TPTP Problem File: NUM925^4.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : NUM925^4 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 192, 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_l192 [Bla11]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.50 v7.5.0, 1.00 v7.2.0, 0.50 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, 0.80 v5.3.0
% Syntax : Number of formulae : 5617 (1975 unt; 405 typ; 0 def)
% Number of atoms : 12219 (3989 equ; 67 cnn)
% Maximal formula atoms : 14 ( 2 avg)
% Number of connectives : 50475 (1036 ~; 222 |; 783 &;42824 @)
% ( 938 <=>;4663 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 1319 (1319 >; 0 *; 0 +; 0 <<)
% Number of symbols : 393 ( 385 usr; 39 con; 0-8 aty)
% ( 0 !!; 9 ??; 0 @@+; 0 @@-)
% Number of variables : 11026 ( 263 ^;10517 !; 246 ?;11026 :)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 20:42:29
%------------------------------------------------------------------------------
%----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 (387)
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_000_Eo,type,
ord_less_eq_o: $o > $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_OatMost_000tc__RealDef__Oreal,type,
ord_atMost_real: real > real > $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_m1____,type,
m1: int ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_r____,type,
r: int ).
thf(sy_v_s1____,type,
s1: int ).
thf(sy_v_s____,type,
s: int ).
thf(sy_v_sa____,type,
sa: int ).
thf(sy_v_t____,type,
t: int ).
thf(sy_v_tn____,type,
tn: nat ).
thf(sy_v_v____,type,
v: int ).
thf(sy_v_w____,type,
w: int ).
thf(sy_v_x____,type,
x: int ).
thf(sy_v_y____,type,
y: int ).
%----Relevant facts (5196)
thf(fact_0_n1pos,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ).
thf(fact_1_t1,axiom,
ord_less_int @ one_one_int @ t ).
thf(fact_2_sum__power2__eq__zero__iff,axiom,
! [X: rat,Y: 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 )
<=> ( ( X = zero_zero_rat )
& ( Y = zero_zero_rat ) ) ) ).
thf(fact_3_sum__power2__eq__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_4_sum__power2__eq__zero__iff,axiom,
! [X: int,Y: 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 )
<=> ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
thf(fact_5_one__power2,axiom,
( ( power_power_rat @ one_one_rat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_rat ) ).
thf(fact_6_one__power2,axiom,
( ( power_881366806de_int @ one_on1684967323de_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_on1684967323de_int ) ).
thf(fact_7_one__power2,axiom,
( ( power_power_complex @ one_one_complex @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_complex ) ).
thf(fact_8_one__power2,axiom,
( ( power_2100829034umeral @ one_on1645066479umeral @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_on1645066479umeral ) ).
thf(fact_9_one__power2,axiom,
( ( power_power_real @ one_one_real @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_real ) ).
thf(fact_10_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_nat ) ).
thf(fact_11_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_int ) ).
thf(fact_12_zero__power2,axiom,
( ( power_power_rat @ zero_zero_rat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_rat ) ).
thf(fact_13_zero__power2,axiom,
( ( power_881366806de_int @ zero_z891286103de_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_z891286103de_int ) ).
thf(fact_14_zero__power2,axiom,
( ( power_power_complex @ zero_zero_complex @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_complex ) ).
thf(fact_15_zero__power2,axiom,
( ( power_2100829034umeral @ zero_z126310315umeral @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_z126310315umeral ) ).
thf(fact_16_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_real ) ).
thf(fact_17_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_nat ) ).
thf(fact_18_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_int ) ).
thf(fact_19_zero__eq__power2,axiom,
! [A_250: rat] :
( ( ( power_power_rat @ A_250 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_rat )
<=> ( A_250 = zero_zero_rat ) ) ).
thf(fact_20_zero__eq__power2,axiom,
! [A_250: complex] :
( ( ( power_power_complex @ A_250 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_complex )
<=> ( A_250 = zero_zero_complex ) ) ).
thf(fact_21_zero__eq__power2,axiom,
! [A_250: real] :
( ( ( power_power_real @ A_250 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_real )
<=> ( A_250 = zero_zero_real ) ) ).
thf(fact_22_zero__eq__power2,axiom,
! [A_250: int] :
( ( ( power_power_int @ A_250 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_zero_int )
<=> ( A_250 = zero_zero_int ) ) ).
thf(fact_23_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_24_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_25_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_26_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_27_add__special_I3_J,axiom,
! [V_25: int] :
( ( plus_plus_rat @ ( number_number_of_rat @ V_25 ) @ one_one_rat )
= ( number_number_of_rat @ ( plus_plus_int @ V_25 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_28_add__special_I3_J,axiom,
! [V_25: int] :
( ( plus_plus_complex @ ( number528085621omplex @ V_25 ) @ one_one_complex )
= ( number528085621omplex @ ( plus_plus_int @ V_25 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_29_add__special_I3_J,axiom,
! [V_25: int] :
( ( plus_plus_real @ ( number267125858f_real @ V_25 ) @ one_one_real )
= ( number267125858f_real @ ( plus_plus_int @ V_25 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_30_add__special_I3_J,axiom,
! [V_25: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_25 ) @ one_one_int )
= ( number_number_of_int @ ( plus_plus_int @ V_25 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_31_one__add__one__is__two,axiom,
( ( plus_plus_rat @ one_one_rat @ one_one_rat )
= ( number_number_of_rat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_32_one__add__one__is__two,axiom,
( ( plus_plus_complex @ one_one_complex @ one_one_complex )
= ( number528085621omplex @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_33_one__add__one__is__two,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_34_one__add__one__is__two,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_35_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_36_semiring__one__add__one__is__two,axiom,
( ( plus_plus_complex @ one_one_complex @ one_one_complex )
= ( number528085621omplex @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_37_semiring__one__add__one__is__two,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_38_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_39_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_40_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_41_power__0__left__number__of,axiom,
! [W_19: int] :
( ( ( ( number_number_of_nat @ W_19 )
= zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ ( number_number_of_nat @ W_19 ) )
= one_one_rat ) )
& ( ( ( number_number_of_nat @ W_19 )
!= zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ ( number_number_of_nat @ W_19 ) )
= zero_zero_rat ) ) ) ).
thf(fact_42_power__0__left__number__of,axiom,
! [W_19: int] :
( ( ( ( number_number_of_nat @ W_19 )
= zero_zero_nat )
=> ( ( power_881366806de_int @ zero_z891286103de_int @ ( number_number_of_nat @ W_19 ) )
= one_on1684967323de_int ) )
& ( ( ( number_number_of_nat @ W_19 )
!= zero_zero_nat )
=> ( ( power_881366806de_int @ zero_z891286103de_int @ ( number_number_of_nat @ W_19 ) )
= zero_z891286103de_int ) ) ) ).
thf(fact_43_power__0__left__number__of,axiom,
! [W_19: int] :
( ( ( ( number_number_of_nat @ W_19 )
= zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ ( number_number_of_nat @ W_19 ) )
= one_one_complex ) )
& ( ( ( number_number_of_nat @ W_19 )
!= zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ ( number_number_of_nat @ W_19 ) )
= zero_zero_complex ) ) ) ).
thf(fact_44_power__0__left__number__of,axiom,
! [W_19: int] :
( ( ( ( number_number_of_nat @ W_19 )
= zero_zero_nat )
=> ( ( power_2100829034umeral @ zero_z126310315umeral @ ( number_number_of_nat @ W_19 ) )
= one_on1645066479umeral ) )
& ( ( ( number_number_of_nat @ W_19 )
!= zero_zero_nat )
=> ( ( power_2100829034umeral @ zero_z126310315umeral @ ( number_number_of_nat @ W_19 ) )
= zero_z126310315umeral ) ) ) ).
thf(fact_45_power__0__left__number__of,axiom,
! [W_19: int] :
( ( ( ( number_number_of_nat @ W_19 )
= zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ ( number_number_of_nat @ W_19 ) )
= one_one_real ) )
& ( ( ( number_number_of_nat @ W_19 )
!= zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ ( number_number_of_nat @ W_19 ) )
= zero_zero_real ) ) ) ).
thf(fact_46_power__0__left__number__of,axiom,
! [W_19: int] :
( ( ( ( number_number_of_nat @ W_19 )
= zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ W_19 ) )
= one_one_nat ) )
& ( ( ( number_number_of_nat @ W_19 )
!= zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ ( number_number_of_nat @ W_19 ) )
= zero_zero_nat ) ) ) ).
thf(fact_47_power__0__left__number__of,axiom,
! [W_19: int] :
( ( ( ( number_number_of_nat @ W_19 )
= zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ W_19 ) )
= one_one_int ) )
& ( ( ( number_number_of_nat @ W_19 )
!= zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ ( number_number_of_nat @ W_19 ) )
= zero_zero_int ) ) ) ).
thf(fact_48_semiring__norm_I110_J,axiom,
( one_one_rat
= ( number_number_of_rat @ ( bit1 @ pls ) ) ) ).
thf(fact_49_semiring__norm_I110_J,axiom,
( one_one_complex
= ( number528085621omplex @ ( bit1 @ pls ) ) ) ).
thf(fact_50_semiring__norm_I110_J,axiom,
( one_one_real
= ( number267125858f_real @ ( bit1 @ pls ) ) ) ).
thf(fact_51_semiring__norm_I110_J,axiom,
( one_one_int
= ( number_number_of_int @ ( bit1 @ pls ) ) ) ).
thf(fact_52_numeral__1__eq__1,axiom,
( ( number_number_of_rat @ ( bit1 @ pls ) )
= one_one_rat ) ).
thf(fact_53_numeral__1__eq__1,axiom,
( ( number528085621omplex @ ( bit1 @ pls ) )
= one_one_complex ) ).
thf(fact_54_numeral__1__eq__1,axiom,
( ( number267125858f_real @ ( bit1 @ pls ) )
= one_one_real ) ).
thf(fact_55_numeral__1__eq__1,axiom,
( ( number_number_of_int @ ( bit1 @ pls ) )
= one_one_int ) ).
thf(fact_56_n0,axiom,
ord_less_nat @ zero_zero_nat @ n ).
thf(fact_57_zless__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
thf(fact_58_less__number__of__int__code,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( number_number_of_int @ K ) @ ( number_number_of_int @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_59_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_60_less__number__of,axiom,
! [X_100: int,Y_82: int] :
( ( ord_less_rat @ ( number_number_of_rat @ X_100 ) @ ( number_number_of_rat @ Y_82 ) )
<=> ( ord_less_int @ X_100 @ Y_82 ) ) ).
thf(fact_61_less__number__of,axiom,
! [X_100: int,Y_82: int] :
( ( ord_less_real @ ( number267125858f_real @ X_100 ) @ ( number267125858f_real @ Y_82 ) )
<=> ( ord_less_int @ X_100 @ Y_82 ) ) ).
thf(fact_62_less__number__of,axiom,
! [X_100: int,Y_82: int] :
( ( ord_less_int @ ( number_number_of_int @ X_100 ) @ ( number_number_of_int @ Y_82 ) )
<=> ( ord_less_int @ X_100 @ Y_82 ) ) ).
thf(fact_63_zero__is__num__zero,axiom,
( zero_zero_int
= ( number_number_of_int @ pls ) ) ).
thf(fact_64_zpower__int,axiom,
! [M: nat,N: nat] :
( ( power_power_int @ ( semiri1621563631at_int @ M ) @ N )
= ( semiri1621563631at_int @ ( power_power_nat @ M @ N ) ) ) ).
thf(fact_65_int__power,axiom,
! [M: nat,N: nat] :
( ( semiri1621563631at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri1621563631at_int @ M ) @ N ) ) ).
thf(fact_66_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_67_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_68_int__1,axiom,
( ( semiri1621563631at_int @ one_one_nat )
= one_one_int ) ).
thf(fact_69_nat__number__of__Pls,axiom,
( ( number_number_of_nat @ pls )
= zero_zero_nat ) ).
thf(fact_70_semiring__norm_I113_J,axiom,
( zero_zero_nat
= ( number_number_of_nat @ pls ) ) ).
thf(fact_71_int__eq__0__conv,axiom,
! [N: nat] :
( ( ( semiri1621563631at_int @ N )
= zero_zero_int )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_72_int__0,axiom,
( ( semiri1621563631at_int @ zero_zero_nat )
= zero_zero_int ) ).
thf(fact_73_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_74_less__int__code_I16_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit1 @ K1 ) @ ( bit1 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_75_rel__simps_I17_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_76_rel__simps_I2_J,axiom,
~ ( ord_less_int @ pls @ pls ) ).
thf(fact_77_less__int__code_I13_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit0 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_78_rel__simps_I14_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_79_zadd__strict__right__mono,axiom,
! [K: int,I: int,J: int] :
( ( ord_less_int @ I @ J )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ K ) ) ) ).
thf(fact_80_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_81_one__is__num__one,axiom,
( one_one_int
= ( number_number_of_int @ ( bit1 @ pls ) ) ) ).
thf(fact_82_nat__numeral__1__eq__1,axiom,
( ( number_number_of_nat @ ( bit1 @ pls ) )
= one_one_nat ) ).
thf(fact_83_Numeral1__eq1__nat,axiom,
( one_one_nat
= ( number_number_of_nat @ ( bit1 @ pls ) ) ) ).
thf(fact_84_eq__number__of,axiom,
! [X_99: int,Y_81: int] :
( ( ( number_number_of_rat @ X_99 )
= ( number_number_of_rat @ Y_81 ) )
<=> ( X_99 = Y_81 ) ) ).
thf(fact_85_eq__number__of,axiom,
! [X_99: int,Y_81: int] :
( ( ( number528085621omplex @ X_99 )
= ( number528085621omplex @ Y_81 ) )
<=> ( X_99 = Y_81 ) ) ).
thf(fact_86_eq__number__of,axiom,
! [X_99: int,Y_81: int] :
( ( ( number267125858f_real @ X_99 )
= ( number267125858f_real @ Y_81 ) )
<=> ( X_99 = Y_81 ) ) ).
thf(fact_87_eq__number__of,axiom,
! [X_99: int,Y_81: int] :
( ( ( number_number_of_int @ X_99 )
= ( number_number_of_int @ Y_81 ) )
<=> ( X_99 = Y_81 ) ) ).
thf(fact_88_number__of__reorient,axiom,
! [W_18: int,X_98: rat] :
( ( ( number_number_of_rat @ W_18 )
= X_98 )
<=> ( X_98
= ( number_number_of_rat @ W_18 ) ) ) ).
thf(fact_89_number__of__reorient,axiom,
! [W_18: int,X_98: quickcheck_code_int] :
( ( ( number1226105091de_int @ W_18 )
= X_98 )
<=> ( X_98
= ( number1226105091de_int @ W_18 ) ) ) ).
thf(fact_90_number__of__reorient,axiom,
! [W_18: int,X_98: complex] :
( ( ( number528085621omplex @ W_18 )
= X_98 )
<=> ( X_98
= ( number528085621omplex @ W_18 ) ) ) ).
thf(fact_91_number__of__reorient,axiom,
! [W_18: int,X_98: code_code_numeral] :
( ( ( number1443263063umeral @ W_18 )
= X_98 )
<=> ( X_98
= ( number1443263063umeral @ W_18 ) ) ) ).
thf(fact_92_number__of__reorient,axiom,
! [W_18: int,X_98: real] :
( ( ( number267125858f_real @ W_18 )
= X_98 )
<=> ( X_98
= ( number267125858f_real @ W_18 ) ) ) ).
thf(fact_93_number__of__reorient,axiom,
! [W_18: int,X_98: int] :
( ( ( number_number_of_int @ W_18 )
= X_98 )
<=> ( X_98
= ( number_number_of_int @ W_18 ) ) ) ).
thf(fact_94_number__of__reorient,axiom,
! [W_18: int,X_98: nat] :
( ( ( number_number_of_nat @ W_18 )
= X_98 )
<=> ( X_98
= ( number_number_of_nat @ W_18 ) ) ) ).
thf(fact_95_rel__simps_I51_J,axiom,
! [K: int,L: int] :
( ( ( bit1 @ K )
= ( bit1 @ L ) )
<=> ( K = L ) ) ).
thf(fact_96_rel__simps_I48_J,axiom,
! [K: int,L: int] :
( ( ( bit0 @ K )
= ( bit0 @ L ) )
<=> ( K = L ) ) ).
thf(fact_97_even__less__0__iff,axiom,
! [A_249: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A_249 @ A_249 ) @ zero_zero_rat )
<=> ( ord_less_rat @ A_249 @ zero_zero_rat ) ) ).
thf(fact_98_even__less__0__iff,axiom,
! [A_249: real] :
( ( ord_less_real @ ( plus_plus_real @ A_249 @ A_249 ) @ zero_zero_real )
<=> ( ord_less_real @ A_249 @ zero_zero_real ) ) ).
thf(fact_99_even__less__0__iff,axiom,
! [A_249: int] :
( ( ord_less_int @ ( plus_plus_int @ A_249 @ A_249 ) @ zero_zero_int )
<=> ( ord_less_int @ A_249 @ zero_zero_int ) ) ).
thf(fact_100_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_101_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_102_zadd__commute,axiom,
! [Z_1: int,W: int] :
( ( plus_plus_int @ Z_1 @ W )
= ( plus_plus_int @ W @ Z_1 ) ) ).
thf(fact_103_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1621563631at_int @ M )
= ( semiri1621563631at_int @ N ) )
<=> ( M = N ) ) ).
thf(fact_104_less__special_I3_J,axiom,
! [X_97: int] :
( ( ord_less_rat @ ( number_number_of_rat @ X_97 ) @ zero_zero_rat )
<=> ( ord_less_int @ X_97 @ pls ) ) ).
thf(fact_105_less__special_I3_J,axiom,
! [X_97: int] :
( ( ord_less_real @ ( number267125858f_real @ X_97 ) @ zero_zero_real )
<=> ( ord_less_int @ X_97 @ pls ) ) ).
thf(fact_106_less__special_I3_J,axiom,
! [X_97: int] :
( ( ord_less_int @ ( number_number_of_int @ X_97 ) @ zero_zero_int )
<=> ( ord_less_int @ X_97 @ pls ) ) ).
thf(fact_107_less__special_I1_J,axiom,
! [Y_80: int] :
( ( ord_less_rat @ zero_zero_rat @ ( number_number_of_rat @ Y_80 ) )
<=> ( ord_less_int @ pls @ Y_80 ) ) ).
thf(fact_108_less__special_I1_J,axiom,
! [Y_80: int] :
( ( ord_less_real @ zero_zero_real @ ( number267125858f_real @ Y_80 ) )
<=> ( ord_less_int @ pls @ Y_80 ) ) ).
thf(fact_109_less__special_I1_J,axiom,
! [Y_80: int] :
( ( ord_less_int @ zero_zero_int @ ( number_number_of_int @ Y_80 ) )
<=> ( ord_less_int @ pls @ Y_80 ) ) ).
thf(fact_110_rel__simps_I12_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit1 @ K ) @ pls )
<=> ( ord_less_int @ K @ pls ) ) ).
thf(fact_111_less__int__code_I15_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit1 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_112_rel__simps_I16_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_113_rel__simps_I10_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit0 @ K ) @ pls )
<=> ( ord_less_int @ K @ pls ) ) ).
thf(fact_114_rel__simps_I4_J,axiom,
! [K: int] :
( ( ord_less_int @ pls @ ( bit0 @ K ) )
<=> ( ord_less_int @ pls @ K ) ) ).
thf(fact_115_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_116_bin__less__0__simps_I1_J,axiom,
~ ( ord_less_int @ pls @ zero_zero_int ) ).
thf(fact_117_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_118_int__0__less__1,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
thf(fact_119_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_120_int__less__0__conv,axiom,
! [K: nat] :
~ ( ord_less_int @ ( semiri1621563631at_int @ K ) @ zero_zero_int ) ).
thf(fact_121_less__special_I4_J,axiom,
! [X_96: int] :
( ( ord_less_rat @ ( number_number_of_rat @ X_96 ) @ one_one_rat )
<=> ( ord_less_int @ X_96 @ ( bit1 @ pls ) ) ) ).
thf(fact_122_less__special_I4_J,axiom,
! [X_96: int] :
( ( ord_less_real @ ( number267125858f_real @ X_96 ) @ one_one_real )
<=> ( ord_less_int @ X_96 @ ( bit1 @ pls ) ) ) ).
thf(fact_123_less__special_I4_J,axiom,
! [X_96: int] :
( ( ord_less_int @ ( number_number_of_int @ X_96 ) @ one_one_int )
<=> ( ord_less_int @ X_96 @ ( bit1 @ pls ) ) ) ).
thf(fact_124_less__special_I2_J,axiom,
! [Y_79: int] :
( ( ord_less_rat @ one_one_rat @ ( number_number_of_rat @ Y_79 ) )
<=> ( ord_less_int @ ( bit1 @ pls ) @ Y_79 ) ) ).
thf(fact_125_less__special_I2_J,axiom,
! [Y_79: int] :
( ( ord_less_real @ one_one_real @ ( number267125858f_real @ Y_79 ) )
<=> ( ord_less_int @ ( bit1 @ pls ) @ Y_79 ) ) ).
thf(fact_126_less__special_I2_J,axiom,
! [Y_79: int] :
( ( ord_less_int @ one_one_int @ ( number_number_of_int @ Y_79 ) )
<=> ( ord_less_int @ ( bit1 @ pls ) @ Y_79 ) ) ).
thf(fact_127_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_128_double__eq__0__iff,axiom,
! [A_248: rat] :
( ( ( plus_plus_rat @ A_248 @ A_248 )
= zero_zero_rat )
<=> ( A_248 = zero_zero_rat ) ) ).
thf(fact_129_double__eq__0__iff,axiom,
! [A_248: real] :
( ( ( plus_plus_real @ A_248 @ A_248 )
= zero_zero_real )
<=> ( A_248 = zero_zero_real ) ) ).
thf(fact_130_double__eq__0__iff,axiom,
! [A_248: int] :
( ( ( plus_plus_int @ A_248 @ A_248 )
= zero_zero_int )
<=> ( A_248 = zero_zero_int ) ) ).
thf(fact_131_rel__simps_I46_J,axiom,
! [K: int] :
( ( bit1 @ K )
!= pls ) ).
thf(fact_132_rel__simps_I39_J,axiom,
! [L: int] :
( pls
!= ( bit1 @ L ) ) ).
thf(fact_133_rel__simps_I50_J,axiom,
! [K: int,L: int] :
( ( bit1 @ K )
!= ( bit0 @ L ) ) ).
thf(fact_134_rel__simps_I49_J,axiom,
! [K: int,L: int] :
( ( bit0 @ K )
!= ( bit1 @ L ) ) ).
thf(fact_135_rel__simps_I44_J,axiom,
! [K: int] :
( ( ( bit0 @ K )
= pls )
<=> ( K = pls ) ) ).
thf(fact_136_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls
= ( bit0 @ L ) )
<=> ( pls = L ) ) ).
thf(fact_137_Bit0__Pls,axiom,
( ( bit0 @ pls )
= pls ) ).
thf(fact_138_mem__def,axiom,
! [X_95: produc975137661_int_o,A_247: produc975137661_int_o > $o] :
( ( member1329254762_int_o @ X_95 @ A_247 )
<=> ( A_247 @ X_95 ) ) ).
thf(fact_139_mem__def,axiom,
! [X_95: produc1137372701nt_int,A_247: produc1137372701nt_int > $o] :
( ( member2143287562nt_int @ X_95 @ A_247 )
<=> ( A_247 @ X_95 ) ) ).
thf(fact_140_mem__def,axiom,
! [X_95: real,A_247: real > $o] :
( ( member_real @ X_95 @ A_247 )
<=> ( A_247 @ X_95 ) ) ).
thf(fact_141_mem__def,axiom,
! [X_95: nat,A_247: nat > $o] :
( ( member_nat @ X_95 @ A_247 )
<=> ( A_247 @ X_95 ) ) ).
thf(fact_142_mem__def,axiom,
! [X_95: int > $o,A_247: ( int > $o ) > $o] :
( ( member_int_o @ X_95 @ A_247 )
<=> ( A_247 @ X_95 ) ) ).
thf(fact_143_mem__def,axiom,
! [X_95: produc1322466333at_nat,A_247: produc1322466333at_nat > $o] :
( ( member180897546at_nat @ X_95 @ A_247 )
<=> ( A_247 @ X_95 ) ) ).
thf(fact_144_mem__def,axiom,
! [X_95: int,A_247: int > $o] :
( ( member_int @ X_95 @ A_247 )
<=> ( A_247 @ X_95 ) ) ).
thf(fact_145_Collect__def,axiom,
! [P_9: product_prod_nat_nat > $o] :
( ( collec1979865426at_nat @ P_9 )
= P_9 ) ).
thf(fact_146_Collect__def,axiom,
! [P_9: real > $o] :
( ( collect_real @ P_9 )
= P_9 ) ).
thf(fact_147_Collect__def,axiom,
! [P_9: product_prod_int_int > $o] :
( ( collec1347809874nt_int @ P_9 )
= P_9 ) ).
thf(fact_148_Collect__def,axiom,
! [P_9: produc1137372701nt_int > $o] :
( ( collec50511176nt_int @ P_9 )
= P_9 ) ).
thf(fact_149_Collect__def,axiom,
! [P_9: nat > $o] :
( ( collect_nat @ P_9 )
= P_9 ) ).
thf(fact_150_Collect__def,axiom,
! [P_9: int > $o] :
( ( collect_int @ P_9 )
= P_9 ) ).
thf(fact_151_Pls__def,axiom,
pls = zero_zero_int ).
thf(fact_152_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
thf(fact_153_add__Pls__right,axiom,
! [K: int] :
( ( plus_plus_int @ K @ pls )
= K ) ).
thf(fact_154_add__Pls,axiom,
! [K: int] :
( ( plus_plus_int @ pls @ K )
= K ) ).
thf(fact_155_add__Bit0__Bit0,axiom,
! [K: int,L: int] :
( ( plus_plus_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
= ( bit0 @ ( plus_plus_int @ K @ L ) ) ) ).
thf(fact_156_Bit0__def,axiom,
! [K: int] :
( ( bit0 @ K )
= ( plus_plus_int @ K @ K ) ) ).
thf(fact_157_zadd__0__right,axiom,
! [Z_1: int] :
( ( plus_plus_int @ Z_1 @ zero_zero_int )
= Z_1 ) ).
thf(fact_158_zadd__0,axiom,
! [Z_1: int] :
( ( plus_plus_int @ zero_zero_int @ Z_1 )
= Z_1 ) ).
thf(fact_159_semiring__numeral__0__eq__0,axiom,
( ( number_number_of_rat @ pls )
= zero_zero_rat ) ).
thf(fact_160_semiring__numeral__0__eq__0,axiom,
( ( number528085621omplex @ pls )
= zero_zero_complex ) ).
thf(fact_161_semiring__numeral__0__eq__0,axiom,
( ( number267125858f_real @ pls )
= zero_zero_real ) ).
thf(fact_162_semiring__numeral__0__eq__0,axiom,
( ( number_number_of_nat @ pls )
= zero_zero_nat ) ).
thf(fact_163_semiring__numeral__0__eq__0,axiom,
( ( number_number_of_int @ pls )
= zero_zero_int ) ).
thf(fact_164_number__of__Pls,axiom,
( ( number_number_of_rat @ pls )
= zero_zero_rat ) ).
thf(fact_165_number__of__Pls,axiom,
( ( number528085621omplex @ pls )
= zero_zero_complex ) ).
thf(fact_166_number__of__Pls,axiom,
( ( number267125858f_real @ pls )
= zero_zero_real ) ).
thf(fact_167_number__of__Pls,axiom,
( ( number_number_of_int @ pls )
= zero_zero_int ) ).
thf(fact_168_semiring__norm_I112_J,axiom,
( zero_zero_rat
= ( number_number_of_rat @ pls ) ) ).
thf(fact_169_semiring__norm_I112_J,axiom,
( zero_zero_complex
= ( number528085621omplex @ pls ) ) ).
thf(fact_170_semiring__norm_I112_J,axiom,
( zero_zero_real
= ( number267125858f_real @ pls ) ) ).
thf(fact_171_semiring__norm_I112_J,axiom,
( zero_zero_int
= ( number_number_of_int @ pls ) ) ).
thf(fact_172_add__numeral__0,axiom,
! [A_246: rat] :
( ( plus_plus_rat @ ( number_number_of_rat @ pls ) @ A_246 )
= A_246 ) ).
thf(fact_173_add__numeral__0,axiom,
! [A_246: complex] :
( ( plus_plus_complex @ ( number528085621omplex @ pls ) @ A_246 )
= A_246 ) ).
thf(fact_174_add__numeral__0,axiom,
! [A_246: real] :
( ( plus_plus_real @ ( number267125858f_real @ pls ) @ A_246 )
= A_246 ) ).
thf(fact_175_add__numeral__0,axiom,
! [A_246: int] :
( ( plus_plus_int @ ( number_number_of_int @ pls ) @ A_246 )
= A_246 ) ).
thf(fact_176_add__numeral__0__right,axiom,
! [A_245: rat] :
( ( plus_plus_rat @ A_245 @ ( number_number_of_rat @ pls ) )
= A_245 ) ).
thf(fact_177_add__numeral__0__right,axiom,
! [A_245: complex] :
( ( plus_plus_complex @ A_245 @ ( number528085621omplex @ pls ) )
= A_245 ) ).
thf(fact_178_add__numeral__0__right,axiom,
! [A_245: real] :
( ( plus_plus_real @ A_245 @ ( number267125858f_real @ pls ) )
= A_245 ) ).
thf(fact_179_add__numeral__0__right,axiom,
! [A_245: int] :
( ( plus_plus_int @ A_245 @ ( number_number_of_int @ pls ) )
= A_245 ) ).
thf(fact_180_power__eq__0__iff__number__of,axiom,
! [A_244: rat,W_17: int] :
( ( ( power_power_rat @ A_244 @ ( number_number_of_nat @ W_17 ) )
= zero_zero_rat )
<=> ( ( A_244 = zero_zero_rat )
& ( ( number_number_of_nat @ W_17 )
!= zero_zero_nat ) ) ) ).
thf(fact_181_power__eq__0__iff__number__of,axiom,
! [A_244: quickcheck_code_int,W_17: int] :
( ( ( power_881366806de_int @ A_244 @ ( number_number_of_nat @ W_17 ) )
= zero_z891286103de_int )
<=> ( ( A_244 = zero_z891286103de_int )
& ( ( number_number_of_nat @ W_17 )
!= zero_zero_nat ) ) ) ).
thf(fact_182_power__eq__0__iff__number__of,axiom,
! [A_244: complex,W_17: int] :
( ( ( power_power_complex @ A_244 @ ( number_number_of_nat @ W_17 ) )
= zero_zero_complex )
<=> ( ( A_244 = zero_zero_complex )
& ( ( number_number_of_nat @ W_17 )
!= zero_zero_nat ) ) ) ).
thf(fact_183_power__eq__0__iff__number__of,axiom,
! [A_244: code_code_numeral,W_17: int] :
( ( ( power_2100829034umeral @ A_244 @ ( number_number_of_nat @ W_17 ) )
= zero_z126310315umeral )
<=> ( ( A_244 = zero_z126310315umeral )
& ( ( number_number_of_nat @ W_17 )
!= zero_zero_nat ) ) ) ).
thf(fact_184_power__eq__0__iff__number__of,axiom,
! [A_244: real,W_17: int] :
( ( ( power_power_real @ A_244 @ ( number_number_of_nat @ W_17 ) )
= zero_zero_real )
<=> ( ( A_244 = zero_zero_real )
& ( ( number_number_of_nat @ W_17 )
!= zero_zero_nat ) ) ) ).
thf(fact_185_power__eq__0__iff__number__of,axiom,
! [A_244: nat,W_17: int] :
( ( ( power_power_nat @ A_244 @ ( number_number_of_nat @ W_17 ) )
= zero_zero_nat )
<=> ( ( A_244 = zero_zero_nat )
& ( ( number_number_of_nat @ W_17 )
!= zero_zero_nat ) ) ) ).
thf(fact_186_power__eq__0__iff__number__of,axiom,
! [A_244: int,W_17: int] :
( ( ( power_power_int @ A_244 @ ( number_number_of_nat @ W_17 ) )
= zero_zero_int )
<=> ( ( A_244 = zero_zero_int )
& ( ( number_number_of_nat @ W_17 )
!= zero_zero_nat ) ) ) ).
thf(fact_187_add__number__of__left,axiom,
! [V_24: int,W_16: int,Z_21: rat] :
( ( plus_plus_rat @ ( number_number_of_rat @ V_24 ) @ ( plus_plus_rat @ ( number_number_of_rat @ W_16 ) @ Z_21 ) )
= ( plus_plus_rat @ ( number_number_of_rat @ ( plus_plus_int @ V_24 @ W_16 ) ) @ Z_21 ) ) ).
thf(fact_188_add__number__of__left,axiom,
! [V_24: int,W_16: int,Z_21: complex] :
( ( plus_plus_complex @ ( number528085621omplex @ V_24 ) @ ( plus_plus_complex @ ( number528085621omplex @ W_16 ) @ Z_21 ) )
= ( plus_plus_complex @ ( number528085621omplex @ ( plus_plus_int @ V_24 @ W_16 ) ) @ Z_21 ) ) ).
thf(fact_189_add__number__of__left,axiom,
! [V_24: int,W_16: int,Z_21: real] :
( ( plus_plus_real @ ( number267125858f_real @ V_24 ) @ ( plus_plus_real @ ( number267125858f_real @ W_16 ) @ Z_21 ) )
= ( plus_plus_real @ ( number267125858f_real @ ( plus_plus_int @ V_24 @ W_16 ) ) @ Z_21 ) ) ).
thf(fact_190_add__number__of__left,axiom,
! [V_24: int,W_16: int,Z_21: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_24 ) @ ( plus_plus_int @ ( number_number_of_int @ W_16 ) @ Z_21 ) )
= ( plus_plus_int @ ( number_number_of_int @ ( plus_plus_int @ V_24 @ W_16 ) ) @ Z_21 ) ) ).
thf(fact_191_add__number__of__eq,axiom,
! [V_23: int,W_15: int] :
( ( plus_plus_rat @ ( number_number_of_rat @ V_23 ) @ ( number_number_of_rat @ W_15 ) )
= ( number_number_of_rat @ ( plus_plus_int @ V_23 @ W_15 ) ) ) ).
thf(fact_192_add__number__of__eq,axiom,
! [V_23: int,W_15: int] :
( ( plus_plus_complex @ ( number528085621omplex @ V_23 ) @ ( number528085621omplex @ W_15 ) )
= ( number528085621omplex @ ( plus_plus_int @ V_23 @ W_15 ) ) ) ).
thf(fact_193_add__number__of__eq,axiom,
! [V_23: int,W_15: int] :
( ( plus_plus_real @ ( number267125858f_real @ V_23 ) @ ( number267125858f_real @ W_15 ) )
= ( number267125858f_real @ ( plus_plus_int @ V_23 @ W_15 ) ) ) ).
thf(fact_194_add__number__of__eq,axiom,
! [V_23: int,W_15: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_23 ) @ ( number_number_of_int @ W_15 ) )
= ( number_number_of_int @ ( plus_plus_int @ V_23 @ W_15 ) ) ) ).
thf(fact_195_number__of__add,axiom,
! [V_22: int,W_14: int] :
( ( number_number_of_rat @ ( plus_plus_int @ V_22 @ W_14 ) )
= ( plus_plus_rat @ ( number_number_of_rat @ V_22 ) @ ( number_number_of_rat @ W_14 ) ) ) ).
thf(fact_196_number__of__add,axiom,
! [V_22: int,W_14: int] :
( ( number528085621omplex @ ( plus_plus_int @ V_22 @ W_14 ) )
= ( plus_plus_complex @ ( number528085621omplex @ V_22 ) @ ( number528085621omplex @ W_14 ) ) ) ).
thf(fact_197_number__of__add,axiom,
! [V_22: int,W_14: int] :
( ( number267125858f_real @ ( plus_plus_int @ V_22 @ W_14 ) )
= ( plus_plus_real @ ( number267125858f_real @ V_22 ) @ ( number267125858f_real @ W_14 ) ) ) ).
thf(fact_198_number__of__add,axiom,
! [V_22: int,W_14: int] :
( ( number_number_of_int @ ( plus_plus_int @ V_22 @ W_14 ) )
= ( plus_plus_int @ ( number_number_of_int @ V_22 ) @ ( number_number_of_int @ W_14 ) ) ) ).
thf(fact_199_add__Bit1__Bit0,axiom,
! [K: int,L: int] :
( ( plus_plus_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
= ( bit1 @ ( plus_plus_int @ K @ L ) ) ) ).
thf(fact_200_add__Bit0__Bit1,axiom,
! [K: int,L: int] :
( ( plus_plus_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
= ( bit1 @ ( plus_plus_int @ K @ L ) ) ) ).
thf(fact_201_Bit1__def,axiom,
! [K: int] :
( ( bit1 @ K )
= ( plus_plus_int @ ( plus_plus_int @ one_one_int @ K ) @ K ) ) ).
thf(fact_202_odd__nonzero,axiom,
! [Z_1: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z_1 ) @ Z_1 )
!= zero_zero_int ) ).
thf(fact_203_number__of__int,axiom,
! [N_63: nat] :
( ( number_number_of_rat @ ( semiri1621563631at_int @ N_63 ) )
= ( semiri151668891at_rat @ N_63 ) ) ).
thf(fact_204_number__of__int,axiom,
! [N_63: nat] :
( ( number528085621omplex @ ( semiri1621563631at_int @ N_63 ) )
= ( semiri2020571505omplex @ N_63 ) ) ).
thf(fact_205_number__of__int,axiom,
! [N_63: nat] :
( ( number267125858f_real @ ( semiri1621563631at_int @ N_63 ) )
= ( semiri132038758t_real @ N_63 ) ) ).
thf(fact_206_number__of__int,axiom,
! [N_63: nat] :
( ( number_number_of_int @ ( semiri1621563631at_int @ N_63 ) )
= ( semiri1621563631at_int @ N_63 ) ) ).
thf(fact_207_number__of__int,axiom,
! [N_63: nat] :
( ( number_number_of_nat @ ( semiri1621563631at_int @ N_63 ) )
= ( semiri984289939at_nat @ N_63 ) ) ).
thf(fact_208_zero__less__power2,axiom,
! [A_243: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A_243 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( A_243 != zero_zero_rat ) ) ).
thf(fact_209_zero__less__power2,axiom,
! [A_243: real] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A_243 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( A_243 != zero_zero_real ) ) ).
thf(fact_210_zero__less__power2,axiom,
! [A_243: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A_243 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( A_243 != zero_zero_int ) ) ).
thf(fact_211_power2__less__0,axiom,
! [A_242: rat] :
~ ( ord_less_rat @ ( power_power_rat @ A_242 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_rat ) ).
thf(fact_212_power2__less__0,axiom,
! [A_242: real] :
~ ( ord_less_real @ ( power_power_real @ A_242 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_real ) ).
thf(fact_213_power2__less__0,axiom,
! [A_242: int] :
~ ( ord_less_int @ ( power_power_int @ A_242 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_int ) ).
thf(fact_214_sum__power2__gt__zero__iff,axiom,
! [X_94: rat,Y_78: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X_94 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_rat @ Y_78 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
<=> ( ( X_94 != zero_zero_rat )
| ( Y_78 != zero_zero_rat ) ) ) ).
thf(fact_215_sum__power2__gt__zero__iff,axiom,
! [X_94: real,Y_78: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X_94 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_78 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
<=> ( ( X_94 != zero_zero_real )
| ( Y_78 != zero_zero_real ) ) ) ).
thf(fact_216_sum__power2__gt__zero__iff,axiom,
! [X_94: int,Y_78: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X_94 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_78 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
<=> ( ( X_94 != zero_zero_int )
| ( Y_78 != zero_zero_int ) ) ) ).
thf(fact_217_not__sum__power2__lt__zero,axiom,
! [X_93: rat,Y_77: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X_93 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_rat @ Y_77 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_rat ) ).
thf(fact_218_not__sum__power2__lt__zero,axiom,
! [X_93: real,Y_77: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X_93 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_77 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_real ) ).
thf(fact_219_not__sum__power2__lt__zero,axiom,
! [X_93: int,Y_77: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X_93 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_77 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_int ) ).
thf(fact_220_number__of__Bit0,axiom,
! [W_13: int] :
( ( number_number_of_rat @ ( bit0 @ W_13 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ zero_zero_rat @ ( number_number_of_rat @ W_13 ) ) @ ( number_number_of_rat @ W_13 ) ) ) ).
thf(fact_221_number__of__Bit0,axiom,
! [W_13: int] :
( ( number528085621omplex @ ( bit0 @ W_13 ) )
= ( plus_plus_complex @ ( plus_plus_complex @ zero_zero_complex @ ( number528085621omplex @ W_13 ) ) @ ( number528085621omplex @ W_13 ) ) ) ).
thf(fact_222_number__of__Bit0,axiom,
! [W_13: int] :
( ( number267125858f_real @ ( bit0 @ W_13 ) )
= ( plus_plus_real @ ( plus_plus_real @ zero_zero_real @ ( number267125858f_real @ W_13 ) ) @ ( number267125858f_real @ W_13 ) ) ) ).
thf(fact_223_number__of__Bit0,axiom,
! [W_13: int] :
( ( number_number_of_int @ ( bit0 @ W_13 ) )
= ( plus_plus_int @ ( plus_plus_int @ zero_zero_int @ ( number_number_of_int @ W_13 ) ) @ ( number_number_of_int @ W_13 ) ) ) ).
thf(fact_224_number__of__Bit1,axiom,
! [W_12: int] :
( ( number_number_of_rat @ ( bit1 @ W_12 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ ( number_number_of_rat @ W_12 ) ) @ ( number_number_of_rat @ W_12 ) ) ) ).
thf(fact_225_number__of__Bit1,axiom,
! [W_12: int] :
( ( number528085621omplex @ ( bit1 @ W_12 ) )
= ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ ( number528085621omplex @ W_12 ) ) @ ( number528085621omplex @ W_12 ) ) ) ).
thf(fact_226_number__of__Bit1,axiom,
! [W_12: int] :
( ( number267125858f_real @ ( bit1 @ W_12 ) )
= ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( number267125858f_real @ W_12 ) ) @ ( number267125858f_real @ W_12 ) ) ) ).
thf(fact_227_number__of__Bit1,axiom,
! [W_12: int] :
( ( number_number_of_int @ ( bit1 @ W_12 ) )
= ( plus_plus_int @ ( plus_plus_int @ one_one_int @ ( number_number_of_int @ W_12 ) ) @ ( number_number_of_int @ W_12 ) ) ) ).
thf(fact_228_semiring__numeral__1__eq__1,axiom,
( ( number_number_of_rat @ ( bit1 @ pls ) )
= one_one_rat ) ).
thf(fact_229_semiring__numeral__1__eq__1,axiom,
( ( number528085621omplex @ ( bit1 @ pls ) )
= one_one_complex ) ).
thf(fact_230_semiring__numeral__1__eq__1,axiom,
( ( number267125858f_real @ ( bit1 @ pls ) )
= one_one_real ) ).
thf(fact_231_semiring__numeral__1__eq__1,axiom,
( ( number_number_of_nat @ ( bit1 @ pls ) )
= one_one_nat ) ).
thf(fact_232_semiring__numeral__1__eq__1,axiom,
( ( number_number_of_int @ ( bit1 @ pls ) )
= one_one_int ) ).
thf(fact_233_mn,axiom,
ord_less_int @ m1 @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ).
thf(fact_234_of__nat__less__two__power,axiom,
! [N_62: nat] : ( ord_less_real @ ( semiri132038758t_real @ N_62 ) @ ( power_power_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_62 ) ) ).
thf(fact_235_of__nat__less__two__power,axiom,
! [N_62: nat] : ( ord_less_int @ ( semiri1621563631at_int @ N_62 ) @ ( power_power_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_62 ) ) ).
thf(fact_236_of__nat__less__two__power,axiom,
! [N_62: nat] : ( ord_less_rat @ ( semiri151668891at_rat @ N_62 ) @ ( power_power_rat @ ( number_number_of_rat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_62 ) ) ).
thf(fact_237_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_238_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_239_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_240_of__nat__0__less__iff,axiom,
! [N_61: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( semiri151668891at_rat @ N_61 ) )
<=> ( ord_less_nat @ zero_zero_nat @ N_61 ) ) ).
thf(fact_241_of__nat__0__less__iff,axiom,
! [N_61: nat] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ ( semiri1424489471de_int @ N_61 ) )
<=> ( ord_less_nat @ zero_zero_nat @ N_61 ) ) ).
thf(fact_242_of__nat__0__less__iff,axiom,
! [N_61: nat] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ ( semiri1619134803umeral @ N_61 ) )
<=> ( ord_less_nat @ zero_zero_nat @ N_61 ) ) ).
thf(fact_243_of__nat__0__less__iff,axiom,
! [N_61: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri132038758t_real @ N_61 ) )
<=> ( ord_less_nat @ zero_zero_nat @ N_61 ) ) ).
thf(fact_244_of__nat__0__less__iff,axiom,
! [N_61: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri984289939at_nat @ N_61 ) )
<=> ( ord_less_nat @ zero_zero_nat @ N_61 ) ) ).
thf(fact_245_of__nat__0__less__iff,axiom,
! [N_61: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1621563631at_int @ N_61 ) )
<=> ( ord_less_nat @ zero_zero_nat @ N_61 ) ) ).
thf(fact_246_one__less__power,axiom,
! [N_60: nat,A_241: rat] :
( ( ord_less_rat @ one_one_rat @ A_241 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_60 )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A_241 @ N_60 ) ) ) ) ).
thf(fact_247_one__less__power,axiom,
! [N_60: nat,A_241: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ A_241 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_60 )
=> ( ord_le1860547276de_int @ one_on1684967323de_int @ ( power_881366806de_int @ A_241 @ N_60 ) ) ) ) ).
thf(fact_248_one__less__power,axiom,
! [N_60: nat,A_241: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ A_241 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_60 )
=> ( ord_le1304079648umeral @ one_on1645066479umeral @ ( power_2100829034umeral @ A_241 @ N_60 ) ) ) ) ).
thf(fact_249_one__less__power,axiom,
! [N_60: nat,A_241: real] :
( ( ord_less_real @ one_one_real @ A_241 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_60 )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A_241 @ N_60 ) ) ) ) ).
thf(fact_250_one__less__power,axiom,
! [N_60: nat,A_241: nat] :
( ( ord_less_nat @ one_one_nat @ A_241 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_60 )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A_241 @ N_60 ) ) ) ) ).
thf(fact_251_one__less__power,axiom,
! [N_60: nat,A_241: int] :
( ( ord_less_int @ one_one_int @ A_241 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_60 )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A_241 @ N_60 ) ) ) ) ).
thf(fact_252_power__0__left,axiom,
! [N_59: nat] :
( ( ( N_59 = zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N_59 )
= one_one_rat ) )
& ( ( N_59 != zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N_59 )
= zero_zero_rat ) ) ) ).
thf(fact_253_power__0__left,axiom,
! [N_59: nat] :
( ( ( N_59 = zero_zero_nat )
=> ( ( power_881366806de_int @ zero_z891286103de_int @ N_59 )
= one_on1684967323de_int ) )
& ( ( N_59 != zero_zero_nat )
=> ( ( power_881366806de_int @ zero_z891286103de_int @ N_59 )
= zero_z891286103de_int ) ) ) ).
thf(fact_254_power__0__left,axiom,
! [N_59: nat] :
( ( ( N_59 = zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N_59 )
= one_one_complex ) )
& ( ( N_59 != zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N_59 )
= zero_zero_complex ) ) ) ).
thf(fact_255_power__0__left,axiom,
! [N_59: nat] :
( ( ( N_59 = zero_zero_nat )
=> ( ( power_2100829034umeral @ zero_z126310315umeral @ N_59 )
= one_on1645066479umeral ) )
& ( ( N_59 != zero_zero_nat )
=> ( ( power_2100829034umeral @ zero_z126310315umeral @ N_59 )
= zero_z126310315umeral ) ) ) ).
thf(fact_256_power__0__left,axiom,
! [N_59: nat] :
( ( ( N_59 = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N_59 )
= one_one_real ) )
& ( ( N_59 != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N_59 )
= zero_zero_real ) ) ) ).
thf(fact_257_power__0__left,axiom,
! [N_59: nat] :
( ( ( N_59 = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N_59 )
= one_one_nat ) )
& ( ( N_59 != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N_59 )
= zero_zero_nat ) ) ) ).
thf(fact_258_power__0__left,axiom,
! [N_59: nat] :
( ( ( N_59 = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N_59 )
= one_one_int ) )
& ( ( N_59 != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N_59 )
= zero_zero_int ) ) ) ).
thf(fact_259_power__strict__decreasing,axiom,
! [A_240: rat,N_58: nat,N_57: nat] :
( ( ord_less_nat @ N_58 @ N_57 )
=> ( ( ord_less_rat @ zero_zero_rat @ A_240 )
=> ( ( ord_less_rat @ A_240 @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A_240 @ N_57 ) @ ( power_power_rat @ A_240 @ N_58 ) ) ) ) ) ).
thf(fact_260_power__strict__decreasing,axiom,
! [A_240: quickcheck_code_int,N_58: nat,N_57: nat] :
( ( ord_less_nat @ N_58 @ N_57 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_240 )
=> ( ( ord_le1860547276de_int @ A_240 @ one_on1684967323de_int )
=> ( ord_le1860547276de_int @ ( power_881366806de_int @ A_240 @ N_57 ) @ ( power_881366806de_int @ A_240 @ N_58 ) ) ) ) ) ).
thf(fact_261_power__strict__decreasing,axiom,
! [A_240: code_code_numeral,N_58: nat,N_57: nat] :
( ( ord_less_nat @ N_58 @ N_57 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_240 )
=> ( ( ord_le1304079648umeral @ A_240 @ one_on1645066479umeral )
=> ( ord_le1304079648umeral @ ( power_2100829034umeral @ A_240 @ N_57 ) @ ( power_2100829034umeral @ A_240 @ N_58 ) ) ) ) ) ).
thf(fact_262_power__strict__decreasing,axiom,
! [A_240: real,N_58: nat,N_57: nat] :
( ( ord_less_nat @ N_58 @ N_57 )
=> ( ( ord_less_real @ zero_zero_real @ A_240 )
=> ( ( ord_less_real @ A_240 @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A_240 @ N_57 ) @ ( power_power_real @ A_240 @ N_58 ) ) ) ) ) ).
thf(fact_263_power__strict__decreasing,axiom,
! [A_240: nat,N_58: nat,N_57: nat] :
( ( ord_less_nat @ N_58 @ N_57 )
=> ( ( ord_less_nat @ zero_zero_nat @ A_240 )
=> ( ( ord_less_nat @ A_240 @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A_240 @ N_57 ) @ ( power_power_nat @ A_240 @ N_58 ) ) ) ) ) ).
thf(fact_264_power__strict__decreasing,axiom,
! [A_240: int,N_58: nat,N_57: nat] :
( ( ord_less_nat @ N_58 @ N_57 )
=> ( ( ord_less_int @ zero_zero_int @ A_240 )
=> ( ( ord_less_int @ A_240 @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A_240 @ N_57 ) @ ( power_power_int @ A_240 @ N_58 ) ) ) ) ) ).
thf(fact_265_zero__less__two,axiom,
ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
thf(fact_266_zero__less__two,axiom,
ord_le1860547276de_int @ zero_z891286103de_int @ ( plus_p1446045655de_int @ one_on1684967323de_int @ one_on1684967323de_int ) ).
thf(fact_267_zero__less__two,axiom,
ord_le1304079648umeral @ zero_z126310315umeral @ ( plus_p1627245867umeral @ one_on1645066479umeral @ one_on1645066479umeral ) ).
thf(fact_268_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
thf(fact_269_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
thf(fact_270_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
thf(fact_271_int__gr__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I_1: int] :
( ( ord_less_int @ K @ I_1 )
=> ( ( P @ I_1 )
=> ( P @ ( plus_plus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_272_transfer__int__nat__numerals_I1_J,axiom,
( zero_zero_int
= ( semiri1621563631at_int @ zero_zero_nat ) ) ).
thf(fact_273_tn0,axiom,
ord_less_nat @ zero_zero_nat @ tn ).
thf(fact_274_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_275_real__zero__not__eq__one,axiom,
zero_zero_real != one_one_real ).
thf(fact_276_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_277_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
thf(fact_278_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
thf(fact_279_number__of__is__id,axiom,
! [K: int] :
( ( number_number_of_int @ K )
= K ) ).
thf(fact_280_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
<=> ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
thf(fact_281_nat__add__commute,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ N )
= ( plus_plus_nat @ N @ M ) ) ).
thf(fact_282_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_283_nat__add__assoc,axiom,
! [M: nat,N: nat,K: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ M @ ( plus_plus_nat @ N @ K ) ) ) ).
thf(fact_284_nat__add__left__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( plus_plus_nat @ K @ M )
= ( plus_plus_nat @ K @ N ) )
<=> ( M = N ) ) ).
thf(fact_285_nat__add__right__cancel,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( plus_plus_nat @ M @ K )
= ( plus_plus_nat @ N @ K ) )
<=> ( M = N ) ) ).
thf(fact_286_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_287_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
thf(fact_288_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_289_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
thf(fact_290_less__not__refl3,axiom,
! [S_1: nat,T: nat] :
( ( ord_less_nat @ S_1 @ T )
=> ( S_1 != T ) ) ).
thf(fact_291_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_292_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_293_add__less__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_294_add__less__mono,axiom,
! [K: nat,L: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
thf(fact_295_less__add__eq__less,axiom,
! [M: nat,N: nat,K: nat,L: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_296_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
thf(fact_297_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_298_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_299_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
thf(fact_300_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_301_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_302_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_303_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_304_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
<=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_305_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_306_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_307_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_308_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_309_zless__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_310_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_311_linorder__neqE__linordered__idom,axiom,
! [X_92: rat,Y_76: rat] :
( ( X_92 != Y_76 )
=> ( ~ ( ord_less_rat @ X_92 @ Y_76 )
=> ( ord_less_rat @ Y_76 @ X_92 ) ) ) ).
thf(fact_312_linorder__neqE__linordered__idom,axiom,
! [X_92: real,Y_76: real] :
( ( X_92 != Y_76 )
=> ( ~ ( ord_less_real @ X_92 @ Y_76 )
=> ( ord_less_real @ Y_76 @ X_92 ) ) ) ).
thf(fact_313_linorder__neqE__linordered__idom,axiom,
! [X_92: int,Y_76: int] :
( ( X_92 != Y_76 )
=> ( ~ ( ord_less_int @ X_92 @ Y_76 )
=> ( ord_less_int @ Y_76 @ X_92 ) ) ) ).
thf(fact_314_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
thf(fact_315_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_316_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_317_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
thf(fact_318_power__one__right,axiom,
! [A_239: complex] :
( ( power_power_complex @ A_239 @ one_one_nat )
= A_239 ) ).
thf(fact_319_power__one__right,axiom,
! [A_239: real] :
( ( power_power_real @ A_239 @ one_one_nat )
= A_239 ) ).
thf(fact_320_power__one__right,axiom,
! [A_239: nat] :
( ( power_power_nat @ A_239 @ one_one_nat )
= A_239 ) ).
thf(fact_321_power__one__right,axiom,
! [A_239: int] :
( ( power_power_int @ A_239 @ one_one_nat )
= A_239 ) ).
thf(fact_322_of__nat__eq__iff,axiom,
! [M_33: nat,N_56: nat] :
( ( ( semiri151668891at_rat @ M_33 )
= ( semiri151668891at_rat @ N_56 ) )
<=> ( M_33 = N_56 ) ) ).
thf(fact_323_of__nat__eq__iff,axiom,
! [M_33: nat,N_56: nat] :
( ( ( semiri2020571505omplex @ M_33 )
= ( semiri2020571505omplex @ N_56 ) )
<=> ( M_33 = N_56 ) ) ).
thf(fact_324_of__nat__eq__iff,axiom,
! [M_33: nat,N_56: nat] :
( ( ( semiri132038758t_real @ M_33 )
= ( semiri132038758t_real @ N_56 ) )
<=> ( M_33 = N_56 ) ) ).
thf(fact_325_of__nat__eq__iff,axiom,
! [M_33: nat,N_56: nat] :
( ( ( semiri1621563631at_int @ M_33 )
= ( semiri1621563631at_int @ N_56 ) )
<=> ( M_33 = N_56 ) ) ).
thf(fact_326_Nat__Transfer_Otransfer__int__nat__relations_I1_J,axiom,
! [X: nat,Y: nat] :
( ( ( semiri1621563631at_int @ X )
= ( semiri1621563631at_int @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_327_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_328_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_329_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_330_one__neq__zero,axiom,
one_one_rat != zero_zero_rat ).
thf(fact_331_one__neq__zero,axiom,
one_on1684967323de_int != zero_z891286103de_int ).
thf(fact_332_one__neq__zero,axiom,
one_one_complex != zero_zero_complex ).
thf(fact_333_one__neq__zero,axiom,
one_on1645066479umeral != zero_z126310315umeral ).
thf(fact_334_one__neq__zero,axiom,
one_one_real != zero_zero_real ).
thf(fact_335_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
thf(fact_336_one__neq__zero,axiom,
one_one_int != zero_zero_int ).
thf(fact_337_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
thf(fact_338_zero__neq__one,axiom,
zero_z891286103de_int != one_on1684967323de_int ).
thf(fact_339_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
thf(fact_340_zero__neq__one,axiom,
zero_z126310315umeral != one_on1645066479umeral ).
thf(fact_341_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
thf(fact_342_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
thf(fact_343_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
thf(fact_344_field__power__not__zero,axiom,
! [N_55: nat,A_238: rat] :
( ( A_238 != zero_zero_rat )
=> ( ( power_power_rat @ A_238 @ N_55 )
!= zero_zero_rat ) ) ).
thf(fact_345_field__power__not__zero,axiom,
! [N_55: nat,A_238: complex] :
( ( A_238 != zero_zero_complex )
=> ( ( power_power_complex @ A_238 @ N_55 )
!= zero_zero_complex ) ) ).
thf(fact_346_field__power__not__zero,axiom,
! [N_55: nat,A_238: real] :
( ( A_238 != zero_zero_real )
=> ( ( power_power_real @ A_238 @ N_55 )
!= zero_zero_real ) ) ).
thf(fact_347_field__power__not__zero,axiom,
! [N_55: nat,A_238: int] :
( ( A_238 != zero_zero_int )
=> ( ( power_power_int @ A_238 @ N_55 )
!= zero_zero_int ) ) ).
thf(fact_348_power__one,axiom,
! [N_54: nat] :
( ( power_power_rat @ one_one_rat @ N_54 )
= one_one_rat ) ).
thf(fact_349_power__one,axiom,
! [N_54: nat] :
( ( power_881366806de_int @ one_on1684967323de_int @ N_54 )
= one_on1684967323de_int ) ).
thf(fact_350_power__one,axiom,
! [N_54: nat] :
( ( power_power_complex @ one_one_complex @ N_54 )
= one_one_complex ) ).
thf(fact_351_power__one,axiom,
! [N_54: nat] :
( ( power_2100829034umeral @ one_on1645066479umeral @ N_54 )
= one_on1645066479umeral ) ).
thf(fact_352_power__one,axiom,
! [N_54: nat] :
( ( power_power_real @ one_one_real @ N_54 )
= one_one_real ) ).
thf(fact_353_power__one,axiom,
! [N_54: nat] :
( ( power_power_nat @ one_one_nat @ N_54 )
= one_one_nat ) ).
thf(fact_354_power__one,axiom,
! [N_54: nat] :
( ( power_power_int @ one_one_int @ N_54 )
= one_one_int ) ).
thf(fact_355_of__nat__less__iff,axiom,
! [M_32: nat,N_53: nat] :
( ( ord_less_rat @ ( semiri151668891at_rat @ M_32 ) @ ( semiri151668891at_rat @ N_53 ) )
<=> ( ord_less_nat @ M_32 @ N_53 ) ) ).
thf(fact_356_of__nat__less__iff,axiom,
! [M_32: nat,N_53: nat] :
( ( ord_le1860547276de_int @ ( semiri1424489471de_int @ M_32 ) @ ( semiri1424489471de_int @ N_53 ) )
<=> ( ord_less_nat @ M_32 @ N_53 ) ) ).
thf(fact_357_of__nat__less__iff,axiom,
! [M_32: nat,N_53: nat] :
( ( ord_less_real @ ( semiri132038758t_real @ M_32 ) @ ( semiri132038758t_real @ N_53 ) )
<=> ( ord_less_nat @ M_32 @ N_53 ) ) ).
thf(fact_358_of__nat__less__iff,axiom,
! [M_32: nat,N_53: nat] :
( ( ord_less_nat @ ( semiri984289939at_nat @ M_32 ) @ ( semiri984289939at_nat @ N_53 ) )
<=> ( ord_less_nat @ M_32 @ N_53 ) ) ).
thf(fact_359_of__nat__less__iff,axiom,
! [M_32: nat,N_53: nat] :
( ( ord_less_int @ ( semiri1621563631at_int @ M_32 ) @ ( semiri1621563631at_int @ N_53 ) )
<=> ( ord_less_nat @ M_32 @ N_53 ) ) ).
thf(fact_360_less__imp__of__nat__less,axiom,
! [M_31: nat,N_52: nat] :
( ( ord_less_nat @ M_31 @ N_52 )
=> ( ord_less_rat @ ( semiri151668891at_rat @ M_31 ) @ ( semiri151668891at_rat @ N_52 ) ) ) ).
thf(fact_361_less__imp__of__nat__less,axiom,
! [M_31: nat,N_52: nat] :
( ( ord_less_nat @ M_31 @ N_52 )
=> ( ord_le1860547276de_int @ ( semiri1424489471de_int @ M_31 ) @ ( semiri1424489471de_int @ N_52 ) ) ) ).
thf(fact_362_less__imp__of__nat__less,axiom,
! [M_31: nat,N_52: nat] :
( ( ord_less_nat @ M_31 @ N_52 )
=> ( ord_less_real @ ( semiri132038758t_real @ M_31 ) @ ( semiri132038758t_real @ N_52 ) ) ) ).
thf(fact_363_less__imp__of__nat__less,axiom,
! [M_31: nat,N_52: nat] :
( ( ord_less_nat @ M_31 @ N_52 )
=> ( ord_less_nat @ ( semiri984289939at_nat @ M_31 ) @ ( semiri984289939at_nat @ N_52 ) ) ) ).
thf(fact_364_less__imp__of__nat__less,axiom,
! [M_31: nat,N_52: nat] :
( ( ord_less_nat @ M_31 @ N_52 )
=> ( ord_less_int @ ( semiri1621563631at_int @ M_31 ) @ ( semiri1621563631at_int @ N_52 ) ) ) ).
thf(fact_365_of__nat__less__imp__less,axiom,
! [M_30: nat,N_51: nat] :
( ( ord_less_rat @ ( semiri151668891at_rat @ M_30 ) @ ( semiri151668891at_rat @ N_51 ) )
=> ( ord_less_nat @ M_30 @ N_51 ) ) ).
thf(fact_366_of__nat__less__imp__less,axiom,
! [M_30: nat,N_51: nat] :
( ( ord_le1860547276de_int @ ( semiri1424489471de_int @ M_30 ) @ ( semiri1424489471de_int @ N_51 ) )
=> ( ord_less_nat @ M_30 @ N_51 ) ) ).
thf(fact_367_of__nat__less__imp__less,axiom,
! [M_30: nat,N_51: nat] :
( ( ord_less_real @ ( semiri132038758t_real @ M_30 ) @ ( semiri132038758t_real @ N_51 ) )
=> ( ord_less_nat @ M_30 @ N_51 ) ) ).
thf(fact_368_of__nat__less__imp__less,axiom,
! [M_30: nat,N_51: nat] :
( ( ord_less_nat @ ( semiri984289939at_nat @ M_30 ) @ ( semiri984289939at_nat @ N_51 ) )
=> ( ord_less_nat @ M_30 @ N_51 ) ) ).
thf(fact_369_of__nat__less__imp__less,axiom,
! [M_30: nat,N_51: nat] :
( ( ord_less_int @ ( semiri1621563631at_int @ M_30 ) @ ( semiri1621563631at_int @ N_51 ) )
=> ( ord_less_nat @ M_30 @ N_51 ) ) ).
thf(fact_370_of__nat__add,axiom,
! [M_29: nat,N_50: nat] :
( ( semiri151668891at_rat @ ( plus_plus_nat @ M_29 @ N_50 ) )
= ( plus_plus_rat @ ( semiri151668891at_rat @ M_29 ) @ ( semiri151668891at_rat @ N_50 ) ) ) ).
thf(fact_371_of__nat__add,axiom,
! [M_29: nat,N_50: nat] :
( ( semiri1424489471de_int @ ( plus_plus_nat @ M_29 @ N_50 ) )
= ( plus_p1446045655de_int @ ( semiri1424489471de_int @ M_29 ) @ ( semiri1424489471de_int @ N_50 ) ) ) ).
thf(fact_372_of__nat__add,axiom,
! [M_29: nat,N_50: nat] :
( ( semiri2020571505omplex @ ( plus_plus_nat @ M_29 @ N_50 ) )
= ( plus_plus_complex @ ( semiri2020571505omplex @ M_29 ) @ ( semiri2020571505omplex @ N_50 ) ) ) ).
thf(fact_373_of__nat__add,axiom,
! [M_29: nat,N_50: nat] :
( ( semiri132038758t_real @ ( plus_plus_nat @ M_29 @ N_50 ) )
= ( plus_plus_real @ ( semiri132038758t_real @ M_29 ) @ ( semiri132038758t_real @ N_50 ) ) ) ).
thf(fact_374_of__nat__add,axiom,
! [M_29: nat,N_50: nat] :
( ( semiri984289939at_nat @ ( plus_plus_nat @ M_29 @ N_50 ) )
= ( plus_plus_nat @ ( semiri984289939at_nat @ M_29 ) @ ( semiri984289939at_nat @ N_50 ) ) ) ).
thf(fact_375_of__nat__add,axiom,
! [M_29: nat,N_50: nat] :
( ( semiri1621563631at_int @ ( plus_plus_nat @ M_29 @ N_50 ) )
= ( plus_plus_int @ ( semiri1621563631at_int @ M_29 ) @ ( semiri1621563631at_int @ N_50 ) ) ) ).
thf(fact_376_of__nat__1,axiom,
( ( semiri151668891at_rat @ one_one_nat )
= one_one_rat ) ).
thf(fact_377_of__nat__1,axiom,
( ( semiri1424489471de_int @ one_one_nat )
= one_on1684967323de_int ) ).
thf(fact_378_of__nat__1,axiom,
( ( semiri2020571505omplex @ one_one_nat )
= one_one_complex ) ).
thf(fact_379_of__nat__1,axiom,
( ( semiri1619134803umeral @ one_one_nat )
= one_on1645066479umeral ) ).
thf(fact_380_of__nat__1,axiom,
( ( semiri132038758t_real @ one_one_nat )
= one_one_real ) ).
thf(fact_381_of__nat__1,axiom,
( ( semiri984289939at_nat @ one_one_nat )
= one_one_nat ) ).
thf(fact_382_of__nat__1,axiom,
( ( semiri1621563631at_int @ one_one_nat )
= one_one_int ) ).
thf(fact_383_of__nat__power,axiom,
! [M_28: nat,N_49: nat] :
( ( semiri2020571505omplex @ ( power_power_nat @ M_28 @ N_49 ) )
= ( power_power_complex @ ( semiri2020571505omplex @ M_28 ) @ N_49 ) ) ).
