powr @ X @ ( minus_minus_real @ A @ B ) ) ) ).
thf(fact_4414_powr__minus,axiom,
! [X: real,A: real] :
( ( powr @ X @ ( uminus_uminus_real @ A ) )
= ( inverse_inverse_real @ ( powr @ X @ A ) ) ) ).
thf(fact_4415_real__of__int__div4,axiom,
! [N: int,X: int] : ( ord_less_eq_real @ ( real_int @ ( div_div_int @ N @ X ) ) @ ( inverse_divide_real @ ( real_int @ N ) @ ( real_int @ X ) ) ) ).
thf(fact_4416_floor__less__eq,axiom,
! [X: real,A: int] :
( ( ord_less_int @ ( archim1246769320r_real @ X ) @ A )
<=> ( ord_less_real @ X @ ( real_int @ A ) ) ) ).
thf(fact_4417_le__floor,axiom,
! [A: int,X: real] :
( ( ord_less_eq_real @ ( real_int @ A ) @ X )
=> ( ord_less_eq_int @ A @ ( archim1246769320r_real @ X ) ) ) ).
thf(fact_4418_real__le__floor,axiom,
! [A: int,X: real] :
( ( ord_less_eq_int @ A @ ( archim1246769320r_real @ X ) )
=> ( ord_less_eq_real @ ( real_int @ A ) @ X ) ) ).
thf(fact_4419_le__floor__eq,axiom,
! [A: int,X: real] :
( ( ord_less_eq_int @ A @ ( archim1246769320r_real @ X ) )
<=> ( ord_less_eq_real @ ( real_int @ A ) @ X ) ) ).
thf(fact_4420_floor__add,axiom,
! [X: real,A: int] :
( ( archim1246769320r_real @ ( plus_plus_real @ X @ ( real_int @ A ) ) )
= ( plus_plus_int @ ( archim1246769320r_real @ X ) @ A ) ) ).
thf(fact_4421_less__ceiling__eq,axiom,
! [A: int,X: real] :
( ( ord_less_int @ A @ ( archim856651990g_real @ X ) )
<=> ( ord_less_real @ ( real_int @ A ) @ X ) ) ).
thf(fact_4422_ceiling__le__eq,axiom,
! [X: real,A: int] :
( ( ord_less_eq_int @ ( archim856651990g_real @ X ) @ A )
<=> ( ord_less_eq_real @ X @ ( real_int @ A ) ) ) ).
thf(fact_4423_ceiling__le,axiom,
! [X: real,A: int] :
( ( ord_less_eq_real @ X @ ( real_int @ A ) )
=> ( ord_less_eq_int @ ( archim856651990g_real @ X ) @ A ) ) ).
thf(fact_4424_ceiling__le__real,axiom,
! [X: real,A: int] :
( ( ord_less_eq_int @ ( archim856651990g_real @ X ) @ A )
=> ( ord_less_eq_real @ X @ ( real_int @ A ) ) ) ).
thf(fact_4425_floor__power,axiom,
! [N: nat,X: real] :
( ( X
= ( real_int @ ( archim1246769320r_real @ X ) ) )
=> ( ( archim1246769320r_real @ ( power_power_real @ X @ N ) )
= ( power_power_int @ ( archim1246769320r_real @ X ) @ N ) ) ) ).
thf(fact_4426_floor__subtract,axiom,
! [X: real,A: int] :
( ( archim1246769320r_real @ ( minus_minus_real @ X @ ( real_int @ A ) ) )
= ( minus_minus_int @ ( archim1246769320r_real @ X ) @ A ) ) ).
thf(fact_4427_ceiling__add,axiom,
! [X: real,A: int] :
( ( archim856651990g_real @ ( plus_plus_real @ X @ ( real_int @ A ) ) )
= ( plus_plus_int @ ( archim856651990g_real @ X ) @ A ) ) ).
thf(fact_4428_floor__minus__real__of__int,axiom,
! [N: int] :
( ( archim1246769320r_real @ ( uminus_uminus_real @ ( real_int @ N ) ) )
= ( uminus_uminus_int @ N ) ) ).
thf(fact_4429_ceiling__subtract,axiom,
! [X: real,A: int] :
( ( archim856651990g_real @ ( minus_minus_real @ X @ ( real_int @ A ) ) )
= ( minus_minus_int @ ( archim856651990g_real @ X ) @ A ) ) ).
thf(fact_4430_powr__mono2,axiom,
! [Y: real,X: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( powr @ X @ A ) @ ( powr @ Y @ A ) ) ) ) ) ).
thf(fact_4431_powr__mult,axiom,
! [A: real,Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( powr @ ( times_times_real @ X @ Y ) @ A )
= ( times_times_real @ ( powr @ X @ A ) @ ( powr @ Y @ A ) ) ) ) ) ).
thf(fact_4432_ge__one__powr__ge__zero,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( powr @ X @ A ) ) ) ) ).
thf(fact_4433_powr__le__cancel__iff,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ ( powr @ X @ A ) @ ( powr @ X @ B ) )
<=> ( ord_less_eq_real @ A @ B ) ) ) ).
thf(fact_4434_powr__one__gt__zero__iff,axiom,
! [X: real] :
( ( ( powr @ X @ one_one_real )
= X )
<=> ( ord_less_real @ zero_zero_real @ X ) ) ).
thf(fact_4435_powr__divide,axiom,
! [A: real,Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( powr @ ( inverse_divide_real @ X @ Y ) @ A )
= ( inverse_divide_real @ ( powr @ X @ A ) @ ( powr @ Y @ A ) ) ) ) ) ).
thf(fact_4436_powr__minus__divide,axiom,
! [X: real,A: real] :
( ( powr @ X @ ( uminus_uminus_real @ A ) )
= ( inverse_divide_real @ one_one_real @ ( powr @ X @ A ) ) ) ).
thf(fact_4437_powr__def,axiom,
! [X: real,A: real] :
( ( powr @ X @ A )
= ( exp_real @ ( times_times_real @ A @ ( ln @ X ) ) ) ) ).
thf(fact_4438_real__of__int__lt__zero__cancel__iff,axiom,
! [N: int] :
( ( ord_less_real @ ( real_int @ N ) @ zero_zero_real )
<=> ( ord_less_int @ N @ zero_zero_int ) ) ).
thf(fact_4439_real__of__int__gt__zero__cancel__iff,axiom,
! [N: int] :
( ( ord_less_real @ zero_zero_real @ ( real_int @ N ) )
<=> ( ord_less_int @ zero_zero_int @ N ) ) ).
thf(fact_4440_real__of__int__ge__zero__cancel__iff,axiom,
! [N: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( real_int @ N ) )
<=> ( ord_less_eq_int @ zero_zero_int @ N ) ) ).
thf(fact_4441_real__of__int__le__zero__cancel__iff,axiom,
! [N: int] :
( ( ord_less_eq_real @ ( real_int @ N ) @ zero_zero_real )
<=> ( ord_less_eq_int @ N @ zero_zero_int ) ) ).
thf(fact_4442_number__of__less__real__of__int__iff2,axiom,
! [M: int,N: int] :
( ( ord_less_real @ ( real_int @ M ) @ ( number267125858f_real @ N ) )
<=> ( ord_less_int @ M @ ( number_number_of_int @ N ) ) ) ).
thf(fact_4443_number__of__less__real__of__int__iff,axiom,
! [N: int,M: int] :
( ( ord_less_real @ ( number267125858f_real @ N ) @ ( real_int @ M ) )
<=> ( ord_less_int @ ( number_number_of_int @ N ) @ M ) ) ).
thf(fact_4444_number__of__le__real__of__int__iff2,axiom,
! [M: int,N: int] :
( ( ord_less_eq_real @ ( real_int @ M ) @ ( number267125858f_real @ N ) )
<=> ( ord_less_eq_int @ M @ ( number_number_of_int @ N ) ) ) ).
thf(fact_4445_number__of__le__real__of__int__iff,axiom,
! [N: int,M: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ N ) @ ( real_int @ M ) )
<=> ( ord_less_eq_int @ ( number_number_of_int @ N ) @ M ) ) ).
thf(fact_4446_int__less__real__le,axiom,
! [N: int,M: int] :
( ( ord_less_int @ N @ M )
<=> ( ord_less_eq_real @ ( plus_plus_real @ ( real_int @ N ) @ one_one_real ) @ ( real_int @ M ) ) ) ).
thf(fact_4447_lemma__floor2,axiom,
! [N: int,X: int] :
( ( ord_less_real @ ( real_int @ N ) @ ( plus_plus_real @ ( real_int @ X ) @ one_one_real ) )
=> ( ord_less_eq_int @ N @ X ) ) ).
thf(fact_4448_int__le__real__less,axiom,
! [N: int,M: int] :
( ( ord_less_eq_int @ N @ M )
<=> ( ord_less_real @ ( real_int @ N ) @ ( plus_plus_real @ ( real_int @ M ) @ one_one_real ) ) ) ).
thf(fact_4449_real__nat__eq__real,axiom,
! [X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( real_nat @ ( nat_1 @ X ) )
= ( real_int @ X ) ) ) ).
thf(fact_4450_real__of__int__div,axiom,
! [N: int,D: int] :
( ( D != zero_zero_int )
=> ( ( dvd_dvd_int @ D @ N )
=> ( ( real_int @ ( div_div_int @ N @ D ) )
= ( inverse_divide_real @ ( real_int @ N ) @ ( real_int @ D ) ) ) ) ) ).
thf(fact_4451_real__of__int__floor__add__one__ge,axiom,
! [R_1: real] : ( ord_less_eq_real @ R_1 @ ( plus_plus_real @ ( real_int @ ( archim1246769320r_real @ R_1 ) ) @ one_one_real ) ) ).
thf(fact_4452_real__of__int__floor__add__one__gt,axiom,
! [R_1: real] : ( ord_less_real @ R_1 @ ( plus_plus_real @ ( real_int @ ( archim1246769320r_real @ R_1 ) ) @ one_one_real ) ) ).
thf(fact_4453_floor__eq,axiom,
! [N: int,X: real] :
( ( ord_less_real @ ( real_int @ N ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( real_int @ N ) @ one_one_real ) )
=> ( ( archim1246769320r_real @ X )
= N ) ) ) ).
thf(fact_4454_tan__periodic__int,axiom,
! [X: real,I: int] :
( ( tan @ ( plus_plus_real @ X @ ( times_times_real @ ( real_int @ I ) @ pi ) ) )
= ( tan @ X ) ) ).
thf(fact_4455_real__of__int__floor__ge__diff__one,axiom,
! [R_1: real] : ( ord_less_eq_real @ ( minus_minus_real @ R_1 @ one_one_real ) @ ( real_int @ ( archim1246769320r_real @ R_1 ) ) ) ).
thf(fact_4456_real__of__int__floor__gt__diff__one,axiom,
! [R_1: real] : ( ord_less_real @ ( minus_minus_real @ R_1 @ one_one_real ) @ ( real_int @ ( archim1246769320r_real @ R_1 ) ) ) ).
thf(fact_4457_real__of__int__ceiling__le__add__one,axiom,
! [R_1: real] : ( ord_less_eq_real @ ( real_int @ ( archim856651990g_real @ R_1 ) ) @ ( plus_plus_real @ R_1 @ one_one_real ) ) ).
thf(fact_4458_real__of__int__ceiling__diff__one__le,axiom,
! [R_1: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_int @ ( archim856651990g_real @ R_1 ) ) @ one_one_real ) @ R_1 ) ).
thf(fact_4459_powr__realpow,axiom,
! [N: nat,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr @ X @ ( real_nat @ N ) )
= ( power_power_real @ X @ N ) ) ) ).
thf(fact_4460_ln__powr,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ln @ ( powr @ X @ Y ) )
= ( times_times_real @ Y @ ( ln @ X ) ) ) ) ) ).
thf(fact_4461_powr__log__cancel,axiom,
! [X: real,A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr @ A @ ( log @ A @ X ) )
= X ) ) ) ) ).
thf(fact_4462_log__powr__cancel,axiom,
! [Y: real,A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ ( powr @ A @ Y ) )
= Y ) ) ) ).
thf(fact_4463_lemma__floor,axiom,
! [N: int,M: int,R_1: real] :
( ( ord_less_eq_real @ ( real_int @ M ) @ R_1 )
=> ( ( ord_less_real @ R_1 @ ( plus_plus_real @ ( real_int @ N ) @ one_one_real ) )
=> ( ord_less_eq_int @ M @ N ) ) ) ).
thf(fact_4464_real__of__int__div2,axiom,
! [N: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( inverse_divide_real @ ( real_int @ N ) @ ( real_int @ X ) ) @ ( real_int @ ( div_div_int @ N @ X ) ) ) ) ).
thf(fact_4465_real__of__int__div3,axiom,
! [N: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( inverse_divide_real @ ( real_int @ N ) @ ( real_int @ X ) ) @ ( real_int @ ( div_div_int @ N @ X ) ) ) @ one_one_real ) ).
thf(fact_4466_real__of__int__div__aux,axiom,
! [X: int,D: int] :
( ( D != zero_zero_int )
=> ( ( inverse_divide_real @ ( real_int @ X ) @ ( real_int @ D ) )
= ( plus_plus_real @ ( real_int @ ( div_div_int @ X @ D ) ) @ ( inverse_divide_real @ ( real_int @ ( div_mod_int @ X @ D ) ) @ ( real_int @ D ) ) ) ) ) ).
thf(fact_4467_floor__eq2,axiom,
! [N: int,X: real] :
( ( ord_less_eq_real @ ( real_int @ N ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( real_int @ N ) @ one_one_real ) )
=> ( ( archim1246769320r_real @ X )
= N ) ) ) ).
thf(fact_4468_less__floor__eq,axiom,
! [A: int,X: real] :
( ( ord_less_int @ A @ ( archim1246769320r_real @ X ) )
<=> ( ord_less_eq_real @ ( plus_plus_real @ ( real_int @ A ) @ one_one_real ) @ X ) ) ).
thf(fact_4469_floor__le__eq,axiom,
! [X: real,A: int] :
( ( ord_less_eq_int @ ( archim1246769320r_real @ X ) @ A )
<=> ( ord_less_real @ X @ ( plus_plus_real @ ( real_int @ A ) @ one_one_real ) ) ) ).
thf(fact_4470_ceiling__eq3,axiom,
! [N: int,X: real] :
( ( ord_less_real @ ( minus_minus_real @ ( real_int @ N ) @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ ( real_int @ N ) )
=> ( ( archim856651990g_real @ X )
= N ) ) ) ).
thf(fact_4471_ceiling__less__eq,axiom,
! [X: real,A: int] :
( ( ord_less_int @ ( archim856651990g_real @ X ) @ A )
<=> ( ord_less_eq_real @ X @ ( minus_minus_real @ ( real_int @ A ) @ one_one_real ) ) ) ).
thf(fact_4472_le__ceiling__eq,axiom,
! [A: int,X: real] :
( ( ord_less_eq_int @ A @ ( archim856651990g_real @ X ) )
<=> ( ord_less_real @ ( minus_minus_real @ ( real_int @ A ) @ one_one_real ) @ X ) ) ).
thf(fact_4473_log__powr,axiom,
! [B: real,Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( log @ B @ ( powr @ X @ Y ) )
= ( times_times_real @ Y @ ( log @ B @ X ) ) ) ) ) ).
thf(fact_4474_ceiling__eq,axiom,
! [N: int,X: real] :
( ( ord_less_real @ ( real_int @ N ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( real_int @ N ) @ one_one_real ) )
=> ( ( archim856651990g_real @ X )
= ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
thf(fact_4475_powr__realpow2,axiom,
! [N: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( X = zero_zero_real )
=> ( ( power_power_real @ X @ N )
= zero_zero_real ) )
& ( ( X != zero_zero_real )
=> ( ( power_power_real @ X @ N )
= ( powr @ X @ ( real_nat @ N ) ) ) ) ) ) ) ).
thf(fact_4476_ln__powr__bound2,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( powr @ ( ln @ X ) @ A ) @ ( times_times_real @ ( powr @ A @ A ) @ X ) ) ) ) ).
thf(fact_4477_reals__Archimedean__6b__int,axiom,
! [R_1: real] :
( ( ord_less_eq_real @ zero_zero_real @ R_1 )
=> ? [N_1: int] :
( ( ord_less_eq_real @ ( real_int @ N_1 ) @ R_1 )
& ( ord_less_real @ R_1 @ ( real_int @ ( plus_plus_int @ N_1 @ one_one_int ) ) ) ) ) ).
thf(fact_4478_reals__Archimedean__6c__int,axiom,
! [R_1: real] :
( ( ord_less_real @ R_1 @ zero_zero_real )
=> ? [N_1: int] :
( ( ord_less_eq_real @ ( real_int @ N_1 ) @ R_1 )
& ( ord_less_real @ R_1 @ ( real_int @ ( plus_plus_int @ N_1 @ one_one_int ) ) ) ) ) ).
thf(fact_4479_real__lb__ub__int,axiom,
! [R_1: real] :
? [N_1: int] :
( ( ord_less_eq_real @ ( real_int @ N_1 ) @ R_1 )
& ( ord_less_real @ R_1 @ ( real_int @ ( plus_plus_int @ N_1 @ one_one_int ) ) ) ) ).
thf(fact_4480_coprime__pow,axiom,
! [C: nat,N: nat,A: nat,B: nat] :
( ( coprime @ A @ B )
=> ( ( ( times_times_nat @ A @ B )
= ( power_power_nat @ C @ N ) )
=> ? [R: nat,S_2: nat] :
( ( A
= ( power_power_nat @ R @ N ) )
& ( B
= ( power_power_nat @ S_2 @ N ) ) ) ) ) ).
thf(fact_4481_prime__power__mult,axiom,
! [X: nat,Y: nat,K_1: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( ( times_times_nat @ X @ Y )
= ( power_power_nat @ P_3 @ K_1 ) )
=> ? [I_1: nat,J_1: nat] :
( ( X
= ( power_power_nat @ P_3 @ I_1 ) )
& ( Y
= ( power_power_nat @ P_3 @ J_1 ) ) ) ) ) ).
thf(fact_4482_Euler__Fermat,axiom,
! [X: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( legacy_zgcd @ X @ M )
= one_one_int )
=> ( zcong @ ( power_power_int @ X @ ( phi @ M ) ) @ one_one_int @ M ) ) ) ).
thf(fact_4483_zgcd__zdvd1,axiom,
! [I: int,J: int] : ( dvd_dvd_int @ ( legacy_zgcd @ I @ J ) @ I ) ).
thf(fact_4484_zgcd__zdvd2,axiom,
! [I: int,J: int] : ( dvd_dvd_int @ ( legacy_zgcd @ I @ J ) @ J ) ).
thf(fact_4485_zgcd__zmult__cancel,axiom,
! [M: int,K_1: int,N: int] :
( ( ( legacy_zgcd @ K_1 @ N )
= one_one_int )
=> ( ( legacy_zgcd @ ( times_times_int @ K_1 @ M ) @ N )
= ( legacy_zgcd @ M @ N ) ) ) ).
thf(fact_4486_zgcd__zgcd__zmult,axiom,
! [N: int,K_1: int,M: int] :
( ( ( legacy_zgcd @ K_1 @ M )
= one_one_int )
=> ( ( ( legacy_zgcd @ N @ M )
= one_one_int )
=> ( ( legacy_zgcd @ ( times_times_int @ K_1 @ N ) @ M )
= one_one_int ) ) ) ).
thf(fact_4487_zgcd__eq,axiom,
! [M: int,N: int] :
( ( legacy_zgcd @ M @ N )
= ( legacy_zgcd @ N @ ( div_mod_int @ M @ N ) ) ) ).
thf(fact_4488_zgcd__1,axiom,
! [M: int] :
( ( legacy_zgcd @ M @ one_one_int )
= one_one_int ) ).
thf(fact_4489_zgcd__1__left,axiom,
! [M: int] :
( ( legacy_zgcd @ one_one_int @ M )
= one_one_int ) ).
thf(fact_4490_zgcd0,axiom,
! [I: int,J: int] :
( ( ( legacy_zgcd @ I @ J )
= zero_zero_int )
<=> ( ( I = zero_zero_int )
& ( J = zero_zero_int ) ) ) ).
thf(fact_4491_zgcd__greatest__iff,axiom,
! [K_1: int,M: int,N: int] :
( ( dvd_dvd_int @ K_1 @ ( legacy_zgcd @ M @ N ) )
<=> ( ( dvd_dvd_int @ K_1 @ M )
& ( dvd_dvd_int @ K_1 @ N ) ) ) ).
thf(fact_4492_zgcd__greatest,axiom,
! [N: int,K_1: int,M: int] :
( ( dvd_dvd_int @ K_1 @ M )
=> ( ( dvd_dvd_int @ K_1 @ N )
=> ( dvd_dvd_int @ K_1 @ ( legacy_zgcd @ M @ N ) ) ) ) ).
thf(fact_4493_zgcd__assoc,axiom,
! [K_1: int,M: int,N: int] :
( ( legacy_zgcd @ ( legacy_zgcd @ K_1 @ M ) @ N )
= ( legacy_zgcd @ K_1 @ ( legacy_zgcd @ M @ N ) ) ) ).
thf(fact_4494_zgcd__left__commute,axiom,
! [K_1: int,M: int,N: int] :
( ( legacy_zgcd @ K_1 @ ( legacy_zgcd @ M @ N ) )
= ( legacy_zgcd @ M @ ( legacy_zgcd @ K_1 @ N ) ) ) ).
thf(fact_4495_zgcd__commute,axiom,
! [I: int,J: int] :
( ( legacy_zgcd @ I @ J )
= ( legacy_zgcd @ J @ I ) ) ).
thf(fact_4496_zgcd__power__distrib,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( legacy_zgcd @ A @ B ) @ N )
= ( legacy_zgcd @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
thf(fact_4497_zgcd__zminus,axiom,
! [I: int,J: int] :
( ( legacy_zgcd @ ( uminus_uminus_int @ I ) @ J )
= ( legacy_zgcd @ I @ J ) ) ).
thf(fact_4498_zgcd__zminus2,axiom,
! [I: int,J: int] :
( ( legacy_zgcd @ I @ ( uminus_uminus_int @ J ) )
= ( legacy_zgcd @ I @ J ) ) ).
thf(fact_4499_zgcd__self,axiom,
! [M: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( legacy_zgcd @ M @ M )
= M ) ) ).
thf(fact_4500_zgcd__pos,axiom,
! [I: int,J: int] : ( ord_less_eq_int @ zero_zero_int @ ( legacy_zgcd @ I @ J ) ) ).
thf(fact_4501_zgcd__geq__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( legacy_zgcd @ X @ Y ) ) ).
thf(fact_4502_zgcd__zadd__zmult,axiom,
! [M: int,N: int,K_1: int] :
( ( legacy_zgcd @ ( plus_plus_int @ M @ ( times_times_int @ N @ K_1 ) ) @ N )
= ( legacy_zgcd @ M @ N ) ) ).
thf(fact_4503_zgcd__zmult__distrib2__abs,axiom,
! [K_1: int,M: int,N: int] :
( ( legacy_zgcd @ ( times_times_int @ K_1 @ M ) @ ( times_times_int @ K_1 @ N ) )
= ( times_times_int @ ( abs_abs_int @ K_1 ) @ ( legacy_zgcd @ M @ N ) ) ) ).
thf(fact_4504_zgcd__1__power__left__distrib,axiom,
! [N: nat,A: int,B: int] :
( ( ( legacy_zgcd @ A @ B )
= one_one_int )
=> ( ( legacy_zgcd @ ( power_power_int @ A @ N ) @ B )
= one_one_int ) ) ).
thf(fact_4505_zgcd__1__power__distrib,axiom,
! [N: nat,A: int,B: int] :
( ( ( legacy_zgcd @ A @ B )
= one_one_int )
=> ( ( legacy_zgcd @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= one_one_int ) ) ).
thf(fact_4506_zgcd__0__left,axiom,
! [M: int] :
( ( legacy_zgcd @ zero_zero_int @ M )
= ( abs_abs_int @ M ) ) ).
thf(fact_4507_zgcd__0,axiom,
! [M: int] :
( ( legacy_zgcd @ M @ zero_zero_int )
= ( abs_abs_int @ M ) ) ).
thf(fact_4508_zgcd__zdvd__zgcd__zmult,axiom,
! [M: int,N: int,K_1: int] : ( dvd_dvd_int @ ( legacy_zgcd @ M @ N ) @ ( legacy_zgcd @ ( times_times_int @ K_1 @ M ) @ N ) ) ).
thf(fact_4509_zgcd__zmult__eq__self,axiom,
! [N: int,K_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ( ( legacy_zgcd @ K_1 @ ( times_times_int @ K_1 @ N ) )
= K_1 ) ) ).
thf(fact_4510_zgcd__zmult__distrib2,axiom,
! [M: int,N: int,K_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ( ( times_times_int @ K_1 @ ( legacy_zgcd @ M @ N ) )
= ( legacy_zgcd @ ( times_times_int @ K_1 @ M ) @ ( times_times_int @ K_1 @ N ) ) ) ) ).
thf(fact_4511_zgcd__zmult__eq__self2,axiom,
! [N: int,K_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ( ( legacy_zgcd @ ( times_times_int @ K_1 @ N ) @ K_1 )
= K_1 ) ) ).
thf(fact_4512_zdvd__iff__zgcd,axiom,
! [N: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( dvd_dvd_int @ M @ N )
<=> ( ( legacy_zgcd @ N @ M )
= M ) ) ) ).
thf(fact_4513_div__zgcd__relprime,axiom,
! [B: int,A: int] :
( ( ( A != zero_zero_int )
| ( B != zero_zero_int ) )
=> ( ( legacy_zgcd @ ( div_div_int @ A @ ( legacy_zgcd @ A @ B ) ) @ ( div_div_int @ B @ ( legacy_zgcd @ A @ B ) ) )
= one_one_int ) ) ).
thf(fact_4514_zgcd__non__0,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ N )
=> ( ( legacy_zgcd @ M @ N )
= ( legacy_zgcd @ N @ ( div_mod_int @ M @ N ) ) ) ) ).
thf(fact_4515_zcong__zgcd__zmult__zmod,axiom,
! [N: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ A @ B @ N )
=> ( ( ( legacy_zgcd @ M @ N )
= one_one_int )
=> ( zcong @ A @ B @ ( times_times_int @ M @ N ) ) ) ) ) ).
thf(fact_4516_zgcd__zmult__zdvd__zgcd,axiom,
! [M: int,K_1: int,N: int] :
( ( ( legacy_zgcd @ K_1 @ N )
= one_one_int )
=> ( dvd_dvd_int @ ( legacy_zgcd @ ( times_times_int @ K_1 @ M ) @ N ) @ ( legacy_zgcd @ M @ N ) ) ) ).
thf(fact_4517_zrelprime__zdvd__zmult,axiom,
! [M: int,N: int,K_1: int] :
( ( ( legacy_zgcd @ N @ K_1 )
= one_one_int )
=> ( ( dvd_dvd_int @ K_1 @ ( times_times_int @ M @ N ) )
=> ( dvd_dvd_int @ K_1 @ M ) ) ) ).
thf(fact_4518_zrelprime__dvd__mult,axiom,
! [K_1: int,I: int,J: int] :
( ( ( legacy_zgcd @ I @ J )
= one_one_int )
=> ( ( dvd_dvd_int @ I @ ( times_times_int @ K_1 @ J ) )
=> ( dvd_dvd_int @ I @ K_1 ) ) ) ).
thf(fact_4519_zgcd__0__1__iff,axiom,
! [M: int] :
( ( ( legacy_zgcd @ zero_zero_int @ M )
= one_one_int )
<=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
thf(fact_4520_zgcd__code,axiom,
! [K_1: int,L: int] :
( ( legacy_zgcd @ K_1 @ L )
= ( abs_abs_int @ ( if_int @ ( L = zero_zero_int ) @ K_1 @ ( legacy_zgcd @ L @ ( div_mod_int @ ( abs_abs_int @ K_1 ) @ ( abs_abs_int @ L ) ) ) ) ) ) ).
thf(fact_4521_zprime__imp__zrelprime,axiom,
! [N: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ~ ( dvd_dvd_int @ P_3 @ N )
=> ( ( legacy_zgcd @ N @ P_3 )
= one_one_int ) ) ) ).
thf(fact_4522_zgcd1__iff__no__common__primedivisor,axiom,
! [A: int,B: int] :
( ( ( legacy_zgcd @ A @ B )
= one_one_int )
<=> ~ ? [P_4: int] :
( ( zprime @ P_4 )
& ( dvd_dvd_int @ P_4 @ A )
& ( dvd_dvd_int @ P_4 @ B ) ) ) ).
thf(fact_4523_zgcd__zcong__zgcd,axiom,
! [B: int,A: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( legacy_zgcd @ A @ M )
= one_one_int )
=> ( ( zcong @ A @ B @ M )
=> ( ( legacy_zgcd @ B @ M )
= one_one_int ) ) ) ) ).
thf(fact_4524_zless__zprime__imp__zrelprime,axiom,
! [N: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ord_less_int @ N @ P_3 )
=> ( ( legacy_zgcd @ N @ P_3 )
= one_one_int ) ) ) ) ).
thf(fact_4525_zcong__cancel,axiom,
! [A: int,B: int,K_1: int,M: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ( legacy_zgcd @ K_1 @ M )
= one_one_int )
=> ( ( zcong @ ( times_times_int @ A @ K_1 ) @ ( times_times_int @ B @ K_1 ) @ M )
<=> ( zcong @ A @ B @ M ) ) ) ) ).
thf(fact_4526_zcong__cancel2,axiom,
! [A: int,B: int,K_1: int,M: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ( legacy_zgcd @ K_1 @ M )
= one_one_int )
=> ( ( zcong @ ( times_times_int @ K_1 @ A ) @ ( times_times_int @ K_1 @ B ) @ M )
<=> ( zcong @ A @ B @ M ) ) ) ) ).