thf(fact_384_of__nat__power,axiom,
! [M_28: nat,N_49: nat] :
( ( semiri132038758t_real @ ( power_power_nat @ M_28 @ N_49 ) )
= ( power_power_real @ ( semiri132038758t_real @ M_28 ) @ N_49 ) ) ).
thf(fact_385_of__nat__power,axiom,
! [M_28: nat,N_49: nat] :
( ( semiri984289939at_nat @ ( power_power_nat @ M_28 @ N_49 ) )
= ( power_power_nat @ ( semiri984289939at_nat @ M_28 ) @ N_49 ) ) ).
thf(fact_386_of__nat__power,axiom,
! [M_28: nat,N_49: nat] :
( ( semiri1621563631at_int @ ( power_power_nat @ M_28 @ N_49 ) )
= ( power_power_int @ ( semiri1621563631at_int @ M_28 ) @ N_49 ) ) ).
thf(fact_387_of__nat__power,axiom,
! [M_28: nat,N_49: nat] :
( ( semiri151668891at_rat @ ( power_power_nat @ M_28 @ N_49 ) )
= ( power_power_rat @ ( semiri151668891at_rat @ M_28 ) @ N_49 ) ) ).
thf(fact_388_transfer__int__nat__numerals_I2_J,axiom,
( one_one_int
= ( semiri1621563631at_int @ one_one_nat ) ) ).
thf(fact_389_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_390_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_391_pos__add__strict,axiom,
! [B_167: rat,C_101: rat,A_237: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_237 )
=> ( ( ord_less_rat @ B_167 @ C_101 )
=> ( ord_less_rat @ B_167 @ ( plus_plus_rat @ A_237 @ C_101 ) ) ) ) ).
thf(fact_392_pos__add__strict,axiom,
! [B_167: quickcheck_code_int,C_101: quickcheck_code_int,A_237: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_237 )
=> ( ( ord_le1860547276de_int @ B_167 @ C_101 )
=> ( ord_le1860547276de_int @ B_167 @ ( plus_p1446045655de_int @ A_237 @ C_101 ) ) ) ) ).
thf(fact_393_pos__add__strict,axiom,
! [B_167: real,C_101: real,A_237: real] :
( ( ord_less_real @ zero_zero_real @ A_237 )
=> ( ( ord_less_real @ B_167 @ C_101 )
=> ( ord_less_real @ B_167 @ ( plus_plus_real @ A_237 @ C_101 ) ) ) ) ).
thf(fact_394_pos__add__strict,axiom,
! [B_167: nat,C_101: nat,A_237: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_237 )
=> ( ( ord_less_nat @ B_167 @ C_101 )
=> ( ord_less_nat @ B_167 @ ( plus_plus_nat @ A_237 @ C_101 ) ) ) ) ).
thf(fact_395_pos__add__strict,axiom,
! [B_167: int,C_101: int,A_237: int] :
( ( ord_less_int @ zero_zero_int @ A_237 )
=> ( ( ord_less_int @ B_167 @ C_101 )
=> ( ord_less_int @ B_167 @ ( plus_plus_int @ A_237 @ C_101 ) ) ) ) ).
thf(fact_396_pos__add__strict,axiom,
! [B_167: code_code_numeral,C_101: code_code_numeral,A_237: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_237 )
=> ( ( ord_le1304079648umeral @ B_167 @ C_101 )
=> ( ord_le1304079648umeral @ B_167 @ ( plus_p1627245867umeral @ A_237 @ C_101 ) ) ) ) ).
thf(fact_397_not__one__less__zero,axiom,
~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
thf(fact_398_not__one__less__zero,axiom,
~ ( ord_le1860547276de_int @ one_on1684967323de_int @ zero_z891286103de_int ) ).
thf(fact_399_not__one__less__zero,axiom,
~ ( ord_le1304079648umeral @ one_on1645066479umeral @ zero_z126310315umeral ) ).
thf(fact_400_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
thf(fact_401_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
thf(fact_402_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
thf(fact_403_zero__less__one,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
thf(fact_404_zero__less__one,axiom,
ord_le1860547276de_int @ zero_z891286103de_int @ one_on1684967323de_int ).
thf(fact_405_zero__less__one,axiom,
ord_le1304079648umeral @ zero_z126310315umeral @ one_on1645066479umeral ).
thf(fact_406_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
thf(fact_407_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
thf(fact_408_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
thf(fact_409_zero__less__power,axiom,
! [N_48: nat,A_236: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_236 )
=> ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A_236 @ N_48 ) ) ) ).
thf(fact_410_zero__less__power,axiom,
! [N_48: nat,A_236: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_236 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ ( power_881366806de_int @ A_236 @ N_48 ) ) ) ).
thf(fact_411_zero__less__power,axiom,
! [N_48: nat,A_236: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_236 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ ( power_2100829034umeral @ A_236 @ N_48 ) ) ) ).
thf(fact_412_zero__less__power,axiom,
! [N_48: nat,A_236: real] :
( ( ord_less_real @ zero_zero_real @ A_236 )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A_236 @ N_48 ) ) ) ).
thf(fact_413_zero__less__power,axiom,
! [N_48: nat,A_236: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_236 )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A_236 @ N_48 ) ) ) ).
thf(fact_414_zero__less__power,axiom,
! [N_48: nat,A_236: int] :
( ( ord_less_int @ zero_zero_int @ A_236 )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A_236 @ N_48 ) ) ) ).
thf(fact_415_less__add__one,axiom,
! [A_235: rat] : ( ord_less_rat @ A_235 @ ( plus_plus_rat @ A_235 @ one_one_rat ) ) ).
thf(fact_416_less__add__one,axiom,
! [A_235: quickcheck_code_int] : ( ord_le1860547276de_int @ A_235 @ ( plus_p1446045655de_int @ A_235 @ one_on1684967323de_int ) ) ).
thf(fact_417_less__add__one,axiom,
! [A_235: code_code_numeral] : ( ord_le1304079648umeral @ A_235 @ ( plus_p1627245867umeral @ A_235 @ one_on1645066479umeral ) ) ).
thf(fact_418_less__add__one,axiom,
! [A_235: real] : ( ord_less_real @ A_235 @ ( plus_plus_real @ A_235 @ one_one_real ) ) ).
thf(fact_419_less__add__one,axiom,
! [A_235: nat] : ( ord_less_nat @ A_235 @ ( plus_plus_nat @ A_235 @ one_one_nat ) ) ).
thf(fact_420_less__add__one,axiom,
! [A_235: int] : ( ord_less_int @ A_235 @ ( plus_plus_int @ A_235 @ one_one_int ) ) ).
thf(fact_421_power__inject__exp,axiom,
! [M_27: nat,N_47: nat,A_234: rat] :
( ( ord_less_rat @ one_one_rat @ A_234 )
=> ( ( ( power_power_rat @ A_234 @ M_27 )
= ( power_power_rat @ A_234 @ N_47 ) )
<=> ( M_27 = N_47 ) ) ) ).
thf(fact_422_power__inject__exp,axiom,
! [M_27: nat,N_47: nat,A_234: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ A_234 )
=> ( ( ( power_881366806de_int @ A_234 @ M_27 )
= ( power_881366806de_int @ A_234 @ N_47 ) )
<=> ( M_27 = N_47 ) ) ) ).
thf(fact_423_power__inject__exp,axiom,
! [M_27: nat,N_47: nat,A_234: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ A_234 )
=> ( ( ( power_2100829034umeral @ A_234 @ M_27 )
= ( power_2100829034umeral @ A_234 @ N_47 ) )
<=> ( M_27 = N_47 ) ) ) ).
thf(fact_424_power__inject__exp,axiom,
! [M_27: nat,N_47: nat,A_234: real] :
( ( ord_less_real @ one_one_real @ A_234 )
=> ( ( ( power_power_real @ A_234 @ M_27 )
= ( power_power_real @ A_234 @ N_47 ) )
<=> ( M_27 = N_47 ) ) ) ).
thf(fact_425_power__inject__exp,axiom,
! [M_27: nat,N_47: nat,A_234: nat] :
( ( ord_less_nat @ one_one_nat @ A_234 )
=> ( ( ( power_power_nat @ A_234 @ M_27 )
= ( power_power_nat @ A_234 @ N_47 ) )
<=> ( M_27 = N_47 ) ) ) ).
thf(fact_426_power__inject__exp,axiom,
! [M_27: nat,N_47: nat,A_234: int] :
( ( ord_less_int @ one_one_int @ A_234 )
=> ( ( ( power_power_int @ A_234 @ M_27 )
= ( power_power_int @ A_234 @ N_47 ) )
<=> ( M_27 = N_47 ) ) ) ).
thf(fact_427_power__strict__increasing__iff,axiom,
! [X_91: nat,Y_75: nat,B_166: rat] :
( ( ord_less_rat @ one_one_rat @ B_166 )
=> ( ( ord_less_rat @ ( power_power_rat @ B_166 @ X_91 ) @ ( power_power_rat @ B_166 @ Y_75 ) )
<=> ( ord_less_nat @ X_91 @ Y_75 ) ) ) ).
thf(fact_428_power__strict__increasing__iff,axiom,
! [X_91: nat,Y_75: nat,B_166: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ B_166 )
=> ( ( ord_le1860547276de_int @ ( power_881366806de_int @ B_166 @ X_91 ) @ ( power_881366806de_int @ B_166 @ Y_75 ) )
<=> ( ord_less_nat @ X_91 @ Y_75 ) ) ) ).
thf(fact_429_power__strict__increasing__iff,axiom,
! [X_91: nat,Y_75: nat,B_166: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ B_166 )
=> ( ( ord_le1304079648umeral @ ( power_2100829034umeral @ B_166 @ X_91 ) @ ( power_2100829034umeral @ B_166 @ Y_75 ) )
<=> ( ord_less_nat @ X_91 @ Y_75 ) ) ) ).
thf(fact_430_power__strict__increasing__iff,axiom,
! [X_91: nat,Y_75: nat,B_166: real] :
( ( ord_less_real @ one_one_real @ B_166 )
=> ( ( ord_less_real @ ( power_power_real @ B_166 @ X_91 ) @ ( power_power_real @ B_166 @ Y_75 ) )
<=> ( ord_less_nat @ X_91 @ Y_75 ) ) ) ).
thf(fact_431_power__strict__increasing__iff,axiom,
! [X_91: nat,Y_75: nat,B_166: nat] :
( ( ord_less_nat @ one_one_nat @ B_166 )
=> ( ( ord_less_nat @ ( power_power_nat @ B_166 @ X_91 ) @ ( power_power_nat @ B_166 @ Y_75 ) )
<=> ( ord_less_nat @ X_91 @ Y_75 ) ) ) ).
thf(fact_432_power__strict__increasing__iff,axiom,
! [X_91: nat,Y_75: nat,B_166: int] :
( ( ord_less_int @ one_one_int @ B_166 )
=> ( ( ord_less_int @ ( power_power_int @ B_166 @ X_91 ) @ ( power_power_int @ B_166 @ Y_75 ) )
<=> ( ord_less_nat @ X_91 @ Y_75 ) ) ) ).
thf(fact_433_power__less__imp__less__exp,axiom,
! [M_26: nat,N_46: nat,A_233: rat] :
( ( ord_less_rat @ one_one_rat @ A_233 )
=> ( ( ord_less_rat @ ( power_power_rat @ A_233 @ M_26 ) @ ( power_power_rat @ A_233 @ N_46 ) )
=> ( ord_less_nat @ M_26 @ N_46 ) ) ) ).
thf(fact_434_power__less__imp__less__exp,axiom,
! [M_26: nat,N_46: nat,A_233: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ A_233 )
=> ( ( ord_le1860547276de_int @ ( power_881366806de_int @ A_233 @ M_26 ) @ ( power_881366806de_int @ A_233 @ N_46 ) )
=> ( ord_less_nat @ M_26 @ N_46 ) ) ) ).
thf(fact_435_power__less__imp__less__exp,axiom,
! [M_26: nat,N_46: nat,A_233: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ A_233 )
=> ( ( ord_le1304079648umeral @ ( power_2100829034umeral @ A_233 @ M_26 ) @ ( power_2100829034umeral @ A_233 @ N_46 ) )
=> ( ord_less_nat @ M_26 @ N_46 ) ) ) ).
thf(fact_436_power__less__imp__less__exp,axiom,
! [M_26: nat,N_46: nat,A_233: real] :
( ( ord_less_real @ one_one_real @ A_233 )
=> ( ( ord_less_real @ ( power_power_real @ A_233 @ M_26 ) @ ( power_power_real @ A_233 @ N_46 ) )
=> ( ord_less_nat @ M_26 @ N_46 ) ) ) ).
thf(fact_437_power__less__imp__less__exp,axiom,
! [M_26: nat,N_46: nat,A_233: nat] :
( ( ord_less_nat @ one_one_nat @ A_233 )
=> ( ( ord_less_nat @ ( power_power_nat @ A_233 @ M_26 ) @ ( power_power_nat @ A_233 @ N_46 ) )
=> ( ord_less_nat @ M_26 @ N_46 ) ) ) ).
thf(fact_438_power__less__imp__less__exp,axiom,
! [M_26: nat,N_46: nat,A_233: int] :
( ( ord_less_int @ one_one_int @ A_233 )
=> ( ( ord_less_int @ ( power_power_int @ A_233 @ M_26 ) @ ( power_power_int @ A_233 @ N_46 ) )
=> ( ord_less_nat @ M_26 @ N_46 ) ) ) ).
thf(fact_439_power__strict__increasing,axiom,
! [A_232: rat,N_45: nat,N_44: nat] :
( ( ord_less_nat @ N_45 @ N_44 )
=> ( ( ord_less_rat @ one_one_rat @ A_232 )
=> ( ord_less_rat @ ( power_power_rat @ A_232 @ N_45 ) @ ( power_power_rat @ A_232 @ N_44 ) ) ) ) ).
thf(fact_440_power__strict__increasing,axiom,
! [A_232: quickcheck_code_int,N_45: nat,N_44: nat] :
( ( ord_less_nat @ N_45 @ N_44 )
=> ( ( ord_le1860547276de_int @ one_on1684967323de_int @ A_232 )
=> ( ord_le1860547276de_int @ ( power_881366806de_int @ A_232 @ N_45 ) @ ( power_881366806de_int @ A_232 @ N_44 ) ) ) ) ).
thf(fact_441_power__strict__increasing,axiom,
! [A_232: code_code_numeral,N_45: nat,N_44: nat] :
( ( ord_less_nat @ N_45 @ N_44 )
=> ( ( ord_le1304079648umeral @ one_on1645066479umeral @ A_232 )
=> ( ord_le1304079648umeral @ ( power_2100829034umeral @ A_232 @ N_45 ) @ ( power_2100829034umeral @ A_232 @ N_44 ) ) ) ) ).
thf(fact_442_power__strict__increasing,axiom,
! [A_232: real,N_45: nat,N_44: nat] :
( ( ord_less_nat @ N_45 @ N_44 )
=> ( ( ord_less_real @ one_one_real @ A_232 )
=> ( ord_less_real @ ( power_power_real @ A_232 @ N_45 ) @ ( power_power_real @ A_232 @ N_44 ) ) ) ) ).
thf(fact_443_power__strict__increasing,axiom,
! [A_232: nat,N_45: nat,N_44: nat] :
( ( ord_less_nat @ N_45 @ N_44 )
=> ( ( ord_less_nat @ one_one_nat @ A_232 )
=> ( ord_less_nat @ ( power_power_nat @ A_232 @ N_45 ) @ ( power_power_nat @ A_232 @ N_44 ) ) ) ) ).
thf(fact_444_power__strict__increasing,axiom,
! [A_232: int,N_45: nat,N_44: nat] :
( ( ord_less_nat @ N_45 @ N_44 )
=> ( ( ord_less_int @ one_one_int @ A_232 )
=> ( ord_less_int @ ( power_power_int @ A_232 @ N_45 ) @ ( power_power_int @ A_232 @ N_44 ) ) ) ) ).
thf(fact_445_power__eq__0__iff,axiom,
! [A_231: rat,N_43: nat] :
( ( ( power_power_rat @ A_231 @ N_43 )
= zero_zero_rat )
<=> ( ( A_231 = zero_zero_rat )
& ( N_43 != zero_zero_nat ) ) ) ).
thf(fact_446_power__eq__0__iff,axiom,
! [A_231: quickcheck_code_int,N_43: nat] :
( ( ( power_881366806de_int @ A_231 @ N_43 )
= zero_z891286103de_int )
<=> ( ( A_231 = zero_z891286103de_int )
& ( N_43 != zero_zero_nat ) ) ) ).
thf(fact_447_power__eq__0__iff,axiom,
! [A_231: complex,N_43: nat] :
( ( ( power_power_complex @ A_231 @ N_43 )
= zero_zero_complex )
<=> ( ( A_231 = zero_zero_complex )
& ( N_43 != zero_zero_nat ) ) ) ).
thf(fact_448_power__eq__0__iff,axiom,
! [A_231: code_code_numeral,N_43: nat] :
( ( ( power_2100829034umeral @ A_231 @ N_43 )
= zero_z126310315umeral )
<=> ( ( A_231 = zero_z126310315umeral )
& ( N_43 != zero_zero_nat ) ) ) ).
thf(fact_449_power__eq__0__iff,axiom,
! [A_231: real,N_43: nat] :
( ( ( power_power_real @ A_231 @ N_43 )
= zero_zero_real )
<=> ( ( A_231 = zero_zero_real )
& ( N_43 != zero_zero_nat ) ) ) ).
thf(fact_450_power__eq__0__iff,axiom,
! [A_231: nat,N_43: nat] :
( ( ( power_power_nat @ A_231 @ N_43 )
= zero_zero_nat )
<=> ( ( A_231 = zero_zero_nat )
& ( N_43 != zero_zero_nat ) ) ) ).
thf(fact_451_power__eq__0__iff,axiom,
! [A_231: int,N_43: nat] :
( ( ( power_power_int @ A_231 @ N_43 )
= zero_zero_int )
<=> ( ( A_231 = zero_zero_int )
& ( N_43 != zero_zero_nat ) ) ) ).
thf(fact_452_of__nat__less__0__iff,axiom,
! [M_25: nat] :
~ ( ord_less_rat @ ( semiri151668891at_rat @ M_25 ) @ zero_zero_rat ) ).
thf(fact_453_of__nat__less__0__iff,axiom,
! [M_25: nat] :
~ ( ord_le1860547276de_int @ ( semiri1424489471de_int @ M_25 ) @ zero_z891286103de_int ) ).
thf(fact_454_of__nat__less__0__iff,axiom,
! [M_25: nat] :
~ ( ord_le1304079648umeral @ ( semiri1619134803umeral @ M_25 ) @ zero_z126310315umeral ) ).
thf(fact_455_of__nat__less__0__iff,axiom,
! [M_25: nat] :
~ ( ord_less_real @ ( semiri132038758t_real @ M_25 ) @ zero_zero_real ) ).
thf(fact_456_of__nat__less__0__iff,axiom,
! [M_25: nat] :
~ ( ord_less_nat @ ( semiri984289939at_nat @ M_25 ) @ zero_zero_nat ) ).
thf(fact_457_of__nat__less__0__iff,axiom,
! [M_25: nat] :
~ ( ord_less_int @ ( semiri1621563631at_int @ M_25 ) @ zero_zero_int ) ).
thf(fact_458_power__0,axiom,
! [A_230: rat] :
( ( power_power_rat @ A_230 @ zero_zero_nat )
= one_one_rat ) ).
thf(fact_459_power__0,axiom,
! [A_230: quickcheck_code_int] :
( ( power_881366806de_int @ A_230 @ zero_zero_nat )
= one_on1684967323de_int ) ).
thf(fact_460_power__0,axiom,
! [A_230: complex] :
( ( power_power_complex @ A_230 @ zero_zero_nat )
= one_one_complex ) ).
thf(fact_461_power__0,axiom,
! [A_230: code_code_numeral] :
( ( power_2100829034umeral @ A_230 @ zero_zero_nat )
= one_on1645066479umeral ) ).
thf(fact_462_power__0,axiom,
! [A_230: real] :
( ( power_power_real @ A_230 @ zero_zero_nat )
= one_one_real ) ).
thf(fact_463_power__0,axiom,
! [A_230: nat] :
( ( power_power_nat @ A_230 @ zero_zero_nat )
= one_one_nat ) ).
thf(fact_464_power__0,axiom,
! [A_230: int] :
( ( power_power_int @ A_230 @ zero_zero_nat )
= one_one_int ) ).
thf(fact_465_of__nat__0,axiom,
( ( semiri151668891at_rat @ zero_zero_nat )
= zero_zero_rat ) ).
thf(fact_466_of__nat__0,axiom,
( ( semiri1424489471de_int @ zero_zero_nat )
= zero_z891286103de_int ) ).
thf(fact_467_of__nat__0,axiom,
( ( semiri2020571505omplex @ zero_zero_nat )
= zero_zero_complex ) ).
thf(fact_468_of__nat__0,axiom,
( ( semiri1619134803umeral @ zero_zero_nat )
= zero_z126310315umeral ) ).
thf(fact_469_of__nat__0,axiom,
( ( semiri132038758t_real @ zero_zero_nat )
= zero_zero_real ) ).
thf(fact_470_of__nat__0,axiom,
( ( semiri984289939at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_471_of__nat__0,axiom,
( ( semiri1621563631at_int @ zero_zero_nat )
= zero_zero_int ) ).
thf(fact_472_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_473_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_1 )
& ( K
= ( semiri1621563631at_int @ N_1 ) ) ) ) ).
thf(fact_474_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
& ( ( plus_plus_nat @ I @ K_1 )
= J ) ) ) ).
thf(fact_475_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_476_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_90: rat] :
( ( power_power_rat @ X_90 @ zero_zero_nat )
= one_one_rat ) ).
thf(fact_477_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_90: quickcheck_code_int] :
( ( power_881366806de_int @ X_90 @ zero_zero_nat )
= one_on1684967323de_int ) ).
thf(fact_478_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_90: complex] :
( ( power_power_complex @ X_90 @ zero_zero_nat )
= one_one_complex ) ).
thf(fact_479_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_90: code_code_numeral] :
( ( power_2100829034umeral @ X_90 @ zero_zero_nat )
= one_on1645066479umeral ) ).
thf(fact_480_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_90: real] :
( ( power_power_real @ X_90 @ zero_zero_nat )
= one_one_real ) ).
thf(fact_481_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_90: nat] :
( ( power_power_nat @ X_90 @ zero_zero_nat )
= one_one_nat ) ).
thf(fact_482_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_90: int] :
( ( power_power_int @ X_90 @ zero_zero_nat )
= one_one_int ) ).
thf(fact_483_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_229: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A_229 @ A_229 ) )
<=> ( ord_less_rat @ zero_zero_rat @ A_229 ) ) ).
thf(fact_484_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_229: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A_229 @ A_229 ) )
<=> ( ord_less_real @ zero_zero_real @ A_229 ) ) ).
thf(fact_485_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_229: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_229 @ A_229 ) )
<=> ( ord_less_int @ zero_zero_int @ A_229 ) ) ).
thf(fact_486_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_228: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A_228 @ A_228 ) @ zero_zero_rat )
<=> ( ord_less_rat @ A_228 @ zero_zero_rat ) ) ).
thf(fact_487_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_228: real] :
( ( ord_less_real @ ( plus_plus_real @ A_228 @ A_228 ) @ zero_zero_real )
<=> ( ord_less_real @ A_228 @ zero_zero_real ) ) ).
thf(fact_488_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_228: int] :
( ( ord_less_int @ ( plus_plus_int @ A_228 @ A_228 ) @ zero_zero_int )
<=> ( ord_less_int @ A_228 @ zero_zero_int ) ) ).
thf(fact_489_add__pos__pos,axiom,
! [B_165: rat,A_227: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_227 )
=> ( ( ord_less_rat @ zero_zero_rat @ B_165 )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A_227 @ B_165 ) ) ) ) ).
thf(fact_490_add__pos__pos,axiom,
! [B_165: quickcheck_code_int,A_227: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_227 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ B_165 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ ( plus_p1446045655de_int @ A_227 @ B_165 ) ) ) ) ).
thf(fact_491_add__pos__pos,axiom,
! [B_165: real,A_227: real] :
( ( ord_less_real @ zero_zero_real @ A_227 )
=> ( ( ord_less_real @ zero_zero_real @ B_165 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A_227 @ B_165 ) ) ) ) ).
thf(fact_492_add__pos__pos,axiom,
! [B_165: nat,A_227: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_227 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_165 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A_227 @ B_165 ) ) ) ) ).
thf(fact_493_add__pos__pos,axiom,
! [B_165: int,A_227: int] :
( ( ord_less_int @ zero_zero_int @ A_227 )
=> ( ( ord_less_int @ zero_zero_int @ B_165 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_227 @ B_165 ) ) ) ) ).
thf(fact_494_add__pos__pos,axiom,
! [B_165: code_code_numeral,A_227: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_227 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ B_165 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ ( plus_p1627245867umeral @ A_227 @ B_165 ) ) ) ) ).
thf(fact_495_add__neg__neg,axiom,
! [B_164: rat,A_226: rat] :
( ( ord_less_rat @ A_226 @ zero_zero_rat )
=> ( ( ord_less_rat @ B_164 @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A_226 @ B_164 ) @ zero_zero_rat ) ) ) ).
thf(fact_496_add__neg__neg,axiom,
! [B_164: quickcheck_code_int,A_226: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_226 @ zero_z891286103de_int )
=> ( ( ord_le1860547276de_int @ B_164 @ zero_z891286103de_int )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_226 @ B_164 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_497_add__neg__neg,axiom,
! [B_164: real,A_226: real] :
( ( ord_less_real @ A_226 @ zero_zero_real )
=> ( ( ord_less_real @ B_164 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A_226 @ B_164 ) @ zero_zero_real ) ) ) ).
thf(fact_498_add__neg__neg,axiom,
! [B_164: nat,A_226: nat] :
( ( ord_less_nat @ A_226 @ zero_zero_nat )
=> ( ( ord_less_nat @ B_164 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A_226 @ B_164 ) @ zero_zero_nat ) ) ) ).
thf(fact_499_add__neg__neg,axiom,
! [B_164: int,A_226: int] :
( ( ord_less_int @ A_226 @ zero_zero_int )
=> ( ( ord_less_int @ B_164 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A_226 @ B_164 ) @ zero_zero_int ) ) ) ).
thf(fact_500_add__neg__neg,axiom,
! [B_164: code_code_numeral,A_226: code_code_numeral] :
( ( ord_le1304079648umeral @ A_226 @ zero_z126310315umeral )
=> ( ( ord_le1304079648umeral @ B_164 @ zero_z126310315umeral )
=> ( ord_le1304079648umeral @ ( plus_p1627245867umeral @ A_226 @ B_164 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_501_tpos,axiom,
ord_less_eq_int @ one_one_int @ t ).
thf(fact_502_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_503_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_504_zle__trans,axiom,
! [K: int,I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ord_less_eq_int @ I @ K ) ) ) ).
thf(fact_505_zle__linear,axiom,
! [Z_1: int,W: int] :
( ( ord_less_eq_int @ Z_1 @ W )
| ( ord_less_eq_int @ W @ Z_1 ) ) ).
thf(fact_506_zle__refl,axiom,
! [W: int] : ( ord_less_eq_int @ W @ W ) ).
thf(fact_507_le__number__of,axiom,
! [X_89: int,Y_74: int] :
( ( ord_less_eq_rat @ ( number_number_of_rat @ X_89 ) @ ( number_number_of_rat @ Y_74 ) )
<=> ( ord_less_eq_int @ X_89 @ Y_74 ) ) ).
thf(fact_508_le__number__of,axiom,
! [X_89: int,Y_74: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_89 ) @ ( number267125858f_real @ Y_74 ) )
<=> ( ord_less_eq_int @ X_89 @ Y_74 ) ) ).
thf(fact_509_le__number__of,axiom,
! [X_89: int,Y_74: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_89 ) @ ( number_number_of_int @ Y_74 ) )
<=> ( ord_less_eq_int @ X_89 @ Y_74 ) ) ).
thf(fact_510_of__nat__le__iff,axiom,
! [M_24: nat,N_42: nat] :
( ( ord_less_eq_rat @ ( semiri151668891at_rat @ M_24 ) @ ( semiri151668891at_rat @ N_42 ) )
<=> ( ord_less_eq_nat @ M_24 @ N_42 ) ) ).
thf(fact_511_of__nat__le__iff,axiom,
! [M_24: nat,N_42: nat] :
( ( ord_less_eq_real @ ( semiri132038758t_real @ M_24 ) @ ( semiri132038758t_real @ N_42 ) )
<=> ( ord_less_eq_nat @ M_24 @ N_42 ) ) ).
thf(fact_512_of__nat__le__iff,axiom,
! [M_24: nat,N_42: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ M_24 ) @ ( semiri1621563631at_int @ N_42 ) )
<=> ( ord_less_eq_nat @ M_24 @ N_42 ) ) ).
thf(fact_513_of__nat__le__iff,axiom,
! [M_24: nat,N_42: nat] :
( ( ord_le258702272de_int @ ( semiri1424489471de_int @ M_24 ) @ ( semiri1424489471de_int @ N_42 ) )
<=> ( ord_less_eq_nat @ M_24 @ N_42 ) ) ).
thf(fact_514_of__nat__le__iff,axiom,
! [M_24: nat,N_42: nat] :
( ( ord_less_eq_nat @ ( semiri984289939at_nat @ M_24 ) @ ( semiri984289939at_nat @ N_42 ) )
<=> ( ord_less_eq_nat @ M_24 @ N_42 ) ) ).
thf(fact_515_add__le__imp__le__left,axiom,
! [C_100: rat,A_225: rat,B_163: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C_100 @ A_225 ) @ ( plus_plus_rat @ C_100 @ B_163 ) )
=> ( ord_less_eq_rat @ A_225 @ B_163 ) ) ).
thf(fact_516_add__le__imp__le__left,axiom,
! [C_100: quickcheck_code_int,A_225: quickcheck_code_int,B_163: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( plus_p1446045655de_int @ C_100 @ A_225 ) @ ( plus_p1446045655de_int @ C_100 @ B_163 ) )
=> ( ord_le258702272de_int @ A_225 @ B_163 ) ) ).
thf(fact_517_add__le__imp__le__left,axiom,
! [C_100: real,A_225: real,B_163: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C_100 @ A_225 ) @ ( plus_plus_real @ C_100 @ B_163 ) )
=> ( ord_less_eq_real @ A_225 @ B_163 ) ) ).
thf(fact_518_add__le__imp__le__left,axiom,
! [C_100: nat,A_225: nat,B_163: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_100 @ A_225 ) @ ( plus_plus_nat @ C_100 @ B_163 ) )
=> ( ord_less_eq_nat @ A_225 @ B_163 ) ) ).
thf(fact_519_add__le__imp__le__left,axiom,
! [C_100: int,A_225: int,B_163: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_100 @ A_225 ) @ ( plus_plus_int @ C_100 @ B_163 ) )
=> ( ord_less_eq_int @ A_225 @ B_163 ) ) ).
thf(fact_520_add__le__imp__le__right,axiom,
! [A_224: rat,C_99: rat,B_162: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A_224 @ C_99 ) @ ( plus_plus_rat @ B_162 @ C_99 ) )
=> ( ord_less_eq_rat @ A_224 @ B_162 ) ) ).
thf(fact_521_add__le__imp__le__right,axiom,
! [A_224: quickcheck_code_int,C_99: quickcheck_code_int,B_162: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( plus_p1446045655de_int @ A_224 @ C_99 ) @ ( plus_p1446045655de_int @ B_162 @ C_99 ) )
=> ( ord_le258702272de_int @ A_224 @ B_162 ) ) ).
thf(fact_522_add__le__imp__le__right,axiom,
! [A_224: real,C_99: real,B_162: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A_224 @ C_99 ) @ ( plus_plus_real @ B_162 @ C_99 ) )
=> ( ord_less_eq_real @ A_224 @ B_162 ) ) ).
thf(fact_523_add__le__imp__le__right,axiom,
! [A_224: nat,C_99: nat,B_162: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_224 @ C_99 ) @ ( plus_plus_nat @ B_162 @ C_99 ) )
=> ( ord_less_eq_nat @ A_224 @ B_162 ) ) ).
thf(fact_524_add__le__imp__le__right,axiom,
! [A_224: int,C_99: int,B_162: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_224 @ C_99 ) @ ( plus_plus_int @ B_162 @ C_99 ) )
=> ( ord_less_eq_int @ A_224 @ B_162 ) ) ).
thf(fact_525_add__mono,axiom,
! [C_98: rat,D_35: rat,A_223: rat,B_161: rat] :
( ( ord_less_eq_rat @ A_223 @ B_161 )
=> ( ( ord_less_eq_rat @ C_98 @ D_35 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A_223 @ C_98 ) @ ( plus_plus_rat @ B_161 @ D_35 ) ) ) ) ).
thf(fact_526_add__mono,axiom,
! [C_98: quickcheck_code_int,D_35: quickcheck_code_int,A_223: quickcheck_code_int,B_161: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_223 @ B_161 )
=> ( ( ord_le258702272de_int @ C_98 @ D_35 )
=> ( ord_le258702272de_int @ ( plus_p1446045655de_int @ A_223 @ C_98 ) @ ( plus_p1446045655de_int @ B_161 @ D_35 ) ) ) ) ).
thf(fact_527_add__mono,axiom,
! [C_98: real,D_35: real,A_223: real,B_161: real] :
( ( ord_less_eq_real @ A_223 @ B_161 )
=> ( ( ord_less_eq_real @ C_98 @ D_35 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A_223 @ C_98 ) @ ( plus_plus_real @ B_161 @ D_35 ) ) ) ) ).
thf(fact_528_add__mono,axiom,
! [C_98: nat,D_35: nat,A_223: nat,B_161: nat] :
( ( ord_less_eq_nat @ A_223 @ B_161 )
=> ( ( ord_less_eq_nat @ C_98 @ D_35 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_223 @ C_98 ) @ ( plus_plus_nat @ B_161 @ D_35 ) ) ) ) ).
thf(fact_529_add__mono,axiom,
! [C_98: int,D_35: int,A_223: int,B_161: int] :
( ( ord_less_eq_int @ A_223 @ B_161 )
=> ( ( ord_less_eq_int @ C_98 @ D_35 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_223 @ C_98 ) @ ( plus_plus_int @ B_161 @ D_35 ) ) ) ) ).
thf(fact_530_add__left__mono,axiom,
! [C_97: rat,A_222: rat,B_160: rat] :
( ( ord_less_eq_rat @ A_222 @ B_160 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ C_97 @ A_222 ) @ ( plus_plus_rat @ C_97 @ B_160 ) ) ) ).
thf(fact_531_add__left__mono,axiom,
! [C_97: quickcheck_code_int,A_222: quickcheck_code_int,B_160: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_222 @ B_160 )
=> ( ord_le258702272de_int @ ( plus_p1446045655de_int @ C_97 @ A_222 ) @ ( plus_p1446045655de_int @ C_97 @ B_160 ) ) ) ).
thf(fact_532_add__left__mono,axiom,
! [C_97: real,A_222: real,B_160: real] :
( ( ord_less_eq_real @ A_222 @ B_160 )
=> ( ord_less_eq_real @ ( plus_plus_real @ C_97 @ A_222 ) @ ( plus_plus_real @ C_97 @ B_160 ) ) ) ).
thf(fact_533_add__left__mono,axiom,
! [C_97: nat,A_222: nat,B_160: nat] :
( ( ord_less_eq_nat @ A_222 @ B_160 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C_97 @ A_222 ) @ ( plus_plus_nat @ C_97 @ B_160 ) ) ) ).
thf(fact_534_add__left__mono,axiom,
! [C_97: int,A_222: int,B_160: int] :
( ( ord_less_eq_int @ A_222 @ B_160 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C_97 @ A_222 ) @ ( plus_plus_int @ C_97 @ B_160 ) ) ) ).
thf(fact_535_add__right__mono,axiom,
! [C_96: rat,A_221: rat,B_159: rat] :
( ( ord_less_eq_rat @ A_221 @ B_159 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A_221 @ C_96 ) @ ( plus_plus_rat @ B_159 @ C_96 ) ) ) ).
thf(fact_536_add__right__mono,axiom,
! [C_96: quickcheck_code_int,A_221: quickcheck_code_int,B_159: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_221 @ B_159 )
=> ( ord_le258702272de_int @ ( plus_p1446045655de_int @ A_221 @ C_96 ) @ ( plus_p1446045655de_int @ B_159 @ C_96 ) ) ) ).
thf(fact_537_add__right__mono,axiom,
! [C_96: real,A_221: real,B_159: real] :
( ( ord_less_eq_real @ A_221 @ B_159 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A_221 @ C_96 ) @ ( plus_plus_real @ B_159 @ C_96 ) ) ) ).
thf(fact_538_add__right__mono,axiom,
! [C_96: nat,A_221: nat,B_159: nat] :
( ( ord_less_eq_nat @ A_221 @ B_159 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_221 @ C_96 ) @ ( plus_plus_nat @ B_159 @ C_96 ) ) ) ).
thf(fact_539_add__right__mono,axiom,
! [C_96: int,A_221: int,B_159: int] :
( ( ord_less_eq_int @ A_221 @ B_159 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_221 @ C_96 ) @ ( plus_plus_int @ B_159 @ C_96 ) ) ) ).
thf(fact_540_add__le__cancel__left,axiom,
! [C_95: rat,A_220: rat,B_158: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C_95 @ A_220 ) @ ( plus_plus_rat @ C_95 @ B_158 ) )
<=> ( ord_less_eq_rat @ A_220 @ B_158 ) ) ).
thf(fact_541_add__le__cancel__left,axiom,
! [C_95: quickcheck_code_int,A_220: quickcheck_code_int,B_158: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( plus_p1446045655de_int @ C_95 @ A_220 ) @ ( plus_p1446045655de_int @ C_95 @ B_158 ) )
<=> ( ord_le258702272de_int @ A_220 @ B_158 ) ) ).
thf(fact_542_add__le__cancel__left,axiom,
! [C_95: real,A_220: real,B_158: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C_95 @ A_220 ) @ ( plus_plus_real @ C_95 @ B_158 ) )
<=> ( ord_less_eq_real @ A_220 @ B_158 ) ) ).
thf(fact_543_add__le__cancel__left,axiom,
! [C_95: nat,A_220: nat,B_158: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_95 @ A_220 ) @ ( plus_plus_nat @ C_95 @ B_158 ) )
<=> ( ord_less_eq_nat @ A_220 @ B_158 ) ) ).
thf(fact_544_add__le__cancel__left,axiom,
! [C_95: int,A_220: int,B_158: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_95 @ A_220 ) @ ( plus_plus_int @ C_95 @ B_158 ) )
<=> ( ord_less_eq_int @ A_220 @ B_158 ) ) ).
thf(fact_545_add__le__cancel__right,axiom,
! [A_219: rat,C_94: rat,B_157: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A_219 @ C_94 ) @ ( plus_plus_rat @ B_157 @ C_94 ) )
<=> ( ord_less_eq_rat @ A_219 @ B_157 ) ) ).
thf(fact_546_add__le__cancel__right,axiom,
! [A_219: quickcheck_code_int,C_94: quickcheck_code_int,B_157: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( plus_p1446045655de_int @ A_219 @ C_94 ) @ ( plus_p1446045655de_int @ B_157 @ C_94 ) )
<=> ( ord_le258702272de_int @ A_219 @ B_157 ) ) ).
thf(fact_547_add__le__cancel__right,axiom,
! [A_219: real,C_94: real,B_157: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A_219 @ C_94 ) @ ( plus_plus_real @ B_157 @ C_94 ) )
<=> ( ord_less_eq_real @ A_219 @ B_157 ) ) ).
thf(fact_548_add__le__cancel__right,axiom,
! [A_219: nat,C_94: nat,B_157: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_219 @ C_94 ) @ ( plus_plus_nat @ B_157 @ C_94 ) )
<=> ( ord_less_eq_nat @ A_219 @ B_157 ) ) ).
thf(fact_549_add__le__cancel__right,axiom,
! [A_219: int,C_94: int,B_157: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_219 @ C_94 ) @ ( plus_plus_int @ B_157 @ C_94 ) )
<=> ( ord_less_eq_int @ A_219 @ B_157 ) ) ).
thf(fact_550_power__increasing,axiom,
! [A_218: rat,N_41: nat,N_40: nat] :
( ( ord_less_eq_nat @ N_41 @ N_40 )
=> ( ( ord_less_eq_rat @ one_one_rat @ A_218 )
=> ( ord_less_eq_rat @ ( power_power_rat @ A_218 @ N_41 ) @ ( power_power_rat @ A_218 @ N_40 ) ) ) ) ).
thf(fact_551_power__increasing,axiom,
! [A_218: quickcheck_code_int,N_41: nat,N_40: nat] :
( ( ord_less_eq_nat @ N_41 @ N_40 )
=> ( ( ord_le258702272de_int @ one_on1684967323de_int @ A_218 )
=> ( ord_le258702272de_int @ ( power_881366806de_int @ A_218 @ N_41 ) @ ( power_881366806de_int @ A_218 @ N_40 ) ) ) ) ).
thf(fact_552_power__increasing,axiom,
! [A_218: code_code_numeral,N_41: nat,N_40: nat] :
( ( ord_less_eq_nat @ N_41 @ N_40 )
=> ( ( ord_le565307924umeral @ one_on1645066479umeral @ A_218 )
=> ( ord_le565307924umeral @ ( power_2100829034umeral @ A_218 @ N_41 ) @ ( power_2100829034umeral @ A_218 @ N_40 ) ) ) ) ).
thf(fact_553_power__increasing,axiom,
! [A_218: real,N_41: nat,N_40: nat] :
( ( ord_less_eq_nat @ N_41 @ N_40 )
=> ( ( ord_less_eq_real @ one_one_real @ A_218 )
=> ( ord_less_eq_real @ ( power_power_real @ A_218 @ N_41 ) @ ( power_power_real @ A_218 @ N_40 ) ) ) ) ).
thf(fact_554_power__increasing,axiom,
! [A_218: nat,N_41: nat,N_40: nat] :
( ( ord_less_eq_nat @ N_41 @ N_40 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A_218 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A_218 @ N_41 ) @ ( power_power_nat @ A_218 @ N_40 ) ) ) ) ).
thf(fact_555_power__increasing,axiom,
! [A_218: int,N_41: nat,N_40: nat] :
( ( ord_less_eq_nat @ N_41 @ N_40 )
=> ( ( ord_less_eq_int @ one_one_int @ A_218 )
=> ( ord_less_eq_int @ ( power_power_int @ A_218 @ N_41 ) @ ( power_power_int @ A_218 @ N_40 ) ) ) ) ).
thf(fact_556_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_557_rel__simps_I34_J,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_558_rel__simps_I19_J,axiom,
ord_less_eq_int @ pls @ pls ).
thf(fact_559_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_560_rel__simps_I31_J,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_561_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
thf(fact_562_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_563_zadd__left__mono,axiom,
! [K: int,I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ord_less_eq_int @ ( plus_plus_int @ K @ I ) @ ( plus_plus_int @ K @ J ) ) ) ).
thf(fact_564_less__eq__number__of__int__code,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ K ) @ ( number_number_of_int @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_565_add__nonpos__nonpos,axiom,
! [B_156: rat,A_217: rat] :
( ( ord_less_eq_rat @ A_217 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B_156 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A_217 @ B_156 ) @ zero_zero_rat ) ) ) ).
thf(fact_566_add__nonpos__nonpos,axiom,
! [B_156: quickcheck_code_int,A_217: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_217 @ zero_z891286103de_int )
=> ( ( ord_le258702272de_int @ B_156 @ zero_z891286103de_int )
=> ( ord_le258702272de_int @ ( plus_p1446045655de_int @ A_217 @ B_156 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_567_add__nonpos__nonpos,axiom,
! [B_156: real,A_217: real] :
( ( ord_less_eq_real @ A_217 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B_156 @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A_217 @ B_156 ) @ zero_zero_real ) ) ) ).
thf(fact_568_add__nonpos__nonpos,axiom,
! [B_156: nat,A_217: nat] :
( ( ord_less_eq_nat @ A_217 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B_156 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_217 @ B_156 ) @ zero_zero_nat ) ) ) ).
thf(fact_569_add__nonpos__nonpos,axiom,
! [B_156: int,A_217: int] :
( ( ord_less_eq_int @ A_217 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B_156 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_217 @ B_156 ) @ zero_zero_int ) ) ) ).
thf(fact_570_add__nonpos__nonpos,axiom,
! [B_156: code_code_numeral,A_217: code_code_numeral] :
( ( ord_le565307924umeral @ A_217 @ zero_z126310315umeral )
=> ( ( ord_le565307924umeral @ B_156 @ zero_z126310315umeral )
=> ( ord_le565307924umeral @ ( plus_p1627245867umeral @ A_217 @ B_156 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_571_add__increasing2,axiom,
! [B_155: rat,A_216: rat,C_93: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ C_93 )
=> ( ( ord_less_eq_rat @ B_155 @ A_216 )
=> ( ord_less_eq_rat @ B_155 @ ( plus_plus_rat @ A_216 @ C_93 ) ) ) ) ).
thf(fact_572_add__increasing2,axiom,
! [B_155: quickcheck_code_int,A_216: quickcheck_code_int,C_93: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ C_93 )
=> ( ( ord_le258702272de_int @ B_155 @ A_216 )
=> ( ord_le258702272de_int @ B_155 @ ( plus_p1446045655de_int @ A_216 @ C_93 ) ) ) ) ).
thf(fact_573_add__increasing2,axiom,
! [B_155: real,A_216: real,C_93: real] :
( ( ord_less_eq_real @ zero_zero_real @ C_93 )
=> ( ( ord_less_eq_real @ B_155 @ A_216 )
=> ( ord_less_eq_real @ B_155 @ ( plus_plus_real @ A_216 @ C_93 ) ) ) ) ).
thf(fact_574_add__increasing2,axiom,
! [B_155: nat,A_216: nat,C_93: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C_93 )
=> ( ( ord_less_eq_nat @ B_155 @ A_216 )
=> ( ord_less_eq_nat @ B_155 @ ( plus_plus_nat @ A_216 @ C_93 ) ) ) ) ).
thf(fact_575_add__increasing2,axiom,
! [B_155: int,A_216: int,C_93: int] :
( ( ord_less_eq_int @ zero_zero_int @ C_93 )
=> ( ( ord_less_eq_int @ B_155 @ A_216 )
=> ( ord_less_eq_int @ B_155 @ ( plus_plus_int @ A_216 @ C_93 ) ) ) ) ).
thf(fact_576_add__increasing2,axiom,
! [B_155: code_code_numeral,A_216: code_code_numeral,C_93: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ C_93 )
=> ( ( ord_le565307924umeral @ B_155 @ A_216 )
=> ( ord_le565307924umeral @ B_155 @ ( plus_p1627245867umeral @ A_216 @ C_93 ) ) ) ) ).
thf(fact_577_add__increasing,axiom,
! [B_154: rat,C_92: rat,A_215: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_215 )
=> ( ( ord_less_eq_rat @ B_154 @ C_92 )
=> ( ord_less_eq_rat @ B_154 @ ( plus_plus_rat @ A_215 @ C_92 ) ) ) ) ).
thf(fact_578_add__increasing,axiom,
! [B_154: quickcheck_code_int,C_92: quickcheck_code_int,A_215: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_215 )
=> ( ( ord_le258702272de_int @ B_154 @ C_92 )
=> ( ord_le258702272de_int @ B_154 @ ( plus_p1446045655de_int @ A_215 @ C_92 ) ) ) ) ).
thf(fact_579_add__increasing,axiom,
! [B_154: real,C_92: real,A_215: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_215 )
=> ( ( ord_less_eq_real @ B_154 @ C_92 )
=> ( ord_less_eq_real @ B_154 @ ( plus_plus_real @ A_215 @ C_92 ) ) ) ) ).
thf(fact_580_add__increasing,axiom,
! [B_154: nat,C_92: nat,A_215: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_215 )
=> ( ( ord_less_eq_nat @ B_154 @ C_92 )
=> ( ord_less_eq_nat @ B_154 @ ( plus_plus_nat @ A_215 @ C_92 ) ) ) ) ).
thf(fact_581_add__increasing,axiom,
! [B_154: int,C_92: int,A_215: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_215 )
=> ( ( ord_less_eq_int @ B_154 @ C_92 )
=> ( ord_less_eq_int @ B_154 @ ( plus_plus_int @ A_215 @ C_92 ) ) ) ) ).
thf(fact_582_add__increasing,axiom,
! [B_154: code_code_numeral,C_92: code_code_numeral,A_215: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_215 )
=> ( ( ord_le565307924umeral @ B_154 @ C_92 )
=> ( ord_le565307924umeral @ B_154 @ ( plus_p1627245867umeral @ A_215 @ C_92 ) ) ) ) ).
thf(fact_583_add__nonneg__eq__0__iff,axiom,
! [Y_73: rat,X_88: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X_88 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y_73 )
=> ( ( ( plus_plus_rat @ X_88 @ Y_73 )
= zero_zero_rat )
<=> ( ( X_88 = zero_zero_rat )
& ( Y_73 = zero_zero_rat ) ) ) ) ) ).
thf(fact_584_add__nonneg__eq__0__iff,axiom,
! [Y_73: quickcheck_code_int,X_88: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ X_88 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ Y_73 )
=> ( ( ( plus_p1446045655de_int @ X_88 @ Y_73 )
= zero_z891286103de_int )
<=> ( ( X_88 = zero_z891286103de_int )
& ( Y_73 = zero_z891286103de_int ) ) ) ) ) ).
thf(fact_585_add__nonneg__eq__0__iff,axiom,
! [Y_73: real,X_88: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_88 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_73 )
=> ( ( ( plus_plus_real @ X_88 @ Y_73 )
= zero_zero_real )
<=> ( ( X_88 = zero_zero_real )
& ( Y_73 = zero_zero_real ) ) ) ) ) ).
thf(fact_586_add__nonneg__eq__0__iff,axiom,
! [Y_73: nat,X_88: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X_88 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_73 )
=> ( ( ( plus_plus_nat @ X_88 @ Y_73 )
= zero_zero_nat )
<=> ( ( X_88 = zero_zero_nat )
& ( Y_73 = zero_zero_nat ) ) ) ) ) ).
thf(fact_587_add__nonneg__eq__0__iff,axiom,
! [Y_73: int,X_88: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_88 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_73 )
=> ( ( ( plus_plus_int @ X_88 @ Y_73 )
= zero_zero_int )
<=> ( ( X_88 = zero_zero_int )
& ( Y_73 = zero_zero_int ) ) ) ) ) ).
thf(fact_588_add__nonneg__eq__0__iff,axiom,
! [Y_73: code_code_numeral,X_88: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ X_88 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ Y_73 )
=> ( ( ( plus_p1627245867umeral @ X_88 @ Y_73 )
= zero_z126310315umeral )
<=> ( ( X_88 = zero_z126310315umeral )
& ( Y_73 = zero_z126310315umeral ) ) ) ) ) ).
thf(fact_589_add__nonneg__nonneg,axiom,
! [B_153: rat,A_214: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_214 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_153 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A_214 @ B_153 ) ) ) ) ).
thf(fact_590_add__nonneg__nonneg,axiom,
! [B_153: quickcheck_code_int,A_214: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_214 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_153 )
=> ( ord_le258702272de_int @ zero_z891286103de_int @ ( plus_p1446045655de_int @ A_214 @ B_153 ) ) ) ) ).
thf(fact_591_add__nonneg__nonneg,axiom,
! [B_153: real,A_214: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_214 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_153 )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A_214 @ B_153 ) ) ) ) ).
thf(fact_592_add__nonneg__nonneg,axiom,
! [B_153: nat,A_214: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_214 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_153 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A_214 @ B_153 ) ) ) ) ).
thf(fact_593_add__nonneg__nonneg,axiom,
! [B_153: int,A_214: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_214 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_153 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A_214 @ B_153 ) ) ) ) ).
thf(fact_594_add__nonneg__nonneg,axiom,
! [B_153: code_code_numeral,A_214: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_214 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_153 )
=> ( ord_le565307924umeral @ zero_z126310315umeral @ ( plus_p1627245867umeral @ A_214 @ B_153 ) ) ) ) ).
thf(fact_595_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A_213: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A_213 @ A_213 ) @ zero_zero_rat )
<=> ( ord_less_eq_rat @ A_213 @ zero_zero_rat ) ) ).
thf(fact_596_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A_213: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A_213 @ A_213 ) @ zero_zero_real )
<=> ( ord_less_eq_real @ A_213 @ zero_zero_real ) ) ).
thf(fact_597_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A_213: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_213 @ A_213 ) @ zero_zero_int )
<=> ( ord_less_eq_int @ A_213 @ zero_zero_int ) ) ).
thf(fact_598_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A_212: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A_212 @ A_212 ) )
<=> ( ord_less_eq_rat @ zero_zero_rat @ A_212 ) ) ).
thf(fact_599_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A_212: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A_212 @ A_212 ) )
<=> ( ord_less_eq_real @ zero_zero_real @ A_212 ) ) ).
thf(fact_600_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A_212: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A_212 @ A_212 ) )
<=> ( ord_less_eq_int @ zero_zero_int @ A_212 ) ) ).
thf(fact_601_add__le__less__mono,axiom,
! [C_91: rat,D_34: rat,A_211: rat,B_152: rat] :
( ( ord_less_eq_rat @ A_211 @ B_152 )
=> ( ( ord_less_rat @ C_91 @ D_34 )
=> ( ord_less_rat @ ( plus_plus_rat @ A_211 @ C_91 ) @ ( plus_plus_rat @ B_152 @ D_34 ) ) ) ) ).
thf(fact_602_add__le__less__mono,axiom,
! [C_91: quickcheck_code_int,D_34: quickcheck_code_int,A_211: quickcheck_code_int,B_152: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_211 @ B_152 )
=> ( ( ord_le1860547276de_int @ C_91 @ D_34 )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_211 @ C_91 ) @ ( plus_p1446045655de_int @ B_152 @ D_34 ) ) ) ) ).
thf(fact_603_add__le__less__mono,axiom,
! [C_91: real,D_34: real,A_211: real,B_152: real] :
( ( ord_less_eq_real @ A_211 @ B_152 )
=> ( ( ord_less_real @ C_91 @ D_34 )
=> ( ord_less_real @ ( plus_plus_real @ A_211 @ C_91 ) @ ( plus_plus_real @ B_152 @ D_34 ) ) ) ) ).
thf(fact_604_add__le__less__mono,axiom,
! [C_91: nat,D_34: nat,A_211: nat,B_152: nat] :
( ( ord_less_eq_nat @ A_211 @ B_152 )
=> ( ( ord_less_nat @ C_91 @ D_34 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_211 @ C_91 ) @ ( plus_plus_nat @ B_152 @ D_34 ) ) ) ) ).
thf(fact_605_add__le__less__mono,axiom,
! [C_91: int,D_34: int,A_211: int,B_152: int] :
( ( ord_less_eq_int @ A_211 @ B_152 )
=> ( ( ord_less_int @ C_91 @ D_34 )
=> ( ord_less_int @ ( plus_plus_int @ A_211 @ C_91 ) @ ( plus_plus_int @ B_152 @ D_34 ) ) ) ) ).
thf(fact_606_add__less__le__mono,axiom,
! [C_90: rat,D_33: rat,A_210: rat,B_151: rat] :
( ( ord_less_rat @ A_210 @ B_151 )
=> ( ( ord_less_eq_rat @ C_90 @ D_33 )
=> ( ord_less_rat @ ( plus_plus_rat @ A_210 @ C_90 ) @ ( plus_plus_rat @ B_151 @ D_33 ) ) ) ) ).
thf(fact_607_add__less__le__mono,axiom,
! [C_90: quickcheck_code_int,D_33: quickcheck_code_int,A_210: quickcheck_code_int,B_151: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_210 @ B_151 )
=> ( ( ord_le258702272de_int @ C_90 @ D_33 )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_210 @ C_90 ) @ ( plus_p1446045655de_int @ B_151 @ D_33 ) ) ) ) ).
thf(fact_608_add__less__le__mono,axiom,
! [C_90: real,D_33: real,A_210: real,B_151: real] :
( ( ord_less_real @ A_210 @ B_151 )
=> ( ( ord_less_eq_real @ C_90 @ D_33 )
=> ( ord_less_real @ ( plus_plus_real @ A_210 @ C_90 ) @ ( plus_plus_real @ B_151 @ D_33 ) ) ) ) ).
thf(fact_609_add__less__le__mono,axiom,
! [C_90: nat,D_33: nat,A_210: nat,B_151: nat] :
( ( ord_less_nat @ A_210 @ B_151 )
=> ( ( ord_less_eq_nat @ C_90 @ D_33 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_210 @ C_90 ) @ ( plus_plus_nat @ B_151 @ D_33 ) ) ) ) ).
thf(fact_610_add__less__le__mono,axiom,
! [C_90: int,D_33: int,A_210: int,B_151: int] :
( ( ord_less_int @ A_210 @ B_151 )
=> ( ( ord_less_eq_int @ C_90 @ D_33 )
=> ( ord_less_int @ ( plus_plus_int @ A_210 @ C_90 ) @ ( plus_plus_int @ B_151 @ D_33 ) ) ) ) ).
thf(fact_611_le__special_I3_J,axiom,
! [X_87: int] :
( ( ord_less_eq_rat @ ( number_number_of_rat @ X_87 ) @ zero_zero_rat )
<=> ( ord_less_eq_int @ X_87 @ pls ) ) ).
thf(fact_612_le__special_I3_J,axiom,
! [X_87: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_87 ) @ zero_zero_real )
<=> ( ord_less_eq_int @ X_87 @ pls ) ) ).
thf(fact_613_le__special_I3_J,axiom,
! [X_87: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_87 ) @ zero_zero_int )
<=> ( ord_less_eq_int @ X_87 @ pls ) ) ).
thf(fact_614_le__special_I1_J,axiom,
! [Y_72: int] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( number_number_of_rat @ Y_72 ) )
<=> ( ord_less_eq_int @ pls @ Y_72 ) ) ).
thf(fact_615_le__special_I1_J,axiom,
! [Y_72: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( number267125858f_real @ Y_72 ) )
<=> ( ord_less_eq_int @ pls @ Y_72 ) ) ).
thf(fact_616_le__special_I1_J,axiom,
! [Y_72: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ Y_72 ) )
<=> ( ord_less_eq_int @ pls @ Y_72 ) ) ).
thf(fact_617_power__decreasing,axiom,
! [A_209: rat,N_39: nat,N_38: nat] :
( ( ord_less_eq_nat @ N_39 @ N_38 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A_209 )
=> ( ( ord_less_eq_rat @ A_209 @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A_209 @ N_38 ) @ ( power_power_rat @ A_209 @ N_39 ) ) ) ) ) ).
thf(fact_618_power__decreasing,axiom,
! [A_209: quickcheck_code_int,N_39: nat,N_38: nat] :
( ( ord_less_eq_nat @ N_39 @ N_38 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_209 )
=> ( ( ord_le258702272de_int @ A_209 @ one_on1684967323de_int )
=> ( ord_le258702272de_int @ ( power_881366806de_int @ A_209 @ N_38 ) @ ( power_881366806de_int @ A_209 @ N_39 ) ) ) ) ) ).
thf(fact_619_power__decreasing,axiom,
! [A_209: code_code_numeral,N_39: nat,N_38: nat] :
( ( ord_less_eq_nat @ N_39 @ N_38 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_209 )
=> ( ( ord_le565307924umeral @ A_209 @ one_on1645066479umeral )
=> ( ord_le565307924umeral @ ( power_2100829034umeral @ A_209 @ N_38 ) @ ( power_2100829034umeral @ A_209 @ N_39 ) ) ) ) ) ).
thf(fact_620_power__decreasing,axiom,
! [A_209: real,N_39: nat,N_38: nat] :
( ( ord_less_eq_nat @ N_39 @ N_38 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_209 )
=> ( ( ord_less_eq_real @ A_209 @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A_209 @ N_38 ) @ ( power_power_real @ A_209 @ N_39 ) ) ) ) ) ).
thf(fact_621_power__decreasing,axiom,
! [A_209: nat,N_39: nat,N_38: nat] :
( ( ord_less_eq_nat @ N_39 @ N_38 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_209 )
=> ( ( ord_less_eq_nat @ A_209 @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A_209 @ N_38 ) @ ( power_power_nat @ A_209 @ N_39 ) ) ) ) ) ).
thf(fact_622_power__decreasing,axiom,
! [A_209: int,N_39: nat,N_38: nat] :
( ( ord_less_eq_nat @ N_39 @ N_38 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_209 )
=> ( ( ord_less_eq_int @ A_209 @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A_209 @ N_38 ) @ ( power_power_int @ A_209 @ N_39 ) ) ) ) ) ).
thf(fact_623_power__le__imp__le__exp,axiom,
! [M_23: nat,N_37: nat,A_208: rat] :
( ( ord_less_rat @ one_one_rat @ A_208 )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ A_208 @ M_23 ) @ ( power_power_rat @ A_208 @ N_37 ) )
=> ( ord_less_eq_nat @ M_23 @ N_37 ) ) ) ).
thf(fact_624_power__le__imp__le__exp,axiom,
! [M_23: nat,N_37: nat,A_208: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ A_208 )
=> ( ( ord_le258702272de_int @ ( power_881366806de_int @ A_208 @ M_23 ) @ ( power_881366806de_int @ A_208 @ N_37 ) )
=> ( ord_less_eq_nat @ M_23 @ N_37 ) ) ) ).
thf(fact_625_power__le__imp__le__exp,axiom,
! [M_23: nat,N_37: nat,A_208: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ A_208 )
=> ( ( ord_le565307924umeral @ ( power_2100829034umeral @ A_208 @ M_23 ) @ ( power_2100829034umeral @ A_208 @ N_37 ) )
=> ( ord_less_eq_nat @ M_23 @ N_37 ) ) ) ).
thf(fact_626_power__le__imp__le__exp,axiom,
! [M_23: nat,N_37: nat,A_208: real] :
( ( ord_less_real @ one_one_real @ A_208 )
=> ( ( ord_less_eq_real @ ( power_power_real @ A_208 @ M_23 ) @ ( power_power_real @ A_208 @ N_37 ) )
=> ( ord_less_eq_nat @ M_23 @ N_37 ) ) ) ).
thf(fact_627_power__le__imp__le__exp,axiom,
! [M_23: nat,N_37: nat,A_208: nat] :
( ( ord_less_nat @ one_one_nat @ A_208 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A_208 @ M_23 ) @ ( power_power_nat @ A_208 @ N_37 ) )
=> ( ord_less_eq_nat @ M_23 @ N_37 ) ) ) ).
thf(fact_628_power__le__imp__le__exp,axiom,
! [M_23: nat,N_37: nat,A_208: int] :
( ( ord_less_int @ one_one_int @ A_208 )
=> ( ( ord_less_eq_int @ ( power_power_int @ A_208 @ M_23 ) @ ( power_power_int @ A_208 @ N_37 ) )
=> ( ord_less_eq_nat @ M_23 @ N_37 ) ) ) ).
thf(fact_629_power__increasing__iff,axiom,
! [X_86: nat,Y_71: nat,B_150: rat] :
( ( ord_less_rat @ one_one_rat @ B_150 )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ B_150 @ X_86 ) @ ( power_power_rat @ B_150 @ Y_71 ) )
<=> ( ord_less_eq_nat @ X_86 @ Y_71 ) ) ) ).
thf(fact_630_power__increasing__iff,axiom,
! [X_86: nat,Y_71: nat,B_150: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ B_150 )
=> ( ( ord_le258702272de_int @ ( power_881366806de_int @ B_150 @ X_86 ) @ ( power_881366806de_int @ B_150 @ Y_71 ) )
<=> ( ord_less_eq_nat @ X_86 @ Y_71 ) ) ) ).
thf(fact_631_power__increasing__iff,axiom,
! [X_86: nat,Y_71: nat,B_150: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ B_150 )
=> ( ( ord_le565307924umeral @ ( power_2100829034umeral @ B_150 @ X_86 ) @ ( power_2100829034umeral @ B_150 @ Y_71 ) )
<=> ( ord_less_eq_nat @ X_86 @ Y_71 ) ) ) ).
thf(fact_632_power__increasing__iff,axiom,
! [X_86: nat,Y_71: nat,B_150: real] :
( ( ord_less_real @ one_one_real @ B_150 )
=> ( ( ord_less_eq_real @ ( power_power_real @ B_150 @ X_86 ) @ ( power_power_real @ B_150 @ Y_71 ) )
<=> ( ord_less_eq_nat @ X_86 @ Y_71 ) ) ) ).
thf(fact_633_power__increasing__iff,axiom,
! [X_86: nat,Y_71: nat,B_150: nat] :
( ( ord_less_nat @ one_one_nat @ B_150 )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B_150 @ X_86 ) @ ( power_power_nat @ B_150 @ Y_71 ) )
<=> ( ord_less_eq_nat @ X_86 @ Y_71 ) ) ) ).
thf(fact_634_power__increasing__iff,axiom,
! [X_86: nat,Y_71: nat,B_150: int] :
( ( ord_less_int @ one_one_int @ B_150 )
=> ( ( ord_less_eq_int @ ( power_power_int @ B_150 @ X_86 ) @ ( power_power_int @ B_150 @ Y_71 ) )
<=> ( ord_less_eq_nat @ X_86 @ Y_71 ) ) ) ).
thf(fact_635_le__special_I4_J,axiom,
! [X_85: int] :
( ( ord_less_eq_rat @ ( number_number_of_rat @ X_85 ) @ one_one_rat )
<=> ( ord_less_eq_int @ X_85 @ ( bit1 @ pls ) ) ) ).
thf(fact_636_le__special_I4_J,axiom,
! [X_85: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_85 ) @ one_one_real )
<=> ( ord_less_eq_int @ X_85 @ ( bit1 @ pls ) ) ) ).
thf(fact_637_le__special_I4_J,axiom,
! [X_85: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_85 ) @ one_one_int )
<=> ( ord_less_eq_int @ X_85 @ ( bit1 @ pls ) ) ) ).
thf(fact_638_le__special_I2_J,axiom,
! [Y_70: int] :
( ( ord_less_eq_rat @ one_one_rat @ ( number_number_of_rat @ Y_70 ) )
<=> ( ord_less_eq_int @ ( bit1 @ pls ) @ Y_70 ) ) ).
thf(fact_639_le__special_I2_J,axiom,
! [Y_70: int] :
( ( ord_less_eq_real @ one_one_real @ ( number267125858f_real @ Y_70 ) )
<=> ( ord_less_eq_int @ ( bit1 @ pls ) @ Y_70 ) ) ).
thf(fact_640_le__special_I2_J,axiom,
! [Y_70: int] :
( ( ord_less_eq_int @ one_one_int @ ( number_number_of_int @ Y_70 ) )
<=> ( ord_less_eq_int @ ( bit1 @ pls ) @ Y_70 ) ) ).
thf(fact_641_add__nonpos__neg,axiom,
! [B_149: rat,A_207: rat] :
( ( ord_less_eq_rat @ A_207 @ zero_zero_rat )
=> ( ( ord_less_rat @ B_149 @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A_207 @ B_149 ) @ zero_zero_rat ) ) ) ).
thf(fact_642_add__nonpos__neg,axiom,
! [B_149: quickcheck_code_int,A_207: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_207 @ zero_z891286103de_int )
=> ( ( ord_le1860547276de_int @ B_149 @ zero_z891286103de_int )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_207 @ B_149 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_643_add__nonpos__neg,axiom,
! [B_149: real,A_207: real] :
( ( ord_less_eq_real @ A_207 @ zero_zero_real )
=> ( ( ord_less_real @ B_149 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A_207 @ B_149 ) @ zero_zero_real ) ) ) ).
thf(fact_644_add__nonpos__neg,axiom,
! [B_149: nat,A_207: nat] :
( ( ord_less_eq_nat @ A_207 @ zero_zero_nat )
=> ( ( ord_less_nat @ B_149 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A_207 @ B_149 ) @ zero_zero_nat ) ) ) ).
thf(fact_645_add__nonpos__neg,axiom,
! [B_149: int,A_207: int] :
( ( ord_less_eq_int @ A_207 @ zero_zero_int )
=> ( ( ord_less_int @ B_149 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A_207 @ B_149 ) @ zero_zero_int ) ) ) ).
thf(fact_646_add__nonpos__neg,axiom,
! [B_149: code_code_numeral,A_207: code_code_numeral] :
( ( ord_le565307924umeral @ A_207 @ zero_z126310315umeral )
=> ( ( ord_le1304079648umeral @ B_149 @ zero_z126310315umeral )
=> ( ord_le1304079648umeral @ ( plus_p1627245867umeral @ A_207 @ B_149 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_647_add__neg__nonpos,axiom,
! [B_148: rat,A_206: rat] :
( ( ord_less_rat @ A_206 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B_148 @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A_206 @ B_148 ) @ zero_zero_rat ) ) ) ).
thf(fact_648_add__neg__nonpos,axiom,
! [B_148: quickcheck_code_int,A_206: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_206 @ zero_z891286103de_int )
=> ( ( ord_le258702272de_int @ B_148 @ zero_z891286103de_int )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_206 @ B_148 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_649_add__neg__nonpos,axiom,
! [B_148: real,A_206: real] :
( ( ord_less_real @ A_206 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B_148 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A_206 @ B_148 ) @ zero_zero_real ) ) ) ).
thf(fact_650_add__neg__nonpos,axiom,
! [B_148: nat,A_206: nat] :
( ( ord_less_nat @ A_206 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B_148 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A_206 @ B_148 ) @ zero_zero_nat ) ) ) ).
thf(fact_651_add__neg__nonpos,axiom,
! [B_148: int,A_206: int] :
( ( ord_less_int @ A_206 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B_148 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A_206 @ B_148 ) @ zero_zero_int ) ) ) ).
thf(fact_652_add__neg__nonpos,axiom,
! [B_148: code_code_numeral,A_206: code_code_numeral] :
( ( ord_le1304079648umeral @ A_206 @ zero_z126310315umeral )
=> ( ( ord_le565307924umeral @ B_148 @ zero_z126310315umeral )
=> ( ord_le1304079648umeral @ ( plus_p1627245867umeral @ A_206 @ B_148 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_653_add__strict__increasing2,axiom,
! [B_147: rat,C_89: rat,A_205: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_205 )
=> ( ( ord_less_rat @ B_147 @ C_89 )
=> ( ord_less_rat @ B_147 @ ( plus_plus_rat @ A_205 @ C_89 ) ) ) ) ).
thf(fact_654_add__strict__increasing2,axiom,
! [B_147: quickcheck_code_int,C_89: quickcheck_code_int,A_205: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_205 )
=> ( ( ord_le1860547276de_int @ B_147 @ C_89 )
=> ( ord_le1860547276de_int @ B_147 @ ( plus_p1446045655de_int @ A_205 @ C_89 ) ) ) ) ).
thf(fact_655_add__strict__increasing2,axiom,
! [B_147: real,C_89: real,A_205: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_205 )
=> ( ( ord_less_real @ B_147 @ C_89 )
=> ( ord_less_real @ B_147 @ ( plus_plus_real @ A_205 @ C_89 ) ) ) ) ).
thf(fact_656_add__strict__increasing2,axiom,
! [B_147: nat,C_89: nat,A_205: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_205 )
=> ( ( ord_less_nat @ B_147 @ C_89 )
=> ( ord_less_nat @ B_147 @ ( plus_plus_nat @ A_205 @ C_89 ) ) ) ) ).
thf(fact_657_add__strict__increasing2,axiom,
! [B_147: int,C_89: int,A_205: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_205 )
=> ( ( ord_less_int @ B_147 @ C_89 )
=> ( ord_less_int @ B_147 @ ( plus_plus_int @ A_205 @ C_89 ) ) ) ) ).
thf(fact_658_add__strict__increasing2,axiom,
! [B_147: code_code_numeral,C_89: code_code_numeral,A_205: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_205 )
=> ( ( ord_le1304079648umeral @ B_147 @ C_89 )
=> ( ord_le1304079648umeral @ B_147 @ ( plus_p1627245867umeral @ A_205 @ C_89 ) ) ) ) ).
thf(fact_659_add__strict__increasing,axiom,
! [B_146: rat,C_88: rat,A_204: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_204 )
=> ( ( ord_less_eq_rat @ B_146 @ C_88 )
=> ( ord_less_rat @ B_146 @ ( plus_plus_rat @ A_204 @ C_88 ) ) ) ) ).
thf(fact_660_add__strict__increasing,axiom,
! [B_146: quickcheck_code_int,C_88: quickcheck_code_int,A_204: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_204 )
=> ( ( ord_le258702272de_int @ B_146 @ C_88 )
=> ( ord_le1860547276de_int @ B_146 @ ( plus_p1446045655de_int @ A_204 @ C_88 ) ) ) ) ).
thf(fact_661_add__strict__increasing,axiom,
! [B_146: real,C_88: real,A_204: real] :
( ( ord_less_real @ zero_zero_real @ A_204 )
=> ( ( ord_less_eq_real @ B_146 @ C_88 )
=> ( ord_less_real @ B_146 @ ( plus_plus_real @ A_204 @ C_88 ) ) ) ) ).
thf(fact_662_add__strict__increasing,axiom,
! [B_146: nat,C_88: nat,A_204: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_204 )
=> ( ( ord_less_eq_nat @ B_146 @ C_88 )
=> ( ord_less_nat @ B_146 @ ( plus_plus_nat @ A_204 @ C_88 ) ) ) ) ).
thf(fact_663_add__strict__increasing,axiom,
! [B_146: int,C_88: int,A_204: int] :
( ( ord_less_int @ zero_zero_int @ A_204 )
=> ( ( ord_less_eq_int @ B_146 @ C_88 )
=> ( ord_less_int @ B_146 @ ( plus_plus_int @ A_204 @ C_88 ) ) ) ) ).
thf(fact_664_add__strict__increasing,axiom,
! [B_146: code_code_numeral,C_88: code_code_numeral,A_204: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_204 )
=> ( ( ord_le565307924umeral @ B_146 @ C_88 )
=> ( ord_le1304079648umeral @ B_146 @ ( plus_p1627245867umeral @ A_204 @ C_88 ) ) ) ) ).
thf(fact_665_add__nonneg__pos,axiom,
! [B_145: rat,A_203: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_203 )
=> ( ( ord_less_rat @ zero_zero_rat @ B_145 )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A_203 @ B_145 ) ) ) ) ).
thf(fact_666_add__nonneg__pos,axiom,
! [B_145: quickcheck_code_int,A_203: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_203 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ B_145 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ ( plus_p1446045655de_int @ A_203 @ B_145 ) ) ) ) ).
thf(fact_667_add__nonneg__pos,axiom,
! [B_145: real,A_203: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_203 )
=> ( ( ord_less_real @ zero_zero_real @ B_145 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A_203 @ B_145 ) ) ) ) ).
thf(fact_668_add__nonneg__pos,axiom,
! [B_145: nat,A_203: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_203 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_145 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A_203 @ B_145 ) ) ) ) ).
thf(fact_669_add__nonneg__pos,axiom,
! [B_145: int,A_203: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_203 )
=> ( ( ord_less_int @ zero_zero_int @ B_145 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_203 @ B_145 ) ) ) ) ).
thf(fact_670_add__nonneg__pos,axiom,
! [B_145: code_code_numeral,A_203: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_203 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ B_145 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ ( plus_p1627245867umeral @ A_203 @ B_145 ) ) ) ) ).
thf(fact_671_add__pos__nonneg,axiom,
! [B_144: rat,A_202: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_202 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_144 )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A_202 @ B_144 ) ) ) ) ).
thf(fact_672_add__pos__nonneg,axiom,
! [B_144: quickcheck_code_int,A_202: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_202 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_144 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ ( plus_p1446045655de_int @ A_202 @ B_144 ) ) ) ) ).
thf(fact_673_add__pos__nonneg,axiom,
! [B_144: real,A_202: real] :
( ( ord_less_real @ zero_zero_real @ A_202 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_144 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A_202 @ B_144 ) ) ) ) ).
thf(fact_674_add__pos__nonneg,axiom,
! [B_144: nat,A_202: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_202 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_144 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A_202 @ B_144 ) ) ) ) ).
thf(fact_675_add__pos__nonneg,axiom,
! [B_144: int,A_202: int] :
( ( ord_less_int @ zero_zero_int @ A_202 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_144 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_202 @ B_144 ) ) ) ) ).
thf(fact_676_add__pos__nonneg,axiom,
! [B_144: code_code_numeral,A_202: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_202 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_144 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ ( plus_p1627245867umeral @ A_202 @ B_144 ) ) ) ) ).
thf(fact_677_zero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
thf(fact_678_zero__le__one,axiom,
ord_le258702272de_int @ zero_z891286103de_int @ one_on1684967323de_int ).
thf(fact_679_zero__le__one,axiom,
ord_le565307924umeral @ zero_z126310315umeral @ one_on1645066479umeral ).
thf(fact_680_zero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
thf(fact_681_zero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
thf(fact_682_zero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
thf(fact_683_not__one__le__zero,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
thf(fact_684_not__one__le__zero,axiom,
~ ( ord_le258702272de_int @ one_on1684967323de_int @ zero_z891286103de_int ) ).
thf(fact_685_not__one__le__zero,axiom,
~ ( ord_le565307924umeral @ one_on1645066479umeral @ zero_z126310315umeral ) ).
thf(fact_686_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
thf(fact_687_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
thf(fact_688_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
thf(fact_689_le__number__of__eq__not__less,axiom,
! [V_21: int,W_11: int] :
( ( ord_less_eq_rat @ ( number_number_of_rat @ V_21 ) @ ( number_number_of_rat @ W_11 ) )
<=> ~ ( ord_less_rat @ ( number_number_of_rat @ W_11 ) @ ( number_number_of_rat @ V_21 ) ) ) ).
thf(fact_690_le__number__of__eq__not__less,axiom,
! [V_21: int,W_11: int] :
( ( ord_le258702272de_int @ ( number1226105091de_int @ V_21 ) @ ( number1226105091de_int @ W_11 ) )
<=> ~ ( ord_le1860547276de_int @ ( number1226105091de_int @ W_11 ) @ ( number1226105091de_int @ V_21 ) ) ) ).
thf(fact_691_le__number__of__eq__not__less,axiom,
! [V_21: int,W_11: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ V_21 ) @ ( number267125858f_real @ W_11 ) )
<=> ~ ( ord_less_real @ ( number267125858f_real @ W_11 ) @ ( number267125858f_real @ V_21 ) ) ) ).
thf(fact_692_le__number__of__eq__not__less,axiom,
! [V_21: int,W_11: int] :
( ( ord_less_eq_nat @ ( number_number_of_nat @ V_21 ) @ ( number_number_of_nat @ W_11 ) )
<=> ~ ( ord_less_nat @ ( number_number_of_nat @ W_11 ) @ ( number_number_of_nat @ V_21 ) ) ) ).
thf(fact_693_le__number__of__eq__not__less,axiom,
! [V_21: int,W_11: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ V_21 ) @ ( number_number_of_int @ W_11 ) )
<=> ~ ( ord_less_int @ ( number_number_of_int @ W_11 ) @ ( number_number_of_int @ V_21 ) ) ) ).
thf(fact_694_le__number__of__eq__not__less,axiom,
! [V_21: int,W_11: int] :
( ( ord_le565307924umeral @ ( number1443263063umeral @ V_21 ) @ ( number1443263063umeral @ W_11 ) )
<=> ~ ( ord_le1304079648umeral @ ( number1443263063umeral @ W_11 ) @ ( number1443263063umeral @ V_21 ) ) ) ).
thf(fact_695_power__mono,axiom,
! [N_36: nat,A_201: rat,B_143: rat] :
( ( ord_less_eq_rat @ A_201 @ B_143 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A_201 )
=> ( ord_less_eq_rat @ ( power_power_rat @ A_201 @ N_36 ) @ ( power_power_rat @ B_143 @ N_36 ) ) ) ) ).
thf(fact_696_power__mono,axiom,
! [N_36: nat,A_201: quickcheck_code_int,B_143: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_201 @ B_143 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_201 )
=> ( ord_le258702272de_int @ ( power_881366806de_int @ A_201 @ N_36 ) @ ( power_881366806de_int @ B_143 @ N_36 ) ) ) ) ).
thf(fact_697_power__mono,axiom,
! [N_36: nat,A_201: code_code_numeral,B_143: code_code_numeral] :
( ( ord_le565307924umeral @ A_201 @ B_143 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_201 )
=> ( ord_le565307924umeral @ ( power_2100829034umeral @ A_201 @ N_36 ) @ ( power_2100829034umeral @ B_143 @ N_36 ) ) ) ) ).
thf(fact_698_power__mono,axiom,
! [N_36: nat,A_201: real,B_143: real] :
( ( ord_less_eq_real @ A_201 @ B_143 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_201 )
=> ( ord_less_eq_real @ ( power_power_real @ A_201 @ N_36 ) @ ( power_power_real @ B_143 @ N_36 ) ) ) ) ).
thf(fact_699_power__mono,axiom,
! [N_36: nat,A_201: nat,B_143: nat] :
( ( ord_less_eq_nat @ A_201 @ B_143 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_201 )
=> ( ord_less_eq_nat @ ( power_power_nat @ A_201 @ N_36 ) @ ( power_power_nat @ B_143 @ N_36 ) ) ) ) ).
thf(fact_700_power__mono,axiom,
! [N_36: nat,A_201: int,B_143: int] :
( ( ord_less_eq_int @ A_201 @ B_143 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_201 )
=> ( ord_less_eq_int @ ( power_power_int @ A_201 @ N_36 ) @ ( power_power_int @ B_143 @ N_36 ) ) ) ) ).
thf(fact_701_zero__le__power,axiom,
! [N_35: nat,A_200: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_200 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A_200 @ N_35 ) ) ) ).
thf(fact_702_zero__le__power,axiom,
! [N_35: nat,A_200: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_200 )
=> ( ord_le258702272de_int @ zero_z891286103de_int @ ( power_881366806de_int @ A_200 @ N_35 ) ) ) ).
thf(fact_703_zero__le__power,axiom,
! [N_35: nat,A_200: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_200 )
=> ( ord_le565307924umeral @ zero_z126310315umeral @ ( power_2100829034umeral @ A_200 @ N_35 ) ) ) ).
thf(fact_704_zero__le__power,axiom,
! [N_35: nat,A_200: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_200 )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A_200 @ N_35 ) ) ) ).
thf(fact_705_zero__le__power,axiom,
! [N_35: nat,A_200: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_200 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A_200 @ N_35 ) ) ) ).
thf(fact_706_zero__le__power,axiom,
! [N_35: nat,A_200: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_200 )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A_200 @ N_35 ) ) ) ).
thf(fact_707_one__le__power,axiom,
! [N_34: nat,A_199: rat] :
( ( ord_less_eq_rat @ one_one_rat @ A_199 )
=> ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A_199 @ N_34 ) ) ) ).
thf(fact_708_one__le__power,axiom,
! [N_34: nat,A_199: quickcheck_code_int] :
( ( ord_le258702272de_int @ one_on1684967323de_int @ A_199 )
=> ( ord_le258702272de_int @ one_on1684967323de_int @ ( power_881366806de_int @ A_199 @ N_34 ) ) ) ).
thf(fact_709_one__le__power,axiom,
! [N_34: nat,A_199: code_code_numeral] :
( ( ord_le565307924umeral @ one_on1645066479umeral @ A_199 )
=> ( ord_le565307924umeral @ one_on1645066479umeral @ ( power_2100829034umeral @ A_199 @ N_34 ) ) ) ).
thf(fact_710_one__le__power,axiom,
! [N_34: nat,A_199: real] :
( ( ord_less_eq_real @ one_one_real @ A_199 )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A_199 @ N_34 ) ) ) ).
thf(fact_711_one__le__power,axiom,
! [N_34: nat,A_199: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A_199 )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A_199 @ N_34 ) ) ) ).
thf(fact_712_one__le__power,axiom,
! [N_34: nat,A_199: int] :
( ( ord_less_eq_int @ one_one_int @ A_199 )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A_199 @ N_34 ) ) ) ).
thf(fact_713_of__nat__0__le__iff,axiom,
! [N_33: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri151668891at_rat @ N_33 ) ) ).
thf(fact_714_of__nat__0__le__iff,axiom,
! [N_33: nat] : ( ord_le258702272de_int @ zero_z891286103de_int @ ( semiri1424489471de_int @ N_33 ) ) ).
thf(fact_715_of__nat__0__le__iff,axiom,
! [N_33: nat] : ( ord_le565307924umeral @ zero_z126310315umeral @ ( semiri1619134803umeral @ N_33 ) ) ).
thf(fact_716_of__nat__0__le__iff,axiom,
! [N_33: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri132038758t_real @ N_33 ) ) ).
thf(fact_717_of__nat__0__le__iff,axiom,
! [N_33: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri984289939at_nat @ N_33 ) ) ).
thf(fact_718_of__nat__0__le__iff,axiom,
! [N_33: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ N_33 ) ) ).
thf(fact_719_zero__le__imp__of__nat,axiom,
! [M_22: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri151668891at_rat @ M_22 ) ) ).
thf(fact_720_zero__le__imp__of__nat,axiom,
! [M_22: nat] : ( ord_le258702272de_int @ zero_z891286103de_int @ ( semiri1424489471de_int @ M_22 ) ) ).
thf(fact_721_zero__le__imp__of__nat,axiom,
! [M_22: nat] : ( ord_le565307924umeral @ zero_z126310315umeral @ ( semiri1619134803umeral @ M_22 ) ) ).
thf(fact_722_zero__le__imp__of__nat,axiom,
! [M_22: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri132038758t_real @ M_22 ) ) ).
thf(fact_723_zero__le__imp__of__nat,axiom,
! [M_22: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri984289939at_nat @ M_22 ) ) ).
thf(fact_724_zero__le__imp__of__nat,axiom,
! [M_22: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ M_22 ) ) ).
thf(fact_725_rel__simps_I22_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ pls @ ( bit1 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
thf(fact_726_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_727_rel__simps_I32_J,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_728_rel__simps_I27_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit0 @ K ) @ pls )
<=> ( ord_less_eq_int @ K @ pls ) ) ).
thf(fact_729_rel__simps_I21_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ pls @ ( bit0 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
thf(fact_730_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
thf(fact_731_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_732_zadd__zless__mono,axiom,
! [Z_3: int,Z_1: int,W_10: int,W: int] :
( ( ord_less_int @ W_10 @ W )
=> ( ( ord_less_eq_int @ Z_3 @ Z_1 )
=> ( ord_less_int @ ( plus_plus_int @ W_10 @ Z_3 ) @ ( plus_plus_int @ W @ Z_1 ) ) ) ) ).
thf(fact_733_zero__zle__int,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ N ) ) ).
thf(fact_734_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_735_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_736_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_737_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_738_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_739_semiring__number__of__add__1,axiom,
! [V_20: int] :
( ( ord_less_eq_int @ pls @ V_20 )
=> ( ( plus_plus_rat @ ( number_number_of_rat @ V_20 ) @ one_one_rat )
= ( number_number_of_rat @ ( succ @ V_20 ) ) ) ) ).
thf(fact_740_semiring__number__of__add__1,axiom,
! [V_20: int] :
( ( ord_less_eq_int @ pls @ V_20 )
=> ( ( plus_plus_complex @ ( number528085621omplex @ V_20 ) @ one_one_complex )
= ( number528085621omplex @ ( succ @ V_20 ) ) ) ) ).
thf(fact_741_semiring__number__of__add__1,axiom,
! [V_20: int] :
( ( ord_less_eq_int @ pls @ V_20 )
=> ( ( plus_plus_real @ ( number267125858f_real @ V_20 ) @ one_one_real )
= ( number267125858f_real @ ( succ @ V_20 ) ) ) ) ).
thf(fact_742_semiring__number__of__add__1,axiom,
! [V_20: int] :
( ( ord_less_eq_int @ pls @ V_20 )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V_20 ) @ one_one_nat )
= ( number_number_of_nat @ ( succ @ V_20 ) ) ) ) ).
thf(fact_743_semiring__number__of__add__1,axiom,
! [V_20: int] :
( ( ord_less_eq_int @ pls @ V_20 )
=> ( ( plus_plus_int @ ( number_number_of_int @ V_20 ) @ one_one_int )
= ( number_number_of_int @ ( succ @ V_20 ) ) ) ) ).
thf(fact_744_semiring__1__add__number__of,axiom,
! [V_19: int] :
( ( ord_less_eq_int @ pls @ V_19 )
=> ( ( plus_plus_rat @ one_one_rat @ ( number_number_of_rat @ V_19 ) )
= ( number_number_of_rat @ ( succ @ V_19 ) ) ) ) ).
thf(fact_745_semiring__1__add__number__of,axiom,
! [V_19: int] :
( ( ord_less_eq_int @ pls @ V_19 )
=> ( ( plus_plus_complex @ one_one_complex @ ( number528085621omplex @ V_19 ) )
= ( number528085621omplex @ ( succ @ V_19 ) ) ) ) ).
thf(fact_746_semiring__1__add__number__of,axiom,
! [V_19: int] :
( ( ord_less_eq_int @ pls @ V_19 )
=> ( ( plus_plus_real @ one_one_real @ ( number267125858f_real @ V_19 ) )
= ( number267125858f_real @ ( succ @ V_19 ) ) ) ) ).
thf(fact_747_semiring__1__add__number__of,axiom,
! [V_19: int] :
( ( ord_less_eq_int @ pls @ V_19 )
=> ( ( plus_plus_nat @ one_one_nat @ ( number_number_of_nat @ V_19 ) )
= ( number_number_of_nat @ ( succ @ V_19 ) ) ) ) ).
thf(fact_748_semiring__1__add__number__of,axiom,
! [V_19: int] :
( ( ord_less_eq_int @ pls @ V_19 )
=> ( ( plus_plus_int @ one_one_int @ ( number_number_of_int @ V_19 ) )
= ( number_number_of_int @ ( succ @ V_19 ) ) ) ) ).
thf(fact_749_power__less__imp__less__base,axiom,
! [A_198: rat,N_32: nat,B_142: rat] :
( ( ord_less_rat @ ( power_power_rat @ A_198 @ N_32 ) @ ( power_power_rat @ B_142 @ N_32 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_142 )
=> ( ord_less_rat @ A_198 @ B_142 ) ) ) ).
thf(fact_750_power__less__imp__less__base,axiom,
! [A_198: quickcheck_code_int,N_32: nat,B_142: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( power_881366806de_int @ A_198 @ N_32 ) @ ( power_881366806de_int @ B_142 @ N_32 ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_142 )
=> ( ord_le1860547276de_int @ A_198 @ B_142 ) ) ) ).
thf(fact_751_power__less__imp__less__base,axiom,
! [A_198: code_code_numeral,N_32: nat,B_142: code_code_numeral] :
( ( ord_le1304079648umeral @ ( power_2100829034umeral @ A_198 @ N_32 ) @ ( power_2100829034umeral @ B_142 @ N_32 ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_142 )
=> ( ord_le1304079648umeral @ A_198 @ B_142 ) ) ) ).
thf(fact_752_power__less__imp__less__base,axiom,
! [A_198: real,N_32: nat,B_142: real] :
( ( ord_less_real @ ( power_power_real @ A_198 @ N_32 ) @ ( power_power_real @ B_142 @ N_32 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_142 )
=> ( ord_less_real @ A_198 @ B_142 ) ) ) ).
thf(fact_753_power__less__imp__less__base,axiom,
! [A_198: nat,N_32: nat,B_142: nat] :
( ( ord_less_nat @ ( power_power_nat @ A_198 @ N_32 ) @ ( power_power_nat @ B_142 @ N_32 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_142 )
=> ( ord_less_nat @ A_198 @ B_142 ) ) ) ).
thf(fact_754_power__less__imp__less__base,axiom,
! [A_198: int,N_32: nat,B_142: int] :
( ( ord_less_int @ ( power_power_int @ A_198 @ N_32 ) @ ( power_power_int @ B_142 @ N_32 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_142 )
=> ( ord_less_int @ A_198 @ B_142 ) ) ) ).
thf(fact_755_eq__number__of__0,axiom,
! [V: int] :
( ( ( number_number_of_nat @ V )
= zero_zero_nat )
<=> ( ord_less_eq_int @ V @ pls ) ) ).
thf(fact_756_eq__0__number__of,axiom,
! [V: int] :
( ( zero_zero_nat
= ( number_number_of_nat @ V ) )
<=> ( ord_less_eq_int @ V @ pls ) ) ).
thf(fact_757_rel__simps_I29_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ pls )
<=> ( ord_less_int @ K @ pls ) ) ).
thf(fact_758_rel__simps_I5_J,axiom,
! [K: int] :
( ( ord_less_int @ pls @ ( bit1 @ K ) )
<=> ( ord_less_eq_int @ pls @ K ) ) ).
thf(fact_759_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_760_rel__simps_I33_J,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K @ L ) ) ).
thf(fact_761_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_762_rel__simps_I15_J,axiom,
! [K: int,L: int] :
( ( ord_less_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
<=> ( ord_less_eq_int @ K @ L ) ) ).
thf(fact_763_zero__reorient,axiom,
! [X_84: rat] :
( ( zero_zero_rat = X_84 )
<=> ( X_84 = zero_zero_rat ) ) ).
thf(fact_764_zero__reorient,axiom,
! [X_84: quickcheck_code_int] :
( ( zero_z891286103de_int = X_84 )
<=> ( X_84 = zero_z891286103de_int ) ) ).
thf(fact_765_zero__reorient,axiom,
! [X_84: complex] :
( ( zero_zero_complex = X_84 )
<=> ( X_84 = zero_zero_complex ) ) ).
thf(fact_766_zero__reorient,axiom,
! [X_84: code_code_numeral] :
( ( zero_z126310315umeral = X_84 )
<=> ( X_84 = zero_z126310315umeral ) ) ).
thf(fact_767_zero__reorient,axiom,
! [X_84: real] :
( ( zero_zero_real = X_84 )
<=> ( X_84 = zero_zero_real ) ) ).
thf(fact_768_zero__reorient,axiom,
! [X_84: nat] :
( ( zero_zero_nat = X_84 )
<=> ( X_84 = zero_zero_nat ) ) ).
thf(fact_769_zero__reorient,axiom,
! [X_84: int] :
( ( zero_zero_int = X_84 )
<=> ( X_84 = zero_zero_int ) ) ).
thf(fact_770_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_771_int__le__0__conv,axiom,
! [N: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ N ) @ zero_zero_int )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_772_succ__Pls,axiom,
( ( succ @ pls )
= ( bit1 @ pls ) ) ).
thf(fact_773_succ__Bit1,axiom,
! [K: int] :
( ( succ @ ( bit1 @ K ) )
= ( bit0 @ ( succ @ K ) ) ) ).
thf(fact_774_succ__Bit0,axiom,
! [K: int] :
( ( succ @ ( bit0 @ K ) )
= ( bit1 @ K ) ) ).
thf(fact_775_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_776_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_777_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_778_add__right__imp__eq,axiom,
! [B_141: rat,A_197: rat,C_87: rat] :
( ( ( plus_plus_rat @ B_141 @ A_197 )
= ( plus_plus_rat @ C_87 @ A_197 ) )
=> ( B_141 = C_87 ) ) ).
thf(fact_779_add__right__imp__eq,axiom,
! [B_141: quickcheck_code_int,A_197: quickcheck_code_int,C_87: quickcheck_code_int] :
( ( ( plus_p1446045655de_int @ B_141 @ A_197 )
= ( plus_p1446045655de_int @ C_87 @ A_197 ) )
=> ( B_141 = C_87 ) ) ).
thf(fact_780_add__right__imp__eq,axiom,
! [B_141: complex,A_197: complex,C_87: complex] :
( ( ( plus_plus_complex @ B_141 @ A_197 )
= ( plus_plus_complex @ C_87 @ A_197 ) )
=> ( B_141 = C_87 ) ) ).
thf(fact_781_add__right__imp__eq,axiom,
! [B_141: real,A_197: real,C_87: real] :
( ( ( plus_plus_real @ B_141 @ A_197 )
= ( plus_plus_real @ C_87 @ A_197 ) )
=> ( B_141 = C_87 ) ) ).
thf(fact_782_add__right__imp__eq,axiom,
! [B_141: nat,A_197: nat,C_87: nat] :
( ( ( plus_plus_nat @ B_141 @ A_197 )
= ( plus_plus_nat @ C_87 @ A_197 ) )
=> ( B_141 = C_87 ) ) ).
thf(fact_783_add__right__imp__eq,axiom,
! [B_141: int,A_197: int,C_87: int] :
( ( ( plus_plus_int @ B_141 @ A_197 )
= ( plus_plus_int @ C_87 @ A_197 ) )
=> ( B_141 = C_87 ) ) ).
thf(fact_784_add__imp__eq,axiom,
! [A_196: rat,B_140: rat,C_86: rat] :
( ( ( plus_plus_rat @ A_196 @ B_140 )
= ( plus_plus_rat @ A_196 @ C_86 ) )
=> ( B_140 = C_86 ) ) ).
thf(fact_785_add__imp__eq,axiom,
! [A_196: quickcheck_code_int,B_140: quickcheck_code_int,C_86: quickcheck_code_int] :
( ( ( plus_p1446045655de_int @ A_196 @ B_140 )
= ( plus_p1446045655de_int @ A_196 @ C_86 ) )
=> ( B_140 = C_86 ) ) ).
thf(fact_786_add__imp__eq,axiom,
! [A_196: complex,B_140: complex,C_86: complex] :
( ( ( plus_plus_complex @ A_196 @ B_140 )
= ( plus_plus_complex @ A_196 @ C_86 ) )
=> ( B_140 = C_86 ) ) ).
thf(fact_787_add__imp__eq,axiom,
! [A_196: real,B_140: real,C_86: real] :
( ( ( plus_plus_real @ A_196 @ B_140 )
= ( plus_plus_real @ A_196 @ C_86 ) )
=> ( B_140 = C_86 ) ) ).
thf(fact_788_add__imp__eq,axiom,
! [A_196: nat,B_140: nat,C_86: nat] :
( ( ( plus_plus_nat @ A_196 @ B_140 )
= ( plus_plus_nat @ A_196 @ C_86 ) )
=> ( B_140 = C_86 ) ) ).
thf(fact_789_add__imp__eq,axiom,
! [A_196: int,B_140: int,C_86: int] :
( ( ( plus_plus_int @ A_196 @ B_140 )
= ( plus_plus_int @ A_196 @ C_86 ) )
=> ( B_140 = C_86 ) ) ).
thf(fact_790_add__left__imp__eq,axiom,
! [A_195: rat,B_139: rat,C_85: rat] :
( ( ( plus_plus_rat @ A_195 @ B_139 )
= ( plus_plus_rat @ A_195 @ C_85 ) )
=> ( B_139 = C_85 ) ) ).
thf(fact_791_add__left__imp__eq,axiom,
! [A_195: quickcheck_code_int,B_139: quickcheck_code_int,C_85: quickcheck_code_int] :
( ( ( plus_p1446045655de_int @ A_195 @ B_139 )
= ( plus_p1446045655de_int @ A_195 @ C_85 ) )
=> ( B_139 = C_85 ) ) ).
thf(fact_792_add__left__imp__eq,axiom,
! [A_195: complex,B_139: complex,C_85: complex] :
( ( ( plus_plus_complex @ A_195 @ B_139 )
= ( plus_plus_complex @ A_195 @ C_85 ) )
=> ( B_139 = C_85 ) ) ).
thf(fact_793_add__left__imp__eq,axiom,
! [A_195: real,B_139: real,C_85: real] :
( ( ( plus_plus_real @ A_195 @ B_139 )
= ( plus_plus_real @ A_195 @ C_85 ) )
=> ( B_139 = C_85 ) ) ).
thf(fact_794_add__left__imp__eq,axiom,
! [A_195: nat,B_139: nat,C_85: nat] :
( ( ( plus_plus_nat @ A_195 @ B_139 )
= ( plus_plus_nat @ A_195 @ C_85 ) )
=> ( B_139 = C_85 ) ) ).
thf(fact_795_add__left__imp__eq,axiom,
! [A_195: int,B_139: int,C_85: int] :
( ( ( plus_plus_int @ A_195 @ B_139 )
= ( plus_plus_int @ A_195 @ C_85 ) )
=> ( B_139 = C_85 ) ) ).
thf(fact_796_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_194: rat,B_138: rat,C_84: rat,D_32: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A_194 @ B_138 ) @ ( plus_plus_rat @ C_84 @ D_32 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ A_194 @ C_84 ) @ ( plus_plus_rat @ B_138 @ D_32 ) ) ) ).
thf(fact_797_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_194: quickcheck_code_int,B_138: quickcheck_code_int,C_84: quickcheck_code_int,D_32: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_194 @ B_138 ) @ ( plus_p1446045655de_int @ C_84 @ D_32 ) )
= ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_194 @ C_84 ) @ ( plus_p1446045655de_int @ B_138 @ D_32 ) ) ) ).
thf(fact_798_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_194: complex,B_138: complex,C_84: complex,D_32: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A_194 @ B_138 ) @ ( plus_plus_complex @ C_84 @ D_32 ) )
= ( plus_plus_complex @ ( plus_plus_complex @ A_194 @ C_84 ) @ ( plus_plus_complex @ B_138 @ D_32 ) ) ) ).
thf(fact_799_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_194: real,B_138: real,C_84: real,D_32: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_194 @ B_138 ) @ ( plus_plus_real @ C_84 @ D_32 ) )
= ( plus_plus_real @ ( plus_plus_real @ A_194 @ C_84 ) @ ( plus_plus_real @ B_138 @ D_32 ) ) ) ).
thf(fact_800_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_194: nat,B_138: nat,C_84: nat,D_32: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_194 @ B_138 ) @ ( plus_plus_nat @ C_84 @ D_32 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_194 @ C_84 ) @ ( plus_plus_nat @ B_138 @ D_32 ) ) ) ).
thf(fact_801_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_194: int,B_138: int,C_84: int,D_32: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_194 @ B_138 ) @ ( plus_plus_int @ C_84 @ D_32 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_194 @ C_84 ) @ ( plus_plus_int @ B_138 @ D_32 ) ) ) ).
thf(fact_802_add__right__cancel,axiom,
! [B_137: rat,A_193: rat,C_83: rat] :
( ( ( plus_plus_rat @ B_137 @ A_193 )
= ( plus_plus_rat @ C_83 @ A_193 ) )
<=> ( B_137 = C_83 ) ) ).
thf(fact_803_add__right__cancel,axiom,
! [B_137: quickcheck_code_int,A_193: quickcheck_code_int,C_83: quickcheck_code_int] :
( ( ( plus_p1446045655de_int @ B_137 @ A_193 )
= ( plus_p1446045655de_int @ C_83 @ A_193 ) )
<=> ( B_137 = C_83 ) ) ).
thf(fact_804_add__right__cancel,axiom,
! [B_137: complex,A_193: complex,C_83: complex] :
( ( ( plus_plus_complex @ B_137 @ A_193 )
= ( plus_plus_complex @ C_83 @ A_193 ) )
<=> ( B_137 = C_83 ) ) ).
thf(fact_805_add__right__cancel,axiom,
! [B_137: real,A_193: real,C_83: real] :
( ( ( plus_plus_real @ B_137 @ A_193 )
= ( plus_plus_real @ C_83 @ A_193 ) )
<=> ( B_137 = C_83 ) ) ).
thf(fact_806_add__right__cancel,axiom,
! [B_137: nat,A_193: nat,C_83: nat] :
( ( ( plus_plus_nat @ B_137 @ A_193 )
= ( plus_plus_nat @ C_83 @ A_193 ) )
<=> ( B_137 = C_83 ) ) ).
thf(fact_807_add__right__cancel,axiom,
! [B_137: int,A_193: int,C_83: int] :
( ( ( plus_plus_int @ B_137 @ A_193 )
= ( plus_plus_int @ C_83 @ A_193 ) )
<=> ( B_137 = C_83 ) ) ).
thf(fact_808_add__left__cancel,axiom,
! [A_192: rat,B_136: rat,C_82: rat] :
( ( ( plus_plus_rat @ A_192 @ B_136 )
= ( plus_plus_rat @ A_192 @ C_82 ) )
<=> ( B_136 = C_82 ) ) ).
thf(fact_809_add__left__cancel,axiom,
! [A_192: quickcheck_code_int,B_136: quickcheck_code_int,C_82: quickcheck_code_int] :
( ( ( plus_p1446045655de_int @ A_192 @ B_136 )
= ( plus_p1446045655de_int @ A_192 @ C_82 ) )
<=> ( B_136 = C_82 ) ) ).
thf(fact_810_add__left__cancel,axiom,
! [A_192: complex,B_136: complex,C_82: complex] :
( ( ( plus_plus_complex @ A_192 @ B_136 )
= ( plus_plus_complex @ A_192 @ C_82 ) )
<=> ( B_136 = C_82 ) ) ).
thf(fact_811_add__left__cancel,axiom,
! [A_192: real,B_136: real,C_82: real] :
( ( ( plus_plus_real @ A_192 @ B_136 )
= ( plus_plus_real @ A_192 @ C_82 ) )
<=> ( B_136 = C_82 ) ) ).
thf(fact_812_add__left__cancel,axiom,
! [A_192: nat,B_136: nat,C_82: nat] :
( ( ( plus_plus_nat @ A_192 @ B_136 )
= ( plus_plus_nat @ A_192 @ C_82 ) )
<=> ( B_136 = C_82 ) ) ).
thf(fact_813_add__left__cancel,axiom,
! [A_192: int,B_136: int,C_82: int] :
( ( ( plus_plus_int @ A_192 @ B_136 )
= ( plus_plus_int @ A_192 @ C_82 ) )
<=> ( B_136 = C_82 ) ) ).
thf(fact_814_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_191: rat,B_135: rat,C_81: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A_191 @ B_135 ) @ C_81 )
= ( plus_plus_rat @ ( plus_plus_rat @ A_191 @ C_81 ) @ B_135 ) ) ).
thf(fact_815_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_191: quickcheck_code_int,B_135: quickcheck_code_int,C_81: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_191 @ B_135 ) @ C_81 )
= ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_191 @ C_81 ) @ B_135 ) ) ).
thf(fact_816_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_191: complex,B_135: complex,C_81: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A_191 @ B_135 ) @ C_81 )
= ( plus_plus_complex @ ( plus_plus_complex @ A_191 @ C_81 ) @ B_135 ) ) ).
thf(fact_817_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_191: real,B_135: real,C_81: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_191 @ B_135 ) @ C_81 )
= ( plus_plus_real @ ( plus_plus_real @ A_191 @ C_81 ) @ B_135 ) ) ).
thf(fact_818_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_191: nat,B_135: nat,C_81: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_191 @ B_135 ) @ C_81 )
= ( plus_plus_nat @ ( plus_plus_nat @ A_191 @ C_81 ) @ B_135 ) ) ).
thf(fact_819_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_191: int,B_135: int,C_81: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_191 @ B_135 ) @ C_81 )
= ( plus_plus_int @ ( plus_plus_int @ A_191 @ C_81 ) @ B_135 ) ) ).
thf(fact_820_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_190: rat,B_134: rat,C_80: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A_190 @ B_134 ) @ C_80 )
= ( plus_plus_rat @ A_190 @ ( plus_plus_rat @ B_134 @ C_80 ) ) ) ).
thf(fact_821_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_190: quickcheck_code_int,B_134: quickcheck_code_int,C_80: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_190 @ B_134 ) @ C_80 )
= ( plus_p1446045655de_int @ A_190 @ ( plus_p1446045655de_int @ B_134 @ C_80 ) ) ) ).
thf(fact_822_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_190: complex,B_134: complex,C_80: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A_190 @ B_134 ) @ C_80 )
= ( plus_plus_complex @ A_190 @ ( plus_plus_complex @ B_134 @ C_80 ) ) ) ).
thf(fact_823_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_190: real,B_134: real,C_80: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_190 @ B_134 ) @ C_80 )
= ( plus_plus_real @ A_190 @ ( plus_plus_real @ B_134 @ C_80 ) ) ) ).
thf(fact_824_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_190: nat,B_134: nat,C_80: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_190 @ B_134 ) @ C_80 )
= ( plus_plus_nat @ A_190 @ ( plus_plus_nat @ B_134 @ C_80 ) ) ) ).
thf(fact_825_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_190: int,B_134: int,C_80: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_190 @ B_134 ) @ C_80 )
= ( plus_plus_int @ A_190 @ ( plus_plus_int @ B_134 @ C_80 ) ) ) ).
thf(fact_826_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_189: rat,B_133: rat,C_79: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A_189 @ B_133 ) @ C_79 )
= ( plus_plus_rat @ A_189 @ ( plus_plus_rat @ B_133 @ C_79 ) ) ) ).
thf(fact_827_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_189: quickcheck_code_int,B_133: quickcheck_code_int,C_79: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_189 @ B_133 ) @ C_79 )
= ( plus_p1446045655de_int @ A_189 @ ( plus_p1446045655de_int @ B_133 @ C_79 ) ) ) ).
thf(fact_828_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_189: complex,B_133: complex,C_79: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A_189 @ B_133 ) @ C_79 )
= ( plus_plus_complex @ A_189 @ ( plus_plus_complex @ B_133 @ C_79 ) ) ) ).
thf(fact_829_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_189: real,B_133: real,C_79: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_189 @ B_133 ) @ C_79 )
= ( plus_plus_real @ A_189 @ ( plus_plus_real @ B_133 @ C_79 ) ) ) ).
thf(fact_830_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_189: nat,B_133: nat,C_79: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_189 @ B_133 ) @ C_79 )
= ( plus_plus_nat @ A_189 @ ( plus_plus_nat @ B_133 @ C_79 ) ) ) ).
thf(fact_831_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_189: int,B_133: int,C_79: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_189 @ B_133 ) @ C_79 )
= ( plus_plus_int @ A_189 @ ( plus_plus_int @ B_133 @ C_79 ) ) ) ).
thf(fact_832_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_188: rat,C_78: rat,D_31: rat] :
( ( plus_plus_rat @ A_188 @ ( plus_plus_rat @ C_78 @ D_31 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ A_188 @ C_78 ) @ D_31 ) ) ).
thf(fact_833_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_188: quickcheck_code_int,C_78: quickcheck_code_int,D_31: quickcheck_code_int] :
( ( plus_p1446045655de_int @ A_188 @ ( plus_p1446045655de_int @ C_78 @ D_31 ) )
= ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_188 @ C_78 ) @ D_31 ) ) ).
thf(fact_834_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_188: complex,C_78: complex,D_31: complex] :
( ( plus_plus_complex @ A_188 @ ( plus_plus_complex @ C_78 @ D_31 ) )
= ( plus_plus_complex @ ( plus_plus_complex @ A_188 @ C_78 ) @ D_31 ) ) ).
thf(fact_835_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_188: real,C_78: real,D_31: real] :
( ( plus_plus_real @ A_188 @ ( plus_plus_real @ C_78 @ D_31 ) )
= ( plus_plus_real @ ( plus_plus_real @ A_188 @ C_78 ) @ D_31 ) ) ).
thf(fact_836_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_188: nat,C_78: nat,D_31: nat] :
( ( plus_plus_nat @ A_188 @ ( plus_plus_nat @ C_78 @ D_31 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_188 @ C_78 ) @ D_31 ) ) ).
thf(fact_837_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_188: int,C_78: int,D_31: int] :
( ( plus_plus_int @ A_188 @ ( plus_plus_int @ C_78 @ D_31 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_188 @ C_78 ) @ D_31 ) ) ).
thf(fact_838_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_187: rat,C_77: rat,D_30: rat] :
( ( plus_plus_rat @ A_187 @ ( plus_plus_rat @ C_77 @ D_30 ) )
= ( plus_plus_rat @ C_77 @ ( plus_plus_rat @ A_187 @ D_30 ) ) ) ).
thf(fact_839_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_187: quickcheck_code_int,C_77: quickcheck_code_int,D_30: quickcheck_code_int] :
( ( plus_p1446045655de_int @ A_187 @ ( plus_p1446045655de_int @ C_77 @ D_30 ) )
= ( plus_p1446045655de_int @ C_77 @ ( plus_p1446045655de_int @ A_187 @ D_30 ) ) ) ).
thf(fact_840_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_187: complex,C_77: complex,D_30: complex] :
( ( plus_plus_complex @ A_187 @ ( plus_plus_complex @ C_77 @ D_30 ) )
= ( plus_plus_complex @ C_77 @ ( plus_plus_complex @ A_187 @ D_30 ) ) ) ).
thf(fact_841_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_187: real,C_77: real,D_30: real] :
( ( plus_plus_real @ A_187 @ ( plus_plus_real @ C_77 @ D_30 ) )
= ( plus_plus_real @ C_77 @ ( plus_plus_real @ A_187 @ D_30 ) ) ) ).
thf(fact_842_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_187: nat,C_77: nat,D_30: nat] :
( ( plus_plus_nat @ A_187 @ ( plus_plus_nat @ C_77 @ D_30 ) )
= ( plus_plus_nat @ C_77 @ ( plus_plus_nat @ A_187 @ D_30 ) ) ) ).
thf(fact_843_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_187: int,C_77: int,D_30: int] :
( ( plus_plus_int @ A_187 @ ( plus_plus_int @ C_77 @ D_30 ) )
= ( plus_plus_int @ C_77 @ ( plus_plus_int @ A_187 @ D_30 ) ) ) ).
thf(fact_844_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_186: rat,C_76: rat] :
( ( plus_plus_rat @ A_186 @ C_76 )
= ( plus_plus_rat @ C_76 @ A_186 ) ) ).
thf(fact_845_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_186: quickcheck_code_int,C_76: quickcheck_code_int] :
( ( plus_p1446045655de_int @ A_186 @ C_76 )
= ( plus_p1446045655de_int @ C_76 @ A_186 ) ) ).
thf(fact_846_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_186: complex,C_76: complex] :
( ( plus_plus_complex @ A_186 @ C_76 )
= ( plus_plus_complex @ C_76 @ A_186 ) ) ).
thf(fact_847_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_186: real,C_76: real] :
( ( plus_plus_real @ A_186 @ C_76 )
= ( plus_plus_real @ C_76 @ A_186 ) ) ).
thf(fact_848_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_186: nat,C_76: nat] :
( ( plus_plus_nat @ A_186 @ C_76 )
= ( plus_plus_nat @ C_76 @ A_186 ) ) ).
thf(fact_849_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_186: int,C_76: int] :
( ( plus_plus_int @ A_186 @ C_76 )
= ( plus_plus_int @ C_76 @ A_186 ) ) ).
thf(fact_850_one__reorient,axiom,
! [X_83: rat] :
( ( one_one_rat = X_83 )
<=> ( X_83 = one_one_rat ) ) ).
thf(fact_851_one__reorient,axiom,
! [X_83: quickcheck_code_int] :
( ( one_on1684967323de_int = X_83 )
<=> ( X_83 = one_on1684967323de_int ) ) ).
thf(fact_852_one__reorient,axiom,
! [X_83: complex] :
( ( one_one_complex = X_83 )
<=> ( X_83 = one_one_complex ) ) ).
thf(fact_853_one__reorient,axiom,
! [X_83: code_code_numeral] :
( ( one_on1645066479umeral = X_83 )
<=> ( X_83 = one_on1645066479umeral ) ) ).
thf(fact_854_one__reorient,axiom,
! [X_83: real] :
( ( one_one_real = X_83 )
<=> ( X_83 = one_one_real ) ) ).
thf(fact_855_one__reorient,axiom,
! [X_83: nat] :
( ( one_one_nat = X_83 )
<=> ( X_83 = one_one_nat ) ) ).
thf(fact_856_one__reorient,axiom,
! [X_83: int] :
( ( one_one_int = X_83 )
<=> ( X_83 = one_one_int ) ) ).
thf(fact_857_succ__def,axiom,
! [K: int] :
( ( succ @ K )
= ( plus_plus_int @ K @ one_one_int ) ) ).
thf(fact_858_power__eq__imp__eq__base,axiom,
! [A_185: rat,N_31: nat,B_132: rat] :
( ( ( power_power_rat @ A_185 @ N_31 )
= ( power_power_rat @ B_132 @ N_31 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A_185 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_132 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_31 )
=> ( A_185 = B_132 ) ) ) ) ) ).
thf(fact_859_power__eq__imp__eq__base,axiom,
! [A_185: quickcheck_code_int,N_31: nat,B_132: quickcheck_code_int] :
( ( ( power_881366806de_int @ A_185 @ N_31 )
= ( power_881366806de_int @ B_132 @ N_31 ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_185 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_132 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_31 )
=> ( A_185 = B_132 ) ) ) ) ) ).
thf(fact_860_power__eq__imp__eq__base,axiom,
! [A_185: code_code_numeral,N_31: nat,B_132: code_code_numeral] :
( ( ( power_2100829034umeral @ A_185 @ N_31 )
= ( power_2100829034umeral @ B_132 @ N_31 ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_185 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_132 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_31 )
=> ( A_185 = B_132 ) ) ) ) ) ).
thf(fact_861_power__eq__imp__eq__base,axiom,
! [A_185: real,N_31: nat,B_132: real] :
( ( ( power_power_real @ A_185 @ N_31 )
= ( power_power_real @ B_132 @ N_31 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_185 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_132 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_31 )
=> ( A_185 = B_132 ) ) ) ) ) ).
thf(fact_862_power__eq__imp__eq__base,axiom,
! [A_185: nat,N_31: nat,B_132: nat] :
( ( ( power_power_nat @ A_185 @ N_31 )
= ( power_power_nat @ B_132 @ N_31 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_185 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_132 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_31 )
=> ( A_185 = B_132 ) ) ) ) ) ).
thf(fact_863_power__eq__imp__eq__base,axiom,
! [A_185: int,N_31: nat,B_132: int] :
( ( ( power_power_int @ A_185 @ N_31 )
= ( power_power_int @ B_132 @ N_31 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_185 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_132 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_31 )
=> ( A_185 = B_132 ) ) ) ) ) ).
thf(fact_864_semiring__add__number__of,axiom,
! [V_18: int,V_17: int] :
( ( ord_less_eq_int @ pls @ V_17 )
=> ( ( ord_less_eq_int @ pls @ V_18 )
=> ( ( plus_plus_rat @ ( number_number_of_rat @ V_17 ) @ ( number_number_of_rat @ V_18 ) )
= ( number_number_of_rat @ ( plus_plus_int @ V_17 @ V_18 ) ) ) ) ) ).
thf(fact_865_semiring__add__number__of,axiom,
! [V_18: int,V_17: int] :
( ( ord_less_eq_int @ pls @ V_17 )
=> ( ( ord_less_eq_int @ pls @ V_18 )
=> ( ( plus_plus_complex @ ( number528085621omplex @ V_17 ) @ ( number528085621omplex @ V_18 ) )
= ( number528085621omplex @ ( plus_plus_int @ V_17 @ V_18 ) ) ) ) ) ).
thf(fact_866_semiring__add__number__of,axiom,
! [V_18: int,V_17: int] :
( ( ord_less_eq_int @ pls @ V_17 )
=> ( ( ord_less_eq_int @ pls @ V_18 )
=> ( ( plus_plus_real @ ( number267125858f_real @ V_17 ) @ ( number267125858f_real @ V_18 ) )
= ( number267125858f_real @ ( plus_plus_int @ V_17 @ V_18 ) ) ) ) ) ).
thf(fact_867_semiring__add__number__of,axiom,
! [V_18: int,V_17: int] :
( ( ord_less_eq_int @ pls @ V_17 )
=> ( ( ord_less_eq_int @ pls @ V_18 )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V_17 ) @ ( number_number_of_nat @ V_18 ) )
= ( number_number_of_nat @ ( plus_plus_int @ V_17 @ V_18 ) ) ) ) ) ).
thf(fact_868_semiring__add__number__of,axiom,
! [V_18: int,V_17: int] :
( ( ord_less_eq_int @ pls @ V_17 )
=> ( ( ord_less_eq_int @ pls @ V_18 )
=> ( ( plus_plus_int @ ( number_number_of_int @ V_17 ) @ ( number_number_of_int @ V_18 ) )
= ( number_number_of_int @ ( plus_plus_int @ V_17 @ V_18 ) ) ) ) ) ).
thf(fact_869_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_870_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_871_number__of__succ,axiom,
! [K_9: int] :
( ( number_number_of_rat @ ( succ @ K_9 ) )
= ( plus_plus_rat @ one_one_rat @ ( number_number_of_rat @ K_9 ) ) ) ).
thf(fact_872_number__of__succ,axiom,
! [K_9: int] :
( ( number528085621omplex @ ( succ @ K_9 ) )
= ( plus_plus_complex @ one_one_complex @ ( number528085621omplex @ K_9 ) ) ) ).
thf(fact_873_number__of__succ,axiom,
! [K_9: int] :
( ( number267125858f_real @ ( succ @ K_9 ) )
= ( plus_plus_real @ one_one_real @ ( number267125858f_real @ K_9 ) ) ) ).
thf(fact_874_number__of__succ,axiom,
! [K_9: int] :
( ( number_number_of_int @ ( succ @ K_9 ) )
= ( plus_plus_int @ one_one_int @ ( number_number_of_int @ K_9 ) ) ) ).
thf(fact_875_add__Bit1__Bit1,axiom,
! [K: int,L: int] :
( ( plus_plus_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
= ( bit0 @ ( plus_plus_int @ K @ ( succ @ L ) ) ) ) ).
thf(fact_876_power__strict__mono,axiom,
! [N_30: nat,A_184: rat,B_131: rat] :
( ( ord_less_rat @ A_184 @ B_131 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A_184 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_30 )
=> ( ord_less_rat @ ( power_power_rat @ A_184 @ N_30 ) @ ( power_power_rat @ B_131 @ N_30 ) ) ) ) ) ).
thf(fact_877_power__strict__mono,axiom,
! [N_30: nat,A_184: quickcheck_code_int,B_131: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_184 @ B_131 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_184 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_30 )
=> ( ord_le1860547276de_int @ ( power_881366806de_int @ A_184 @ N_30 ) @ ( power_881366806de_int @ B_131 @ N_30 ) ) ) ) ) ).
thf(fact_878_power__strict__mono,axiom,
! [N_30: nat,A_184: code_code_numeral,B_131: code_code_numeral] :
( ( ord_le1304079648umeral @ A_184 @ B_131 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_184 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_30 )
=> ( ord_le1304079648umeral @ ( power_2100829034umeral @ A_184 @ N_30 ) @ ( power_2100829034umeral @ B_131 @ N_30 ) ) ) ) ) ).
thf(fact_879_power__strict__mono,axiom,
! [N_30: nat,A_184: real,B_131: real] :
( ( ord_less_real @ A_184 @ B_131 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_184 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_30 )
=> ( ord_less_real @ ( power_power_real @ A_184 @ N_30 ) @ ( power_power_real @ B_131 @ N_30 ) ) ) ) ) ).
thf(fact_880_power__strict__mono,axiom,
! [N_30: nat,A_184: nat,B_131: nat] :
( ( ord_less_nat @ A_184 @ B_131 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_184 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_30 )
=> ( ord_less_nat @ ( power_power_nat @ A_184 @ N_30 ) @ ( power_power_nat @ B_131 @ N_30 ) ) ) ) ) ).
thf(fact_881_power__strict__mono,axiom,
! [N_30: nat,A_184: int,B_131: int] :
( ( ord_less_int @ A_184 @ B_131 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_184 )
=> ( ( ord_less_nat @ zero_zero_nat @ N_30 )
=> ( ord_less_int @ ( power_power_int @ A_184 @ N_30 ) @ ( power_power_int @ B_131 @ N_30 ) ) ) ) ) ).
thf(fact_882_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_883_power2__ge__self,axiom,
! [X: int] : ( ord_less_eq_int @ X @ ( power_power_int @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_884_power2__eq__imp__eq,axiom,
! [X_82: rat,Y_69: rat] :
( ( ( power_power_rat @ X_82 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_rat @ Y_69 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ X_82 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y_69 )
=> ( X_82 = Y_69 ) ) ) ) ).
thf(fact_885_power2__eq__imp__eq,axiom,
! [X_82: quickcheck_code_int,Y_69: quickcheck_code_int] :
( ( ( power_881366806de_int @ X_82 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_881366806de_int @ Y_69 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ X_82 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ Y_69 )
=> ( X_82 = Y_69 ) ) ) ) ).
thf(fact_886_power2__eq__imp__eq,axiom,
! [X_82: code_code_numeral,Y_69: code_code_numeral] :
( ( ( power_2100829034umeral @ X_82 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_2100829034umeral @ Y_69 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ X_82 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ Y_69 )
=> ( X_82 = Y_69 ) ) ) ) ).
thf(fact_887_power2__eq__imp__eq,axiom,
! [X_82: real,Y_69: real] :
( ( ( power_power_real @ X_82 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_real @ Y_69 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X_82 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_69 )
=> ( X_82 = Y_69 ) ) ) ) ).
thf(fact_888_power2__eq__imp__eq,axiom,
! [X_82: nat,Y_69: nat] :
( ( ( power_power_nat @ X_82 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_nat @ Y_69 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X_82 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_69 )
=> ( X_82 = Y_69 ) ) ) ) ).
thf(fact_889_power2__eq__imp__eq,axiom,
! [X_82: int,Y_69: int] :
( ( ( power_power_int @ X_82 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_int @ Y_69 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ X_82 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_69 )
=> ( X_82 = Y_69 ) ) ) ) ).
thf(fact_890_power2__le__imp__le,axiom,
! [X_81: rat,Y_68: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ X_81 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_rat @ Y_68 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y_68 )
=> ( ord_less_eq_rat @ X_81 @ Y_68 ) ) ) ).
thf(fact_891_power2__le__imp__le,axiom,
! [X_81: quickcheck_code_int,Y_68: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( power_881366806de_int @ X_81 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_881366806de_int @ Y_68 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ Y_68 )
=> ( ord_le258702272de_int @ X_81 @ Y_68 ) ) ) ).
thf(fact_892_power2__le__imp__le,axiom,
! [X_81: code_code_numeral,Y_68: code_code_numeral] :
( ( ord_le565307924umeral @ ( power_2100829034umeral @ X_81 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_2100829034umeral @ Y_68 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ Y_68 )
=> ( ord_le565307924umeral @ X_81 @ Y_68 ) ) ) ).
thf(fact_893_power2__le__imp__le,axiom,
! [X_81: real,Y_68: real] :
( ( ord_less_eq_real @ ( power_power_real @ X_81 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_68 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_68 )
=> ( ord_less_eq_real @ X_81 @ Y_68 ) ) ) ).
thf(fact_894_power2__le__imp__le,axiom,
! [X_81: nat,Y_68: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X_81 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_nat @ Y_68 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_68 )
=> ( ord_less_eq_nat @ X_81 @ Y_68 ) ) ) ).
thf(fact_895_power2__le__imp__le,axiom,
! [X_81: int,Y_68: int] :
( ( ord_less_eq_int @ ( power_power_int @ X_81 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_68 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_68 )
=> ( ord_less_eq_int @ X_81 @ Y_68 ) ) ) ).
thf(fact_896_zero__le__power2,axiom,
! [A_183: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A_183 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_897_zero__le__power2,axiom,
! [A_183: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A_183 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_898_zero__le__power2,axiom,
! [A_183: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A_183 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_899_of__nat__number__of__lemma,axiom,
! [V_16: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ V_16 ) )
=> ( ( semiri151668891at_rat @ ( number_number_of_nat @ V_16 ) )
= ( number_number_of_rat @ V_16 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ V_16 ) )
=> ( ( semiri151668891at_rat @ ( number_number_of_nat @ V_16 ) )
= zero_zero_rat ) ) ) ).
thf(fact_900_of__nat__number__of__lemma,axiom,
! [V_16: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ V_16 ) )
=> ( ( semiri2020571505omplex @ ( number_number_of_nat @ V_16 ) )
= ( number528085621omplex @ V_16 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ V_16 ) )
=> ( ( semiri2020571505omplex @ ( number_number_of_nat @ V_16 ) )
= zero_zero_complex ) ) ) ).
thf(fact_901_of__nat__number__of__lemma,axiom,
! [V_16: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ V_16 ) )
=> ( ( semiri132038758t_real @ ( number_number_of_nat @ V_16 ) )
= ( number267125858f_real @ V_16 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ V_16 ) )
=> ( ( semiri132038758t_real @ ( number_number_of_nat @ V_16 ) )
= zero_zero_real ) ) ) ).
thf(fact_902_of__nat__number__of__lemma,axiom,
! [V_16: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ V_16 ) )
=> ( ( semiri1621563631at_int @ ( number_number_of_nat @ V_16 ) )
= ( number_number_of_int @ V_16 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ V_16 ) )
=> ( ( semiri1621563631at_int @ ( number_number_of_nat @ V_16 ) )
= zero_zero_int ) ) ) ).
thf(fact_903_add__0__iff,axiom,
! [B_130: rat,A_182: rat] :
( ( B_130
= ( plus_plus_rat @ B_130 @ A_182 ) )
<=> ( A_182 = zero_zero_rat ) ) ).
thf(fact_904_add__0__iff,axiom,
! [B_130: complex,A_182: complex] :
( ( B_130
= ( plus_plus_complex @ B_130 @ A_182 ) )
<=> ( A_182 = zero_zero_complex ) ) ).
thf(fact_905_add__0__iff,axiom,
! [B_130: real,A_182: real] :
( ( B_130
= ( plus_plus_real @ B_130 @ A_182 ) )
<=> ( A_182 = zero_zero_real ) ) ).
thf(fact_906_add__0__iff,axiom,
! [B_130: nat,A_182: nat] :
( ( B_130
= ( plus_plus_nat @ B_130 @ A_182 ) )
<=> ( A_182 = zero_zero_nat ) ) ).
thf(fact_907_add__0__iff,axiom,
! [B_130: int,A_182: int] :
( ( B_130
= ( plus_plus_int @ B_130 @ A_182 ) )
<=> ( A_182 = zero_zero_int ) ) ).
thf(fact_908_add_Ocomm__neutral,axiom,
! [A_181: rat] :
( ( plus_plus_rat @ A_181 @ zero_zero_rat )
= A_181 ) ).
thf(fact_909_add_Ocomm__neutral,axiom,
! [A_181: quickcheck_code_int] :
( ( plus_p1446045655de_int @ A_181 @ zero_z891286103de_int )
= A_181 ) ).
thf(fact_910_add_Ocomm__neutral,axiom,
! [A_181: complex] :
( ( plus_plus_complex @ A_181 @ zero_zero_complex )
= A_181 ) ).
thf(fact_911_add_Ocomm__neutral,axiom,
! [A_181: real] :
( ( plus_plus_real @ A_181 @ zero_zero_real )
= A_181 ) ).
thf(fact_912_add_Ocomm__neutral,axiom,
! [A_181: nat] :
( ( plus_plus_nat @ A_181 @ zero_zero_nat )
= A_181 ) ).
thf(fact_913_add_Ocomm__neutral,axiom,
! [A_181: int] :
( ( plus_plus_int @ A_181 @ zero_zero_int )
= A_181 ) ).
thf(fact_914_add_Ocomm__neutral,axiom,
! [A_181: code_code_numeral] :
( ( plus_p1627245867umeral @ A_181 @ zero_z126310315umeral )
= A_181 ) ).
thf(fact_915_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_180: rat] :
( ( plus_plus_rat @ A_180 @ zero_zero_rat )
= A_180 ) ).
thf(fact_916_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_180: quickcheck_code_int] :
( ( plus_p1446045655de_int @ A_180 @ zero_z891286103de_int )
= A_180 ) ).
thf(fact_917_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_180: complex] :
( ( plus_plus_complex @ A_180 @ zero_zero_complex )
= A_180 ) ).
thf(fact_918_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_180: real] :
( ( plus_plus_real @ A_180 @ zero_zero_real )
= A_180 ) ).
thf(fact_919_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_180: nat] :
( ( plus_plus_nat @ A_180 @ zero_zero_nat )
= A_180 ) ).
thf(fact_920_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_180: int] :
( ( plus_plus_int @ A_180 @ zero_zero_int )
= A_180 ) ).
thf(fact_921_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_180: code_code_numeral] :
( ( plus_p1627245867umeral @ A_180 @ zero_z126310315umeral )
= A_180 ) ).
thf(fact_922_add__0__right,axiom,
! [A_179: rat] :
( ( plus_plus_rat @ A_179 @ zero_zero_rat )
= A_179 ) ).
thf(fact_923_add__0__right,axiom,
! [A_179: quickcheck_code_int] :
( ( plus_p1446045655de_int @ A_179 @ zero_z891286103de_int )
= A_179 ) ).
thf(fact_924_add__0__right,axiom,
! [A_179: complex] :
( ( plus_plus_complex @ A_179 @ zero_zero_complex )
= A_179 ) ).
thf(fact_925_add__0__right,axiom,
! [A_179: real] :
( ( plus_plus_real @ A_179 @ zero_zero_real )
= A_179 ) ).
thf(fact_926_add__0__right,axiom,
! [A_179: nat] :
( ( plus_plus_nat @ A_179 @ zero_zero_nat )
= A_179 ) ).
thf(fact_927_add__0__right,axiom,
! [A_179: int] :
( ( plus_plus_int @ A_179 @ zero_zero_int )
= A_179 ) ).
thf(fact_928_add__0__right,axiom,
! [A_179: code_code_numeral] :
( ( plus_p1627245867umeral @ A_179 @ zero_z126310315umeral )
= A_179 ) ).
thf(fact_929_double__zero__sym,axiom,
! [A_178: rat] :
( ( zero_zero_rat
= ( plus_plus_rat @ A_178 @ A_178 ) )
<=> ( A_178 = zero_zero_rat ) ) ).
thf(fact_930_double__zero__sym,axiom,
! [A_178: real] :
( ( zero_zero_real
= ( plus_plus_real @ A_178 @ A_178 ) )
<=> ( A_178 = zero_zero_real ) ) ).
thf(fact_931_double__zero__sym,axiom,
! [A_178: int] :
( ( zero_zero_int
= ( plus_plus_int @ A_178 @ A_178 ) )
<=> ( A_178 = zero_zero_int ) ) ).
thf(fact_932_add__0,axiom,
! [A_177: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A_177 )
= A_177 ) ).
thf(fact_933_add__0,axiom,
! [A_177: quickcheck_code_int] :
( ( plus_p1446045655de_int @ zero_z891286103de_int @ A_177 )
= A_177 ) ).
thf(fact_934_add__0,axiom,
! [A_177: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A_177 )
= A_177 ) ).
thf(fact_935_add__0,axiom,
! [A_177: real] :
( ( plus_plus_real @ zero_zero_real @ A_177 )
= A_177 ) ).
thf(fact_936_add__0,axiom,
! [A_177: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_177 )
= A_177 ) ).
thf(fact_937_add__0,axiom,
! [A_177: int] :
( ( plus_plus_int @ zero_zero_int @ A_177 )
= A_177 ) ).