thf(fact_4527_zrelprime__zdvd__zmult__aux,axiom,
! [M: int,N: int,K_1: int] :
( ( ( legacy_zgcd @ N @ K_1 )
= one_one_int )
=> ( ( dvd_dvd_int @ K_1 @ ( times_times_int @ M @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( dvd_dvd_int @ K_1 @ M ) ) ) ) ).
thf(fact_4528_Bnor__mem__if,axiom,
! [A: int,B: int,M: int] :
( ( ( legacy_zgcd @ B @ M )
= one_one_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( member_int @ B @ ( bnorRset @ A @ M ) ) ) ) ) ).
thf(fact_4529_int__relprime__odd__power__divisors,axiom,
! [A: int,B: int,C: int,X: int] :
( ( ( times_times_int @ A @ B )
= ( power_power_int @ C @ ( nat_1 @ X ) ) )
=> ( ( ord_less_nat @ one_one_nat @ ( nat_1 @ X ) )
=> ( ( member_int @ X @ zOdd )
=> ( ( ( legacy_zgcd @ A @ B )
= one_one_int )
=> ? [K: int] :
( A
= ( power_power_int @ K @ ( nat_1 @ X ) ) ) ) ) ) ) ).
thf(fact_4530_xzgcd__correct,axiom,
! [M: int,K_1: int,N: int] :
( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( legacy_zgcd @ M @ N )
= K_1 )
<=> ? [S_2: int,T_1: int] :
( ( xzgcd @ M @ N )
= ( produc282740534nt_int @ K_1 @ ( product_Pair_int_int @ S_2 @ T_1 ) ) ) ) ) ).
thf(fact_4531_int__relprime__power__divisors,axiom,
! [A: int,B: int,C: int,N: nat] :
( ( ( times_times_int @ A @ B )
= ( power_power_int @ C @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ( legacy_zgcd @ A @ B )
= one_one_int )
=> ? [K: int] :
( ( abs_abs_int @ A )
= ( power_power_int @ K @ N ) ) ) ) ) ).
thf(fact_4532_xzgcd__correct__aux2,axiom,
! [M: int,N: int,R_3: int,R_1: int,S_3: int,S_1: int,T_2: int,T: int,K_1: int] :
( ? [Sn: int,Tn: int] :
( ( xzgcda @ M @ N @ R_3 @ R_1 @ S_3 @ S_1 @ T_2 @ T )
= ( produc282740534nt_int @ K_1 @ ( product_Pair_int_int @ Sn @ Tn ) ) )
=> ( ( ord_less_int @ zero_zero_int @ R_1 )
=> ( ( legacy_zgcd @ R_3 @ R_1 )
= K_1 ) ) ) ).
thf(fact_4533_zcong__lineq__unique,axiom,
! [B: int,A: int,N: int] :
( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( legacy_zgcd @ A @ N )
= one_one_int )
=> ? [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
& ( ord_less_int @ X_1 @ N )
& ( zcong @ ( times_times_int @ A @ X_1 ) @ B @ N )
& ! [Y_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Y_1 )
& ( ord_less_int @ Y_1 @ N )
& ( zcong @ ( times_times_int @ A @ Y_1 ) @ B @ N ) )
=> ( Y_1 = X_1 ) ) ) ) ) ).
thf(fact_4534_zcong__lineq__ex,axiom,
! [A: int,N: int] :
( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( legacy_zgcd @ A @ N )
= one_one_int )
=> ? [X_1: int] : ( zcong @ ( times_times_int @ A @ X_1 ) @ one_one_int @ N ) ) ) ).
thf(fact_4535_xzgcd__correct__aux1,axiom,
! [M: int,N: int,S_3: int,S_1: int,T_2: int,T: int,R_3: int,R_1: int,K_1: int] :
( ( ( legacy_zgcd @ R_3 @ R_1 )
= K_1 )
=> ( ( ord_less_int @ zero_zero_int @ R_1 )
=> ? [Sn: int,Tn: int] :
( ( xzgcda @ M @ N @ R_3 @ R_1 @ S_3 @ S_1 @ T_2 @ T )
= ( produc282740534nt_int @ K_1 @ ( product_Pair_int_int @ Sn @ Tn ) ) ) ) ) ).
thf(fact_4536_int__triple__relprime__odd__power__divisors,axiom,
! [A: int,B: int,C: int,D: int,X: int] :
( ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( power_power_int @ D @ ( nat_1 @ X ) ) )
=> ( ( ord_less_nat @ one_one_nat @ ( nat_1 @ X ) )
=> ( ( member_int @ X @ zOdd )
=> ( ( ( legacy_zgcd @ A @ B )
= one_one_int )
=> ( ( ( legacy_zgcd @ B @ C )
= one_one_int )
=> ( ( ( legacy_zgcd @ C @ A )
= one_one_int )
=> ? [K: int,L_1: int,M_2: int] :
( ( A
= ( power_power_int @ K @ ( nat_1 @ X ) ) )
& ( B
= ( power_power_int @ L_1 @ ( nat_1 @ X ) ) )
& ( C
= ( power_power_int @ M_2 @ ( nat_1 @ X ) ) ) ) ) ) ) ) ) ) ).
thf(fact_4537_zgcd__ex__linear,axiom,
! [M: int,K_1: int,N: int] :
( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( legacy_zgcd @ M @ N )
= K_1 )
=> ? [S_2: int,T_1: int] :
( K_1
= ( plus_plus_int @ ( times_times_int @ S_2 @ M ) @ ( times_times_int @ T_1 @ N ) ) ) ) ) ).
thf(fact_4538_int__triple__relprime__power__divisors,axiom,
! [A: int,B: int,C: int,D: int,N: nat] :
( ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( power_power_int @ D @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ( legacy_zgcd @ A @ B )
= one_one_int )
=> ( ( ( legacy_zgcd @ B @ C )
= one_one_int )
=> ( ( ( legacy_zgcd @ C @ A )
= one_one_int )
=> ? [K: int,L_1: int,M_2: int] :
( ( ( abs_abs_int @ A )
= ( power_power_int @ K @ N ) )
& ( ( abs_abs_int @ B )
= ( power_power_int @ L_1 @ N ) )
& ( ( abs_abs_int @ C )
= ( power_power_int @ M_2 @ N ) ) ) ) ) ) ) ) ).
thf(fact_4539_make__zrelprime,axiom,
! [B: int,A: int] :
( ( ( A != zero_zero_int )
| ( B != zero_zero_int ) )
=> ? [C_2: int,D_2: int] :
( ( A
= ( times_times_int @ ( legacy_zgcd @ A @ B ) @ C_2 ) )
& ( B
= ( times_times_int @ ( legacy_zgcd @ A @ B ) @ D_2 ) )
& ( ( legacy_zgcd @ C_2 @ D_2 )
= one_one_int ) ) ) ).
thf(fact_4540_pair__lessI2,axiom,
! [S_1: nat,T: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ S_1 @ T )
=> ( member180897546at_nat @ ( produc494345619at_nat @ ( product_Pair_nat_nat @ A @ S_1 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ pair_less ) ) ) ).
thf(fact_4541_pair__lessI1,axiom,
! [S_1: nat,T: nat,A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( member180897546at_nat @ ( produc494345619at_nat @ ( product_Pair_nat_nat @ A @ S_1 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ pair_less ) ) ).
thf(fact_4542_pair__leqI2,axiom,
! [S_1: nat,T: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ S_1 @ T )
=> ( member180897546at_nat @ ( produc494345619at_nat @ ( product_Pair_nat_nat @ A @ S_1 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ pair_leq ) ) ) ).
thf(fact_4543_pair__leqI1,axiom,
! [S_1: nat,T: nat,A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( member180897546at_nat @ ( produc494345619at_nat @ ( product_Pair_nat_nat @ A @ S_1 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ pair_leq ) ) ).
thf(fact_4544_polar__Ex,axiom,
! [Y: real,X: real] :
? [R: real,A_2: real] :
( ( X
= ( times_times_real @ R @ ( cos @ A_2 ) ) )
& ( Y
= ( times_times_real @ R @ ( sin @ A_2 ) ) ) ) ).
thf(fact_4545_dist__real__def,axiom,
! [X: real,Y: real] :
( ( dist_dist_real @ X @ Y )
= ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) ) ).
thf(fact_4546_dist__complex__def,axiom,
! [X: complex,Y: complex] :
( ( dist_dist_complex @ X @ Y )
= ( norm_norm_complex @ ( minus_minus_complex @ X @ Y ) ) ) ).
thf(fact_4547_int__cases,axiom,
! [Z_1: int] :
( ! [N_1: nat] :
( Z_1
!= ( semiri1621563631at_int @ N_1 ) )
=> ~ ! [N_1: nat] :
( Z_1
!= ( uminus_uminus_int @ ( semiri1621563631at_int @ ( suc @ N_1 ) ) ) ) ) ).
thf(fact_4548_int__of__nat__induct,axiom,
! [Z_1: int,P: int > $o] :
( ! [N_1: nat] : ( P @ ( semiri1621563631at_int @ N_1 ) )
=> ( ! [N_1: nat] : ( P @ ( uminus_uminus_int @ ( semiri1621563631at_int @ ( suc @ N_1 ) ) ) )
=> ( P @ Z_1 ) ) ) ).
thf(fact_4549_DERIV__nonpos__imp__nonincreasing,axiom,
! [F: real > real,A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A @ X_1 )
& ( ord_less_eq_real @ X_1 @ B ) )
=> ? [Y_1: real] :
( ( deriv_real @ F @ X_1 @ Y_1 )
& ( ord_less_eq_real @ Y_1 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
thf(fact_4550_DERIV__nonneg__imp__nonincreasing,axiom,
! [F: real > real,A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A @ X_1 )
& ( ord_less_eq_real @ X_1 @ B ) )
=> ? [Y_1: real] :
( ( deriv_real @ F @ X_1 @ Y_1 )
& ( ord_less_eq_real @ zero_zero_real @ Y_1 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
thf(fact_4551_gcd__nat__induct,axiom,
! [M: nat,N: nat,P: nat > nat > $o] :
( ! [M_2: nat] : ( P @ M_2 @ zero_zero_nat )
=> ( ! [M_2: nat,N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_1 )
=> ( ( P @ N_1 @ ( div_mod_nat @ M_2 @ N_1 ) )
=> ( P @ M_2 @ N_1 ) ) )
=> ( P @ M @ N ) ) ) ).
thf(fact_4552_divides__fact,axiom,
! [N: nat,P_3: nat] :
( ( ord_less_eq_nat @ one_one_nat @ P_3 )
=> ( ( ord_less_eq_nat @ P_3 @ N )
=> ( dvd_dvd_nat @ P_3 @ ( fact @ N ) ) ) ) ).
thf(fact_4553_mod__code__int__def,axiom,
! [N: quickcheck_code_int,M: quickcheck_code_int] :
( ( div_mo231679042de_int @ N @ M )
= ( quickcheck_of_int @ ( div_mod_int @ ( quickcheck_int_of @ N ) @ ( quickcheck_int_of @ M ) ) ) ) ).
thf(fact_4554_Quickcheck__Narrowing_Oof__int__inject,axiom,
! [N: int,M: int] :
( ( ( quickcheck_of_int @ N )
= ( quickcheck_of_int @ M ) )
<=> ( N = M ) ) ).
thf(fact_4555_int__of__inverse,axiom,
! [X: quickcheck_code_int] :
( ( quickcheck_of_int @ ( quickcheck_int_of @ X ) )
= X ) ).
thf(fact_4556_of__int__int__of,axiom,
! [K_1: quickcheck_code_int] :
( ( quickcheck_of_int @ ( quickcheck_int_of @ K_1 ) )
= K_1 ) ).
thf(fact_4557_int__of__of__int,axiom,
! [N: int] :
( ( quickcheck_int_of @ ( quickcheck_of_int @ N ) )
= N ) ).
thf(fact_4558_number__of__code__int__def,axiom,
number1226105091de_int = quickcheck_of_int ).
thf(fact_4559_fact__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( fact @ M ) @ ( fact @ N ) ) ) ).
thf(fact_4560_times__code__int__def,axiom,
! [N: quickcheck_code_int,M: quickcheck_code_int] :
( ( times_123202395de_int @ N @ M )
= ( quickcheck_of_int @ ( times_times_int @ ( quickcheck_int_of @ N ) @ ( quickcheck_int_of @ M ) ) ) ) ).
thf(fact_4561_plus__code__int__def,axiom,
! [N: quickcheck_code_int,M: quickcheck_code_int] :
( ( plus_p1446045655de_int @ N @ M )
= ( quickcheck_of_int @ ( plus_plus_int @ ( quickcheck_int_of @ N ) @ ( quickcheck_int_of @ M ) ) ) ) ).
thf(fact_4562_minus__code__int__def,axiom,
! [N: quickcheck_code_int,M: quickcheck_code_int] :
( ( minus_534354567de_int @ N @ M )
= ( quickcheck_of_int @ ( minus_minus_int @ ( quickcheck_int_of @ N ) @ ( quickcheck_int_of @ M ) ) ) ) ).
thf(fact_4563_zero__code__int__def,axiom,
( zero_z891286103de_int
= ( quickcheck_of_int @ zero_zero_int ) ) ).
thf(fact_4564_fact__lt,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( fact @ N ) ) ).
thf(fact_4565_fact_Osimps_I1_J,axiom,
( ( fact @ zero_zero_nat )
= one_one_nat ) ).
thf(fact_4566_fact_Osimps_I2_J,axiom,
! [N: nat] :
( ( fact @ ( suc @ N ) )
= ( times_times_nat @ ( suc @ N ) @ ( fact @ N ) ) ) ).
thf(fact_4567_fact__le,axiom,
! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( fact @ N ) ) ).
thf(fact_4568_one__code__int__def,axiom,
( one_on1684967323de_int
= ( quickcheck_of_int @ one_one_int ) ) ).
thf(fact_4569_div__code__int__def,axiom,
! [N: quickcheck_code_int,M: quickcheck_code_int] :
( ( div_di1430059507de_int @ N @ M )
= ( quickcheck_of_int @ ( div_div_int @ ( quickcheck_int_of @ N ) @ ( quickcheck_int_of @ M ) ) ) ) ).
thf(fact_4570_euclid__bound,axiom,
! [N: nat] :
? [P_4: nat] :
( ( prime @ P_4 )
& ( ord_less_nat @ N @ P_4 )
& ( ord_less_eq_nat @ P_4 @ ( suc @ ( fact @ N ) ) ) ) ).
thf(fact_4571_weak__decr__stable,axiom,
! [F: nat > nat] :
( ! [I_1: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ I_1 ) ) @ ( F @ I_1 ) )
=> ? [I_1: nat] :
! [K: nat] :
( ( F @ ( plus_plus_nat @ I_1 @ K ) )
= ( F @ I_1 ) ) ) ).
thf(fact_4572_norm__frac_Opinduct,axiom,
! [P: int > int > $o,A0: int,A1: int] :
( ( accp_P2006205492nt_int @ norm_frac_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [A_2: int,B_4: int] :
( ( accp_P2006205492nt_int @ norm_frac_rel @ ( product_Pair_int_int @ A_2 @ B_4 ) )
=> ( ( ( ord_less_int @ B_4 @ zero_zero_int )
=> ( P @ ( uminus_uminus_int @ A_2 ) @ ( uminus_uminus_int @ B_4 ) ) )
=> ( P @ A_2 @ B_4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
thf(fact_4573_upto_Opinduct,axiom,
! [P: int > int > $o,A0: int,A1: int] :
( ( accp_P2006205492nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [I_1: int,J_1: int] :
( ( accp_P2006205492nt_int @ upto_rel @ ( product_Pair_int_int @ I_1 @ J_1 ) )
=> ( ( ( ord_less_eq_int @ I_1 @ J_1 )
=> ( P @ ( plus_plus_int @ I_1 @ one_one_int ) @ J_1 ) )
=> ( P @ I_1 @ J_1 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
thf(fact_4574_small__lazy_H_Opinduct,axiom,
! [P: int > int > $o,A0: int,A1: int] :
( ( accp_P2006205492nt_int @ lazy_small_lazy_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [D_2: int,I_1: int] :
( ( accp_P2006205492nt_int @ lazy_small_lazy_rel @ ( product_Pair_int_int @ D_2 @ I_1 ) )
=> ( ( ~ ( ord_less_int @ D_2 @ I_1 )
=> ( P @ D_2 @ ( plus_plus_int @ I_1 @ one_one_int ) ) )
=> ( P @ D_2 @ I_1 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
thf(fact_4575_norRRset__def,axiom,
! [M: int] :
( ( norRRset @ M )
= ( bnorRset @ ( minus_minus_int @ M @ one_one_int ) @ M ) ) ).
thf(fact_4576_real__eq__of__int,axiom,
real_int = ring_1_of_int_real ).
thf(fact_4577_real__of__int__def,axiom,
real_int = ring_1_of_int_real ).
thf(fact_4578_complex__Im__of__int,axiom,
! [Z_1: int] :
( ( im @ ( ring_11397209091omplex @ Z_1 ) )
= zero_zero_real ) ).
thf(fact_4579_complex__Re__of__int,axiom,
! [Z_1: int] :
( ( re @ ( ring_11397209091omplex @ Z_1 ) )
= ( ring_1_of_int_real @ Z_1 ) ) ).
thf(fact_4580_complex__cnj__of__int,axiom,
! [Z_1: int] :
( ( cnj @ ( ring_11397209091omplex @ Z_1 ) )
= ( ring_11397209091omplex @ Z_1 ) ) ).
thf(fact_4581_of__int__int__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_int @ ( semiri1621563631at_int @ N ) )
= ( semiri1621563631at_int @ N ) ) ).
thf(fact_4582_number__of__real__def,axiom,
! [X: int] :
( ( number267125858f_real @ X )
= ( ring_1_of_int_real @ X ) ) ).
thf(fact_4583_int__number__of__def,axiom,
! [W: int] :
( ( number_number_of_int @ W )
= ( ring_1_of_int_int @ W ) ) ).
thf(fact_4584_complex__number__of__def,axiom,
! [W: int] :
( ( number528085621omplex @ W )
= ( ring_11397209091omplex @ W ) ) ).
thf(fact_4585_phi__def,axiom,
! [M: int] :
( ( phi @ M )
= ( finite_card_int @ ( norRRset @ M ) ) ) ).
thf(fact_4586_norR__mem__unique,axiom,
! [A: int,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ( legacy_zgcd @ A @ M )
= one_one_int )
=> ? [X_1: int] :
( ( zcong @ A @ X_1 @ M )
& ( member_int @ X_1 @ ( norRRset @ M ) )
& ! [Y_1: int] :
( ( ( zcong @ A @ Y_1 @ M )
& ( member_int @ Y_1 @ ( norRRset @ M ) ) )
=> ( Y_1 = X_1 ) ) ) ) ) ).
thf(fact_4587_card__nor__eq__noX,axiom,
! [X: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( legacy_zgcd @ X @ M )
= one_one_int )
=> ( ( finite_card_int @ ( noXRRset @ M @ X ) )
= ( finite_card_int @ ( norRRset @ M ) ) ) ) ) ).
thf(fact_4588_noX__is__RRset,axiom,
! [X: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( legacy_zgcd @ X @ M )
= one_one_int )
=> ( is_RRset @ ( noXRRset @ M @ X ) @ M ) ) ) ).
thf(fact_4589_RRset__gcd,axiom,
! [A: int,A_1: int > $o,M: int] :
( ( is_RRset @ A_1 @ M )
=> ( ( member_int @ A @ A_1 )
=> ( ( legacy_zgcd @ A @ M )
= one_one_int ) ) ) ).
thf(fact_4590_RRset__zcong__eq,axiom,
! [A: int,B: int,A_1: int > $o,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( is_RRset @ A_1 @ M )
=> ( ( zcong @ A @ B @ M )
=> ( ( member_int @ A @ A_1 )
=> ( ( member_int @ B @ A_1 )
=> ( A = B ) ) ) ) ) ) ).
thf(fact_4591_RRset2norRR__correct,axiom,
! [A: int,A_1: int > $o,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( is_RRset @ A_1 @ M )
=> ( ( member_int @ A @ A_1 )
=> ( ( zcong @ A @ ( rRset2norRR @ A_1 @ M @ A ) @ M )
& ( member_int @ ( rRset2norRR @ A_1 @ M @ A ) @ ( norRRset @ M ) ) ) ) ) ) ).
thf(fact_4592_RRset2norRR__correct2,axiom,
! [A: int,A_1: int > $o,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( is_RRset @ A_1 @ M )
=> ( ( member_int @ A @ A_1 )
=> ( member_int @ ( rRset2norRR @ A_1 @ M @ A ) @ ( norRRset @ M ) ) ) ) ) ).
thf(fact_4593_RRset2norRR__correct1,axiom,
! [A: int,A_1: int > $o,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( is_RRset @ A_1 @ M )
=> ( ( member_int @ A @ A_1 )
=> ( zcong @ A @ ( rRset2norRR @ A_1 @ M @ A ) @ M ) ) ) ) ).
thf(fact_4594_is__RRset__def,axiom,
! [A_1: int > $o,M: int] :
( ( is_RRset @ A_1 @ M )
<=> ( ( member_int_o @ A_1 @ ( rsetR @ M ) )
& ( ( finite_card_int @ A_1 )
= ( phi @ M ) ) ) ) ).
thf(fact_4595_RsetR__fin,axiom,
! [A_1: int > $o,M: int] :
( ( member_int_o @ A_1 @ ( rsetR @ M ) )
=> ( finite_finite_int @ A_1 ) ) ).
thf(fact_4596_Bnor__in__RsetR,axiom,
! [A: int,M: int] :
( ( ord_less_int @ A @ M )
=> ( member_int_o @ ( bnorRset @ A @ M ) @ ( rsetR @ M ) ) ) ).
thf(fact_4597_RRset2norRR__eq__norR,axiom,
! [A_1: int > $o,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( is_RRset @ A_1 @ M )
=> ( ( image_int_int @ ( rRset2norRR @ A_1 @ M ) @ A_1 )
= ( norRRset @ M ) ) ) ) ).
thf(fact_4598_Nitpick_Oint__lcm__def,axiom,
! [X: int,Y: int] :
( ( int_lcm @ X @ Y )
= ( semiri1621563631at_int @ ( nat_lcm @ ( nat_1 @ ( abs_abs_int @ X ) ) @ ( nat_1 @ ( abs_abs_int @ Y ) ) ) ) ) ).
thf(fact_4599_ResSet__image,axiom,
! [F: int > int,A_1: int > $o,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( resSet @ M @ A_1 )
=> ( ! [X_1: int] :
( ( member_int @ X_1 @ A_1 )
=> ! [Xa: int] :
( ( member_int @ Xa @ A_1 )
=> ( ( zcong @ ( F @ X_1 ) @ ( F @ Xa ) @ M )
=> ( X_1 = Xa ) ) ) )
=> ( resSet @ M @ ( image_int_int @ F @ A_1 ) ) ) ) ) ).
thf(fact_4600_transfer__nat__int__set__relations_I5_J,axiom,
! [A_1: nat > $o,B_1: nat > $o] :
( ( ord_less_eq_nat_o @ A_1 @ B_1 )
<=> ( ord_less_eq_int_o @ ( image_nat_int @ semiri1621563631at_int @ A_1 ) @ ( image_nat_int @ semiri1621563631at_int @ B_1 ) ) ) ).
thf(fact_4601_transfer__nat__int__set__relations_I4_J,axiom,
! [A_1: nat > $o,B_1: nat > $o] :
( ( ord_less_nat_o @ A_1 @ B_1 )
<=> ( ord_less_int_o @ ( image_nat_int @ semiri1621563631at_int @ A_1 ) @ ( image_nat_int @ semiri1621563631at_int @ B_1 ) ) ) ).
thf(fact_4602_transfer__nat__int__set__relations_I3_J,axiom,
! [A_1: nat > $o,B_1: nat > $o] :
( ( A_1 = B_1 )
<=> ( ( image_nat_int @ semiri1621563631at_int @ A_1 )
= ( image_nat_int @ semiri1621563631at_int @ B_1 ) ) ) ).
thf(fact_4603_transfer__nat__int__set__relations_I2_J,axiom,
! [X: nat,A_1: nat > $o] :
( ( member_nat @ X @ A_1 )
<=> ( member_int @ ( semiri1621563631at_int @ X ) @ ( image_nat_int @ semiri1621563631at_int @ A_1 ) ) ) ).
thf(fact_4604_transfer__int__nat__set__return__embed,axiom,
! [A_1: nat > $o] :
( ( image_int_nat @ nat_1 @ ( image_nat_int @ semiri1621563631at_int @ A_1 ) )
= A_1 ) ).
thf(fact_4605_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J,axiom,
! [A_1: nat > $o] :
( ( finite_card_nat @ A_1 )
= ( finite_card_int @ ( image_nat_int @ semiri1621563631at_int @ A_1 ) ) ) ).
thf(fact_4606_transfer__nat__int__set__relations_I1_J,axiom,
! [A_1: nat > $o] :
( ( finite_finite_nat @ A_1 )
<=> ( finite_finite_int @ ( image_nat_int @ semiri1621563631at_int @ A_1 ) ) ) ).
thf(fact_4607_SetS__def,axiom,
! [A: int,P_3: int] :
( ( setS @ A @ P_3 )
= ( image_int_int_o @ ( multInvPair @ A @ P_3 ) @ ( sRStar @ P_3 ) ) ) ).
thf(fact_4608_infinite__nat__iff__unbounded__le,axiom,
! [S: nat > $o] :
( ~ ( finite_finite_nat @ S )
<=> ! [M_2: nat] :
? [N_1: nat] :
( ( ord_less_eq_nat @ M_2 @ N_1 )
& ( member_nat @ N_1 @ S ) ) ) ).
thf(fact_4609_infinite__nat__iff__unbounded,axiom,
! [S: nat > $o] :
( ~ ( finite_finite_nat @ S )
<=> ! [M_2: nat] :
? [N_1: nat] :
( ( ord_less_nat @ M_2 @ N_1 )
& ( member_nat @ N_1 @ S ) ) ) ).
thf(fact_4610_Nitpick_Onat__lcm__def,axiom,
! [X: nat,Y: nat] :
( ( nat_lcm @ X @ Y )
= ( div_div_nat @ ( times_times_nat @ X @ Y ) @ ( nat_gcd @ X @ Y ) ) ) ).
thf(fact_4611_transfer__int__nat__set__relations_I4_J,axiom,
! [B_1: int > $o,A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ( nat_nat_set @ B_1 )
=> ( ( ord_less_int_o @ A_1 @ B_1 )
<=> ( ord_less_nat_o @ ( image_int_nat @ nat_1 @ A_1 ) @ ( image_int_nat @ nat_1 @ B_1 ) ) ) ) ) ).
thf(fact_4612_gcd__eq__nitpick__gcd,axiom,
! [X: nat,Y: nat] :
( ( gcd_gcd_nat @ X @ Y )
= ( nat_gcd @ X @ Y ) ) ).
thf(fact_4613_Nat__Transfer_Otransfer__nat__int__set__function__closures_I6_J,axiom,
! [X: int,A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ( member_int @ X @ A_1 )
=> ( ord_less_eq_int @ zero_zero_int @ X ) ) ) ).
thf(fact_4614_nat__set__def,axiom,
! [S: int > $o] :
( ( nat_nat_set @ S )
<=> ! [X_1: int] :
( ( member_int @ X_1 @ S )
=> ( ord_less_eq_int @ zero_zero_int @ X_1 ) ) ) ).
thf(fact_4615_Nat__Transfer_Otransfer__int__nat__set__function__closures_I5_J,axiom,
! [C_1: nat > $o] : ( nat_nat_set @ ( image_nat_int @ semiri1621563631at_int @ C_1 ) ) ).
thf(fact_4616_transfer__int__nat__set__relations_I3_J,axiom,
! [B_1: int > $o,A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ( nat_nat_set @ B_1 )
=> ( ( A_1 = B_1 )
<=> ( ( image_int_nat @ nat_1 @ A_1 )
= ( image_int_nat @ nat_1 @ B_1 ) ) ) ) ) ).
thf(fact_4617_nat__gcd_Osimps,axiom,
! [X: nat,Y: nat] :
( ( ( Y = zero_zero_nat )
=> ( ( nat_gcd @ X @ Y )
= X ) )
& ( ( Y != zero_zero_nat )
=> ( ( nat_gcd @ X @ Y )
= ( nat_gcd @ Y @ ( div_mod_nat @ X @ Y ) ) ) ) ) ).
thf(fact_4618_transfer__nat__int__set__return__embed,axiom,
! [A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ( image_nat_int @ semiri1621563631at_int @ ( image_int_nat @ nat_1 @ A_1 ) )
= A_1 ) ) ).
thf(fact_4619_transfer__int__nat__set__relations_I1_J,axiom,
! [A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ( finite_finite_int @ A_1 )
<=> ( finite_finite_nat @ ( image_int_nat @ nat_1 @ A_1 ) ) ) ) ).
thf(fact_4620_Nat__Transfer_Otransfer__int__nat__set__functions_I1_J,axiom,
! [A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ( finite_card_int @ A_1 )
= ( finite_card_nat @ ( image_int_nat @ nat_1 @ A_1 ) ) ) ) ).
thf(fact_4621_transfer__int__nat__set__relations_I5_J,axiom,
! [B_1: int > $o,A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ( nat_nat_set @ B_1 )
=> ( ( ord_less_eq_int_o @ A_1 @ B_1 )
<=> ( ord_less_eq_nat_o @ ( image_int_nat @ nat_1 @ A_1 ) @ ( image_int_nat @ nat_1 @ B_1 ) ) ) ) ) ).