thf(fact_938_add__0,axiom,
! [A_177: code_code_numeral] :
( ( plus_p1627245867umeral @ zero_z126310315umeral @ A_177 )
= A_177 ) ).
thf(fact_939_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_176: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A_176 )
= A_176 ) ).
thf(fact_940_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_176: quickcheck_code_int] :
( ( plus_p1446045655de_int @ zero_z891286103de_int @ A_176 )
= A_176 ) ).
thf(fact_941_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_176: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A_176 )
= A_176 ) ).
thf(fact_942_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_176: real] :
( ( plus_plus_real @ zero_zero_real @ A_176 )
= A_176 ) ).
thf(fact_943_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_176: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_176 )
= A_176 ) ).
thf(fact_944_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_176: int] :
( ( plus_plus_int @ zero_zero_int @ A_176 )
= A_176 ) ).
thf(fact_945_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_176: code_code_numeral] :
( ( plus_p1627245867umeral @ zero_z126310315umeral @ A_176 )
= A_176 ) ).
thf(fact_946_add__0__left,axiom,
! [A_175: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A_175 )
= A_175 ) ).
thf(fact_947_add__0__left,axiom,
! [A_175: quickcheck_code_int] :
( ( plus_p1446045655de_int @ zero_z891286103de_int @ A_175 )
= A_175 ) ).
thf(fact_948_add__0__left,axiom,
! [A_175: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A_175 )
= A_175 ) ).
thf(fact_949_add__0__left,axiom,
! [A_175: real] :
( ( plus_plus_real @ zero_zero_real @ A_175 )
= A_175 ) ).
thf(fact_950_add__0__left,axiom,
! [A_175: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_175 )
= A_175 ) ).
thf(fact_951_add__0__left,axiom,
! [A_175: int] :
( ( plus_plus_int @ zero_zero_int @ A_175 )
= A_175 ) ).
thf(fact_952_add__0__left,axiom,
! [A_175: code_code_numeral] :
( ( plus_p1627245867umeral @ zero_z126310315umeral @ A_175 )
= A_175 ) ).
thf(fact_953_add__less__imp__less__left,axiom,
! [C_75: rat,A_174: rat,B_129: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C_75 @ A_174 ) @ ( plus_plus_rat @ C_75 @ B_129 ) )
=> ( ord_less_rat @ A_174 @ B_129 ) ) ).
thf(fact_954_add__less__imp__less__left,axiom,
! [C_75: quickcheck_code_int,A_174: quickcheck_code_int,B_129: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ C_75 @ A_174 ) @ ( plus_p1446045655de_int @ C_75 @ B_129 ) )
=> ( ord_le1860547276de_int @ A_174 @ B_129 ) ) ).
thf(fact_955_add__less__imp__less__left,axiom,
! [C_75: real,A_174: real,B_129: real] :
( ( ord_less_real @ ( plus_plus_real @ C_75 @ A_174 ) @ ( plus_plus_real @ C_75 @ B_129 ) )
=> ( ord_less_real @ A_174 @ B_129 ) ) ).
thf(fact_956_add__less__imp__less__left,axiom,
! [C_75: nat,A_174: nat,B_129: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_75 @ A_174 ) @ ( plus_plus_nat @ C_75 @ B_129 ) )
=> ( ord_less_nat @ A_174 @ B_129 ) ) ).
thf(fact_957_add__less__imp__less__left,axiom,
! [C_75: int,A_174: int,B_129: int] :
( ( ord_less_int @ ( plus_plus_int @ C_75 @ A_174 ) @ ( plus_plus_int @ C_75 @ B_129 ) )
=> ( ord_less_int @ A_174 @ B_129 ) ) ).
thf(fact_958_add__less__imp__less__right,axiom,
! [A_173: rat,C_74: rat,B_128: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A_173 @ C_74 ) @ ( plus_plus_rat @ B_128 @ C_74 ) )
=> ( ord_less_rat @ A_173 @ B_128 ) ) ).
thf(fact_959_add__less__imp__less__right,axiom,
! [A_173: quickcheck_code_int,C_74: quickcheck_code_int,B_128: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_173 @ C_74 ) @ ( plus_p1446045655de_int @ B_128 @ C_74 ) )
=> ( ord_le1860547276de_int @ A_173 @ B_128 ) ) ).
thf(fact_960_add__less__imp__less__right,axiom,
! [A_173: real,C_74: real,B_128: real] :
( ( ord_less_real @ ( plus_plus_real @ A_173 @ C_74 ) @ ( plus_plus_real @ B_128 @ C_74 ) )
=> ( ord_less_real @ A_173 @ B_128 ) ) ).
thf(fact_961_add__less__imp__less__right,axiom,
! [A_173: nat,C_74: nat,B_128: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_173 @ C_74 ) @ ( plus_plus_nat @ B_128 @ C_74 ) )
=> ( ord_less_nat @ A_173 @ B_128 ) ) ).
thf(fact_962_add__less__imp__less__right,axiom,
! [A_173: int,C_74: int,B_128: int] :
( ( ord_less_int @ ( plus_plus_int @ A_173 @ C_74 ) @ ( plus_plus_int @ B_128 @ C_74 ) )
=> ( ord_less_int @ A_173 @ B_128 ) ) ).
thf(fact_963_add__strict__mono,axiom,
! [C_73: rat,D_29: rat,A_172: rat,B_127: rat] :
( ( ord_less_rat @ A_172 @ B_127 )
=> ( ( ord_less_rat @ C_73 @ D_29 )
=> ( ord_less_rat @ ( plus_plus_rat @ A_172 @ C_73 ) @ ( plus_plus_rat @ B_127 @ D_29 ) ) ) ) ).
thf(fact_964_add__strict__mono,axiom,
! [C_73: quickcheck_code_int,D_29: quickcheck_code_int,A_172: quickcheck_code_int,B_127: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_172 @ B_127 )
=> ( ( ord_le1860547276de_int @ C_73 @ D_29 )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_172 @ C_73 ) @ ( plus_p1446045655de_int @ B_127 @ D_29 ) ) ) ) ).
thf(fact_965_add__strict__mono,axiom,
! [C_73: real,D_29: real,A_172: real,B_127: real] :
( ( ord_less_real @ A_172 @ B_127 )
=> ( ( ord_less_real @ C_73 @ D_29 )
=> ( ord_less_real @ ( plus_plus_real @ A_172 @ C_73 ) @ ( plus_plus_real @ B_127 @ D_29 ) ) ) ) ).
thf(fact_966_add__strict__mono,axiom,
! [C_73: nat,D_29: nat,A_172: nat,B_127: nat] :
( ( ord_less_nat @ A_172 @ B_127 )
=> ( ( ord_less_nat @ C_73 @ D_29 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_172 @ C_73 ) @ ( plus_plus_nat @ B_127 @ D_29 ) ) ) ) ).
thf(fact_967_add__strict__mono,axiom,
! [C_73: int,D_29: int,A_172: int,B_127: int] :
( ( ord_less_int @ A_172 @ B_127 )
=> ( ( ord_less_int @ C_73 @ D_29 )
=> ( ord_less_int @ ( plus_plus_int @ A_172 @ C_73 ) @ ( plus_plus_int @ B_127 @ D_29 ) ) ) ) ).
thf(fact_968_add__strict__left__mono,axiom,
! [C_72: rat,A_171: rat,B_126: rat] :
( ( ord_less_rat @ A_171 @ B_126 )
=> ( ord_less_rat @ ( plus_plus_rat @ C_72 @ A_171 ) @ ( plus_plus_rat @ C_72 @ B_126 ) ) ) ).
thf(fact_969_add__strict__left__mono,axiom,
! [C_72: quickcheck_code_int,A_171: quickcheck_code_int,B_126: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_171 @ B_126 )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ C_72 @ A_171 ) @ ( plus_p1446045655de_int @ C_72 @ B_126 ) ) ) ).
thf(fact_970_add__strict__left__mono,axiom,
! [C_72: real,A_171: real,B_126: real] :
( ( ord_less_real @ A_171 @ B_126 )
=> ( ord_less_real @ ( plus_plus_real @ C_72 @ A_171 ) @ ( plus_plus_real @ C_72 @ B_126 ) ) ) ).
thf(fact_971_add__strict__left__mono,axiom,
! [C_72: nat,A_171: nat,B_126: nat] :
( ( ord_less_nat @ A_171 @ B_126 )
=> ( ord_less_nat @ ( plus_plus_nat @ C_72 @ A_171 ) @ ( plus_plus_nat @ C_72 @ B_126 ) ) ) ).
thf(fact_972_add__strict__left__mono,axiom,
! [C_72: int,A_171: int,B_126: int] :
( ( ord_less_int @ A_171 @ B_126 )
=> ( ord_less_int @ ( plus_plus_int @ C_72 @ A_171 ) @ ( plus_plus_int @ C_72 @ B_126 ) ) ) ).
thf(fact_973_add__strict__right__mono,axiom,
! [C_71: rat,A_170: rat,B_125: rat] :
( ( ord_less_rat @ A_170 @ B_125 )
=> ( ord_less_rat @ ( plus_plus_rat @ A_170 @ C_71 ) @ ( plus_plus_rat @ B_125 @ C_71 ) ) ) ).
thf(fact_974_add__strict__right__mono,axiom,
! [C_71: quickcheck_code_int,A_170: quickcheck_code_int,B_125: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_170 @ B_125 )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_170 @ C_71 ) @ ( plus_p1446045655de_int @ B_125 @ C_71 ) ) ) ).
thf(fact_975_add__strict__right__mono,axiom,
! [C_71: real,A_170: real,B_125: real] :
( ( ord_less_real @ A_170 @ B_125 )
=> ( ord_less_real @ ( plus_plus_real @ A_170 @ C_71 ) @ ( plus_plus_real @ B_125 @ C_71 ) ) ) ).
thf(fact_976_add__strict__right__mono,axiom,
! [C_71: nat,A_170: nat,B_125: nat] :
( ( ord_less_nat @ A_170 @ B_125 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_170 @ C_71 ) @ ( plus_plus_nat @ B_125 @ C_71 ) ) ) ).
thf(fact_977_add__strict__right__mono,axiom,
! [C_71: int,A_170: int,B_125: int] :
( ( ord_less_int @ A_170 @ B_125 )
=> ( ord_less_int @ ( plus_plus_int @ A_170 @ C_71 ) @ ( plus_plus_int @ B_125 @ C_71 ) ) ) ).
thf(fact_978_add__less__cancel__left,axiom,
! [C_70: rat,A_169: rat,B_124: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C_70 @ A_169 ) @ ( plus_plus_rat @ C_70 @ B_124 ) )
<=> ( ord_less_rat @ A_169 @ B_124 ) ) ).
thf(fact_979_add__less__cancel__left,axiom,
! [C_70: quickcheck_code_int,A_169: quickcheck_code_int,B_124: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ C_70 @ A_169 ) @ ( plus_p1446045655de_int @ C_70 @ B_124 ) )
<=> ( ord_le1860547276de_int @ A_169 @ B_124 ) ) ).
thf(fact_980_add__less__cancel__left,axiom,
! [C_70: real,A_169: real,B_124: real] :
( ( ord_less_real @ ( plus_plus_real @ C_70 @ A_169 ) @ ( plus_plus_real @ C_70 @ B_124 ) )
<=> ( ord_less_real @ A_169 @ B_124 ) ) ).
thf(fact_981_add__less__cancel__left,axiom,
! [C_70: nat,A_169: nat,B_124: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_70 @ A_169 ) @ ( plus_plus_nat @ C_70 @ B_124 ) )
<=> ( ord_less_nat @ A_169 @ B_124 ) ) ).
thf(fact_982_add__less__cancel__left,axiom,
! [C_70: int,A_169: int,B_124: int] :
( ( ord_less_int @ ( plus_plus_int @ C_70 @ A_169 ) @ ( plus_plus_int @ C_70 @ B_124 ) )
<=> ( ord_less_int @ A_169 @ B_124 ) ) ).
thf(fact_983_add__less__cancel__right,axiom,
! [A_168: rat,C_69: rat,B_123: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A_168 @ C_69 ) @ ( plus_plus_rat @ B_123 @ C_69 ) )
<=> ( ord_less_rat @ A_168 @ B_123 ) ) ).
thf(fact_984_add__less__cancel__right,axiom,
! [A_168: quickcheck_code_int,C_69: quickcheck_code_int,B_123: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_168 @ C_69 ) @ ( plus_p1446045655de_int @ B_123 @ C_69 ) )
<=> ( ord_le1860547276de_int @ A_168 @ B_123 ) ) ).
thf(fact_985_add__less__cancel__right,axiom,
! [A_168: real,C_69: real,B_123: real] :
( ( ord_less_real @ ( plus_plus_real @ A_168 @ C_69 ) @ ( plus_plus_real @ B_123 @ C_69 ) )
<=> ( ord_less_real @ A_168 @ B_123 ) ) ).
thf(fact_986_add__less__cancel__right,axiom,
! [A_168: nat,C_69: nat,B_123: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_168 @ C_69 ) @ ( plus_plus_nat @ B_123 @ C_69 ) )
<=> ( ord_less_nat @ A_168 @ B_123 ) ) ).
thf(fact_987_add__less__cancel__right,axiom,
! [A_168: int,C_69: int,B_123: int] :
( ( ord_less_int @ ( plus_plus_int @ A_168 @ C_69 ) @ ( plus_plus_int @ B_123 @ C_69 ) )
<=> ( ord_less_int @ A_168 @ B_123 ) ) ).
thf(fact_988_power2__less__imp__less,axiom,
! [X_80: rat,Y_67: rat] :
( ( ord_less_rat @ ( power_power_rat @ X_80 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_rat @ Y_67 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y_67 )
=> ( ord_less_rat @ X_80 @ Y_67 ) ) ) ).
thf(fact_989_power2__less__imp__less,axiom,
! [X_80: quickcheck_code_int,Y_67: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( power_881366806de_int @ X_80 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_881366806de_int @ Y_67 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ Y_67 )
=> ( ord_le1860547276de_int @ X_80 @ Y_67 ) ) ) ).
thf(fact_990_power2__less__imp__less,axiom,
! [X_80: code_code_numeral,Y_67: code_code_numeral] :
( ( ord_le1304079648umeral @ ( power_2100829034umeral @ X_80 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_2100829034umeral @ Y_67 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ Y_67 )
=> ( ord_le1304079648umeral @ X_80 @ Y_67 ) ) ) ).
thf(fact_991_power2__less__imp__less,axiom,
! [X_80: real,Y_67: real] :
( ( ord_less_real @ ( power_power_real @ X_80 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_67 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_67 )
=> ( ord_less_real @ X_80 @ Y_67 ) ) ) ).
thf(fact_992_power2__less__imp__less,axiom,
! [X_80: nat,Y_67: nat] :
( ( ord_less_nat @ ( power_power_nat @ X_80 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_nat @ Y_67 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_67 )
=> ( ord_less_nat @ X_80 @ Y_67 ) ) ) ).
thf(fact_993_power2__less__imp__less,axiom,
! [X_80: int,Y_67: int] :
( ( ord_less_int @ ( power_power_int @ X_80 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_67 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_67 )
=> ( ord_less_int @ X_80 @ Y_67 ) ) ) ).
thf(fact_994_sum__power2__le__zero__iff,axiom,
! [X_79: rat,Y_66: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X_79 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_rat @ Y_66 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_rat )
<=> ( ( X_79 = zero_zero_rat )
& ( Y_66 = zero_zero_rat ) ) ) ).
thf(fact_995_sum__power2__le__zero__iff,axiom,
! [X_79: real,Y_66: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X_79 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_66 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_real )
<=> ( ( X_79 = zero_zero_real )
& ( Y_66 = zero_zero_real ) ) ) ).
thf(fact_996_sum__power2__le__zero__iff,axiom,
! [X_79: int,Y_66: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X_79 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_66 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ zero_zero_int )
<=> ( ( X_79 = zero_zero_int )
& ( Y_66 = zero_zero_int ) ) ) ).
thf(fact_997_sum__power2__ge__zero,axiom,
! [X_78: rat,Y_65: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X_78 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_rat @ Y_65 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_998_sum__power2__ge__zero,axiom,
! [X_78: real,Y_65: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X_78 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_65 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_999_sum__power2__ge__zero,axiom,
! [X_78: int,Y_65: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X_78 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_65 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_1000_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_77: complex] :
( ( power_power_complex @ X_77 @ one_one_nat )
= X_77 ) ).
thf(fact_1001_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_77: real] :
( ( power_power_real @ X_77 @ one_one_nat )
= X_77 ) ).
thf(fact_1002_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_77: nat] :
( ( power_power_nat @ X_77 @ one_one_nat )
= X_77 ) ).
thf(fact_1003_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_77: int] :
( ( power_power_int @ X_77 @ one_one_nat )
= X_77 ) ).
thf(fact_1004_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_1005_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_1006_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_1007_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_1008_realpow__pos__nth__unique,axiom,
! [A_1: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A_1 )
=> ? [X_1: real] :
( ( ord_less_real @ zero_zero_real @ X_1 )
& ( ( power_power_real @ X_1 @ N )
= A_1 )
& ! [Y_1: real] :
( ( ( ord_less_real @ zero_zero_real @ Y_1 )
& ( ( power_power_real @ Y_1 @ N )
= A_1 ) )
=> ( Y_1 = X_1 ) ) ) ) ) ).
thf(fact_1009_realpow__pos__nth,axiom,
! [A_1: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A_1 )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ N )
= A_1 ) ) ) ) ).
thf(fact_1010_number__of2,axiom,
ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ pls ) ).
thf(fact_1011_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_1012_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N_1: nat] :
( K
!= ( semiri1621563631at_int @ N_1 ) ) ) ).
thf(fact_1013_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N_1: nat] :
( K
= ( semiri1621563631at_int @ N_1 ) ) ) ).
thf(fact_1014_int__ge__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ K @ I_1 )
=> ( ( P @ I_1 )
=> ( P @ ( plus_plus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_1015_order__refl,axiom,
! [X_76: rat] : ( ord_less_eq_rat @ X_76 @ X_76 ) ).
thf(fact_1016_order__refl,axiom,
! [X_76: int > $o] : ( ord_less_eq_int_o @ X_76 @ X_76 ) ).
thf(fact_1017_order__refl,axiom,
! [X_76: nat > $o] : ( ord_less_eq_nat_o @ X_76 @ X_76 ) ).
thf(fact_1018_order__refl,axiom,
! [X_76: quickcheck_code_int] : ( ord_le258702272de_int @ X_76 @ X_76 ) ).
thf(fact_1019_order__refl,axiom,
! [X_76: real] : ( ord_less_eq_real @ X_76 @ X_76 ) ).
thf(fact_1020_order__refl,axiom,
! [X_76: nat] : ( ord_less_eq_nat @ X_76 @ X_76 ) ).
thf(fact_1021_order__refl,axiom,
! [X_76: int] : ( ord_less_eq_int @ X_76 @ X_76 ) ).
thf(fact_1022_zadd__power3,axiom,
! [A_1: int,B_1: int] :
( ( power_power_int @ ( plus_plus_int @ A_1 @ B_1 ) @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ A_1 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) @ ( power_power_int @ A_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ B_1 ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) @ A_1 ) @ ( power_power_int @ B_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( power_power_int @ B_1 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_1023_zadd__power2,axiom,
! [A_1: int,B_1: int] :
( ( power_power_int @ ( plus_plus_int @ A_1 @ B_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ A_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A_1 ) @ B_1 ) ) @ ( power_power_int @ B_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_1024_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_1025_real__less__def,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
<=> ( ( ord_less_eq_real @ X @ Y )
& ( X != Y ) ) ) ).
thf(fact_1026_less__eq__real__def,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
<=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
thf(fact_1027_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
thf(fact_1028_le__trans,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
thf(fact_1029_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_1030_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_1031_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
thf(fact_1032_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_167: rat,B_122: rat] :
( ( times_times_rat @ A_167 @ B_122 )
= ( times_times_rat @ B_122 @ A_167 ) ) ).
thf(fact_1033_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_167: quickcheck_code_int,B_122: quickcheck_code_int] :
( ( times_123202395de_int @ A_167 @ B_122 )
= ( times_123202395de_int @ B_122 @ A_167 ) ) ).
thf(fact_1034_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_167: complex,B_122: complex] :
( ( times_times_complex @ A_167 @ B_122 )
= ( times_times_complex @ B_122 @ A_167 ) ) ).
thf(fact_1035_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_167: real,B_122: real] :
( ( times_times_real @ A_167 @ B_122 )
= ( times_times_real @ B_122 @ A_167 ) ) ).
thf(fact_1036_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_167: nat,B_122: nat] :
( ( times_times_nat @ A_167 @ B_122 )
= ( times_times_nat @ B_122 @ A_167 ) ) ).
thf(fact_1037_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_167: int,B_122: int] :
( ( times_times_int @ A_167 @ B_122 )
= ( times_times_int @ B_122 @ A_167 ) ) ).
thf(fact_1038_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: rat,Rx_6: rat,Ry_4: rat] :
( ( times_times_rat @ Lx_6 @ ( times_times_rat @ Rx_6 @ Ry_4 ) )
= ( times_times_rat @ Rx_6 @ ( times_times_rat @ Lx_6 @ Ry_4 ) ) ) ).
thf(fact_1039_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: quickcheck_code_int,Rx_6: quickcheck_code_int,Ry_4: quickcheck_code_int] :
( ( times_123202395de_int @ Lx_6 @ ( times_123202395de_int @ Rx_6 @ Ry_4 ) )
= ( times_123202395de_int @ Rx_6 @ ( times_123202395de_int @ Lx_6 @ Ry_4 ) ) ) ).
thf(fact_1040_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: complex,Rx_6: complex,Ry_4: complex] :
( ( times_times_complex @ Lx_6 @ ( times_times_complex @ Rx_6 @ Ry_4 ) )
= ( times_times_complex @ Rx_6 @ ( times_times_complex @ Lx_6 @ Ry_4 ) ) ) ).
thf(fact_1041_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: real,Rx_6: real,Ry_4: real] :
( ( times_times_real @ Lx_6 @ ( times_times_real @ Rx_6 @ Ry_4 ) )
= ( times_times_real @ Rx_6 @ ( times_times_real @ Lx_6 @ Ry_4 ) ) ) ).
thf(fact_1042_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: nat,Rx_6: nat,Ry_4: nat] :
( ( times_times_nat @ Lx_6 @ ( times_times_nat @ Rx_6 @ Ry_4 ) )
= ( times_times_nat @ Rx_6 @ ( times_times_nat @ Lx_6 @ Ry_4 ) ) ) ).
thf(fact_1043_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: int,Rx_6: int,Ry_4: int] :
( ( times_times_int @ Lx_6 @ ( times_times_int @ Rx_6 @ Ry_4 ) )
= ( times_times_int @ Rx_6 @ ( times_times_int @ Lx_6 @ Ry_4 ) ) ) ).
thf(fact_1044_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: rat,Rx_5: rat,Ry_3: rat] :
( ( times_times_rat @ Lx_5 @ ( times_times_rat @ Rx_5 @ Ry_3 ) )
= ( times_times_rat @ ( times_times_rat @ Lx_5 @ Rx_5 ) @ Ry_3 ) ) ).
thf(fact_1045_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: quickcheck_code_int,Rx_5: quickcheck_code_int,Ry_3: quickcheck_code_int] :
( ( times_123202395de_int @ Lx_5 @ ( times_123202395de_int @ Rx_5 @ Ry_3 ) )
= ( times_123202395de_int @ ( times_123202395de_int @ Lx_5 @ Rx_5 ) @ Ry_3 ) ) ).
thf(fact_1046_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: complex,Rx_5: complex,Ry_3: complex] :
( ( times_times_complex @ Lx_5 @ ( times_times_complex @ Rx_5 @ Ry_3 ) )
= ( times_times_complex @ ( times_times_complex @ Lx_5 @ Rx_5 ) @ Ry_3 ) ) ).
thf(fact_1047_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: real,Rx_5: real,Ry_3: real] :
( ( times_times_real @ Lx_5 @ ( times_times_real @ Rx_5 @ Ry_3 ) )
= ( times_times_real @ ( times_times_real @ Lx_5 @ Rx_5 ) @ Ry_3 ) ) ).
thf(fact_1048_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: nat,Rx_5: nat,Ry_3: nat] :
( ( times_times_nat @ Lx_5 @ ( times_times_nat @ Rx_5 @ Ry_3 ) )
= ( times_times_nat @ ( times_times_nat @ Lx_5 @ Rx_5 ) @ Ry_3 ) ) ).
thf(fact_1049_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: int,Rx_5: int,Ry_3: int] :
( ( times_times_int @ Lx_5 @ ( times_times_int @ Rx_5 @ Ry_3 ) )
= ( times_times_int @ ( times_times_int @ Lx_5 @ Rx_5 ) @ Ry_3 ) ) ).
thf(fact_1050_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_166: rat,B_121: rat,C_68: rat] :
( ( times_times_rat @ ( times_times_rat @ A_166 @ B_121 ) @ C_68 )
= ( times_times_rat @ A_166 @ ( times_times_rat @ B_121 @ C_68 ) ) ) ).
thf(fact_1051_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_166: quickcheck_code_int,B_121: quickcheck_code_int,C_68: quickcheck_code_int] :
( ( times_123202395de_int @ ( times_123202395de_int @ A_166 @ B_121 ) @ C_68 )
= ( times_123202395de_int @ A_166 @ ( times_123202395de_int @ B_121 @ C_68 ) ) ) ).
thf(fact_1052_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_166: complex,B_121: complex,C_68: complex] :
( ( times_times_complex @ ( times_times_complex @ A_166 @ B_121 ) @ C_68 )
= ( times_times_complex @ A_166 @ ( times_times_complex @ B_121 @ C_68 ) ) ) ).
thf(fact_1053_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_166: real,B_121: real,C_68: real] :
( ( times_times_real @ ( times_times_real @ A_166 @ B_121 ) @ C_68 )
= ( times_times_real @ A_166 @ ( times_times_real @ B_121 @ C_68 ) ) ) ).
thf(fact_1054_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_166: nat,B_121: nat,C_68: nat] :
( ( times_times_nat @ ( times_times_nat @ A_166 @ B_121 ) @ C_68 )
= ( times_times_nat @ A_166 @ ( times_times_nat @ B_121 @ C_68 ) ) ) ).
thf(fact_1055_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_166: int,B_121: int,C_68: int] :
( ( times_times_int @ ( times_times_int @ A_166 @ B_121 ) @ C_68 )
= ( times_times_int @ A_166 @ ( times_times_int @ B_121 @ C_68 ) ) ) ).
thf(fact_1056_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: rat,Ly_4: rat,Rx_4: rat] :
( ( times_times_rat @ ( times_times_rat @ Lx_4 @ Ly_4 ) @ Rx_4 )
= ( times_times_rat @ Lx_4 @ ( times_times_rat @ Ly_4 @ Rx_4 ) ) ) ).
thf(fact_1057_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: quickcheck_code_int,Ly_4: quickcheck_code_int,Rx_4: quickcheck_code_int] :
( ( times_123202395de_int @ ( times_123202395de_int @ Lx_4 @ Ly_4 ) @ Rx_4 )
= ( times_123202395de_int @ Lx_4 @ ( times_123202395de_int @ Ly_4 @ Rx_4 ) ) ) ).
thf(fact_1058_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: complex,Ly_4: complex,Rx_4: complex] :
( ( times_times_complex @ ( times_times_complex @ Lx_4 @ Ly_4 ) @ Rx_4 )
= ( times_times_complex @ Lx_4 @ ( times_times_complex @ Ly_4 @ Rx_4 ) ) ) ).
thf(fact_1059_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: real,Ly_4: real,Rx_4: real] :
( ( times_times_real @ ( times_times_real @ Lx_4 @ Ly_4 ) @ Rx_4 )
= ( times_times_real @ Lx_4 @ ( times_times_real @ Ly_4 @ Rx_4 ) ) ) ).
thf(fact_1060_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: nat,Ly_4: nat,Rx_4: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_4 @ Ly_4 ) @ Rx_4 )
= ( times_times_nat @ Lx_4 @ ( times_times_nat @ Ly_4 @ Rx_4 ) ) ) ).
thf(fact_1061_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: int,Ly_4: int,Rx_4: int] :
( ( times_times_int @ ( times_times_int @ Lx_4 @ Ly_4 ) @ Rx_4 )
= ( times_times_int @ Lx_4 @ ( times_times_int @ Ly_4 @ Rx_4 ) ) ) ).
thf(fact_1062_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: rat,Ly_3: rat,Rx_3: rat] :
( ( times_times_rat @ ( times_times_rat @ Lx_3 @ Ly_3 ) @ Rx_3 )
= ( times_times_rat @ ( times_times_rat @ Lx_3 @ Rx_3 ) @ Ly_3 ) ) ).
thf(fact_1063_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: quickcheck_code_int,Ly_3: quickcheck_code_int,Rx_3: quickcheck_code_int] :
( ( times_123202395de_int @ ( times_123202395de_int @ Lx_3 @ Ly_3 ) @ Rx_3 )
= ( times_123202395de_int @ ( times_123202395de_int @ Lx_3 @ Rx_3 ) @ Ly_3 ) ) ).
thf(fact_1064_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: complex,Ly_3: complex,Rx_3: complex] :
( ( times_times_complex @ ( times_times_complex @ Lx_3 @ Ly_3 ) @ Rx_3 )
= ( times_times_complex @ ( times_times_complex @ Lx_3 @ Rx_3 ) @ Ly_3 ) ) ).
thf(fact_1065_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: real,Ly_3: real,Rx_3: real] :
( ( times_times_real @ ( times_times_real @ Lx_3 @ Ly_3 ) @ Rx_3 )
= ( times_times_real @ ( times_times_real @ Lx_3 @ Rx_3 ) @ Ly_3 ) ) ).
thf(fact_1066_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: nat,Ly_3: nat,Rx_3: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_3 @ Ly_3 ) @ Rx_3 )
= ( times_times_nat @ ( times_times_nat @ Lx_3 @ Rx_3 ) @ Ly_3 ) ) ).
thf(fact_1067_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: int,Ly_3: int,Rx_3: int] :
( ( times_times_int @ ( times_times_int @ Lx_3 @ Ly_3 ) @ Rx_3 )
= ( times_times_int @ ( times_times_int @ Lx_3 @ Rx_3 ) @ Ly_3 ) ) ).
thf(fact_1068_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: rat,Ly_2: rat,Rx_2: rat,Ry_2: rat] :
( ( times_times_rat @ ( times_times_rat @ Lx_2 @ Ly_2 ) @ ( times_times_rat @ Rx_2 @ Ry_2 ) )
= ( times_times_rat @ Lx_2 @ ( times_times_rat @ Ly_2 @ ( times_times_rat @ Rx_2 @ Ry_2 ) ) ) ) ).
thf(fact_1069_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: quickcheck_code_int,Ly_2: quickcheck_code_int,Rx_2: quickcheck_code_int,Ry_2: quickcheck_code_int] :
( ( times_123202395de_int @ ( times_123202395de_int @ Lx_2 @ Ly_2 ) @ ( times_123202395de_int @ Rx_2 @ Ry_2 ) )
= ( times_123202395de_int @ Lx_2 @ ( times_123202395de_int @ Ly_2 @ ( times_123202395de_int @ Rx_2 @ Ry_2 ) ) ) ) ).
thf(fact_1070_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: complex,Ly_2: complex,Rx_2: complex,Ry_2: complex] :
( ( times_times_complex @ ( times_times_complex @ Lx_2 @ Ly_2 ) @ ( times_times_complex @ Rx_2 @ Ry_2 ) )
= ( times_times_complex @ Lx_2 @ ( times_times_complex @ Ly_2 @ ( times_times_complex @ Rx_2 @ Ry_2 ) ) ) ) ).
thf(fact_1071_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: real,Ly_2: real,Rx_2: real,Ry_2: real] :
( ( times_times_real @ ( times_times_real @ Lx_2 @ Ly_2 ) @ ( times_times_real @ Rx_2 @ Ry_2 ) )
= ( times_times_real @ Lx_2 @ ( times_times_real @ Ly_2 @ ( times_times_real @ Rx_2 @ Ry_2 ) ) ) ) ).
thf(fact_1072_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: nat,Ly_2: nat,Rx_2: nat,Ry_2: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_2 @ Ly_2 ) @ ( times_times_nat @ Rx_2 @ Ry_2 ) )
= ( times_times_nat @ Lx_2 @ ( times_times_nat @ Ly_2 @ ( times_times_nat @ Rx_2 @ Ry_2 ) ) ) ) ).
thf(fact_1073_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: int,Ly_2: int,Rx_2: int,Ry_2: int] :
( ( times_times_int @ ( times_times_int @ Lx_2 @ Ly_2 ) @ ( times_times_int @ Rx_2 @ Ry_2 ) )
= ( times_times_int @ Lx_2 @ ( times_times_int @ Ly_2 @ ( times_times_int @ Rx_2 @ Ry_2 ) ) ) ) ).
thf(fact_1074_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: rat,Ly_1: rat,Rx_1: rat,Ry_1: rat] :
( ( times_times_rat @ ( times_times_rat @ Lx_1 @ Ly_1 ) @ ( times_times_rat @ Rx_1 @ Ry_1 ) )
= ( times_times_rat @ Rx_1 @ ( times_times_rat @ ( times_times_rat @ Lx_1 @ Ly_1 ) @ Ry_1 ) ) ) ).
thf(fact_1075_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: quickcheck_code_int,Ly_1: quickcheck_code_int,Rx_1: quickcheck_code_int,Ry_1: quickcheck_code_int] :
( ( times_123202395de_int @ ( times_123202395de_int @ Lx_1 @ Ly_1 ) @ ( times_123202395de_int @ Rx_1 @ Ry_1 ) )
= ( times_123202395de_int @ Rx_1 @ ( times_123202395de_int @ ( times_123202395de_int @ Lx_1 @ Ly_1 ) @ Ry_1 ) ) ) ).
thf(fact_1076_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: complex,Ly_1: complex,Rx_1: complex,Ry_1: complex] :
( ( times_times_complex @ ( times_times_complex @ Lx_1 @ Ly_1 ) @ ( times_times_complex @ Rx_1 @ Ry_1 ) )
= ( times_times_complex @ Rx_1 @ ( times_times_complex @ ( times_times_complex @ Lx_1 @ Ly_1 ) @ Ry_1 ) ) ) ).
thf(fact_1077_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: real,Ly_1: real,Rx_1: real,Ry_1: real] :
( ( times_times_real @ ( times_times_real @ Lx_1 @ Ly_1 ) @ ( times_times_real @ Rx_1 @ Ry_1 ) )
= ( times_times_real @ Rx_1 @ ( times_times_real @ ( times_times_real @ Lx_1 @ Ly_1 ) @ Ry_1 ) ) ) ).
thf(fact_1078_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: nat,Ly_1: nat,Rx_1: nat,Ry_1: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx_1 @ Ly_1 ) @ ( times_times_nat @ Rx_1 @ Ry_1 ) )
= ( times_times_nat @ Rx_1 @ ( times_times_nat @ ( times_times_nat @ Lx_1 @ Ly_1 ) @ Ry_1 ) ) ) ).
thf(fact_1079_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: int,Ly_1: int,Rx_1: int,Ry_1: int] :
( ( times_times_int @ ( times_times_int @ Lx_1 @ Ly_1 ) @ ( times_times_int @ Rx_1 @ Ry_1 ) )
= ( times_times_int @ Rx_1 @ ( times_times_int @ ( times_times_int @ Lx_1 @ Ly_1 ) @ Ry_1 ) ) ) ).
thf(fact_1080_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: rat,Ly: rat,Rx: rat,Ry: rat] :
( ( times_times_rat @ ( times_times_rat @ Lx @ Ly ) @ ( times_times_rat @ Rx @ Ry ) )
= ( times_times_rat @ ( times_times_rat @ Lx @ Rx ) @ ( times_times_rat @ Ly @ Ry ) ) ) ).
thf(fact_1081_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: quickcheck_code_int,Ly: quickcheck_code_int,Rx: quickcheck_code_int,Ry: quickcheck_code_int] :
( ( times_123202395de_int @ ( times_123202395de_int @ Lx @ Ly ) @ ( times_123202395de_int @ Rx @ Ry ) )
= ( times_123202395de_int @ ( times_123202395de_int @ Lx @ Rx ) @ ( times_123202395de_int @ Ly @ Ry ) ) ) ).
thf(fact_1082_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: complex,Ly: complex,Rx: complex,Ry: complex] :
( ( times_times_complex @ ( times_times_complex @ Lx @ Ly ) @ ( times_times_complex @ Rx @ Ry ) )
= ( times_times_complex @ ( times_times_complex @ Lx @ Rx ) @ ( times_times_complex @ Ly @ Ry ) ) ) ).
thf(fact_1083_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: real,Ly: real,Rx: real,Ry: real] :
( ( times_times_real @ ( times_times_real @ Lx @ Ly ) @ ( times_times_real @ Rx @ Ry ) )
= ( times_times_real @ ( times_times_real @ Lx @ Rx ) @ ( times_times_real @ Ly @ Ry ) ) ) ).
thf(fact_1084_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: nat,Ly: nat,Rx: nat,Ry: nat] :
( ( times_times_nat @ ( times_times_nat @ Lx @ Ly ) @ ( times_times_nat @ Rx @ Ry ) )
= ( times_times_nat @ ( times_times_nat @ Lx @ Rx ) @ ( times_times_nat @ Ly @ Ry ) ) ) ).
thf(fact_1085_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: int,Ly: int,Rx: int,Ry: int] :
( ( times_times_int @ ( times_times_int @ Lx @ Ly ) @ ( times_times_int @ Rx @ Ry ) )
= ( times_times_int @ ( times_times_int @ Lx @ Rx ) @ ( times_times_int @ Ly @ Ry ) ) ) ).
thf(fact_1086_zmult__commute,axiom,
! [Z_1: int,W: int] :
( ( times_times_int @ Z_1 @ W )
= ( times_times_int @ W @ Z_1 ) ) ).
thf(fact_1087_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_1088_mult__number__of__left,axiom,
! [V_15: int,W_9: int,Z_20: rat] :
( ( times_times_rat @ ( number_number_of_rat @ V_15 ) @ ( times_times_rat @ ( number_number_of_rat @ W_9 ) @ Z_20 ) )
= ( times_times_rat @ ( number_number_of_rat @ ( times_times_int @ V_15 @ W_9 ) ) @ Z_20 ) ) ).
thf(fact_1089_mult__number__of__left,axiom,
! [V_15: int,W_9: int,Z_20: complex] :
( ( times_times_complex @ ( number528085621omplex @ V_15 ) @ ( times_times_complex @ ( number528085621omplex @ W_9 ) @ Z_20 ) )
= ( times_times_complex @ ( number528085621omplex @ ( times_times_int @ V_15 @ W_9 ) ) @ Z_20 ) ) ).
thf(fact_1090_mult__number__of__left,axiom,
! [V_15: int,W_9: int,Z_20: real] :
( ( times_times_real @ ( number267125858f_real @ V_15 ) @ ( times_times_real @ ( number267125858f_real @ W_9 ) @ Z_20 ) )
= ( times_times_real @ ( number267125858f_real @ ( times_times_int @ V_15 @ W_9 ) ) @ Z_20 ) ) ).
thf(fact_1091_mult__number__of__left,axiom,
! [V_15: int,W_9: int,Z_20: int] :
( ( times_times_int @ ( number_number_of_int @ V_15 ) @ ( times_times_int @ ( number_number_of_int @ W_9 ) @ Z_20 ) )
= ( times_times_int @ ( number_number_of_int @ ( times_times_int @ V_15 @ W_9 ) ) @ Z_20 ) ) ).
thf(fact_1092_arith__simps_I32_J,axiom,
! [V_14: int,W_8: int] :
( ( times_times_rat @ ( number_number_of_rat @ V_14 ) @ ( number_number_of_rat @ W_8 ) )
= ( number_number_of_rat @ ( times_times_int @ V_14 @ W_8 ) ) ) ).
thf(fact_1093_arith__simps_I32_J,axiom,
! [V_14: int,W_8: int] :
( ( times_times_complex @ ( number528085621omplex @ V_14 ) @ ( number528085621omplex @ W_8 ) )
= ( number528085621omplex @ ( times_times_int @ V_14 @ W_8 ) ) ) ).
thf(fact_1094_arith__simps_I32_J,axiom,
! [V_14: int,W_8: int] :
( ( times_times_real @ ( number267125858f_real @ V_14 ) @ ( number267125858f_real @ W_8 ) )
= ( number267125858f_real @ ( times_times_int @ V_14 @ W_8 ) ) ) ).
thf(fact_1095_arith__simps_I32_J,axiom,
! [V_14: int,W_8: int] :
( ( times_times_int @ ( number_number_of_int @ V_14 ) @ ( number_number_of_int @ W_8 ) )
= ( number_number_of_int @ ( times_times_int @ V_14 @ W_8 ) ) ) ).
thf(fact_1096_number__of__mult,axiom,
! [V_13: int,W_7: int] :
( ( number_number_of_rat @ ( times_times_int @ V_13 @ W_7 ) )
= ( times_times_rat @ ( number_number_of_rat @ V_13 ) @ ( number_number_of_rat @ W_7 ) ) ) ).
thf(fact_1097_number__of__mult,axiom,
! [V_13: int,W_7: int] :
( ( number528085621omplex @ ( times_times_int @ V_13 @ W_7 ) )
= ( times_times_complex @ ( number528085621omplex @ V_13 ) @ ( number528085621omplex @ W_7 ) ) ) ).
thf(fact_1098_number__of__mult,axiom,
! [V_13: int,W_7: int] :
( ( number267125858f_real @ ( times_times_int @ V_13 @ W_7 ) )
= ( times_times_real @ ( number267125858f_real @ V_13 ) @ ( number267125858f_real @ W_7 ) ) ) ).
thf(fact_1099_number__of__mult,axiom,
! [V_13: int,W_7: int] :
( ( number_number_of_int @ ( times_times_int @ V_13 @ W_7 ) )
= ( times_times_int @ ( number_number_of_int @ V_13 ) @ ( number_number_of_int @ W_7 ) ) ) ).
thf(fact_1100_of__nat__mult,axiom,
! [M_21: nat,N_29: nat] :
( ( semiri151668891at_rat @ ( times_times_nat @ M_21 @ N_29 ) )
= ( times_times_rat @ ( semiri151668891at_rat @ M_21 ) @ ( semiri151668891at_rat @ N_29 ) ) ) ).
thf(fact_1101_of__nat__mult,axiom,
! [M_21: nat,N_29: nat] :
( ( semiri1424489471de_int @ ( times_times_nat @ M_21 @ N_29 ) )
= ( times_123202395de_int @ ( semiri1424489471de_int @ M_21 ) @ ( semiri1424489471de_int @ N_29 ) ) ) ).
thf(fact_1102_of__nat__mult,axiom,
! [M_21: nat,N_29: nat] :
( ( semiri2020571505omplex @ ( times_times_nat @ M_21 @ N_29 ) )
= ( times_times_complex @ ( semiri2020571505omplex @ M_21 ) @ ( semiri2020571505omplex @ N_29 ) ) ) ).
thf(fact_1103_of__nat__mult,axiom,
! [M_21: nat,N_29: nat] :
( ( semiri132038758t_real @ ( times_times_nat @ M_21 @ N_29 ) )
= ( times_times_real @ ( semiri132038758t_real @ M_21 ) @ ( semiri132038758t_real @ N_29 ) ) ) ).
thf(fact_1104_of__nat__mult,axiom,
! [M_21: nat,N_29: nat] :
( ( semiri984289939at_nat @ ( times_times_nat @ M_21 @ N_29 ) )
= ( times_times_nat @ ( semiri984289939at_nat @ M_21 ) @ ( semiri984289939at_nat @ N_29 ) ) ) ).
thf(fact_1105_of__nat__mult,axiom,
! [M_21: nat,N_29: nat] :
( ( semiri1621563631at_int @ ( times_times_nat @ M_21 @ N_29 ) )
= ( times_times_int @ ( semiri1621563631at_int @ M_21 ) @ ( semiri1621563631at_int @ N_29 ) ) ) ).
thf(fact_1106_mult__zero__left,axiom,
! [A_165: rat] :
( ( times_times_rat @ zero_zero_rat @ A_165 )
= zero_zero_rat ) ).
thf(fact_1107_mult__zero__left,axiom,
! [A_165: quickcheck_code_int] :
( ( times_123202395de_int @ zero_z891286103de_int @ A_165 )
= zero_z891286103de_int ) ).
thf(fact_1108_mult__zero__left,axiom,
! [A_165: complex] :
( ( times_times_complex @ zero_zero_complex @ A_165 )
= zero_zero_complex ) ).
thf(fact_1109_mult__zero__left,axiom,
! [A_165: code_code_numeral] :
( ( times_1655362735umeral @ zero_z126310315umeral @ A_165 )
= zero_z126310315umeral ) ).
thf(fact_1110_mult__zero__left,axiom,
! [A_165: real] :
( ( times_times_real @ zero_zero_real @ A_165 )
= zero_zero_real ) ).
thf(fact_1111_mult__zero__left,axiom,
! [A_165: nat] :
( ( times_times_nat @ zero_zero_nat @ A_165 )
= zero_zero_nat ) ).
thf(fact_1112_mult__zero__left,axiom,
! [A_165: int] :
( ( times_times_int @ zero_zero_int @ A_165 )
= zero_zero_int ) ).
thf(fact_1113_mult__zero__right,axiom,
! [A_164: rat] :
( ( times_times_rat @ A_164 @ zero_zero_rat )
= zero_zero_rat ) ).
thf(fact_1114_mult__zero__right,axiom,
! [A_164: quickcheck_code_int] :
( ( times_123202395de_int @ A_164 @ zero_z891286103de_int )
= zero_z891286103de_int ) ).
thf(fact_1115_mult__zero__right,axiom,
! [A_164: complex] :
( ( times_times_complex @ A_164 @ zero_zero_complex )
= zero_zero_complex ) ).
thf(fact_1116_mult__zero__right,axiom,
! [A_164: code_code_numeral] :
( ( times_1655362735umeral @ A_164 @ zero_z126310315umeral )
= zero_z126310315umeral ) ).
thf(fact_1117_mult__zero__right,axiom,
! [A_164: real] :
( ( times_times_real @ A_164 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_1118_mult__zero__right,axiom,
! [A_164: nat] :
( ( times_times_nat @ A_164 @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_1119_mult__zero__right,axiom,
! [A_164: int] :
( ( times_times_int @ A_164 @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_1120_mult__eq__0__iff,axiom,
! [A_163: rat,B_120: rat] :
( ( ( times_times_rat @ A_163 @ B_120 )
= zero_zero_rat )
<=> ( ( A_163 = zero_zero_rat )
| ( B_120 = zero_zero_rat ) ) ) ).
thf(fact_1121_mult__eq__0__iff,axiom,
! [A_163: complex,B_120: complex] :
( ( ( times_times_complex @ A_163 @ B_120 )
= zero_zero_complex )
<=> ( ( A_163 = zero_zero_complex )
| ( B_120 = zero_zero_complex ) ) ) ).
thf(fact_1122_mult__eq__0__iff,axiom,
! [A_163: real,B_120: real] :
( ( ( times_times_real @ A_163 @ B_120 )
= zero_zero_real )
<=> ( ( A_163 = zero_zero_real )
| ( B_120 = zero_zero_real ) ) ) ).
thf(fact_1123_mult__eq__0__iff,axiom,
! [A_163: int,B_120: int] :
( ( ( times_times_int @ A_163 @ B_120 )
= zero_zero_int )
<=> ( ( A_163 = zero_zero_int )
| ( B_120 = zero_zero_int ) ) ) ).
thf(fact_1124_no__zero__divisors,axiom,
! [B_119: rat,A_162: rat] :
( ( A_162 != zero_zero_rat )
=> ( ( B_119 != zero_zero_rat )
=> ( ( times_times_rat @ A_162 @ B_119 )
!= zero_zero_rat ) ) ) ).
thf(fact_1125_no__zero__divisors,axiom,
! [B_119: quickcheck_code_int,A_162: quickcheck_code_int] :
( ( A_162 != zero_z891286103de_int )
=> ( ( B_119 != zero_z891286103de_int )
=> ( ( times_123202395de_int @ A_162 @ B_119 )
!= zero_z891286103de_int ) ) ) ).
thf(fact_1126_no__zero__divisors,axiom,
! [B_119: complex,A_162: complex] :
( ( A_162 != zero_zero_complex )
=> ( ( B_119 != zero_zero_complex )
=> ( ( times_times_complex @ A_162 @ B_119 )
!= zero_zero_complex ) ) ) ).
thf(fact_1127_no__zero__divisors,axiom,
! [B_119: code_code_numeral,A_162: code_code_numeral] :
( ( A_162 != zero_z126310315umeral )
=> ( ( B_119 != zero_z126310315umeral )
=> ( ( times_1655362735umeral @ A_162 @ B_119 )
!= zero_z126310315umeral ) ) ) ).
thf(fact_1128_no__zero__divisors,axiom,
! [B_119: real,A_162: real] :
( ( A_162 != zero_zero_real )
=> ( ( B_119 != zero_zero_real )
=> ( ( times_times_real @ A_162 @ B_119 )
!= zero_zero_real ) ) ) ).
thf(fact_1129_no__zero__divisors,axiom,
! [B_119: nat,A_162: nat] :
( ( A_162 != zero_zero_nat )
=> ( ( B_119 != zero_zero_nat )
=> ( ( times_times_nat @ A_162 @ B_119 )
!= zero_zero_nat ) ) ) ).
thf(fact_1130_no__zero__divisors,axiom,
! [B_119: int,A_162: int] :
( ( A_162 != zero_zero_int )
=> ( ( B_119 != zero_zero_int )
=> ( ( times_times_int @ A_162 @ B_119 )
!= zero_zero_int ) ) ) ).
thf(fact_1131_divisors__zero,axiom,
! [A_161: rat,B_118: rat] :
( ( ( times_times_rat @ A_161 @ B_118 )
= zero_zero_rat )
=> ( ( A_161 = zero_zero_rat )
| ( B_118 = zero_zero_rat ) ) ) ).
thf(fact_1132_divisors__zero,axiom,
! [A_161: quickcheck_code_int,B_118: quickcheck_code_int] :
( ( ( times_123202395de_int @ A_161 @ B_118 )
= zero_z891286103de_int )
=> ( ( A_161 = zero_z891286103de_int )
| ( B_118 = zero_z891286103de_int ) ) ) ).
thf(fact_1133_divisors__zero,axiom,
! [A_161: complex,B_118: complex] :
( ( ( times_times_complex @ A_161 @ B_118 )
= zero_zero_complex )
=> ( ( A_161 = zero_zero_complex )
| ( B_118 = zero_zero_complex ) ) ) ).
thf(fact_1134_divisors__zero,axiom,
! [A_161: code_code_numeral,B_118: code_code_numeral] :
( ( ( times_1655362735umeral @ A_161 @ B_118 )
= zero_z126310315umeral )
=> ( ( A_161 = zero_z126310315umeral )
| ( B_118 = zero_z126310315umeral ) ) ) ).
thf(fact_1135_divisors__zero,axiom,
! [A_161: real,B_118: real] :
( ( ( times_times_real @ A_161 @ B_118 )
= zero_zero_real )
=> ( ( A_161 = zero_zero_real )
| ( B_118 = zero_zero_real ) ) ) ).
thf(fact_1136_divisors__zero,axiom,
! [A_161: nat,B_118: nat] :
( ( ( times_times_nat @ A_161 @ B_118 )
= zero_zero_nat )
=> ( ( A_161 = zero_zero_nat )
| ( B_118 = zero_zero_nat ) ) ) ).
thf(fact_1137_divisors__zero,axiom,
! [A_161: int,B_118: int] :
( ( ( times_times_int @ A_161 @ B_118 )
= zero_zero_int )
=> ( ( A_161 = zero_zero_int )
| ( B_118 = zero_zero_int ) ) ) ).
thf(fact_1138_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_160: rat] :
( ( times_times_rat @ zero_zero_rat @ A_160 )
= zero_zero_rat ) ).
thf(fact_1139_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_160: quickcheck_code_int] :
( ( times_123202395de_int @ zero_z891286103de_int @ A_160 )
= zero_z891286103de_int ) ).
thf(fact_1140_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_160: complex] :
( ( times_times_complex @ zero_zero_complex @ A_160 )
= zero_zero_complex ) ).
thf(fact_1141_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_160: code_code_numeral] :
( ( times_1655362735umeral @ zero_z126310315umeral @ A_160 )
= zero_z126310315umeral ) ).
thf(fact_1142_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_160: real] :
( ( times_times_real @ zero_zero_real @ A_160 )
= zero_zero_real ) ).
thf(fact_1143_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_160: nat] :
( ( times_times_nat @ zero_zero_nat @ A_160 )
= zero_zero_nat ) ).
thf(fact_1144_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_160: int] :
( ( times_times_int @ zero_zero_int @ A_160 )
= zero_zero_int ) ).
thf(fact_1145_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_159: rat] :
( ( times_times_rat @ A_159 @ zero_zero_rat )
= zero_zero_rat ) ).
thf(fact_1146_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_159: quickcheck_code_int] :
( ( times_123202395de_int @ A_159 @ zero_z891286103de_int )
= zero_z891286103de_int ) ).
thf(fact_1147_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_159: complex] :
( ( times_times_complex @ A_159 @ zero_zero_complex )
= zero_zero_complex ) ).
thf(fact_1148_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_159: code_code_numeral] :
( ( times_1655362735umeral @ A_159 @ zero_z126310315umeral )
= zero_z126310315umeral ) ).
thf(fact_1149_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_159: real] :
( ( times_times_real @ A_159 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_1150_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_159: nat] :
( ( times_times_nat @ A_159 @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_1151_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_159: int] :
( ( times_times_int @ A_159 @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_1152_comm__semiring__class_Odistrib,axiom,
! [A_158: rat,B_117: rat,C_67: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A_158 @ B_117 ) @ C_67 )
= ( plus_plus_rat @ ( times_times_rat @ A_158 @ C_67 ) @ ( times_times_rat @ B_117 @ C_67 ) ) ) ).
thf(fact_1153_comm__semiring__class_Odistrib,axiom,
! [A_158: quickcheck_code_int,B_117: quickcheck_code_int,C_67: quickcheck_code_int] :
( ( times_123202395de_int @ ( plus_p1446045655de_int @ A_158 @ B_117 ) @ C_67 )
= ( plus_p1446045655de_int @ ( times_123202395de_int @ A_158 @ C_67 ) @ ( times_123202395de_int @ B_117 @ C_67 ) ) ) ).
thf(fact_1154_comm__semiring__class_Odistrib,axiom,
! [A_158: complex,B_117: complex,C_67: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A_158 @ B_117 ) @ C_67 )
= ( plus_plus_complex @ ( times_times_complex @ A_158 @ C_67 ) @ ( times_times_complex @ B_117 @ C_67 ) ) ) ).
thf(fact_1155_comm__semiring__class_Odistrib,axiom,
! [A_158: real,B_117: real,C_67: real] :
( ( times_times_real @ ( plus_plus_real @ A_158 @ B_117 ) @ C_67 )
= ( plus_plus_real @ ( times_times_real @ A_158 @ C_67 ) @ ( times_times_real @ B_117 @ C_67 ) ) ) ).
thf(fact_1156_comm__semiring__class_Odistrib,axiom,
! [A_158: nat,B_117: nat,C_67: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_158 @ B_117 ) @ C_67 )
= ( plus_plus_nat @ ( times_times_nat @ A_158 @ C_67 ) @ ( times_times_nat @ B_117 @ C_67 ) ) ) ).
thf(fact_1157_comm__semiring__class_Odistrib,axiom,
! [A_158: int,B_117: int,C_67: int] :
( ( times_times_int @ ( plus_plus_int @ A_158 @ B_117 ) @ C_67 )
= ( plus_plus_int @ ( times_times_int @ A_158 @ C_67 ) @ ( times_times_int @ B_117 @ C_67 ) ) ) ).
thf(fact_1158_combine__common__factor,axiom,
! [A_157: rat,E_8: rat,B_116: rat,C_66: rat] :
( ( plus_plus_rat @ ( times_times_rat @ A_157 @ E_8 ) @ ( plus_plus_rat @ ( times_times_rat @ B_116 @ E_8 ) @ C_66 ) )
= ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A_157 @ B_116 ) @ E_8 ) @ C_66 ) ) ).
thf(fact_1159_combine__common__factor,axiom,
! [A_157: quickcheck_code_int,E_8: quickcheck_code_int,B_116: quickcheck_code_int,C_66: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( times_123202395de_int @ A_157 @ E_8 ) @ ( plus_p1446045655de_int @ ( times_123202395de_int @ B_116 @ E_8 ) @ C_66 ) )
= ( plus_p1446045655de_int @ ( times_123202395de_int @ ( plus_p1446045655de_int @ A_157 @ B_116 ) @ E_8 ) @ C_66 ) ) ).
thf(fact_1160_combine__common__factor,axiom,
! [A_157: complex,E_8: complex,B_116: complex,C_66: complex] :
( ( plus_plus_complex @ ( times_times_complex @ A_157 @ E_8 ) @ ( plus_plus_complex @ ( times_times_complex @ B_116 @ E_8 ) @ C_66 ) )
= ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A_157 @ B_116 ) @ E_8 ) @ C_66 ) ) ).
thf(fact_1161_combine__common__factor,axiom,
! [A_157: real,E_8: real,B_116: real,C_66: real] :
( ( plus_plus_real @ ( times_times_real @ A_157 @ E_8 ) @ ( plus_plus_real @ ( times_times_real @ B_116 @ E_8 ) @ C_66 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A_157 @ B_116 ) @ E_8 ) @ C_66 ) ) ).
thf(fact_1162_combine__common__factor,axiom,
! [A_157: nat,E_8: nat,B_116: nat,C_66: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_157 @ E_8 ) @ ( plus_plus_nat @ ( times_times_nat @ B_116 @ E_8 ) @ C_66 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A_157 @ B_116 ) @ E_8 ) @ C_66 ) ) ).
thf(fact_1163_combine__common__factor,axiom,
! [A_157: int,E_8: int,B_116: int,C_66: int] :
( ( plus_plus_int @ ( times_times_int @ A_157 @ E_8 ) @ ( plus_plus_int @ ( times_times_int @ B_116 @ E_8 ) @ C_66 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A_157 @ B_116 ) @ E_8 ) @ C_66 ) ) ).
thf(fact_1164_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_75: rat,Y_64: rat,Z_19: rat] :
( ( times_times_rat @ X_75 @ ( plus_plus_rat @ Y_64 @ Z_19 ) )
= ( plus_plus_rat @ ( times_times_rat @ X_75 @ Y_64 ) @ ( times_times_rat @ X_75 @ Z_19 ) ) ) ).
thf(fact_1165_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_75: quickcheck_code_int,Y_64: quickcheck_code_int,Z_19: quickcheck_code_int] :
( ( times_123202395de_int @ X_75 @ ( plus_p1446045655de_int @ Y_64 @ Z_19 ) )
= ( plus_p1446045655de_int @ ( times_123202395de_int @ X_75 @ Y_64 ) @ ( times_123202395de_int @ X_75 @ Z_19 ) ) ) ).
thf(fact_1166_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_75: complex,Y_64: complex,Z_19: complex] :
( ( times_times_complex @ X_75 @ ( plus_plus_complex @ Y_64 @ Z_19 ) )
= ( plus_plus_complex @ ( times_times_complex @ X_75 @ Y_64 ) @ ( times_times_complex @ X_75 @ Z_19 ) ) ) ).
thf(fact_1167_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_75: real,Y_64: real,Z_19: real] :
( ( times_times_real @ X_75 @ ( plus_plus_real @ Y_64 @ Z_19 ) )
= ( plus_plus_real @ ( times_times_real @ X_75 @ Y_64 ) @ ( times_times_real @ X_75 @ Z_19 ) ) ) ).
thf(fact_1168_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_75: nat,Y_64: nat,Z_19: nat] :
( ( times_times_nat @ X_75 @ ( plus_plus_nat @ Y_64 @ Z_19 ) )
= ( plus_plus_nat @ ( times_times_nat @ X_75 @ Y_64 ) @ ( times_times_nat @ X_75 @ Z_19 ) ) ) ).
thf(fact_1169_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_75: int,Y_64: int,Z_19: int] :
( ( times_times_int @ X_75 @ ( plus_plus_int @ Y_64 @ Z_19 ) )
= ( plus_plus_int @ ( times_times_int @ X_75 @ Y_64 ) @ ( times_times_int @ X_75 @ Z_19 ) ) ) ).
thf(fact_1170_crossproduct__noteq,axiom,
! [C_65: rat,D_28: rat,A_156: rat,B_115: rat] :
( ( ( A_156 != B_115 )
& ( C_65 != D_28 ) )
<=> ( ( plus_plus_rat @ ( times_times_rat @ A_156 @ C_65 ) @ ( times_times_rat @ B_115 @ D_28 ) )
!= ( plus_plus_rat @ ( times_times_rat @ A_156 @ D_28 ) @ ( times_times_rat @ B_115 @ C_65 ) ) ) ) ).
thf(fact_1171_crossproduct__noteq,axiom,
! [C_65: complex,D_28: complex,A_156: complex,B_115: complex] :
( ( ( A_156 != B_115 )
& ( C_65 != D_28 ) )
<=> ( ( plus_plus_complex @ ( times_times_complex @ A_156 @ C_65 ) @ ( times_times_complex @ B_115 @ D_28 ) )
!= ( plus_plus_complex @ ( times_times_complex @ A_156 @ D_28 ) @ ( times_times_complex @ B_115 @ C_65 ) ) ) ) ).
thf(fact_1172_crossproduct__noteq,axiom,
! [C_65: real,D_28: real,A_156: real,B_115: real] :
( ( ( A_156 != B_115 )
& ( C_65 != D_28 ) )
<=> ( ( plus_plus_real @ ( times_times_real @ A_156 @ C_65 ) @ ( times_times_real @ B_115 @ D_28 ) )
!= ( plus_plus_real @ ( times_times_real @ A_156 @ D_28 ) @ ( times_times_real @ B_115 @ C_65 ) ) ) ) ).
thf(fact_1173_crossproduct__noteq,axiom,
! [C_65: nat,D_28: nat,A_156: nat,B_115: nat] :
( ( ( A_156 != B_115 )
& ( C_65 != D_28 ) )
<=> ( ( plus_plus_nat @ ( times_times_nat @ A_156 @ C_65 ) @ ( times_times_nat @ B_115 @ D_28 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A_156 @ D_28 ) @ ( times_times_nat @ B_115 @ C_65 ) ) ) ) ).
thf(fact_1174_crossproduct__noteq,axiom,
! [C_65: int,D_28: int,A_156: int,B_115: int] :
( ( ( A_156 != B_115 )
& ( C_65 != D_28 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ A_156 @ C_65 ) @ ( times_times_int @ B_115 @ D_28 ) )
!= ( plus_plus_int @ ( times_times_int @ A_156 @ D_28 ) @ ( times_times_int @ B_115 @ C_65 ) ) ) ) ).
thf(fact_1175_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_155: rat,B_114: rat,C_64: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A_155 @ B_114 ) @ C_64 )
= ( plus_plus_rat @ ( times_times_rat @ A_155 @ C_64 ) @ ( times_times_rat @ B_114 @ C_64 ) ) ) ).
thf(fact_1176_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_155: quickcheck_code_int,B_114: quickcheck_code_int,C_64: quickcheck_code_int] :
( ( times_123202395de_int @ ( plus_p1446045655de_int @ A_155 @ B_114 ) @ C_64 )
= ( plus_p1446045655de_int @ ( times_123202395de_int @ A_155 @ C_64 ) @ ( times_123202395de_int @ B_114 @ C_64 ) ) ) ).
thf(fact_1177_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_155: complex,B_114: complex,C_64: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A_155 @ B_114 ) @ C_64 )
= ( plus_plus_complex @ ( times_times_complex @ A_155 @ C_64 ) @ ( times_times_complex @ B_114 @ C_64 ) ) ) ).
thf(fact_1178_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_155: real,B_114: real,C_64: real] :
( ( times_times_real @ ( plus_plus_real @ A_155 @ B_114 ) @ C_64 )
= ( plus_plus_real @ ( times_times_real @ A_155 @ C_64 ) @ ( times_times_real @ B_114 @ C_64 ) ) ) ).
thf(fact_1179_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_155: nat,B_114: nat,C_64: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_155 @ B_114 ) @ C_64 )
= ( plus_plus_nat @ ( times_times_nat @ A_155 @ C_64 ) @ ( times_times_nat @ B_114 @ C_64 ) ) ) ).
thf(fact_1180_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_155: int,B_114: int,C_64: int] :
( ( times_times_int @ ( plus_plus_int @ A_155 @ B_114 ) @ C_64 )
= ( plus_plus_int @ ( times_times_int @ A_155 @ C_64 ) @ ( times_times_int @ B_114 @ C_64 ) ) ) ).
thf(fact_1181_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_154: rat,M_20: rat,B_113: rat] :
( ( plus_plus_rat @ ( times_times_rat @ A_154 @ M_20 ) @ ( times_times_rat @ B_113 @ M_20 ) )
= ( times_times_rat @ ( plus_plus_rat @ A_154 @ B_113 ) @ M_20 ) ) ).
thf(fact_1182_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_154: quickcheck_code_int,M_20: quickcheck_code_int,B_113: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( times_123202395de_int @ A_154 @ M_20 ) @ ( times_123202395de_int @ B_113 @ M_20 ) )
= ( times_123202395de_int @ ( plus_p1446045655de_int @ A_154 @ B_113 ) @ M_20 ) ) ).
thf(fact_1183_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_154: complex,M_20: complex,B_113: complex] :
( ( plus_plus_complex @ ( times_times_complex @ A_154 @ M_20 ) @ ( times_times_complex @ B_113 @ M_20 ) )
= ( times_times_complex @ ( plus_plus_complex @ A_154 @ B_113 ) @ M_20 ) ) ).
thf(fact_1184_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_154: real,M_20: real,B_113: real] :
( ( plus_plus_real @ ( times_times_real @ A_154 @ M_20 ) @ ( times_times_real @ B_113 @ M_20 ) )
= ( times_times_real @ ( plus_plus_real @ A_154 @ B_113 ) @ M_20 ) ) ).
thf(fact_1185_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_154: nat,M_20: nat,B_113: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_154 @ M_20 ) @ ( times_times_nat @ B_113 @ M_20 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_154 @ B_113 ) @ M_20 ) ) ).
thf(fact_1186_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_154: int,M_20: int,B_113: int] :
( ( plus_plus_int @ ( times_times_int @ A_154 @ M_20 ) @ ( times_times_int @ B_113 @ M_20 ) )
= ( times_times_int @ ( plus_plus_int @ A_154 @ B_113 ) @ M_20 ) ) ).
thf(fact_1187_crossproduct__eq,axiom,
! [W_6: rat,Y_63: rat,X_74: rat,Z_18: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ W_6 @ Y_63 ) @ ( times_times_rat @ X_74 @ Z_18 ) )
= ( plus_plus_rat @ ( times_times_rat @ W_6 @ Z_18 ) @ ( times_times_rat @ X_74 @ Y_63 ) ) )
<=> ( ( W_6 = X_74 )
| ( Y_63 = Z_18 ) ) ) ).
thf(fact_1188_crossproduct__eq,axiom,
! [W_6: complex,Y_63: complex,X_74: complex,Z_18: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ W_6 @ Y_63 ) @ ( times_times_complex @ X_74 @ Z_18 ) )
= ( plus_plus_complex @ ( times_times_complex @ W_6 @ Z_18 ) @ ( times_times_complex @ X_74 @ Y_63 ) ) )
<=> ( ( W_6 = X_74 )
| ( Y_63 = Z_18 ) ) ) ).