thf(fact_4622_nat__gcd_Opsimps,axiom,
! [X: nat,Y: nat] :
( ( accp_P490777396at_nat @ nat_gcd_rel @ ( product_Pair_nat_nat @ X @ Y ) )
=> ( ( ( Y = zero_zero_nat )
=> ( ( nat_gcd @ X @ Y )
= X ) )
& ( ( Y != zero_zero_nat )
=> ( ( nat_gcd @ X @ Y )
= ( nat_gcd @ Y @ ( div_mod_nat @ X @ Y ) ) ) ) ) ) ).
thf(fact_4623_Nitpick_Oint__gcd__def,axiom,
! [X: int,Y: int] :
( ( int_gcd @ X @ Y )
= ( semiri1621563631at_int @ ( nat_gcd @ ( nat_1 @ ( abs_abs_int @ X ) ) @ ( nat_1 @ ( abs_abs_int @ Y ) ) ) ) ) ).
thf(fact_4624_nat__gcd_Opinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ( accp_P490777396at_nat @ nat_gcd_rel @ ( product_Pair_nat_nat @ A0 @ A1 ) )
=> ( ! [X_1: nat,Y_1: nat] :
( ( accp_P490777396at_nat @ nat_gcd_rel @ ( product_Pair_nat_nat @ X_1 @ Y_1 ) )
=> ( ( ( Y_1 != zero_zero_nat )
=> ( P @ Y_1 @ ( div_mod_nat @ X_1 @ Y_1 ) ) )
=> ( P @ X_1 @ Y_1 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
thf(fact_4625_finite__nat__set__iff__bounded__le,axiom,
! [N_3: nat > $o] :
( ( finite_finite_nat @ N_3 )
<=> ? [M_2: nat] :
! [X_1: nat] :
( ( member_nat @ X_1 @ N_3 )
=> ( ord_less_eq_nat @ X_1 @ M_2 ) ) ) ).
thf(fact_4626_finite__nat__set__iff__bounded,axiom,
! [N_3: nat > $o] :
( ( finite_finite_nat @ N_3 )
<=> ? [M_2: nat] :
! [X_1: nat] :
( ( member_nat @ X_1 @ N_3 )
=> ( ord_less_nat @ X_1 @ M_2 ) ) ) ).
thf(fact_4627_bounded__nat__set__is__finite,axiom,
! [N: nat,N_3: nat > $o] :
( ! [X_1: nat] :
( ( member_nat @ X_1 @ N_3 )
=> ( ord_less_nat @ X_1 @ N ) )
=> ( finite_finite_nat @ N_3 ) ) ).
thf(fact_4628_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( ord_gr1297742076an_int @ L @ U ) )
= ( nat_1 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).
thf(fact_4629_finite__greaterThanLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( ord_gr1297742076an_int @ L @ U ) ) ).
thf(fact_4630_Rats__abs__nat__div__natE,axiom,
! [X: real] :
( ( member_real @ X @ field_1210416355s_real )
=> ~ ! [M_2: nat,N_1: nat] :
( ( N_1 != zero_zero_nat )
=> ( ( ( abs_abs_real @ X )
= ( inverse_divide_real @ ( real_nat @ M_2 ) @ ( real_nat @ N_1 ) ) )
=> ( ( gcd_gcd_nat @ M_2 @ N_1 )
!= one_one_nat ) ) ) ) ).
thf(fact_4631_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( ord_gr660468384an_nat @ L @ U ) ) ).
thf(fact_4632_Rats__real__nat,axiom,
! [N: nat] : ( member_real @ ( real_nat @ N ) @ field_1210416355s_real ) ).
thf(fact_4633_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( ord_gr660468384an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
thf(fact_4634_Rats__dense__in__nn__real,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ? [X_1: real] :
( ( member_real @ X_1 @ field_1210416355s_real )
& ( ord_less_real @ X @ X_1 )
& ( ord_less_real @ X_1 @ Y ) ) ) ) ).
thf(fact_4635_DERIV__isconst3,axiom,
! [F: real > real,Y: real,X: real,A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( member_real @ X @ ( ord_gr788844697n_real @ A @ B ) )
=> ( ( member_real @ Y @ ( ord_gr788844697n_real @ A @ B ) )
=> ( ! [X_1: real] :
( ( member_real @ X_1 @ ( ord_gr788844697n_real @ A @ B ) )
=> ( deriv_real @ F @ X_1 @ zero_zero_real ) )
=> ( ( F @ X )
= ( F @ Y ) ) ) ) ) ) ).
thf(fact_4636_iszero__rat,axiom,
! [K_1: int] :
( ( iszero_rat @ ( number_number_of_rat @ K_1 ) )
<=> ( iszero_int @ ( number_number_of_int @ K_1 ) ) ) ).
thf(fact_4637_Rats__dense__in__real,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [X_1: real] :
( ( member_real @ X_1 @ field_1210416355s_real )
& ( ord_less_real @ X @ X_1 )
& ( ord_less_real @ X_1 @ Y ) ) ) ).
thf(fact_4638_Sup__greaterThanLessThan,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ Y @ X )
=> ( ( comple124823625p_real @ ( ord_gr788844697n_real @ Y @ X ) )
= X ) ) ).
thf(fact_4639_Sup__upper__EX,axiom,
! [X: real,X_2: real > $o] :
( ( member_real @ X @ X_2 )
=> ( ? [Z: real] :
! [X_1: real] :
( ( member_real @ X_1 @ X_2 )
=> ( ord_less_eq_real @ X_1 @ Z ) )
=> ( ord_less_eq_real @ X @ ( comple124823625p_real @ X_2 ) ) ) ) ).
thf(fact_4640_SupInf_OSup__upper,axiom,
! [Z_1: real,X: real,X_2: real > $o] :
( ( member_real @ X @ X_2 )
=> ( ! [X_1: real] :
( ( member_real @ X_1 @ X_2 )
=> ( ord_less_eq_real @ X_1 @ Z_1 ) )
=> ( ord_less_eq_real @ X @ ( comple124823625p_real @ X_2 ) ) ) ) ).
thf(fact_4641_Frct__code__post_I4_J,axiom,
! [K_1: int] :
( ( frct @ ( product_Pair_int_int @ ( number_number_of_int @ K_1 ) @ one_one_int ) )
= ( number_number_of_rat @ K_1 ) ) ).
thf(fact_4642_Frct__code__post_I2_J,axiom,
! [K_1: int] :
( ( frct @ ( product_Pair_int_int @ K_1 @ zero_zero_int ) )
= zero_zero_rat ) ).
thf(fact_4643_Frct__code__post_I1_J,axiom,
! [K_1: int] :
( ( frct @ ( product_Pair_int_int @ zero_zero_int @ K_1 ) )
= zero_zero_rat ) ).
thf(fact_4644_Frct__code__post_I3_J,axiom,
( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
= one_one_rat ) ).
thf(fact_4645_Frct__code__post_I6_J,axiom,
! [K_1: int,L: int] :
( ( frct @ ( product_Pair_int_int @ ( number_number_of_int @ K_1 ) @ ( number_number_of_int @ L ) ) )
= ( inverse_divide_rat @ ( number_number_of_rat @ K_1 ) @ ( number_number_of_rat @ L ) ) ) ).
thf(fact_4646_Frct__code__post_I5_J,axiom,
! [K_1: int] :
( ( frct @ ( product_Pair_int_int @ one_one_int @ ( number_number_of_int @ K_1 ) ) )
= ( inverse_divide_rat @ one_one_rat @ ( number_number_of_rat @ K_1 ) ) ) ).
thf(fact_4647_Sup__eq__maximum,axiom,
! [Z_1: real,X_2: real > $o] :
( ( member_real @ Z_1 @ X_2 )
=> ( ! [X_1: real] :
( ( member_real @ X_1 @ X_2 )
=> ( ord_less_eq_real @ X_1 @ Z_1 ) )
=> ( ( comple124823625p_real @ X_2 )
= Z_1 ) ) ) ).
thf(fact_4648_SupInf_OSup__upper2,axiom,
! [Z_1: real,Y: real,X: real,X_2: real > $o] :
( ( member_real @ X @ X_2 )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( ! [X_1: real] :
( ( member_real @ X_1 @ X_2 )
=> ( ord_less_eq_real @ X_1 @ Z_1 ) )
=> ( ord_less_eq_real @ Y @ ( comple124823625p_real @ X_2 ) ) ) ) ) ).
thf(fact_4649_abs__rat__def,axiom,
! [Q: rat] :
( ( ( ord_less_rat @ Q @ zero_zero_rat )
=> ( ( abs_abs_rat @ Q )
= ( uminus_uminus_rat @ Q ) ) )
& ( ~ ( ord_less_rat @ Q @ zero_zero_rat )
=> ( ( abs_abs_rat @ Q )
= Q ) ) ) ).
thf(fact_4650_sgn__rat__def,axiom,
! [Q: rat] :
( ( ( Q = zero_zero_rat )
=> ( ( sgn_sgn_rat @ Q )
= zero_zero_rat ) )
& ( ( Q != zero_zero_rat )
=> ( ( ( ord_less_rat @ zero_zero_rat @ Q )
=> ( ( sgn_sgn_rat @ Q )
= one_one_rat ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ Q )
=> ( ( sgn_sgn_rat @ Q )
= ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ) ).
thf(fact_4651_divide__rat__def,axiom,
! [Q: rat,R_1: rat] :
( ( inverse_divide_rat @ Q @ R_1 )
= ( times_times_rat @ Q @ ( inverse_inverse_rat @ R_1 ) ) ) ).
thf(fact_4652_normalize__negative,axiom,
! [P_3: int,Q: int] :
( ( ord_less_int @ Q @ zero_zero_int )
=> ( ( normalize @ ( product_Pair_int_int @ P_3 @ Q ) )
= ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P_3 ) @ ( uminus_uminus_int @ Q ) ) ) ) ) ).
thf(fact_4653_rat__sgn__code,axiom,
! [P_3: rat] :
( ( quotient_of @ ( sgn_sgn_rat @ P_3 ) )
= ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P_3 ) ) ) @ one_one_int ) ) ).
thf(fact_4654_quotient__of__inject,axiom,
! [A: rat,B: rat] :
( ( ( quotient_of @ A )
= ( quotient_of @ B ) )
=> ( A = B ) ) ).
thf(fact_4655_quotient__of__inject__eq,axiom,
! [A: rat,B: rat] :
( ( ( quotient_of @ A )
= ( quotient_of @ B ) )
<=> ( A = B ) ) ).
thf(fact_4656_diff__rat__def,axiom,
! [Q: rat,R_1: rat] :
( ( minus_minus_rat @ Q @ R_1 )
= ( plus_plus_rat @ Q @ ( uminus_uminus_rat @ R_1 ) ) ) ).
thf(fact_4657_less__rat__def,axiom,
! [Z_1: rat,W: rat] :
( ( ord_less_rat @ Z_1 @ W )
<=> ( ( ord_less_eq_rat @ Z_1 @ W )
& ( Z_1 != W ) ) ) ).
thf(fact_4658_quotient__of__denom__pos,axiom,
! [R_1: rat,P_3: int,Q: int] :
( ( ( quotient_of @ R_1 )
= ( product_Pair_int_int @ P_3 @ Q ) )
=> ( ord_less_int @ zero_zero_int @ Q ) ) ).
thf(fact_4659_rat__one__code,axiom,
( ( quotient_of @ one_one_rat )
= ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
thf(fact_4660_normalize__denom__pos,axiom,
! [R_1: product_prod_int_int,P_3: int,Q: int] :
( ( ( normalize @ R_1 )
= ( product_Pair_int_int @ P_3 @ Q ) )
=> ( ord_less_int @ zero_zero_int @ Q ) ) ).
thf(fact_4661_normalize__denom__zero,axiom,
! [P_3: int] :
( ( normalize @ ( product_Pair_int_int @ P_3 @ zero_zero_int ) )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
thf(fact_4662_normalize__crossproduct,axiom,
! [P_3: int,R_1: int,S_1: int,Q: int] :
( ( Q != zero_zero_int )
=> ( ( S_1 != zero_zero_int )
=> ( ( ( normalize @ ( product_Pair_int_int @ P_3 @ Q ) )
= ( normalize @ ( product_Pair_int_int @ R_1 @ S_1 ) ) )
=> ( ( times_times_int @ P_3 @ S_1 )
= ( times_times_int @ R_1 @ Q ) ) ) ) ) ).
thf(fact_4663_rat__zero__code,axiom,
( ( quotient_of @ zero_zero_rat )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
thf(fact_4664_quotient__of__number_I3_J,axiom,
! [K_1: int] :
( ( quotient_of @ ( number_number_of_rat @ K_1 ) )
= ( product_Pair_int_int @ ( number_number_of_int @ K_1 ) @ one_one_int ) ) ).
thf(fact_4665_obtain__pos__sum,axiom,
! [R_1: rat] :
( ( ord_less_rat @ zero_zero_rat @ R_1 )
=> ~ ! [S_2: rat] :
( ( ord_less_rat @ zero_zero_rat @ S_2 )
=> ! [T_1: rat] :
( ( ord_less_rat @ zero_zero_rat @ T_1 )
=> ( R_1
!= ( plus_plus_rat @ S_2 @ T_1 ) ) ) ) ) ).
thf(fact_4666_ratrel__iff,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] :
( ( member2143287562nt_int @ ( produc883642259nt_int @ X @ Y ) @ ratrel )
<=> ( ( ( product_snd_int_int @ X )
!= zero_zero_int )
& ( ( product_snd_int_int @ Y )
!= zero_zero_int )
& ( ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) )
= ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
thf(fact_4667_Fract__1__number__of,axiom,
! [K_1: int] :
( ( fract @ one_one_int @ ( number_number_of_int @ K_1 ) )
= ( inverse_divide_rat @ one_one_rat @ ( number_number_of_rat @ K_1 ) ) ) ).
thf(fact_4668_Ratreal__number__of__quotient2,axiom,
! [R_1: int,S_1: int] :
( ( ratreal @ ( inverse_divide_rat @ ( number_number_of_rat @ R_1 ) @ ( number_number_of_rat @ S_1 ) ) )
= ( inverse_divide_real @ ( number267125858f_real @ R_1 ) @ ( number267125858f_real @ S_1 ) ) ) ).
thf(fact_4669_quotient__of__eq,axiom,
! [A: int,B: int,P_3: int,Q: int] :
( ( ( quotient_of @ ( fract @ A @ B ) )
= ( product_Pair_int_int @ P_3 @ Q ) )
=> ( ( fract @ P_3 @ Q )
= ( fract @ A @ B ) ) ) ).
thf(fact_4670_eq__rat_I2_J,axiom,
! [A: int] :
( ( fract @ A @ zero_zero_int )
= ( fract @ zero_zero_int @ one_one_int ) ) ).
thf(fact_4671_one__rat,axiom,
( one_one_rat
= ( fract @ one_one_int @ one_one_int ) ) ).
thf(fact_4672_rat__number__collapse_I2_J,axiom,
( ( fract @ one_one_int @ one_one_int )
= one_one_rat ) ).
thf(fact_4673_rat__number__collapse_I1_J,axiom,
! [K_1: int] :
( ( fract @ zero_zero_int @ K_1 )
= zero_zero_rat ) ).
thf(fact_4674_rat__number__collapse_I4_J,axiom,
! [K_1: int] :
( ( fract @ K_1 @ zero_zero_int )
= zero_zero_rat ) ).
thf(fact_4675_eq__rat_I3_J,axiom,
! [A: int,C: int] :
( ( fract @ zero_zero_int @ A )
= ( fract @ zero_zero_int @ C ) ) ).
thf(fact_4676_of__nat__rat,axiom,
! [K_1: nat] :
( ( semiri151668891at_rat @ K_1 )
= ( fract @ ( semiri1621563631at_int @ K_1 ) @ one_one_int ) ) ).
thf(fact_4677_Fract__of__nat__eq,axiom,
! [K_1: nat] :
( ( fract @ ( semiri1621563631at_int @ K_1 ) @ one_one_int )
= ( semiri151668891at_rat @ K_1 ) ) ).
thf(fact_4678_of__int__rat,axiom,
! [K_1: int] :
( ( ring_1_of_int_rat @ K_1 )
= ( fract @ K_1 @ one_one_int ) ) ).
thf(fact_4679_Fract__of__int__eq,axiom,
! [K_1: int] :
( ( fract @ K_1 @ one_one_int )
= ( ring_1_of_int_rat @ K_1 ) ) ).
thf(fact_4680_Fract__of__int__quotient,axiom,
! [K_1: int,L: int] :
( ( fract @ K_1 @ L )
= ( inverse_divide_rat @ ( ring_1_of_int_rat @ K_1 ) @ ( ring_1_of_int_rat @ L ) ) ) ).
thf(fact_4681_rat__number__of__def,axiom,
! [W: int] :
( ( number_number_of_rat @ W )
= ( fract @ W @ one_one_int ) ) ).
thf(fact_4682_real__floor__code,axiom,
! [X: rat] :
( ( archim1246769320r_real @ ( ratreal @ X ) )
= ( archim791455193or_rat @ X ) ) ).
thf(fact_4683_floor__Fract,axiom,
! [A: int,B: int] :
( ( archim791455193or_rat @ ( fract @ A @ B ) )
= ( div_div_int @ A @ B ) ) ).
thf(fact_4684_inverse__rat,axiom,
! [A: int,B: int] :
( ( inverse_inverse_rat @ ( fract @ A @ B ) )
= ( fract @ B @ A ) ) ).
thf(fact_4685_minus__rat__cancel,axiom,
! [A: int,B: int] :
( ( fract @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( fract @ A @ B ) ) ).
thf(fact_4686_mult__rat__cancel,axiom,
! [A: int,B: int,C: int] :
( ( C != zero_zero_int )
=> ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( fract @ A @ B ) ) ) ).
thf(fact_4687_eq__rat_I1_J,axiom,
! [A: int,C: int,D: int,B: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( ( fract @ A @ B )
= ( fract @ C @ D ) )
<=> ( ( times_times_int @ A @ D )
= ( times_times_int @ C @ B ) ) ) ) ) ).
thf(fact_4688_divide__rat,axiom,
! [A: int,B: int,C: int,D: int] :
( ( inverse_divide_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ).
thf(fact_4689_minus__rat,axiom,
! [A: int,B: int] :
( ( uminus_uminus_rat @ ( fract @ A @ B ) )
= ( fract @ ( uminus_uminus_int @ A ) @ B ) ) ).
thf(fact_4690_mult__rat,axiom,
! [A: int,B: int,C: int,D: int] :
( ( times_times_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ).
thf(fact_4691_abs__rat,axiom,
! [A: int,B: int] :
( ( abs_abs_rat @ ( fract @ A @ B ) )
= ( fract @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
thf(fact_4692_normalize__eq,axiom,
! [A: int,B: int,P_3: int,Q: int] :
( ( ( normalize @ ( product_Pair_int_int @ A @ B ) )
= ( product_Pair_int_int @ P_3 @ Q ) )
=> ( ( fract @ P_3 @ Q )
= ( fract @ A @ B ) ) ) ).
thf(fact_4693_real__less__code,axiom,
! [X: rat,Y: rat] :
( ( ord_less_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
<=> ( ord_less_rat @ X @ Y ) ) ).
thf(fact_4694_Ratreal__number__collapse_I1_J,axiom,
( ( ratreal @ zero_zero_rat )
= zero_zero_real ) ).
thf(fact_4695_zero__real__code,axiom,
( zero_zero_real
= ( ratreal @ zero_zero_rat ) ) ).
thf(fact_4696_number__of__real__code,axiom,
! [K_1: int] :
( ( number267125858f_real @ K_1 )
= ( ratreal @ ( number_number_of_rat @ K_1 ) ) ) ).
thf(fact_4697_Ratreal__number__collapse_I3_J,axiom,
! [K_1: int] :
( ( ratreal @ ( number_number_of_rat @ K_1 ) )
= ( number267125858f_real @ K_1 ) ) ).
thf(fact_4698_Ratreal__number__collapse_I2_J,axiom,
( ( ratreal @ one_one_rat )
= one_one_real ) ).
thf(fact_4699_one__real__code,axiom,
( one_one_real
= ( ratreal @ one_one_rat ) ) ).
thf(fact_4700_real__less__eq__code,axiom,
! [X: rat,Y: rat] :
( ( ord_less_eq_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
<=> ( ord_less_eq_rat @ X @ Y ) ) ).
thf(fact_4701_real__plus__code,axiom,
! [X: rat,Y: rat] :
( ( plus_plus_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
= ( ratreal @ ( plus_plus_rat @ X @ Y ) ) ) ).
thf(fact_4702_real__minus__code,axiom,
! [X: rat,Y: rat] :
( ( minus_minus_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
= ( ratreal @ ( minus_minus_rat @ X @ Y ) ) ) ).
thf(fact_4703_real__uminus__code,axiom,
! [X: rat] :
( ( uminus_uminus_real @ ( ratreal @ X ) )
= ( ratreal @ ( uminus_uminus_rat @ X ) ) ) ).
thf(fact_4704_real__divide__code,axiom,
! [X: rat,Y: rat] :
( ( inverse_divide_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
= ( ratreal @ ( inverse_divide_rat @ X @ Y ) ) ) ).
thf(fact_4705_real__times__code,axiom,
! [X: rat,Y: rat] :
( ( times_times_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
= ( ratreal @ ( times_times_rat @ X @ Y ) ) ) ).
thf(fact_4706_real__inverse__code,axiom,
! [X: rat] :
( ( inverse_inverse_real @ ( ratreal @ X ) )
= ( ratreal @ ( inverse_inverse_rat @ X ) ) ) ).
thf(fact_4707_zero__rat,axiom,
( zero_zero_rat
= ( fract @ zero_zero_int @ one_one_int ) ) ).
thf(fact_4708_rat__number__expand_I3_J,axiom,
! [K_1: int] :
( ( number_number_of_rat @ K_1 )
= ( fract @ ( number_number_of_int @ K_1 ) @ one_one_int ) ) ).
thf(fact_4709_rat__number__collapse_I3_J,axiom,
! [K_1: int] :
( ( fract @ ( number_number_of_int @ K_1 ) @ one_one_int )
= ( number_number_of_rat @ K_1 ) ) ).
thf(fact_4710_Fract__number__of__quotient,axiom,
! [K_1: int,L: int] :
( ( fract @ ( number_number_of_int @ K_1 ) @ ( number_number_of_int @ L ) )
= ( inverse_divide_rat @ ( number_number_of_rat @ K_1 ) @ ( number_number_of_rat @ L ) ) ) ).
thf(fact_4711_quotient__of__Fract,axiom,
! [A: int,B: int] :
( ( quotient_of @ ( fract @ A @ B ) )
= ( normalize @ ( product_Pair_int_int @ A @ B ) ) ) ).
thf(fact_4712_Frct__def,axiom,
! [P_3: product_prod_int_int] :
( ( frct @ P_3 )
= ( fract @ ( product_fst_int_int @ P_3 ) @ ( product_snd_int_int @ P_3 ) ) ) ).
thf(fact_4713_less__rat,axiom,
! [A: int,C: int,D: int,B: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
<=> ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
thf(fact_4714_Ratreal__number__of__quotient,axiom,
! [R_1: int,S_1: int] :
( ( inverse_divide_real @ ( ratreal @ ( number_number_of_rat @ R_1 ) ) @ ( ratreal @ ( number_number_of_rat @ S_1 ) ) )
= ( inverse_divide_real @ ( number267125858f_real @ R_1 ) @ ( number267125858f_real @ S_1 ) ) ) ).
thf(fact_4715_sgn__rat,axiom,
! [A: int,B: int] :
( ( sgn_sgn_rat @ ( fract @ A @ B ) )
= ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).
thf(fact_4716_Fract__less__zero__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
<=> ( ord_less_int @ A @ zero_zero_int ) ) ) ).
thf(fact_4717_zero__less__Fract__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
<=> ( ord_less_int @ zero_zero_int @ A ) ) ) ).
thf(fact_4718_le__rat,axiom,
! [A: int,C: int,D: int,B: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
<=> ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
thf(fact_4719_add__rat,axiom,
! [A: int,C: int,D: int,B: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
thf(fact_4720_diff__rat,axiom,
! [A: int,C: int,D: int,B: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
thf(fact_4721_Fract__less__one__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
<=> ( ord_less_int @ A @ B ) ) ) ).
thf(fact_4722_one__less__Fract__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
<=> ( ord_less_int @ B @ A ) ) ) ).
thf(fact_4723_Fract__add__one,axiom,
! [M: int,N: int] :
( ( N != zero_zero_int )
=> ( ( fract @ ( plus_plus_int @ M @ N ) @ N )
= ( plus_plus_rat @ ( fract @ M @ N ) @ one_one_rat ) ) ) ).
thf(fact_4724_zero__le__Fract__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
<=> ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
thf(fact_4725_Fract__le__zero__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
<=> ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
thf(fact_4726_rat__floor__lemma,axiom,
! [A: int,B: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( div_div_int @ A @ B ) ) @ ( fract @ A @ B ) )
& ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( div_div_int @ A @ B ) @ one_one_int ) ) ) ) ).
thf(fact_4727_one__le__Fract__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
<=> ( ord_less_eq_int @ B @ A ) ) ) ).
thf(fact_4728_Fract__le__one__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
<=> ( ord_less_eq_int @ A @ B ) ) ) ).
thf(fact_4729_Rat__induct__pos,axiom,
! [Q: rat,P: rat > $o] :
( ! [A_2: int,B_4: int] :
( ( ord_less_int @ zero_zero_int @ B_4 )
=> ( P @ ( fract @ A_2 @ B_4 ) ) )
=> ( P @ Q ) ) ).
thf(fact_4730_adjust__eq,axiom,
! [B: int,Q: int,R_1: int] :
( ( adjust @ B @ ( product_Pair_int_int @ Q @ R_1 ) )
= ( if_Pro1731782967nt_int @ ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ R_1 @ B ) ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Q ) @ one_one_int ) @ ( minus_minus_int @ R_1 @ B ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Q ) @ R_1 ) ) ) ).
thf(fact_4731_finite__Collect__less__nat,axiom,
! [K_1: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N_1: nat] : ( ord_less_nat @ N_1 @ K_1 ) ) ) ).
thf(fact_4732_finite__Collect__le__nat,axiom,
! [K_1: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N_1: nat] : ( ord_less_eq_nat @ N_1 @ K_1 ) ) ) ).
thf(fact_4733_transfer__morphism__int__nat,axiom,
( nat_tr160667106at_int @ semiri1621563631at_int
@ ^ [N_1: nat] : $true ) ).
thf(fact_4734_Nat__Transfer_Otransfer__int__nat__set__functions_I5_J,axiom,
! [P: int > $o] :
( ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( P @ X_1 ) ) )
= ( image_nat_int @ semiri1621563631at_int
@ ( collect_nat
@ ^ [X_1: nat] : ( P @ ( semiri1621563631at_int @ X_1 ) ) ) ) ) ).
thf(fact_4735_int__card__bdd__int__set__l__le,axiom,
! [N: int] :
( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( semiri1621563631at_int
@ ( finite_card_int
@ ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_int @ zero_zero_int @ X_1 ) @ ( ord_less_eq_int @ X_1 @ N ) ) ) ) )
= N ) ) ).
thf(fact_4736_card__bdd__int__set__l,axiom,
! [N: int] :
( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( finite_card_int
@ ( collect_int
@ ^ [Y_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ Y_1 ) @ ( ord_less_int @ Y_1 @ N ) ) ) )
= ( nat_1 @ N ) ) ) ).
thf(fact_4737_card__bdd__int__set__l__le,axiom,
! [N: int] :
( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( finite_card_int
@ ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_int @ zero_zero_int @ X_1 ) @ ( ord_less_eq_int @ X_1 @ N ) ) ) )
= ( nat_1 @ N ) ) ) ).
thf(fact_4738_bdd__int__set__l__le__finite,axiom,
! [N: int] :
( finite_finite_int
@ ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_int @ zero_zero_int @ X_1 ) @ ( ord_less_eq_int @ X_1 @ N ) ) ) ) ).
thf(fact_4739_bdd__int__set__l__finite,axiom,
! [N: int] :
( finite_finite_int
@ ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( ord_less_int @ X_1 @ N ) ) ) ) ).
thf(fact_4740_DERIV__cos__add,axiom,
! [K_1: real,Xa_1: real] :
( deriv_real
@ ^ [X_1: real] : ( cos @ ( plus_plus_real @ X_1 @ K_1 ) )
@ Xa_1
@ ( uminus_uminus_real @ ( sin @ ( plus_plus_real @ Xa_1 @ K_1 ) ) ) ) ).
thf(fact_4741_less__eq__Suc__le__raw,axiom,
! [X_1: nat] :
( ( ord_less_nat @ X_1 )
= ( ord_less_eq_nat @ ( suc @ X_1 ) ) ) ).
thf(fact_4742_DERIV__sin__sin__mult,axiom,
! [X: real] :
( deriv_real
@ ^ [X_1: real] : ( times_times_real @ ( sin @ X_1 ) @ ( sin @ X_1 ) )
@ X
@ ( plus_plus_real @ ( times_times_real @ ( cos @ X ) @ ( sin @ X ) ) @ ( times_times_real @ ( cos @ X ) @ ( sin @ X ) ) ) ) ).
thf(fact_4743_DERIV__sin__add,axiom,
! [K_1: real,Xa_1: real] :
( deriv_real
@ ^ [X_1: real] : ( sin @ ( plus_plus_real @ X_1 @ K_1 ) )
@ Xa_1
@ ( cos @ ( plus_plus_real @ Xa_1 @ K_1 ) ) ) ).
thf(fact_4744_bdd__nat__set__le__finite,axiom,
! [X: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [Y_1: nat] : ( ord_less_eq_nat @ Y_1 @ X ) ) ) ).
thf(fact_4745_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I_1: nat] : ( ord_less_eq_nat @ I_1 @ N ) ) )
= ( suc @ N ) ) ).
thf(fact_4746_card__bdd__nat__set__le,axiom,
! [X: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [Y_1: nat] : ( ord_less_eq_nat @ Y_1 @ X ) ) )
= ( suc @ X ) ) ).