thf(fact_1189_crossproduct__eq,axiom,
! [W_6: real,Y_63: real,X_74: real,Z_18: real] :
( ( ( plus_plus_real @ ( times_times_real @ W_6 @ Y_63 ) @ ( times_times_real @ X_74 @ Z_18 ) )
= ( plus_plus_real @ ( times_times_real @ W_6 @ Z_18 ) @ ( times_times_real @ X_74 @ Y_63 ) ) )
<=> ( ( W_6 = X_74 )
| ( Y_63 = Z_18 ) ) ) ).
thf(fact_1190_crossproduct__eq,axiom,
! [W_6: nat,Y_63: nat,X_74: nat,Z_18: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W_6 @ Y_63 ) @ ( times_times_nat @ X_74 @ Z_18 ) )
= ( plus_plus_nat @ ( times_times_nat @ W_6 @ Z_18 ) @ ( times_times_nat @ X_74 @ Y_63 ) ) )
<=> ( ( W_6 = X_74 )
| ( Y_63 = Z_18 ) ) ) ).
thf(fact_1191_crossproduct__eq,axiom,
! [W_6: int,Y_63: int,X_74: int,Z_18: int] :
( ( ( plus_plus_int @ ( times_times_int @ W_6 @ Y_63 ) @ ( times_times_int @ X_74 @ Z_18 ) )
= ( plus_plus_int @ ( times_times_int @ W_6 @ Z_18 ) @ ( times_times_int @ X_74 @ Y_63 ) ) )
<=> ( ( W_6 = X_74 )
| ( Y_63 = Z_18 ) ) ) ).
thf(fact_1192_mult__1__left,axiom,
! [A_153: rat] :
( ( times_times_rat @ one_one_rat @ A_153 )
= A_153 ) ).
thf(fact_1193_mult__1__left,axiom,
! [A_153: quickcheck_code_int] :
( ( times_123202395de_int @ one_on1684967323de_int @ A_153 )
= A_153 ) ).
thf(fact_1194_mult__1__left,axiom,
! [A_153: complex] :
( ( times_times_complex @ one_one_complex @ A_153 )
= A_153 ) ).
thf(fact_1195_mult__1__left,axiom,
! [A_153: code_code_numeral] :
( ( times_1655362735umeral @ one_on1645066479umeral @ A_153 )
= A_153 ) ).
thf(fact_1196_mult__1__left,axiom,
! [A_153: real] :
( ( times_times_real @ one_one_real @ A_153 )
= A_153 ) ).
thf(fact_1197_mult__1__left,axiom,
! [A_153: nat] :
( ( times_times_nat @ one_one_nat @ A_153 )
= A_153 ) ).
thf(fact_1198_mult__1__left,axiom,
! [A_153: int] :
( ( times_times_int @ one_one_int @ A_153 )
= A_153 ) ).
thf(fact_1199_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_152: rat] :
( ( times_times_rat @ one_one_rat @ A_152 )
= A_152 ) ).
thf(fact_1200_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_152: quickcheck_code_int] :
( ( times_123202395de_int @ one_on1684967323de_int @ A_152 )
= A_152 ) ).
thf(fact_1201_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_152: complex] :
( ( times_times_complex @ one_one_complex @ A_152 )
= A_152 ) ).
thf(fact_1202_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_152: code_code_numeral] :
( ( times_1655362735umeral @ one_on1645066479umeral @ A_152 )
= A_152 ) ).
thf(fact_1203_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_152: real] :
( ( times_times_real @ one_one_real @ A_152 )
= A_152 ) ).
thf(fact_1204_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_152: nat] :
( ( times_times_nat @ one_one_nat @ A_152 )
= A_152 ) ).
thf(fact_1205_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_152: int] :
( ( times_times_int @ one_one_int @ A_152 )
= A_152 ) ).
thf(fact_1206_mult__1,axiom,
! [A_151: rat] :
( ( times_times_rat @ one_one_rat @ A_151 )
= A_151 ) ).
thf(fact_1207_mult__1,axiom,
! [A_151: quickcheck_code_int] :
( ( times_123202395de_int @ one_on1684967323de_int @ A_151 )
= A_151 ) ).
thf(fact_1208_mult__1,axiom,
! [A_151: complex] :
( ( times_times_complex @ one_one_complex @ A_151 )
= A_151 ) ).
thf(fact_1209_mult__1,axiom,
! [A_151: code_code_numeral] :
( ( times_1655362735umeral @ one_on1645066479umeral @ A_151 )
= A_151 ) ).
thf(fact_1210_mult__1,axiom,
! [A_151: real] :
( ( times_times_real @ one_one_real @ A_151 )
= A_151 ) ).
thf(fact_1211_mult__1,axiom,
! [A_151: nat] :
( ( times_times_nat @ one_one_nat @ A_151 )
= A_151 ) ).
thf(fact_1212_mult__1,axiom,
! [A_151: int] :
( ( times_times_int @ one_one_int @ A_151 )
= A_151 ) ).
thf(fact_1213_mult__1__right,axiom,
! [A_150: rat] :
( ( times_times_rat @ A_150 @ one_one_rat )
= A_150 ) ).
thf(fact_1214_mult__1__right,axiom,
! [A_150: quickcheck_code_int] :
( ( times_123202395de_int @ A_150 @ one_on1684967323de_int )
= A_150 ) ).
thf(fact_1215_mult__1__right,axiom,
! [A_150: complex] :
( ( times_times_complex @ A_150 @ one_one_complex )
= A_150 ) ).
thf(fact_1216_mult__1__right,axiom,
! [A_150: code_code_numeral] :
( ( times_1655362735umeral @ A_150 @ one_on1645066479umeral )
= A_150 ) ).
thf(fact_1217_mult__1__right,axiom,
! [A_150: real] :
( ( times_times_real @ A_150 @ one_one_real )
= A_150 ) ).
thf(fact_1218_mult__1__right,axiom,
! [A_150: nat] :
( ( times_times_nat @ A_150 @ one_one_nat )
= A_150 ) ).
thf(fact_1219_mult__1__right,axiom,
! [A_150: int] :
( ( times_times_int @ A_150 @ one_one_int )
= A_150 ) ).
thf(fact_1220_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_149: rat] :
( ( times_times_rat @ A_149 @ one_one_rat )
= A_149 ) ).
thf(fact_1221_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_149: quickcheck_code_int] :
( ( times_123202395de_int @ A_149 @ one_on1684967323de_int )
= A_149 ) ).
thf(fact_1222_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_149: complex] :
( ( times_times_complex @ A_149 @ one_one_complex )
= A_149 ) ).
thf(fact_1223_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_149: code_code_numeral] :
( ( times_1655362735umeral @ A_149 @ one_on1645066479umeral )
= A_149 ) ).
thf(fact_1224_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_149: real] :
( ( times_times_real @ A_149 @ one_one_real )
= A_149 ) ).
thf(fact_1225_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_149: nat] :
( ( times_times_nat @ A_149 @ one_one_nat )
= A_149 ) ).
thf(fact_1226_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_149: int] :
( ( times_times_int @ A_149 @ one_one_int )
= A_149 ) ).
thf(fact_1227_mult_Ocomm__neutral,axiom,
! [A_148: rat] :
( ( times_times_rat @ A_148 @ one_one_rat )
= A_148 ) ).
thf(fact_1228_mult_Ocomm__neutral,axiom,
! [A_148: quickcheck_code_int] :
( ( times_123202395de_int @ A_148 @ one_on1684967323de_int )
= A_148 ) ).
thf(fact_1229_mult_Ocomm__neutral,axiom,
! [A_148: complex] :
( ( times_times_complex @ A_148 @ one_one_complex )
= A_148 ) ).
thf(fact_1230_mult_Ocomm__neutral,axiom,
! [A_148: code_code_numeral] :
( ( times_1655362735umeral @ A_148 @ one_on1645066479umeral )
= A_148 ) ).
thf(fact_1231_mult_Ocomm__neutral,axiom,
! [A_148: real] :
( ( times_times_real @ A_148 @ one_one_real )
= A_148 ) ).
thf(fact_1232_mult_Ocomm__neutral,axiom,
! [A_148: nat] :
( ( times_times_nat @ A_148 @ one_one_nat )
= A_148 ) ).
thf(fact_1233_mult_Ocomm__neutral,axiom,
! [A_148: int] :
( ( times_times_int @ A_148 @ one_one_int )
= A_148 ) ).
thf(fact_1234_power__commutes,axiom,
! [A_147: complex,N_28: nat] :
( ( times_times_complex @ ( power_power_complex @ A_147 @ N_28 ) @ A_147 )
= ( times_times_complex @ A_147 @ ( power_power_complex @ A_147 @ N_28 ) ) ) ).
thf(fact_1235_power__commutes,axiom,
! [A_147: real,N_28: nat] :
( ( times_times_real @ ( power_power_real @ A_147 @ N_28 ) @ A_147 )
= ( times_times_real @ A_147 @ ( power_power_real @ A_147 @ N_28 ) ) ) ).
thf(fact_1236_power__commutes,axiom,
! [A_147: nat,N_28: nat] :
( ( times_times_nat @ ( power_power_nat @ A_147 @ N_28 ) @ A_147 )
= ( times_times_nat @ A_147 @ ( power_power_nat @ A_147 @ N_28 ) ) ) ).
thf(fact_1237_power__commutes,axiom,
! [A_147: int,N_28: nat] :
( ( times_times_int @ ( power_power_int @ A_147 @ N_28 ) @ A_147 )
= ( times_times_int @ A_147 @ ( power_power_int @ A_147 @ N_28 ) ) ) ).
thf(fact_1238_power__commutes,axiom,
! [A_147: rat,N_28: nat] :
( ( times_times_rat @ ( power_power_rat @ A_147 @ N_28 ) @ A_147 )
= ( times_times_rat @ A_147 @ ( power_power_rat @ A_147 @ N_28 ) ) ) ).
thf(fact_1239_power__commutes,axiom,
! [A_147: quickcheck_code_int,N_28: nat] :
( ( times_123202395de_int @ ( power_881366806de_int @ A_147 @ N_28 ) @ A_147 )
= ( times_123202395de_int @ A_147 @ ( power_881366806de_int @ A_147 @ N_28 ) ) ) ).
thf(fact_1240_power__mult__distrib,axiom,
! [A_146: complex,B_112: complex,N_27: nat] :
( ( power_power_complex @ ( times_times_complex @ A_146 @ B_112 ) @ N_27 )
= ( times_times_complex @ ( power_power_complex @ A_146 @ N_27 ) @ ( power_power_complex @ B_112 @ N_27 ) ) ) ).
thf(fact_1241_power__mult__distrib,axiom,
! [A_146: real,B_112: real,N_27: nat] :
( ( power_power_real @ ( times_times_real @ A_146 @ B_112 ) @ N_27 )
= ( times_times_real @ ( power_power_real @ A_146 @ N_27 ) @ ( power_power_real @ B_112 @ N_27 ) ) ) ).
thf(fact_1242_power__mult__distrib,axiom,
! [A_146: nat,B_112: nat,N_27: nat] :
( ( power_power_nat @ ( times_times_nat @ A_146 @ B_112 ) @ N_27 )
= ( times_times_nat @ ( power_power_nat @ A_146 @ N_27 ) @ ( power_power_nat @ B_112 @ N_27 ) ) ) ).
thf(fact_1243_power__mult__distrib,axiom,
! [A_146: int,B_112: int,N_27: nat] :
( ( power_power_int @ ( times_times_int @ A_146 @ B_112 ) @ N_27 )
= ( times_times_int @ ( power_power_int @ A_146 @ N_27 ) @ ( power_power_int @ B_112 @ N_27 ) ) ) ).
thf(fact_1244_power__mult__distrib,axiom,
! [A_146: rat,B_112: rat,N_27: nat] :
( ( power_power_rat @ ( times_times_rat @ A_146 @ B_112 ) @ N_27 )
= ( times_times_rat @ ( power_power_rat @ A_146 @ N_27 ) @ ( power_power_rat @ B_112 @ N_27 ) ) ) ).
thf(fact_1245_power__mult__distrib,axiom,
! [A_146: quickcheck_code_int,B_112: quickcheck_code_int,N_27: nat] :
( ( power_881366806de_int @ ( times_123202395de_int @ A_146 @ B_112 ) @ N_27 )
= ( times_123202395de_int @ ( power_881366806de_int @ A_146 @ N_27 ) @ ( power_881366806de_int @ B_112 @ N_27 ) ) ) ).
thf(fact_1246_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_73: complex,Y_62: complex,Q_7: nat] :
( ( power_power_complex @ ( times_times_complex @ X_73 @ Y_62 ) @ Q_7 )
= ( times_times_complex @ ( power_power_complex @ X_73 @ Q_7 ) @ ( power_power_complex @ Y_62 @ Q_7 ) ) ) ).
thf(fact_1247_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_73: real,Y_62: real,Q_7: nat] :
( ( power_power_real @ ( times_times_real @ X_73 @ Y_62 ) @ Q_7 )
= ( times_times_real @ ( power_power_real @ X_73 @ Q_7 ) @ ( power_power_real @ Y_62 @ Q_7 ) ) ) ).
thf(fact_1248_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_73: nat,Y_62: nat,Q_7: nat] :
( ( power_power_nat @ ( times_times_nat @ X_73 @ Y_62 ) @ Q_7 )
= ( times_times_nat @ ( power_power_nat @ X_73 @ Q_7 ) @ ( power_power_nat @ Y_62 @ Q_7 ) ) ) ).
thf(fact_1249_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_73: int,Y_62: int,Q_7: nat] :
( ( power_power_int @ ( times_times_int @ X_73 @ Y_62 ) @ Q_7 )
= ( times_times_int @ ( power_power_int @ X_73 @ Q_7 ) @ ( power_power_int @ Y_62 @ Q_7 ) ) ) ).
thf(fact_1250_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_73: rat,Y_62: rat,Q_7: nat] :
( ( power_power_rat @ ( times_times_rat @ X_73 @ Y_62 ) @ Q_7 )
= ( times_times_rat @ ( power_power_rat @ X_73 @ Q_7 ) @ ( power_power_rat @ Y_62 @ Q_7 ) ) ) ).
thf(fact_1251_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_73: quickcheck_code_int,Y_62: quickcheck_code_int,Q_7: nat] :
( ( power_881366806de_int @ ( times_123202395de_int @ X_73 @ Y_62 ) @ Q_7 )
= ( times_123202395de_int @ ( power_881366806de_int @ X_73 @ Q_7 ) @ ( power_881366806de_int @ Y_62 @ Q_7 ) ) ) ).
thf(fact_1252_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_1253_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_1254_mult__Pls,axiom,
! [W: int] :
( ( times_times_int @ pls @ W )
= pls ) ).
thf(fact_1255_mult__Bit0,axiom,
! [K: int,L: int] :
( ( times_times_int @ ( bit0 @ K ) @ L )
= ( bit0 @ ( times_times_int @ K @ L ) ) ) ).
thf(fact_1256_nat__less__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
<=> ( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ).
thf(fact_1257_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_1258_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_1259_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_1260_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_1261_zmult__1,axiom,
! [Z_1: int] :
( ( times_times_int @ one_one_int @ Z_1 )
= Z_1 ) ).
thf(fact_1262_zmult__1__right,axiom,
! [Z_1: int] :
( ( times_times_int @ Z_1 @ one_one_int )
= Z_1 ) ).
thf(fact_1263_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_1264_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_1265_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
thf(fact_1266_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
thf(fact_1267_le__iff__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
<=> ? [K_1: nat] :
( N
= ( plus_plus_nat @ M @ K_1 ) ) ) ).
thf(fact_1268_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_1269_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_1270_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_1271_add__le__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_1272_add__le__mono,axiom,
! [K: nat,L: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
thf(fact_1273_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
thf(fact_1274_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_1275_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
thf(fact_1276_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_1277_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_1278_semiring__mult__number__of,axiom,
! [V_12: int,V_11: int] :
( ( ord_less_eq_int @ pls @ V_11 )
=> ( ( ord_less_eq_int @ pls @ V_12 )
=> ( ( times_times_rat @ ( number_number_of_rat @ V_11 ) @ ( number_number_of_rat @ V_12 ) )
= ( number_number_of_rat @ ( times_times_int @ V_11 @ V_12 ) ) ) ) ) ).
thf(fact_1279_semiring__mult__number__of,axiom,
! [V_12: int,V_11: int] :
( ( ord_less_eq_int @ pls @ V_11 )
=> ( ( ord_less_eq_int @ pls @ V_12 )
=> ( ( times_times_complex @ ( number528085621omplex @ V_11 ) @ ( number528085621omplex @ V_12 ) )
= ( number528085621omplex @ ( times_times_int @ V_11 @ V_12 ) ) ) ) ) ).
thf(fact_1280_semiring__mult__number__of,axiom,
! [V_12: int,V_11: int] :
( ( ord_less_eq_int @ pls @ V_11 )
=> ( ( ord_less_eq_int @ pls @ V_12 )
=> ( ( times_times_real @ ( number267125858f_real @ V_11 ) @ ( number267125858f_real @ V_12 ) )
= ( number267125858f_real @ ( times_times_int @ V_11 @ V_12 ) ) ) ) ) ).
thf(fact_1281_semiring__mult__number__of,axiom,
! [V_12: int,V_11: int] :
( ( ord_less_eq_int @ pls @ V_11 )
=> ( ( ord_less_eq_int @ pls @ V_12 )
=> ( ( times_times_nat @ ( number_number_of_nat @ V_11 ) @ ( number_number_of_nat @ V_12 ) )
= ( number_number_of_nat @ ( times_times_int @ V_11 @ V_12 ) ) ) ) ) ).
thf(fact_1282_semiring__mult__number__of,axiom,
! [V_12: int,V_11: int] :
( ( ord_less_eq_int @ pls @ V_11 )
=> ( ( ord_less_eq_int @ pls @ V_12 )
=> ( ( times_times_int @ ( number_number_of_int @ V_11 ) @ ( number_number_of_int @ V_12 ) )
= ( number_number_of_int @ ( times_times_int @ V_11 @ V_12 ) ) ) ) ) ).
thf(fact_1283_zero__le__square,axiom,
! [A_145: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A_145 @ A_145 ) ) ).
thf(fact_1284_zero__le__square,axiom,
! [A_145: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_145 @ A_145 ) ) ).
thf(fact_1285_zero__le__square,axiom,
! [A_145: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_145 @ A_145 ) ) ).
thf(fact_1286_zero__le__mult__iff,axiom,
! [A_144: rat,B_111: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A_144 @ B_111 ) )
<=> ( ( ( ord_less_eq_rat @ zero_zero_rat @ A_144 )
& ( ord_less_eq_rat @ zero_zero_rat @ B_111 ) )
| ( ( ord_less_eq_rat @ A_144 @ zero_zero_rat )
& ( ord_less_eq_rat @ B_111 @ zero_zero_rat ) ) ) ) ).
thf(fact_1287_zero__le__mult__iff,axiom,
! [A_144: real,B_111: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_144 @ B_111 ) )
<=> ( ( ( ord_less_eq_real @ zero_zero_real @ A_144 )
& ( ord_less_eq_real @ zero_zero_real @ B_111 ) )
| ( ( ord_less_eq_real @ A_144 @ zero_zero_real )
& ( ord_less_eq_real @ B_111 @ zero_zero_real ) ) ) ) ).
thf(fact_1288_zero__le__mult__iff,axiom,
! [A_144: int,B_111: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_144 @ B_111 ) )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ A_144 )
& ( ord_less_eq_int @ zero_zero_int @ B_111 ) )
| ( ( ord_less_eq_int @ A_144 @ zero_zero_int )
& ( ord_less_eq_int @ B_111 @ zero_zero_int ) ) ) ) ).
thf(fact_1289_mult__le__0__iff,axiom,
! [A_143: rat,B_110: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A_143 @ B_110 ) @ zero_zero_rat )
<=> ( ( ( ord_less_eq_rat @ zero_zero_rat @ A_143 )
& ( ord_less_eq_rat @ B_110 @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A_143 @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B_110 ) ) ) ) ).
thf(fact_1290_mult__le__0__iff,axiom,
! [A_143: real,B_110: real] :
( ( ord_less_eq_real @ ( times_times_real @ A_143 @ B_110 ) @ zero_zero_real )
<=> ( ( ( ord_less_eq_real @ zero_zero_real @ A_143 )
& ( ord_less_eq_real @ B_110 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A_143 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B_110 ) ) ) ) ).
thf(fact_1291_mult__le__0__iff,axiom,
! [A_143: int,B_110: int] :
( ( ord_less_eq_int @ ( times_times_int @ A_143 @ B_110 ) @ zero_zero_int )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ A_143 )
& ( ord_less_eq_int @ B_110 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A_143 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B_110 ) ) ) ) ).
thf(fact_1292_mult__nonneg__nonneg,axiom,
! [B_109: rat,A_142: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_142 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_109 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A_142 @ B_109 ) ) ) ) ).
thf(fact_1293_mult__nonneg__nonneg,axiom,
! [B_109: quickcheck_code_int,A_142: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_142 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_109 )
=> ( ord_le258702272de_int @ zero_z891286103de_int @ ( times_123202395de_int @ A_142 @ B_109 ) ) ) ) ).
thf(fact_1294_mult__nonneg__nonneg,axiom,
! [B_109: code_code_numeral,A_142: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_142 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_109 )
=> ( ord_le565307924umeral @ zero_z126310315umeral @ ( times_1655362735umeral @ A_142 @ B_109 ) ) ) ) ).
thf(fact_1295_mult__nonneg__nonneg,axiom,
! [B_109: real,A_142: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_142 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_109 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_142 @ B_109 ) ) ) ) ).
thf(fact_1296_mult__nonneg__nonneg,axiom,
! [B_109: nat,A_142: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_142 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_109 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A_142 @ B_109 ) ) ) ) ).
thf(fact_1297_mult__nonneg__nonneg,axiom,
! [B_109: int,A_142: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_142 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_109 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_142 @ B_109 ) ) ) ) ).
thf(fact_1298_mult__nonneg__nonpos,axiom,
! [B_108: rat,A_141: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_141 )
=> ( ( ord_less_eq_rat @ B_108 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_141 @ B_108 ) @ zero_zero_rat ) ) ) ).
thf(fact_1299_mult__nonneg__nonpos,axiom,
! [B_108: quickcheck_code_int,A_141: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_141 )
=> ( ( ord_le258702272de_int @ B_108 @ zero_z891286103de_int )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ A_141 @ B_108 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1300_mult__nonneg__nonpos,axiom,
! [B_108: code_code_numeral,A_141: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_141 )
=> ( ( ord_le565307924umeral @ B_108 @ zero_z126310315umeral )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ A_141 @ B_108 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1301_mult__nonneg__nonpos,axiom,
! [B_108: real,A_141: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_141 )
=> ( ( ord_less_eq_real @ B_108 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A_141 @ B_108 ) @ zero_zero_real ) ) ) ).
thf(fact_1302_mult__nonneg__nonpos,axiom,
! [B_108: nat,A_141: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_141 )
=> ( ( ord_less_eq_nat @ B_108 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_141 @ B_108 ) @ zero_zero_nat ) ) ) ).
thf(fact_1303_mult__nonneg__nonpos,axiom,
! [B_108: int,A_141: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_141 )
=> ( ( ord_less_eq_int @ B_108 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A_141 @ B_108 ) @ zero_zero_int ) ) ) ).
thf(fact_1304_mult__nonneg__nonpos2,axiom,
! [B_107: rat,A_140: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_140 )
=> ( ( ord_less_eq_rat @ B_107 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ B_107 @ A_140 ) @ zero_zero_rat ) ) ) ).
thf(fact_1305_mult__nonneg__nonpos2,axiom,
! [B_107: quickcheck_code_int,A_140: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_140 )
=> ( ( ord_le258702272de_int @ B_107 @ zero_z891286103de_int )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ B_107 @ A_140 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1306_mult__nonneg__nonpos2,axiom,
! [B_107: code_code_numeral,A_140: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_140 )
=> ( ( ord_le565307924umeral @ B_107 @ zero_z126310315umeral )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ B_107 @ A_140 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1307_mult__nonneg__nonpos2,axiom,
! [B_107: real,A_140: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_140 )
=> ( ( ord_less_eq_real @ B_107 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B_107 @ A_140 ) @ zero_zero_real ) ) ) ).
thf(fact_1308_mult__nonneg__nonpos2,axiom,
! [B_107: nat,A_140: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_140 )
=> ( ( ord_less_eq_nat @ B_107 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B_107 @ A_140 ) @ zero_zero_nat ) ) ) ).
thf(fact_1309_mult__nonneg__nonpos2,axiom,
! [B_107: int,A_140: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_140 )
=> ( ( ord_less_eq_int @ B_107 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B_107 @ A_140 ) @ zero_zero_int ) ) ) ).
thf(fact_1310_mult__nonpos__nonneg,axiom,
! [B_106: rat,A_139: rat] :
( ( ord_less_eq_rat @ A_139 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_106 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_139 @ B_106 ) @ zero_zero_rat ) ) ) ).
thf(fact_1311_mult__nonpos__nonneg,axiom,
! [B_106: quickcheck_code_int,A_139: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_139 @ zero_z891286103de_int )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_106 )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ A_139 @ B_106 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1312_mult__nonpos__nonneg,axiom,
! [B_106: code_code_numeral,A_139: code_code_numeral] :
( ( ord_le565307924umeral @ A_139 @ zero_z126310315umeral )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_106 )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ A_139 @ B_106 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1313_mult__nonpos__nonneg,axiom,
! [B_106: real,A_139: real] :
( ( ord_less_eq_real @ A_139 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_106 )
=> ( ord_less_eq_real @ ( times_times_real @ A_139 @ B_106 ) @ zero_zero_real ) ) ) ).
thf(fact_1314_mult__nonpos__nonneg,axiom,
! [B_106: nat,A_139: nat] :
( ( ord_less_eq_nat @ A_139 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_106 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_139 @ B_106 ) @ zero_zero_nat ) ) ) ).
thf(fact_1315_mult__nonpos__nonneg,axiom,
! [B_106: int,A_139: int] :
( ( ord_less_eq_int @ A_139 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_106 )
=> ( ord_less_eq_int @ ( times_times_int @ A_139 @ B_106 ) @ zero_zero_int ) ) ) ).
thf(fact_1316_mult__nonpos__nonpos,axiom,
! [B_105: rat,A_138: rat] :
( ( ord_less_eq_rat @ A_138 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B_105 @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A_138 @ B_105 ) ) ) ) ).
thf(fact_1317_mult__nonpos__nonpos,axiom,
! [B_105: real,A_138: real] :
( ( ord_less_eq_real @ A_138 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B_105 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_138 @ B_105 ) ) ) ) ).
thf(fact_1318_mult__nonpos__nonpos,axiom,
! [B_105: int,A_138: int] :
( ( ord_less_eq_int @ A_138 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B_105 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_138 @ B_105 ) ) ) ) ).
thf(fact_1319_mult__right__mono,axiom,
! [C_63: rat,A_137: rat,B_104: rat] :
( ( ord_less_eq_rat @ A_137 @ B_104 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_63 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_137 @ C_63 ) @ ( times_times_rat @ B_104 @ C_63 ) ) ) ) ).
thf(fact_1320_mult__right__mono,axiom,
! [C_63: quickcheck_code_int,A_137: quickcheck_code_int,B_104: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_137 @ B_104 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_63 )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ A_137 @ C_63 ) @ ( times_123202395de_int @ B_104 @ C_63 ) ) ) ) ).
thf(fact_1321_mult__right__mono,axiom,
! [C_63: code_code_numeral,A_137: code_code_numeral,B_104: code_code_numeral] :
( ( ord_le565307924umeral @ A_137 @ B_104 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_63 )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ A_137 @ C_63 ) @ ( times_1655362735umeral @ B_104 @ C_63 ) ) ) ) ).
thf(fact_1322_mult__right__mono,axiom,
! [C_63: real,A_137: real,B_104: real] :
( ( ord_less_eq_real @ A_137 @ B_104 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_63 )
=> ( ord_less_eq_real @ ( times_times_real @ A_137 @ C_63 ) @ ( times_times_real @ B_104 @ C_63 ) ) ) ) ).
thf(fact_1323_mult__right__mono,axiom,
! [C_63: nat,A_137: nat,B_104: nat] :
( ( ord_less_eq_nat @ A_137 @ B_104 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_63 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_137 @ C_63 ) @ ( times_times_nat @ B_104 @ C_63 ) ) ) ) ).
thf(fact_1324_mult__right__mono,axiom,
! [C_63: int,A_137: int,B_104: int] :
( ( ord_less_eq_int @ A_137 @ B_104 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_63 )
=> ( ord_less_eq_int @ ( times_times_int @ A_137 @ C_63 ) @ ( times_times_int @ B_104 @ C_63 ) ) ) ) ).
thf(fact_1325_mult__left__mono,axiom,
! [C_62: rat,A_136: rat,B_103: rat] :
( ( ord_less_eq_rat @ A_136 @ B_103 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_62 )
=> ( ord_less_eq_rat @ ( times_times_rat @ C_62 @ A_136 ) @ ( times_times_rat @ C_62 @ B_103 ) ) ) ) ).
thf(fact_1326_mult__left__mono,axiom,
! [C_62: quickcheck_code_int,A_136: quickcheck_code_int,B_103: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_136 @ B_103 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_62 )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ C_62 @ A_136 ) @ ( times_123202395de_int @ C_62 @ B_103 ) ) ) ) ).
thf(fact_1327_mult__left__mono,axiom,
! [C_62: code_code_numeral,A_136: code_code_numeral,B_103: code_code_numeral] :
( ( ord_le565307924umeral @ A_136 @ B_103 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_62 )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ C_62 @ A_136 ) @ ( times_1655362735umeral @ C_62 @ B_103 ) ) ) ) ).
thf(fact_1328_mult__left__mono,axiom,
! [C_62: real,A_136: real,B_103: real] :
( ( ord_less_eq_real @ A_136 @ B_103 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_62 )
=> ( ord_less_eq_real @ ( times_times_real @ C_62 @ A_136 ) @ ( times_times_real @ C_62 @ B_103 ) ) ) ) ).
thf(fact_1329_mult__left__mono,axiom,
! [C_62: nat,A_136: nat,B_103: nat] :
( ( ord_less_eq_nat @ A_136 @ B_103 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_62 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C_62 @ A_136 ) @ ( times_times_nat @ C_62 @ B_103 ) ) ) ) ).
thf(fact_1330_mult__left__mono,axiom,
! [C_62: int,A_136: int,B_103: int] :
( ( ord_less_eq_int @ A_136 @ B_103 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_62 )
=> ( ord_less_eq_int @ ( times_times_int @ C_62 @ A_136 ) @ ( times_times_int @ C_62 @ B_103 ) ) ) ) ).
thf(fact_1331_comm__mult__left__mono,axiom,
! [C_61: rat,A_135: rat,B_102: rat] :
( ( ord_less_eq_rat @ A_135 @ B_102 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_61 )
=> ( ord_less_eq_rat @ ( times_times_rat @ C_61 @ A_135 ) @ ( times_times_rat @ C_61 @ B_102 ) ) ) ) ).
thf(fact_1332_comm__mult__left__mono,axiom,
! [C_61: quickcheck_code_int,A_135: quickcheck_code_int,B_102: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_135 @ B_102 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_61 )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ C_61 @ A_135 ) @ ( times_123202395de_int @ C_61 @ B_102 ) ) ) ) ).
thf(fact_1333_comm__mult__left__mono,axiom,
! [C_61: code_code_numeral,A_135: code_code_numeral,B_102: code_code_numeral] :
( ( ord_le565307924umeral @ A_135 @ B_102 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_61 )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ C_61 @ A_135 ) @ ( times_1655362735umeral @ C_61 @ B_102 ) ) ) ) ).
thf(fact_1334_comm__mult__left__mono,axiom,
! [C_61: real,A_135: real,B_102: real] :
( ( ord_less_eq_real @ A_135 @ B_102 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_61 )
=> ( ord_less_eq_real @ ( times_times_real @ C_61 @ A_135 ) @ ( times_times_real @ C_61 @ B_102 ) ) ) ) ).
thf(fact_1335_comm__mult__left__mono,axiom,
! [C_61: nat,A_135: nat,B_102: nat] :
( ( ord_less_eq_nat @ A_135 @ B_102 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_61 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C_61 @ A_135 ) @ ( times_times_nat @ C_61 @ B_102 ) ) ) ) ).
thf(fact_1336_comm__mult__left__mono,axiom,
! [C_61: int,A_135: int,B_102: int] :
( ( ord_less_eq_int @ A_135 @ B_102 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_61 )
=> ( ord_less_eq_int @ ( times_times_int @ C_61 @ A_135 ) @ ( times_times_int @ C_61 @ B_102 ) ) ) ) ).
thf(fact_1337_mult__right__mono__neg,axiom,
! [C_60: rat,B_101: rat,A_134: rat] :
( ( ord_less_eq_rat @ B_101 @ A_134 )
=> ( ( ord_less_eq_rat @ C_60 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_134 @ C_60 ) @ ( times_times_rat @ B_101 @ C_60 ) ) ) ) ).
thf(fact_1338_mult__right__mono__neg,axiom,
! [C_60: real,B_101: real,A_134: real] :
( ( ord_less_eq_real @ B_101 @ A_134 )
=> ( ( ord_less_eq_real @ C_60 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A_134 @ C_60 ) @ ( times_times_real @ B_101 @ C_60 ) ) ) ) ).
thf(fact_1339_mult__right__mono__neg,axiom,
! [C_60: int,B_101: int,A_134: int] :
( ( ord_less_eq_int @ B_101 @ A_134 )
=> ( ( ord_less_eq_int @ C_60 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A_134 @ C_60 ) @ ( times_times_int @ B_101 @ C_60 ) ) ) ) ).
thf(fact_1340_mult__left__mono__neg,axiom,
! [C_59: rat,B_100: rat,A_133: rat] :
( ( ord_less_eq_rat @ B_100 @ A_133 )
=> ( ( ord_less_eq_rat @ C_59 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ C_59 @ A_133 ) @ ( times_times_rat @ C_59 @ B_100 ) ) ) ) ).
thf(fact_1341_mult__left__mono__neg,axiom,
! [C_59: real,B_100: real,A_133: real] :
( ( ord_less_eq_real @ B_100 @ A_133 )
=> ( ( ord_less_eq_real @ C_59 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C_59 @ A_133 ) @ ( times_times_real @ C_59 @ B_100 ) ) ) ) ).
thf(fact_1342_mult__left__mono__neg,axiom,
! [C_59: int,B_100: int,A_133: int] :
( ( ord_less_eq_int @ B_100 @ A_133 )
=> ( ( ord_less_eq_int @ C_59 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C_59 @ A_133 ) @ ( times_times_int @ C_59 @ B_100 ) ) ) ) ).
thf(fact_1343_mult__mono_H,axiom,
! [C_58: rat,D_27: rat,A_132: rat,B_99: rat] :
( ( ord_less_eq_rat @ A_132 @ B_99 )
=> ( ( ord_less_eq_rat @ C_58 @ D_27 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A_132 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_58 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_132 @ C_58 ) @ ( times_times_rat @ B_99 @ D_27 ) ) ) ) ) ) ).
thf(fact_1344_mult__mono_H,axiom,
! [C_58: quickcheck_code_int,D_27: quickcheck_code_int,A_132: quickcheck_code_int,B_99: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_132 @ B_99 )
=> ( ( ord_le258702272de_int @ C_58 @ D_27 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_132 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_58 )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ A_132 @ C_58 ) @ ( times_123202395de_int @ B_99 @ D_27 ) ) ) ) ) ) ).
thf(fact_1345_mult__mono_H,axiom,
! [C_58: code_code_numeral,D_27: code_code_numeral,A_132: code_code_numeral,B_99: code_code_numeral] :
( ( ord_le565307924umeral @ A_132 @ B_99 )
=> ( ( ord_le565307924umeral @ C_58 @ D_27 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_132 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_58 )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ A_132 @ C_58 ) @ ( times_1655362735umeral @ B_99 @ D_27 ) ) ) ) ) ) ).
thf(fact_1346_mult__mono_H,axiom,
! [C_58: real,D_27: real,A_132: real,B_99: real] :
( ( ord_less_eq_real @ A_132 @ B_99 )
=> ( ( ord_less_eq_real @ C_58 @ D_27 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_132 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_58 )
=> ( ord_less_eq_real @ ( times_times_real @ A_132 @ C_58 ) @ ( times_times_real @ B_99 @ D_27 ) ) ) ) ) ) ).
thf(fact_1347_mult__mono_H,axiom,
! [C_58: nat,D_27: nat,A_132: nat,B_99: nat] :
( ( ord_less_eq_nat @ A_132 @ B_99 )
=> ( ( ord_less_eq_nat @ C_58 @ D_27 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_132 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_58 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_132 @ C_58 ) @ ( times_times_nat @ B_99 @ D_27 ) ) ) ) ) ) ).
thf(fact_1348_mult__mono_H,axiom,
! [C_58: int,D_27: int,A_132: int,B_99: int] :
( ( ord_less_eq_int @ A_132 @ B_99 )
=> ( ( ord_less_eq_int @ C_58 @ D_27 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_132 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_58 )
=> ( ord_less_eq_int @ ( times_times_int @ A_132 @ C_58 ) @ ( times_times_int @ B_99 @ D_27 ) ) ) ) ) ) ).
thf(fact_1349_mult__mono,axiom,
! [C_57: rat,D_26: rat,A_131: rat,B_98: rat] :
( ( ord_less_eq_rat @ A_131 @ B_98 )
=> ( ( ord_less_eq_rat @ C_57 @ D_26 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_98 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_57 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_131 @ C_57 ) @ ( times_times_rat @ B_98 @ D_26 ) ) ) ) ) ) ).
thf(fact_1350_mult__mono,axiom,
! [C_57: quickcheck_code_int,D_26: quickcheck_code_int,A_131: quickcheck_code_int,B_98: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_131 @ B_98 )
=> ( ( ord_le258702272de_int @ C_57 @ D_26 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_98 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_57 )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ A_131 @ C_57 ) @ ( times_123202395de_int @ B_98 @ D_26 ) ) ) ) ) ) ).
thf(fact_1351_mult__mono,axiom,
! [C_57: code_code_numeral,D_26: code_code_numeral,A_131: code_code_numeral,B_98: code_code_numeral] :
( ( ord_le565307924umeral @ A_131 @ B_98 )
=> ( ( ord_le565307924umeral @ C_57 @ D_26 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_98 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_57 )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ A_131 @ C_57 ) @ ( times_1655362735umeral @ B_98 @ D_26 ) ) ) ) ) ) ).
thf(fact_1352_mult__mono,axiom,
! [C_57: real,D_26: real,A_131: real,B_98: real] :
( ( ord_less_eq_real @ A_131 @ B_98 )
=> ( ( ord_less_eq_real @ C_57 @ D_26 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_98 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_57 )
=> ( ord_less_eq_real @ ( times_times_real @ A_131 @ C_57 ) @ ( times_times_real @ B_98 @ D_26 ) ) ) ) ) ) ).
thf(fact_1353_mult__mono,axiom,
! [C_57: nat,D_26: nat,A_131: nat,B_98: nat] :
( ( ord_less_eq_nat @ A_131 @ B_98 )
=> ( ( ord_less_eq_nat @ C_57 @ D_26 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_98 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_57 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_131 @ C_57 ) @ ( times_times_nat @ B_98 @ D_26 ) ) ) ) ) ) ).
thf(fact_1354_mult__mono,axiom,
! [C_57: int,D_26: int,A_131: int,B_98: int] :
( ( ord_less_eq_int @ A_131 @ B_98 )
=> ( ( ord_less_eq_int @ C_57 @ D_26 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_98 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_57 )
=> ( ord_less_eq_int @ ( times_times_int @ A_131 @ C_57 ) @ ( times_times_int @ B_98 @ D_26 ) ) ) ) ) ) ).
thf(fact_1355_split__mult__pos__le,axiom,
! [B_97: rat,A_130: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A_130 )
& ( ord_less_eq_rat @ zero_zero_rat @ B_97 ) )
| ( ( ord_less_eq_rat @ A_130 @ zero_zero_rat )
& ( ord_less_eq_rat @ B_97 @ zero_zero_rat ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A_130 @ B_97 ) ) ) ).
thf(fact_1356_split__mult__pos__le,axiom,
! [B_97: real,A_130: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A_130 )
& ( ord_less_eq_real @ zero_zero_real @ B_97 ) )
| ( ( ord_less_eq_real @ A_130 @ zero_zero_real )
& ( ord_less_eq_real @ B_97 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_130 @ B_97 ) ) ) ).
thf(fact_1357_split__mult__pos__le,axiom,
! [B_97: int,A_130: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A_130 )
& ( ord_less_eq_int @ zero_zero_int @ B_97 ) )
| ( ( ord_less_eq_int @ A_130 @ zero_zero_int )
& ( ord_less_eq_int @ B_97 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_130 @ B_97 ) ) ) ).
thf(fact_1358_split__mult__neg__le,axiom,
! [B_96: rat,A_129: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A_129 )
& ( ord_less_eq_rat @ B_96 @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A_129 @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B_96 ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_129 @ B_96 ) @ zero_zero_rat ) ) ).
thf(fact_1359_split__mult__neg__le,axiom,
! [B_96: quickcheck_code_int,A_129: quickcheck_code_int] :
( ( ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_129 )
& ( ord_le258702272de_int @ B_96 @ zero_z891286103de_int ) )
| ( ( ord_le258702272de_int @ A_129 @ zero_z891286103de_int )
& ( ord_le258702272de_int @ zero_z891286103de_int @ B_96 ) ) )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ A_129 @ B_96 ) @ zero_z891286103de_int ) ) ).
thf(fact_1360_split__mult__neg__le,axiom,
! [B_96: code_code_numeral,A_129: code_code_numeral] :
( ( ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_129 )
& ( ord_le565307924umeral @ B_96 @ zero_z126310315umeral ) )
| ( ( ord_le565307924umeral @ A_129 @ zero_z126310315umeral )
& ( ord_le565307924umeral @ zero_z126310315umeral @ B_96 ) ) )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ A_129 @ B_96 ) @ zero_z126310315umeral ) ) ).
thf(fact_1361_split__mult__neg__le,axiom,
! [B_96: real,A_129: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A_129 )
& ( ord_less_eq_real @ B_96 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A_129 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B_96 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A_129 @ B_96 ) @ zero_zero_real ) ) ).
thf(fact_1362_split__mult__neg__le,axiom,
! [B_96: nat,A_129: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A_129 )
& ( ord_less_eq_nat @ B_96 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A_129 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B_96 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_129 @ B_96 ) @ zero_zero_nat ) ) ).
thf(fact_1363_split__mult__neg__le,axiom,
! [B_96: int,A_129: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A_129 )
& ( ord_less_eq_int @ B_96 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A_129 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B_96 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A_129 @ B_96 ) @ zero_zero_int ) ) ).
thf(fact_1364_not__square__less__zero,axiom,
! [A_128: rat] :
~ ( ord_less_rat @ ( times_times_rat @ A_128 @ A_128 ) @ zero_zero_rat ) ).
thf(fact_1365_not__square__less__zero,axiom,
! [A_128: real] :
~ ( ord_less_real @ ( times_times_real @ A_128 @ A_128 ) @ zero_zero_real ) ).
thf(fact_1366_not__square__less__zero,axiom,
! [A_128: int] :
~ ( ord_less_int @ ( times_times_int @ A_128 @ A_128 ) @ zero_zero_int ) ).
thf(fact_1367_mult__less__cancel__right__disj,axiom,
! [A_127: rat,C_56: rat,B_95: rat] :
( ( ord_less_rat @ ( times_times_rat @ A_127 @ C_56 ) @ ( times_times_rat @ B_95 @ C_56 ) )
<=> ( ( ( ord_less_rat @ zero_zero_rat @ C_56 )
& ( ord_less_rat @ A_127 @ B_95 ) )
| ( ( ord_less_rat @ C_56 @ zero_zero_rat )
& ( ord_less_rat @ B_95 @ A_127 ) ) ) ) ).
thf(fact_1368_mult__less__cancel__right__disj,axiom,
! [A_127: real,C_56: real,B_95: real] :
( ( ord_less_real @ ( times_times_real @ A_127 @ C_56 ) @ ( times_times_real @ B_95 @ C_56 ) )
<=> ( ( ( ord_less_real @ zero_zero_real @ C_56 )
& ( ord_less_real @ A_127 @ B_95 ) )
| ( ( ord_less_real @ C_56 @ zero_zero_real )
& ( ord_less_real @ B_95 @ A_127 ) ) ) ) ).
thf(fact_1369_mult__less__cancel__right__disj,axiom,
! [A_127: int,C_56: int,B_95: int] :
( ( ord_less_int @ ( times_times_int @ A_127 @ C_56 ) @ ( times_times_int @ B_95 @ C_56 ) )
<=> ( ( ( ord_less_int @ zero_zero_int @ C_56 )
& ( ord_less_int @ A_127 @ B_95 ) )
| ( ( ord_less_int @ C_56 @ zero_zero_int )
& ( ord_less_int @ B_95 @ A_127 ) ) ) ) ).
thf(fact_1370_mult__less__cancel__left__disj,axiom,
! [C_55: rat,A_126: rat,B_94: rat] :
( ( ord_less_rat @ ( times_times_rat @ C_55 @ A_126 ) @ ( times_times_rat @ C_55 @ B_94 ) )
<=> ( ( ( ord_less_rat @ zero_zero_rat @ C_55 )
& ( ord_less_rat @ A_126 @ B_94 ) )
| ( ( ord_less_rat @ C_55 @ zero_zero_rat )
& ( ord_less_rat @ B_94 @ A_126 ) ) ) ) ).
thf(fact_1371_mult__less__cancel__left__disj,axiom,
! [C_55: real,A_126: real,B_94: real] :
( ( ord_less_real @ ( times_times_real @ C_55 @ A_126 ) @ ( times_times_real @ C_55 @ B_94 ) )
<=> ( ( ( ord_less_real @ zero_zero_real @ C_55 )
& ( ord_less_real @ A_126 @ B_94 ) )
| ( ( ord_less_real @ C_55 @ zero_zero_real )
& ( ord_less_real @ B_94 @ A_126 ) ) ) ) ).
thf(fact_1372_mult__less__cancel__left__disj,axiom,
! [C_55: int,A_126: int,B_94: int] :
( ( ord_less_int @ ( times_times_int @ C_55 @ A_126 ) @ ( times_times_int @ C_55 @ B_94 ) )
<=> ( ( ( ord_less_int @ zero_zero_int @ C_55 )
& ( ord_less_int @ A_126 @ B_94 ) )
| ( ( ord_less_int @ C_55 @ zero_zero_int )
& ( ord_less_int @ B_94 @ A_126 ) ) ) ) ).
thf(fact_1373_mult__less__cancel__left__pos,axiom,
! [A_125: rat,B_93: rat,C_54: rat] :
( ( ord_less_rat @ zero_zero_rat @ C_54 )
=> ( ( ord_less_rat @ ( times_times_rat @ C_54 @ A_125 ) @ ( times_times_rat @ C_54 @ B_93 ) )
<=> ( ord_less_rat @ A_125 @ B_93 ) ) ) ).
thf(fact_1374_mult__less__cancel__left__pos,axiom,
! [A_125: real,B_93: real,C_54: real] :
( ( ord_less_real @ zero_zero_real @ C_54 )
=> ( ( ord_less_real @ ( times_times_real @ C_54 @ A_125 ) @ ( times_times_real @ C_54 @ B_93 ) )
<=> ( ord_less_real @ A_125 @ B_93 ) ) ) ).
thf(fact_1375_mult__less__cancel__left__pos,axiom,
! [A_125: int,B_93: int,C_54: int] :
( ( ord_less_int @ zero_zero_int @ C_54 )
=> ( ( ord_less_int @ ( times_times_int @ C_54 @ A_125 ) @ ( times_times_int @ C_54 @ B_93 ) )
<=> ( ord_less_int @ A_125 @ B_93 ) ) ) ).
thf(fact_1376_mult__pos__pos,axiom,
! [B_92: rat,A_124: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_124 )
=> ( ( ord_less_rat @ zero_zero_rat @ B_92 )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A_124 @ B_92 ) ) ) ) ).
thf(fact_1377_mult__pos__pos,axiom,
! [B_92: quickcheck_code_int,A_124: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_124 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ B_92 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ ( times_123202395de_int @ A_124 @ B_92 ) ) ) ) ).
thf(fact_1378_mult__pos__pos,axiom,
! [B_92: code_code_numeral,A_124: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_124 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ B_92 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ ( times_1655362735umeral @ A_124 @ B_92 ) ) ) ) ).
thf(fact_1379_mult__pos__pos,axiom,
! [B_92: real,A_124: real] :
( ( ord_less_real @ zero_zero_real @ A_124 )
=> ( ( ord_less_real @ zero_zero_real @ B_92 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A_124 @ B_92 ) ) ) ) ).
thf(fact_1380_mult__pos__pos,axiom,
! [B_92: nat,A_124: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_124 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_92 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A_124 @ B_92 ) ) ) ) ).
thf(fact_1381_mult__pos__pos,axiom,
! [B_92: int,A_124: int] :
( ( ord_less_int @ zero_zero_int @ A_124 )
=> ( ( ord_less_int @ zero_zero_int @ B_92 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_124 @ B_92 ) ) ) ) ).
thf(fact_1382_mult__pos__neg,axiom,
! [B_91: rat,A_123: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_123 )
=> ( ( ord_less_rat @ B_91 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A_123 @ B_91 ) @ zero_zero_rat ) ) ) ).
thf(fact_1383_mult__pos__neg,axiom,
! [B_91: quickcheck_code_int,A_123: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_123 )
=> ( ( ord_le1860547276de_int @ B_91 @ zero_z891286103de_int )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_123 @ B_91 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1384_mult__pos__neg,axiom,
! [B_91: code_code_numeral,A_123: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_123 )
=> ( ( ord_le1304079648umeral @ B_91 @ zero_z126310315umeral )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_123 @ B_91 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1385_mult__pos__neg,axiom,
! [B_91: real,A_123: real] :
( ( ord_less_real @ zero_zero_real @ A_123 )
=> ( ( ord_less_real @ B_91 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A_123 @ B_91 ) @ zero_zero_real ) ) ) ).
thf(fact_1386_mult__pos__neg,axiom,
! [B_91: nat,A_123: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_123 )
=> ( ( ord_less_nat @ B_91 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A_123 @ B_91 ) @ zero_zero_nat ) ) ) ).
thf(fact_1387_mult__pos__neg,axiom,
! [B_91: int,A_123: int] :
( ( ord_less_int @ zero_zero_int @ A_123 )
=> ( ( ord_less_int @ B_91 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A_123 @ B_91 ) @ zero_zero_int ) ) ) ).
thf(fact_1388_mult__pos__neg2,axiom,
! [B_90: rat,A_122: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_122 )
=> ( ( ord_less_rat @ B_90 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ B_90 @ A_122 ) @ zero_zero_rat ) ) ) ).
thf(fact_1389_mult__pos__neg2,axiom,
! [B_90: quickcheck_code_int,A_122: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_122 )
=> ( ( ord_le1860547276de_int @ B_90 @ zero_z891286103de_int )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ B_90 @ A_122 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1390_mult__pos__neg2,axiom,
! [B_90: code_code_numeral,A_122: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_122 )
=> ( ( ord_le1304079648umeral @ B_90 @ zero_z126310315umeral )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ B_90 @ A_122 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1391_mult__pos__neg2,axiom,
! [B_90: real,A_122: real] :
( ( ord_less_real @ zero_zero_real @ A_122 )
=> ( ( ord_less_real @ B_90 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B_90 @ A_122 ) @ zero_zero_real ) ) ) ).
thf(fact_1392_mult__pos__neg2,axiom,
! [B_90: nat,A_122: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_122 )
=> ( ( ord_less_nat @ B_90 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B_90 @ A_122 ) @ zero_zero_nat ) ) ) ).
thf(fact_1393_mult__pos__neg2,axiom,
! [B_90: int,A_122: int] :
( ( ord_less_int @ zero_zero_int @ A_122 )
=> ( ( ord_less_int @ B_90 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B_90 @ A_122 ) @ zero_zero_int ) ) ) ).
thf(fact_1394_zero__less__mult__pos,axiom,
! [A_121: rat,B_89: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A_121 @ B_89 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A_121 )
=> ( ord_less_rat @ zero_zero_rat @ B_89 ) ) ) ).
thf(fact_1395_zero__less__mult__pos,axiom,
! [A_121: quickcheck_code_int,B_89: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ ( times_123202395de_int @ A_121 @ B_89 ) )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_121 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ B_89 ) ) ) ).
thf(fact_1396_zero__less__mult__pos,axiom,
! [A_121: code_code_numeral,B_89: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ ( times_1655362735umeral @ A_121 @ B_89 ) )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_121 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ B_89 ) ) ) ).
thf(fact_1397_zero__less__mult__pos,axiom,
! [A_121: real,B_89: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A_121 @ B_89 ) )
=> ( ( ord_less_real @ zero_zero_real @ A_121 )
=> ( ord_less_real @ zero_zero_real @ B_89 ) ) ) ).
thf(fact_1398_zero__less__mult__pos,axiom,
! [A_121: nat,B_89: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A_121 @ B_89 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A_121 )
=> ( ord_less_nat @ zero_zero_nat @ B_89 ) ) ) ).
thf(fact_1399_zero__less__mult__pos,axiom,
! [A_121: int,B_89: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_121 @ B_89 ) )
=> ( ( ord_less_int @ zero_zero_int @ A_121 )
=> ( ord_less_int @ zero_zero_int @ B_89 ) ) ) ).
thf(fact_1400_zero__less__mult__pos2,axiom,
! [B_88: rat,A_120: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B_88 @ A_120 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A_120 )
=> ( ord_less_rat @ zero_zero_rat @ B_88 ) ) ) ).
thf(fact_1401_zero__less__mult__pos2,axiom,
! [B_88: quickcheck_code_int,A_120: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ ( times_123202395de_int @ B_88 @ A_120 ) )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_120 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ B_88 ) ) ) ).
thf(fact_1402_zero__less__mult__pos2,axiom,
! [B_88: code_code_numeral,A_120: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ ( times_1655362735umeral @ B_88 @ A_120 ) )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_120 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ B_88 ) ) ) ).
thf(fact_1403_zero__less__mult__pos2,axiom,
! [B_88: real,A_120: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B_88 @ A_120 ) )
=> ( ( ord_less_real @ zero_zero_real @ A_120 )
=> ( ord_less_real @ zero_zero_real @ B_88 ) ) ) ).
thf(fact_1404_zero__less__mult__pos2,axiom,
! [B_88: nat,A_120: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B_88 @ A_120 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A_120 )
=> ( ord_less_nat @ zero_zero_nat @ B_88 ) ) ) ).
thf(fact_1405_zero__less__mult__pos2,axiom,
! [B_88: int,A_120: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B_88 @ A_120 ) )
=> ( ( ord_less_int @ zero_zero_int @ A_120 )
=> ( ord_less_int @ zero_zero_int @ B_88 ) ) ) ).
thf(fact_1406_mult__less__cancel__left__neg,axiom,
! [A_119: rat,B_87: rat,C_53: rat] :
( ( ord_less_rat @ C_53 @ zero_zero_rat )
=> ( ( ord_less_rat @ ( times_times_rat @ C_53 @ A_119 ) @ ( times_times_rat @ C_53 @ B_87 ) )
<=> ( ord_less_rat @ B_87 @ A_119 ) ) ) ).
thf(fact_1407_mult__less__cancel__left__neg,axiom,
! [A_119: real,B_87: real,C_53: real] :
( ( ord_less_real @ C_53 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C_53 @ A_119 ) @ ( times_times_real @ C_53 @ B_87 ) )
<=> ( ord_less_real @ B_87 @ A_119 ) ) ) ).
thf(fact_1408_mult__less__cancel__left__neg,axiom,
! [A_119: int,B_87: int,C_53: int] :
( ( ord_less_int @ C_53 @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C_53 @ A_119 ) @ ( times_times_int @ C_53 @ B_87 ) )
<=> ( ord_less_int @ B_87 @ A_119 ) ) ) ).
thf(fact_1409_mult__neg__pos,axiom,
! [B_86: rat,A_118: rat] :
( ( ord_less_rat @ A_118 @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ B_86 )
=> ( ord_less_rat @ ( times_times_rat @ A_118 @ B_86 ) @ zero_zero_rat ) ) ) ).
thf(fact_1410_mult__neg__pos,axiom,
! [B_86: quickcheck_code_int,A_118: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_118 @ zero_z891286103de_int )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ B_86 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_118 @ B_86 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1411_mult__neg__pos,axiom,
! [B_86: code_code_numeral,A_118: code_code_numeral] :
( ( ord_le1304079648umeral @ A_118 @ zero_z126310315umeral )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ B_86 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_118 @ B_86 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1412_mult__neg__pos,axiom,
! [B_86: real,A_118: real] :
( ( ord_less_real @ A_118 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B_86 )
=> ( ord_less_real @ ( times_times_real @ A_118 @ B_86 ) @ zero_zero_real ) ) ) ).
thf(fact_1413_mult__neg__pos,axiom,
! [B_86: nat,A_118: nat] :
( ( ord_less_nat @ A_118 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B_86 )
=> ( ord_less_nat @ ( times_times_nat @ A_118 @ B_86 ) @ zero_zero_nat ) ) ) ).
thf(fact_1414_mult__neg__pos,axiom,
! [B_86: int,A_118: int] :
( ( ord_less_int @ A_118 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B_86 )
=> ( ord_less_int @ ( times_times_int @ A_118 @ B_86 ) @ zero_zero_int ) ) ) ).
thf(fact_1415_mult__neg__neg,axiom,
! [B_85: rat,A_117: rat] :
( ( ord_less_rat @ A_117 @ zero_zero_rat )
=> ( ( ord_less_rat @ B_85 @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A_117 @ B_85 ) ) ) ) ).
thf(fact_1416_mult__neg__neg,axiom,
! [B_85: real,A_117: real] :
( ( ord_less_real @ A_117 @ zero_zero_real )
=> ( ( ord_less_real @ B_85 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A_117 @ B_85 ) ) ) ) ).
thf(fact_1417_mult__neg__neg,axiom,
! [B_85: int,A_117: int] :
( ( ord_less_int @ A_117 @ zero_zero_int )
=> ( ( ord_less_int @ B_85 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_117 @ B_85 ) ) ) ) ).
thf(fact_1418_mult__strict__right__mono,axiom,
! [C_52: rat,A_116: rat,B_84: rat] :
( ( ord_less_rat @ A_116 @ B_84 )
=> ( ( ord_less_rat @ zero_zero_rat @ C_52 )
=> ( ord_less_rat @ ( times_times_rat @ A_116 @ C_52 ) @ ( times_times_rat @ B_84 @ C_52 ) ) ) ) ).
thf(fact_1419_mult__strict__right__mono,axiom,
! [C_52: quickcheck_code_int,A_116: quickcheck_code_int,B_84: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_116 @ B_84 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ C_52 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_116 @ C_52 ) @ ( times_123202395de_int @ B_84 @ C_52 ) ) ) ) ).
thf(fact_1420_mult__strict__right__mono,axiom,
! [C_52: code_code_numeral,A_116: code_code_numeral,B_84: code_code_numeral] :
( ( ord_le1304079648umeral @ A_116 @ B_84 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ C_52 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_116 @ C_52 ) @ ( times_1655362735umeral @ B_84 @ C_52 ) ) ) ) ).
thf(fact_1421_mult__strict__right__mono,axiom,
! [C_52: real,A_116: real,B_84: real] :
( ( ord_less_real @ A_116 @ B_84 )
=> ( ( ord_less_real @ zero_zero_real @ C_52 )
=> ( ord_less_real @ ( times_times_real @ A_116 @ C_52 ) @ ( times_times_real @ B_84 @ C_52 ) ) ) ) ).
thf(fact_1422_mult__strict__right__mono,axiom,
! [C_52: nat,A_116: nat,B_84: nat] :
( ( ord_less_nat @ A_116 @ B_84 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_52 )
=> ( ord_less_nat @ ( times_times_nat @ A_116 @ C_52 ) @ ( times_times_nat @ B_84 @ C_52 ) ) ) ) ).
thf(fact_1423_mult__strict__right__mono,axiom,
! [C_52: int,A_116: int,B_84: int] :
( ( ord_less_int @ A_116 @ B_84 )
=> ( ( ord_less_int @ zero_zero_int @ C_52 )
=> ( ord_less_int @ ( times_times_int @ A_116 @ C_52 ) @ ( times_times_int @ B_84 @ C_52 ) ) ) ) ).
thf(fact_1424_mult__strict__left__mono,axiom,
! [C_51: rat,A_115: rat,B_83: rat] :
( ( ord_less_rat @ A_115 @ B_83 )
=> ( ( ord_less_rat @ zero_zero_rat @ C_51 )
=> ( ord_less_rat @ ( times_times_rat @ C_51 @ A_115 ) @ ( times_times_rat @ C_51 @ B_83 ) ) ) ) ).
thf(fact_1425_mult__strict__left__mono,axiom,
! [C_51: quickcheck_code_int,A_115: quickcheck_code_int,B_83: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_115 @ B_83 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ C_51 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ C_51 @ A_115 ) @ ( times_123202395de_int @ C_51 @ B_83 ) ) ) ) ).
thf(fact_1426_mult__strict__left__mono,axiom,
! [C_51: code_code_numeral,A_115: code_code_numeral,B_83: code_code_numeral] :
( ( ord_le1304079648umeral @ A_115 @ B_83 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ C_51 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ C_51 @ A_115 ) @ ( times_1655362735umeral @ C_51 @ B_83 ) ) ) ) ).
thf(fact_1427_mult__strict__left__mono,axiom,
! [C_51: real,A_115: real,B_83: real] :
( ( ord_less_real @ A_115 @ B_83 )
=> ( ( ord_less_real @ zero_zero_real @ C_51 )
=> ( ord_less_real @ ( times_times_real @ C_51 @ A_115 ) @ ( times_times_real @ C_51 @ B_83 ) ) ) ) ).
thf(fact_1428_mult__strict__left__mono,axiom,
! [C_51: nat,A_115: nat,B_83: nat] :
( ( ord_less_nat @ A_115 @ B_83 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_51 )
=> ( ord_less_nat @ ( times_times_nat @ C_51 @ A_115 ) @ ( times_times_nat @ C_51 @ B_83 ) ) ) ) ).
thf(fact_1429_mult__strict__left__mono,axiom,
! [C_51: int,A_115: int,B_83: int] :
( ( ord_less_int @ A_115 @ B_83 )
=> ( ( ord_less_int @ zero_zero_int @ C_51 )
=> ( ord_less_int @ ( times_times_int @ C_51 @ A_115 ) @ ( times_times_int @ C_51 @ B_83 ) ) ) ) ).
thf(fact_1430_comm__mult__strict__left__mono,axiom,
! [C_50: rat,A_114: rat,B_82: rat] :
( ( ord_less_rat @ A_114 @ B_82 )
=> ( ( ord_less_rat @ zero_zero_rat @ C_50 )
=> ( ord_less_rat @ ( times_times_rat @ C_50 @ A_114 ) @ ( times_times_rat @ C_50 @ B_82 ) ) ) ) ).
thf(fact_1431_comm__mult__strict__left__mono,axiom,
! [C_50: quickcheck_code_int,A_114: quickcheck_code_int,B_82: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_114 @ B_82 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ C_50 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ C_50 @ A_114 ) @ ( times_123202395de_int @ C_50 @ B_82 ) ) ) ) ).
thf(fact_1432_comm__mult__strict__left__mono,axiom,
! [C_50: code_code_numeral,A_114: code_code_numeral,B_82: code_code_numeral] :
( ( ord_le1304079648umeral @ A_114 @ B_82 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ C_50 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ C_50 @ A_114 ) @ ( times_1655362735umeral @ C_50 @ B_82 ) ) ) ) ).
thf(fact_1433_comm__mult__strict__left__mono,axiom,
! [C_50: real,A_114: real,B_82: real] :
( ( ord_less_real @ A_114 @ B_82 )
=> ( ( ord_less_real @ zero_zero_real @ C_50 )
=> ( ord_less_real @ ( times_times_real @ C_50 @ A_114 ) @ ( times_times_real @ C_50 @ B_82 ) ) ) ) ).
thf(fact_1434_comm__mult__strict__left__mono,axiom,
! [C_50: nat,A_114: nat,B_82: nat] :
( ( ord_less_nat @ A_114 @ B_82 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_50 )
=> ( ord_less_nat @ ( times_times_nat @ C_50 @ A_114 ) @ ( times_times_nat @ C_50 @ B_82 ) ) ) ) ).
thf(fact_1435_comm__mult__strict__left__mono,axiom,
! [C_50: int,A_114: int,B_82: int] :
( ( ord_less_int @ A_114 @ B_82 )
=> ( ( ord_less_int @ zero_zero_int @ C_50 )
=> ( ord_less_int @ ( times_times_int @ C_50 @ A_114 ) @ ( times_times_int @ C_50 @ B_82 ) ) ) ) ).
thf(fact_1436_mult__strict__right__mono__neg,axiom,
! [C_49: rat,B_81: rat,A_113: rat] :
( ( ord_less_rat @ B_81 @ A_113 )
=> ( ( ord_less_rat @ C_49 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A_113 @ C_49 ) @ ( times_times_rat @ B_81 @ C_49 ) ) ) ) ).
thf(fact_1437_mult__strict__right__mono__neg,axiom,
! [C_49: real,B_81: real,A_113: real] :
( ( ord_less_real @ B_81 @ A_113 )
=> ( ( ord_less_real @ C_49 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A_113 @ C_49 ) @ ( times_times_real @ B_81 @ C_49 ) ) ) ) ).
thf(fact_1438_mult__strict__right__mono__neg,axiom,
! [C_49: int,B_81: int,A_113: int] :
( ( ord_less_int @ B_81 @ A_113 )
=> ( ( ord_less_int @ C_49 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A_113 @ C_49 ) @ ( times_times_int @ B_81 @ C_49 ) ) ) ) ).
thf(fact_1439_mult__strict__left__mono__neg,axiom,
! [C_48: rat,B_80: rat,A_112: rat] :
( ( ord_less_rat @ B_80 @ A_112 )
=> ( ( ord_less_rat @ C_48 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ C_48 @ A_112 ) @ ( times_times_rat @ C_48 @ B_80 ) ) ) ) ).
thf(fact_1440_mult__strict__left__mono__neg,axiom,
! [C_48: real,B_80: real,A_112: real] :
( ( ord_less_real @ B_80 @ A_112 )
=> ( ( ord_less_real @ C_48 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C_48 @ A_112 ) @ ( times_times_real @ C_48 @ B_80 ) ) ) ) ).
thf(fact_1441_mult__strict__left__mono__neg,axiom,
! [C_48: int,B_80: int,A_112: int] :
( ( ord_less_int @ B_80 @ A_112 )
=> ( ( ord_less_int @ C_48 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C_48 @ A_112 ) @ ( times_times_int @ C_48 @ B_80 ) ) ) ) ).
thf(fact_1442_sum__squares__eq__zero__iff,axiom,
! [X_72: rat,Y_61: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ X_72 @ X_72 ) @ ( times_times_rat @ Y_61 @ Y_61 ) )
= zero_zero_rat )
<=> ( ( X_72 = zero_zero_rat )
& ( Y_61 = zero_zero_rat ) ) ) ).