thf(fact_4747_nat__number__of__Bit0,axiom,
! [W: int] :
( ( number_number_of_nat @ ( bit0 @ W ) )
= ( plus_plus_nat @ ( number_number_of_nat @ W ) @ ( number_number_of_nat @ W ) ) ) ).
thf(fact_4748_Nat__Transfer_Otransfer__int__nat__set__function__closures_I4_J,axiom,
! [P: int > $o] :
( nat_nat_set
@ ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( P @ X_1 ) ) ) ) ).
thf(fact_4749_Nat__Transfer_Otransfer__nat__int__set__functions_I5_J,axiom,
! [P: nat > $o] :
( ( collect_nat @ P )
= ( image_int_nat @ nat_1
@ ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( P @ ( nat_1 @ X_1 ) ) ) ) ) ) ).
thf(fact_4750_Bseq__realpow,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( bseq_real @ ( power_power_real @ X ) ) ) ) ).
thf(fact_4751_bdd__int__set__le__finite,axiom,
! [N: int] :
( finite_finite_int
@ ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( ord_less_eq_int @ X_1 @ N ) ) ) ) ).
thf(fact_4752_diff__nat__eq__if,axiom,
! [Z_1: int,Z_3: int] :
( ( ( nat_neg @ Z_3 )
=> ( ( minus_minus_nat @ ( nat_1 @ Z_1 ) @ ( nat_1 @ Z_3 ) )
= ( nat_1 @ Z_1 ) ) )
& ( ~ ( nat_neg @ Z_3 )
=> ( ( minus_minus_nat @ ( nat_1 @ Z_1 ) @ ( nat_1 @ Z_3 ) )
= ( if_nat @ ( nat_neg @ ( minus_minus_int @ Z_1 @ Z_3 ) ) @ zero_zero_nat @ ( nat_1 @ ( minus_minus_int @ Z_1 @ Z_3 ) ) ) ) ) ) ).
thf(fact_4753_finite__divisors__int,axiom,
! [I: int] :
( ( I != zero_zero_int )
=> ( finite_finite_int
@ ( collect_int
@ ^ [D_2: int] : ( dvd_dvd_int @ D_2 @ I ) ) ) ) ).
thf(fact_4754_finite__divisors__nat,axiom,
! [M: nat] :
( ( M != zero_zero_nat )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [D_2: nat] : ( dvd_dvd_nat @ D_2 @ M ) ) ) ) ).
thf(fact_4755_primes__infinite,axiom,
~ ( finite_finite_nat @ ( collect_nat @ prime ) ) ).
thf(fact_4756_cnj_OCauchy,axiom,
! [X_2: nat > complex] :
( ( cauchy_complex @ X_2 )
=> ( cauchy_complex
@ ^ [N_1: nat] : ( cnj @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_4757_Im_OCauchy,axiom,
! [X_2: nat > complex] :
( ( cauchy_complex @ X_2 )
=> ( cauchy_real
@ ^ [N_1: nat] : ( im @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_4758_Re_OCauchy,axiom,
! [X_2: nat > complex] :
( ( cauchy_complex @ X_2 )
=> ( cauchy_real
@ ^ [N_1: nat] : ( re @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_4759_Ints__real__of__nat,axiom,
! [N: nat] : ( member_real @ ( real_nat @ N ) @ ring_1_Ints_real ) ).
thf(fact_4760_Ints__real__of__int,axiom,
! [X: int] : ( member_real @ ( real_int @ X ) @ ring_1_Ints_real ) ).
thf(fact_4761_DERIV__fun__cos,axiom,
! [G: real > real,X: real,M: real] :
( ( deriv_real @ G @ X @ M )
=> ( deriv_real
@ ^ [X_1: real] : ( cos @ ( G @ X_1 ) )
@ X
@ ( times_times_real @ ( uminus_uminus_real @ ( sin @ ( G @ X ) ) ) @ M ) ) ) ).
thf(fact_4762_DERIV__fun__sin,axiom,
! [G: real > real,X: real,M: real] :
( ( deriv_real @ G @ X @ M )
=> ( deriv_real
@ ^ [X_1: real] : ( sin @ ( G @ X_1 ) )
@ X
@ ( times_times_real @ ( cos @ ( G @ X ) ) @ M ) ) ) ).
thf(fact_4763_bdd__nat__set__l__finite,axiom,
! [X: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [Y_1: nat] : ( ord_less_nat @ Y_1 @ X ) ) ) ).
thf(fact_4764_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I_1: nat] : ( ord_less_nat @ I_1 @ N ) ) )
= N ) ).
thf(fact_4765_card__bdd__nat__set__l,axiom,
! [X: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [Y_1: nat] : ( ord_less_nat @ Y_1 @ X ) ) )
= X ) ).
thf(fact_4766_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K: nat] : ( (&) @ ( P @ K ) @ ( ord_less_nat @ K @ I ) ) ) ) ).
thf(fact_4767_zpower__number__of__even,axiom,
! [Z_1: int,W: int] :
( ( power_power_int @ Z_1 @ ( number_number_of_nat @ ( bit0 @ W ) ) )
= ( times_times_int @ ( power_power_int @ Z_1 @ ( number_number_of_nat @ W ) ) @ ( power_power_int @ Z_1 @ ( number_number_of_nat @ W ) ) ) ) ).
thf(fact_4768_card__less__Suc2,axiom,
! [I: nat,M_3: nat > $o] :
( ~ ( member_nat @ zero_zero_nat @ M_3 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] : ( (&) @ ( member_nat @ ( suc @ K ) @ M_3 ) @ ( ord_less_nat @ K @ I ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] : ( (&) @ ( member_nat @ K @ M_3 ) @ ( ord_less_nat @ K @ ( suc @ I ) ) ) ) ) ) ) ).
thf(fact_4769_card__less,axiom,
! [I: nat,M_3: nat > $o] :
( ( member_nat @ zero_zero_nat @ M_3 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] : ( (&) @ ( member_nat @ K @ M_3 ) @ ( ord_less_nat @ K @ ( suc @ I ) ) ) ) )
!= zero_zero_nat ) ) ).
thf(fact_4770_card__less__Suc,axiom,
! [I: nat,M_3: nat > $o] :
( ( member_nat @ zero_zero_nat @ M_3 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] : ( (&) @ ( member_nat @ ( suc @ K ) @ M_3 ) @ ( ord_less_nat @ K @ I ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] : ( (&) @ ( member_nat @ K @ M_3 ) @ ( ord_less_nat @ K @ ( suc @ I ) ) ) ) ) ) ) ).
thf(fact_4771_bdd__int__set__l__l__finite,axiom,
! [N: int] :
( finite_finite_int
@ ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_int @ zero_zero_int @ X_1 ) @ ( ord_less_int @ X_1 @ N ) ) ) ) ).
thf(fact_4772_noXRRset__def,axiom,
! [M: int,X: int] :
( ( noXRRset @ M @ X )
= ( image_int_int
@ ^ [A_2: int] : ( times_times_int @ A_2 @ X )
@ ( norRRset @ M ) ) ) ).
thf(fact_4773_DERIV__fun__pow,axiom,
! [N: nat,G: real > real,X: real,M: real] :
( ( deriv_real @ G @ X @ M )
=> ( deriv_real
@ ^ [X_1: real] : ( power_power_real @ ( G @ X_1 ) @ N )
@ X
@ ( times_times_real @ ( times_times_real @ ( real_nat @ N ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M ) ) ) ).
thf(fact_4774_card__bdd__int__set__le,axiom,
! [N: int] :
( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( finite_card_int
@ ( collect_int
@ ^ [Y_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ Y_1 ) @ ( ord_less_eq_int @ Y_1 @ N ) ) ) )
= ( plus_plus_nat @ ( nat_1 @ N ) @ one_one_nat ) ) ) ).
thf(fact_4775_int__card__bdd__int__set__l__l,axiom,
! [N: int] :
( ( ord_less_int @ zero_zero_int @ N )
=> ( ( semiri1621563631at_int
@ ( finite_card_int
@ ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_int @ zero_zero_int @ X_1 ) @ ( ord_less_int @ X_1 @ N ) ) ) ) )
= ( minus_minus_int @ N @ one_one_int ) ) ) ).
thf(fact_4776_card__bdd__int__set__l__l,axiom,
! [N: int] :
( ( ord_less_int @ zero_zero_int @ N )
=> ( ( finite_card_int
@ ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_int @ zero_zero_int @ X_1 ) @ ( ord_less_int @ X_1 @ N ) ) ) )
= ( minus_minus_nat @ ( nat_1 @ N ) @ one_one_nat ) ) ) ).
thf(fact_4777_DERIV__cos__cos__mult,axiom,
! [X: real] :
( deriv_real
@ ^ [X_1: real] : ( times_times_real @ ( cos @ X_1 ) @ ( cos @ X_1 ) )
@ X
@ ( plus_plus_real @ ( times_times_real @ ( uminus_uminus_real @ ( sin @ X ) ) @ ( cos @ X ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( sin @ X ) ) @ ( cos @ X ) ) ) ) ).
thf(fact_4778_Suc__eq__number__of,axiom,
! [N: nat,V: int] :
( ( ( suc @ N )
= ( number_number_of_nat @ V ) )
<=> ( ~ ( nat_neg @ ( number_number_of_int @ ( pred @ V ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ ( pred @ V ) ) )
=> ( ( nat_1 @ ( number_number_of_int @ ( pred @ V ) ) )
= N ) ) ) ) ).
thf(fact_4779_eq__number__of__Suc,axiom,
! [V: int,N: nat] :
( ( ( number_number_of_nat @ V )
= ( suc @ N ) )
<=> ( ~ ( nat_neg @ ( number_number_of_int @ ( pred @ V ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ ( pred @ V ) ) )
=> ( ( nat_1 @ ( number_number_of_int @ ( pred @ V ) ) )
= N ) ) ) ) ).
thf(fact_4780_nat__number__of__Bit1,axiom,
! [W: int] :
( ( ( nat_neg @ ( number_number_of_int @ W ) )
=> ( ( number_number_of_nat @ ( bit1 @ W ) )
= zero_zero_nat ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ W ) )
=> ( ( number_number_of_nat @ ( bit1 @ W ) )
= ( suc @ ( plus_plus_nat @ ( number_number_of_nat @ W ) @ ( number_number_of_nat @ W ) ) ) ) ) ) ).
thf(fact_4781_DERIV__pow,axiom,
! [N: nat,X: real] :
( deriv_real
@ ^ [X_1: real] : ( power_power_real @ X_1 @ N )
@ X
@ ( times_times_real @ ( real_nat @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ).
thf(fact_4782_DERIV__cos__cos__mult2,axiom,
! [X: real] :
( deriv_real
@ ^ [X_1: real] : ( times_times_real @ ( cos @ X_1 ) @ ( cos @ X_1 ) )
@ X
@ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ min ) ) @ ( cos @ X ) ) @ ( sin @ X ) ) ) ).
thf(fact_4783_less__number__of__Suc,axiom,
! [V: int,N: nat] :
( ( ord_less_nat @ ( number_number_of_nat @ V ) @ ( suc @ N ) )
<=> ( ~ ( nat_neg @ ( number_number_of_int @ ( pred @ V ) ) )
=> ( ord_less_nat @ ( nat_1 @ ( number_number_of_int @ ( pred @ V ) ) ) @ N ) ) ) ).
thf(fact_4784_less__Suc__number__of,axiom,
! [N: nat,V: int] :
( ( ord_less_nat @ ( suc @ N ) @ ( number_number_of_nat @ V ) )
<=> ( ~ ( nat_neg @ ( number_number_of_int @ ( pred @ V ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ ( pred @ V ) ) )
=> ( ord_less_nat @ N @ ( nat_1 @ ( number_number_of_int @ ( pred @ V ) ) ) ) ) ) ) ).
thf(fact_4785_le__Suc__number__of,axiom,
! [N: nat,V: int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( number_number_of_nat @ V ) )
<=> ( ~ ( nat_neg @ ( number_number_of_int @ ( pred @ V ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ ( pred @ V ) ) )
=> ( ord_less_eq_nat @ N @ ( nat_1 @ ( number_number_of_int @ ( pred @ V ) ) ) ) ) ) ) ).
thf(fact_4786_le__number__of__Suc,axiom,
! [V: int,N: nat] :
( ( ord_less_eq_nat @ ( number_number_of_nat @ V ) @ ( suc @ N ) )
<=> ( ~ ( nat_neg @ ( number_number_of_int @ ( pred @ V ) ) )
=> ( ord_less_eq_nat @ ( nat_1 @ ( number_number_of_int @ ( pred @ V ) ) ) @ N ) ) ) ).
thf(fact_4787_SR__pos,axiom,
! [X_2: int > $o,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ord_less_eq_int_o @ ( image_int_int @ ( standardRes @ M ) @ X_2 )
@ ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( ord_less_int @ X_1 @ M ) ) ) ) ) ).
thf(fact_4788_RsetR__zmult__mono,axiom,
! [X: int,A_1: int > $o,M: int] :
( ( member_int_o @ A_1 @ ( rsetR @ M ) )
=> ( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( legacy_zgcd @ X @ M )
= one_one_int )
=> ( member_int_o
@ ( image_int_int
@ ^ [A_2: int] : ( times_times_int @ A_2 @ X )
@ A_1 )
@ ( rsetR @ M ) ) ) ) ) ).
thf(fact_4789_zpower__number__of__odd,axiom,
! [Z_1: int,W: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ W ) )
=> ( ( power_power_int @ Z_1 @ ( number_number_of_nat @ ( bit1 @ W ) ) )
= ( times_times_int @ ( times_times_int @ Z_1 @ ( power_power_int @ Z_1 @ ( number_number_of_nat @ W ) ) ) @ ( power_power_int @ Z_1 @ ( number_number_of_nat @ W ) ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ W ) )
=> ( ( power_power_int @ Z_1 @ ( number_number_of_nat @ ( bit1 @ W ) ) )
= one_one_int ) ) ) ).
thf(fact_4790_DERIV__sin__sin__mult2,axiom,
! [X: real] :
( deriv_real
@ ^ [X_1: real] : ( times_times_real @ ( sin @ X_1 ) @ ( sin @ X_1 ) )
@ X
@ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( cos @ X ) ) @ ( sin @ X ) ) ) ).
thf(fact_4791_DERIV__log,axiom,
! [B: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( deriv_real @ ( log @ B ) @ X @ ( inverse_divide_real @ one_one_real @ ( times_times_real @ ( ln @ B ) @ X ) ) ) ) ).
thf(fact_4792_DERIV__cos__cos__mult3,axiom,
! [X: real] :
( deriv_real
@ ^ [X_1: real] : ( times_times_real @ ( cos @ X_1 ) @ ( cos @ X_1 ) )
@ X
@ ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( cos @ X ) ) @ ( sin @ X ) ) ) ) ).
thf(fact_4793_DERIV__sin__circle__all__zero,axiom,
! [X_1: real] :
( deriv_real
@ ^ [Y_1: real] : ( plus_plus_real @ ( power_power_real @ ( sin @ Y_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( cos @ Y_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
@ X_1
@ zero_zero_real ) ).
thf(fact_4794_DERIV__sin__realpow2a,axiom,
! [X: real] :
( deriv_real
@ ^ [X_1: real] : ( power_power_real @ ( sin @ X_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
@ X
@ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( cos @ X ) ) @ ( sin @ X ) ) ) ).
thf(fact_4795_DERIV__sin__realpow2,axiom,
! [X: real] :
( deriv_real
@ ^ [X_1: real] : ( power_power_real @ ( sin @ X_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
@ X
@ ( plus_plus_real @ ( times_times_real @ ( cos @ X ) @ ( sin @ X ) ) @ ( times_times_real @ ( cos @ X ) @ ( sin @ X ) ) ) ) ).
thf(fact_4796_diff__nat__number__of,axiom,
! [V: int,V_1: int] :
( ( ( ord_less_int @ V_1 @ pls )
=> ( ( minus_minus_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
= ( number_number_of_nat @ V ) ) )
& ( ~ ( ord_less_int @ V_1 @ pls )
=> ( ( minus_minus_nat @ ( number_number_of_nat @ V ) @ ( number_number_of_nat @ V_1 ) )
= ( if_nat @ ( nat_neg @ ( number_number_of_int @ ( plus_plus_int @ V @ ( uminus_uminus_int @ V_1 ) ) ) ) @ zero_zero_nat @ ( nat_1 @ ( number_number_of_int @ ( plus_plus_int @ V @ ( uminus_uminus_int @ V_1 ) ) ) ) ) ) ) ) ).
thf(fact_4797_DERIV__cos__realpow2a,axiom,
! [X: real] :
( deriv_real
@ ^ [X_1: real] : ( power_power_real @ ( cos @ X_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
@ X
@ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ min ) ) @ ( cos @ X ) ) @ ( sin @ X ) ) ) ).
thf(fact_4798_DERIV__cos__realpow2b,axiom,
! [X: real] :
( deriv_real
@ ^ [X_1: real] : ( power_power_real @ ( cos @ X_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
@ X
@ ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( cos @ X ) ) @ ( sin @ X ) ) ) ) ).
thf(fact_4799_DERIV__cos__realpow2,axiom,
! [X: real] :
( deriv_real
@ ^ [X_1: real] : ( power_power_real @ ( cos @ X_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
@ X
@ ( plus_plus_real @ ( times_times_real @ ( uminus_uminus_real @ ( sin @ X ) ) @ ( cos @ X ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( sin @ X ) ) @ ( cos @ X ) ) ) ) ).
thf(fact_4800_lemma__DERIV__tan,axiom,
! [X: real] :
( ( ( cos @ X )
!= zero_zero_real )
=> ( deriv_real
@ ^ [X_1: real] : ( inverse_divide_real @ ( sin @ X_1 ) @ ( cos @ X_1 ) )
@ X
@ ( inverse_inverse_real @ ( power_power_real @ ( cos @ X ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_4801_lemma__DERIV__sin__cos__add,axiom,
! [Y: real,X_1: real] :
( deriv_real
@ ^ [Y_1: real] : ( plus_plus_real @ ( power_power_real @ ( minus_minus_real @ ( sin @ ( plus_plus_real @ Y_1 @ Y ) ) @ ( plus_plus_real @ ( times_times_real @ ( sin @ Y_1 ) @ ( cos @ Y ) ) @ ( times_times_real @ ( cos @ Y_1 ) @ ( sin @ Y ) ) ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ ( cos @ ( plus_plus_real @ Y_1 @ Y ) ) @ ( minus_minus_real @ ( times_times_real @ ( cos @ Y_1 ) @ ( cos @ Y ) ) @ ( times_times_real @ ( sin @ Y_1 ) @ ( sin @ Y ) ) ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
@ X_1
@ zero_zero_real ) ).
thf(fact_4802_DERIV__sin__circle__all,axiom,
! [X_1: real] :
( deriv_real
@ ^ [Y_1: real] : ( plus_plus_real @ ( power_power_real @ ( sin @ Y_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( cos @ Y_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
@ X_1
@ ( minus_minus_real @ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( cos @ X_1 ) ) @ ( sin @ X_1 ) ) @ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( cos @ X_1 ) ) @ ( sin @ X_1 ) ) ) ) ).
thf(fact_4803_lemma__DERIV__sin__cos__minus,axiom,
! [X_1: real] :
( deriv_real
@ ^ [Y_1: real] : ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ ( sin @ ( uminus_uminus_real @ Y_1 ) ) @ ( sin @ Y_1 ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ ( cos @ ( uminus_uminus_real @ Y_1 ) ) @ ( cos @ Y_1 ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
@ X_1
@ zero_zero_real ) ).
thf(fact_4804_norm__frac_Opsimps,axiom,
! [A: int,B: int] :
( ( accp_P2006205492nt_int @ norm_frac_rel @ ( product_Pair_int_int @ A @ B ) )
=> ( ( ( ord_less_int @ B @ zero_zero_int )
=> ( ( norm_frac @ A @ B )
= ( norm_frac @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) )
& ( ~ ( ord_less_int @ B @ zero_zero_int )
=> ( ( ( ( A = zero_zero_int )
| ( B = zero_zero_int ) )
=> ( ( norm_frac @ A @ B )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) )
& ( ~ ( ( A = zero_zero_int )
| ( B = zero_zero_int ) )
=> ( ( norm_frac @ A @ B )
= ( product_Pair_int_int @ ( div_div_int @ A @ ( int_gcd @ A @ B ) ) @ ( div_div_int @ B @ ( int_gcd @ A @ B ) ) ) ) ) ) ) ) ) ).
thf(fact_4805_monoseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( monoseq_real
@ ^ [N_1: nat] : ( times_times_real @ ( inverse_divide_real @ one_one_real @ ( real_nat @ ( plus_plus_nat @ ( times_times_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) ) ) ).
thf(fact_4806_monoseq__realpow,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( monoseq_real @ ( power_power_real @ X ) ) ) ) ).
thf(fact_4807_norm__frac_Osimps,axiom,
! [A: int,B: int] :
( ( ( ord_less_int @ B @ zero_zero_int )
=> ( ( norm_frac @ A @ B )
= ( norm_frac @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) )
& ( ~ ( ord_less_int @ B @ zero_zero_int )
=> ( ( ( ( A = zero_zero_int )
| ( B = zero_zero_int ) )
=> ( ( norm_frac @ A @ B )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) )
& ( ~ ( ( A = zero_zero_int )
| ( B = zero_zero_int ) )
=> ( ( norm_frac @ A @ B )
= ( product_Pair_int_int @ ( div_div_int @ A @ ( int_gcd @ A @ B ) ) @ ( div_div_int @ B @ ( int_gcd @ A @ B ) ) ) ) ) ) ) ) ).
thf(fact_4808_arg__def,axiom,
! [Z_1: complex] :
( ( arg @ Z_1 )
= ( hilbert_Eps_real
@ ^ [A_2: real] :
( (&)
@ ( ( re @ ( sgn_sgn_complex @ Z_1 ) )
= ( cos @ A_2 ) )
@ ( (&)
@ ( ( im @ ( sgn_sgn_complex @ Z_1 ) )
= ( sin @ A_2 ) )
@ ( (&) @ ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A_2 ) @ ( ord_less_eq_real @ A_2 @ pi ) ) ) ) ) ) ).
thf(fact_4809_Union__SetS__setprod__prop1,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A )
=> ( zcong
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( comple1092985777_int_o @ ( setS @ A @ P_3 ) ) )
@ ( power_power_int @ A @ ( nat_1 @ ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
@ P_3 ) ) ) ) ) ).
thf(fact_4810_d22set__prod__zfact,axiom,
! [A: int] :
( ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( d22set @ A ) )
= ( zfact @ A ) ) ).
thf(fact_4811_Bnor__prod__zgcd,axiom,
! [A: int,M: int] :
( ( ord_less_int @ A @ M )
=> ( ( legacy_zgcd
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( bnorRset @ A @ M ) )
@ M )
= one_one_int ) ) ).
thf(fact_4812_Bnor__prod__power,axiom,
! [A: int,M: int,X: int] :
( ( X != zero_zero_int )
=> ( ( ord_less_int @ A @ M )
=> ( ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( image_int_int
@ ^ [A_2: int] : ( times_times_int @ A_2 @ X )
@ ( bnorRset @ A @ M ) ) )
= ( times_times_int
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( bnorRset @ A @ M ) )
@ ( power_power_int @ X @ ( finite_card_int @ ( bnorRset @ A @ M ) ) ) ) ) ) ) ).
thf(fact_4813_SetS__setprod__prop,axiom,
! [X: int > $o,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A )
=> ( ( member_int_o @ X @ ( setS @ A @ P_3 ) )
=> ( zcong
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ X )
@ A
@ P_3 ) ) ) ) ) ) ).
thf(fact_4814_wset__zcong__prod__1,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_eq_int @ ( number_number_of_int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P_3 )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( zcong
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( wset @ A @ P_3 ) )
@ one_one_int
@ P_3 ) ) ) ) ).
thf(fact_4815_Union__SetS__setprod__prop2,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( comple1092985777_int_o @ ( setS @ A @ P_3 ) ) )
= ( zfact @ ( minus_minus_int @ P_3 @ one_one_int ) ) ) ) ) ) ).
thf(fact_4816_transfer__nat__int__sum__prod__closure_I2_J,axiom,
! [F: int > int,A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ! [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X_1 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( big_co1548731110nt_int @ F @ A_1 ) ) ) ) ).
thf(fact_4817_arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( arctan @ X )
= ( suminf_real
@ ^ [K: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ K ) @ ( times_times_real @ ( inverse_divide_real @ one_one_real @ ( real_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
thf(fact_4818_DERIV__arctan__series,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( deriv_real
@ ^ [X_3: real] :
( suminf_real
@ ^ [K: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ K ) @ ( times_times_real @ ( inverse_divide_real @ one_one_real @ ( real_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X_3 @ ( plus_plus_nat @ ( times_times_nat @ K @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) ) )
@ X
@ ( suminf_real
@ ^ [K: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ K ) @ ( power_power_real @ X @ ( times_times_nat @ K @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_4819_lemma__sin__ext,axiom,
! [X_1: real] :
( ( sin @ X_1 )
= ( suminf_real
@ ^ [N_1: nat] : ( times_times_real @ ( sin_coeff @ N_1 ) @ ( power_power_real @ X_1 @ N_1 ) ) ) ) ).
thf(fact_4820_sin__def,axiom,
! [X: real] :
( ( sin @ X )
= ( suminf_real
@ ^ [N_1: nat] : ( times_times_real @ ( sin_coeff @ N_1 ) @ ( power_power_real @ X @ N_1 ) ) ) ) ).
thf(fact_4821_lemma__cos__ext,axiom,
! [X_1: real] :
( ( cos @ X_1 )
= ( suminf_real
@ ^ [N_1: nat] : ( times_times_real @ ( cos_coeff @ N_1 ) @ ( power_power_real @ X_1 @ N_1 ) ) ) ) ).
thf(fact_4822_cos__def,axiom,
! [X: real] :
( ( cos @ X )
= ( suminf_real
@ ^ [N_1: nat] : ( times_times_real @ ( cos_coeff @ N_1 ) @ ( power_power_real @ X @ N_1 ) ) ) ) ).
thf(fact_4823_lemma__sin__minus,axiom,
! [X: real] :
( ( uminus_uminus_real @ ( sin @ X ) )
= ( suminf_real
@ ^ [N_1: nat] : ( uminus_uminus_real @ ( times_times_real @ ( sin_coeff @ N_1 ) @ ( power_power_real @ X @ N_1 ) ) ) ) ) ).
thf(fact_4824_exp__first__two__terms,axiom,
! [X: real] :
( ( exp_real @ X )
= ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X )
@ ( suminf_real
@ ^ [N_1: nat] : ( times_times_real @ ( inverse_inverse_real @ ( real_nat @ ( fact_fact_nat @ ( plus_plus_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_4825_pi__series,axiom,
( ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( suminf_real
@ ^ [K: nat] : ( inverse_divide_real @ ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ K ) @ one_one_real ) @ ( real_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) ) ) ).
thf(fact_4826_ln__series,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
=> ( ( ln @ X )
= ( suminf_real
@ ^ [N_1: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ N_1 ) @ ( inverse_divide_real @ one_one_real @ ( real_nat @ ( plus_plus_nat @ N_1 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N_1 ) ) ) ) ) ) ) ).
thf(fact_4827_summable__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( summable_real
@ ^ [K: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ K ) @ ( times_times_real @ ( inverse_divide_real @ one_one_real @ ( real_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) ) ) ) ).
thf(fact_4828_summable__rabs__cancel,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N_1: nat] : ( abs_abs_real @ ( F @ N_1 ) ) )
=> ( summable_real @ F ) ) ).
thf(fact_4829_summable__rabs__comparison__test,axiom,
! [F: nat > real,G: nat > real] :
( ? [N_2: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ N_2 @ N_1 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N_1 ) ) @ ( G @ N_1 ) ) )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N_1: nat] : ( abs_abs_real @ ( F @ N_1 ) ) ) ) ) ).
thf(fact_4830_summable__rabs,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N_1: nat] : ( abs_abs_real @ ( F @ N_1 ) ) )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
@ ( suminf_real
@ ^ [N_1: nat] : ( abs_abs_real @ ( F @ N_1 ) ) ) ) ) ).
thf(fact_4831_summable__sin,axiom,
! [X: real] :
( summable_real
@ ^ [N_1: nat] : ( times_times_real @ ( sin_coeff @ N_1 ) @ ( power_power_real @ X @ N_1 ) ) ) ).
thf(fact_4832_summable__cos,axiom,
! [X: real] :
( summable_real
@ ^ [N_1: nat] : ( times_times_real @ ( cos_coeff @ N_1 ) @ ( power_power_real @ X @ N_1 ) ) ) ).
thf(fact_4833_rat__floor__code,axiom,
! [P_3: rat] :
( ( archim791455193or_rat @ P_3 )
= ( produc1298267108nt_int @ div_div_int @ ( quotient_of @ P_3 ) ) ) ).
thf(fact_4834_rat__uminus__code,axiom,
! [P_3: rat] :
( ( quotient_of @ ( uminus_uminus_rat @ P_3 ) )
= ( produc1518849193nt_int
@ ^ [A_2: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A_2 ) )
@ ( quotient_of @ P_3 ) ) ) ).
thf(fact_4835_summable__exp,axiom,
! [X: real] :
( summable_real
@ ^ [N_1: nat] : ( times_times_real @ ( inverse_inverse_real @ ( real_nat @ ( fact_fact_nat @ N_1 ) ) ) @ ( power_power_real @ X @ N_1 ) ) ) ).
thf(fact_4836_rat__abs__code,axiom,
! [P_3: rat] :
( ( quotient_of @ ( abs_abs_rat @ P_3 ) )
= ( produc1518849193nt_int
@ ^ [A_2: int] : ( product_Pair_int_int @ ( abs_abs_int @ A_2 ) )
@ ( quotient_of @ P_3 ) ) ) ).
thf(fact_4837_divmod__int__rel__def,axiom,
! [A: int,B: int] :
( ( divmod_int_rel @ A @ B )
= ( produc450523309_int_o
@ ^ [Q_2: int,R: int] :
( (&)
@ ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q_2 ) @ R ) )
@ ( (&) @ ( (=>) @ ( ord_less_int @ zero_zero_int @ B ) @ ( (&) @ ( ord_less_eq_int @ zero_zero_int @ R ) @ ( ord_less_int @ R @ B ) ) ) @ ( (=>) @ ( (~) @ ( ord_less_int @ zero_zero_int @ B ) ) @ ( (&) @ ( ord_less_int @ B @ R ) @ ( ord_less_eq_int @ R @ zero_zero_int ) ) ) ) ) ) ) ).
thf(fact_4838_sum2sq__def,axiom,
( twoSqu2072599593sum2sq
= ( produc1298267108nt_int
@ ^ [A_2: int,B_4: int] : ( plus_plus_int @ ( power_power_int @ A_2 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ B_4 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_4839_rat__inverse__code,axiom,
! [P_3: rat] :
( ( quotient_of @ ( inverse_inverse_rat @ P_3 ) )
= ( produc1518849193nt_int
@ ^ [A_2: int,B_4: int] : ( if_Pro1731782967nt_int @ ( A_2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A_2 ) @ B_4 ) @ ( abs_abs_int @ A_2 ) ) )
@ ( quotient_of @ P_3 ) ) ) ).
thf(fact_4840_exp__tail__after__first__two__terms__summable,axiom,
! [X: real] :
( summable_real
@ ^ [N_1: nat] : ( times_times_real @ ( inverse_inverse_real @ ( real_nat @ ( fact_fact_nat @ ( plus_plus_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_4841_adjust__def,axiom,
! [B: int] :
( ( adjust @ B )
= ( produc1518849193nt_int
@ ^ [Q_2: int,R: int] : ( if_Pro1731782967nt_int @ ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ R @ B ) ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Q_2 ) @ one_one_int ) @ ( minus_minus_int @ R @ B ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Q_2 ) @ R ) ) ) ) ).
thf(fact_4842_DERIV__power__series_H,axiom,
! [X0: real,F: nat > real,R_2: real] :
( ! [X_1: real] :
( ( member_real @ X_1 @ ( ord_gr788844697n_real @ ( uminus_uminus_real @ R_2 ) @ R_2 ) )
=> ( summable_real
@ ^ [N_1: nat] : ( times_times_real @ ( times_times_real @ ( F @ N_1 ) @ ( real_nat @ ( suc @ N_1 ) ) ) @ ( power_power_real @ X_1 @ N_1 ) ) ) )
=> ( ( member_real @ X0 @ ( ord_gr788844697n_real @ ( uminus_uminus_real @ R_2 ) @ R_2 ) )
=> ( ( ord_less_real @ zero_zero_real @ R_2 )
=> ( deriv_real
@ ^ [X_1: real] :
( suminf_real
@ ^ [N_1: nat] : ( times_times_real @ ( F @ N_1 ) @ ( power_power_real @ X_1 @ ( suc @ N_1 ) ) ) )
@ X0
@ ( suminf_real
@ ^ [N_1: nat] : ( times_times_real @ ( times_times_real @ ( F @ N_1 ) @ ( real_nat @ ( suc @ N_1 ) ) ) @ ( power_power_real @ X0 @ N_1 ) ) ) ) ) ) ) ).
thf(fact_4843_Nitpick_OFrac__def,axiom,
( frac
= ( produc450523309_int_o
@ ^ [A_2: int,B_4: int] :
( (&) @ ( ord_less_int @ zero_zero_int @ B_4 )
@ ( ( int_gcd @ A_2 @ B_4 )
= one_one_int ) ) ) ) ).
thf(fact_4844_cnj_Osuminf,axiom,
! [X_2: nat > complex] :
( ( summable_complex @ X_2 )
=> ( ( cnj @ ( suminf_complex @ X_2 ) )
= ( suminf_complex
@ ^ [N_1: nat] : ( cnj @ ( X_2 @ N_1 ) ) ) ) ) ).
thf(fact_4845_cnj_Osummable,axiom,
! [X_2: nat > complex] :
( ( summable_complex @ X_2 )
=> ( summable_complex
@ ^ [N_1: nat] : ( cnj @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_4846_Re_Osummable,axiom,
! [X_2: nat > complex] :
( ( summable_complex @ X_2 )
=> ( summable_real
@ ^ [N_1: nat] : ( re @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_4847_Im_Osummable,axiom,
! [X_2: nat > complex] :
( ( summable_complex @ X_2 )
=> ( summable_real
@ ^ [N_1: nat] : ( im @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_4848_Re_Osuminf,axiom,
! [X_2: nat > complex] :
( ( summable_complex @ X_2 )
=> ( ( re @ ( suminf_complex @ X_2 ) )
= ( suminf_real
@ ^ [N_1: nat] : ( re @ ( X_2 @ N_1 ) ) ) ) ) ).
thf(fact_4849_Im_Osuminf,axiom,
! [X_2: nat > complex] :
( ( summable_complex @ X_2 )
=> ( ( im @ ( suminf_complex @ X_2 ) )
= ( suminf_real
@ ^ [N_1: nat] : ( im @ ( X_2 @ N_1 ) ) ) ) ) ).
thf(fact_4850_ratrel__def,axiom,
( ratrel
= ( collec50511176nt_int
@ ( produc141074865_int_o
@ ^ [X_1: product_prod_int_int,Y_1: product_prod_int_int] :
( (&)
@ ( (~)
@ ( ( product_snd_int_int @ X_1 )
= zero_zero_int ) )
@ ( (&)
@ ( (~)
@ ( ( product_snd_int_int @ Y_1 )
= zero_zero_int ) )
@ ( ( times_times_int @ ( product_fst_int_int @ X_1 ) @ ( product_snd_int_int @ Y_1 ) )
= ( times_times_int @ ( product_fst_int_int @ Y_1 ) @ ( product_snd_int_int @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_4851_divmod__nat__if,axiom,
! [M: nat,N: nat] :
( ( ( ( N = zero_zero_nat )
| ( ord_less_nat @ M @ N ) )
=> ( ( divmod_nat @ M @ N )
= ( product_Pair_nat_nat @ zero_zero_nat @ M ) ) )
& ( ~ ( ( N = zero_zero_nat )
| ( ord_less_nat @ M @ N ) )
=> ( ( divmod_nat @ M @ N )
= ( produc1391996073at_nat
@ ^ [Q_2: nat] : ( product_Pair_nat_nat @ ( suc @ Q_2 ) )
@ ( divmod_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
thf(fact_4852_Bolzano__bisect_Osimps_I2_J,axiom,
! [P: produc914805421l_real > $o,A: real,B: real,N: nat] :
( ( bolzano_bisect @ P @ A @ B @ ( suc @ N ) )
= ( produc595218619l_real
@ ^ [X_1: real,Y_1: real] : ( if_Pro313124157l_real @ ( P @ ( produc865579683l_real @ X_1 @ ( inverse_divide_real @ ( plus_plus_real @ X_1 @ Y_1 ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( produc865579683l_real @ ( inverse_divide_real @ ( plus_plus_real @ X_1 @ Y_1 ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ Y_1 ) @ ( produc865579683l_real @ X_1 @ ( inverse_divide_real @ ( plus_plus_real @ X_1 @ Y_1 ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
@ ( bolzano_bisect @ P @ A @ B @ N ) ) ) ).
thf(fact_4853_suminf__exist__split,axiom,
! [A: nat > real,R_1: real] :
( ( ord_less_real @ zero_zero_real @ R_1 )
=> ( ( summable_real @ A )
=> ? [N_2: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ N_2 @ N_1 )
=> ( ord_less_real
@ ( abs_abs_real
@ ( suminf_real
@ ^ [I_1: nat] : ( A @ ( plus_plus_nat @ I_1 @ N_1 ) ) ) )
@ R_1 ) ) ) ) ).
thf(fact_4854_divmod__int__pdivmod,axiom,
! [L: int,K_1: int] :
( ( ( K_1 = zero_zero_int )
=> ( ( divmod_int @ K_1 @ L )
= ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) )
& ( ( K_1 != zero_zero_int )
=> ( ( ( L = zero_zero_int )
=> ( ( divmod_int @ K_1 @ L )
= ( product_Pair_int_int @ zero_zero_int @ K_1 ) ) )
& ( ( L != zero_zero_int )
=> ( ( divmod_int @ K_1 @ L )
= ( produc713050258nt_int @ ( times_times_int @ ( sgn_sgn_int @ L ) )
@ ( if_Pro1731782967nt_int @ ( (|) @ ( (&) @ ( ord_less_int @ zero_zero_int @ L ) @ ( ord_less_eq_int @ zero_zero_int @ K_1 ) ) @ ( (&) @ ( ord_less_int @ L @ zero_zero_int ) @ ( ord_less_int @ K_1 @ zero_zero_int ) ) ) @ ( pdivmod @ K_1 @ L )
@ ( produc1518849193nt_int
@ ^ [R: int,S_2: int] : ( if_Pro1731782967nt_int @ ( S_2 = zero_zero_int ) @ ( product_Pair_int_int @ ( uminus_uminus_int @ R ) @ zero_zero_int ) @ ( product_Pair_int_int @ ( minus_minus_int @ ( uminus_uminus_int @ R ) @ one_one_int ) @ ( minus_minus_int @ ( abs_abs_int @ L ) @ S_2 ) ) )
@ ( pdivmod @ K_1 @ L ) ) ) ) ) ) ) ) ) ).
thf(fact_4855_negateSnd__def,axiom,
( negateSnd
= ( produc713050258nt_int @ uminus_uminus_int ) ) ).
thf(fact_4856_divmod__int__code,axiom,
! [L: int,K_1: int] :
( ( ( K_1 = zero_zero_int )
=> ( ( divmod_int @ K_1 @ L )
= ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) )
& ( ( K_1 != zero_zero_int )
=> ( ( ( L = zero_zero_int )
=> ( ( divmod_int @ K_1 @ L )
= ( product_Pair_int_int @ zero_zero_int @ K_1 ) ) )
& ( ( L != zero_zero_int )
=> ( ( divmod_int @ K_1 @ L )
= ( produc713050258nt_int @ ( times_times_int @ ( sgn_sgn_int @ L ) )
@ ( if_Pro1731782967nt_int
@ ( ( sgn_sgn_int @ K_1 )
= ( sgn_sgn_int @ L ) )
@ ( pdivmod @ K_1 @ L )
@ ( produc1518849193nt_int
@ ^ [R: int,S_2: int] : ( if_Pro1731782967nt_int @ ( S_2 = zero_zero_int ) @ ( product_Pair_int_int @ ( uminus_uminus_int @ R ) @ zero_zero_int ) @ ( product_Pair_int_int @ ( minus_minus_int @ ( uminus_uminus_int @ R ) @ one_one_int ) @ ( minus_minus_int @ ( abs_abs_int @ L ) @ S_2 ) ) )
@ ( pdivmod @ K_1 @ L ) ) ) ) ) ) ) ) ) ).
thf(fact_4857_summable__le2,axiom,
! [F: nat > real,G: nat > real] :
( ! [N_1: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N_1 ) ) @ ( G @ N_1 ) )
=> ( ( summable_real @ G )
=> ( ( summable_real @ F )
& ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
thf(fact_4858_suminf__gt__zero,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N_1: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N_1 ) )
=> ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
thf(fact_4859_suminf__ge__zero,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N_1 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
thf(fact_4860_suminf__0__le,axiom,
! [F: nat > real] :
( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N_1 ) )
=> ( ( summable_real @ F )
=> ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
thf(fact_4861_int__ge__less__than__def,axiom,
! [D: int] :
( ( int_ge_less_than @ D )
= ( collec1347809874nt_int
@ ( produc450523309_int_o
@ ^ [Z_2: int,Z: int] : ( (&) @ ( ord_less_eq_int @ D @ Z_2 ) @ ( ord_less_int @ Z_2 @ Z ) ) ) ) ) ).
thf(fact_4862_int__ge__less__than2__def,axiom,
! [D: int] :
( ( int_ge_less_than2 @ D )
= ( collec1347809874nt_int
@ ( produc450523309_int_o
@ ^ [Z_2: int,Z: int] : ( (&) @ ( ord_less_eq_int @ D @ Z ) @ ( ord_less_int @ Z_2 @ Z ) ) ) ) ) ).
thf(fact_4863_sin__paired,axiom,
! [X: real] :
( sums_real
@ ^ [N_1: nat] : ( times_times_real @ ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ N_1 ) @ ( real_nat @ ( fact_fact_nat @ ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) @ one_one_nat ) ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) @ one_one_nat ) ) )
@ ( sin @ X ) ) ).
thf(fact_4864_sin__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N_1: nat] : ( times_times_real @ ( sin_coeff @ N_1 ) @ ( power_power_real @ X @ N_1 ) )
@ ( sin @ X ) ) ).
thf(fact_4865_cos__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N_1: nat] : ( times_times_real @ ( cos_coeff @ N_1 ) @ ( power_power_real @ X @ N_1 ) )
@ ( cos @ X ) ) ).
thf(fact_4866_power__half__series,axiom,
( sums_real
@ ^ [N_1: nat] : ( power_power_real @ ( inverse_divide_real @ one_one_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( suc @ N_1 ) )
@ one_one_real ) ).
thf(fact_4867_Ln_Oaux2,axiom,
! [X: real] :
( sums_real
@ ^ [N_1: nat] : ( times_times_real @ ( inverse_divide_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( inverse_divide_real @ one_one_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ N_1 ) )
@ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_4868_sums__if_H,axiom,
! [G: nat > real,X: real] :
( ( sums_real @ G @ X )
=> ( sums_real
@ ^ [N_1: nat] : ( if_real @ ( even_odd_even_nat @ N_1 ) @ zero_zero_real @ ( G @ ( div_div_nat @ ( minus_minus_nat @ N_1 @ one_one_nat ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
@ X ) ) ).
thf(fact_4869_sums__if,axiom,
! [F: nat > real,Y: real,G: nat > real,X: real] :
( ( sums_real @ G @ X )
=> ( ( sums_real @ F @ Y )
=> ( sums_real
@ ^ [N_1: nat] : ( if_real @ ( even_odd_even_nat @ N_1 ) @ ( F @ ( div_div_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( G @ ( div_div_nat @ ( minus_minus_nat @ N_1 @ one_one_nat ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
@ ( plus_plus_real @ X @ Y ) ) ) ) ).
thf(fact_4870_cos__paired,axiom,
! [X: real] :
( sums_real
@ ^ [N_1: nat] : ( times_times_real @ ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ N_1 ) @ ( real_nat @ ( fact_fact_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) ) )
@ ( cos @ X ) ) ).
thf(fact_4871_cnj_Osums,axiom,
! [X_2: nat > complex,A: complex] :
( ( sums_complex @ X_2 @ A )
=> ( sums_complex
@ ^ [N_1: nat] : ( cnj @ ( X_2 @ N_1 ) )
@ ( cnj @ A ) ) ) ).
thf(fact_4872_Re_Osums,axiom,
! [X_2: nat > complex,A: complex] :
( ( sums_complex @ X_2 @ A )
=> ( sums_real
@ ^ [N_1: nat] : ( re @ ( X_2 @ N_1 ) )
@ ( re @ A ) ) ) ).
thf(fact_4873_Im_Osums,axiom,
! [X_2: nat > complex,A: complex] :
( ( sums_complex @ X_2 @ A )
=> ( sums_real
@ ^ [N_1: nat] : ( im @ ( X_2 @ N_1 ) )
@ ( im @ A ) ) ) ).
thf(fact_4874_sin__fdiffs2,axiom,
! [N: nat] :
( ( diffs_real @ sin_coeff @ N )
= ( cos_coeff @ N ) ) ).
thf(fact_4875_sin__fdiffs,axiom,
( ( diffs_real @ sin_coeff )
= cos_coeff ) ).
thf(fact_4876_exp__fdiffs,axiom,
! [X_1: nat] :
( ( diffs_real
@ ^ [N_1: nat] : ( inverse_inverse_real @ ( real_nat @ ( fact_fact_nat @ N_1 ) ) )
@ X_1 )
= ( inverse_inverse_real @ ( real_nat @ ( fact_fact_nat @ X_1 ) ) ) ) ).
thf(fact_4877_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
<=> ( nat_case_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
thf(fact_4878_cos__fdiffs2,axiom,
! [N: nat] :
( ( diffs_real @ cos_coeff @ N )
= ( uminus_uminus_real @ ( sin_coeff @ N ) ) ) ).
thf(fact_4879_cos__fdiffs,axiom,
! [X_1: nat] :
( ( diffs_real @ cos_coeff @ X_1 )
= ( uminus_uminus_real @ ( sin_coeff @ X_1 ) ) ) ).
thf(fact_4880_diff__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( nat_case_nat @ zero_zero_nat
@ ^ [K: nat] : K
@ ( minus_minus_nat @ M @ N ) ) ) ).
thf(fact_4881_summable__le,axiom,
! [F: nat > real,G: nat > real] :
( ! [N_1: nat] : ( ord_less_eq_real @ ( F @ N_1 ) @ ( G @ N_1 ) )
=> ( ( summable_real @ F )
=> ( ( summable_real @ G )
=> ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
thf(fact_4882_arctan__def,axiom,
! [Y: real] :
( ( arctan @ Y )
= ( the_real
@ ^ [X_1: real] :
( (&) @ ( ord_less_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ X_1 )
@ ( (&) @ ( ord_less_real @ X_1 @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
@ ( ( tan @ X_1 )
= Y ) ) ) ) ) ).
thf(fact_4883_arcsin__def,axiom,
! [Y: real] :
( ( arcsin @ Y )
= ( the_real
@ ^ [X_1: real] :
( (&) @ ( ord_less_eq_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ X_1 )
@ ( (&) @ ( ord_less_eq_real @ X_1 @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
@ ( ( sin @ X_1 )
= Y ) ) ) ) ) ).
thf(fact_4884_ln__def,axiom,
! [X: real] :
( ( ln @ X )
= ( the_real
@ ^ [U_1: real] :
( ( exp_real @ U_1 )
= X ) ) ) ).
thf(fact_4885_root__def,axiom,
! [N: nat,X: real] :
( ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( root @ N @ X )
= ( the_real
@ ^ [U_1: real] :
( (&) @ ( ord_less_real @ zero_zero_real @ U_1 )
@ ( ( power_power_real @ U_1 @ N )
= X ) ) ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ord_less_real @ X @ zero_zero_real )
=> ( ( root @ N @ X )
= ( uminus_uminus_real
@ ( the_real
@ ^ [U_1: real] :
( (&) @ ( ord_less_real @ zero_zero_real @ U_1 )
@ ( ( power_power_real @ U_1 @ N )
= ( uminus_uminus_real @ X ) ) ) ) ) ) )
& ( ~ ( ord_less_real @ X @ zero_zero_real )
=> ( ( root @ N @ X )
= zero_zero_real ) ) ) ) ) ).
thf(fact_4886_arccos__def,axiom,
! [Y: real] :
( ( arccos @ Y )
= ( the_real
@ ^ [X_1: real] :
( (&) @ ( ord_less_eq_real @ zero_zero_real @ X_1 )
@ ( (&) @ ( ord_less_eq_real @ X_1 @ pi )
@ ( ( cos @ X_1 )
= Y ) ) ) ) ) ).
thf(fact_4887_pi__def,axiom,
( pi
= ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) )
@ ( the_real
@ ^ [X_1: real] :
( (&) @ ( ord_less_eq_real @ zero_zero_real @ X_1 )
@ ( (&) @ ( ord_less_eq_real @ X_1 @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
@ ( ( cos @ X_1 )
= zero_zero_real ) ) ) ) ) ) ).
thf(fact_4888_pi__half,axiom,
( ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( the_real
@ ^ [X_1: real] :
( (&) @ ( ord_less_eq_real @ zero_zero_real @ X_1 )
@ ( (&) @ ( ord_less_eq_real @ X_1 @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
@ ( ( cos @ X_1 )
= zero_zero_real ) ) ) ) ) ).
thf(fact_4889_divmod__nat__def,axiom,
! [M: nat,N: nat] :
( ( divmod_nat @ M @ N )
= ( the_Pr588456374at_nat @ ( divmod_nat_rel @ M @ N ) ) ) ).
thf(fact_4890_floor__real__def,axiom,
! [X: real] :
( ( archim1246769320r_real @ X )
= ( the_int
@ ^ [Z: int] : ( (&) @ ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X ) @ ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z @ one_one_int ) ) ) ) ) ) ).
thf(fact_4891_floor__rat__def,axiom,
! [X: rat] :
( ( archim791455193or_rat @ X )
= ( the_int
@ ^ [Z: int] : ( (&) @ ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X ) @ ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z @ one_one_int ) ) ) ) ) ) ).
thf(fact_4892_RRset2norRR__def,axiom,
! [A: int,A_1: int > $o,M: int] :
( ( ( ( ord_less_int @ one_one_int @ M )
& ( is_RRset @ A_1 @ M )
& ( member_int @ A @ A_1 ) )
=> ( ( rRset2norRR @ A_1 @ M @ A )
= ( hilbert_Eps_int
@ ^ [B_4: int] : ( (&) @ ( zcong @ A @ B_4 @ M ) @ ( member_int @ B_4 @ ( norRRset @ M ) ) ) ) ) )
& ( ~ ( ( ord_less_int @ one_one_int @ M )
& ( is_RRset @ A_1 @ M )
& ( member_int @ A @ A_1 ) )
=> ( ( rRset2norRR @ A_1 @ M @ A )
= zero_zero_int ) ) ) ).
thf(fact_4893_aux__some,axiom,
! [A: int,A_1: int > $o,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( is_RRset @ A_1 @ M )
=> ( ( member_int @ A @ A_1 )
=> ( ( zcong @ A
@ ( hilbert_Eps_int
@ ^ [B_4: int] : ( (&) @ ( zcong @ A @ B_4 @ M ) @ ( member_int @ B_4 @ ( norRRset @ M ) ) ) )
@ M )
& ( member_int
@ ( hilbert_Eps_int
@ ^ [B_4: int] : ( (&) @ ( zcong @ A @ B_4 @ M ) @ ( member_int @ B_4 @ ( norRRset @ M ) ) ) )
@ ( norRRset @ M ) ) ) ) ) ) ).
thf(fact_4894_Maclaurin__sin__bound,axiom,
! [X: real,N: nat] :
( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real @ ( sin @ X )
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( if_real @ ( even_odd_even_nat @ M_2 ) @ zero_zero_real @ ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ ( div_div_nat @ ( minus_minus_nat @ M_2 @ ( suc @ zero_zero_nat ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) ) @ ( power_power_real @ X @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) ) ) )
@ ( times_times_real @ ( inverse_inverse_real @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).
thf(fact_4895_finite__less__ub,axiom,
! [U: nat,F: nat > nat] :
( ! [N_1: nat] : ( ord_less_eq_nat @ N_1 @ ( F @ N_1 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N_1: nat] : ( ord_less_eq_nat @ ( F @ N_1 ) @ U ) ) ) ) ).
thf(fact_4896_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( ord_at4362885an_nat @ L @ U ) ) ).
thf(fact_4897_sumr__diff__mult__const,axiom,
! [F: nat > real,N: nat,R_1: real] :
( ( minus_minus_real @ ( big_co604158596t_real @ F @ ( ord_at4362885an_nat @ zero_zero_nat @ N ) ) @ ( times_times_real @ ( real_nat @ N ) @ R_1 ) )
= ( big_co604158596t_real
@ ^ [I_1: nat] : ( minus_minus_real @ ( F @ I_1 ) @ R_1 )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) ) ) ).
thf(fact_4898_sumr__offset2,axiom,
! [K_1: nat,N: nat,F_2: nat > real] :
( ( big_co604158596t_real
@ ^ [M_2: nat] : ( F_2 @ ( plus_plus_nat @ M_2 @ K_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
= ( minus_minus_real @ ( big_co604158596t_real @ F_2 @ ( ord_at4362885an_nat @ zero_zero_nat @ ( plus_plus_nat @ N @ K_1 ) ) ) @ ( big_co604158596t_real @ F_2 @ ( ord_at4362885an_nat @ zero_zero_nat @ K_1 ) ) ) ) ).
thf(fact_4899_sumr__offset4,axiom,
! [K_1: nat,N_1: nat,F_2: nat > real] :
( ( big_co604158596t_real @ F_2 @ ( ord_at4362885an_nat @ zero_zero_nat @ ( plus_plus_nat @ N_1 @ K_1 ) ) )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( F_2 @ ( plus_plus_nat @ M_2 @ K_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N_1 ) )
@ ( big_co604158596t_real @ F_2 @ ( ord_at4362885an_nat @ zero_zero_nat @ K_1 ) ) ) ) ).
thf(fact_4900_sumr__one__lb__realpow__zero,axiom,
! [F: nat > real,N: nat] :
( ( big_co604158596t_real
@ ^ [N_1: nat] : ( times_times_real @ ( F @ N_1 ) @ ( power_power_real @ zero_zero_real @ N_1 ) )
@ ( ord_at4362885an_nat @ ( suc @ zero_zero_nat ) @ N ) )
= zero_zero_real ) ).
thf(fact_4901_ex__nat__less__eq,axiom,
! [P: nat > $o,N: nat] :
( ? [M_2: nat] :
( ( ord_less_nat @ M_2 @ N )
& ( P @ M_2 ) )
<=> ? [X_1: nat] :
( ( member_nat @ X_1 @ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
& ( P @ X_1 ) ) ) ).
thf(fact_4902_all__nat__less__eq,axiom,
! [P: nat > $o,N: nat] :
( ! [M_2: nat] :
( ( ord_less_nat @ M_2 @ N )
=> ( P @ M_2 ) )
<=> ! [X_1: nat] :
( ( member_nat @ X_1 @ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
=> ( P @ X_1 ) ) ) ).
thf(fact_4903_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( ord_at4362885an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
thf(fact_4904_sumr__geometric,axiom,
! [N: nat,X: real] :
( ( X != one_one_real )
=> ( ( big_co604158596t_real @ ( power_power_real @ X ) @ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
= ( inverse_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ N ) @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ) ).
thf(fact_4905_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( ord_at4362885an_nat @ ( suc @ L ) @ U )
= ( ord_gr660468384an_nat @ L @ U ) ) ).
thf(fact_4906_image__add__atLeastLessThan,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( image_nat_nat
@ ^ [N_1: nat] : ( plus_plus_nat @ N_1 @ K_1 )
@ ( ord_at4362885an_nat @ I @ J ) )
= ( ord_at4362885an_nat @ ( plus_plus_nat @ I @ K_1 ) @ ( plus_plus_nat @ J @ K_1 ) ) ) ).
thf(fact_4907_image__Suc__atLeastLessThan,axiom,
! [I: nat,J: nat] :
( ( image_nat_nat @ suc @ ( ord_at4362885an_nat @ I @ J ) )
= ( ord_at4362885an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
thf(fact_4908_Maclaurin__zero,axiom,
! [Diff: nat > real > real,N: nat,X: real] :
( ( X = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ M_2 @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ ( power_power_real @ X @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
= ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).
thf(fact_4909_subset__card__intvl__is__intvl,axiom,
! [A_1: nat > $o,K_1: nat] :
( ( ord_less_eq_nat_o @ A_1 @ ( ord_at4362885an_nat @ K_1 @ ( plus_plus_nat @ K_1 @ ( finite_card_nat @ A_1 ) ) ) )
=> ( A_1
= ( ord_at4362885an_nat @ K_1 @ ( plus_plus_nat @ K_1 @ ( finite_card_nat @ A_1 ) ) ) ) ) ).
thf(fact_4910_sum__split__even__odd,axiom,
! [F: nat > real,G: nat > real,N: nat] :
( ( big_co604158596t_real
@ ^ [I_1: nat] : ( if_real @ ( even_odd_even_nat @ I_1 ) @ ( F @ I_1 ) @ ( G @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( F @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ I_1 ) @ one_one_nat ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) ) ) ) ).
thf(fact_4911_sumr__minus__one__realpow__zero,axiom,
! [N: nat] :
( ( big_co604158596t_real
@ ^ [I_1: nat] : ( power_power_real @ ( number267125858f_real @ min ) @ ( suc @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) )
= zero_zero_real ) ).
thf(fact_4912_lemma__STAR__cos2,axiom,
! [N: nat] :
( ( big_co604158596t_real
@ ^ [N_1: nat] : ( if_real @ ( even_odd_even_nat @ N_1 ) @ ( times_times_real @ ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ ( div_div_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( real_nat @ ( fact_fact_nat @ N_1 ) ) ) @ ( power_power_real @ zero_zero_real @ N_1 ) ) @ zero_zero_real )
@ ( ord_at4362885an_nat @ one_one_nat @ N ) )
= zero_zero_real ) ).
thf(fact_4913_sumr__cos__zero__one,axiom,
! [N: nat] :
( ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( if_real @ ( even_odd_even_nat @ M_2 ) @ ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ ( div_div_nat @ M_2 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ zero_zero_real ) @ ( power_power_real @ zero_zero_real @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= one_one_real ) ).