thf(fact_1443_sum__squares__eq__zero__iff,axiom,
! [X_72: real,Y_61: real] :
( ( ( plus_plus_real @ ( times_times_real @ X_72 @ X_72 ) @ ( times_times_real @ Y_61 @ Y_61 ) )
= zero_zero_real )
<=> ( ( X_72 = zero_zero_real )
& ( Y_61 = zero_zero_real ) ) ) ).
thf(fact_1444_sum__squares__eq__zero__iff,axiom,
! [X_72: int,Y_61: int] :
( ( ( plus_plus_int @ ( times_times_int @ X_72 @ X_72 ) @ ( times_times_int @ Y_61 @ Y_61 ) )
= zero_zero_int )
<=> ( ( X_72 = zero_zero_int )
& ( Y_61 = zero_zero_int ) ) ) ).
thf(fact_1445_add__scale__eq__noteq,axiom,
! [C_47: rat,D_25: rat,A_111: rat,B_79: rat,R_5: rat] :
( ( R_5 != zero_zero_rat )
=> ( ( ( A_111 = B_79 )
& ( C_47 != D_25 ) )
=> ( ( plus_plus_rat @ A_111 @ ( times_times_rat @ R_5 @ C_47 ) )
!= ( plus_plus_rat @ B_79 @ ( times_times_rat @ R_5 @ D_25 ) ) ) ) ) ).
thf(fact_1446_add__scale__eq__noteq,axiom,
! [C_47: complex,D_25: complex,A_111: complex,B_79: complex,R_5: complex] :
( ( R_5 != zero_zero_complex )
=> ( ( ( A_111 = B_79 )
& ( C_47 != D_25 ) )
=> ( ( plus_plus_complex @ A_111 @ ( times_times_complex @ R_5 @ C_47 ) )
!= ( plus_plus_complex @ B_79 @ ( times_times_complex @ R_5 @ D_25 ) ) ) ) ) ).
thf(fact_1447_add__scale__eq__noteq,axiom,
! [C_47: real,D_25: real,A_111: real,B_79: real,R_5: real] :
( ( R_5 != zero_zero_real )
=> ( ( ( A_111 = B_79 )
& ( C_47 != D_25 ) )
=> ( ( plus_plus_real @ A_111 @ ( times_times_real @ R_5 @ C_47 ) )
!= ( plus_plus_real @ B_79 @ ( times_times_real @ R_5 @ D_25 ) ) ) ) ) ).
thf(fact_1448_add__scale__eq__noteq,axiom,
! [C_47: nat,D_25: nat,A_111: nat,B_79: nat,R_5: nat] :
( ( R_5 != zero_zero_nat )
=> ( ( ( A_111 = B_79 )
& ( C_47 != D_25 ) )
=> ( ( plus_plus_nat @ A_111 @ ( times_times_nat @ R_5 @ C_47 ) )
!= ( plus_plus_nat @ B_79 @ ( times_times_nat @ R_5 @ D_25 ) ) ) ) ) ).
thf(fact_1449_add__scale__eq__noteq,axiom,
! [C_47: int,D_25: int,A_111: int,B_79: int,R_5: int] :
( ( R_5 != zero_zero_int )
=> ( ( ( A_111 = B_79 )
& ( C_47 != D_25 ) )
=> ( ( plus_plus_int @ A_111 @ ( times_times_int @ R_5 @ C_47 ) )
!= ( plus_plus_int @ B_79 @ ( times_times_int @ R_5 @ D_25 ) ) ) ) ) ).
thf(fact_1450_less__1__mult,axiom,
! [N_26: rat,M_19: rat] :
( ( ord_less_rat @ one_one_rat @ M_19 )
=> ( ( ord_less_rat @ one_one_rat @ N_26 )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M_19 @ N_26 ) ) ) ) ).
thf(fact_1451_less__1__mult,axiom,
! [N_26: quickcheck_code_int,M_19: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ M_19 )
=> ( ( ord_le1860547276de_int @ one_on1684967323de_int @ N_26 )
=> ( ord_le1860547276de_int @ one_on1684967323de_int @ ( times_123202395de_int @ M_19 @ N_26 ) ) ) ) ).
thf(fact_1452_less__1__mult,axiom,
! [N_26: code_code_numeral,M_19: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ M_19 )
=> ( ( ord_le1304079648umeral @ one_on1645066479umeral @ N_26 )
=> ( ord_le1304079648umeral @ one_on1645066479umeral @ ( times_1655362735umeral @ M_19 @ N_26 ) ) ) ) ).
thf(fact_1453_less__1__mult,axiom,
! [N_26: real,M_19: real] :
( ( ord_less_real @ one_one_real @ M_19 )
=> ( ( ord_less_real @ one_one_real @ N_26 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M_19 @ N_26 ) ) ) ) ).
thf(fact_1454_less__1__mult,axiom,
! [N_26: nat,M_19: nat] :
( ( ord_less_nat @ one_one_nat @ M_19 )
=> ( ( ord_less_nat @ one_one_nat @ N_26 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M_19 @ N_26 ) ) ) ) ).
thf(fact_1455_less__1__mult,axiom,
! [N_26: int,M_19: int] :
( ( ord_less_int @ one_one_int @ M_19 )
=> ( ( ord_less_int @ one_one_int @ N_26 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M_19 @ N_26 ) ) ) ) ).
thf(fact_1456_right__distrib__number__of,axiom,
! [V_10: int,B_78: rat,C_46: rat] :
( ( times_times_rat @ ( number_number_of_rat @ V_10 ) @ ( plus_plus_rat @ B_78 @ C_46 ) )
= ( plus_plus_rat @ ( times_times_rat @ ( number_number_of_rat @ V_10 ) @ B_78 ) @ ( times_times_rat @ ( number_number_of_rat @ V_10 ) @ C_46 ) ) ) ).
thf(fact_1457_right__distrib__number__of,axiom,
! [V_10: int,B_78: quickcheck_code_int,C_46: quickcheck_code_int] :
( ( times_123202395de_int @ ( number1226105091de_int @ V_10 ) @ ( plus_p1446045655de_int @ B_78 @ C_46 ) )
= ( plus_p1446045655de_int @ ( times_123202395de_int @ ( number1226105091de_int @ V_10 ) @ B_78 ) @ ( times_123202395de_int @ ( number1226105091de_int @ V_10 ) @ C_46 ) ) ) ).
thf(fact_1458_right__distrib__number__of,axiom,
! [V_10: int,B_78: complex,C_46: complex] :
( ( times_times_complex @ ( number528085621omplex @ V_10 ) @ ( plus_plus_complex @ B_78 @ C_46 ) )
= ( plus_plus_complex @ ( times_times_complex @ ( number528085621omplex @ V_10 ) @ B_78 ) @ ( times_times_complex @ ( number528085621omplex @ V_10 ) @ C_46 ) ) ) ).
thf(fact_1459_right__distrib__number__of,axiom,
! [V_10: int,B_78: real,C_46: real] :
( ( times_times_real @ ( number267125858f_real @ V_10 ) @ ( plus_plus_real @ B_78 @ C_46 ) )
= ( plus_plus_real @ ( times_times_real @ ( number267125858f_real @ V_10 ) @ B_78 ) @ ( times_times_real @ ( number267125858f_real @ V_10 ) @ C_46 ) ) ) ).
thf(fact_1460_right__distrib__number__of,axiom,
! [V_10: int,B_78: nat,C_46: nat] :
( ( times_times_nat @ ( number_number_of_nat @ V_10 ) @ ( plus_plus_nat @ B_78 @ C_46 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ V_10 ) @ B_78 ) @ ( times_times_nat @ ( number_number_of_nat @ V_10 ) @ C_46 ) ) ) ).
thf(fact_1461_right__distrib__number__of,axiom,
! [V_10: int,B_78: int,C_46: int] :
( ( times_times_int @ ( number_number_of_int @ V_10 ) @ ( plus_plus_int @ B_78 @ C_46 ) )
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ V_10 ) @ B_78 ) @ ( times_times_int @ ( number_number_of_int @ V_10 ) @ C_46 ) ) ) ).
thf(fact_1462_right__distrib__number__of,axiom,
! [V_10: int,B_78: code_code_numeral,C_46: code_code_numeral] :
( ( times_1655362735umeral @ ( number1443263063umeral @ V_10 ) @ ( plus_p1627245867umeral @ B_78 @ C_46 ) )
= ( plus_p1627245867umeral @ ( times_1655362735umeral @ ( number1443263063umeral @ V_10 ) @ B_78 ) @ ( times_1655362735umeral @ ( number1443263063umeral @ V_10 ) @ C_46 ) ) ) ).
thf(fact_1463_left__distrib__number__of,axiom,
! [A_110: rat,B_77: rat,V_9: int] :
( ( times_times_rat @ ( plus_plus_rat @ A_110 @ B_77 ) @ ( number_number_of_rat @ V_9 ) )
= ( plus_plus_rat @ ( times_times_rat @ A_110 @ ( number_number_of_rat @ V_9 ) ) @ ( times_times_rat @ B_77 @ ( number_number_of_rat @ V_9 ) ) ) ) ).
thf(fact_1464_left__distrib__number__of,axiom,
! [A_110: quickcheck_code_int,B_77: quickcheck_code_int,V_9: int] :
( ( times_123202395de_int @ ( plus_p1446045655de_int @ A_110 @ B_77 ) @ ( number1226105091de_int @ V_9 ) )
= ( plus_p1446045655de_int @ ( times_123202395de_int @ A_110 @ ( number1226105091de_int @ V_9 ) ) @ ( times_123202395de_int @ B_77 @ ( number1226105091de_int @ V_9 ) ) ) ) ).
thf(fact_1465_left__distrib__number__of,axiom,
! [A_110: complex,B_77: complex,V_9: int] :
( ( times_times_complex @ ( plus_plus_complex @ A_110 @ B_77 ) @ ( number528085621omplex @ V_9 ) )
= ( plus_plus_complex @ ( times_times_complex @ A_110 @ ( number528085621omplex @ V_9 ) ) @ ( times_times_complex @ B_77 @ ( number528085621omplex @ V_9 ) ) ) ) ).
thf(fact_1466_left__distrib__number__of,axiom,
! [A_110: real,B_77: real,V_9: int] :
( ( times_times_real @ ( plus_plus_real @ A_110 @ B_77 ) @ ( number267125858f_real @ V_9 ) )
= ( plus_plus_real @ ( times_times_real @ A_110 @ ( number267125858f_real @ V_9 ) ) @ ( times_times_real @ B_77 @ ( number267125858f_real @ V_9 ) ) ) ) ).
thf(fact_1467_left__distrib__number__of,axiom,
! [A_110: nat,B_77: nat,V_9: int] :
( ( times_times_nat @ ( plus_plus_nat @ A_110 @ B_77 ) @ ( number_number_of_nat @ V_9 ) )
= ( plus_plus_nat @ ( times_times_nat @ A_110 @ ( number_number_of_nat @ V_9 ) ) @ ( times_times_nat @ B_77 @ ( number_number_of_nat @ V_9 ) ) ) ) ).
thf(fact_1468_left__distrib__number__of,axiom,
! [A_110: int,B_77: int,V_9: int] :
( ( times_times_int @ ( plus_plus_int @ A_110 @ B_77 ) @ ( number_number_of_int @ V_9 ) )
= ( plus_plus_int @ ( times_times_int @ A_110 @ ( number_number_of_int @ V_9 ) ) @ ( times_times_int @ B_77 @ ( number_number_of_int @ V_9 ) ) ) ) ).
thf(fact_1469_left__distrib__number__of,axiom,
! [A_110: code_code_numeral,B_77: code_code_numeral,V_9: int] :
( ( times_1655362735umeral @ ( plus_p1627245867umeral @ A_110 @ B_77 ) @ ( number1443263063umeral @ V_9 ) )
= ( plus_p1627245867umeral @ ( times_1655362735umeral @ A_110 @ ( number1443263063umeral @ V_9 ) ) @ ( times_1655362735umeral @ B_77 @ ( number1443263063umeral @ V_9 ) ) ) ) ).
thf(fact_1470_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_18: rat] :
( ( plus_plus_rat @ M_18 @ M_18 )
= ( times_times_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) @ M_18 ) ) ).
thf(fact_1471_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_18: quickcheck_code_int] :
( ( plus_p1446045655de_int @ M_18 @ M_18 )
= ( times_123202395de_int @ ( plus_p1446045655de_int @ one_on1684967323de_int @ one_on1684967323de_int ) @ M_18 ) ) ).
thf(fact_1472_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_18: complex] :
( ( plus_plus_complex @ M_18 @ M_18 )
= ( times_times_complex @ ( plus_plus_complex @ one_one_complex @ one_one_complex ) @ M_18 ) ) ).
thf(fact_1473_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_18: code_code_numeral] :
( ( plus_p1627245867umeral @ M_18 @ M_18 )
= ( times_1655362735umeral @ ( plus_p1627245867umeral @ one_on1645066479umeral @ one_on1645066479umeral ) @ M_18 ) ) ).
thf(fact_1474_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_18: real] :
( ( plus_plus_real @ M_18 @ M_18 )
= ( times_times_real @ ( plus_plus_real @ one_one_real @ one_one_real ) @ M_18 ) ) ).
thf(fact_1475_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_18: nat] :
( ( plus_plus_nat @ M_18 @ M_18 )
= ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) @ M_18 ) ) ).
thf(fact_1476_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_18: int] :
( ( plus_plus_int @ M_18 @ M_18 )
= ( times_times_int @ ( plus_plus_int @ one_one_int @ one_one_int ) @ M_18 ) ) ).
thf(fact_1477_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_17: rat,A_109: rat] :
( ( plus_plus_rat @ M_17 @ ( times_times_rat @ A_109 @ M_17 ) )
= ( times_times_rat @ ( plus_plus_rat @ A_109 @ one_one_rat ) @ M_17 ) ) ).
thf(fact_1478_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_17: quickcheck_code_int,A_109: quickcheck_code_int] :
( ( plus_p1446045655de_int @ M_17 @ ( times_123202395de_int @ A_109 @ M_17 ) )
= ( times_123202395de_int @ ( plus_p1446045655de_int @ A_109 @ one_on1684967323de_int ) @ M_17 ) ) ).
thf(fact_1479_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_17: complex,A_109: complex] :
( ( plus_plus_complex @ M_17 @ ( times_times_complex @ A_109 @ M_17 ) )
= ( times_times_complex @ ( plus_plus_complex @ A_109 @ one_one_complex ) @ M_17 ) ) ).
thf(fact_1480_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_17: code_code_numeral,A_109: code_code_numeral] :
( ( plus_p1627245867umeral @ M_17 @ ( times_1655362735umeral @ A_109 @ M_17 ) )
= ( times_1655362735umeral @ ( plus_p1627245867umeral @ A_109 @ one_on1645066479umeral ) @ M_17 ) ) ).
thf(fact_1481_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_17: real,A_109: real] :
( ( plus_plus_real @ M_17 @ ( times_times_real @ A_109 @ M_17 ) )
= ( times_times_real @ ( plus_plus_real @ A_109 @ one_one_real ) @ M_17 ) ) ).
thf(fact_1482_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_17: nat,A_109: nat] :
( ( plus_plus_nat @ M_17 @ ( times_times_nat @ A_109 @ M_17 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_109 @ one_one_nat ) @ M_17 ) ) ).
thf(fact_1483_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_17: int,A_109: int] :
( ( plus_plus_int @ M_17 @ ( times_times_int @ A_109 @ M_17 ) )
= ( times_times_int @ ( plus_plus_int @ A_109 @ one_one_int ) @ M_17 ) ) ).
thf(fact_1484_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_108: rat,M_16: rat] :
( ( plus_plus_rat @ ( times_times_rat @ A_108 @ M_16 ) @ M_16 )
= ( times_times_rat @ ( plus_plus_rat @ A_108 @ one_one_rat ) @ M_16 ) ) ).
thf(fact_1485_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_108: quickcheck_code_int,M_16: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( times_123202395de_int @ A_108 @ M_16 ) @ M_16 )
= ( times_123202395de_int @ ( plus_p1446045655de_int @ A_108 @ one_on1684967323de_int ) @ M_16 ) ) ).
thf(fact_1486_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_108: complex,M_16: complex] :
( ( plus_plus_complex @ ( times_times_complex @ A_108 @ M_16 ) @ M_16 )
= ( times_times_complex @ ( plus_plus_complex @ A_108 @ one_one_complex ) @ M_16 ) ) ).
thf(fact_1487_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_108: code_code_numeral,M_16: code_code_numeral] :
( ( plus_p1627245867umeral @ ( times_1655362735umeral @ A_108 @ M_16 ) @ M_16 )
= ( times_1655362735umeral @ ( plus_p1627245867umeral @ A_108 @ one_on1645066479umeral ) @ M_16 ) ) ).
thf(fact_1488_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_108: real,M_16: real] :
( ( plus_plus_real @ ( times_times_real @ A_108 @ M_16 ) @ M_16 )
= ( times_times_real @ ( plus_plus_real @ A_108 @ one_one_real ) @ M_16 ) ) ).
thf(fact_1489_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_108: nat,M_16: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_108 @ M_16 ) @ M_16 )
= ( times_times_nat @ ( plus_plus_nat @ A_108 @ one_one_nat ) @ M_16 ) ) ).
thf(fact_1490_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_108: int,M_16: int] :
( ( plus_plus_int @ ( times_times_int @ A_108 @ M_16 ) @ M_16 )
= ( times_times_int @ ( plus_plus_int @ A_108 @ one_one_int ) @ M_16 ) ) ).
thf(fact_1491_power__add,axiom,
! [A_107: complex,M_15: nat,N_25: nat] :
( ( power_power_complex @ A_107 @ ( plus_plus_nat @ M_15 @ N_25 ) )
= ( times_times_complex @ ( power_power_complex @ A_107 @ M_15 ) @ ( power_power_complex @ A_107 @ N_25 ) ) ) ).
thf(fact_1492_power__add,axiom,
! [A_107: real,M_15: nat,N_25: nat] :
( ( power_power_real @ A_107 @ ( plus_plus_nat @ M_15 @ N_25 ) )
= ( times_times_real @ ( power_power_real @ A_107 @ M_15 ) @ ( power_power_real @ A_107 @ N_25 ) ) ) ).
thf(fact_1493_power__add,axiom,
! [A_107: nat,M_15: nat,N_25: nat] :
( ( power_power_nat @ A_107 @ ( plus_plus_nat @ M_15 @ N_25 ) )
= ( times_times_nat @ ( power_power_nat @ A_107 @ M_15 ) @ ( power_power_nat @ A_107 @ N_25 ) ) ) ).
thf(fact_1494_power__add,axiom,
! [A_107: int,M_15: nat,N_25: nat] :
( ( power_power_int @ A_107 @ ( plus_plus_nat @ M_15 @ N_25 ) )
= ( times_times_int @ ( power_power_int @ A_107 @ M_15 ) @ ( power_power_int @ A_107 @ N_25 ) ) ) ).
thf(fact_1495_power__add,axiom,
! [A_107: rat,M_15: nat,N_25: nat] :
( ( power_power_rat @ A_107 @ ( plus_plus_nat @ M_15 @ N_25 ) )
= ( times_times_rat @ ( power_power_rat @ A_107 @ M_15 ) @ ( power_power_rat @ A_107 @ N_25 ) ) ) ).
thf(fact_1496_power__add,axiom,
! [A_107: quickcheck_code_int,M_15: nat,N_25: nat] :
( ( power_881366806de_int @ A_107 @ ( plus_plus_nat @ M_15 @ N_25 ) )
= ( times_123202395de_int @ ( power_881366806de_int @ A_107 @ M_15 ) @ ( power_881366806de_int @ A_107 @ N_25 ) ) ) ).
thf(fact_1497_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_71: complex,P_8: nat,Q_6: nat] :
( ( times_times_complex @ ( power_power_complex @ X_71 @ P_8 ) @ ( power_power_complex @ X_71 @ Q_6 ) )
= ( power_power_complex @ X_71 @ ( plus_plus_nat @ P_8 @ Q_6 ) ) ) ).
thf(fact_1498_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_71: real,P_8: nat,Q_6: nat] :
( ( times_times_real @ ( power_power_real @ X_71 @ P_8 ) @ ( power_power_real @ X_71 @ Q_6 ) )
= ( power_power_real @ X_71 @ ( plus_plus_nat @ P_8 @ Q_6 ) ) ) ).
thf(fact_1499_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_71: nat,P_8: nat,Q_6: nat] :
( ( times_times_nat @ ( power_power_nat @ X_71 @ P_8 ) @ ( power_power_nat @ X_71 @ Q_6 ) )
= ( power_power_nat @ X_71 @ ( plus_plus_nat @ P_8 @ Q_6 ) ) ) ).
thf(fact_1500_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_71: int,P_8: nat,Q_6: nat] :
( ( times_times_int @ ( power_power_int @ X_71 @ P_8 ) @ ( power_power_int @ X_71 @ Q_6 ) )
= ( power_power_int @ X_71 @ ( plus_plus_nat @ P_8 @ Q_6 ) ) ) ).
thf(fact_1501_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_71: rat,P_8: nat,Q_6: nat] :
( ( times_times_rat @ ( power_power_rat @ X_71 @ P_8 ) @ ( power_power_rat @ X_71 @ Q_6 ) )
= ( power_power_rat @ X_71 @ ( plus_plus_nat @ P_8 @ Q_6 ) ) ) ).
thf(fact_1502_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_71: quickcheck_code_int,P_8: nat,Q_6: nat] :
( ( times_123202395de_int @ ( power_881366806de_int @ X_71 @ P_8 ) @ ( power_881366806de_int @ X_71 @ Q_6 ) )
= ( power_881366806de_int @ X_71 @ ( plus_plus_nat @ P_8 @ Q_6 ) ) ) ).
thf(fact_1503_pos__zmult__pos,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_int @ zero_zero_int @ A_1 )
=> ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_1 @ B_1 ) )
=> ( ord_less_int @ zero_zero_int @ B_1 ) ) ) ).
thf(fact_1504_zmult__zless__mono2,axiom,
! [K: int,I: int,J: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
thf(fact_1505_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_1506_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_1507_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_1508_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_1509_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_70: complex,N_24: nat] :
( ( power_power_complex @ X_70 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_24 ) )
= ( times_times_complex @ ( power_power_complex @ X_70 @ N_24 ) @ ( power_power_complex @ X_70 @ N_24 ) ) ) ).
thf(fact_1510_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_70: real,N_24: nat] :
( ( power_power_real @ X_70 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_24 ) )
= ( times_times_real @ ( power_power_real @ X_70 @ N_24 ) @ ( power_power_real @ X_70 @ N_24 ) ) ) ).
thf(fact_1511_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_70: nat,N_24: nat] :
( ( power_power_nat @ X_70 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_24 ) )
= ( times_times_nat @ ( power_power_nat @ X_70 @ N_24 ) @ ( power_power_nat @ X_70 @ N_24 ) ) ) ).
thf(fact_1512_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_70: int,N_24: nat] :
( ( power_power_int @ X_70 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_24 ) )
= ( times_times_int @ ( power_power_int @ X_70 @ N_24 ) @ ( power_power_int @ X_70 @ N_24 ) ) ) ).
thf(fact_1513_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_70: rat,N_24: nat] :
( ( power_power_rat @ X_70 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_24 ) )
= ( times_times_rat @ ( power_power_rat @ X_70 @ N_24 ) @ ( power_power_rat @ X_70 @ N_24 ) ) ) ).
thf(fact_1514_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_70: quickcheck_code_int,N_24: nat] :
( ( power_881366806de_int @ X_70 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_24 ) )
= ( times_123202395de_int @ ( power_881366806de_int @ X_70 @ N_24 ) @ ( power_881366806de_int @ X_70 @ N_24 ) ) ) ).
thf(fact_1515_mult__left__le__imp__le,axiom,
! [C_45: rat,A_106: rat,B_76: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C_45 @ A_106 ) @ ( times_times_rat @ C_45 @ B_76 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C_45 )
=> ( ord_less_eq_rat @ A_106 @ B_76 ) ) ) ).
thf(fact_1516_mult__left__le__imp__le,axiom,
! [C_45: quickcheck_code_int,A_106: quickcheck_code_int,B_76: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( times_123202395de_int @ C_45 @ A_106 ) @ ( times_123202395de_int @ C_45 @ B_76 ) )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ C_45 )
=> ( ord_le258702272de_int @ A_106 @ B_76 ) ) ) ).
thf(fact_1517_mult__left__le__imp__le,axiom,
! [C_45: code_code_numeral,A_106: code_code_numeral,B_76: code_code_numeral] :
( ( ord_le565307924umeral @ ( times_1655362735umeral @ C_45 @ A_106 ) @ ( times_1655362735umeral @ C_45 @ B_76 ) )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ C_45 )
=> ( ord_le565307924umeral @ A_106 @ B_76 ) ) ) ).
thf(fact_1518_mult__left__le__imp__le,axiom,
! [C_45: real,A_106: real,B_76: real] :
( ( ord_less_eq_real @ ( times_times_real @ C_45 @ A_106 ) @ ( times_times_real @ C_45 @ B_76 ) )
=> ( ( ord_less_real @ zero_zero_real @ C_45 )
=> ( ord_less_eq_real @ A_106 @ B_76 ) ) ) ).
thf(fact_1519_mult__left__le__imp__le,axiom,
! [C_45: nat,A_106: nat,B_76: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C_45 @ A_106 ) @ ( times_times_nat @ C_45 @ B_76 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C_45 )
=> ( ord_less_eq_nat @ A_106 @ B_76 ) ) ) ).
thf(fact_1520_mult__left__le__imp__le,axiom,
! [C_45: int,A_106: int,B_76: int] :
( ( ord_less_eq_int @ ( times_times_int @ C_45 @ A_106 ) @ ( times_times_int @ C_45 @ B_76 ) )
=> ( ( ord_less_int @ zero_zero_int @ C_45 )
=> ( ord_less_eq_int @ A_106 @ B_76 ) ) ) ).
thf(fact_1521_mult__right__le__imp__le,axiom,
! [A_105: rat,C_44: rat,B_75: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A_105 @ C_44 ) @ ( times_times_rat @ B_75 @ C_44 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C_44 )
=> ( ord_less_eq_rat @ A_105 @ B_75 ) ) ) ).
thf(fact_1522_mult__right__le__imp__le,axiom,
! [A_105: quickcheck_code_int,C_44: quickcheck_code_int,B_75: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( times_123202395de_int @ A_105 @ C_44 ) @ ( times_123202395de_int @ B_75 @ C_44 ) )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ C_44 )
=> ( ord_le258702272de_int @ A_105 @ B_75 ) ) ) ).
thf(fact_1523_mult__right__le__imp__le,axiom,
! [A_105: code_code_numeral,C_44: code_code_numeral,B_75: code_code_numeral] :
( ( ord_le565307924umeral @ ( times_1655362735umeral @ A_105 @ C_44 ) @ ( times_1655362735umeral @ B_75 @ C_44 ) )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ C_44 )
=> ( ord_le565307924umeral @ A_105 @ B_75 ) ) ) ).
thf(fact_1524_mult__right__le__imp__le,axiom,
! [A_105: real,C_44: real,B_75: real] :
( ( ord_less_eq_real @ ( times_times_real @ A_105 @ C_44 ) @ ( times_times_real @ B_75 @ C_44 ) )
=> ( ( ord_less_real @ zero_zero_real @ C_44 )
=> ( ord_less_eq_real @ A_105 @ B_75 ) ) ) ).
thf(fact_1525_mult__right__le__imp__le,axiom,
! [A_105: nat,C_44: nat,B_75: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A_105 @ C_44 ) @ ( times_times_nat @ B_75 @ C_44 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C_44 )
=> ( ord_less_eq_nat @ A_105 @ B_75 ) ) ) ).
thf(fact_1526_mult__right__le__imp__le,axiom,
! [A_105: int,C_44: int,B_75: int] :
( ( ord_less_eq_int @ ( times_times_int @ A_105 @ C_44 ) @ ( times_times_int @ B_75 @ C_44 ) )
=> ( ( ord_less_int @ zero_zero_int @ C_44 )
=> ( ord_less_eq_int @ A_105 @ B_75 ) ) ) ).
thf(fact_1527_mult__less__imp__less__left,axiom,
! [C_43: rat,A_104: rat,B_74: rat] :
( ( ord_less_rat @ ( times_times_rat @ C_43 @ A_104 ) @ ( times_times_rat @ C_43 @ B_74 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_43 )
=> ( ord_less_rat @ A_104 @ B_74 ) ) ) ).
thf(fact_1528_mult__less__imp__less__left,axiom,
! [C_43: quickcheck_code_int,A_104: quickcheck_code_int,B_74: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( times_123202395de_int @ C_43 @ A_104 ) @ ( times_123202395de_int @ C_43 @ B_74 ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_43 )
=> ( ord_le1860547276de_int @ A_104 @ B_74 ) ) ) ).
thf(fact_1529_mult__less__imp__less__left,axiom,
! [C_43: code_code_numeral,A_104: code_code_numeral,B_74: code_code_numeral] :
( ( ord_le1304079648umeral @ ( times_1655362735umeral @ C_43 @ A_104 ) @ ( times_1655362735umeral @ C_43 @ B_74 ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_43 )
=> ( ord_le1304079648umeral @ A_104 @ B_74 ) ) ) ).
thf(fact_1530_mult__less__imp__less__left,axiom,
! [C_43: real,A_104: real,B_74: real] :
( ( ord_less_real @ ( times_times_real @ C_43 @ A_104 ) @ ( times_times_real @ C_43 @ B_74 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_43 )
=> ( ord_less_real @ A_104 @ B_74 ) ) ) ).
thf(fact_1531_mult__less__imp__less__left,axiom,
! [C_43: nat,A_104: nat,B_74: nat] :
( ( ord_less_nat @ ( times_times_nat @ C_43 @ A_104 ) @ ( times_times_nat @ C_43 @ B_74 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_43 )
=> ( ord_less_nat @ A_104 @ B_74 ) ) ) ).
thf(fact_1532_mult__less__imp__less__left,axiom,
! [C_43: int,A_104: int,B_74: int] :
( ( ord_less_int @ ( times_times_int @ C_43 @ A_104 ) @ ( times_times_int @ C_43 @ B_74 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_43 )
=> ( ord_less_int @ A_104 @ B_74 ) ) ) ).
thf(fact_1533_mult__left__less__imp__less,axiom,
! [C_42: rat,A_103: rat,B_73: rat] :
( ( ord_less_rat @ ( times_times_rat @ C_42 @ A_103 ) @ ( times_times_rat @ C_42 @ B_73 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_42 )
=> ( ord_less_rat @ A_103 @ B_73 ) ) ) ).
thf(fact_1534_mult__left__less__imp__less,axiom,
! [C_42: quickcheck_code_int,A_103: quickcheck_code_int,B_73: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( times_123202395de_int @ C_42 @ A_103 ) @ ( times_123202395de_int @ C_42 @ B_73 ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_42 )
=> ( ord_le1860547276de_int @ A_103 @ B_73 ) ) ) ).
thf(fact_1535_mult__left__less__imp__less,axiom,
! [C_42: code_code_numeral,A_103: code_code_numeral,B_73: code_code_numeral] :
( ( ord_le1304079648umeral @ ( times_1655362735umeral @ C_42 @ A_103 ) @ ( times_1655362735umeral @ C_42 @ B_73 ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_42 )
=> ( ord_le1304079648umeral @ A_103 @ B_73 ) ) ) ).
thf(fact_1536_mult__left__less__imp__less,axiom,
! [C_42: real,A_103: real,B_73: real] :
( ( ord_less_real @ ( times_times_real @ C_42 @ A_103 ) @ ( times_times_real @ C_42 @ B_73 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_42 )
=> ( ord_less_real @ A_103 @ B_73 ) ) ) ).
thf(fact_1537_mult__left__less__imp__less,axiom,
! [C_42: nat,A_103: nat,B_73: nat] :
( ( ord_less_nat @ ( times_times_nat @ C_42 @ A_103 ) @ ( times_times_nat @ C_42 @ B_73 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_42 )
=> ( ord_less_nat @ A_103 @ B_73 ) ) ) ).
thf(fact_1538_mult__left__less__imp__less,axiom,
! [C_42: int,A_103: int,B_73: int] :
( ( ord_less_int @ ( times_times_int @ C_42 @ A_103 ) @ ( times_times_int @ C_42 @ B_73 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_42 )
=> ( ord_less_int @ A_103 @ B_73 ) ) ) ).
thf(fact_1539_mult__less__imp__less__right,axiom,
! [A_102: rat,C_41: rat,B_72: rat] :
( ( ord_less_rat @ ( times_times_rat @ A_102 @ C_41 ) @ ( times_times_rat @ B_72 @ C_41 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_41 )
=> ( ord_less_rat @ A_102 @ B_72 ) ) ) ).
thf(fact_1540_mult__less__imp__less__right,axiom,
! [A_102: quickcheck_code_int,C_41: quickcheck_code_int,B_72: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( times_123202395de_int @ A_102 @ C_41 ) @ ( times_123202395de_int @ B_72 @ C_41 ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_41 )
=> ( ord_le1860547276de_int @ A_102 @ B_72 ) ) ) ).
thf(fact_1541_mult__less__imp__less__right,axiom,
! [A_102: code_code_numeral,C_41: code_code_numeral,B_72: code_code_numeral] :
( ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_102 @ C_41 ) @ ( times_1655362735umeral @ B_72 @ C_41 ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_41 )
=> ( ord_le1304079648umeral @ A_102 @ B_72 ) ) ) ).
thf(fact_1542_mult__less__imp__less__right,axiom,
! [A_102: real,C_41: real,B_72: real] :
( ( ord_less_real @ ( times_times_real @ A_102 @ C_41 ) @ ( times_times_real @ B_72 @ C_41 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_41 )
=> ( ord_less_real @ A_102 @ B_72 ) ) ) ).
thf(fact_1543_mult__less__imp__less__right,axiom,
! [A_102: nat,C_41: nat,B_72: nat] :
( ( ord_less_nat @ ( times_times_nat @ A_102 @ C_41 ) @ ( times_times_nat @ B_72 @ C_41 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_41 )
=> ( ord_less_nat @ A_102 @ B_72 ) ) ) ).
thf(fact_1544_mult__less__imp__less__right,axiom,
! [A_102: int,C_41: int,B_72: int] :
( ( ord_less_int @ ( times_times_int @ A_102 @ C_41 ) @ ( times_times_int @ B_72 @ C_41 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_41 )
=> ( ord_less_int @ A_102 @ B_72 ) ) ) ).
thf(fact_1545_mult__right__less__imp__less,axiom,
! [A_101: rat,C_40: rat,B_71: rat] :
( ( ord_less_rat @ ( times_times_rat @ A_101 @ C_40 ) @ ( times_times_rat @ B_71 @ C_40 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_40 )
=> ( ord_less_rat @ A_101 @ B_71 ) ) ) ).
thf(fact_1546_mult__right__less__imp__less,axiom,
! [A_101: quickcheck_code_int,C_40: quickcheck_code_int,B_71: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( times_123202395de_int @ A_101 @ C_40 ) @ ( times_123202395de_int @ B_71 @ C_40 ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_40 )
=> ( ord_le1860547276de_int @ A_101 @ B_71 ) ) ) ).
thf(fact_1547_mult__right__less__imp__less,axiom,
! [A_101: code_code_numeral,C_40: code_code_numeral,B_71: code_code_numeral] :
( ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_101 @ C_40 ) @ ( times_1655362735umeral @ B_71 @ C_40 ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_40 )
=> ( ord_le1304079648umeral @ A_101 @ B_71 ) ) ) ).
thf(fact_1548_mult__right__less__imp__less,axiom,
! [A_101: real,C_40: real,B_71: real] :
( ( ord_less_real @ ( times_times_real @ A_101 @ C_40 ) @ ( times_times_real @ B_71 @ C_40 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_40 )
=> ( ord_less_real @ A_101 @ B_71 ) ) ) ).
thf(fact_1549_mult__right__less__imp__less,axiom,
! [A_101: nat,C_40: nat,B_71: nat] :
( ( ord_less_nat @ ( times_times_nat @ A_101 @ C_40 ) @ ( times_times_nat @ B_71 @ C_40 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_40 )
=> ( ord_less_nat @ A_101 @ B_71 ) ) ) ).
thf(fact_1550_mult__right__less__imp__less,axiom,
! [A_101: int,C_40: int,B_71: int] :
( ( ord_less_int @ ( times_times_int @ A_101 @ C_40 ) @ ( times_times_int @ B_71 @ C_40 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_40 )
=> ( ord_less_int @ A_101 @ B_71 ) ) ) ).
thf(fact_1551_mult__le__less__imp__less,axiom,
! [C_39: rat,D_24: rat,A_100: rat,B_70: rat] :
( ( ord_less_eq_rat @ A_100 @ B_70 )
=> ( ( ord_less_rat @ C_39 @ D_24 )
=> ( ( ord_less_rat @ zero_zero_rat @ A_100 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_39 )
=> ( ord_less_rat @ ( times_times_rat @ A_100 @ C_39 ) @ ( times_times_rat @ B_70 @ D_24 ) ) ) ) ) ) ).
thf(fact_1552_mult__le__less__imp__less,axiom,
! [C_39: quickcheck_code_int,D_24: quickcheck_code_int,A_100: quickcheck_code_int,B_70: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_100 @ B_70 )
=> ( ( ord_le1860547276de_int @ C_39 @ D_24 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_100 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_39 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_100 @ C_39 ) @ ( times_123202395de_int @ B_70 @ D_24 ) ) ) ) ) ) ).
thf(fact_1553_mult__le__less__imp__less,axiom,
! [C_39: code_code_numeral,D_24: code_code_numeral,A_100: code_code_numeral,B_70: code_code_numeral] :
( ( ord_le565307924umeral @ A_100 @ B_70 )
=> ( ( ord_le1304079648umeral @ C_39 @ D_24 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_100 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_39 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_100 @ C_39 ) @ ( times_1655362735umeral @ B_70 @ D_24 ) ) ) ) ) ) ).
thf(fact_1554_mult__le__less__imp__less,axiom,
! [C_39: real,D_24: real,A_100: real,B_70: real] :
( ( ord_less_eq_real @ A_100 @ B_70 )
=> ( ( ord_less_real @ C_39 @ D_24 )
=> ( ( ord_less_real @ zero_zero_real @ A_100 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_39 )
=> ( ord_less_real @ ( times_times_real @ A_100 @ C_39 ) @ ( times_times_real @ B_70 @ D_24 ) ) ) ) ) ) ).
thf(fact_1555_mult__le__less__imp__less,axiom,
! [C_39: nat,D_24: nat,A_100: nat,B_70: nat] :
( ( ord_less_eq_nat @ A_100 @ B_70 )
=> ( ( ord_less_nat @ C_39 @ D_24 )
=> ( ( ord_less_nat @ zero_zero_nat @ A_100 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_39 )
=> ( ord_less_nat @ ( times_times_nat @ A_100 @ C_39 ) @ ( times_times_nat @ B_70 @ D_24 ) ) ) ) ) ) ).
thf(fact_1556_mult__le__less__imp__less,axiom,
! [C_39: int,D_24: int,A_100: int,B_70: int] :
( ( ord_less_eq_int @ A_100 @ B_70 )
=> ( ( ord_less_int @ C_39 @ D_24 )
=> ( ( ord_less_int @ zero_zero_int @ A_100 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_39 )
=> ( ord_less_int @ ( times_times_int @ A_100 @ C_39 ) @ ( times_times_int @ B_70 @ D_24 ) ) ) ) ) ) ).
thf(fact_1557_mult__less__le__imp__less,axiom,
! [C_38: rat,D_23: rat,A_99: rat,B_69: rat] :
( ( ord_less_rat @ A_99 @ B_69 )
=> ( ( ord_less_eq_rat @ C_38 @ D_23 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A_99 )
=> ( ( ord_less_rat @ zero_zero_rat @ C_38 )
=> ( ord_less_rat @ ( times_times_rat @ A_99 @ C_38 ) @ ( times_times_rat @ B_69 @ D_23 ) ) ) ) ) ) ).
thf(fact_1558_mult__less__le__imp__less,axiom,
! [C_38: quickcheck_code_int,D_23: quickcheck_code_int,A_99: quickcheck_code_int,B_69: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_99 @ B_69 )
=> ( ( ord_le258702272de_int @ C_38 @ D_23 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_99 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ C_38 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_99 @ C_38 ) @ ( times_123202395de_int @ B_69 @ D_23 ) ) ) ) ) ) ).
thf(fact_1559_mult__less__le__imp__less,axiom,
! [C_38: code_code_numeral,D_23: code_code_numeral,A_99: code_code_numeral,B_69: code_code_numeral] :
( ( ord_le1304079648umeral @ A_99 @ B_69 )
=> ( ( ord_le565307924umeral @ C_38 @ D_23 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_99 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ C_38 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_99 @ C_38 ) @ ( times_1655362735umeral @ B_69 @ D_23 ) ) ) ) ) ) ).
thf(fact_1560_mult__less__le__imp__less,axiom,
! [C_38: real,D_23: real,A_99: real,B_69: real] :
( ( ord_less_real @ A_99 @ B_69 )
=> ( ( ord_less_eq_real @ C_38 @ D_23 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_99 )
=> ( ( ord_less_real @ zero_zero_real @ C_38 )
=> ( ord_less_real @ ( times_times_real @ A_99 @ C_38 ) @ ( times_times_real @ B_69 @ D_23 ) ) ) ) ) ) ).
thf(fact_1561_mult__less__le__imp__less,axiom,
! [C_38: nat,D_23: nat,A_99: nat,B_69: nat] :
( ( ord_less_nat @ A_99 @ B_69 )
=> ( ( ord_less_eq_nat @ C_38 @ D_23 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_99 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_38 )
=> ( ord_less_nat @ ( times_times_nat @ A_99 @ C_38 ) @ ( times_times_nat @ B_69 @ D_23 ) ) ) ) ) ) ).
thf(fact_1562_mult__less__le__imp__less,axiom,
! [C_38: int,D_23: int,A_99: int,B_69: int] :
( ( ord_less_int @ A_99 @ B_69 )
=> ( ( ord_less_eq_int @ C_38 @ D_23 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_99 )
=> ( ( ord_less_int @ zero_zero_int @ C_38 )
=> ( ord_less_int @ ( times_times_int @ A_99 @ C_38 ) @ ( times_times_int @ B_69 @ D_23 ) ) ) ) ) ) ).
thf(fact_1563_mult__strict__mono_H,axiom,
! [C_37: rat,D_22: rat,A_98: rat,B_68: rat] :
( ( ord_less_rat @ A_98 @ B_68 )
=> ( ( ord_less_rat @ C_37 @ D_22 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A_98 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_37 )
=> ( ord_less_rat @ ( times_times_rat @ A_98 @ C_37 ) @ ( times_times_rat @ B_68 @ D_22 ) ) ) ) ) ) ).
thf(fact_1564_mult__strict__mono_H,axiom,
! [C_37: quickcheck_code_int,D_22: quickcheck_code_int,A_98: quickcheck_code_int,B_68: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_98 @ B_68 )
=> ( ( ord_le1860547276de_int @ C_37 @ D_22 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_98 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_37 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_98 @ C_37 ) @ ( times_123202395de_int @ B_68 @ D_22 ) ) ) ) ) ) ).
thf(fact_1565_mult__strict__mono_H,axiom,
! [C_37: code_code_numeral,D_22: code_code_numeral,A_98: code_code_numeral,B_68: code_code_numeral] :
( ( ord_le1304079648umeral @ A_98 @ B_68 )
=> ( ( ord_le1304079648umeral @ C_37 @ D_22 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_98 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_37 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_98 @ C_37 ) @ ( times_1655362735umeral @ B_68 @ D_22 ) ) ) ) ) ) ).
thf(fact_1566_mult__strict__mono_H,axiom,
! [C_37: real,D_22: real,A_98: real,B_68: real] :
( ( ord_less_real @ A_98 @ B_68 )
=> ( ( ord_less_real @ C_37 @ D_22 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_98 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_37 )
=> ( ord_less_real @ ( times_times_real @ A_98 @ C_37 ) @ ( times_times_real @ B_68 @ D_22 ) ) ) ) ) ) ).
thf(fact_1567_mult__strict__mono_H,axiom,
! [C_37: nat,D_22: nat,A_98: nat,B_68: nat] :
( ( ord_less_nat @ A_98 @ B_68 )
=> ( ( ord_less_nat @ C_37 @ D_22 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_98 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_37 )
=> ( ord_less_nat @ ( times_times_nat @ A_98 @ C_37 ) @ ( times_times_nat @ B_68 @ D_22 ) ) ) ) ) ) ).
thf(fact_1568_mult__strict__mono_H,axiom,
! [C_37: int,D_22: int,A_98: int,B_68: int] :
( ( ord_less_int @ A_98 @ B_68 )
=> ( ( ord_less_int @ C_37 @ D_22 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_98 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_37 )
=> ( ord_less_int @ ( times_times_int @ A_98 @ C_37 ) @ ( times_times_int @ B_68 @ D_22 ) ) ) ) ) ) ).
thf(fact_1569_mult__strict__mono,axiom,
! [C_36: rat,D_21: rat,A_97: rat,B_67: rat] :
( ( ord_less_rat @ A_97 @ B_67 )
=> ( ( ord_less_rat @ C_36 @ D_21 )
=> ( ( ord_less_rat @ zero_zero_rat @ B_67 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_36 )
=> ( ord_less_rat @ ( times_times_rat @ A_97 @ C_36 ) @ ( times_times_rat @ B_67 @ D_21 ) ) ) ) ) ) ).
thf(fact_1570_mult__strict__mono,axiom,
! [C_36: quickcheck_code_int,D_21: quickcheck_code_int,A_97: quickcheck_code_int,B_67: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_97 @ B_67 )
=> ( ( ord_le1860547276de_int @ C_36 @ D_21 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ B_67 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_36 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_97 @ C_36 ) @ ( times_123202395de_int @ B_67 @ D_21 ) ) ) ) ) ) ).
thf(fact_1571_mult__strict__mono,axiom,
! [C_36: code_code_numeral,D_21: code_code_numeral,A_97: code_code_numeral,B_67: code_code_numeral] :
( ( ord_le1304079648umeral @ A_97 @ B_67 )
=> ( ( ord_le1304079648umeral @ C_36 @ D_21 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ B_67 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_36 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_97 @ C_36 ) @ ( times_1655362735umeral @ B_67 @ D_21 ) ) ) ) ) ) ).
thf(fact_1572_mult__strict__mono,axiom,
! [C_36: real,D_21: real,A_97: real,B_67: real] :
( ( ord_less_real @ A_97 @ B_67 )
=> ( ( ord_less_real @ C_36 @ D_21 )
=> ( ( ord_less_real @ zero_zero_real @ B_67 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_36 )
=> ( ord_less_real @ ( times_times_real @ A_97 @ C_36 ) @ ( times_times_real @ B_67 @ D_21 ) ) ) ) ) ) ).
thf(fact_1573_mult__strict__mono,axiom,
! [C_36: nat,D_21: nat,A_97: nat,B_67: nat] :
( ( ord_less_nat @ A_97 @ B_67 )
=> ( ( ord_less_nat @ C_36 @ D_21 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_67 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_36 )
=> ( ord_less_nat @ ( times_times_nat @ A_97 @ C_36 ) @ ( times_times_nat @ B_67 @ D_21 ) ) ) ) ) ) ).
thf(fact_1574_mult__strict__mono,axiom,
! [C_36: int,D_21: int,A_97: int,B_67: int] :
( ( ord_less_int @ A_97 @ B_67 )
=> ( ( ord_less_int @ C_36 @ D_21 )
=> ( ( ord_less_int @ zero_zero_int @ B_67 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_36 )
=> ( ord_less_int @ ( times_times_int @ A_97 @ C_36 ) @ ( times_times_int @ B_67 @ D_21 ) ) ) ) ) ) ).
thf(fact_1575_mult__le__cancel__left__neg,axiom,
! [A_96: rat,B_66: rat,C_35: rat] :
( ( ord_less_rat @ C_35 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C_35 @ A_96 ) @ ( times_times_rat @ C_35 @ B_66 ) )
<=> ( ord_less_eq_rat @ B_66 @ A_96 ) ) ) ).
thf(fact_1576_mult__le__cancel__left__neg,axiom,
! [A_96: real,B_66: real,C_35: real] :
( ( ord_less_real @ C_35 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C_35 @ A_96 ) @ ( times_times_real @ C_35 @ B_66 ) )
<=> ( ord_less_eq_real @ B_66 @ A_96 ) ) ) ).
thf(fact_1577_mult__le__cancel__left__neg,axiom,
! [A_96: int,B_66: int,C_35: int] :
( ( ord_less_int @ C_35 @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C_35 @ A_96 ) @ ( times_times_int @ C_35 @ B_66 ) )
<=> ( ord_less_eq_int @ B_66 @ A_96 ) ) ) ).
thf(fact_1578_mult__le__cancel__left__pos,axiom,
! [A_95: rat,B_65: rat,C_34: rat] :
( ( ord_less_rat @ zero_zero_rat @ C_34 )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C_34 @ A_95 ) @ ( times_times_rat @ C_34 @ B_65 ) )
<=> ( ord_less_eq_rat @ A_95 @ B_65 ) ) ) ).
thf(fact_1579_mult__le__cancel__left__pos,axiom,
! [A_95: real,B_65: real,C_34: real] :
( ( ord_less_real @ zero_zero_real @ C_34 )
=> ( ( ord_less_eq_real @ ( times_times_real @ C_34 @ A_95 ) @ ( times_times_real @ C_34 @ B_65 ) )
<=> ( ord_less_eq_real @ A_95 @ B_65 ) ) ) ).
thf(fact_1580_mult__le__cancel__left__pos,axiom,
! [A_95: int,B_65: int,C_34: int] :
( ( ord_less_int @ zero_zero_int @ C_34 )
=> ( ( ord_less_eq_int @ ( times_times_int @ C_34 @ A_95 ) @ ( times_times_int @ C_34 @ B_65 ) )
<=> ( ord_less_eq_int @ A_95 @ B_65 ) ) ) ).
thf(fact_1581_sum__squares__ge__zero,axiom,
! [X_69: rat,Y_60: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X_69 @ X_69 ) @ ( times_times_rat @ Y_60 @ Y_60 ) ) ) ).
thf(fact_1582_sum__squares__ge__zero,axiom,
! [X_69: real,Y_60: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X_69 @ X_69 ) @ ( times_times_real @ Y_60 @ Y_60 ) ) ) ).
thf(fact_1583_sum__squares__ge__zero,axiom,
! [X_69: int,Y_60: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X_69 @ X_69 ) @ ( times_times_int @ Y_60 @ Y_60 ) ) ) ).
thf(fact_1584_sum__squares__le__zero__iff,axiom,
! [X_68: rat,Y_59: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X_68 @ X_68 ) @ ( times_times_rat @ Y_59 @ Y_59 ) ) @ zero_zero_rat )
<=> ( ( X_68 = zero_zero_rat )
& ( Y_59 = zero_zero_rat ) ) ) ).
thf(fact_1585_sum__squares__le__zero__iff,axiom,
! [X_68: real,Y_59: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X_68 @ X_68 ) @ ( times_times_real @ Y_59 @ Y_59 ) ) @ zero_zero_real )
<=> ( ( X_68 = zero_zero_real )
& ( Y_59 = zero_zero_real ) ) ) ).
thf(fact_1586_sum__squares__le__zero__iff,axiom,
! [X_68: int,Y_59: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X_68 @ X_68 ) @ ( times_times_int @ Y_59 @ Y_59 ) ) @ zero_zero_int )
<=> ( ( X_68 = zero_zero_int )
& ( Y_59 = zero_zero_int ) ) ) ).
thf(fact_1587_not__sum__squares__lt__zero,axiom,
! [X_67: rat,Y_58: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X_67 @ X_67 ) @ ( times_times_rat @ Y_58 @ Y_58 ) ) @ zero_zero_rat ) ).
thf(fact_1588_not__sum__squares__lt__zero,axiom,
! [X_67: real,Y_58: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X_67 @ X_67 ) @ ( times_times_real @ Y_58 @ Y_58 ) ) @ zero_zero_real ) ).
thf(fact_1589_not__sum__squares__lt__zero,axiom,
! [X_67: int,Y_58: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X_67 @ X_67 ) @ ( times_times_int @ Y_58 @ Y_58 ) ) @ zero_zero_int ) ).
thf(fact_1590_sum__squares__gt__zero__iff,axiom,
! [X_66: rat,Y_57: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X_66 @ X_66 ) @ ( times_times_rat @ Y_57 @ Y_57 ) ) )
<=> ( ( X_66 != zero_zero_rat )
| ( Y_57 != zero_zero_rat ) ) ) ).
thf(fact_1591_sum__squares__gt__zero__iff,axiom,
! [X_66: real,Y_57: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X_66 @ X_66 ) @ ( times_times_real @ Y_57 @ Y_57 ) ) )
<=> ( ( X_66 != zero_zero_real )
| ( Y_57 != zero_zero_real ) ) ) ).
thf(fact_1592_sum__squares__gt__zero__iff,axiom,
! [X_66: int,Y_57: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X_66 @ X_66 ) @ ( times_times_int @ Y_57 @ Y_57 ) ) )
<=> ( ( X_66 != zero_zero_int )
| ( Y_57 != zero_zero_int ) ) ) ).
thf(fact_1593_mult__right__le__one__le,axiom,
! [Y_56: rat,X_65: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X_65 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y_56 )
=> ( ( ord_less_eq_rat @ Y_56 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ X_65 @ Y_56 ) @ X_65 ) ) ) ) ).
thf(fact_1594_mult__right__le__one__le,axiom,
! [Y_56: real,X_65: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_65 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_56 )
=> ( ( ord_less_eq_real @ Y_56 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X_65 @ Y_56 ) @ X_65 ) ) ) ) ).
thf(fact_1595_mult__right__le__one__le,axiom,
! [Y_56: int,X_65: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_65 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_56 )
=> ( ( ord_less_eq_int @ Y_56 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X_65 @ Y_56 ) @ X_65 ) ) ) ) ).
thf(fact_1596_mult__left__le__one__le,axiom,
! [Y_55: rat,X_64: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X_64 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y_55 )
=> ( ( ord_less_eq_rat @ Y_55 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Y_55 @ X_64 ) @ X_64 ) ) ) ) ).
thf(fact_1597_mult__left__le__one__le,axiom,
! [Y_55: real,X_64: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_64 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_55 )
=> ( ( ord_less_eq_real @ Y_55 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y_55 @ X_64 ) @ X_64 ) ) ) ) ).
thf(fact_1598_mult__left__le__one__le,axiom,
! [Y_55: int,X_64: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_64 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_55 )
=> ( ( ord_less_eq_int @ Y_55 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y_55 @ X_64 ) @ X_64 ) ) ) ) ).
thf(fact_1599_power__gt1__lemma,axiom,
! [N_23: nat,A_94: rat] :
( ( ord_less_rat @ one_one_rat @ A_94 )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A_94 @ ( power_power_rat @ A_94 @ N_23 ) ) ) ) ).
thf(fact_1600_power__gt1__lemma,axiom,
! [N_23: nat,A_94: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ A_94 )
=> ( ord_le1860547276de_int @ one_on1684967323de_int @ ( times_123202395de_int @ A_94 @ ( power_881366806de_int @ A_94 @ N_23 ) ) ) ) ).
thf(fact_1601_power__gt1__lemma,axiom,
! [N_23: nat,A_94: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ A_94 )
=> ( ord_le1304079648umeral @ one_on1645066479umeral @ ( times_1655362735umeral @ A_94 @ ( power_2100829034umeral @ A_94 @ N_23 ) ) ) ) ).
thf(fact_1602_power__gt1__lemma,axiom,
! [N_23: nat,A_94: real] :
( ( ord_less_real @ one_one_real @ A_94 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A_94 @ ( power_power_real @ A_94 @ N_23 ) ) ) ) ).
thf(fact_1603_power__gt1__lemma,axiom,
! [N_23: nat,A_94: nat] :
( ( ord_less_nat @ one_one_nat @ A_94 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A_94 @ ( power_power_nat @ A_94 @ N_23 ) ) ) ) ).
thf(fact_1604_power__gt1__lemma,axiom,
! [N_23: nat,A_94: int] :
( ( ord_less_int @ one_one_int @ A_94 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A_94 @ ( power_power_int @ A_94 @ N_23 ) ) ) ) ).
thf(fact_1605_power__less__power__Suc,axiom,
! [N_22: nat,A_93: rat] :
( ( ord_less_rat @ one_one_rat @ A_93 )
=> ( ord_less_rat @ ( power_power_rat @ A_93 @ N_22 ) @ ( times_times_rat @ A_93 @ ( power_power_rat @ A_93 @ N_22 ) ) ) ) ).
thf(fact_1606_power__less__power__Suc,axiom,
! [N_22: nat,A_93: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ A_93 )
=> ( ord_le1860547276de_int @ ( power_881366806de_int @ A_93 @ N_22 ) @ ( times_123202395de_int @ A_93 @ ( power_881366806de_int @ A_93 @ N_22 ) ) ) ) ).
thf(fact_1607_power__less__power__Suc,axiom,
! [N_22: nat,A_93: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ A_93 )
=> ( ord_le1304079648umeral @ ( power_2100829034umeral @ A_93 @ N_22 ) @ ( times_1655362735umeral @ A_93 @ ( power_2100829034umeral @ A_93 @ N_22 ) ) ) ) ).
thf(fact_1608_power__less__power__Suc,axiom,
! [N_22: nat,A_93: real] :
( ( ord_less_real @ one_one_real @ A_93 )
=> ( ord_less_real @ ( power_power_real @ A_93 @ N_22 ) @ ( times_times_real @ A_93 @ ( power_power_real @ A_93 @ N_22 ) ) ) ) ).
thf(fact_1609_power__less__power__Suc,axiom,
! [N_22: nat,A_93: nat] :
( ( ord_less_nat @ one_one_nat @ A_93 )
=> ( ord_less_nat @ ( power_power_nat @ A_93 @ N_22 ) @ ( times_times_nat @ A_93 @ ( power_power_nat @ A_93 @ N_22 ) ) ) ) ).
thf(fact_1610_power__less__power__Suc,axiom,
! [N_22: nat,A_93: int] :
( ( ord_less_int @ one_one_int @ A_93 )
=> ( ord_less_int @ ( power_power_int @ A_93 @ N_22 ) @ ( times_times_int @ A_93 @ ( power_power_int @ A_93 @ N_22 ) ) ) ) ).
thf(fact_1611_mult__numeral__1__right,axiom,
! [A_92: rat] :
( ( times_times_rat @ A_92 @ ( number_number_of_rat @ ( bit1 @ pls ) ) )
= A_92 ) ).
thf(fact_1612_mult__numeral__1__right,axiom,
! [A_92: complex] :
( ( times_times_complex @ A_92 @ ( number528085621omplex @ ( bit1 @ pls ) ) )
= A_92 ) ).
thf(fact_1613_mult__numeral__1__right,axiom,
! [A_92: real] :
( ( times_times_real @ A_92 @ ( number267125858f_real @ ( bit1 @ pls ) ) )
= A_92 ) ).
thf(fact_1614_mult__numeral__1__right,axiom,
! [A_92: int] :
( ( times_times_int @ A_92 @ ( number_number_of_int @ ( bit1 @ pls ) ) )
= A_92 ) ).
thf(fact_1615_mult__numeral__1,axiom,
! [A_91: rat] :
( ( times_times_rat @ ( number_number_of_rat @ ( bit1 @ pls ) ) @ A_91 )
= A_91 ) ).
thf(fact_1616_mult__numeral__1,axiom,
! [A_91: complex] :
( ( times_times_complex @ ( number528085621omplex @ ( bit1 @ pls ) ) @ A_91 )
= A_91 ) ).
thf(fact_1617_mult__numeral__1,axiom,
! [A_91: real] :
( ( times_times_real @ ( number267125858f_real @ ( bit1 @ pls ) ) @ A_91 )
= A_91 ) ).
thf(fact_1618_mult__numeral__1,axiom,
! [A_91: int] :
( ( times_times_int @ ( number_number_of_int @ ( bit1 @ pls ) ) @ A_91 )
= A_91 ) ).
thf(fact_1619_mult__Bit1,axiom,
! [K: int,L: int] :
( ( times_times_int @ ( bit1 @ K ) @ L )
= ( plus_plus_int @ ( bit0 @ ( times_times_int @ K @ L ) ) @ L ) ) ).
thf(fact_1620_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_1621_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_1622_linorder__le__cases,axiom,
! [X_63: rat,Y_54: rat] :
( ~ ( ord_less_eq_rat @ X_63 @ Y_54 )
=> ( ord_less_eq_rat @ Y_54 @ X_63 ) ) ).
thf(fact_1623_linorder__le__cases,axiom,
! [X_63: quickcheck_code_int,Y_54: quickcheck_code_int] :
( ~ ( ord_le258702272de_int @ X_63 @ Y_54 )
=> ( ord_le258702272de_int @ Y_54 @ X_63 ) ) ).
thf(fact_1624_linorder__le__cases,axiom,
! [X_63: real,Y_54: real] :
( ~ ( ord_less_eq_real @ X_63 @ Y_54 )
=> ( ord_less_eq_real @ Y_54 @ X_63 ) ) ).
thf(fact_1625_linorder__le__cases,axiom,
! [X_63: nat,Y_54: nat] :
( ~ ( ord_less_eq_nat @ X_63 @ Y_54 )
=> ( ord_less_eq_nat @ Y_54 @ X_63 ) ) ).
thf(fact_1626_linorder__le__cases,axiom,
! [X_63: int,Y_54: int] :
( ~ ( ord_less_eq_int @ X_63 @ Y_54 )
=> ( ord_less_eq_int @ Y_54 @ X_63 ) ) ).
thf(fact_1627_le__funE,axiom,
! [X_62: int,F_6: int > $o,G_4: int > $o] :
( ( ord_less_eq_int_o @ F_6 @ G_4 )
=> ( ord_less_eq_o @ ( F_6 @ X_62 ) @ ( G_4 @ X_62 ) ) ) ).
thf(fact_1628_le__funE,axiom,
! [X_62: nat,F_6: nat > $o,G_4: nat > $o] :
( ( ord_less_eq_nat_o @ F_6 @ G_4 )
=> ( ord_less_eq_o @ ( F_6 @ X_62 ) @ ( G_4 @ X_62 ) ) ) ).
thf(fact_1629_xt1_I6_J,axiom,
! [Z_17: rat,Y_53: rat,X_61: rat] :
( ( ord_less_eq_rat @ Y_53 @ X_61 )
=> ( ( ord_less_eq_rat @ Z_17 @ Y_53 )
=> ( ord_less_eq_rat @ Z_17 @ X_61 ) ) ) ).
thf(fact_1630_xt1_I6_J,axiom,
! [Z_17: int > $o,Y_53: int > $o,X_61: int > $o] :
( ( ord_less_eq_int_o @ Y_53 @ X_61 )
=> ( ( ord_less_eq_int_o @ Z_17 @ Y_53 )
=> ( ord_less_eq_int_o @ Z_17 @ X_61 ) ) ) ).
thf(fact_1631_xt1_I6_J,axiom,
! [Z_17: nat > $o,Y_53: nat > $o,X_61: nat > $o] :
( ( ord_less_eq_nat_o @ Y_53 @ X_61 )
=> ( ( ord_less_eq_nat_o @ Z_17 @ Y_53 )
=> ( ord_less_eq_nat_o @ Z_17 @ X_61 ) ) ) ).
thf(fact_1632_xt1_I6_J,axiom,
! [Z_17: quickcheck_code_int,Y_53: quickcheck_code_int,X_61: quickcheck_code_int] :
( ( ord_le258702272de_int @ Y_53 @ X_61 )
=> ( ( ord_le258702272de_int @ Z_17 @ Y_53 )
=> ( ord_le258702272de_int @ Z_17 @ X_61 ) ) ) ).
thf(fact_1633_xt1_I6_J,axiom,
! [Z_17: real,Y_53: real,X_61: real] :
( ( ord_less_eq_real @ Y_53 @ X_61 )
=> ( ( ord_less_eq_real @ Z_17 @ Y_53 )
=> ( ord_less_eq_real @ Z_17 @ X_61 ) ) ) ).
thf(fact_1634_xt1_I6_J,axiom,
! [Z_17: nat,Y_53: nat,X_61: nat] :
( ( ord_less_eq_nat @ Y_53 @ X_61 )
=> ( ( ord_less_eq_nat @ Z_17 @ Y_53 )
=> ( ord_less_eq_nat @ Z_17 @ X_61 ) ) ) ).
thf(fact_1635_xt1_I6_J,axiom,
! [Z_17: int,Y_53: int,X_61: int] :
( ( ord_less_eq_int @ Y_53 @ X_61 )
=> ( ( ord_less_eq_int @ Z_17 @ Y_53 )
=> ( ord_less_eq_int @ Z_17 @ X_61 ) ) ) ).
thf(fact_1636_xt1_I5_J,axiom,
! [Y_52: rat,X_60: rat] :
( ( ord_less_eq_rat @ Y_52 @ X_60 )
=> ( ( ord_less_eq_rat @ X_60 @ Y_52 )
=> ( X_60 = Y_52 ) ) ) ).
thf(fact_1637_xt1_I5_J,axiom,
! [Y_52: int > $o,X_60: int > $o] :
( ( ord_less_eq_int_o @ Y_52 @ X_60 )
=> ( ( ord_less_eq_int_o @ X_60 @ Y_52 )
=> ( X_60 = Y_52 ) ) ) ).
thf(fact_1638_xt1_I5_J,axiom,
! [Y_52: nat > $o,X_60: nat > $o] :
( ( ord_less_eq_nat_o @ Y_52 @ X_60 )
=> ( ( ord_less_eq_nat_o @ X_60 @ Y_52 )
=> ( X_60 = Y_52 ) ) ) ).
thf(fact_1639_xt1_I5_J,axiom,
! [Y_52: quickcheck_code_int,X_60: quickcheck_code_int] :
( ( ord_le258702272de_int @ Y_52 @ X_60 )
=> ( ( ord_le258702272de_int @ X_60 @ Y_52 )
=> ( X_60 = Y_52 ) ) ) ).
thf(fact_1640_xt1_I5_J,axiom,
! [Y_52: real,X_60: real] :
( ( ord_less_eq_real @ Y_52 @ X_60 )
=> ( ( ord_less_eq_real @ X_60 @ Y_52 )
=> ( X_60 = Y_52 ) ) ) ).
thf(fact_1641_xt1_I5_J,axiom,
! [Y_52: nat,X_60: nat] :
( ( ord_less_eq_nat @ Y_52 @ X_60 )
=> ( ( ord_less_eq_nat @ X_60 @ Y_52 )
=> ( X_60 = Y_52 ) ) ) ).
thf(fact_1642_xt1_I5_J,axiom,
! [Y_52: int,X_60: int] :
( ( ord_less_eq_int @ Y_52 @ X_60 )
=> ( ( ord_less_eq_int @ X_60 @ Y_52 )
=> ( X_60 = Y_52 ) ) ) ).
thf(fact_1643_order__trans,axiom,
! [Z_16: rat,X_59: rat,Y_51: rat] :
( ( ord_less_eq_rat @ X_59 @ Y_51 )
=> ( ( ord_less_eq_rat @ Y_51 @ Z_16 )
=> ( ord_less_eq_rat @ X_59 @ Z_16 ) ) ) ).
thf(fact_1644_order__trans,axiom,
! [Z_16: int > $o,X_59: int > $o,Y_51: int > $o] :
( ( ord_less_eq_int_o @ X_59 @ Y_51 )
=> ( ( ord_less_eq_int_o @ Y_51 @ Z_16 )
=> ( ord_less_eq_int_o @ X_59 @ Z_16 ) ) ) ).