thf(fact_4914_Maclaurin__sin__expansion2,axiom,
! [N: nat,X: real] :
? [T_1: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T_1 ) @ ( abs_abs_real @ X ) )
& ( ( sin @ X )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( if_real @ ( even_odd_even_nat @ M_2 ) @ zero_zero_real @ ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ ( div_div_nat @ ( minus_minus_nat @ M_2 @ ( suc @ zero_zero_nat ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) ) @ ( power_power_real @ X @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( sin @ ( plus_plus_real @ T_1 @ ( times_times_real @ ( times_times_real @ ( inverse_divide_real @ one_one_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( real_nat @ N ) ) @ pi ) ) ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
thf(fact_4915_Maclaurin__cos__expansion,axiom,
! [N: nat,X: real] :
? [T_1: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T_1 ) @ ( abs_abs_real @ X ) )
& ( ( cos @ X )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( if_real @ ( even_odd_even_nat @ M_2 ) @ ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ ( div_div_nat @ M_2 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ zero_zero_real ) @ ( power_power_real @ X @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( cos @ ( plus_plus_real @ T_1 @ ( times_times_real @ ( times_times_real @ ( inverse_divide_real @ one_one_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( real_nat @ N ) ) @ pi ) ) ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
thf(fact_4916_finite__atLeastLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( ord_at641636577an_int @ L @ U ) ) ).
thf(fact_4917_Sup__atLeastLessThan,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ Y @ X )
=> ( ( comple124823625p_real @ ( ord_at1496968948n_real @ Y @ X ) )
= X ) ) ).
thf(fact_4918_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite_int @ ( ord_at641636577an_int @ zero_zero_int @ U ) ) ).
thf(fact_4919_card__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( finite_card_int @ ( ord_at641636577an_int @ zero_zero_int @ U ) )
= ( nat_1 @ U ) ) ).
thf(fact_4920_card__atLeastLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( ord_at641636577an_int @ L @ U ) )
= ( nat_1 @ ( minus_minus_int @ U @ L ) ) ) ).
thf(fact_4921_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( ord_at641636577an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
= ( ord_gr1297742076an_int @ L @ U ) ) ).
thf(fact_4922_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image_int_int
@ ^ [X_1: int] : ( plus_plus_int @ X_1 @ L )
@ ( ord_at641636577an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
= ( ord_at641636577an_int @ L @ U ) ) ).
thf(fact_4923_Maclaurin__sin__expansion,axiom,
! [X: real,N: nat] :
? [T_1: real] :
( ( sin @ X )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( if_real @ ( even_odd_even_nat @ M_2 ) @ zero_zero_real @ ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ ( div_div_nat @ ( minus_minus_nat @ M_2 @ ( suc @ zero_zero_nat ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) ) @ ( power_power_real @ X @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( sin @ ( plus_plus_real @ T_1 @ ( times_times_real @ ( times_times_real @ ( inverse_divide_real @ one_one_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( real_nat @ N ) ) @ pi ) ) ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ X @ N ) ) ) ) ).
thf(fact_4924_Maclaurin__lemma2,axiom,
! [B_1: real,K_1: nat,Diff: nat > real > real,H: real,N: nat] :
( ! [M_2: nat,T_1: real] :
( ( ( ord_less_nat @ M_2 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T_1 )
& ( ord_less_eq_real @ T_1 @ H ) )
=> ( deriv_real @ ( Diff @ M_2 ) @ T_1 @ ( Diff @ ( suc @ M_2 ) @ T_1 ) ) )
=> ( ( N
= ( suc @ K_1 ) )
=> ! [M_2: nat,T_1: real] :
( ( ( ord_less_nat @ M_2 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T_1 )
& ( ord_less_eq_real @ T_1 @ H ) )
=> ( deriv_real
@ ^ [U_1: real] :
( minus_minus_real @ ( Diff @ M_2 @ U_1 )
@ ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [P_4: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ ( plus_plus_nat @ M_2 @ P_4 ) @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ P_4 ) ) ) @ ( power_power_real @ U_1 @ P_4 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M_2 ) ) )
@ ( times_times_real @ B_1 @ ( inverse_divide_real @ ( power_power_real @ U_1 @ ( minus_minus_nat @ N @ M_2 ) ) @ ( real_nat @ ( fact_fact_nat @ ( minus_minus_nat @ N @ M_2 ) ) ) ) ) ) )
@ T_1
@ ( minus_minus_real @ ( Diff @ ( suc @ M_2 ) @ T_1 )
@ ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [P_4: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M_2 ) @ P_4 ) @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ P_4 ) ) ) @ ( power_power_real @ T_1 @ P_4 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ ( suc @ M_2 ) ) ) )
@ ( times_times_real @ B_1 @ ( inverse_divide_real @ ( power_power_real @ T_1 @ ( minus_minus_nat @ N @ ( suc @ M_2 ) ) ) @ ( real_nat @ ( fact_fact_nat @ ( minus_minus_nat @ N @ ( suc @ M_2 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(fact_4925_Maclaurin__exp__lt,axiom,
! [N: nat,X: real] :
( ( X != zero_zero_real )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [T_1: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T_1 ) )
& ( ord_less_real @ ( abs_abs_real @ T_1 ) @ ( abs_abs_real @ X ) )
& ( ( exp_real @ X )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( inverse_divide_real @ ( power_power_real @ X @ M_2 ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( exp_real @ T_1 ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
thf(fact_4926_Maclaurin__exp__le,axiom,
! [N: nat,X: real] :
? [T_1: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T_1 ) @ ( abs_abs_real @ X ) )
& ( ( exp_real @ X )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( inverse_divide_real @ ( power_power_real @ X @ M_2 ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( exp_real @ T_1 ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
thf(fact_4927_sumr__pos__lt__pair,axiom,
! [K_1: nat,F: nat > real] :
( ( summable_real @ F )
=> ( ! [D_2: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K_1 @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D_2 ) ) ) @ ( F @ ( plus_plus_nat @ K_1 @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D_2 ) @ one_one_nat ) ) ) ) )
=> ( ord_less_real @ ( big_co604158596t_real @ F @ ( ord_at4362885an_nat @ zero_zero_nat @ K_1 ) ) @ ( suminf_real @ F ) ) ) ) ).
thf(fact_4928_Maclaurin__lemma,axiom,
! [F: real > real,J: nat > real,N: nat,H: real] :
( ( ord_less_real @ zero_zero_real @ H )
=> ? [B_6: real] :
( ( F @ H )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( inverse_divide_real @ ( J @ M_2 ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ ( power_power_real @ H @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ B_6 @ ( inverse_divide_real @ ( power_power_real @ H @ N ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) ) ) ) ) ).
thf(fact_4929_series__pos__less,axiom,
! [N: nat,F: nat > real] :
( ( summable_real @ F )
=> ( ! [M_2: nat] :
( ( ord_less_eq_nat @ N @ M_2 )
=> ( ord_less_real @ zero_zero_real @ ( F @ M_2 ) ) )
=> ( ord_less_real @ ( big_co604158596t_real @ F @ ( ord_at4362885an_nat @ zero_zero_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).
thf(fact_4930_series__pos__le,axiom,
! [N: nat,F: nat > real] :
( ( summable_real @ F )
=> ( ! [M_2: nat] :
( ( ord_less_eq_nat @ N @ M_2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ M_2 ) ) )
=> ( ord_less_eq_real @ ( big_co604158596t_real @ F @ ( ord_at4362885an_nat @ zero_zero_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).
thf(fact_4931_Maclaurin__all__lt,axiom,
! [X: real,N: nat,Diff: nat > real > real,F: real > real] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( X != zero_zero_real )
=> ( ! [M_2: nat,X_1: real] : ( deriv_real @ ( Diff @ M_2 ) @ X_1 @ ( Diff @ ( suc @ M_2 ) @ X_1 ) )
=> ? [T_1: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T_1 ) )
& ( ord_less_real @ ( abs_abs_real @ T_1 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ M_2 @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ ( power_power_real @ X @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( Diff @ N @ T_1 ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).
thf(fact_4932_Maclaurin__all__lt__objl,axiom,
! [N: nat,X: real,Diff: nat > real > real,F: real > real] :
( ( ( ( Diff @ zero_zero_nat )
= F )
& ! [M_2: nat,X_1: real] : ( deriv_real @ ( Diff @ M_2 ) @ X_1 @ ( Diff @ ( suc @ M_2 ) @ X_1 ) )
& ( X != zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N ) )
=> ? [T_1: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T_1 ) )
& ( ord_less_real @ ( abs_abs_real @ T_1 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ M_2 @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ ( power_power_real @ X @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( Diff @ N @ T_1 ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
thf(fact_4933_Maclaurin__bi__le,axiom,
! [X: real,N: nat,Diff: nat > real > real,F: real > real] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M_2: nat,T_1: real] :
( ( ( ord_less_nat @ M_2 @ N )
& ( ord_less_eq_real @ ( abs_abs_real @ T_1 ) @ ( abs_abs_real @ X ) ) )
=> ( deriv_real @ ( Diff @ M_2 ) @ T_1 @ ( Diff @ ( suc @ M_2 ) @ T_1 ) ) )
=> ? [T_1: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T_1 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ M_2 @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ ( power_power_real @ X @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( Diff @ N @ T_1 ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
thf(fact_4934_Maclaurin__objl,axiom,
! [Diff: nat > real > real,F: real > real,N: nat,H: real] :
( ( ( ord_less_real @ zero_zero_real @ H )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( ( Diff @ zero_zero_nat )
= F )
& ! [M_2: nat,T_1: real] :
( ( ( ord_less_nat @ M_2 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T_1 )
& ( ord_less_eq_real @ T_1 @ H ) )
=> ( deriv_real @ ( Diff @ M_2 ) @ T_1 @ ( Diff @ ( suc @ M_2 ) @ T_1 ) ) ) )
=> ? [T_1: real] :
( ( ord_less_real @ zero_zero_real @ T_1 )
& ( ord_less_real @ T_1 @ H )
& ( ( F @ H )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ M_2 @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ ( power_power_real @ H @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( Diff @ N @ T_1 ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ).
thf(fact_4935_Maclaurin2__objl,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,H: real] :
( ( ( ord_less_real @ zero_zero_real @ H )
& ( ( Diff @ zero_zero_nat )
= F )
& ! [M_2: nat,T_1: real] :
( ( ( ord_less_nat @ M_2 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T_1 )
& ( ord_less_eq_real @ T_1 @ H ) )
=> ( deriv_real @ ( Diff @ M_2 ) @ T_1 @ ( Diff @ ( suc @ M_2 ) @ T_1 ) ) ) )
=> ? [T_1: real] :
( ( ord_less_real @ zero_zero_real @ T_1 )
& ( ord_less_eq_real @ T_1 @ H )
& ( ( F @ H )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ M_2 @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ ( power_power_real @ H @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( Diff @ N @ T_1 ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ).
thf(fact_4936_Maclaurin__minus,axiom,
! [Diff: nat > real > real,F: real > real,N: nat,H: real] :
( ( ord_less_real @ H @ zero_zero_real )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M_2: nat,T_1: real] :
( ( ( ord_less_nat @ M_2 @ N )
& ( ord_less_eq_real @ H @ T_1 )
& ( ord_less_eq_real @ T_1 @ zero_zero_real ) )
=> ( deriv_real @ ( Diff @ M_2 ) @ T_1 @ ( Diff @ ( suc @ M_2 ) @ T_1 ) ) )
=> ? [T_1: real] :
( ( ord_less_real @ H @ T_1 )
& ( ord_less_real @ T_1 @ zero_zero_real )
& ( ( F @ H )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ M_2 @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ ( power_power_real @ H @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( Diff @ N @ T_1 ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).
thf(fact_4937_Maclaurin2,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,H: real] :
( ( ord_less_real @ zero_zero_real @ H )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M_2: nat,T_1: real] :
( ( ( ord_less_nat @ M_2 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T_1 )
& ( ord_less_eq_real @ T_1 @ H ) )
=> ( deriv_real @ ( Diff @ M_2 ) @ T_1 @ ( Diff @ ( suc @ M_2 ) @ T_1 ) ) )
=> ? [T_1: real] :
( ( ord_less_real @ zero_zero_real @ T_1 )
& ( ord_less_eq_real @ T_1 @ H )
& ( ( F @ H )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ M_2 @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ ( power_power_real @ H @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( Diff @ N @ T_1 ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ).
thf(fact_4938_Maclaurin__minus__objl,axiom,
! [Diff: nat > real > real,F: real > real,N: nat,H: real] :
( ( ( ord_less_real @ H @ zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( ( Diff @ zero_zero_nat )
= F )
& ! [M_2: nat,T_1: real] :
( ( ( ord_less_nat @ M_2 @ N )
& ( ord_less_eq_real @ H @ T_1 )
& ( ord_less_eq_real @ T_1 @ zero_zero_real ) )
=> ( deriv_real @ ( Diff @ M_2 ) @ T_1 @ ( Diff @ ( suc @ M_2 ) @ T_1 ) ) ) )
=> ? [T_1: real] :
( ( ord_less_real @ H @ T_1 )
& ( ord_less_real @ T_1 @ zero_zero_real )
& ( ( F @ H )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ M_2 @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ ( power_power_real @ H @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( Diff @ N @ T_1 ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ).
thf(fact_4939_Maclaurin,axiom,
! [Diff: nat > real > real,F: real > real,N: nat,H: real] :
( ( ord_less_real @ zero_zero_real @ H )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M_2: nat,T_1: real] :
( ( ( ord_less_nat @ M_2 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T_1 )
& ( ord_less_eq_real @ T_1 @ H ) )
=> ( deriv_real @ ( Diff @ M_2 ) @ T_1 @ ( Diff @ ( suc @ M_2 ) @ T_1 ) ) )
=> ? [T_1: real] :
( ( ord_less_real @ zero_zero_real @ T_1 )
& ( ord_less_real @ T_1 @ H )
& ( ( F @ H )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ M_2 @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ ( power_power_real @ H @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( Diff @ N @ T_1 ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).
thf(fact_4940_Maclaurin__all__le__objl,axiom,
! [N: nat,X: real,Diff: nat > real > real,F: real > real] :
( ( ( ( Diff @ zero_zero_nat )
= F )
& ! [M_2: nat,X_1: real] : ( deriv_real @ ( Diff @ M_2 ) @ X_1 @ ( Diff @ ( suc @ M_2 ) @ X_1 ) ) )
=> ? [T_1: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T_1 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ M_2 @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ ( power_power_real @ X @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( Diff @ N @ T_1 ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
thf(fact_4941_Maclaurin__all__le,axiom,
! [N: nat,X: real,Diff: nat > real > real,F: real > real] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M_2: nat,X_1: real] : ( deriv_real @ ( Diff @ M_2 ) @ X_1 @ ( Diff @ ( suc @ M_2 ) @ X_1 ) )
=> ? [T_1: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T_1 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ M_2 @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ ( power_power_real @ X @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( Diff @ N @ T_1 ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
thf(fact_4942_real__setsum__nat__ivl__bounded,axiom,
! [F: nat > real,K_2: real,N: nat] :
( ! [P_4: nat] :
( ( ord_less_nat @ P_4 @ N )
=> ( ord_less_eq_real @ ( F @ P_4 ) @ K_2 ) )
=> ( ord_less_eq_real @ ( big_co604158596t_real @ F @ ( ord_at4362885an_nat @ zero_zero_nat @ N ) ) @ ( times_times_real @ ( real_nat @ N ) @ K_2 ) ) ) ).
thf(fact_4943_DERIV__sumr,axiom,
! [F: nat > real > real,X: real,F_1: nat > real > real,N: nat,M: nat] :
( ! [R: nat] :
( ( ( ord_less_eq_nat @ M @ R )
& ( ord_less_nat @ R @ ( plus_plus_nat @ M @ N ) ) )
=> ( deriv_real @ ( F @ R ) @ X @ ( F_1 @ R @ X ) ) )
=> ( deriv_real
@ ^ [X_1: real] :
( big_co604158596t_real
@ ^ [N_1: nat] : ( F @ N_1 @ X_1 )
@ ( ord_at4362885an_nat @ M @ N ) )
@ X
@ ( big_co604158596t_real
@ ^ [R: nat] : ( F_1 @ R @ X )
@ ( ord_at4362885an_nat @ M @ N ) ) ) ) ).
thf(fact_4944_suminf__le,axiom,
! [X: real,F: nat > real] :
( ( summable_real @ F )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( big_co604158596t_real @ F @ ( ord_at4362885an_nat @ zero_zero_nat @ N_1 ) ) @ X )
=> ( ord_less_eq_real @ ( suminf_real @ F ) @ X ) ) ) ).
thf(fact_4945_transfer__nat__int__sum__prod__closure_I1_J,axiom,
! [F: int > int,A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ! [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X_1 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( big_co230513141nt_int @ F @ A_1 ) ) ) ) ).
thf(fact_4946_pos__summable,axiom,
! [X: real,F: nat > real] :
( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N_1 ) )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( big_co604158596t_real @ F @ ( ord_at4362885an_nat @ zero_zero_nat @ N_1 ) ) @ X )
=> ( summable_real @ F ) ) ) ).
thf(fact_4947_arith__series__int,axiom,
! [A: int,D: int,N: nat] :
( ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) )
@ ( big_co1024481617at_int
@ ^ [I_1: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1621563631at_int @ I_1 ) @ D ) )
@ ( ord_lessThan_nat @ N ) ) )
= ( times_times_int @ ( semiri1621563631at_int @ N ) @ ( plus_plus_int @ A @ ( plus_plus_int @ A @ ( times_times_int @ ( semiri1621563631at_int @ ( minus_minus_nat @ N @ one_one_nat ) ) @ D ) ) ) ) ) ).
thf(fact_4948_BseqI2,axiom,
! [K_2: real,K_1: real,F: nat > real] :
( ! [N_1: nat] :
( ( ord_less_eq_real @ K_1 @ ( F @ N_1 ) )
& ( ord_less_eq_real @ ( F @ N_1 ) @ K_2 ) )
=> ( bseq_real @ F ) ) ).
thf(fact_4949_finite__lessThan,axiom,
! [K_1: nat] : ( finite_finite_nat @ ( ord_lessThan_nat @ K_1 ) ) ).
thf(fact_4950_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( ord_lessThan_nat @ U ) )
= U ) ).
thf(fact_4951_atLeast0LessThan,axiom,
! [N: nat] :
( ( ord_at4362885an_nat @ zero_zero_nat @ N )
= ( ord_lessThan_nat @ N ) ) ).
thf(fact_4952_finite__nat__iff__bounded,axiom,
! [S: nat > $o] :
( ( finite_finite_nat @ S )
<=> ? [K: nat] : ( ord_less_eq_nat_o @ S @ ( ord_lessThan_nat @ K ) ) ) ).
thf(fact_4953_arith__series__nat,axiom,
! [A: nat,D: nat,N: nat] :
( ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) )
@ ( big_co387207925at_nat
@ ^ [I_1: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I_1 @ D ) )
@ ( ord_lessThan_nat @ N ) ) )
= ( times_times_nat @ N @ ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ ( times_times_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ D ) ) ) ) ) ).
thf(fact_4954_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_at641636577an_int @ zero_zero_int @ U )
= ( image_nat_int @ semiri1621563631at_int @ ( ord_lessThan_nat @ ( nat_1 @ U ) ) ) ) ) ).
thf(fact_4955_finite__nat__bounded,axiom,
! [S: nat > $o] :
( ( finite_finite_nat @ S )
=> ? [K: nat] : ( ord_less_eq_nat_o @ S @ ( ord_lessThan_nat @ K ) ) ) ).
thf(fact_4956_sum__diff__distrib,axiom,
! [N: nat,Q_1: nat > nat,P: nat > nat] :
( ! [X_1: nat] : ( ord_less_eq_nat @ ( Q_1 @ X_1 ) @ ( P @ X_1 ) )
=> ( ( minus_minus_nat @ ( big_co387207925at_nat @ P @ ( ord_lessThan_nat @ N ) ) @ ( big_co387207925at_nat @ Q_1 @ ( ord_lessThan_nat @ N ) ) )
= ( big_co387207925at_nat
@ ^ [X_1: nat] : ( minus_minus_nat @ ( P @ X_1 ) @ ( Q_1 @ X_1 ) )
@ ( ord_lessThan_nat @ N ) ) ) ) ).
thf(fact_4957_Sup__lessThan,axiom,
! [X: real] :
( ( comple124823625p_real @ ( ord_lessThan_real @ X ) )
= X ) ).
thf(fact_4958_transfer__nat__int__set__cong,axiom,
! [P_1: int > $o,P: int > $o] :
( ! [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( ( P @ X_1 )
<=> ( P_1 @ X_1 ) ) )
=> ( ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( P @ X_1 ) ) )
= ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( P_1 @ X_1 ) ) ) ) ) ).
thf(fact_4959_normalize__def,axiom,
! [P_3: product_prod_int_int] :
( ( ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P_3 ) )
=> ( ( normalize @ P_3 )
= ( product_Pair_int_int @ ( div_div_int @ ( product_fst_int_int @ P_3 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P_3 ) @ ( product_snd_int_int @ P_3 ) ) ) @ ( div_div_int @ ( product_snd_int_int @ P_3 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P_3 ) @ ( product_snd_int_int @ P_3 ) ) ) ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P_3 ) )
=> ( ( ( ( product_snd_int_int @ P_3 )
= zero_zero_int )
=> ( ( normalize @ P_3 )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) )
& ( ( ( product_snd_int_int @ P_3 )
!= zero_zero_int )
=> ( ( normalize @ P_3 )
= ( product_Pair_int_int @ ( div_div_int @ ( product_fst_int_int @ P_3 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P_3 ) @ ( product_snd_int_int @ P_3 ) ) ) ) @ ( div_div_int @ ( product_snd_int_int @ P_3 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P_3 ) @ ( product_snd_int_int @ P_3 ) ) ) ) ) ) ) ) ) ) ).
thf(fact_4960_gcd__dvd1__int,axiom,
! [X: int,Y: int] : ( dvd_dvd_int @ ( gcd_gcd_int @ X @ Y ) @ X ) ).
thf(fact_4961_gcd__dvd2__int,axiom,
! [X: int,Y: int] : ( dvd_dvd_int @ ( gcd_gcd_int @ X @ Y ) @ Y ) ).
thf(fact_4962_coprime__exp2__int,axiom,
! [N: nat,M: nat,A: int,B: int] :
( ( ( gcd_gcd_int @ A @ B )
= one_one_int )
=> ( ( gcd_gcd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ M ) )
= one_one_int ) ) ).
thf(fact_4963_invertible__coprime__int,axiom,
! [X: int,Y: int,M: int] :
( ( ( div_mod_int @ ( times_times_int @ X @ Y ) @ M )
= one_one_int )
=> ( ( gcd_gcd_int @ X @ M )
= one_one_int ) ) ).
thf(fact_4964_gcd__non__0__int,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Y )
=> ( ( gcd_gcd_int @ X @ Y )
= ( gcd_gcd_int @ Y @ ( div_mod_int @ X @ Y ) ) ) ) ).
thf(fact_4965_gcd__coprime__int,axiom,
! [B_5: int,A_5: int,A: int,B: int] :
( ( ( gcd_gcd_int @ A @ B )
!= zero_zero_int )
=> ( ( A
= ( times_times_int @ A_5 @ ( gcd_gcd_int @ A @ B ) ) )
=> ( ( B
= ( times_times_int @ B_5 @ ( gcd_gcd_int @ A @ B ) ) )
=> ( ( gcd_gcd_int @ A_5 @ B_5 )
= one_one_int ) ) ) ) ).
thf(fact_4966_transfer__int__nat__gcd_I1_J,axiom,
! [X: nat,Y: nat] :
( ( gcd_gcd_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) )
= ( semiri1621563631at_int @ ( gcd_gcd_nat @ X @ Y ) ) ) ).
thf(fact_4967_gcd__add__mult__int,axiom,
! [M: int,K_1: int,N: int] :
( ( gcd_gcd_int @ M @ ( plus_plus_int @ ( times_times_int @ K_1 @ M ) @ N ) )
= ( gcd_gcd_int @ M @ N ) ) ).
thf(fact_4968_gcd__add1__int,axiom,
! [M: int,N: int] :
( ( gcd_gcd_int @ ( plus_plus_int @ M @ N ) @ N )
= ( gcd_gcd_int @ M @ N ) ) ).
thf(fact_4969_gcd__add2__int,axiom,
! [M: int,N: int] :
( ( gcd_gcd_int @ M @ ( plus_plus_int @ M @ N ) )
= ( gcd_gcd_int @ M @ N ) ) ).
thf(fact_4970_coprime__plus__one__int,axiom,
! [N: int] :
( ( gcd_gcd_int @ ( plus_plus_int @ N @ one_one_int ) @ N )
= one_one_int ) ).
thf(fact_4971_coprime__minus__one__int,axiom,
! [N: int] :
( ( gcd_gcd_int @ ( minus_minus_int @ N @ one_one_int ) @ N )
= one_one_int ) ).
thf(fact_4972_coprime__divisors__nat,axiom,
! [E_1: int,B: int,D: int,A: int] :
( ( dvd_dvd_int @ D @ A )
=> ( ( dvd_dvd_int @ E_1 @ B )
=> ( ( ( gcd_gcd_int @ A @ B )
= one_one_int )
=> ( ( gcd_gcd_int @ D @ E_1 )
= one_one_int ) ) ) ) ).
thf(fact_4973_coprime__exp__int,axiom,
! [N: nat,D: int,A: int] :
( ( ( gcd_gcd_int @ D @ A )
= one_one_int )
=> ( ( gcd_gcd_int @ D @ ( power_power_int @ A @ N ) )
= one_one_int ) ) ).
thf(fact_4974_gcd__1__int,axiom,
! [M: int] :
( ( gcd_gcd_int @ M @ one_one_int )
= one_one_int ) ).
thf(fact_4975_gcd__zero__int,axiom,
! [M: int,N: int] :
( ( ( gcd_gcd_int @ M @ N )
= zero_zero_int )
<=> ( ( M = zero_zero_int )
& ( N = zero_zero_int ) ) ) ).
thf(fact_4976_gcd__exp__int,axiom,
! [A: int,N: nat,B: int] :
( ( gcd_gcd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( power_power_int @ ( gcd_gcd_int @ A @ B ) @ N ) ) ).
thf(fact_4977_gcd__commute__int,axiom,
! [A: int,B: int] :
( ( gcd_gcd_int @ A @ B )
= ( gcd_gcd_int @ B @ A ) ) ).
thf(fact_4978_gcd__int_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( gcd_gcd_int @ B @ ( gcd_gcd_int @ A @ C ) )
= ( gcd_gcd_int @ A @ ( gcd_gcd_int @ B @ C ) ) ) ).
thf(fact_4979_gcd__assoc__int,axiom,
! [A: int,B: int,C: int] :
( ( gcd_gcd_int @ ( gcd_gcd_int @ A @ B ) @ C )
= ( gcd_gcd_int @ A @ ( gcd_gcd_int @ B @ C ) ) ) ).
thf(fact_4980_dvd__gcd__D2__int,axiom,
! [I: int,M: int,N: int] :
( ( dvd_dvd_int @ I @ ( gcd_gcd_int @ M @ N ) )
=> ( dvd_dvd_int @ I @ N ) ) ).
thf(fact_4981_dvd__gcd__D1__int,axiom,
! [I: int,M: int,N: int] :
( ( dvd_dvd_int @ I @ ( gcd_gcd_int @ M @ N ) )
=> ( dvd_dvd_int @ I @ M ) ) ).
thf(fact_4982_gcd__greatest__int,axiom,
! [N: int,K_1: int,M: int] :
( ( dvd_dvd_int @ K_1 @ M )
=> ( ( dvd_dvd_int @ K_1 @ N )
=> ( dvd_dvd_int @ K_1 @ ( gcd_gcd_int @ M @ N ) ) ) ) ).
thf(fact_4983_gcd__greatest__iff__int,axiom,
! [K_1: int,M: int,N: int] :
( ( dvd_dvd_int @ K_1 @ ( gcd_gcd_int @ M @ N ) )
<=> ( ( dvd_dvd_int @ K_1 @ M )
& ( dvd_dvd_int @ K_1 @ N ) ) ) ).
thf(fact_4984_gcd__red__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ X @ Y )
= ( gcd_gcd_int @ Y @ ( div_mod_int @ X @ Y ) ) ) ).
thf(fact_4985_abs__gcd__int,axiom,
! [X: int,Y: int] :
( ( abs_abs_int @ ( gcd_gcd_int @ X @ Y ) )
= ( gcd_gcd_int @ X @ Y ) ) ).
thf(fact_4986_gcd__idem__int,axiom,
! [X: int] :
( ( gcd_gcd_int @ X @ X )
= ( abs_abs_int @ X ) ) ).
thf(fact_4987_gcd__abs__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ X @ Y )
= ( gcd_gcd_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) ) ) ).
thf(fact_4988_gcd__abs2__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ X @ ( abs_abs_int @ Y ) )
= ( gcd_gcd_int @ X @ Y ) ) ).
thf(fact_4989_gcd__abs1__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ ( abs_abs_int @ X ) @ Y )
= ( gcd_gcd_int @ X @ Y ) ) ).
thf(fact_4990_gcd__neg1__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y )
= ( gcd_gcd_int @ X @ Y ) ) ).
thf(fact_4991_gcd__neg2__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) )
= ( gcd_gcd_int @ X @ Y ) ) ).
thf(fact_4992_gcd__proj1__if__dvd__int,axiom,
! [X: int,Y: int] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( gcd_gcd_int @ X @ Y )
= ( abs_abs_int @ X ) ) ) ).
thf(fact_4993_gcd__proj2__if__dvd__int,axiom,
! [Y: int,X: int] :
( ( dvd_dvd_int @ Y @ X )
=> ( ( gcd_gcd_int @ X @ Y )
= ( abs_abs_int @ Y ) ) ) ).
thf(fact_4994_gcd__mult__distrib__int,axiom,
! [K_1: int,M: int,N: int] :
( ( times_times_int @ ( abs_abs_int @ K_1 ) @ ( gcd_gcd_int @ M @ N ) )
= ( gcd_gcd_int @ ( times_times_int @ K_1 @ M ) @ ( times_times_int @ K_1 @ N ) ) ) ).
thf(fact_4995_gcd__pos__int,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
<=> ( ( M != zero_zero_int )
| ( N != zero_zero_int ) ) ) ).
thf(fact_4996_coprime__rmult__int,axiom,
! [D: int,A: int,B: int] :
( ( ( gcd_gcd_int @ D @ ( times_times_int @ A @ B ) )
= one_one_int )
=> ( ( gcd_gcd_int @ D @ B )
= one_one_int ) ) ).
thf(fact_4997_coprime__lmult__int,axiom,
! [D: int,A: int,B: int] :
( ( ( gcd_gcd_int @ D @ ( times_times_int @ A @ B ) )
= one_one_int )
=> ( ( gcd_gcd_int @ D @ A )
= one_one_int ) ) ).
thf(fact_4998_coprime__mult__int,axiom,
! [B: int,D: int,A: int] :
( ( ( gcd_gcd_int @ D @ A )
= one_one_int )
=> ( ( ( gcd_gcd_int @ D @ B )
= one_one_int )
=> ( ( gcd_gcd_int @ D @ ( times_times_int @ A @ B ) )
= one_one_int ) ) ) ).
thf(fact_4999_gcd__mult__cancel__int,axiom,
! [M: int,K_1: int,N: int] :
( ( ( gcd_gcd_int @ K_1 @ N )
= one_one_int )
=> ( ( gcd_gcd_int @ ( times_times_int @ K_1 @ M ) @ N )
= ( gcd_gcd_int @ M @ N ) ) ) ).
thf(fact_5000_coprime__mul__eq__int,axiom,
! [D: int,A: int,B: int] :
( ( ( gcd_gcd_int @ D @ ( times_times_int @ A @ B ) )
= one_one_int )
<=> ( ( ( gcd_gcd_int @ D @ A )
= one_one_int )
& ( ( gcd_gcd_int @ D @ B )
= one_one_int ) ) ) ).
thf(fact_5001_gcd__0__left__int,axiom,
! [X: int] :
( ( gcd_gcd_int @ zero_zero_int @ X )
= ( abs_abs_int @ X ) ) ).
thf(fact_5002_gcd__0__int,axiom,
! [X: int] :
( ( gcd_gcd_int @ X @ zero_zero_int )
= ( abs_abs_int @ X ) ) ).
thf(fact_5003_Fract__coprime,axiom,
! [A: int,B: int] :
( ( fract @ ( div_div_int @ A @ ( gcd_gcd_int @ A @ B ) ) @ ( div_div_int @ B @ ( gcd_gcd_int @ A @ B ) ) )
= ( fract @ A @ B ) ) ).
thf(fact_5004_div__gcd__coprime__int,axiom,
! [B: int,A: int] :
( ( ( A != zero_zero_int )
| ( B != zero_zero_int ) )
=> ( ( gcd_gcd_int @ ( div_div_int @ A @ ( gcd_gcd_int @ A @ B ) ) @ ( div_div_int @ B @ ( gcd_gcd_int @ A @ B ) ) )
= one_one_int ) ) ).
thf(fact_5005_coprime__dvd__mult__int,axiom,
! [M: int,K_1: int,N: int] :
( ( ( gcd_gcd_int @ K_1 @ N )
= one_one_int )
=> ( ( dvd_dvd_int @ K_1 @ ( times_times_int @ M @ N ) )
=> ( dvd_dvd_int @ K_1 @ M ) ) ) ).
thf(fact_5006_coprime__dvd__mult__iff__int,axiom,
! [M: int,K_1: int,N: int] :
( ( ( gcd_gcd_int @ K_1 @ N )
= one_one_int )
=> ( ( dvd_dvd_int @ K_1 @ ( times_times_int @ M @ N ) )
<=> ( dvd_dvd_int @ K_1 @ M ) ) ) ).
thf(fact_5007_divides__mult__int,axiom,
! [N: int,M: int,R_1: int] :
( ( dvd_dvd_int @ M @ R_1 )
=> ( ( dvd_dvd_int @ N @ R_1 )
=> ( ( ( gcd_gcd_int @ M @ N )
= one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ R_1 ) ) ) ) ).
thf(fact_5008_gcd__cases__int,axiom,
! [P: int > $o,Y: int,X: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( P @ ( gcd_gcd_int @ X @ Y ) ) ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( P @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) ) ) ) )
=> ( ( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y ) ) ) )
=> ( ( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) )
=> ( P @ ( gcd_gcd_int @ X @ Y ) ) ) ) ) ) ).
thf(fact_5009_gcd__unique__int,axiom,
! [B: int,A: int,D: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ D )
& ( dvd_dvd_int @ D @ A )
& ( dvd_dvd_int @ D @ B )
& ! [E: int] :
( ( ( dvd_dvd_int @ E @ A )
& ( dvd_dvd_int @ E @ B ) )
=> ( dvd_dvd_int @ E @ D ) ) )
<=> ( D
= ( gcd_gcd_int @ A @ B ) ) ) ).
thf(fact_5010_gcd__ge__0__int,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y ) ) ).
thf(fact_5011_transfer__nat__int__gcd__closures_I1_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y ) ) ) ) ).
thf(fact_5012_gcd__le2__int,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
thf(fact_5013_gcd__le1__int,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
thf(fact_5014_coprime__crossproduct__int,axiom,
! [B: int,C: int,A: int,D: int] :
( ( ( gcd_gcd_int @ A @ D )
= one_one_int )
=> ( ( ( gcd_gcd_int @ B @ C )
= one_one_int )
=> ( ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ C ) )
= ( times_times_int @ ( abs_abs_int @ B ) @ ( abs_abs_int @ D ) ) )
<=> ( ( ( abs_abs_int @ A )
= ( abs_abs_int @ B ) )
& ( ( abs_abs_int @ C )
= ( abs_abs_int @ D ) ) ) ) ) ) ).
thf(fact_5015_gcd__code__int,axiom,
! [K_1: int,L: int] :
( ( gcd_gcd_int @ K_1 @ L )
= ( abs_abs_int @ ( if_int @ ( L = zero_zero_int ) @ K_1 @ ( gcd_gcd_int @ L @ ( div_mod_int @ ( abs_abs_int @ K_1 ) @ ( abs_abs_int @ L ) ) ) ) ) ) ).
thf(fact_5016_coprime__common__divisor__int,axiom,
! [X: int,A: int,B: int] :
( ( ( gcd_gcd_int @ A @ B )
= one_one_int )
=> ( ( dvd_dvd_int @ X @ A )
=> ( ( dvd_dvd_int @ X @ B )
=> ( ( abs_abs_int @ X )
= one_one_int ) ) ) ) ).
thf(fact_5017_quotient__of__coprime,axiom,
! [R_1: rat,P_3: int,Q: int] :
( ( ( quotient_of @ R_1 )
= ( product_Pair_int_int @ P_3 @ Q ) )
=> ( ( gcd_gcd_int @ P_3 @ Q )
= one_one_int ) ) ).
thf(fact_5018_normalize__coprime,axiom,
! [R_1: product_prod_int_int,P_3: int,Q: int] :
( ( ( normalize @ R_1 )
= ( product_Pair_int_int @ P_3 @ Q ) )
=> ( ( gcd_gcd_int @ P_3 @ Q )
= one_one_int ) ) ).
thf(fact_5019_coprime__int,axiom,
! [A: int,B: int] :
( ( ( gcd_gcd_int @ A @ B )
= one_one_int )
<=> ! [D_2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ D_2 )
& ( dvd_dvd_int @ D_2 @ A )
& ( dvd_dvd_int @ D_2 @ B ) )
<=> ( D_2 = one_one_int ) ) ) ).
thf(fact_5020_transfer__nat__int__gcd_I1_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( gcd_gcd_nat @ ( nat_1 @ X ) @ ( nat_1 @ Y ) )
= ( nat_1 @ ( gcd_gcd_int @ X @ Y ) ) ) ) ) ).
thf(fact_5021_gcd__int__def,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ X @ Y )
= ( semiri1621563631at_int @ ( gcd_gcd_nat @ ( nat_1 @ ( abs_abs_int @ X ) ) @ ( nat_1 @ ( abs_abs_int @ Y ) ) ) ) ) ).
thf(fact_5022_normalize__stable,axiom,
! [P_3: int,Q: int] :
( ( ord_less_int @ zero_zero_int @ Q )
=> ( ( ( gcd_gcd_int @ P_3 @ Q )
= one_one_int )
=> ( ( normalize @ ( product_Pair_int_int @ P_3 @ Q ) )
= ( product_Pair_int_int @ P_3 @ Q ) ) ) ) ).
thf(fact_5023_quotient__of__def,axiom,
! [X: rat] :
( ( quotient_of @ X )
= ( the_Pr2103884470nt_int
@ ^ [Pair: product_prod_int_int] :
( (&)
@ ( X
= ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
@ ( (&) @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
@ ( ( gcd_gcd_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) )
= one_one_int ) ) ) ) ) ).
thf(fact_5024_quotient__of__unique,axiom,
! [R_1: rat] :
? [X_1: product_prod_int_int] :
( ( R_1
= ( fract @ ( product_fst_int_int @ X_1 ) @ ( product_snd_int_int @ X_1 ) ) )
& ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X_1 ) )
& ( ( gcd_gcd_int @ ( product_fst_int_int @ X_1 ) @ ( product_snd_int_int @ X_1 ) )
= one_one_int )
& ! [Y_1: product_prod_int_int] :
( ( ( R_1
= ( fract @ ( product_fst_int_int @ Y_1 ) @ ( product_snd_int_int @ Y_1 ) ) )
& ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y_1 ) )
& ( ( gcd_gcd_int @ ( product_fst_int_int @ Y_1 ) @ ( product_snd_int_int @ Y_1 ) )
= one_one_int ) )
=> ( Y_1 = X_1 ) ) ) ).
thf(fact_5025_Rat__cases__nonzero,axiom,
! [Q: rat] :
( ! [A_2: int,B_4: int] :
( ( Q
= ( fract @ A_2 @ B_4 ) )
=> ( ( ord_less_int @ zero_zero_int @ B_4 )
=> ( ( A_2 != zero_zero_int )
=> ( ( gcd_gcd_int @ A_2 @ B_4 )
!= one_one_int ) ) ) )
=> ( Q = zero_zero_rat ) ) ).
thf(fact_5026_Rat__cases,axiom,
! [Q: rat] :
~ ! [A_2: int,B_4: int] :
( ( Q
= ( fract @ A_2 @ B_4 ) )
=> ( ( ord_less_int @ zero_zero_int @ B_4 )
=> ( ( gcd_gcd_int @ A_2 @ B_4 )
!= one_one_int ) ) ) ).
thf(fact_5027_Rat__induct,axiom,
! [Q: rat,P: rat > $o] :
( ! [A_2: int,B_4: int] :
( ( ord_less_int @ zero_zero_int @ B_4 )
=> ( ( ( gcd_gcd_int @ A_2 @ B_4 )
= one_one_int )
=> ( P @ ( fract @ A_2 @ B_4 ) ) ) )
=> ( P @ Q ) ) ).
thf(fact_5028_card__setsum__aux,axiom,
! [N: nat,S: ( int > $o ) > $o] :
( ( finite_finite_int_o @ S )
=> ( ! [X_1: int > $o] :
( ( member_int_o @ X_1 @ S )
=> ( finite_finite_int @ X_1 ) )
=> ( ! [X_1: int > $o] :
( ( member_int_o @ X_1 @ S )
=> ( ( finite_card_int @ X_1 )
= N ) )
=> ( ( big_co1971440592_o_nat @ finite_card_int @ S )
= ( big_co1971440592_o_nat
@ ^ [X_1: int > $o] : N
@ S ) ) ) ) ) ).
thf(fact_5029_zOdd__def,axiom,
( zOdd
= ( collect_int
@ ^ [X_1: int] :
? [K: int] :
( X_1
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ K ) @ one_one_int ) ) ) ) ).
thf(fact_5030_Rats__eq__int__div__int,axiom,
( field_1210416355s_real
= ( collect_real
@ ^ [Uu: real] :
? [I_1: int,J_1: int] :
( (&)
@ ( Uu
= ( inverse_divide_real @ ( real_int @ I_1 ) @ ( real_int @ J_1 ) ) )
@ ( (~) @ ( J_1 = zero_zero_int ) ) ) ) ) ).
thf(fact_5031_zEven__def,axiom,
( zEven
= ( collect_int
@ ^ [X_1: int] :
? [K: int] :
( X_1
= ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ K ) ) ) ) ).
thf(fact_5032_summable__Leibniz_I5_J,axiom,
! [A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ( monoseq_real @ A )
=> ( tendsto_nat_real
@ ^ [N_1: nat] :
( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) @ one_one_nat ) ) )
@ ( suminf_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) ) )
@ sequentially ) ) ) ).
thf(fact_5033_LIMSEQ__realpow__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( tendsto_nat_real @ ( power_power_real @ X ) @ zero_zero_real @ sequentially ) ) ) ).
thf(fact_5034_LIMSEQ__le,axiom,
! [Y_2: nat > real,Y: real,X_2: nat > real,X: real] :
( ( tendsto_nat_real @ X_2 @ X @ sequentially )
=> ( ( tendsto_nat_real @ Y_2 @ Y @ sequentially )
=> ( ? [N_2: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ N_2 @ N_1 )
=> ( ord_less_eq_real @ ( X_2 @ N_1 ) @ ( Y_2 @ N_1 ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ) ) ).
thf(fact_5035_LIMSEQ__le__const2,axiom,
! [A: real,X_2: nat > real,X: real] :
( ( tendsto_nat_real @ X_2 @ X @ sequentially )
=> ( ? [N_2: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ N_2 @ N_1 )
=> ( ord_less_eq_real @ ( X_2 @ N_1 ) @ A ) )
=> ( ord_less_eq_real @ X @ A ) ) ) ).
thf(fact_5036_LIMSEQ__le__const,axiom,
! [A: real,X_2: nat > real,X: real] :
( ( tendsto_nat_real @ X_2 @ X @ sequentially )
=> ( ? [N_2: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ N_2 @ N_1 )
=> ( ord_less_eq_real @ A @ ( X_2 @ N_1 ) ) )
=> ( ord_less_eq_real @ A @ X ) ) ) ).
thf(fact_5037_monoseq__le,axiom,
! [X: real,A: nat > real] :
( ( monoseq_real @ A )
=> ( ( tendsto_nat_real @ A @ X @ sequentially )
=> ( ( ! [N_1: nat] : ( ord_less_eq_real @ ( A @ N_1 ) @ X )
& ! [M_2: nat,N_1: nat] :
( ( ord_less_eq_nat @ M_2 @ N_1 )
=> ( ord_less_eq_real @ ( A @ M_2 ) @ ( A @ N_1 ) ) ) )
| ( ! [N_1: nat] : ( ord_less_eq_real @ X @ ( A @ N_1 ) )
& ! [M_2: nat,N_1: nat] :
( ( ord_less_eq_nat @ M_2 @ N_1 )
=> ( ord_less_eq_real @ ( A @ N_1 ) @ ( A @ M_2 ) ) ) ) ) ) ) ).
thf(fact_5038_LIMSEQ__inverse__realpow__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( tendsto_nat_real
@ ^ [N_1: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N_1 ) )
@ zero_zero_real
@ sequentially ) ) ).
thf(fact_5039_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R_1: real] :
( tendsto_nat_real
@ ^ [N_1: nat] : ( plus_plus_real @ R_1 @ ( inverse_inverse_real @ ( real_nat @ ( suc @ N_1 ) ) ) )
@ R_1
@ sequentially ) ).
thf(fact_5040_LIMSEQ__imp__rabs,axiom,
! [F: nat > real,L: real] :
( ( tendsto_nat_real @ F @ L @ sequentially )
=> ( tendsto_nat_real
@ ^ [N_1: nat] : ( abs_abs_real @ ( F @ N_1 ) )
@ ( abs_abs_real @ L )
@ sequentially ) ) ).
thf(fact_5041_LIMSEQ__rabs__zero,axiom,
! [F: nat > real] :
( ( tendsto_nat_real
@ ^ [N_1: nat] : ( abs_abs_real @ ( F @ N_1 ) )
@ zero_zero_real
@ sequentially )
<=> ( tendsto_nat_real @ F @ zero_zero_real @ sequentially ) ) ).
thf(fact_5042_LIMSEQ__inverse__real__of__nat,axiom,
( tendsto_nat_real
@ ^ [N_1: nat] : ( inverse_inverse_real @ ( real_nat @ ( suc @ N_1 ) ) )
@ zero_zero_real
@ sequentially ) ).
thf(fact_5043_LIMSEQ__divide__realpow__zero,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( tendsto_nat_real
@ ^ [N_1: nat] : ( inverse_divide_real @ A @ ( power_power_real @ X @ N_1 ) )
@ zero_zero_real
@ sequentially ) ) ).
thf(fact_5044_LIMSEQ__rabs__realpow__zero,axiom,
! [C: real] :
( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
=> ( tendsto_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ zero_zero_real @ sequentially ) ) ).
thf(fact_5045_LIMSEQ__rabs__realpow__zero2,axiom,
! [C: real] :
( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
=> ( tendsto_nat_real @ ( power_power_real @ C ) @ zero_zero_real @ sequentially ) ) ).
thf(fact_5046_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R_1: real] :
( tendsto_nat_real
@ ^ [N_1: nat] : ( plus_plus_real @ R_1 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( real_nat @ ( suc @ N_1 ) ) ) ) )
@ R_1
@ sequentially ) ).
thf(fact_5047_LIMSEQ__neg__powr,axiom,
! [S_1: real] :
( ( ord_less_real @ zero_zero_real @ S_1 )
=> ( tendsto_nat_real
@ ^ [X_1: nat] : ( powr @ ( real_nat @ X_1 ) @ ( uminus_uminus_real @ S_1 ) )
@ zero_zero_real
@ sequentially ) ) ).
thf(fact_5048_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R_1: real] :
( tendsto_nat_real
@ ^ [N_1: nat] : ( times_times_real @ R_1 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( real_nat @ ( suc @ N_1 ) ) ) ) ) )
@ R_1
@ sequentially ) ).
thf(fact_5049_summable__Leibniz_I1_J,axiom,
! [A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ( monoseq_real @ A )
=> ( summable_real
@ ^ [N_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ N_1 ) @ ( A @ N_1 ) ) ) ) ) ).
thf(fact_5050_zeroseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( tendsto_nat_real
@ ^ [N_1: nat] : ( times_times_real @ ( inverse_divide_real @ one_one_real @ ( real_nat @ ( plus_plus_nat @ ( times_times_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) )
@ zero_zero_real
@ sequentially ) ) ).
thf(fact_5051_Rats__eq__int__div__nat,axiom,
( field_1210416355s_real
= ( collect_real
@ ^ [Uu: real] :
? [I_1: int,N_1: nat] :
( (&)
@ ( Uu
= ( inverse_divide_real @ ( real_int @ I_1 ) @ ( real_nat @ N_1 ) ) )
@ ( (~) @ ( N_1 = zero_zero_nat ) ) ) ) ) ).
thf(fact_5052_summable__Leibniz_I4_J,axiom,
! [A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ( monoseq_real @ A )
=> ( tendsto_nat_real
@ ^ [N_1: nat] :
( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) ) )
@ ( suminf_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) ) )
@ sequentially ) ) ) ).
thf(fact_5053_summable__Leibniz_H_I5_J,axiom,
! [A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N_1 ) )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N_1 ) ) @ ( A @ N_1 ) )
=> ( tendsto_nat_real
@ ^ [N_1: nat] :
( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) @ one_one_nat ) ) )
@ ( suminf_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) ) )
@ sequentially ) ) ) ) ).
thf(fact_5054_summable__Leibniz_H_I4_J,axiom,
! [N: nat,A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N_1 ) )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N_1 ) ) @ ( A @ N_1 ) )
=> ( ord_less_eq_real
@ ( suminf_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) ) )
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
thf(fact_5055_cnj_OLIMSEQ,axiom,
! [X_2: nat > complex,A: complex] :
( ( tendsto_nat_complex @ X_2 @ A @ sequentially )
=> ( tendsto_nat_complex
@ ^ [N_1: nat] : ( cnj @ ( X_2 @ N_1 ) )
@ ( cnj @ A )
@ sequentially ) ) ).
thf(fact_5056_Re_OLIMSEQ,axiom,
! [X_2: nat > complex,A: complex] :
( ( tendsto_nat_complex @ X_2 @ A @ sequentially )
=> ( tendsto_nat_real
@ ^ [N_1: nat] : ( re @ ( X_2 @ N_1 ) )
@ ( re @ A )
@ sequentially ) ) ).
thf(fact_5057_Im_OLIMSEQ,axiom,
! [X_2: nat > complex,A: complex] :
( ( tendsto_nat_complex @ X_2 @ A @ sequentially )
=> ( tendsto_nat_real
@ ^ [N_1: nat] : ( im @ ( X_2 @ N_1 ) )
@ ( im @ A )
@ sequentially ) ) ).
thf(fact_5058_LIMSEQ__Complex,axiom,
! [Y_2: nat > real,B: real,X_2: nat > real,A: real] :
( ( tendsto_nat_real @ X_2 @ A @ sequentially )
=> ( ( tendsto_nat_real @ Y_2 @ B @ sequentially )
=> ( tendsto_nat_complex
@ ^ [N_1: nat] : ( complex_1 @ ( X_2 @ N_1 ) @ ( Y_2 @ N_1 ) )
@ ( complex_1 @ A @ B )
@ sequentially ) ) ) ).
thf(fact_5059_Bolzano__nest__unique,axiom,
! [P_2: produc914805421l_real > $o,A_4: real,B_3: real] :
( ( ord_less_eq_real @ A_4 @ B_3 )
=> ( ( ord_less_eq_real @ A_4 @ B_3 )
=> ( ( ord_less_eq_real @ A_4 @ B_3 )
=> ( ( tendsto_nat_real
@ ^ [N_1: nat] : ( minus_minus_real @ ( produc1935615926l_real @ ( bolzano_bisect @ P_2 @ A_4 @ B_3 @ N_1 ) ) @ ( produc556554744l_real @ ( bolzano_bisect @ P_2 @ A_4 @ B_3 @ N_1 ) ) )
@ zero_zero_real
@ sequentially )
=> ? [L_1: real] :
( ! [N_1: nat] : ( ord_less_eq_real @ ( produc1935615926l_real @ ( bolzano_bisect @ P_2 @ A_4 @ B_3 @ N_1 ) ) @ L_1 )
& ( tendsto_nat_real
@ ^ [N_1: nat] : ( produc1935615926l_real @ ( bolzano_bisect @ P_2 @ A_4 @ B_3 @ N_1 ) )
@ L_1
@ sequentially )
& ! [N_1: nat] : ( ord_less_eq_real @ L_1 @ ( produc556554744l_real @ ( bolzano_bisect @ P_2 @ A_4 @ B_3 @ N_1 ) ) )
& ( tendsto_nat_real
@ ^ [N_1: nat] : ( produc556554744l_real @ ( bolzano_bisect @ P_2 @ A_4 @ B_3 @ N_1 ) )
@ L_1
@ sequentially ) ) ) ) ) ) ).
thf(fact_5060_summable__Leibniz_H_I3_J,axiom,
! [A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N_1 ) )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N_1 ) ) @ ( A @ N_1 ) )
=> ( tendsto_nat_real
@ ^ [N_1: nat] :
( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) ) )
@ ( suminf_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) ) )
@ sequentially ) ) ) ) ).
thf(fact_5061_summable__Leibniz_H_I2_J,axiom,
! [N: nat,A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N_1 ) )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N_1 ) ) @ ( A @ N_1 ) )
=> ( ord_less_eq_real
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) )
@ ( suminf_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) ) ) ) ) ) ) ).
thf(fact_5062_summable__Leibniz_I3_J,axiom,
! [A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ( monoseq_real @ A )
=> ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
=> ! [N_1: nat] :
( member_real
@ ( suminf_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) ) )
@ ( ord_at1589558736t_real
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) @ one_one_nat ) ) )
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) ) ) ) ) ) ) ) ).
thf(fact_5063_summable__Leibniz_I2_J,axiom,
! [A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ( monoseq_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
=> ! [N_1: nat] :
( member_real
@ ( suminf_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) ) )
@ ( ord_at1589558736t_real
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) ) )
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
thf(fact_5064_Sup__atLeastAtMost,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( comple124823625p_real @ ( ord_at1589558736t_real @ Y @ X ) )
= X ) ) ).
thf(fact_5065_summable,axiom,
! [A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N_1 ) )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N_1 ) ) @ ( A @ N_1 ) )
=> ( summable_real
@ ^ [N_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ N_1 ) @ ( A @ N_1 ) ) ) ) ) ) ).
thf(fact_5066_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( ord_at238088361st_nat @ L @ U ) ) ).
thf(fact_5067_finite__atLeastAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( ord_at875362053st_int @ L @ U ) ) ).
thf(fact_5068_SetInterval_Otransfer__nat__int__set__function__closures,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( nat_nat_set @ ( ord_at875362053st_int @ X @ Y ) ) ) ).
thf(fact_5069_all__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ! [M_2: nat] :
( ( ord_less_eq_nat @ M_2 @ N )
=> ( P @ M_2 ) )
<=> ! [X_1: nat] :
( ( member_nat @ X_1 @ ( ord_at238088361st_nat @ zero_zero_nat @ N ) )
=> ( P @ X_1 ) ) ) ).
thf(fact_5070_ex__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ? [M_2: nat] :
( ( ord_less_eq_nat @ M_2 @ N )
& ( P @ M_2 ) )
<=> ? [X_1: nat] :
( ( member_nat @ X_1 @ ( ord_at238088361st_nat @ zero_zero_nat @ N ) )
& ( P @ X_1 ) ) ) ).
thf(fact_5071_SetInterval_Otransfer__nat__int__set__functions_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_at238088361st_nat @ M @ N )
= ( image_int_nat @ nat_1 @ ( ord_at875362053st_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) ) ) ) ).
thf(fact_5072_image__Suc__atLeastAtMost,axiom,
! [I: nat,J: nat] :
( ( image_nat_nat @ suc @ ( ord_at238088361st_nat @ I @ J ) )
= ( ord_at238088361st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
thf(fact_5073_image__add__atLeastAtMost,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( image_nat_nat
@ ^ [N_1: nat] : ( plus_plus_nat @ N_1 @ K_1 )
@ ( ord_at238088361st_nat @ I @ J ) )
= ( ord_at238088361st_nat @ ( plus_plus_nat @ I @ K_1 ) @ ( plus_plus_nat @ J @ K_1 ) ) ) ).
thf(fact_5074_fact__altdef__nat,axiom,
! [N: nat] :
( ( fact_fact_nat @ N )
= ( big_co1705425894at_nat
@ ^ [I_1: nat] : I_1
@ ( ord_at238088361st_nat @ one_one_nat @ N ) ) ) ).
thf(fact_5075_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( ord_at4362885an_nat @ L @ ( suc @ U ) )
= ( ord_at238088361st_nat @ L @ U ) ) ).
thf(fact_5076_Re_Ocont,axiom,
! [A: complex] : ( tendsto_complex_real @ re @ ( re @ A ) @ ( at_complex @ A ) ) ).
thf(fact_5077_Im_Ocont,axiom,
! [A: complex] : ( tendsto_complex_real @ im @ ( im @ A ) @ ( at_complex @ A ) ) ).
thf(fact_5078_cnj_Ocont,axiom,
! [A: complex] : ( tendst1507391555omplex @ cnj @ ( cnj @ A ) @ ( at_complex @ A ) ) ).
thf(fact_5079_card__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( ord_at238088361st_nat @ L @ U ) )
= ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
thf(fact_5080_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( ord_at641636577an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
= ( ord_at875362053st_int @ L @ U ) ) ).
thf(fact_5081_setsum__shift__lb__Suc0__0,axiom,
! [K_1: nat,F: nat > nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( big_co387207925at_nat @ F @ ( ord_at238088361st_nat @ ( suc @ zero_zero_nat ) @ K_1 ) )
= ( big_co387207925at_nat @ F @ ( ord_at238088361st_nat @ zero_zero_nat @ K_1 ) ) ) ) ).
thf(fact_5082_card__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( ord_at875362053st_int @ L @ U ) )
= ( nat_1 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).
thf(fact_5083_fact__div__fact,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( div_div_nat @ ( fact_fact_nat @ M ) @ ( fact_fact_nat @ N ) )
= ( big_co1705425894at_nat
@ ^ [X_1: nat] : X_1
@ ( ord_at238088361st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
thf(fact_5084_fact__altdef__int,axiom,
! [N: int] :
( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( fact_fact_int @ N )
= ( big_co1548731110nt_int
@ ^ [I_1: int] : I_1
@ ( ord_at875362053st_int @ one_one_int @ N ) ) ) ) ).
thf(fact_5085_LIM__cos__div__sin,axiom,
( tendsto_real_real
@ ^ [X_1: real] : ( inverse_divide_real @ ( cos @ X_1 ) @ ( sin @ X_1 ) )
@ zero_zero_real
@ ( at_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_5086_LIM__fun__not__zero,axiom,
! [F: real > real,L: real,C: real] :
( ( tendsto_real_real @ F @ L @ ( at_real @ C ) )
=> ( ( L != zero_zero_real )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ! [X_1: real] :
( ( ( X_1 != C )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X_1 ) ) @ R ) )
=> ( ( F @ X_1 )
!= zero_zero_real ) ) ) ) ) ).
thf(fact_5087_LIM__fun__gt__zero,axiom,
! [F: real > real,L: real,C: real] :
( ( tendsto_real_real @ F @ L @ ( at_real @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ! [X_1: real] :
( ( ( X_1 != C )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X_1 ) ) @ R ) )
=> ( ord_less_real @ zero_zero_real @ ( F @ X_1 ) ) ) ) ) ) ).
thf(fact_5088_LIM__fun__less__zero,axiom,
! [F: real > real,L: real,C: real] :
( ( tendsto_real_real @ F @ L @ ( at_real @ C ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ! [X_1: real] :
( ( ( X_1 != C )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X_1 ) ) @ R ) )
=> ( ord_less_real @ ( F @ X_1 ) @ zero_zero_real ) ) ) ) ) ).