thf(fact_1645_order__trans,axiom,
! [Z_16: nat > $o,X_59: nat > $o,Y_51: nat > $o] :
( ( ord_less_eq_nat_o @ X_59 @ Y_51 )
=> ( ( ord_less_eq_nat_o @ Y_51 @ Z_16 )
=> ( ord_less_eq_nat_o @ X_59 @ Z_16 ) ) ) ).
thf(fact_1646_order__trans,axiom,
! [Z_16: quickcheck_code_int,X_59: quickcheck_code_int,Y_51: quickcheck_code_int] :
( ( ord_le258702272de_int @ X_59 @ Y_51 )
=> ( ( ord_le258702272de_int @ Y_51 @ Z_16 )
=> ( ord_le258702272de_int @ X_59 @ Z_16 ) ) ) ).
thf(fact_1647_order__trans,axiom,
! [Z_16: real,X_59: real,Y_51: real] :
( ( ord_less_eq_real @ X_59 @ Y_51 )
=> ( ( ord_less_eq_real @ Y_51 @ Z_16 )
=> ( ord_less_eq_real @ X_59 @ Z_16 ) ) ) ).
thf(fact_1648_order__trans,axiom,
! [Z_16: nat,X_59: nat,Y_51: nat] :
( ( ord_less_eq_nat @ X_59 @ Y_51 )
=> ( ( ord_less_eq_nat @ Y_51 @ Z_16 )
=> ( ord_less_eq_nat @ X_59 @ Z_16 ) ) ) ).
thf(fact_1649_order__trans,axiom,
! [Z_16: int,X_59: int,Y_51: int] :
( ( ord_less_eq_int @ X_59 @ Y_51 )
=> ( ( ord_less_eq_int @ Y_51 @ Z_16 )
=> ( ord_less_eq_int @ X_59 @ Z_16 ) ) ) ).
thf(fact_1650_order__antisym,axiom,
! [X_58: rat,Y_50: rat] :
( ( ord_less_eq_rat @ X_58 @ Y_50 )
=> ( ( ord_less_eq_rat @ Y_50 @ X_58 )
=> ( X_58 = Y_50 ) ) ) ).
thf(fact_1651_order__antisym,axiom,
! [X_58: int > $o,Y_50: int > $o] :
( ( ord_less_eq_int_o @ X_58 @ Y_50 )
=> ( ( ord_less_eq_int_o @ Y_50 @ X_58 )
=> ( X_58 = Y_50 ) ) ) ).
thf(fact_1652_order__antisym,axiom,
! [X_58: nat > $o,Y_50: nat > $o] :
( ( ord_less_eq_nat_o @ X_58 @ Y_50 )
=> ( ( ord_less_eq_nat_o @ Y_50 @ X_58 )
=> ( X_58 = Y_50 ) ) ) ).
thf(fact_1653_order__antisym,axiom,
! [X_58: quickcheck_code_int,Y_50: quickcheck_code_int] :
( ( ord_le258702272de_int @ X_58 @ Y_50 )
=> ( ( ord_le258702272de_int @ Y_50 @ X_58 )
=> ( X_58 = Y_50 ) ) ) ).
thf(fact_1654_order__antisym,axiom,
! [X_58: real,Y_50: real] :
( ( ord_less_eq_real @ X_58 @ Y_50 )
=> ( ( ord_less_eq_real @ Y_50 @ X_58 )
=> ( X_58 = Y_50 ) ) ) ).
thf(fact_1655_order__antisym,axiom,
! [X_58: nat,Y_50: nat] :
( ( ord_less_eq_nat @ X_58 @ Y_50 )
=> ( ( ord_less_eq_nat @ Y_50 @ X_58 )
=> ( X_58 = Y_50 ) ) ) ).
thf(fact_1656_order__antisym,axiom,
! [X_58: int,Y_50: int] :
( ( ord_less_eq_int @ X_58 @ Y_50 )
=> ( ( ord_less_eq_int @ Y_50 @ X_58 )
=> ( X_58 = Y_50 ) ) ) ).
thf(fact_1657_xt1_I4_J,axiom,
! [C_33: rat,B_64: rat,A_90: rat] :
( ( ord_less_eq_rat @ B_64 @ A_90 )
=> ( ( B_64 = C_33 )
=> ( ord_less_eq_rat @ C_33 @ A_90 ) ) ) ).
thf(fact_1658_xt1_I4_J,axiom,
! [C_33: int > $o,B_64: int > $o,A_90: int > $o] :
( ( ord_less_eq_int_o @ B_64 @ A_90 )
=> ( ( B_64 = C_33 )
=> ( ord_less_eq_int_o @ C_33 @ A_90 ) ) ) ).
thf(fact_1659_xt1_I4_J,axiom,
! [C_33: nat > $o,B_64: nat > $o,A_90: nat > $o] :
( ( ord_less_eq_nat_o @ B_64 @ A_90 )
=> ( ( B_64 = C_33 )
=> ( ord_less_eq_nat_o @ C_33 @ A_90 ) ) ) ).
thf(fact_1660_xt1_I4_J,axiom,
! [C_33: quickcheck_code_int,B_64: quickcheck_code_int,A_90: quickcheck_code_int] :
( ( ord_le258702272de_int @ B_64 @ A_90 )
=> ( ( B_64 = C_33 )
=> ( ord_le258702272de_int @ C_33 @ A_90 ) ) ) ).
thf(fact_1661_xt1_I4_J,axiom,
! [C_33: real,B_64: real,A_90: real] :
( ( ord_less_eq_real @ B_64 @ A_90 )
=> ( ( B_64 = C_33 )
=> ( ord_less_eq_real @ C_33 @ A_90 ) ) ) ).
thf(fact_1662_xt1_I4_J,axiom,
! [C_33: nat,B_64: nat,A_90: nat] :
( ( ord_less_eq_nat @ B_64 @ A_90 )
=> ( ( B_64 = C_33 )
=> ( ord_less_eq_nat @ C_33 @ A_90 ) ) ) ).
thf(fact_1663_xt1_I4_J,axiom,
! [C_33: int,B_64: int,A_90: int] :
( ( ord_less_eq_int @ B_64 @ A_90 )
=> ( ( B_64 = C_33 )
=> ( ord_less_eq_int @ C_33 @ A_90 ) ) ) ).
thf(fact_1664_ord__le__eq__trans,axiom,
! [C_32: rat,A_89: rat,B_63: rat] :
( ( ord_less_eq_rat @ A_89 @ B_63 )
=> ( ( B_63 = C_32 )
=> ( ord_less_eq_rat @ A_89 @ C_32 ) ) ) ).
thf(fact_1665_ord__le__eq__trans,axiom,
! [C_32: int > $o,A_89: int > $o,B_63: int > $o] :
( ( ord_less_eq_int_o @ A_89 @ B_63 )
=> ( ( B_63 = C_32 )
=> ( ord_less_eq_int_o @ A_89 @ C_32 ) ) ) ).
thf(fact_1666_ord__le__eq__trans,axiom,
! [C_32: nat > $o,A_89: nat > $o,B_63: nat > $o] :
( ( ord_less_eq_nat_o @ A_89 @ B_63 )
=> ( ( B_63 = C_32 )
=> ( ord_less_eq_nat_o @ A_89 @ C_32 ) ) ) ).
thf(fact_1667_ord__le__eq__trans,axiom,
! [C_32: quickcheck_code_int,A_89: quickcheck_code_int,B_63: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_89 @ B_63 )
=> ( ( B_63 = C_32 )
=> ( ord_le258702272de_int @ A_89 @ C_32 ) ) ) ).
thf(fact_1668_ord__le__eq__trans,axiom,
! [C_32: real,A_89: real,B_63: real] :
( ( ord_less_eq_real @ A_89 @ B_63 )
=> ( ( B_63 = C_32 )
=> ( ord_less_eq_real @ A_89 @ C_32 ) ) ) ).
thf(fact_1669_ord__le__eq__trans,axiom,
! [C_32: nat,A_89: nat,B_63: nat] :
( ( ord_less_eq_nat @ A_89 @ B_63 )
=> ( ( B_63 = C_32 )
=> ( ord_less_eq_nat @ A_89 @ C_32 ) ) ) ).
thf(fact_1670_ord__le__eq__trans,axiom,
! [C_32: int,A_89: int,B_63: int] :
( ( ord_less_eq_int @ A_89 @ B_63 )
=> ( ( B_63 = C_32 )
=> ( ord_less_eq_int @ A_89 @ C_32 ) ) ) ).
thf(fact_1671_xt1_I3_J,axiom,
! [C_31: rat,A_88: rat,B_62: rat] :
( ( A_88 = B_62 )
=> ( ( ord_less_eq_rat @ C_31 @ B_62 )
=> ( ord_less_eq_rat @ C_31 @ A_88 ) ) ) ).
thf(fact_1672_xt1_I3_J,axiom,
! [C_31: int > $o,A_88: int > $o,B_62: int > $o] :
( ( A_88 = B_62 )
=> ( ( ord_less_eq_int_o @ C_31 @ B_62 )
=> ( ord_less_eq_int_o @ C_31 @ A_88 ) ) ) ).
thf(fact_1673_xt1_I3_J,axiom,
! [C_31: nat > $o,A_88: nat > $o,B_62: nat > $o] :
( ( A_88 = B_62 )
=> ( ( ord_less_eq_nat_o @ C_31 @ B_62 )
=> ( ord_less_eq_nat_o @ C_31 @ A_88 ) ) ) ).
thf(fact_1674_xt1_I3_J,axiom,
! [C_31: quickcheck_code_int,A_88: quickcheck_code_int,B_62: quickcheck_code_int] :
( ( A_88 = B_62 )
=> ( ( ord_le258702272de_int @ C_31 @ B_62 )
=> ( ord_le258702272de_int @ C_31 @ A_88 ) ) ) ).
thf(fact_1675_xt1_I3_J,axiom,
! [C_31: real,A_88: real,B_62: real] :
( ( A_88 = B_62 )
=> ( ( ord_less_eq_real @ C_31 @ B_62 )
=> ( ord_less_eq_real @ C_31 @ A_88 ) ) ) ).
thf(fact_1676_xt1_I3_J,axiom,
! [C_31: nat,A_88: nat,B_62: nat] :
( ( A_88 = B_62 )
=> ( ( ord_less_eq_nat @ C_31 @ B_62 )
=> ( ord_less_eq_nat @ C_31 @ A_88 ) ) ) ).
thf(fact_1677_xt1_I3_J,axiom,
! [C_31: int,A_88: int,B_62: int] :
( ( A_88 = B_62 )
=> ( ( ord_less_eq_int @ C_31 @ B_62 )
=> ( ord_less_eq_int @ C_31 @ A_88 ) ) ) ).
thf(fact_1678_ord__eq__le__trans,axiom,
! [C_30: rat,A_87: rat,B_61: rat] :
( ( A_87 = B_61 )
=> ( ( ord_less_eq_rat @ B_61 @ C_30 )
=> ( ord_less_eq_rat @ A_87 @ C_30 ) ) ) ).
thf(fact_1679_ord__eq__le__trans,axiom,
! [C_30: int > $o,A_87: int > $o,B_61: int > $o] :
( ( A_87 = B_61 )
=> ( ( ord_less_eq_int_o @ B_61 @ C_30 )
=> ( ord_less_eq_int_o @ A_87 @ C_30 ) ) ) ).
thf(fact_1680_ord__eq__le__trans,axiom,
! [C_30: nat > $o,A_87: nat > $o,B_61: nat > $o] :
( ( A_87 = B_61 )
=> ( ( ord_less_eq_nat_o @ B_61 @ C_30 )
=> ( ord_less_eq_nat_o @ A_87 @ C_30 ) ) ) ).
thf(fact_1681_ord__eq__le__trans,axiom,
! [C_30: quickcheck_code_int,A_87: quickcheck_code_int,B_61: quickcheck_code_int] :
( ( A_87 = B_61 )
=> ( ( ord_le258702272de_int @ B_61 @ C_30 )
=> ( ord_le258702272de_int @ A_87 @ C_30 ) ) ) ).
thf(fact_1682_ord__eq__le__trans,axiom,
! [C_30: real,A_87: real,B_61: real] :
( ( A_87 = B_61 )
=> ( ( ord_less_eq_real @ B_61 @ C_30 )
=> ( ord_less_eq_real @ A_87 @ C_30 ) ) ) ).
thf(fact_1683_ord__eq__le__trans,axiom,
! [C_30: nat,A_87: nat,B_61: nat] :
( ( A_87 = B_61 )
=> ( ( ord_less_eq_nat @ B_61 @ C_30 )
=> ( ord_less_eq_nat @ A_87 @ C_30 ) ) ) ).
thf(fact_1684_ord__eq__le__trans,axiom,
! [C_30: int,A_87: int,B_61: int] :
( ( A_87 = B_61 )
=> ( ( ord_less_eq_int @ B_61 @ C_30 )
=> ( ord_less_eq_int @ A_87 @ C_30 ) ) ) ).
thf(fact_1685_order__antisym__conv,axiom,
! [Y_49: rat,X_57: rat] :
( ( ord_less_eq_rat @ Y_49 @ X_57 )
=> ( ( ord_less_eq_rat @ X_57 @ Y_49 )
<=> ( X_57 = Y_49 ) ) ) ).
thf(fact_1686_order__antisym__conv,axiom,
! [Y_49: int > $o,X_57: int > $o] :
( ( ord_less_eq_int_o @ Y_49 @ X_57 )
=> ( ( ord_less_eq_int_o @ X_57 @ Y_49 )
<=> ( X_57 = Y_49 ) ) ) ).
thf(fact_1687_order__antisym__conv,axiom,
! [Y_49: nat > $o,X_57: nat > $o] :
( ( ord_less_eq_nat_o @ Y_49 @ X_57 )
=> ( ( ord_less_eq_nat_o @ X_57 @ Y_49 )
<=> ( X_57 = Y_49 ) ) ) ).
thf(fact_1688_order__antisym__conv,axiom,
! [Y_49: quickcheck_code_int,X_57: quickcheck_code_int] :
( ( ord_le258702272de_int @ Y_49 @ X_57 )
=> ( ( ord_le258702272de_int @ X_57 @ Y_49 )
<=> ( X_57 = Y_49 ) ) ) ).
thf(fact_1689_order__antisym__conv,axiom,
! [Y_49: real,X_57: real] :
( ( ord_less_eq_real @ Y_49 @ X_57 )
=> ( ( ord_less_eq_real @ X_57 @ Y_49 )
<=> ( X_57 = Y_49 ) ) ) ).
thf(fact_1690_order__antisym__conv,axiom,
! [Y_49: nat,X_57: nat] :
( ( ord_less_eq_nat @ Y_49 @ X_57 )
=> ( ( ord_less_eq_nat @ X_57 @ Y_49 )
<=> ( X_57 = Y_49 ) ) ) ).
thf(fact_1691_order__antisym__conv,axiom,
! [Y_49: int,X_57: int] :
( ( ord_less_eq_int @ Y_49 @ X_57 )
=> ( ( ord_less_eq_int @ X_57 @ Y_49 )
<=> ( X_57 = Y_49 ) ) ) ).
thf(fact_1692_le__funD,axiom,
! [X_56: int,F_5: int > $o,G_3: int > $o] :
( ( ord_less_eq_int_o @ F_5 @ G_3 )
=> ( ord_less_eq_o @ ( F_5 @ X_56 ) @ ( G_3 @ X_56 ) ) ) ).
thf(fact_1693_le__funD,axiom,
! [X_56: nat,F_5: nat > $o,G_3: nat > $o] :
( ( ord_less_eq_nat_o @ F_5 @ G_3 )
=> ( ord_less_eq_o @ ( F_5 @ X_56 ) @ ( G_3 @ X_56 ) ) ) ).
thf(fact_1694_order__eq__refl,axiom,
! [X_55: rat,Y_48: rat] :
( ( X_55 = Y_48 )
=> ( ord_less_eq_rat @ X_55 @ Y_48 ) ) ).
thf(fact_1695_order__eq__refl,axiom,
! [X_55: int > $o,Y_48: int > $o] :
( ( X_55 = Y_48 )
=> ( ord_less_eq_int_o @ X_55 @ Y_48 ) ) ).
thf(fact_1696_order__eq__refl,axiom,
! [X_55: nat > $o,Y_48: nat > $o] :
( ( X_55 = Y_48 )
=> ( ord_less_eq_nat_o @ X_55 @ Y_48 ) ) ).
thf(fact_1697_order__eq__refl,axiom,
! [X_55: quickcheck_code_int,Y_48: quickcheck_code_int] :
( ( X_55 = Y_48 )
=> ( ord_le258702272de_int @ X_55 @ Y_48 ) ) ).
thf(fact_1698_order__eq__refl,axiom,
! [X_55: real,Y_48: real] :
( ( X_55 = Y_48 )
=> ( ord_less_eq_real @ X_55 @ Y_48 ) ) ).
thf(fact_1699_order__eq__refl,axiom,
! [X_55: nat,Y_48: nat] :
( ( X_55 = Y_48 )
=> ( ord_less_eq_nat @ X_55 @ Y_48 ) ) ).
thf(fact_1700_order__eq__refl,axiom,
! [X_55: int,Y_48: int] :
( ( X_55 = Y_48 )
=> ( ord_less_eq_int @ X_55 @ Y_48 ) ) ).
thf(fact_1701_order__eq__iff,axiom,
! [X_54: rat,Y_47: rat] :
( ( X_54 = Y_47 )
<=> ( ( ord_less_eq_rat @ X_54 @ Y_47 )
& ( ord_less_eq_rat @ Y_47 @ X_54 ) ) ) ).
thf(fact_1702_order__eq__iff,axiom,
! [X_54: int > $o,Y_47: int > $o] :
( ( X_54 = Y_47 )
<=> ( ( ord_less_eq_int_o @ X_54 @ Y_47 )
& ( ord_less_eq_int_o @ Y_47 @ X_54 ) ) ) ).
thf(fact_1703_order__eq__iff,axiom,
! [X_54: nat > $o,Y_47: nat > $o] :
( ( X_54 = Y_47 )
<=> ( ( ord_less_eq_nat_o @ X_54 @ Y_47 )
& ( ord_less_eq_nat_o @ Y_47 @ X_54 ) ) ) ).
thf(fact_1704_order__eq__iff,axiom,
! [X_54: quickcheck_code_int,Y_47: quickcheck_code_int] :
( ( X_54 = Y_47 )
<=> ( ( ord_le258702272de_int @ X_54 @ Y_47 )
& ( ord_le258702272de_int @ Y_47 @ X_54 ) ) ) ).
thf(fact_1705_order__eq__iff,axiom,
! [X_54: real,Y_47: real] :
( ( X_54 = Y_47 )
<=> ( ( ord_less_eq_real @ X_54 @ Y_47 )
& ( ord_less_eq_real @ Y_47 @ X_54 ) ) ) ).
thf(fact_1706_order__eq__iff,axiom,
! [X_54: nat,Y_47: nat] :
( ( X_54 = Y_47 )
<=> ( ( ord_less_eq_nat @ X_54 @ Y_47 )
& ( ord_less_eq_nat @ Y_47 @ X_54 ) ) ) ).
thf(fact_1707_order__eq__iff,axiom,
! [X_54: int,Y_47: int] :
( ( X_54 = Y_47 )
<=> ( ( ord_less_eq_int @ X_54 @ Y_47 )
& ( ord_less_eq_int @ Y_47 @ X_54 ) ) ) ).
thf(fact_1708_linorder__linear,axiom,
! [X_53: rat,Y_46: rat] :
( ( ord_less_eq_rat @ X_53 @ Y_46 )
| ( ord_less_eq_rat @ Y_46 @ X_53 ) ) ).
thf(fact_1709_linorder__linear,axiom,
! [X_53: quickcheck_code_int,Y_46: quickcheck_code_int] :
( ( ord_le258702272de_int @ X_53 @ Y_46 )
| ( ord_le258702272de_int @ Y_46 @ X_53 ) ) ).
thf(fact_1710_linorder__linear,axiom,
! [X_53: real,Y_46: real] :
( ( ord_less_eq_real @ X_53 @ Y_46 )
| ( ord_less_eq_real @ Y_46 @ X_53 ) ) ).
thf(fact_1711_linorder__linear,axiom,
! [X_53: nat,Y_46: nat] :
( ( ord_less_eq_nat @ X_53 @ Y_46 )
| ( ord_less_eq_nat @ Y_46 @ X_53 ) ) ).
thf(fact_1712_linorder__linear,axiom,
! [X_53: int,Y_46: int] :
( ( ord_less_eq_int @ X_53 @ Y_46 )
| ( ord_less_eq_int @ Y_46 @ X_53 ) ) ).
thf(fact_1713_le__fun__def,axiom,
! [F_4: int > $o,G_2: int > $o] :
( ( ord_less_eq_int_o @ F_4 @ G_2 )
<=> ! [X_1: int] : ( ord_less_eq_o @ ( F_4 @ X_1 ) @ ( G_2 @ X_1 ) ) ) ).
thf(fact_1714_le__fun__def,axiom,
! [F_4: nat > $o,G_2: nat > $o] :
( ( ord_less_eq_nat_o @ F_4 @ G_2 )
<=> ! [X_1: nat] : ( ord_less_eq_o @ ( F_4 @ X_1 ) @ ( G_2 @ X_1 ) ) ) ).
thf(fact_1715_linorder__cases,axiom,
! [X_52: rat,Y_45: rat] :
( ~ ( ord_less_rat @ X_52 @ Y_45 )
=> ( ( X_52 != Y_45 )
=> ( ord_less_rat @ Y_45 @ X_52 ) ) ) ).
thf(fact_1716_linorder__cases,axiom,
! [X_52: quickcheck_code_int,Y_45: quickcheck_code_int] :
( ~ ( ord_le1860547276de_int @ X_52 @ Y_45 )
=> ( ( X_52 != Y_45 )
=> ( ord_le1860547276de_int @ Y_45 @ X_52 ) ) ) ).
thf(fact_1717_linorder__cases,axiom,
! [X_52: real,Y_45: real] :
( ~ ( ord_less_real @ X_52 @ Y_45 )
=> ( ( X_52 != Y_45 )
=> ( ord_less_real @ Y_45 @ X_52 ) ) ) ).
thf(fact_1718_linorder__cases,axiom,
! [X_52: nat,Y_45: nat] :
( ~ ( ord_less_nat @ X_52 @ Y_45 )
=> ( ( X_52 != Y_45 )
=> ( ord_less_nat @ Y_45 @ X_52 ) ) ) ).
thf(fact_1719_linorder__cases,axiom,
! [X_52: int,Y_45: int] :
( ~ ( ord_less_int @ X_52 @ Y_45 )
=> ( ( X_52 != Y_45 )
=> ( ord_less_int @ Y_45 @ X_52 ) ) ) ).
thf(fact_1720_order__less__asym,axiom,
! [X_51: rat,Y_44: rat] :
( ( ord_less_rat @ X_51 @ Y_44 )
=> ~ ( ord_less_rat @ Y_44 @ X_51 ) ) ).
thf(fact_1721_order__less__asym,axiom,
! [X_51: int > $o,Y_44: int > $o] :
( ( ord_less_int_o @ X_51 @ Y_44 )
=> ~ ( ord_less_int_o @ Y_44 @ X_51 ) ) ).
thf(fact_1722_order__less__asym,axiom,
! [X_51: nat > $o,Y_44: nat > $o] :
( ( ord_less_nat_o @ X_51 @ Y_44 )
=> ~ ( ord_less_nat_o @ Y_44 @ X_51 ) ) ).
thf(fact_1723_order__less__asym,axiom,
! [X_51: quickcheck_code_int,Y_44: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_51 @ Y_44 )
=> ~ ( ord_le1860547276de_int @ Y_44 @ X_51 ) ) ).
thf(fact_1724_order__less__asym,axiom,
! [X_51: real,Y_44: real] :
( ( ord_less_real @ X_51 @ Y_44 )
=> ~ ( ord_less_real @ Y_44 @ X_51 ) ) ).
thf(fact_1725_order__less__asym,axiom,
! [X_51: nat,Y_44: nat] :
( ( ord_less_nat @ X_51 @ Y_44 )
=> ~ ( ord_less_nat @ Y_44 @ X_51 ) ) ).
thf(fact_1726_order__less__asym,axiom,
! [X_51: int,Y_44: int] :
( ( ord_less_int @ X_51 @ Y_44 )
=> ~ ( ord_less_int @ Y_44 @ X_51 ) ) ).
thf(fact_1727_xt1_I10_J,axiom,
! [Z_15: rat,Y_43: rat,X_50: rat] :
( ( ord_less_rat @ Y_43 @ X_50 )
=> ( ( ord_less_rat @ Z_15 @ Y_43 )
=> ( ord_less_rat @ Z_15 @ X_50 ) ) ) ).
thf(fact_1728_xt1_I10_J,axiom,
! [Z_15: int > $o,Y_43: int > $o,X_50: int > $o] :
( ( ord_less_int_o @ Y_43 @ X_50 )
=> ( ( ord_less_int_o @ Z_15 @ Y_43 )
=> ( ord_less_int_o @ Z_15 @ X_50 ) ) ) ).
thf(fact_1729_xt1_I10_J,axiom,
! [Z_15: nat > $o,Y_43: nat > $o,X_50: nat > $o] :
( ( ord_less_nat_o @ Y_43 @ X_50 )
=> ( ( ord_less_nat_o @ Z_15 @ Y_43 )
=> ( ord_less_nat_o @ Z_15 @ X_50 ) ) ) ).
thf(fact_1730_xt1_I10_J,axiom,
! [Z_15: quickcheck_code_int,Y_43: quickcheck_code_int,X_50: quickcheck_code_int] :
( ( ord_le1860547276de_int @ Y_43 @ X_50 )
=> ( ( ord_le1860547276de_int @ Z_15 @ Y_43 )
=> ( ord_le1860547276de_int @ Z_15 @ X_50 ) ) ) ).
thf(fact_1731_xt1_I10_J,axiom,
! [Z_15: real,Y_43: real,X_50: real] :
( ( ord_less_real @ Y_43 @ X_50 )
=> ( ( ord_less_real @ Z_15 @ Y_43 )
=> ( ord_less_real @ Z_15 @ X_50 ) ) ) ).
thf(fact_1732_xt1_I10_J,axiom,
! [Z_15: nat,Y_43: nat,X_50: nat] :
( ( ord_less_nat @ Y_43 @ X_50 )
=> ( ( ord_less_nat @ Z_15 @ Y_43 )
=> ( ord_less_nat @ Z_15 @ X_50 ) ) ) ).
thf(fact_1733_xt1_I10_J,axiom,
! [Z_15: int,Y_43: int,X_50: int] :
( ( ord_less_int @ Y_43 @ X_50 )
=> ( ( ord_less_int @ Z_15 @ Y_43 )
=> ( ord_less_int @ Z_15 @ X_50 ) ) ) ).
thf(fact_1734_order__less__trans,axiom,
! [Z_14: rat,X_49: rat,Y_42: rat] :
( ( ord_less_rat @ X_49 @ Y_42 )
=> ( ( ord_less_rat @ Y_42 @ Z_14 )
=> ( ord_less_rat @ X_49 @ Z_14 ) ) ) ).
thf(fact_1735_order__less__trans,axiom,
! [Z_14: int > $o,X_49: int > $o,Y_42: int > $o] :
( ( ord_less_int_o @ X_49 @ Y_42 )
=> ( ( ord_less_int_o @ Y_42 @ Z_14 )
=> ( ord_less_int_o @ X_49 @ Z_14 ) ) ) ).
thf(fact_1736_order__less__trans,axiom,
! [Z_14: nat > $o,X_49: nat > $o,Y_42: nat > $o] :
( ( ord_less_nat_o @ X_49 @ Y_42 )
=> ( ( ord_less_nat_o @ Y_42 @ Z_14 )
=> ( ord_less_nat_o @ X_49 @ Z_14 ) ) ) ).
thf(fact_1737_order__less__trans,axiom,
! [Z_14: quickcheck_code_int,X_49: quickcheck_code_int,Y_42: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_49 @ Y_42 )
=> ( ( ord_le1860547276de_int @ Y_42 @ Z_14 )
=> ( ord_le1860547276de_int @ X_49 @ Z_14 ) ) ) ).
thf(fact_1738_order__less__trans,axiom,
! [Z_14: real,X_49: real,Y_42: real] :
( ( ord_less_real @ X_49 @ Y_42 )
=> ( ( ord_less_real @ Y_42 @ Z_14 )
=> ( ord_less_real @ X_49 @ Z_14 ) ) ) ).
thf(fact_1739_order__less__trans,axiom,
! [Z_14: nat,X_49: nat,Y_42: nat] :
( ( ord_less_nat @ X_49 @ Y_42 )
=> ( ( ord_less_nat @ Y_42 @ Z_14 )
=> ( ord_less_nat @ X_49 @ Z_14 ) ) ) ).
thf(fact_1740_order__less__trans,axiom,
! [Z_14: int,X_49: int,Y_42: int] :
( ( ord_less_int @ X_49 @ Y_42 )
=> ( ( ord_less_int @ Y_42 @ Z_14 )
=> ( ord_less_int @ X_49 @ Z_14 ) ) ) ).
thf(fact_1741_xt1_I2_J,axiom,
! [C_29: rat,B_60: rat,A_86: rat] :
( ( ord_less_rat @ B_60 @ A_86 )
=> ( ( B_60 = C_29 )
=> ( ord_less_rat @ C_29 @ A_86 ) ) ) ).
thf(fact_1742_xt1_I2_J,axiom,
! [C_29: int > $o,B_60: int > $o,A_86: int > $o] :
( ( ord_less_int_o @ B_60 @ A_86 )
=> ( ( B_60 = C_29 )
=> ( ord_less_int_o @ C_29 @ A_86 ) ) ) ).
thf(fact_1743_xt1_I2_J,axiom,
! [C_29: nat > $o,B_60: nat > $o,A_86: nat > $o] :
( ( ord_less_nat_o @ B_60 @ A_86 )
=> ( ( B_60 = C_29 )
=> ( ord_less_nat_o @ C_29 @ A_86 ) ) ) ).
thf(fact_1744_xt1_I2_J,axiom,
! [C_29: quickcheck_code_int,B_60: quickcheck_code_int,A_86: quickcheck_code_int] :
( ( ord_le1860547276de_int @ B_60 @ A_86 )
=> ( ( B_60 = C_29 )
=> ( ord_le1860547276de_int @ C_29 @ A_86 ) ) ) ).
thf(fact_1745_xt1_I2_J,axiom,
! [C_29: real,B_60: real,A_86: real] :
( ( ord_less_real @ B_60 @ A_86 )
=> ( ( B_60 = C_29 )
=> ( ord_less_real @ C_29 @ A_86 ) ) ) ).
thf(fact_1746_xt1_I2_J,axiom,
! [C_29: nat,B_60: nat,A_86: nat] :
( ( ord_less_nat @ B_60 @ A_86 )
=> ( ( B_60 = C_29 )
=> ( ord_less_nat @ C_29 @ A_86 ) ) ) ).
thf(fact_1747_xt1_I2_J,axiom,
! [C_29: int,B_60: int,A_86: int] :
( ( ord_less_int @ B_60 @ A_86 )
=> ( ( B_60 = C_29 )
=> ( ord_less_int @ C_29 @ A_86 ) ) ) ).
thf(fact_1748_ord__less__eq__trans,axiom,
! [C_28: rat,A_85: rat,B_59: rat] :
( ( ord_less_rat @ A_85 @ B_59 )
=> ( ( B_59 = C_28 )
=> ( ord_less_rat @ A_85 @ C_28 ) ) ) ).
thf(fact_1749_ord__less__eq__trans,axiom,
! [C_28: int > $o,A_85: int > $o,B_59: int > $o] :
( ( ord_less_int_o @ A_85 @ B_59 )
=> ( ( B_59 = C_28 )
=> ( ord_less_int_o @ A_85 @ C_28 ) ) ) ).
thf(fact_1750_ord__less__eq__trans,axiom,
! [C_28: nat > $o,A_85: nat > $o,B_59: nat > $o] :
( ( ord_less_nat_o @ A_85 @ B_59 )
=> ( ( B_59 = C_28 )
=> ( ord_less_nat_o @ A_85 @ C_28 ) ) ) ).
thf(fact_1751_ord__less__eq__trans,axiom,
! [C_28: quickcheck_code_int,A_85: quickcheck_code_int,B_59: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_85 @ B_59 )
=> ( ( B_59 = C_28 )
=> ( ord_le1860547276de_int @ A_85 @ C_28 ) ) ) ).
thf(fact_1752_ord__less__eq__trans,axiom,
! [C_28: real,A_85: real,B_59: real] :
( ( ord_less_real @ A_85 @ B_59 )
=> ( ( B_59 = C_28 )
=> ( ord_less_real @ A_85 @ C_28 ) ) ) ).
thf(fact_1753_ord__less__eq__trans,axiom,
! [C_28: nat,A_85: nat,B_59: nat] :
( ( ord_less_nat @ A_85 @ B_59 )
=> ( ( B_59 = C_28 )
=> ( ord_less_nat @ A_85 @ C_28 ) ) ) ).
thf(fact_1754_ord__less__eq__trans,axiom,
! [C_28: int,A_85: int,B_59: int] :
( ( ord_less_int @ A_85 @ B_59 )
=> ( ( B_59 = C_28 )
=> ( ord_less_int @ A_85 @ C_28 ) ) ) ).
thf(fact_1755_xt1_I1_J,axiom,
! [C_27: rat,A_84: rat,B_58: rat] :
( ( A_84 = B_58 )
=> ( ( ord_less_rat @ C_27 @ B_58 )
=> ( ord_less_rat @ C_27 @ A_84 ) ) ) ).
thf(fact_1756_xt1_I1_J,axiom,
! [C_27: int > $o,A_84: int > $o,B_58: int > $o] :
( ( A_84 = B_58 )
=> ( ( ord_less_int_o @ C_27 @ B_58 )
=> ( ord_less_int_o @ C_27 @ A_84 ) ) ) ).
thf(fact_1757_xt1_I1_J,axiom,
! [C_27: nat > $o,A_84: nat > $o,B_58: nat > $o] :
( ( A_84 = B_58 )
=> ( ( ord_less_nat_o @ C_27 @ B_58 )
=> ( ord_less_nat_o @ C_27 @ A_84 ) ) ) ).
thf(fact_1758_xt1_I1_J,axiom,
! [C_27: quickcheck_code_int,A_84: quickcheck_code_int,B_58: quickcheck_code_int] :
( ( A_84 = B_58 )
=> ( ( ord_le1860547276de_int @ C_27 @ B_58 )
=> ( ord_le1860547276de_int @ C_27 @ A_84 ) ) ) ).
thf(fact_1759_xt1_I1_J,axiom,
! [C_27: real,A_84: real,B_58: real] :
( ( A_84 = B_58 )
=> ( ( ord_less_real @ C_27 @ B_58 )
=> ( ord_less_real @ C_27 @ A_84 ) ) ) ).
thf(fact_1760_xt1_I1_J,axiom,
! [C_27: nat,A_84: nat,B_58: nat] :
( ( A_84 = B_58 )
=> ( ( ord_less_nat @ C_27 @ B_58 )
=> ( ord_less_nat @ C_27 @ A_84 ) ) ) ).
thf(fact_1761_xt1_I1_J,axiom,
! [C_27: int,A_84: int,B_58: int] :
( ( A_84 = B_58 )
=> ( ( ord_less_int @ C_27 @ B_58 )
=> ( ord_less_int @ C_27 @ A_84 ) ) ) ).
thf(fact_1762_ord__eq__less__trans,axiom,
! [C_26: rat,A_83: rat,B_57: rat] :
( ( A_83 = B_57 )
=> ( ( ord_less_rat @ B_57 @ C_26 )
=> ( ord_less_rat @ A_83 @ C_26 ) ) ) ).
thf(fact_1763_ord__eq__less__trans,axiom,
! [C_26: int > $o,A_83: int > $o,B_57: int > $o] :
( ( A_83 = B_57 )
=> ( ( ord_less_int_o @ B_57 @ C_26 )
=> ( ord_less_int_o @ A_83 @ C_26 ) ) ) ).
thf(fact_1764_ord__eq__less__trans,axiom,
! [C_26: nat > $o,A_83: nat > $o,B_57: nat > $o] :
( ( A_83 = B_57 )
=> ( ( ord_less_nat_o @ B_57 @ C_26 )
=> ( ord_less_nat_o @ A_83 @ C_26 ) ) ) ).
thf(fact_1765_ord__eq__less__trans,axiom,
! [C_26: quickcheck_code_int,A_83: quickcheck_code_int,B_57: quickcheck_code_int] :
( ( A_83 = B_57 )
=> ( ( ord_le1860547276de_int @ B_57 @ C_26 )
=> ( ord_le1860547276de_int @ A_83 @ C_26 ) ) ) ).
thf(fact_1766_ord__eq__less__trans,axiom,
! [C_26: real,A_83: real,B_57: real] :
( ( A_83 = B_57 )
=> ( ( ord_less_real @ B_57 @ C_26 )
=> ( ord_less_real @ A_83 @ C_26 ) ) ) ).
thf(fact_1767_ord__eq__less__trans,axiom,
! [C_26: nat,A_83: nat,B_57: nat] :
( ( A_83 = B_57 )
=> ( ( ord_less_nat @ B_57 @ C_26 )
=> ( ord_less_nat @ A_83 @ C_26 ) ) ) ).
thf(fact_1768_ord__eq__less__trans,axiom,
! [C_26: int,A_83: int,B_57: int] :
( ( A_83 = B_57 )
=> ( ( ord_less_int @ B_57 @ C_26 )
=> ( ord_less_int @ A_83 @ C_26 ) ) ) ).
thf(fact_1769_xt1_I9_J,axiom,
! [B_56: rat,A_82: rat] :
( ( ord_less_rat @ B_56 @ A_82 )
=> ~ ( ord_less_rat @ A_82 @ B_56 ) ) ).
thf(fact_1770_xt1_I9_J,axiom,
! [B_56: int > $o,A_82: int > $o] :
( ( ord_less_int_o @ B_56 @ A_82 )
=> ~ ( ord_less_int_o @ A_82 @ B_56 ) ) ).
thf(fact_1771_xt1_I9_J,axiom,
! [B_56: nat > $o,A_82: nat > $o] :
( ( ord_less_nat_o @ B_56 @ A_82 )
=> ~ ( ord_less_nat_o @ A_82 @ B_56 ) ) ).
thf(fact_1772_xt1_I9_J,axiom,
! [B_56: quickcheck_code_int,A_82: quickcheck_code_int] :
( ( ord_le1860547276de_int @ B_56 @ A_82 )
=> ~ ( ord_le1860547276de_int @ A_82 @ B_56 ) ) ).
thf(fact_1773_xt1_I9_J,axiom,
! [B_56: real,A_82: real] :
( ( ord_less_real @ B_56 @ A_82 )
=> ~ ( ord_less_real @ A_82 @ B_56 ) ) ).
thf(fact_1774_xt1_I9_J,axiom,
! [B_56: nat,A_82: nat] :
( ( ord_less_nat @ B_56 @ A_82 )
=> ~ ( ord_less_nat @ A_82 @ B_56 ) ) ).
thf(fact_1775_xt1_I9_J,axiom,
! [B_56: int,A_82: int] :
( ( ord_less_int @ B_56 @ A_82 )
=> ~ ( ord_less_int @ A_82 @ B_56 ) ) ).
thf(fact_1776_order__less__asym_H,axiom,
! [A_81: rat,B_55: rat] :
( ( ord_less_rat @ A_81 @ B_55 )
=> ~ ( ord_less_rat @ B_55 @ A_81 ) ) ).
thf(fact_1777_order__less__asym_H,axiom,
! [A_81: int > $o,B_55: int > $o] :
( ( ord_less_int_o @ A_81 @ B_55 )
=> ~ ( ord_less_int_o @ B_55 @ A_81 ) ) ).
thf(fact_1778_order__less__asym_H,axiom,
! [A_81: nat > $o,B_55: nat > $o] :
( ( ord_less_nat_o @ A_81 @ B_55 )
=> ~ ( ord_less_nat_o @ B_55 @ A_81 ) ) ).
thf(fact_1779_order__less__asym_H,axiom,
! [A_81: quickcheck_code_int,B_55: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_81 @ B_55 )
=> ~ ( ord_le1860547276de_int @ B_55 @ A_81 ) ) ).
thf(fact_1780_order__less__asym_H,axiom,
! [A_81: real,B_55: real] :
( ( ord_less_real @ A_81 @ B_55 )
=> ~ ( ord_less_real @ B_55 @ A_81 ) ) ).
thf(fact_1781_order__less__asym_H,axiom,
! [A_81: nat,B_55: nat] :
( ( ord_less_nat @ A_81 @ B_55 )
=> ~ ( ord_less_nat @ B_55 @ A_81 ) ) ).
thf(fact_1782_order__less__asym_H,axiom,
! [A_81: int,B_55: int] :
( ( ord_less_int @ A_81 @ B_55 )
=> ~ ( ord_less_int @ B_55 @ A_81 ) ) ).
thf(fact_1783_order__less__imp__triv,axiom,
! [P_7: $o,X_48: rat,Y_41: rat] :
( ( ord_less_rat @ X_48 @ Y_41 )
=> ( ( ord_less_rat @ Y_41 @ X_48 )
=> P_7 ) ) ).
thf(fact_1784_order__less__imp__triv,axiom,
! [P_7: $o,X_48: int > $o,Y_41: int > $o] :
( ( ord_less_int_o @ X_48 @ Y_41 )
=> ( ( ord_less_int_o @ Y_41 @ X_48 )
=> P_7 ) ) ).
thf(fact_1785_order__less__imp__triv,axiom,
! [P_7: $o,X_48: nat > $o,Y_41: nat > $o] :
( ( ord_less_nat_o @ X_48 @ Y_41 )
=> ( ( ord_less_nat_o @ Y_41 @ X_48 )
=> P_7 ) ) ).
thf(fact_1786_order__less__imp__triv,axiom,
! [P_7: $o,X_48: quickcheck_code_int,Y_41: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_48 @ Y_41 )
=> ( ( ord_le1860547276de_int @ Y_41 @ X_48 )
=> P_7 ) ) ).
thf(fact_1787_order__less__imp__triv,axiom,
! [P_7: $o,X_48: real,Y_41: real] :
( ( ord_less_real @ X_48 @ Y_41 )
=> ( ( ord_less_real @ Y_41 @ X_48 )
=> P_7 ) ) ).
thf(fact_1788_order__less__imp__triv,axiom,
! [P_7: $o,X_48: nat,Y_41: nat] :
( ( ord_less_nat @ X_48 @ Y_41 )
=> ( ( ord_less_nat @ Y_41 @ X_48 )
=> P_7 ) ) ).
thf(fact_1789_order__less__imp__triv,axiom,
! [P_7: $o,X_48: int,Y_41: int] :
( ( ord_less_int @ X_48 @ Y_41 )
=> ( ( ord_less_int @ Y_41 @ X_48 )
=> P_7 ) ) ).
thf(fact_1790_order__less__imp__not__eq2,axiom,
! [X_47: rat,Y_40: rat] :
( ( ord_less_rat @ X_47 @ Y_40 )
=> ( Y_40 != X_47 ) ) ).
thf(fact_1791_order__less__imp__not__eq2,axiom,
! [X_47: int > $o,Y_40: int > $o] :
( ( ord_less_int_o @ X_47 @ Y_40 )
=> ( Y_40 != X_47 ) ) ).
thf(fact_1792_order__less__imp__not__eq2,axiom,
! [X_47: nat > $o,Y_40: nat > $o] :
( ( ord_less_nat_o @ X_47 @ Y_40 )
=> ( Y_40 != X_47 ) ) ).
thf(fact_1793_order__less__imp__not__eq2,axiom,
! [X_47: quickcheck_code_int,Y_40: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_47 @ Y_40 )
=> ( Y_40 != X_47 ) ) ).
thf(fact_1794_order__less__imp__not__eq2,axiom,
! [X_47: real,Y_40: real] :
( ( ord_less_real @ X_47 @ Y_40 )
=> ( Y_40 != X_47 ) ) ).
thf(fact_1795_order__less__imp__not__eq2,axiom,
! [X_47: nat,Y_40: nat] :
( ( ord_less_nat @ X_47 @ Y_40 )
=> ( Y_40 != X_47 ) ) ).
thf(fact_1796_order__less__imp__not__eq2,axiom,
! [X_47: int,Y_40: int] :
( ( ord_less_int @ X_47 @ Y_40 )
=> ( Y_40 != X_47 ) ) ).
thf(fact_1797_order__less__imp__not__eq,axiom,
! [X_46: rat,Y_39: rat] :
( ( ord_less_rat @ X_46 @ Y_39 )
=> ( X_46 != Y_39 ) ) ).
thf(fact_1798_order__less__imp__not__eq,axiom,
! [X_46: int > $o,Y_39: int > $o] :
( ( ord_less_int_o @ X_46 @ Y_39 )
=> ( X_46 != Y_39 ) ) ).
thf(fact_1799_order__less__imp__not__eq,axiom,
! [X_46: nat > $o,Y_39: nat > $o] :
( ( ord_less_nat_o @ X_46 @ Y_39 )
=> ( X_46 != Y_39 ) ) ).
thf(fact_1800_order__less__imp__not__eq,axiom,
! [X_46: quickcheck_code_int,Y_39: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_46 @ Y_39 )
=> ( X_46 != Y_39 ) ) ).
thf(fact_1801_order__less__imp__not__eq,axiom,
! [X_46: real,Y_39: real] :
( ( ord_less_real @ X_46 @ Y_39 )
=> ( X_46 != Y_39 ) ) ).
thf(fact_1802_order__less__imp__not__eq,axiom,
! [X_46: nat,Y_39: nat] :
( ( ord_less_nat @ X_46 @ Y_39 )
=> ( X_46 != Y_39 ) ) ).
thf(fact_1803_order__less__imp__not__eq,axiom,
! [X_46: int,Y_39: int] :
( ( ord_less_int @ X_46 @ Y_39 )
=> ( X_46 != Y_39 ) ) ).
thf(fact_1804_order__less__imp__not__less,axiom,
! [X_45: rat,Y_38: rat] :
( ( ord_less_rat @ X_45 @ Y_38 )
=> ~ ( ord_less_rat @ Y_38 @ X_45 ) ) ).
thf(fact_1805_order__less__imp__not__less,axiom,
! [X_45: int > $o,Y_38: int > $o] :
( ( ord_less_int_o @ X_45 @ Y_38 )
=> ~ ( ord_less_int_o @ Y_38 @ X_45 ) ) ).
thf(fact_1806_order__less__imp__not__less,axiom,
! [X_45: nat > $o,Y_38: nat > $o] :
( ( ord_less_nat_o @ X_45 @ Y_38 )
=> ~ ( ord_less_nat_o @ Y_38 @ X_45 ) ) ).
thf(fact_1807_order__less__imp__not__less,axiom,
! [X_45: quickcheck_code_int,Y_38: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_45 @ Y_38 )
=> ~ ( ord_le1860547276de_int @ Y_38 @ X_45 ) ) ).
thf(fact_1808_order__less__imp__not__less,axiom,
! [X_45: real,Y_38: real] :
( ( ord_less_real @ X_45 @ Y_38 )
=> ~ ( ord_less_real @ Y_38 @ X_45 ) ) ).
thf(fact_1809_order__less__imp__not__less,axiom,
! [X_45: nat,Y_38: nat] :
( ( ord_less_nat @ X_45 @ Y_38 )
=> ~ ( ord_less_nat @ Y_38 @ X_45 ) ) ).
thf(fact_1810_order__less__imp__not__less,axiom,
! [X_45: int,Y_38: int] :
( ( ord_less_int @ X_45 @ Y_38 )
=> ~ ( ord_less_int @ Y_38 @ X_45 ) ) ).
thf(fact_1811_order__less__not__sym,axiom,
! [X_44: rat,Y_37: rat] :
( ( ord_less_rat @ X_44 @ Y_37 )
=> ~ ( ord_less_rat @ Y_37 @ X_44 ) ) ).
thf(fact_1812_order__less__not__sym,axiom,
! [X_44: int > $o,Y_37: int > $o] :
( ( ord_less_int_o @ X_44 @ Y_37 )
=> ~ ( ord_less_int_o @ Y_37 @ X_44 ) ) ).
thf(fact_1813_order__less__not__sym,axiom,
! [X_44: nat > $o,Y_37: nat > $o] :
( ( ord_less_nat_o @ X_44 @ Y_37 )
=> ~ ( ord_less_nat_o @ Y_37 @ X_44 ) ) ).
thf(fact_1814_order__less__not__sym,axiom,
! [X_44: quickcheck_code_int,Y_37: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_44 @ Y_37 )
=> ~ ( ord_le1860547276de_int @ Y_37 @ X_44 ) ) ).
thf(fact_1815_order__less__not__sym,axiom,
! [X_44: real,Y_37: real] :
( ( ord_less_real @ X_44 @ Y_37 )
=> ~ ( ord_less_real @ Y_37 @ X_44 ) ) ).
thf(fact_1816_order__less__not__sym,axiom,
! [X_44: nat,Y_37: nat] :
( ( ord_less_nat @ X_44 @ Y_37 )
=> ~ ( ord_less_nat @ Y_37 @ X_44 ) ) ).
thf(fact_1817_order__less__not__sym,axiom,
! [X_44: int,Y_37: int] :
( ( ord_less_int @ X_44 @ Y_37 )
=> ~ ( ord_less_int @ Y_37 @ X_44 ) ) ).
thf(fact_1818_less__imp__neq,axiom,
! [X_43: rat,Y_36: rat] :
( ( ord_less_rat @ X_43 @ Y_36 )
=> ( X_43 != Y_36 ) ) ).
thf(fact_1819_less__imp__neq,axiom,
! [X_43: int > $o,Y_36: int > $o] :
( ( ord_less_int_o @ X_43 @ Y_36 )
=> ( X_43 != Y_36 ) ) ).
thf(fact_1820_less__imp__neq,axiom,
! [X_43: nat > $o,Y_36: nat > $o] :
( ( ord_less_nat_o @ X_43 @ Y_36 )
=> ( X_43 != Y_36 ) ) ).
thf(fact_1821_less__imp__neq,axiom,
! [X_43: quickcheck_code_int,Y_36: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_43 @ Y_36 )
=> ( X_43 != Y_36 ) ) ).
thf(fact_1822_less__imp__neq,axiom,
! [X_43: real,Y_36: real] :
( ( ord_less_real @ X_43 @ Y_36 )
=> ( X_43 != Y_36 ) ) ).
thf(fact_1823_less__imp__neq,axiom,
! [X_43: nat,Y_36: nat] :
( ( ord_less_nat @ X_43 @ Y_36 )
=> ( X_43 != Y_36 ) ) ).
thf(fact_1824_less__imp__neq,axiom,
! [X_43: int,Y_36: int] :
( ( ord_less_int @ X_43 @ Y_36 )
=> ( X_43 != Y_36 ) ) ).
thf(fact_1825_linorder__neqE,axiom,
! [X_42: rat,Y_35: rat] :
( ( X_42 != Y_35 )
=> ( ~ ( ord_less_rat @ X_42 @ Y_35 )
=> ( ord_less_rat @ Y_35 @ X_42 ) ) ) ).
thf(fact_1826_linorder__neqE,axiom,
! [X_42: quickcheck_code_int,Y_35: quickcheck_code_int] :
( ( X_42 != Y_35 )
=> ( ~ ( ord_le1860547276de_int @ X_42 @ Y_35 )
=> ( ord_le1860547276de_int @ Y_35 @ X_42 ) ) ) ).
thf(fact_1827_linorder__neqE,axiom,
! [X_42: real,Y_35: real] :
( ( X_42 != Y_35 )
=> ( ~ ( ord_less_real @ X_42 @ Y_35 )
=> ( ord_less_real @ Y_35 @ X_42 ) ) ) ).
thf(fact_1828_linorder__neqE,axiom,
! [X_42: nat,Y_35: nat] :
( ( X_42 != Y_35 )
=> ( ~ ( ord_less_nat @ X_42 @ Y_35 )
=> ( ord_less_nat @ Y_35 @ X_42 ) ) ) ).
thf(fact_1829_linorder__neqE,axiom,
! [X_42: int,Y_35: int] :
( ( X_42 != Y_35 )
=> ( ~ ( ord_less_int @ X_42 @ Y_35 )
=> ( ord_less_int @ Y_35 @ X_42 ) ) ) ).
thf(fact_1830_linorder__antisym__conv3,axiom,
! [Y_34: rat,X_41: rat] :
( ~ ( ord_less_rat @ Y_34 @ X_41 )
=> ( ~ ( ord_less_rat @ X_41 @ Y_34 )
<=> ( X_41 = Y_34 ) ) ) ).
thf(fact_1831_linorder__antisym__conv3,axiom,
! [Y_34: quickcheck_code_int,X_41: quickcheck_code_int] :
( ~ ( ord_le1860547276de_int @ Y_34 @ X_41 )
=> ( ~ ( ord_le1860547276de_int @ X_41 @ Y_34 )
<=> ( X_41 = Y_34 ) ) ) ).
thf(fact_1832_linorder__antisym__conv3,axiom,
! [Y_34: real,X_41: real] :
( ~ ( ord_less_real @ Y_34 @ X_41 )
=> ( ~ ( ord_less_real @ X_41 @ Y_34 )
<=> ( X_41 = Y_34 ) ) ) ).
thf(fact_1833_linorder__antisym__conv3,axiom,
! [Y_34: nat,X_41: nat] :
( ~ ( ord_less_nat @ Y_34 @ X_41 )
=> ( ~ ( ord_less_nat @ X_41 @ Y_34 )
<=> ( X_41 = Y_34 ) ) ) ).
thf(fact_1834_linorder__antisym__conv3,axiom,
! [Y_34: int,X_41: int] :
( ~ ( ord_less_int @ Y_34 @ X_41 )
=> ( ~ ( ord_less_int @ X_41 @ Y_34 )
<=> ( X_41 = Y_34 ) ) ) ).
thf(fact_1835_linorder__less__linear,axiom,
! [X_40: rat,Y_33: rat] :
( ( ord_less_rat @ X_40 @ Y_33 )
| ( X_40 = Y_33 )
| ( ord_less_rat @ Y_33 @ X_40 ) ) ).
thf(fact_1836_linorder__less__linear,axiom,
! [X_40: quickcheck_code_int,Y_33: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_40 @ Y_33 )
| ( X_40 = Y_33 )
| ( ord_le1860547276de_int @ Y_33 @ X_40 ) ) ).
thf(fact_1837_linorder__less__linear,axiom,
! [X_40: real,Y_33: real] :
( ( ord_less_real @ X_40 @ Y_33 )
| ( X_40 = Y_33 )
| ( ord_less_real @ Y_33 @ X_40 ) ) ).
thf(fact_1838_linorder__less__linear,axiom,
! [X_40: nat,Y_33: nat] :
( ( ord_less_nat @ X_40 @ Y_33 )
| ( X_40 = Y_33 )
| ( ord_less_nat @ Y_33 @ X_40 ) ) ).
thf(fact_1839_linorder__less__linear,axiom,
! [X_40: int,Y_33: int] :
( ( ord_less_int @ X_40 @ Y_33 )
| ( X_40 = Y_33 )
| ( ord_less_int @ Y_33 @ X_40 ) ) ).
thf(fact_1840_not__less__iff__gr__or__eq,axiom,
! [X_39: rat,Y_32: rat] :
( ~ ( ord_less_rat @ X_39 @ Y_32 )
<=> ( ( ord_less_rat @ Y_32 @ X_39 )
| ( X_39 = Y_32 ) ) ) ).
thf(fact_1841_not__less__iff__gr__or__eq,axiom,
! [X_39: quickcheck_code_int,Y_32: quickcheck_code_int] :
( ~ ( ord_le1860547276de_int @ X_39 @ Y_32 )
<=> ( ( ord_le1860547276de_int @ Y_32 @ X_39 )
| ( X_39 = Y_32 ) ) ) ).
thf(fact_1842_not__less__iff__gr__or__eq,axiom,
! [X_39: real,Y_32: real] :
( ~ ( ord_less_real @ X_39 @ Y_32 )
<=> ( ( ord_less_real @ Y_32 @ X_39 )
| ( X_39 = Y_32 ) ) ) ).
thf(fact_1843_not__less__iff__gr__or__eq,axiom,
! [X_39: nat,Y_32: nat] :
( ~ ( ord_less_nat @ X_39 @ Y_32 )
<=> ( ( ord_less_nat @ Y_32 @ X_39 )
| ( X_39 = Y_32 ) ) ) ).
thf(fact_1844_not__less__iff__gr__or__eq,axiom,
! [X_39: int,Y_32: int] :
( ~ ( ord_less_int @ X_39 @ Y_32 )
<=> ( ( ord_less_int @ Y_32 @ X_39 )
| ( X_39 = Y_32 ) ) ) ).
thf(fact_1845_linorder__neq__iff,axiom,
! [X_38: rat,Y_31: rat] :
( ( X_38 != Y_31 )
<=> ( ( ord_less_rat @ X_38 @ Y_31 )
| ( ord_less_rat @ Y_31 @ X_38 ) ) ) ).
thf(fact_1846_linorder__neq__iff,axiom,
! [X_38: quickcheck_code_int,Y_31: quickcheck_code_int] :
( ( X_38 != Y_31 )
<=> ( ( ord_le1860547276de_int @ X_38 @ Y_31 )
| ( ord_le1860547276de_int @ Y_31 @ X_38 ) ) ) ).
thf(fact_1847_linorder__neq__iff,axiom,
! [X_38: real,Y_31: real] :
( ( X_38 != Y_31 )
<=> ( ( ord_less_real @ X_38 @ Y_31 )
| ( ord_less_real @ Y_31 @ X_38 ) ) ) ).
thf(fact_1848_linorder__neq__iff,axiom,
! [X_38: nat,Y_31: nat] :
( ( X_38 != Y_31 )
<=> ( ( ord_less_nat @ X_38 @ Y_31 )
| ( ord_less_nat @ Y_31 @ X_38 ) ) ) ).
thf(fact_1849_linorder__neq__iff,axiom,
! [X_38: int,Y_31: int] :
( ( X_38 != Y_31 )
<=> ( ( ord_less_int @ X_38 @ Y_31 )
| ( ord_less_int @ Y_31 @ X_38 ) ) ) ).
thf(fact_1850_order__less__irrefl,axiom,
! [X_37: rat] :
~ ( ord_less_rat @ X_37 @ X_37 ) ).
thf(fact_1851_order__less__irrefl,axiom,
! [X_37: int > $o] :
~ ( ord_less_int_o @ X_37 @ X_37 ) ).
thf(fact_1852_order__less__irrefl,axiom,
! [X_37: nat > $o] :
~ ( ord_less_nat_o @ X_37 @ X_37 ) ).
thf(fact_1853_order__less__irrefl,axiom,
! [X_37: quickcheck_code_int] :
~ ( ord_le1860547276de_int @ X_37 @ X_37 ) ).
thf(fact_1854_order__less__irrefl,axiom,
! [X_37: real] :
~ ( ord_less_real @ X_37 @ X_37 ) ).
thf(fact_1855_order__less__irrefl,axiom,
! [X_37: nat] :
~ ( ord_less_nat @ X_37 @ X_37 ) ).
thf(fact_1856_order__less__irrefl,axiom,
! [X_37: int] :
~ ( ord_less_int @ X_37 @ X_37 ) ).
thf(fact_1857_convex__bound__le,axiom,
! [V_8: rat,U_3: rat,Y_30: rat,X_36: rat,A_80: rat] :
( ( ord_less_eq_rat @ X_36 @ A_80 )
=> ( ( ord_less_eq_rat @ Y_30 @ A_80 )
=> ( ( 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_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U_3 @ X_36 ) @ ( times_times_rat @ V_8 @ Y_30 ) ) @ A_80 ) ) ) ) ) ) ).
thf(fact_1858_convex__bound__le,axiom,
! [V_8: real,U_3: real,Y_30: real,X_36: real,A_80: real] :
( ( ord_less_eq_real @ X_36 @ A_80 )
=> ( ( ord_less_eq_real @ Y_30 @ A_80 )
=> ( ( 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_eq_real @ ( plus_plus_real @ ( times_times_real @ U_3 @ X_36 ) @ ( times_times_real @ V_8 @ Y_30 ) ) @ A_80 ) ) ) ) ) ) ).
thf(fact_1859_convex__bound__le,axiom,
! [V_8: int,U_3: int,Y_30: int,X_36: int,A_80: int] :
( ( ord_less_eq_int @ X_36 @ A_80 )
=> ( ( ord_less_eq_int @ Y_30 @ A_80 )
=> ( ( 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_eq_int @ ( plus_plus_int @ ( times_times_int @ U_3 @ X_36 ) @ ( times_times_int @ V_8 @ Y_30 ) ) @ A_80 ) ) ) ) ) ) ).
thf(fact_1860_power__Suc__less,axiom,
! [N_21: nat,A_79: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_79 )
=> ( ( ord_less_rat @ A_79 @ one_one_rat )
=> ( ord_less_rat @ ( times_times_rat @ A_79 @ ( power_power_rat @ A_79 @ N_21 ) ) @ ( power_power_rat @ A_79 @ N_21 ) ) ) ) ).
thf(fact_1861_power__Suc__less,axiom,
! [N_21: nat,A_79: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_79 )
=> ( ( ord_le1860547276de_int @ A_79 @ one_on1684967323de_int )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_79 @ ( power_881366806de_int @ A_79 @ N_21 ) ) @ ( power_881366806de_int @ A_79 @ N_21 ) ) ) ) ).
thf(fact_1862_power__Suc__less,axiom,
! [N_21: nat,A_79: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_79 )
=> ( ( ord_le1304079648umeral @ A_79 @ one_on1645066479umeral )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_79 @ ( power_2100829034umeral @ A_79 @ N_21 ) ) @ ( power_2100829034umeral @ A_79 @ N_21 ) ) ) ) ).
thf(fact_1863_power__Suc__less,axiom,
! [N_21: nat,A_79: real] :
( ( ord_less_real @ zero_zero_real @ A_79 )
=> ( ( ord_less_real @ A_79 @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A_79 @ ( power_power_real @ A_79 @ N_21 ) ) @ ( power_power_real @ A_79 @ N_21 ) ) ) ) ).
thf(fact_1864_power__Suc__less,axiom,
! [N_21: nat,A_79: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_79 )
=> ( ( ord_less_nat @ A_79 @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A_79 @ ( power_power_nat @ A_79 @ N_21 ) ) @ ( power_power_nat @ A_79 @ N_21 ) ) ) ) ).
thf(fact_1865_power__Suc__less,axiom,
! [N_21: nat,A_79: int] :
( ( ord_less_int @ zero_zero_int @ A_79 )
=> ( ( ord_less_int @ A_79 @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A_79 @ ( power_power_int @ A_79 @ N_21 ) ) @ ( power_power_int @ A_79 @ N_21 ) ) ) ) ).
thf(fact_1866_double__number__of__Bit0,axiom,
! [W_5: int] :
( ( times_times_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) @ ( number_number_of_rat @ W_5 ) )
= ( number_number_of_rat @ ( bit0 @ W_5 ) ) ) ).
thf(fact_1867_double__number__of__Bit0,axiom,
! [W_5: int] :
( ( times_times_complex @ ( plus_plus_complex @ one_one_complex @ one_one_complex ) @ ( number528085621omplex @ W_5 ) )
= ( number528085621omplex @ ( bit0 @ W_5 ) ) ) ).
thf(fact_1868_double__number__of__Bit0,axiom,
! [W_5: int] :
( ( times_times_real @ ( plus_plus_real @ one_one_real @ one_one_real ) @ ( number267125858f_real @ W_5 ) )
= ( number267125858f_real @ ( bit0 @ W_5 ) ) ) ).
thf(fact_1869_double__number__of__Bit0,axiom,
! [W_5: int] :
( ( times_times_int @ ( plus_plus_int @ one_one_int @ one_one_int ) @ ( number_number_of_int @ W_5 ) )
= ( number_number_of_int @ ( bit0 @ W_5 ) ) ) ).
thf(fact_1870_power3__eq__cube,axiom,
! [A_78: complex] :
( ( power_power_complex @ A_78 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_times_complex @ ( times_times_complex @ A_78 @ A_78 ) @ A_78 ) ) ).
thf(fact_1871_power3__eq__cube,axiom,
! [A_78: real] :
( ( power_power_real @ A_78 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_times_real @ ( times_times_real @ A_78 @ A_78 ) @ A_78 ) ) ).
thf(fact_1872_power3__eq__cube,axiom,
! [A_78: nat] :
( ( power_power_nat @ A_78 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_times_nat @ ( times_times_nat @ A_78 @ A_78 ) @ A_78 ) ) ).
thf(fact_1873_power3__eq__cube,axiom,
! [A_78: int] :
( ( power_power_int @ A_78 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_times_int @ ( times_times_int @ A_78 @ A_78 ) @ A_78 ) ) ).
thf(fact_1874_power3__eq__cube,axiom,
! [A_78: rat] :
( ( power_power_rat @ A_78 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_times_rat @ ( times_times_rat @ A_78 @ A_78 ) @ A_78 ) ) ).
thf(fact_1875_power3__eq__cube,axiom,
! [A_78: quickcheck_code_int] :
( ( power_881366806de_int @ A_78 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( times_123202395de_int @ ( times_123202395de_int @ A_78 @ A_78 ) @ A_78 ) ) ).
thf(fact_1876_zmult__zless__mono2__lemma,axiom,
! [K: nat,I: int,J: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1621563631at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1621563631at_int @ K ) @ J ) ) ) ) ).
thf(fact_1877_convex__bound__lt,axiom,
! [V_7: rat,U_2: rat,Y_29: rat,X_35: rat,A_77: rat] :
( ( ord_less_rat @ X_35 @ A_77 )
=> ( ( ord_less_rat @ Y_29 @ A_77 )
=> ( ( 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_rat @ ( plus_plus_rat @ ( times_times_rat @ U_2 @ X_35 ) @ ( times_times_rat @ V_7 @ Y_29 ) ) @ A_77 ) ) ) ) ) ) ).
thf(fact_1878_convex__bound__lt,axiom,
! [V_7: real,U_2: real,Y_29: real,X_35: real,A_77: real] :
( ( ord_less_real @ X_35 @ A_77 )
=> ( ( ord_less_real @ Y_29 @ A_77 )
=> ( ( 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_real @ ( plus_plus_real @ ( times_times_real @ U_2 @ X_35 ) @ ( times_times_real @ V_7 @ Y_29 ) ) @ A_77 ) ) ) ) ) ) ).
thf(fact_1879_convex__bound__lt,axiom,
! [V_7: int,U_2: int,Y_29: int,X_35: int,A_77: int] :
( ( ord_less_int @ X_35 @ A_77 )
=> ( ( ord_less_int @ Y_29 @ A_77 )
=> ( ( 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_int @ ( plus_plus_int @ ( times_times_int @ U_2 @ X_35 ) @ ( times_times_int @ V_7 @ Y_29 ) ) @ A_77 ) ) ) ) ) ) ).
thf(fact_1880_mult__2__right,axiom,
! [Z_13: rat] :
( ( times_times_rat @ Z_13 @ ( number_number_of_rat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_rat @ Z_13 @ Z_13 ) ) ).
thf(fact_1881_mult__2__right,axiom,
! [Z_13: complex] :
( ( times_times_complex @ Z_13 @ ( number528085621omplex @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_complex @ Z_13 @ Z_13 ) ) ).