thf(fact_5089_aset_I6_J,axiom,
! [T: int,A_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A_1 )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ X_1 @ T )
=> ( ord_less_eq_int @ ( plus_plus_int @ X_1 @ D_1 ) @ T ) ) ) ) ) ).
thf(fact_5090_aset_I8_J,axiom,
! [T: int,A_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A_1 )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ T @ X_1 )
=> ( ord_less_eq_int @ T @ ( plus_plus_int @ X_1 @ D_1 ) ) ) ) ) ).
thf(fact_5091_bset_I6_J,axiom,
! [T: int,B_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B_1 )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ X_1 @ T )
=> ( ord_less_eq_int @ ( minus_minus_int @ X_1 @ D_1 ) @ T ) ) ) ) ).
thf(fact_5092_bset_I8_J,axiom,
! [T: int,B_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B_1 )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ T @ X_1 )
=> ( ord_less_eq_int @ T @ ( minus_minus_int @ X_1 @ D_1 ) ) ) ) ) ) ).
thf(fact_5093_aset_I7_J,axiom,
! [T: int,A_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A_1 )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ T @ X_1 )
=> ( ord_less_int @ T @ ( plus_plus_int @ X_1 @ D_1 ) ) ) ) ) ).
thf(fact_5094_bset_I5_J,axiom,
! [T: int,B_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B_1 )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ X_1 @ T )
=> ( ord_less_int @ ( minus_minus_int @ X_1 @ D_1 ) @ T ) ) ) ) ).
thf(fact_5095_bset_I3_J,axiom,
! [T: int,B_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B_1 )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( X_1 = T )
=> ( ( minus_minus_int @ X_1 @ D_1 )
= T ) ) ) ) ) ).
thf(fact_5096_aset_I3_J,axiom,
! [T: int,A_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A_1 )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( X_1 = T )
=> ( ( plus_plus_int @ X_1 @ D_1 )
= T ) ) ) ) ) ).
thf(fact_5097_aset_I4_J,axiom,
! [T: int,A_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ( member_int @ T @ A_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A_1 )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( X_1 != T )
=> ( ( plus_plus_int @ X_1 @ D_1 )
!= T ) ) ) ) ) ).
thf(fact_5098_bset_I4_J,axiom,
! [T: int,B_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ( member_int @ T @ B_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B_1 )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( X_1 != T )
=> ( ( minus_minus_int @ X_1 @ D_1 )
!= T ) ) ) ) ) ).
thf(fact_5099_aset_I5_J,axiom,
! [T: int,A_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ( member_int @ T @ A_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A_1 )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ X_1 @ T )
=> ( ord_less_int @ ( plus_plus_int @ X_1 @ D_1 ) @ T ) ) ) ) ) ).
thf(fact_5100_bset_I7_J,axiom,
! [T: int,B_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ( member_int @ T @ B_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B_1 )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ T @ X_1 )
=> ( ord_less_int @ T @ ( minus_minus_int @ X_1 @ D_1 ) ) ) ) ) ) ).
thf(fact_5101_periodic__finite__ex,axiom,
! [P: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int,K: int] :
( ( P @ X_1 )
<=> ( P @ ( minus_minus_int @ X_1 @ ( times_times_int @ K @ D ) ) ) )
=> ( ( ?? @ int @ P )
<=> ? [X_1: int] :
( ( member_int @ X_1 @ ( ord_at875362053st_int @ one_one_int @ D ) )
& ( P @ X_1 ) ) ) ) ) ).
thf(fact_5102_aset_I10_J,axiom,
! [T: int,A_1: int > $o,D: int,D_1: int] :
( ( dvd_dvd_int @ D @ D_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A_1 )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X_1 @ T ) )
=> ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X_1 @ D_1 ) @ T ) ) ) ) ) ).
thf(fact_5103_bset_I10_J,axiom,
! [T: int,B_1: int > $o,D: int,D_1: int] :
( ( dvd_dvd_int @ D @ D_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B_1 )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X_1 @ T ) )
=> ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X_1 @ D_1 ) @ T ) ) ) ) ) ).
thf(fact_5104_bset_I9_J,axiom,
! [T: int,B_1: int > $o,D: int,D_1: int] :
( ( dvd_dvd_int @ D @ D_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B_1 )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X_1 @ T ) )
=> ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X_1 @ D_1 ) @ T ) ) ) ) ) ).
thf(fact_5105_aset_I9_J,axiom,
! [T: int,A_1: int > $o,D: int,D_1: int] :
( ( dvd_dvd_int @ D @ D_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A_1 )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X_1 @ T ) )
=> ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X_1 @ D_1 ) @ T ) ) ) ) ) ).
thf(fact_5106_LIMSEQ__inverse__zero,axiom,
! [X_2: nat > real] :
( ! [R: real] :
? [N_2: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ N_2 @ N_1 )
=> ( ord_less_real @ R @ ( X_2 @ N_1 ) ) )
=> ( tendsto_nat_real
@ ^ [N_1: nat] : ( inverse_inverse_real @ ( X_2 @ N_1 ) )
@ zero_zero_real
@ sequentially ) ) ).
thf(fact_5107_real__scaleR__def,axiom,
! [A: real,X: real] :
( ( scaleR_scaleR_real @ A @ X )
= ( times_times_real @ A @ X ) ) ).
thf(fact_5108_complex__Im__scaleR,axiom,
! [R_1: real,X: complex] :
( ( im @ ( scaleR1652505878omplex @ R_1 @ X ) )
= ( times_times_real @ R_1 @ ( im @ X ) ) ) ).
thf(fact_5109_complex__Re__scaleR,axiom,
! [R_1: real,X: complex] :
( ( re @ ( scaleR1652505878omplex @ R_1 @ X ) )
= ( times_times_real @ R_1 @ ( re @ X ) ) ) ).
thf(fact_5110_complex__scaleR,axiom,
! [R_1: real,A: real,B: real] :
( ( scaleR1652505878omplex @ R_1 @ ( complex_1 @ A @ B ) )
= ( complex_1 @ ( times_times_real @ R_1 @ A ) @ ( times_times_real @ R_1 @ B ) ) ) ).
thf(fact_5111_Re_OscaleR,axiom,
! [R_1: real,X: complex] :
( ( re @ ( scaleR1652505878omplex @ R_1 @ X ) )
= ( scaleR_scaleR_real @ R_1 @ ( re @ X ) ) ) ).
thf(fact_5112_Im_OscaleR,axiom,
! [R_1: real,X: complex] :
( ( im @ ( scaleR1652505878omplex @ R_1 @ X ) )
= ( scaleR_scaleR_real @ R_1 @ ( im @ X ) ) ) ).
thf(fact_5113_cnj_OscaleR,axiom,
! [R_1: real,X: complex] :
( ( cnj @ ( scaleR1652505878omplex @ R_1 @ X ) )
= ( scaleR1652505878omplex @ R_1 @ ( cnj @ X ) ) ) ).
thf(fact_5114_complex__sgn__def,axiom,
! [X: complex] :
( ( sgn_sgn_complex @ X )
= ( scaleR1652505878omplex @ ( inverse_inverse_real @ ( norm_norm_complex @ X ) ) @ X ) ) ).
thf(fact_5115_complex__scaleR__def,axiom,
! [R_1: real,X: complex] :
( ( scaleR1652505878omplex @ R_1 @ X )
= ( complex_1 @ ( times_times_real @ R_1 @ ( re @ X ) ) @ ( times_times_real @ R_1 @ ( im @ X ) ) ) ) ).
thf(fact_5116_trivial__limit__sequentially,axiom,
~ ( trivial_limit_nat @ sequentially ) ).
thf(fact_5117_pred__nat__def,axiom,
( pred_nat
= ( collec1979865426at_nat
@ ( produc1038563245_nat_o
@ ^ [M_2: nat,N_1: nat] :
( N_1
= ( suc @ M_2 ) ) ) ) ) ).
thf(fact_5118_gcd__coprime__exists__int,axiom,
! [A: int,B: int] :
( ( ( gcd_gcd_int @ A @ B )
!= zero_zero_int )
=> ? [A_3: int,B_2: int] :
( ( A
= ( times_times_int @ A_3 @ ( gcd_gcd_int @ A @ B ) ) )
& ( B
= ( times_times_int @ B_2 @ ( gcd_gcd_int @ A @ B ) ) )
& ( ( gcd_gcd_int @ A_3 @ B_2 )
= one_one_int ) ) ) ).
thf(fact_5119_min__number__of__Suc,axiom,
! [N: nat,V: int] :
( ( ord_min_nat @ ( suc @ N ) @ ( number_number_of_nat @ V ) )
= ( if_nat @ ( nat_neg @ ( number_number_of_int @ ( pred @ V ) ) ) @ zero_zero_nat @ ( suc @ ( ord_min_nat @ N @ ( nat_1 @ ( number_number_of_int @ ( pred @ V ) ) ) ) ) ) ) ).
thf(fact_5120_min__Suc__number__of,axiom,
! [V: int,N: nat] :
( ( ord_min_nat @ ( number_number_of_nat @ V ) @ ( suc @ N ) )
= ( if_nat @ ( nat_neg @ ( number_number_of_int @ ( pred @ V ) ) ) @ zero_zero_nat @ ( suc @ ( ord_min_nat @ ( nat_1 @ ( number_number_of_int @ ( pred @ V ) ) ) @ N ) ) ) ) ).
thf(fact_5121_min__diff,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
= ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
thf(fact_5122_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_5123_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_5124_min__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
thf(fact_5125_min__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ M @ ( suc @ N ) )
= ( nat_case_nat @ zero_zero_nat
@ ^ [M_1: nat] : ( suc @ ( ord_min_nat @ M_1 @ N ) )
@ M ) ) ).
thf(fact_5126_min__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_min_nat @ ( suc @ N ) @ M )
= ( nat_case_nat @ zero_zero_nat
@ ^ [M_1: nat] : ( suc @ ( ord_min_nat @ N @ M_1 ) )
@ M ) ) ).
thf(fact_5127_vanishes__mult__bounded,axiom,
! [Y_2: nat > rat,X_2: nat > rat] :
( ? [A_2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_2 )
& ! [N_1: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X_2 @ N_1 ) ) @ A_2 ) )
=> ( ( vanishes @ Y_2 )
=> ( vanishes
@ ^ [N_1: nat] : ( times_times_rat @ ( X_2 @ N_1 ) @ ( Y_2 @ N_1 ) ) ) ) ) ).
thf(fact_5128_SetInterval_Otransfer__int__nat__set__functions,axiom,
! [N: int,M: int] :
( ( nat_is_nat @ M )
=> ( ( nat_is_nat @ N )
=> ( ( ord_at875362053st_int @ M @ N )
= ( image_nat_int @ semiri1621563631at_int @ ( ord_at238088361st_nat @ ( nat_1 @ M ) @ ( nat_1 @ N ) ) ) ) ) ) ).
thf(fact_5129_is__nat__def,axiom,
! [X: int] :
( ( nat_is_nat @ X )
<=> ( ord_less_eq_int @ zero_zero_int @ X ) ) ).
thf(fact_5130_vanishes__diff,axiom,
! [Y_2: nat > rat,X_2: nat > rat] :
( ( vanishes @ X_2 )
=> ( ( vanishes @ Y_2 )
=> ( vanishes
@ ^ [N_1: nat] : ( minus_minus_rat @ ( X_2 @ N_1 ) @ ( Y_2 @ N_1 ) ) ) ) ) ).
thf(fact_5131_vanishes__add,axiom,
! [Y_2: nat > rat,X_2: nat > rat] :
( ( vanishes @ X_2 )
=> ( ( vanishes @ Y_2 )
=> ( vanishes
@ ^ [N_1: nat] : ( plus_plus_rat @ ( X_2 @ N_1 ) @ ( Y_2 @ N_1 ) ) ) ) ) ).
thf(fact_5132_vanishes__const,axiom,
! [C: rat] :
( ( vanishes
@ ^ [N_1: nat] : C )
<=> ( C = zero_zero_rat ) ) ).
thf(fact_5133_vanishes__minus,axiom,
! [X_2: nat > rat] :
( ( vanishes @ X_2 )
=> ( vanishes
@ ^ [N_1: nat] : ( uminus_uminus_rat @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_5134_Nat__Transfer_Otransfer__int__nat__function__closures_I2_J,axiom,
! [Y: int,X: int] :
( ( nat_is_nat @ X )
=> ( ( nat_is_nat @ Y )
=> ( nat_is_nat @ ( times_times_int @ X @ Y ) ) ) ) ).
thf(fact_5135_Nat__Transfer_Otransfer__int__nat__function__closures_I4_J,axiom,
! [N: nat,X: int] :
( ( nat_is_nat @ X )
=> ( nat_is_nat @ ( power_power_int @ X @ N ) ) ) ).
thf(fact_5136_Divides_Otransfer__int__nat__function__closures_I2_J,axiom,
! [Y: int,X: int] :
( ( nat_is_nat @ X )
=> ( ( nat_is_nat @ Y )
=> ( nat_is_nat @ ( div_mod_int @ X @ Y ) ) ) ) ).
thf(fact_5137_Divides_Otransfer__int__nat__function__closures_I1_J,axiom,
! [Y: int,X: int] :
( ( nat_is_nat @ X )
=> ( ( nat_is_nat @ Y )
=> ( nat_is_nat @ ( div_div_int @ X @ Y ) ) ) ) ).
thf(fact_5138_Nat__Transfer_Otransfer__int__nat__function__closures_I3_J,axiom,
! [Y: int,X: int] :
( ( nat_is_nat @ X )
=> ( ( nat_is_nat @ Y )
=> ( nat_is_nat @ ( nat_tsub @ X @ Y ) ) ) ) ).
thf(fact_5139_Nat__Transfer_Otransfer__int__nat__set__function__closures_I6_J,axiom,
! [X: int,A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ( member_int @ X @ A_1 )
=> ( nat_is_nat @ X ) ) ) ).
thf(fact_5140_Nat__Transfer_Otransfer__int__nat__function__closures_I5_J,axiom,
nat_is_nat @ zero_zero_int ).
thf(fact_5141_Nat__Transfer_Otransfer__int__nat__function__closures_I6_J,axiom,
nat_is_nat @ one_one_int ).
thf(fact_5142_Nat__Transfer_Otransfer__int__nat__function__closures_I1_J,axiom,
! [Y: int,X: int] :
( ( nat_is_nat @ X )
=> ( ( nat_is_nat @ Y )
=> ( nat_is_nat @ ( plus_plus_int @ X @ Y ) ) ) ) ).
thf(fact_5143_Nat__Transfer_Otransfer__int__nat__function__closures_I9_J,axiom,
! [Z_1: nat] : ( nat_is_nat @ ( semiri1621563631at_int @ Z_1 ) ) ).
thf(fact_5144_SetInterval_Otransfer__int__nat__set__function__closures,axiom,
! [Y: int,X: int] :
( ( nat_is_nat @ X )
=> ( nat_nat_set @ ( ord_at875362053st_int @ X @ Y ) ) ) ).
thf(fact_5145_Nat__Transfer_Otransfer__int__nat__function__closures_I8_J,axiom,
nat_is_nat @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) ).
thf(fact_5146_transfer__int__nat__gcd__closures_I1_J,axiom,
! [Y: int,X: int] :
( ( nat_is_nat @ X )
=> ( ( nat_is_nat @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y ) ) ) ) ).
thf(fact_5147_transfer__int__nat__factorial__closure,axiom,
! [X: int] :
( ( nat_is_nat @ X )
=> ( ord_less_eq_int @ zero_zero_int @ ( fact_fact_int @ X ) ) ) ).
thf(fact_5148_transfer__int__nat__set__relations_I2_J,axiom,
! [A_1: int > $o,X: int] :
( ( nat_is_nat @ X )
=> ( ( nat_nat_set @ A_1 )
=> ( ( member_int @ X @ A_1 )
<=> ( member_nat @ ( nat_1 @ X ) @ ( image_int_nat @ nat_1 @ A_1 ) ) ) ) ) ).
thf(fact_5149_Nat__Transfer_Otransfer__int__nat__function__closures_I7_J,axiom,
nat_is_nat @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_5150_vanishes__def,axiom,
! [X_2: nat > rat] :
( ( vanishes @ X_2 )
<=> ! [R: rat] :
( ( ord_less_rat @ zero_zero_rat @ R )
=> ? [K: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ K @ N_1 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X_2 @ N_1 ) ) @ R ) ) ) ) ).
thf(fact_5151_vanishesD,axiom,
! [R_1: rat,X_2: nat > rat] :
( ( vanishes @ X_2 )
=> ( ( ord_less_rat @ zero_zero_rat @ R_1 )
=> ? [K: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ K @ N_1 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X_2 @ N_1 ) ) @ R_1 ) ) ) ) ).
thf(fact_5152_vanishesI,axiom,
! [X_2: nat > rat] :
( ! [R: rat] :
( ( ord_less_rat @ zero_zero_rat @ R )
=> ? [K: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ K @ N_1 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X_2 @ N_1 ) ) @ R ) ) )
=> ( vanishes @ X_2 ) ) ).
thf(fact_5153_zcongm__def,axiom,
! [M: int,X_1: int,Xa: int] :
( ( zcongm @ M @ X_1 @ Xa )
<=> ( zcong @ X_1 @ Xa @ M ) ) ).
thf(fact_5154_cpmi,axiom,
! [B_1: int > $o,P_1: int > $o,P: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ? [Z: int] :
! [X_1: int] :
( ( ord_less_int @ X_1 @ Z )
=> ( ( P @ X_1 )
<=> ( P_1 @ X_1 ) ) )
=> ( ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B_1 )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X_1 )
=> ( P @ ( minus_minus_int @ X_1 @ D_1 ) ) ) )
=> ( ! [X_1: int,K: int] :
( ( P_1 @ X_1 )
<=> ( P_1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K @ D_1 ) ) ) )
=> ( ( ?? @ int @ P )
<=> ( ? [X_1: int] :
( ( member_int @ X_1 @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
& ( P_1 @ X_1 ) )
| ? [X_1: int] :
( ( member_int @ X_1 @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
& ? [Xa: int] :
( ( member_int @ Xa @ B_1 )
& ( P @ ( plus_plus_int @ Xa @ X_1 ) ) ) ) ) ) ) ) ) ) ).
thf(fact_5155_bijzcong__zcong__prod,axiom,
! [A_1: int > $o,B_1: int > $o,M: int] :
( ( member1329254762_int_o @ ( produc398918003_int_o @ A_1 @ B_1 ) @ ( bijR_int_int @ ( zcongm @ M ) ) )
=> ( zcong
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ A_1 )
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ B_1 )
@ M ) ) ).
thf(fact_5156_cppi,axiom,
! [A_1: int > $o,P_1: int > $o,P: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ? [Z: int] :
! [X_1: int] :
( ( ord_less_int @ Z @ X_1 )
=> ( ( P @ X_1 )
<=> ( P_1 @ X_1 ) ) )
=> ( ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A_1 )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X_1 )
=> ( P @ ( plus_plus_int @ X_1 @ D_1 ) ) ) )
=> ( ! [X_1: int,K: int] :
( ( P_1 @ X_1 )
<=> ( P_1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K @ D_1 ) ) ) )
=> ( ( ?? @ int @ P )
<=> ( ? [X_1: int] :
( ( member_int @ X_1 @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
& ( P_1 @ X_1 ) )
| ? [X_1: int] :
( ( member_int @ X_1 @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
& ? [Xa: int] :
( ( member_int @ Xa @ A_1 )
& ( P @ ( minus_minus_int @ Xa @ X_1 ) ) ) ) ) ) ) ) ) ) ).
thf(fact_5157_increasing__LIMSEQ,axiom,
! [L: real,F: nat > real] :
( ! [N_1: nat] : ( ord_less_eq_real @ ( F @ N_1 ) @ ( F @ ( suc @ N_1 ) ) )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( F @ N_1 ) @ L )
=> ( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [N_1: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N_1 ) @ E ) ) )
=> ( tendsto_nat_real @ F @ L @ sequentially ) ) ) ) ).
thf(fact_5158_isCont__ln,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( isCont_real_real @ ln @ X ) ) ).
thf(fact_5159_isCont__real__root,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( isCont_real_real @ ( root @ N ) @ X ) ) ).
thf(fact_5160_isCont__tan,axiom,
! [X: real] :
( ( ( cos @ X )
!= zero_zero_real )
=> ( isCont_real_real @ tan @ X ) ) ).
thf(fact_5161_isCont__cos,axiom,
! [X: real] : ( isCont_real_real @ cos @ X ) ).
thf(fact_5162_isCont__arctan,axiom,
! [X: real] : ( isCont_real_real @ arctan @ X ) ).
thf(fact_5163_isCont__real__sqrt,axiom,
! [X: real] : ( isCont_real_real @ sqrt @ X ) ).
thf(fact_5164_isCont__sin,axiom,
! [X: real] : ( isCont_real_real @ sin @ X ) ).
thf(fact_5165_cnj_OisCont,axiom,
! [A: complex] : ( isCont156215680omplex @ cnj @ A ) ).
thf(fact_5166_Im_OisCont,axiom,
! [A: complex] : ( isCont_complex_real @ im @ A ) ).
thf(fact_5167_Re_OisCont,axiom,
! [A: complex] : ( isCont_complex_real @ re @ A ) ).
thf(fact_5168_isCont__abs,axiom,
! [A: real] : ( isCont_real_real @ abs_abs_real @ A ) ).
thf(fact_5169_isCont__root__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( isCont_real_real @ ( root @ N ) @ zero_zero_real ) ) ).
thf(fact_5170_isCont__root__neg,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( isCont_real_real @ ( root @ N ) @ X ) ) ) ).
thf(fact_5171_isCont__root__pos,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( isCont_real_real @ ( root @ N ) @ X ) ) ) ).
thf(fact_5172_isCont__arccos,axiom,
! [X: real] :
( ( ord_less_real @ ( number267125858f_real @ min ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( isCont_real_real @ arccos @ X ) ) ) ).
thf(fact_5173_isCont__arcsin,axiom,
! [X: real] :
( ( ord_less_real @ ( number267125858f_real @ min ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( isCont_real_real @ arcsin @ X ) ) ) ).
thf(fact_5174_LIM__less__bound,axiom,
! [F: real > real,B: real,X: real] :
( ( ord_less_real @ B @ X )
=> ( ! [X_1: real] :
( ( member_real @ X_1 @ ( ord_gr788844697n_real @ B @ X ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X_1 ) ) )
=> ( ( isCont_real_real @ F @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).
thf(fact_5175_DERIV__inverse__function,axiom,
! [B: real,A: real,F: real > real,G: real > real,X: real,D_1: real] :
( ( deriv_real @ F @ ( G @ X ) @ D_1 )
=> ( ( D_1 != zero_zero_real )
=> ( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ( ! [Y_1: real] :
( ( ( ord_less_real @ A @ Y_1 )
& ( ord_less_real @ Y_1 @ B ) )
=> ( ( F @ ( G @ Y_1 ) )
= Y_1 ) )
=> ( ( isCont_real_real @ G @ X )
=> ( deriv_real @ G @ X @ ( inverse_inverse_real @ D_1 ) ) ) ) ) ) ) ) ).
thf(fact_5176_isCont__inv__fun,axiom,
! [G: real > real,F: real > real,X: real,D: real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
=> ( ( G @ ( F @ Z ) )
= Z ) )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
=> ( isCont_real_real @ F @ Z ) )
=> ( isCont_real_real @ G @ ( F @ X ) ) ) ) ) ).
thf(fact_5177_DERIV__isconst2,axiom,
! [X: real,F: real > real,A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A @ X_1 )
& ( ord_less_eq_real @ X_1 @ B ) )
=> ( isCont_real_real @ F @ X_1 ) )
=> ( ! [X_1: real] :
( ( ( ord_less_real @ A @ X_1 )
& ( ord_less_real @ X_1 @ B ) )
=> ( deriv_real @ F @ X_1 @ zero_zero_real ) )
=> ( ( ord_less_eq_real @ A @ X )
=> ( ( ord_less_eq_real @ X @ B )
=> ( ( F @ X )
= ( F @ A ) ) ) ) ) ) ) ).
thf(fact_5178_DERIV__isconst__end,axiom,
! [F: real > real,A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A @ X_1 )
& ( ord_less_eq_real @ X_1 @ B ) )
=> ( isCont_real_real @ F @ X_1 ) )
=> ( ! [X_1: real] :
( ( ( ord_less_real @ A @ X_1 )
& ( ord_less_real @ X_1 @ B ) )
=> ( deriv_real @ F @ X_1 @ zero_zero_real ) )
=> ( ( F @ B )
= ( F @ A ) ) ) ) ) ).
thf(fact_5179_DERIV__isconst1,axiom,
! [F: real > real,A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A @ X_1 )
& ( ord_less_eq_real @ X_1 @ B ) )
=> ( isCont_real_real @ F @ X_1 ) )
=> ( ! [X_1: real] :
( ( ( ord_less_real @ A @ X_1 )
& ( ord_less_real @ X_1 @ B ) )
=> ( deriv_real @ F @ X_1 @ zero_zero_real ) )
=> ! [X_1: real] :
( ( ( ord_less_eq_real @ A @ X_1 )
& ( ord_less_eq_real @ X_1 @ B ) )
=> ( ( F @ X_1 )
= ( F @ A ) ) ) ) ) ) ).
thf(fact_5180_isCont__inv__fun__inv,axiom,
! [G: real > real,F: real > real,X: real,D: real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
=> ( ( G @ ( F @ Z ) )
= Z ) )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
=> ( isCont_real_real @ F @ Z ) )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ! [Y_1: real] :
( ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ ( minus_minus_real @ Y_1 @ ( F @ X ) ) ) )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ Y_1 @ ( F @ X ) ) ) @ E ) )
=> ( ( F @ ( G @ Y_1 ) )
= Y_1 ) ) ) ) ) ) ).
thf(fact_5181_lemma__isCont__inj2,axiom,
! [G: real > real,F: real > real,X: real,D: real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
=> ( ( G @ ( F @ Z ) )
= Z ) )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
=> ( isCont_real_real @ F @ Z ) )
=> ? [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
& ( ord_less_real @ ( F @ Z ) @ ( F @ X ) ) ) ) ) ) ).
thf(fact_5182_lemma__isCont__inj,axiom,
! [G: real > real,F: real > real,X: real,D: real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
=> ( ( G @ ( F @ Z ) )
= Z ) )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
=> ( isCont_real_real @ F @ Z ) )
=> ? [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
& ( ord_less_real @ ( F @ X ) @ ( F @ Z ) ) ) ) ) ) ).
thf(fact_5183_SetInterval_Otransfer__nat__int__set__functions_I1_J,axiom,
! [N: nat] :
( ( ord_atMost_nat @ N )
= ( image_int_nat @ nat_1 @ ( ord_at875362053st_int @ zero_zero_int @ ( semiri1621563631at_int @ N ) ) ) ) ).
thf(fact_5184_max__number__of__Suc,axiom,
! [N: nat,V: int] :
( ( ord_max_nat @ ( suc @ N ) @ ( number_number_of_nat @ V ) )
= ( if_nat @ ( nat_neg @ ( number_number_of_int @ ( pred @ V ) ) ) @ ( suc @ N ) @ ( suc @ ( ord_max_nat @ N @ ( nat_1 @ ( number_number_of_int @ ( pred @ V ) ) ) ) ) ) ) ).
thf(fact_5185_finite__atMost,axiom,
! [K_1: nat] : ( finite_finite_nat @ ( ord_atMost_nat @ K_1 ) ) ).
thf(fact_5186_atLeast0AtMost,axiom,
! [N: nat] :
( ( ord_at238088361st_nat @ zero_zero_nat @ N )
= ( ord_atMost_nat @ N ) ) ).
thf(fact_5187_lessThan__Suc__atMost,axiom,
! [K_1: nat] :
( ( ord_lessThan_nat @ ( suc @ K_1 ) )
= ( ord_atMost_nat @ K_1 ) ) ).
thf(fact_5188_max__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_max_nat @ ( suc @ N ) @ M )
= ( nat_case_nat @ ( suc @ N )
@ ^ [M_1: nat] : ( suc @ ( ord_max_nat @ N @ M_1 ) )
@ M ) ) ).
thf(fact_5189_max__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ M @ ( suc @ N ) )
= ( nat_case_nat @ ( suc @ N )
@ ^ [M_1: nat] : ( suc @ ( ord_max_nat @ M_1 @ N ) )
@ M ) ) ).
thf(fact_5190_nat__minus__add__max,axiom,
! [N: nat,M: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
= ( ord_max_nat @ N @ M ) ) ).
thf(fact_5191_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
thf(fact_5192_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
thf(fact_5193_max__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
thf(fact_5194_card__atMost,axiom,
! [U: nat] :
( ( finite_card_nat @ ( ord_atMost_nat @ U ) )
= ( suc @ U ) ) ).
thf(fact_5195_finite__nat__iff__bounded__le,axiom,
! [S: nat > $o] :
( ( finite_finite_nat @ S )
<=> ? [K: nat] : ( ord_less_eq_nat_o @ S @ ( ord_atMost_nat @ K ) ) ) ).
%----Helper facts (15)
thf(help_If_1_1_If_000tc__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_000tc__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__Int__Oint_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_1_1_If_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_1_1_If_000tc__RealDef__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_000tc__RealDef__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__RealDef__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_1_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] :
( ( if_Pro1731782967nt_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] :
( ( if_Pro1731782967nt_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_1_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T,axiom,
! [X: produc914805421l_real,Y: produc914805421l_real] :
( ( if_Pro313124157l_real @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T,axiom,
! [X: produc914805421l_real,Y: produc914805421l_real] :
( ( if_Pro313124157l_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
ord_less_int @ ( plus_plus_int @ ( power_power_int @ s @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_int ) @ zero_zero_int ).
%------------------------------------------------------------------------------