thf(fact_1882_mult__2__right,axiom,
! [Z_13: real] :
( ( times_times_real @ Z_13 @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_real @ Z_13 @ Z_13 ) ) ).
thf(fact_1883_mult__2__right,axiom,
! [Z_13: int] :
( ( times_times_int @ Z_13 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ Z_13 @ Z_13 ) ) ).
thf(fact_1884_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_1885_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_1886_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_1887_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_1888_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_1889_mult__2,axiom,
! [Z_11: rat] :
( ( times_times_rat @ ( number_number_of_rat @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_11 )
= ( plus_plus_rat @ Z_11 @ Z_11 ) ) ).
thf(fact_1890_mult__2,axiom,
! [Z_11: complex] :
( ( times_times_complex @ ( number528085621omplex @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_11 )
= ( plus_plus_complex @ Z_11 @ Z_11 ) ) ).
thf(fact_1891_mult__2,axiom,
! [Z_11: real] :
( ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_11 )
= ( plus_plus_real @ Z_11 @ Z_11 ) ) ).
thf(fact_1892_mult__2,axiom,
! [Z_11: int] :
( ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_11 )
= ( plus_plus_int @ Z_11 @ Z_11 ) ) ).
thf(fact_1893_semiring__mult__2,axiom,
! [Z_10: rat] :
( ( times_times_rat @ ( number_number_of_rat @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_10 )
= ( plus_plus_rat @ Z_10 @ Z_10 ) ) ).
thf(fact_1894_semiring__mult__2,axiom,
! [Z_10: complex] :
( ( times_times_complex @ ( number528085621omplex @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_10 )
= ( plus_plus_complex @ Z_10 @ Z_10 ) ) ).
thf(fact_1895_semiring__mult__2,axiom,
! [Z_10: real] :
( ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_10 )
= ( plus_plus_real @ Z_10 @ Z_10 ) ) ).
thf(fact_1896_semiring__mult__2,axiom,
! [Z_10: nat] :
( ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_10 )
= ( plus_plus_nat @ Z_10 @ Z_10 ) ) ).
thf(fact_1897_semiring__mult__2,axiom,
! [Z_10: int] :
( ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Z_10 )
= ( plus_plus_int @ Z_10 @ Z_10 ) ) ).
thf(fact_1898_power2__eq__square,axiom,
! [A_76: complex] :
( ( power_power_complex @ A_76 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_complex @ A_76 @ A_76 ) ) ).
thf(fact_1899_power2__eq__square,axiom,
! [A_76: real] :
( ( power_power_real @ A_76 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_real @ A_76 @ A_76 ) ) ).
thf(fact_1900_power2__eq__square,axiom,
! [A_76: nat] :
( ( power_power_nat @ A_76 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_nat @ A_76 @ A_76 ) ) ).
thf(fact_1901_power2__eq__square,axiom,
! [A_76: int] :
( ( power_power_int @ A_76 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_int @ A_76 @ A_76 ) ) ).
thf(fact_1902_power2__eq__square,axiom,
! [A_76: rat] :
( ( power_power_rat @ A_76 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_rat @ A_76 @ A_76 ) ) ).
thf(fact_1903_power2__eq__square,axiom,
! [A_76: quickcheck_code_int] :
( ( power_881366806de_int @ A_76 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_123202395de_int @ A_76 @ A_76 ) ) ).
thf(fact_1904_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_34: complex] :
( ( times_times_complex @ X_34 @ X_34 )
= ( power_power_complex @ X_34 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_1905_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_34: real] :
( ( times_times_real @ X_34 @ X_34 )
= ( power_power_real @ X_34 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_1906_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_34: nat] :
( ( times_times_nat @ X_34 @ X_34 )
= ( power_power_nat @ X_34 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_1907_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_34: int] :
( ( times_times_int @ X_34 @ X_34 )
= ( power_power_int @ X_34 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_1908_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_34: rat] :
( ( times_times_rat @ X_34 @ X_34 )
= ( power_power_rat @ X_34 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_1909_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_34: quickcheck_code_int] :
( ( times_123202395de_int @ X_34 @ X_34 )
= ( power_881366806de_int @ X_34 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_1910_cube__square,axiom,
! [A_1: int] :
( ( times_times_int @ A_1 @ ( power_power_int @ A_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( power_power_int @ A_1 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_1911_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_1912_power2__eq__square__number__of,axiom,
! [W_4: int] :
( ( power_power_complex @ ( number528085621omplex @ W_4 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_complex @ ( number528085621omplex @ W_4 ) @ ( number528085621omplex @ W_4 ) ) ) ).
thf(fact_1913_power2__eq__square__number__of,axiom,
! [W_4: int] :
( ( power_power_real @ ( number267125858f_real @ W_4 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_real @ ( number267125858f_real @ W_4 ) @ ( number267125858f_real @ W_4 ) ) ) ).
thf(fact_1914_power2__eq__square__number__of,axiom,
! [W_4: int] :
( ( power_power_nat @ ( number_number_of_nat @ W_4 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_nat @ ( number_number_of_nat @ W_4 ) @ ( number_number_of_nat @ W_4 ) ) ) ).
thf(fact_1915_power2__eq__square__number__of,axiom,
! [W_4: int] :
( ( power_power_int @ ( number_number_of_int @ W_4 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_int @ ( number_number_of_int @ W_4 ) @ ( number_number_of_int @ W_4 ) ) ) ).
thf(fact_1916_power2__eq__square__number__of,axiom,
! [W_4: int] :
( ( power_power_rat @ ( number_number_of_rat @ W_4 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_times_rat @ ( number_number_of_rat @ W_4 ) @ ( number_number_of_rat @ W_4 ) ) ) ).
thf(fact_1917_power2__eq__square__number__of,axiom,
! [W_4: int] :
( ( power_881366806de_int @ ( number1226105091de_int @ W_4 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_123202395de_int @ ( number1226105091de_int @ W_4 ) @ ( number1226105091de_int @ W_4 ) ) ) ).
thf(fact_1918_power2__eq__square__number__of,axiom,
! [W_4: int] :
( ( power_2100829034umeral @ ( number1443263063umeral @ W_4 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( times_1655362735umeral @ ( number1443263063umeral @ W_4 ) @ ( number1443263063umeral @ W_4 ) ) ) ).
thf(fact_1919_xt1_I8_J,axiom,
! [Z_9: rat,Y_28: rat,X_33: rat] :
( ( ord_less_eq_rat @ Y_28 @ X_33 )
=> ( ( ord_less_rat @ Z_9 @ Y_28 )
=> ( ord_less_rat @ Z_9 @ X_33 ) ) ) ).
thf(fact_1920_xt1_I8_J,axiom,
! [Z_9: int > $o,Y_28: int > $o,X_33: int > $o] :
( ( ord_less_eq_int_o @ Y_28 @ X_33 )
=> ( ( ord_less_int_o @ Z_9 @ Y_28 )
=> ( ord_less_int_o @ Z_9 @ X_33 ) ) ) ).
thf(fact_1921_xt1_I8_J,axiom,
! [Z_9: nat > $o,Y_28: nat > $o,X_33: nat > $o] :
( ( ord_less_eq_nat_o @ Y_28 @ X_33 )
=> ( ( ord_less_nat_o @ Z_9 @ Y_28 )
=> ( ord_less_nat_o @ Z_9 @ X_33 ) ) ) ).
thf(fact_1922_xt1_I8_J,axiom,
! [Z_9: quickcheck_code_int,Y_28: quickcheck_code_int,X_33: quickcheck_code_int] :
( ( ord_le258702272de_int @ Y_28 @ X_33 )
=> ( ( ord_le1860547276de_int @ Z_9 @ Y_28 )
=> ( ord_le1860547276de_int @ Z_9 @ X_33 ) ) ) ).
thf(fact_1923_xt1_I8_J,axiom,
! [Z_9: real,Y_28: real,X_33: real] :
( ( ord_less_eq_real @ Y_28 @ X_33 )
=> ( ( ord_less_real @ Z_9 @ Y_28 )
=> ( ord_less_real @ Z_9 @ X_33 ) ) ) ).
thf(fact_1924_xt1_I8_J,axiom,
! [Z_9: nat,Y_28: nat,X_33: nat] :
( ( ord_less_eq_nat @ Y_28 @ X_33 )
=> ( ( ord_less_nat @ Z_9 @ Y_28 )
=> ( ord_less_nat @ Z_9 @ X_33 ) ) ) ).
thf(fact_1925_xt1_I8_J,axiom,
! [Z_9: int,Y_28: int,X_33: int] :
( ( ord_less_eq_int @ Y_28 @ X_33 )
=> ( ( ord_less_int @ Z_9 @ Y_28 )
=> ( ord_less_int @ Z_9 @ X_33 ) ) ) ).
thf(fact_1926_order__le__less__trans,axiom,
! [Z_8: rat,X_32: rat,Y_27: rat] :
( ( ord_less_eq_rat @ X_32 @ Y_27 )
=> ( ( ord_less_rat @ Y_27 @ Z_8 )
=> ( ord_less_rat @ X_32 @ Z_8 ) ) ) ).
thf(fact_1927_order__le__less__trans,axiom,
! [Z_8: int > $o,X_32: int > $o,Y_27: int > $o] :
( ( ord_less_eq_int_o @ X_32 @ Y_27 )
=> ( ( ord_less_int_o @ Y_27 @ Z_8 )
=> ( ord_less_int_o @ X_32 @ Z_8 ) ) ) ).
thf(fact_1928_order__le__less__trans,axiom,
! [Z_8: nat > $o,X_32: nat > $o,Y_27: nat > $o] :
( ( ord_less_eq_nat_o @ X_32 @ Y_27 )
=> ( ( ord_less_nat_o @ Y_27 @ Z_8 )
=> ( ord_less_nat_o @ X_32 @ Z_8 ) ) ) ).
thf(fact_1929_order__le__less__trans,axiom,
! [Z_8: quickcheck_code_int,X_32: quickcheck_code_int,Y_27: quickcheck_code_int] :
( ( ord_le258702272de_int @ X_32 @ Y_27 )
=> ( ( ord_le1860547276de_int @ Y_27 @ Z_8 )
=> ( ord_le1860547276de_int @ X_32 @ Z_8 ) ) ) ).
thf(fact_1930_order__le__less__trans,axiom,
! [Z_8: real,X_32: real,Y_27: real] :
( ( ord_less_eq_real @ X_32 @ Y_27 )
=> ( ( ord_less_real @ Y_27 @ Z_8 )
=> ( ord_less_real @ X_32 @ Z_8 ) ) ) ).
thf(fact_1931_order__le__less__trans,axiom,
! [Z_8: nat,X_32: nat,Y_27: nat] :
( ( ord_less_eq_nat @ X_32 @ Y_27 )
=> ( ( ord_less_nat @ Y_27 @ Z_8 )
=> ( ord_less_nat @ X_32 @ Z_8 ) ) ) ).
thf(fact_1932_order__le__less__trans,axiom,
! [Z_8: int,X_32: int,Y_27: int] :
( ( ord_less_eq_int @ X_32 @ Y_27 )
=> ( ( ord_less_int @ Y_27 @ Z_8 )
=> ( ord_less_int @ X_32 @ Z_8 ) ) ) ).
thf(fact_1933_xt1_I7_J,axiom,
! [Z_7: rat,Y_26: rat,X_31: rat] :
( ( ord_less_rat @ Y_26 @ X_31 )
=> ( ( ord_less_eq_rat @ Z_7 @ Y_26 )
=> ( ord_less_rat @ Z_7 @ X_31 ) ) ) ).
thf(fact_1934_xt1_I7_J,axiom,
! [Z_7: int > $o,Y_26: int > $o,X_31: int > $o] :
( ( ord_less_int_o @ Y_26 @ X_31 )
=> ( ( ord_less_eq_int_o @ Z_7 @ Y_26 )
=> ( ord_less_int_o @ Z_7 @ X_31 ) ) ) ).
thf(fact_1935_xt1_I7_J,axiom,
! [Z_7: nat > $o,Y_26: nat > $o,X_31: nat > $o] :
( ( ord_less_nat_o @ Y_26 @ X_31 )
=> ( ( ord_less_eq_nat_o @ Z_7 @ Y_26 )
=> ( ord_less_nat_o @ Z_7 @ X_31 ) ) ) ).
thf(fact_1936_xt1_I7_J,axiom,
! [Z_7: quickcheck_code_int,Y_26: quickcheck_code_int,X_31: quickcheck_code_int] :
( ( ord_le1860547276de_int @ Y_26 @ X_31 )
=> ( ( ord_le258702272de_int @ Z_7 @ Y_26 )
=> ( ord_le1860547276de_int @ Z_7 @ X_31 ) ) ) ).
thf(fact_1937_xt1_I7_J,axiom,
! [Z_7: real,Y_26: real,X_31: real] :
( ( ord_less_real @ Y_26 @ X_31 )
=> ( ( ord_less_eq_real @ Z_7 @ Y_26 )
=> ( ord_less_real @ Z_7 @ X_31 ) ) ) ).
thf(fact_1938_xt1_I7_J,axiom,
! [Z_7: nat,Y_26: nat,X_31: nat] :
( ( ord_less_nat @ Y_26 @ X_31 )
=> ( ( ord_less_eq_nat @ Z_7 @ Y_26 )
=> ( ord_less_nat @ Z_7 @ X_31 ) ) ) ).
thf(fact_1939_xt1_I7_J,axiom,
! [Z_7: int,Y_26: int,X_31: int] :
( ( ord_less_int @ Y_26 @ X_31 )
=> ( ( ord_less_eq_int @ Z_7 @ Y_26 )
=> ( ord_less_int @ Z_7 @ X_31 ) ) ) ).
thf(fact_1940_order__less__le__trans,axiom,
! [Z_6: rat,X_30: rat,Y_25: rat] :
( ( ord_less_rat @ X_30 @ Y_25 )
=> ( ( ord_less_eq_rat @ Y_25 @ Z_6 )
=> ( ord_less_rat @ X_30 @ Z_6 ) ) ) ).
thf(fact_1941_order__less__le__trans,axiom,
! [Z_6: int > $o,X_30: int > $o,Y_25: int > $o] :
( ( ord_less_int_o @ X_30 @ Y_25 )
=> ( ( ord_less_eq_int_o @ Y_25 @ Z_6 )
=> ( ord_less_int_o @ X_30 @ Z_6 ) ) ) ).
thf(fact_1942_order__less__le__trans,axiom,
! [Z_6: nat > $o,X_30: nat > $o,Y_25: nat > $o] :
( ( ord_less_nat_o @ X_30 @ Y_25 )
=> ( ( ord_less_eq_nat_o @ Y_25 @ Z_6 )
=> ( ord_less_nat_o @ X_30 @ Z_6 ) ) ) ).
thf(fact_1943_order__less__le__trans,axiom,
! [Z_6: quickcheck_code_int,X_30: quickcheck_code_int,Y_25: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_30 @ Y_25 )
=> ( ( ord_le258702272de_int @ Y_25 @ Z_6 )
=> ( ord_le1860547276de_int @ X_30 @ Z_6 ) ) ) ).
thf(fact_1944_order__less__le__trans,axiom,
! [Z_6: real,X_30: real,Y_25: real] :
( ( ord_less_real @ X_30 @ Y_25 )
=> ( ( ord_less_eq_real @ Y_25 @ Z_6 )
=> ( ord_less_real @ X_30 @ Z_6 ) ) ) ).
thf(fact_1945_order__less__le__trans,axiom,
! [Z_6: nat,X_30: nat,Y_25: nat] :
( ( ord_less_nat @ X_30 @ Y_25 )
=> ( ( ord_less_eq_nat @ Y_25 @ Z_6 )
=> ( ord_less_nat @ X_30 @ Z_6 ) ) ) ).
thf(fact_1946_order__less__le__trans,axiom,
! [Z_6: int,X_30: int,Y_25: int] :
( ( ord_less_int @ X_30 @ Y_25 )
=> ( ( ord_less_eq_int @ Y_25 @ Z_6 )
=> ( ord_less_int @ X_30 @ Z_6 ) ) ) ).
thf(fact_1947_xt1_I11_J,axiom,
! [B_54: rat,A_75: rat] :
( ( ord_less_eq_rat @ B_54 @ A_75 )
=> ( ( A_75 != B_54 )
=> ( ord_less_rat @ B_54 @ A_75 ) ) ) ).
thf(fact_1948_xt1_I11_J,axiom,
! [B_54: int > $o,A_75: int > $o] :
( ( ord_less_eq_int_o @ B_54 @ A_75 )
=> ( ( A_75 != B_54 )
=> ( ord_less_int_o @ B_54 @ A_75 ) ) ) ).
thf(fact_1949_xt1_I11_J,axiom,
! [B_54: nat > $o,A_75: nat > $o] :
( ( ord_less_eq_nat_o @ B_54 @ A_75 )
=> ( ( A_75 != B_54 )
=> ( ord_less_nat_o @ B_54 @ A_75 ) ) ) ).
thf(fact_1950_xt1_I11_J,axiom,
! [B_54: quickcheck_code_int,A_75: quickcheck_code_int] :
( ( ord_le258702272de_int @ B_54 @ A_75 )
=> ( ( A_75 != B_54 )
=> ( ord_le1860547276de_int @ B_54 @ A_75 ) ) ) ).
thf(fact_1951_xt1_I11_J,axiom,
! [B_54: real,A_75: real] :
( ( ord_less_eq_real @ B_54 @ A_75 )
=> ( ( A_75 != B_54 )
=> ( ord_less_real @ B_54 @ A_75 ) ) ) ).
thf(fact_1952_xt1_I11_J,axiom,
! [B_54: nat,A_75: nat] :
( ( ord_less_eq_nat @ B_54 @ A_75 )
=> ( ( A_75 != B_54 )
=> ( ord_less_nat @ B_54 @ A_75 ) ) ) ).
thf(fact_1953_xt1_I11_J,axiom,
! [B_54: int,A_75: int] :
( ( ord_less_eq_int @ B_54 @ A_75 )
=> ( ( A_75 != B_54 )
=> ( ord_less_int @ B_54 @ A_75 ) ) ) ).
thf(fact_1954_order__le__neq__trans,axiom,
! [A_74: rat,B_53: rat] :
( ( ord_less_eq_rat @ A_74 @ B_53 )
=> ( ( A_74 != B_53 )
=> ( ord_less_rat @ A_74 @ B_53 ) ) ) ).
thf(fact_1955_order__le__neq__trans,axiom,
! [A_74: int > $o,B_53: int > $o] :
( ( ord_less_eq_int_o @ A_74 @ B_53 )
=> ( ( A_74 != B_53 )
=> ( ord_less_int_o @ A_74 @ B_53 ) ) ) ).
thf(fact_1956_order__le__neq__trans,axiom,
! [A_74: nat > $o,B_53: nat > $o] :
( ( ord_less_eq_nat_o @ A_74 @ B_53 )
=> ( ( A_74 != B_53 )
=> ( ord_less_nat_o @ A_74 @ B_53 ) ) ) ).
thf(fact_1957_order__le__neq__trans,axiom,
! [A_74: quickcheck_code_int,B_53: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_74 @ B_53 )
=> ( ( A_74 != B_53 )
=> ( ord_le1860547276de_int @ A_74 @ B_53 ) ) ) ).
thf(fact_1958_order__le__neq__trans,axiom,
! [A_74: real,B_53: real] :
( ( ord_less_eq_real @ A_74 @ B_53 )
=> ( ( A_74 != B_53 )
=> ( ord_less_real @ A_74 @ B_53 ) ) ) ).
thf(fact_1959_order__le__neq__trans,axiom,
! [A_74: nat,B_53: nat] :
( ( ord_less_eq_nat @ A_74 @ B_53 )
=> ( ( A_74 != B_53 )
=> ( ord_less_nat @ A_74 @ B_53 ) ) ) ).
thf(fact_1960_order__le__neq__trans,axiom,
! [A_74: int,B_53: int] :
( ( ord_less_eq_int @ A_74 @ B_53 )
=> ( ( A_74 != B_53 )
=> ( ord_less_int @ A_74 @ B_53 ) ) ) ).
thf(fact_1961_order__le__imp__less__or__eq,axiom,
! [X_29: rat,Y_24: rat] :
( ( ord_less_eq_rat @ X_29 @ Y_24 )
=> ( ( ord_less_rat @ X_29 @ Y_24 )
| ( X_29 = Y_24 ) ) ) ).
thf(fact_1962_order__le__imp__less__or__eq,axiom,
! [X_29: int > $o,Y_24: int > $o] :
( ( ord_less_eq_int_o @ X_29 @ Y_24 )
=> ( ( ord_less_int_o @ X_29 @ Y_24 )
| ( X_29 = Y_24 ) ) ) ).
thf(fact_1963_order__le__imp__less__or__eq,axiom,
! [X_29: nat > $o,Y_24: nat > $o] :
( ( ord_less_eq_nat_o @ X_29 @ Y_24 )
=> ( ( ord_less_nat_o @ X_29 @ Y_24 )
| ( X_29 = Y_24 ) ) ) ).
thf(fact_1964_order__le__imp__less__or__eq,axiom,
! [X_29: quickcheck_code_int,Y_24: quickcheck_code_int] :
( ( ord_le258702272de_int @ X_29 @ Y_24 )
=> ( ( ord_le1860547276de_int @ X_29 @ Y_24 )
| ( X_29 = Y_24 ) ) ) ).
thf(fact_1965_order__le__imp__less__or__eq,axiom,
! [X_29: real,Y_24: real] :
( ( ord_less_eq_real @ X_29 @ Y_24 )
=> ( ( ord_less_real @ X_29 @ Y_24 )
| ( X_29 = Y_24 ) ) ) ).
thf(fact_1966_order__le__imp__less__or__eq,axiom,
! [X_29: nat,Y_24: nat] :
( ( ord_less_eq_nat @ X_29 @ Y_24 )
=> ( ( ord_less_nat @ X_29 @ Y_24 )
| ( X_29 = Y_24 ) ) ) ).
thf(fact_1967_order__le__imp__less__or__eq,axiom,
! [X_29: int,Y_24: int] :
( ( ord_less_eq_int @ X_29 @ Y_24 )
=> ( ( ord_less_int @ X_29 @ Y_24 )
| ( X_29 = Y_24 ) ) ) ).
thf(fact_1968_linorder__antisym__conv2,axiom,
! [X_28: rat,Y_23: rat] :
( ( ord_less_eq_rat @ X_28 @ Y_23 )
=> ( ~ ( ord_less_rat @ X_28 @ Y_23 )
<=> ( X_28 = Y_23 ) ) ) ).
thf(fact_1969_linorder__antisym__conv2,axiom,
! [X_28: quickcheck_code_int,Y_23: quickcheck_code_int] :
( ( ord_le258702272de_int @ X_28 @ Y_23 )
=> ( ~ ( ord_le1860547276de_int @ X_28 @ Y_23 )
<=> ( X_28 = Y_23 ) ) ) ).
thf(fact_1970_linorder__antisym__conv2,axiom,
! [X_28: real,Y_23: real] :
( ( ord_less_eq_real @ X_28 @ Y_23 )
=> ( ~ ( ord_less_real @ X_28 @ Y_23 )
<=> ( X_28 = Y_23 ) ) ) ).
thf(fact_1971_linorder__antisym__conv2,axiom,
! [X_28: nat,Y_23: nat] :
( ( ord_less_eq_nat @ X_28 @ Y_23 )
=> ( ~ ( ord_less_nat @ X_28 @ Y_23 )
<=> ( X_28 = Y_23 ) ) ) ).
thf(fact_1972_linorder__antisym__conv2,axiom,
! [X_28: int,Y_23: int] :
( ( ord_less_eq_int @ X_28 @ Y_23 )
=> ( ~ ( ord_less_int @ X_28 @ Y_23 )
<=> ( X_28 = Y_23 ) ) ) ).
thf(fact_1973_order__less__imp__le,axiom,
! [X_27: rat,Y_22: rat] :
( ( ord_less_rat @ X_27 @ Y_22 )
=> ( ord_less_eq_rat @ X_27 @ Y_22 ) ) ).
thf(fact_1974_order__less__imp__le,axiom,
! [X_27: int > $o,Y_22: int > $o] :
( ( ord_less_int_o @ X_27 @ Y_22 )
=> ( ord_less_eq_int_o @ X_27 @ Y_22 ) ) ).
thf(fact_1975_order__less__imp__le,axiom,
! [X_27: nat > $o,Y_22: nat > $o] :
( ( ord_less_nat_o @ X_27 @ Y_22 )
=> ( ord_less_eq_nat_o @ X_27 @ Y_22 ) ) ).
thf(fact_1976_order__less__imp__le,axiom,
! [X_27: quickcheck_code_int,Y_22: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_27 @ Y_22 )
=> ( ord_le258702272de_int @ X_27 @ Y_22 ) ) ).
thf(fact_1977_order__less__imp__le,axiom,
! [X_27: real,Y_22: real] :
( ( ord_less_real @ X_27 @ Y_22 )
=> ( ord_less_eq_real @ X_27 @ Y_22 ) ) ).
thf(fact_1978_order__less__imp__le,axiom,
! [X_27: nat,Y_22: nat] :
( ( ord_less_nat @ X_27 @ Y_22 )
=> ( ord_less_eq_nat @ X_27 @ Y_22 ) ) ).
thf(fact_1979_order__less__imp__le,axiom,
! [X_27: int,Y_22: int] :
( ( ord_less_int @ X_27 @ Y_22 )
=> ( ord_less_eq_int @ X_27 @ Y_22 ) ) ).
thf(fact_1980_leD,axiom,
! [Y_21: rat,X_26: rat] :
( ( ord_less_eq_rat @ Y_21 @ X_26 )
=> ~ ( ord_less_rat @ X_26 @ Y_21 ) ) ).
thf(fact_1981_leD,axiom,
! [Y_21: quickcheck_code_int,X_26: quickcheck_code_int] :
( ( ord_le258702272de_int @ Y_21 @ X_26 )
=> ~ ( ord_le1860547276de_int @ X_26 @ Y_21 ) ) ).
thf(fact_1982_leD,axiom,
! [Y_21: real,X_26: real] :
( ( ord_less_eq_real @ Y_21 @ X_26 )
=> ~ ( ord_less_real @ X_26 @ Y_21 ) ) ).
thf(fact_1983_leD,axiom,
! [Y_21: nat,X_26: nat] :
( ( ord_less_eq_nat @ Y_21 @ X_26 )
=> ~ ( ord_less_nat @ X_26 @ Y_21 ) ) ).
thf(fact_1984_leD,axiom,
! [Y_21: int,X_26: int] :
( ( ord_less_eq_int @ Y_21 @ X_26 )
=> ~ ( ord_less_int @ X_26 @ Y_21 ) ) ).
thf(fact_1985_xt1_I12_J,axiom,
! [A_73: rat,B_52: rat] :
( ( A_73 != B_52 )
=> ( ( ord_less_eq_rat @ B_52 @ A_73 )
=> ( ord_less_rat @ B_52 @ A_73 ) ) ) ).
thf(fact_1986_xt1_I12_J,axiom,
! [A_73: int > $o,B_52: int > $o] :
( ( A_73 != B_52 )
=> ( ( ord_less_eq_int_o @ B_52 @ A_73 )
=> ( ord_less_int_o @ B_52 @ A_73 ) ) ) ).
thf(fact_1987_xt1_I12_J,axiom,
! [A_73: nat > $o,B_52: nat > $o] :
( ( A_73 != B_52 )
=> ( ( ord_less_eq_nat_o @ B_52 @ A_73 )
=> ( ord_less_nat_o @ B_52 @ A_73 ) ) ) ).
thf(fact_1988_xt1_I12_J,axiom,
! [A_73: quickcheck_code_int,B_52: quickcheck_code_int] :
( ( A_73 != B_52 )
=> ( ( ord_le258702272de_int @ B_52 @ A_73 )
=> ( ord_le1860547276de_int @ B_52 @ A_73 ) ) ) ).
thf(fact_1989_xt1_I12_J,axiom,
! [A_73: real,B_52: real] :
( ( A_73 != B_52 )
=> ( ( ord_less_eq_real @ B_52 @ A_73 )
=> ( ord_less_real @ B_52 @ A_73 ) ) ) ).
thf(fact_1990_xt1_I12_J,axiom,
! [A_73: nat,B_52: nat] :
( ( A_73 != B_52 )
=> ( ( ord_less_eq_nat @ B_52 @ A_73 )
=> ( ord_less_nat @ B_52 @ A_73 ) ) ) ).
thf(fact_1991_xt1_I12_J,axiom,
! [A_73: int,B_52: int] :
( ( A_73 != B_52 )
=> ( ( ord_less_eq_int @ B_52 @ A_73 )
=> ( ord_less_int @ B_52 @ A_73 ) ) ) ).
thf(fact_1992_order__neq__le__trans,axiom,
! [A_72: rat,B_51: rat] :
( ( A_72 != B_51 )
=> ( ( ord_less_eq_rat @ A_72 @ B_51 )
=> ( ord_less_rat @ A_72 @ B_51 ) ) ) ).
thf(fact_1993_order__neq__le__trans,axiom,
! [A_72: int > $o,B_51: int > $o] :
( ( A_72 != B_51 )
=> ( ( ord_less_eq_int_o @ A_72 @ B_51 )
=> ( ord_less_int_o @ A_72 @ B_51 ) ) ) ).
thf(fact_1994_order__neq__le__trans,axiom,
! [A_72: nat > $o,B_51: nat > $o] :
( ( A_72 != B_51 )
=> ( ( ord_less_eq_nat_o @ A_72 @ B_51 )
=> ( ord_less_nat_o @ A_72 @ B_51 ) ) ) ).
thf(fact_1995_order__neq__le__trans,axiom,
! [A_72: quickcheck_code_int,B_51: quickcheck_code_int] :
( ( A_72 != B_51 )
=> ( ( ord_le258702272de_int @ A_72 @ B_51 )
=> ( ord_le1860547276de_int @ A_72 @ B_51 ) ) ) ).
thf(fact_1996_order__neq__le__trans,axiom,
! [A_72: real,B_51: real] :
( ( A_72 != B_51 )
=> ( ( ord_less_eq_real @ A_72 @ B_51 )
=> ( ord_less_real @ A_72 @ B_51 ) ) ) ).
thf(fact_1997_order__neq__le__trans,axiom,
! [A_72: nat,B_51: nat] :
( ( A_72 != B_51 )
=> ( ( ord_less_eq_nat @ A_72 @ B_51 )
=> ( ord_less_nat @ A_72 @ B_51 ) ) ) ).
thf(fact_1998_order__neq__le__trans,axiom,
! [A_72: int,B_51: int] :
( ( A_72 != B_51 )
=> ( ( ord_less_eq_int @ A_72 @ B_51 )
=> ( ord_less_int @ A_72 @ B_51 ) ) ) ).
thf(fact_1999_linorder__antisym__conv1,axiom,
! [X_25: rat,Y_20: rat] :
( ~ ( ord_less_rat @ X_25 @ Y_20 )
=> ( ( ord_less_eq_rat @ X_25 @ Y_20 )
<=> ( X_25 = Y_20 ) ) ) ).
thf(fact_2000_linorder__antisym__conv1,axiom,
! [X_25: quickcheck_code_int,Y_20: quickcheck_code_int] :
( ~ ( ord_le1860547276de_int @ X_25 @ Y_20 )
=> ( ( ord_le258702272de_int @ X_25 @ Y_20 )
<=> ( X_25 = Y_20 ) ) ) ).
thf(fact_2001_linorder__antisym__conv1,axiom,
! [X_25: real,Y_20: real] :
( ~ ( ord_less_real @ X_25 @ Y_20 )
=> ( ( ord_less_eq_real @ X_25 @ Y_20 )
<=> ( X_25 = Y_20 ) ) ) ).
thf(fact_2002_linorder__antisym__conv1,axiom,
! [X_25: nat,Y_20: nat] :
( ~ ( ord_less_nat @ X_25 @ Y_20 )
=> ( ( ord_less_eq_nat @ X_25 @ Y_20 )
<=> ( X_25 = Y_20 ) ) ) ).
thf(fact_2003_linorder__antisym__conv1,axiom,
! [X_25: int,Y_20: int] :
( ~ ( ord_less_int @ X_25 @ Y_20 )
=> ( ( ord_less_eq_int @ X_25 @ Y_20 )
<=> ( X_25 = Y_20 ) ) ) ).
thf(fact_2004_not__leE,axiom,
! [Y_19: rat,X_24: rat] :
( ~ ( ord_less_eq_rat @ Y_19 @ X_24 )
=> ( ord_less_rat @ X_24 @ Y_19 ) ) ).
thf(fact_2005_not__leE,axiom,
! [Y_19: quickcheck_code_int,X_24: quickcheck_code_int] :
( ~ ( ord_le258702272de_int @ Y_19 @ X_24 )
=> ( ord_le1860547276de_int @ X_24 @ Y_19 ) ) ).
thf(fact_2006_not__leE,axiom,
! [Y_19: real,X_24: real] :
( ~ ( ord_less_eq_real @ Y_19 @ X_24 )
=> ( ord_less_real @ X_24 @ Y_19 ) ) ).
thf(fact_2007_not__leE,axiom,
! [Y_19: nat,X_24: nat] :
( ~ ( ord_less_eq_nat @ Y_19 @ X_24 )
=> ( ord_less_nat @ X_24 @ Y_19 ) ) ).
thf(fact_2008_not__leE,axiom,
! [Y_19: int,X_24: int] :
( ~ ( ord_less_eq_int @ Y_19 @ X_24 )
=> ( ord_less_int @ X_24 @ Y_19 ) ) ).
thf(fact_2009_leI,axiom,
! [X_23: rat,Y_18: rat] :
( ~ ( ord_less_rat @ X_23 @ Y_18 )
=> ( ord_less_eq_rat @ Y_18 @ X_23 ) ) ).
thf(fact_2010_leI,axiom,
! [X_23: quickcheck_code_int,Y_18: quickcheck_code_int] :
( ~ ( ord_le1860547276de_int @ X_23 @ Y_18 )
=> ( ord_le258702272de_int @ Y_18 @ X_23 ) ) ).
thf(fact_2011_leI,axiom,
! [X_23: real,Y_18: real] :
( ~ ( ord_less_real @ X_23 @ Y_18 )
=> ( ord_less_eq_real @ Y_18 @ X_23 ) ) ).
thf(fact_2012_leI,axiom,
! [X_23: nat,Y_18: nat] :
( ~ ( ord_less_nat @ X_23 @ Y_18 )
=> ( ord_less_eq_nat @ Y_18 @ X_23 ) ) ).
thf(fact_2013_leI,axiom,
! [X_23: int,Y_18: int] :
( ~ ( ord_less_int @ X_23 @ Y_18 )
=> ( ord_less_eq_int @ Y_18 @ X_23 ) ) ).
thf(fact_2014_order__le__less,axiom,
! [X_22: rat,Y_17: rat] :
( ( ord_less_eq_rat @ X_22 @ Y_17 )
<=> ( ( ord_less_rat @ X_22 @ Y_17 )
| ( X_22 = Y_17 ) ) ) ).
thf(fact_2015_order__le__less,axiom,
! [X_22: int > $o,Y_17: int > $o] :
( ( ord_less_eq_int_o @ X_22 @ Y_17 )
<=> ( ( ord_less_int_o @ X_22 @ Y_17 )
| ( X_22 = Y_17 ) ) ) ).
thf(fact_2016_order__le__less,axiom,
! [X_22: nat > $o,Y_17: nat > $o] :
( ( ord_less_eq_nat_o @ X_22 @ Y_17 )
<=> ( ( ord_less_nat_o @ X_22 @ Y_17 )
| ( X_22 = Y_17 ) ) ) ).
thf(fact_2017_order__le__less,axiom,
! [X_22: quickcheck_code_int,Y_17: quickcheck_code_int] :
( ( ord_le258702272de_int @ X_22 @ Y_17 )
<=> ( ( ord_le1860547276de_int @ X_22 @ Y_17 )
| ( X_22 = Y_17 ) ) ) ).
thf(fact_2018_order__le__less,axiom,
! [X_22: real,Y_17: real] :
( ( ord_less_eq_real @ X_22 @ Y_17 )
<=> ( ( ord_less_real @ X_22 @ Y_17 )
| ( X_22 = Y_17 ) ) ) ).
thf(fact_2019_order__le__less,axiom,
! [X_22: nat,Y_17: nat] :
( ( ord_less_eq_nat @ X_22 @ Y_17 )
<=> ( ( ord_less_nat @ X_22 @ Y_17 )
| ( X_22 = Y_17 ) ) ) ).
thf(fact_2020_order__le__less,axiom,
! [X_22: int,Y_17: int] :
( ( ord_less_eq_int @ X_22 @ Y_17 )
<=> ( ( ord_less_int @ X_22 @ Y_17 )
| ( X_22 = Y_17 ) ) ) ).
thf(fact_2021_less__le__not__le,axiom,
! [X_21: rat,Y_16: rat] :
( ( ord_less_rat @ X_21 @ Y_16 )
<=> ( ( ord_less_eq_rat @ X_21 @ Y_16 )
& ~ ( ord_less_eq_rat @ Y_16 @ X_21 ) ) ) ).
thf(fact_2022_less__le__not__le,axiom,
! [X_21: int > $o,Y_16: int > $o] :
( ( ord_less_int_o @ X_21 @ Y_16 )
<=> ( ( ord_less_eq_int_o @ X_21 @ Y_16 )
& ~ ( ord_less_eq_int_o @ Y_16 @ X_21 ) ) ) ).
thf(fact_2023_less__le__not__le,axiom,
! [X_21: nat > $o,Y_16: nat > $o] :
( ( ord_less_nat_o @ X_21 @ Y_16 )
<=> ( ( ord_less_eq_nat_o @ X_21 @ Y_16 )
& ~ ( ord_less_eq_nat_o @ Y_16 @ X_21 ) ) ) ).
thf(fact_2024_less__le__not__le,axiom,
! [X_21: quickcheck_code_int,Y_16: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_21 @ Y_16 )
<=> ( ( ord_le258702272de_int @ X_21 @ Y_16 )
& ~ ( ord_le258702272de_int @ Y_16 @ X_21 ) ) ) ).
thf(fact_2025_less__le__not__le,axiom,
! [X_21: real,Y_16: real] :
( ( ord_less_real @ X_21 @ Y_16 )
<=> ( ( ord_less_eq_real @ X_21 @ Y_16 )
& ~ ( ord_less_eq_real @ Y_16 @ X_21 ) ) ) ).
thf(fact_2026_less__le__not__le,axiom,
! [X_21: nat,Y_16: nat] :
( ( ord_less_nat @ X_21 @ Y_16 )
<=> ( ( ord_less_eq_nat @ X_21 @ Y_16 )
& ~ ( ord_less_eq_nat @ Y_16 @ X_21 ) ) ) ).
thf(fact_2027_less__le__not__le,axiom,
! [X_21: int,Y_16: int] :
( ( ord_less_int @ X_21 @ Y_16 )
<=> ( ( ord_less_eq_int @ X_21 @ Y_16 )
& ~ ( ord_less_eq_int @ Y_16 @ X_21 ) ) ) ).
thf(fact_2028_order__less__le,axiom,
! [X_20: rat,Y_15: rat] :
( ( ord_less_rat @ X_20 @ Y_15 )
<=> ( ( ord_less_eq_rat @ X_20 @ Y_15 )
& ( X_20 != Y_15 ) ) ) ).
thf(fact_2029_order__less__le,axiom,
! [X_20: int > $o,Y_15: int > $o] :
( ( ord_less_int_o @ X_20 @ Y_15 )
<=> ( ( ord_less_eq_int_o @ X_20 @ Y_15 )
& ( X_20 != Y_15 ) ) ) ).
thf(fact_2030_order__less__le,axiom,
! [X_20: nat > $o,Y_15: nat > $o] :
( ( ord_less_nat_o @ X_20 @ Y_15 )
<=> ( ( ord_less_eq_nat_o @ X_20 @ Y_15 )
& ( X_20 != Y_15 ) ) ) ).
thf(fact_2031_order__less__le,axiom,
! [X_20: quickcheck_code_int,Y_15: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_20 @ Y_15 )
<=> ( ( ord_le258702272de_int @ X_20 @ Y_15 )
& ( X_20 != Y_15 ) ) ) ).
thf(fact_2032_order__less__le,axiom,
! [X_20: real,Y_15: real] :
( ( ord_less_real @ X_20 @ Y_15 )
<=> ( ( ord_less_eq_real @ X_20 @ Y_15 )
& ( X_20 != Y_15 ) ) ) ).
thf(fact_2033_order__less__le,axiom,
! [X_20: nat,Y_15: nat] :
( ( ord_less_nat @ X_20 @ Y_15 )
<=> ( ( ord_less_eq_nat @ X_20 @ Y_15 )
& ( X_20 != Y_15 ) ) ) ).
thf(fact_2034_order__less__le,axiom,
! [X_20: int,Y_15: int] :
( ( ord_less_int @ X_20 @ Y_15 )
<=> ( ( ord_less_eq_int @ X_20 @ Y_15 )
& ( X_20 != Y_15 ) ) ) ).
thf(fact_2035_linorder__le__less__linear,axiom,
! [X_19: rat,Y_14: rat] :
( ( ord_less_eq_rat @ X_19 @ Y_14 )
| ( ord_less_rat @ Y_14 @ X_19 ) ) ).
thf(fact_2036_linorder__le__less__linear,axiom,
! [X_19: quickcheck_code_int,Y_14: quickcheck_code_int] :
( ( ord_le258702272de_int @ X_19 @ Y_14 )
| ( ord_le1860547276de_int @ Y_14 @ X_19 ) ) ).
thf(fact_2037_linorder__le__less__linear,axiom,
! [X_19: real,Y_14: real] :
( ( ord_less_eq_real @ X_19 @ Y_14 )
| ( ord_less_real @ Y_14 @ X_19 ) ) ).
thf(fact_2038_linorder__le__less__linear,axiom,
! [X_19: nat,Y_14: nat] :
( ( ord_less_eq_nat @ X_19 @ Y_14 )
| ( ord_less_nat @ Y_14 @ X_19 ) ) ).
thf(fact_2039_linorder__le__less__linear,axiom,
! [X_19: int,Y_14: int] :
( ( ord_less_eq_int @ X_19 @ Y_14 )
| ( ord_less_int @ Y_14 @ X_19 ) ) ).
thf(fact_2040_linorder__not__le,axiom,
! [X_18: rat,Y_13: rat] :
( ~ ( ord_less_eq_rat @ X_18 @ Y_13 )
<=> ( ord_less_rat @ Y_13 @ X_18 ) ) ).
thf(fact_2041_linorder__not__le,axiom,
! [X_18: quickcheck_code_int,Y_13: quickcheck_code_int] :
( ~ ( ord_le258702272de_int @ X_18 @ Y_13 )
<=> ( ord_le1860547276de_int @ Y_13 @ X_18 ) ) ).
thf(fact_2042_linorder__not__le,axiom,
! [X_18: real,Y_13: real] :
( ~ ( ord_less_eq_real @ X_18 @ Y_13 )
<=> ( ord_less_real @ Y_13 @ X_18 ) ) ).
thf(fact_2043_linorder__not__le,axiom,
! [X_18: nat,Y_13: nat] :
( ~ ( ord_less_eq_nat @ X_18 @ Y_13 )
<=> ( ord_less_nat @ Y_13 @ X_18 ) ) ).
thf(fact_2044_linorder__not__le,axiom,
! [X_18: int,Y_13: int] :
( ~ ( ord_less_eq_int @ X_18 @ Y_13 )
<=> ( ord_less_int @ Y_13 @ X_18 ) ) ).
thf(fact_2045_linorder__not__less,axiom,
! [X_17: rat,Y_12: rat] :
( ~ ( ord_less_rat @ X_17 @ Y_12 )
<=> ( ord_less_eq_rat @ Y_12 @ X_17 ) ) ).
thf(fact_2046_linorder__not__less,axiom,
! [X_17: quickcheck_code_int,Y_12: quickcheck_code_int] :
( ~ ( ord_le1860547276de_int @ X_17 @ Y_12 )
<=> ( ord_le258702272de_int @ Y_12 @ X_17 ) ) ).
thf(fact_2047_linorder__not__less,axiom,
! [X_17: real,Y_12: real] :
( ~ ( ord_less_real @ X_17 @ Y_12 )
<=> ( ord_less_eq_real @ Y_12 @ X_17 ) ) ).
thf(fact_2048_linorder__not__less,axiom,
! [X_17: nat,Y_12: nat] :
( ~ ( ord_less_nat @ X_17 @ Y_12 )
<=> ( ord_less_eq_nat @ Y_12 @ X_17 ) ) ).
thf(fact_2049_linorder__not__less,axiom,
! [X_17: int,Y_12: int] :
( ~ ( ord_less_int @ X_17 @ Y_12 )
<=> ( ord_less_eq_int @ Y_12 @ X_17 ) ) ).
thf(fact_2050_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_2051_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_2052_power2__sum,axiom,
! [X_16: rat,Y_11: rat] :
( ( power_power_rat @ ( plus_plus_rat @ X_16 @ Y_11 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X_16 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_rat @ Y_11 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( number_number_of_rat @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_16 ) @ Y_11 ) ) ) ).
thf(fact_2053_power2__sum,axiom,
! [X_16: complex,Y_11: complex] :
( ( power_power_complex @ ( plus_plus_complex @ X_16 @ Y_11 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X_16 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_complex @ Y_11 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( number528085621omplex @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_16 ) @ Y_11 ) ) ) ).
thf(fact_2054_power2__sum,axiom,
! [X_16: real,Y_11: real] :
( ( power_power_real @ ( plus_plus_real @ X_16 @ Y_11 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X_16 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_11 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_16 ) @ Y_11 ) ) ) ).
thf(fact_2055_power2__sum,axiom,
! [X_16: nat,Y_11: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X_16 @ Y_11 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X_16 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_nat @ Y_11 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_16 ) @ Y_11 ) ) ) ).
thf(fact_2056_power2__sum,axiom,
! [X_16: int,Y_11: int] :
( ( power_power_int @ ( plus_plus_int @ X_16 @ Y_11 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X_16 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_11 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_16 ) @ Y_11 ) ) ) ).
thf(fact_2057_self__quotient__aux2,axiom,
! [R_1: int,Q: int,A_1: int] :
( ( ord_less_int @ zero_zero_int @ A_1 )
=> ( ( A_1
= ( plus_plus_int @ R_1 @ ( times_times_int @ A_1 @ Q ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ord_less_eq_int @ Q @ one_one_int ) ) ) ) ).
thf(fact_2058_self__quotient__aux1,axiom,
! [R_1: int,Q: int,A_1: int] :
( ( ord_less_int @ zero_zero_int @ A_1 )
=> ( ( A_1
= ( plus_plus_int @ R_1 @ ( times_times_int @ A_1 @ Q ) ) )
=> ( ( ord_less_int @ R_1 @ A_1 )
=> ( ord_less_eq_int @ one_one_int @ Q ) ) ) ) ).
thf(fact_2059_ex__least__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ( P @ N )
=> ? [K_1: nat] :
( ( ord_less_nat @ K_1 @ N )
& ! [I_1: nat] :
( ( ord_less_eq_nat @ I_1 @ K_1 )
=> ~ ( P @ I_1 ) )
& ( P @ ( plus_plus_nat @ K_1 @ one_one_nat ) ) ) ) ) ).
thf(fact_2060_incr__mult__lemma,axiom,
! [K: int,P: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( plus_plus_int @ X_1 @ D_1 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( plus_plus_int @ X_1 @ ( times_times_int @ K @ D_1 ) ) ) ) ) ) ) ).
thf(fact_2061_smaller_I2_J,axiom,
~ ( ( ord_less_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
=> ~ ( twoSqu1152398899sum2sq @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) ) ).
thf(fact_2062_q__pos__lemma,axiom,
! [B_5: int,Q_3: int,R_3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_5 @ Q_3 ) @ 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_3 ) ) ) ) ).
thf(fact_2063_q__neg__lemma,axiom,
! [B_5: int,Q_3: int,R_3: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_5 @ Q_3 ) @ 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_3 @ zero_zero_int ) ) ) ) ).
thf(fact_2064_unique__quotient__lemma,axiom,
! [B_1: int,Q_3: int,R_3: int,Q: int,R_1: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q_3 ) @ R_3 ) @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R_1 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_3 )
=> ( ( ord_less_int @ R_3 @ B_1 )
=> ( ( ord_less_int @ R_1 @ B_1 )
=> ( ord_less_eq_int @ Q_3 @ Q ) ) ) ) ) ).
thf(fact_2065_zdiv__mono2__lemma,axiom,
! [B_1: int,Q: int,R_1: int,B_5: int,Q_3: int,R_3: int] :
( ( ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R_1 )
= ( plus_plus_int @ ( times_times_int @ B_5 @ Q_3 ) @ R_3 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_5 @ Q_3 ) @ 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_1 )
=> ( ord_less_eq_int @ Q @ Q_3 ) ) ) ) ) ) ) ).
thf(fact_2066_unique__quotient__lemma__neg,axiom,
! [B_1: int,Q_3: int,R_3: int,Q: int,R_1: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q_3 ) @ R_3 ) @ ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R_1 ) )
=> ( ( ord_less_eq_int @ R_1 @ zero_zero_int )
=> ( ( ord_less_int @ B_1 @ R_1 )
=> ( ( ord_less_int @ B_1 @ R_3 )
=> ( ord_less_eq_int @ Q @ Q_3 ) ) ) ) ) ).
thf(fact_2067_nQ1,axiom,
~ ( twoSqu1152398899sum2sq @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ zero_zero_nat ) ) ) ) ).
thf(fact_2068_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_2069_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_2070_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_2071_IH,axiom,
( ( ord_less_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) )
& ( twoSqu1152398899sum2sq @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) ) ).
thf(fact_2072__096_B_Bthesis_O_A_I_B_Bx_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_061_A_I4_,axiom,
~ ! [X_1: int,Y_1: int] :
( ( plus_plus_int @ ( power_power_int @ X_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
!= ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) ).
thf(fact_2073_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_2074_p,axiom,
zprime @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) ).
thf(fact_2075__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A,axiom,
( ( t = one_one_int )
=> ? [X_1: int,Y_1: int] :
( ( plus_plus_int @ ( power_power_int @ X_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_1 @ ( 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_2076_less__fun__def,axiom,
! [F_3: int > $o,G_1: int > $o] :
( ( ord_less_int_o @ F_3 @ G_1 )
<=> ( ( ord_less_eq_int_o @ F_3 @ G_1 )
& ~ ( ord_less_eq_int_o @ G_1 @ F_3 ) ) ) ).
thf(fact_2077_less__fun__def,axiom,
! [F_3: nat > $o,G_1: nat > $o] :
( ( ord_less_nat_o @ F_3 @ G_1 )
<=> ( ( ord_less_eq_nat_o @ F_3 @ G_1 )
& ~ ( ord_less_eq_nat_o @ G_1 @ F_3 ) ) ) ).
thf(fact_2078_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_2079_real__le__trans,axiom,
! [K: real,I: real,J: real] :
( ( ord_less_eq_real @ I @ J )
=> ( ( ord_less_eq_real @ J @ K )
=> ( ord_less_eq_real @ I @ K ) ) ) ).
thf(fact_2080_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_2081_real__le__refl,axiom,
! [W: real] : ( ord_less_eq_real @ W @ W ) ).
thf(fact_2082_nat__mult__commute,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ N )
= ( times_times_nat @ N @ M ) ) ).
thf(fact_2083_nat__mult__assoc,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( times_times_nat @ M @ N ) @ K )
= ( times_times_nat @ M @ ( times_times_nat @ N @ K ) ) ) ).
thf(fact_2084_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_2085_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_2086_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_2087_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
<=> ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
thf(fact_2088_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
<=> ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
thf(fact_2089_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_15: complex,P_6: nat,Q_5: nat] :
( ( power_power_complex @ ( power_power_complex @ X_15 @ P_6 ) @ Q_5 )
= ( power_power_complex @ X_15 @ ( times_times_nat @ P_6 @ Q_5 ) ) ) ).
thf(fact_2090_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_15: real,P_6: nat,Q_5: nat] :
( ( power_power_real @ ( power_power_real @ X_15 @ P_6 ) @ Q_5 )
= ( power_power_real @ X_15 @ ( times_times_nat @ P_6 @ Q_5 ) ) ) ).
thf(fact_2091_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_15: nat,P_6: nat,Q_5: nat] :
( ( power_power_nat @ ( power_power_nat @ X_15 @ P_6 ) @ Q_5 )
= ( power_power_nat @ X_15 @ ( times_times_nat @ P_6 @ Q_5 ) ) ) ).
thf(fact_2092_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_15: int,P_6: nat,Q_5: nat] :
( ( power_power_int @ ( power_power_int @ X_15 @ P_6 ) @ Q_5 )
= ( power_power_int @ X_15 @ ( times_times_nat @ P_6 @ Q_5 ) ) ) ).
thf(fact_2093_power__mult,axiom,
! [A_71: complex,M_14: nat,N_20: nat] :
( ( power_power_complex @ A_71 @ ( times_times_nat @ M_14 @ N_20 ) )
= ( power_power_complex @ ( power_power_complex @ A_71 @ M_14 ) @ N_20 ) ) ).
thf(fact_2094_power__mult,axiom,
! [A_71: real,M_14: nat,N_20: nat] :
( ( power_power_real @ A_71 @ ( times_times_nat @ M_14 @ N_20 ) )
= ( power_power_real @ ( power_power_real @ A_71 @ M_14 ) @ N_20 ) ) ).
thf(fact_2095_power__mult,axiom,
! [A_71: nat,M_14: nat,N_20: nat] :
( ( power_power_nat @ A_71 @ ( times_times_nat @ M_14 @ N_20 ) )
= ( power_power_nat @ ( power_power_nat @ A_71 @ M_14 ) @ N_20 ) ) ).
thf(fact_2096_power__mult,axiom,
! [A_71: int,M_14: nat,N_20: nat] :
( ( power_power_int @ A_71 @ ( times_times_nat @ M_14 @ N_20 ) )
= ( power_power_int @ ( power_power_int @ A_71 @ M_14 ) @ N_20 ) ) ).
thf(fact_2097_real__mult__left__cancel,axiom,
! [A_1: real,B_1: real,C: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A_1 )
= ( times_times_real @ C @ B_1 ) )
<=> ( A_1 = B_1 ) ) ) ).
thf(fact_2098_real__mult__right__cancel,axiom,
! [A_1: real,B_1: real,C: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A_1 @ C )
= ( times_times_real @ B_1 @ C ) )
<=> ( A_1 = B_1 ) ) ) ).
thf(fact_2099_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
thf(fact_2100_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
thf(fact_2101_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
thf(fact_2102_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
thf(fact_2103_mult__le__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
thf(fact_2104_mult__le__mono2,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
thf(fact_2105_mult__le__mono,axiom,
! [K: nat,L: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
thf(fact_2106_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_2107_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_2108_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
thf(fact_2109_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_2110_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
thf(fact_2111_real__mult__1,axiom,
! [Z_1: real] :
( ( times_times_real @ one_one_real @ Z_1 )
= Z_1 ) ).
thf(fact_2112_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_2113_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_2114_mult__less__mono2,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
thf(fact_2115_mult__less__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
thf(fact_2116_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_2117_mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_2118_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_2119_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_2120_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_2121_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_2122_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_2123_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_2124_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_2125_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_2126_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_2127_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_2128_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_2129_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_2130_is__mult__sum2sq,axiom,
! [Y: int,X: int] :
( ( twoSqu1152398899sum2sq @ X )
=> ( ( twoSqu1152398899sum2sq @ Y )
=> ( twoSqu1152398899sum2sq @ ( times_times_int @ X @ Y ) ) ) ) ).
thf(fact_2131_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_2132_mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_2133_power__even__eq,axiom,
! [A_70: complex,N_19: nat] :
( ( power_power_complex @ A_70 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_19 ) )
= ( power_power_complex @ ( power_power_complex @ A_70 @ N_19 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_2134_power__even__eq,axiom,
! [A_70: real,N_19: nat] :
( ( power_power_real @ A_70 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_19 ) )
= ( power_power_real @ ( power_power_real @ A_70 @ N_19 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_2135_power__even__eq,axiom,
! [A_70: nat,N_19: nat] :
( ( power_power_nat @ A_70 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_19 ) )
= ( power_power_nat @ ( power_power_nat @ A_70 @ N_19 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_2136_power__even__eq,axiom,
! [A_70: int,N_19: nat] :
( ( power_power_int @ A_70 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_19 ) )
= ( power_power_int @ ( power_power_int @ A_70 @ N_19 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_2137_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_2138_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_2139_nat__number__of__mult__left,axiom,
! [V_1: int,K: 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 ) )
= 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 ) )
= ( times_times_nat @ ( number_number_of_nat @ ( times_times_int @ V @ V_1 ) ) @ K ) ) ) ) ).
thf(fact_2140_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_2141_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_2142_even__power__le__0__imp__0,axiom,
! [A_69: rat,K_8: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A_69 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_8 ) ) @ zero_zero_rat )
=> ( A_69 = zero_zero_rat ) ) ).
thf(fact_2143_even__power__le__0__imp__0,axiom,
! [A_69: real,K_8: nat] :
( ( ord_less_eq_real @ ( power_power_real @ A_69 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_8 ) ) @ zero_zero_real )
=> ( A_69 = zero_zero_real ) ) ).
thf(fact_2144_even__power__le__0__imp__0,axiom,
! [A_69: int,K_8: nat] :
( ( ord_less_eq_int @ ( power_power_int @ A_69 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_8 ) ) @ zero_zero_int )
=> ( A_69 = zero_zero_int ) ) ).
thf(fact_2145_zero__le__even__power_H,axiom,
! [A_68: rat,N_18: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A_68 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_18 ) ) ) ).
thf(fact_2146_zero__le__even__power_H,axiom,
! [A_68: real,N_18: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A_68 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_18 ) ) ) ).
thf(fact_2147_zero__le__even__power_H,axiom,
! [A_68: int,N_18: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A_68 @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_18 ) ) ) ).
thf(fact_2148_zdiv__mono2__neg__lemma,axiom,
! [B_1: int,Q: int,R_1: int,B_5: int,Q_3: int,R_3: int] :
( ( ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R_1 )
= ( plus_plus_int @ ( times_times_int @ B_5 @ Q_3 ) @ R_3 ) )
=> ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_5 @ Q_3 ) @ R_3 ) @ zero_zero_int )
=> ( ( ord_less_int @ R_1 @ B_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_3 )
=> ( ( ord_less_int @ zero_zero_int @ B_5 )
=> ( ( ord_less_eq_int @ B_5 @ B_1 )
=> ( ord_less_eq_int @ Q_3 @ Q ) ) ) ) ) ) ) ).
thf(fact_2149_xy,axiom,
( ( 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 ) ) ) ) )
= ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) ).
thf(fact_2150_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_2151__096_B_Bthesis_O_A_I_B_Bt_O_As_____A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A,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_2152_nat__mult__le__cancel1,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_2153_mono__nat__linear__lb,axiom,
! [M: nat,K: 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 ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
thf(fact_2154_ex__least__nat__le,axiom,
! [N: nat,P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ( P @ N )
=> ? [K_1: nat] :
( ( ord_less_eq_nat @ K_1 @ N )
& ! [I_1: nat] :
( ( ord_less_nat @ I_1 @ K_1 )
=> ~ ( P @ I_1 ) )
& ( P @ K_1 ) ) ) ) ).
thf(fact_2155__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_2156_mult__left__cancel,axiom,
! [N: nat,M: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ N )
= ( times_times_nat @ K @ M ) )
=> ( N = M ) ) ) ).
thf(fact_2157_nat__mult__less__cancel1,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_2158_real__mult__commute,axiom,
! [Z_1: real,W: real] :
( ( times_times_real @ Z_1 @ W )
= ( times_times_real @ W @ Z_1 ) ) ).
thf(fact_2159_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_2160_Bit1__Min,axiom,
( ( bit1 @ min )
= min ) ).
thf(fact_2161_rel__simps_I43_J,axiom,
! [L: int] :
( ( min
= ( bit1 @ L ) )
<=> ( min = L ) ) ).
thf(fact_2162_rel__simps_I47_J,axiom,
! [K: int] :
( ( ( bit1 @ K )
= min )
<=> ( K = min ) ) ).
thf(fact_2163_rel__simps_I40_J,axiom,
min != pls ).
thf(fact_2164_rel__simps_I37_J,axiom,
pls != min ).
thf(fact_2165_rel__simps_I42_J,axiom,
! [L: int] :
( min
!= ( bit0 @ L ) ) ).
thf(fact_2166_rel__simps_I45_J,axiom,
! [K: int] :
( ( bit0 @ K )
!= min ) ).
thf(fact_2167_rel__simps_I7_J,axiom,
~ ( ord_less_int @ min @ min ) ).
thf(fact_2168_rel__simps_I24_J,axiom,
ord_less_eq_int @ min @ min ).
thf(fact_2169_rel__simps_I9_J,axiom,
! [K: int] :
( ( ord_less_int @ min @ ( bit1 @ K ) )
<=> ( ord_less_int @ min @ K ) ) ).
thf(fact_2170_rel__simps_I13_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit1 @ K ) @ min )
<=> ( ord_less_int @ K @ min ) ) ).
thf(fact_2171_rel__simps_I6_J,axiom,
ord_less_int @ min @ pls ).
thf(fact_2172_rel__simps_I3_J,axiom,
~ ( ord_less_int @ pls @ min ) ).
thf(fact_2173_rel__simps_I8_J,axiom,
! [K: int] :
( ( ord_less_int @ min @ ( bit0 @ K ) )
<=> ( ord_less_int @ min @ K ) ) ).
thf(fact_2174_rel__simps_I30_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit1 @ K ) @ min )
<=> ( ord_less_eq_int @ K @ min ) ) ).
thf(fact_2175_rel__simps_I26_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ min @ ( bit1 @ K ) )
<=> ( ord_less_eq_int @ min @ K ) ) ).
thf(fact_2176_bin__less__0__simps_I2_J,axiom,
ord_less_int @ min @ zero_zero_int ).
thf(fact_2177_rel__simps_I20_J,axiom,
~ ( ord_less_eq_int @ pls @ min ) ).
thf(fact_2178_rel__simps_I23_J,axiom,
ord_less_eq_int @ min @ pls ).
thf(fact_2179_rel__simps_I28_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ ( bit0 @ K ) @ min )
<=> ( ord_less_eq_int @ K @ min ) ) ).
thf(fact_2180_eq__number__of__Pls__Min,axiom,
( ( number_number_of_int @ pls )
!= ( number_number_of_int @ min ) ) ).
thf(fact_2181_succ__Min,axiom,
( ( succ @ min )
= pls ) ).
thf(fact_2182_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_2183_rel__simps_I11_J,axiom,
! [K: int] :
( ( ord_less_int @ ( bit0 @ K ) @ min )
<=> ( ord_less_eq_int @ K @ min ) ) ).
thf(fact_2184_rel__simps_I25_J,axiom,
! [K: int] :
( ( ord_less_eq_int @ min @ ( bit0 @ K ) )
<=> ( ord_less_int @ min @ K ) ) ).
thf(fact_2185_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_2186_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_2187_zprime__2,axiom,
zprime @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_2188_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
<=> ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
thf(fact_2189_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
thf(fact_2190_power__m1__even,axiom,
! [N_17: nat] :
( ( power_power_rat @ ( number_number_of_rat @ min ) @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_17 ) )
= one_one_rat ) ).
thf(fact_2191_power__m1__even,axiom,
! [N_17: nat] :
( ( power_power_complex @ ( number528085621omplex @ min ) @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_17 ) )
= one_one_complex ) ).
thf(fact_2192_power__m1__even,axiom,
! [N_17: nat] :
( ( power_power_real @ ( number267125858f_real @ min ) @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_17 ) )
= one_one_real ) ).
thf(fact_2193_power__m1__even,axiom,
! [N_17: nat] :
( ( power_power_int @ ( number_number_of_int @ min ) @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_17 ) )
= one_one_int ) ).
thf(fact_2194_nat__mult__eq__cancel1,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
<=> ( M = N ) ) ) ).
thf(fact_2195__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_2196__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_2197_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_2198__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_2199__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_2200_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_2201_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_2202_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_2203__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_2204_dvd__0__right,axiom,
! [A_67: nat] : ( dvd_dvd_nat @ A_67 @ zero_zero_nat ) ).
thf(fact_2205_dvd__0__right,axiom,
! [A_67: int] : ( dvd_dvd_int @ A_67 @ zero_zero_int ) ).
thf(fact_2206_dvd__0__right,axiom,
! [A_67: rat] : ( dvd_dvd_rat @ A_67 @ zero_zero_rat ) ).
thf(fact_2207_dvd__0__right,axiom,
! [A_67: quickcheck_code_int] : ( dvd_dv1760642554de_int @ A_67 @ zero_z891286103de_int ) ).
thf(fact_2208_dvd__0__right,axiom,
! [A_67: complex] : ( dvd_dvd_complex @ A_67 @ zero_zero_complex ) ).
thf(fact_2209_dvd__0__right,axiom,
! [A_67: code_code_numeral] : ( dvd_dv174992974umeral @ A_67 @ zero_z126310315umeral ) ).
thf(fact_2210_dvd__0__right,axiom,
! [A_67: real] : ( dvd_dvd_real @ A_67 @ zero_zero_real ) ).
thf(fact_2211__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_2212__096_091s_____A_094_A2_A_061_As1_A_094_A2_093_A_Imod_A4_A_K_Am_A_L_A1_,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_2213__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_2214__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s,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_2215_dvd__diff,axiom,
! [Z_5: int,X_14: int,Y_10: int] :
( ( dvd_dvd_int @ X_14 @ Y_10 )
=> ( ( dvd_dvd_int @ X_14 @ Z_5 )
=> ( dvd_dvd_int @ X_14 @ ( minus_minus_int @ Y_10 @ Z_5 ) ) ) ) ).
thf(fact_2216_dvd__diff,axiom,
! [Z_5: rat,X_14: rat,Y_10: rat] :
( ( dvd_dvd_rat @ X_14 @ Y_10 )
=> ( ( dvd_dvd_rat @ X_14 @ Z_5 )
=> ( dvd_dvd_rat @ X_14 @ ( minus_minus_rat @ Y_10 @ Z_5 ) ) ) ) ).
thf(fact_2217_dvd__diff,axiom,
! [Z_5: complex,X_14: complex,Y_10: complex] :
( ( dvd_dvd_complex @ X_14 @ Y_10 )
=> ( ( dvd_dvd_complex @ X_14 @ Z_5 )
=> ( dvd_dvd_complex @ X_14 @ ( minus_minus_complex @ Y_10 @ Z_5 ) ) ) ) ).
thf(fact_2218_dvd__diff,axiom,
! [Z_5: real,X_14: real,Y_10: real] :
( ( dvd_dvd_real @ X_14 @ Y_10 )
=> ( ( dvd_dvd_real @ X_14 @ Z_5 )
=> ( dvd_dvd_real @ X_14 @ ( minus_minus_real @ Y_10 @ Z_5 ) ) ) ) ).
thf(fact_2219_dvd__trans,axiom,
! [C_25: nat,A_66: nat,B_50: nat] :
( ( dvd_dvd_nat @ A_66 @ B_50 )
=> ( ( dvd_dvd_nat @ B_50 @ C_25 )
=> ( dvd_dvd_nat @ A_66 @ C_25 ) ) ) ).
thf(fact_2220_dvd__trans,axiom,
! [C_25: int,A_66: int,B_50: int] :
( ( dvd_dvd_int @ A_66 @ B_50 )
=> ( ( dvd_dvd_int @ B_50 @ C_25 )
=> ( dvd_dvd_int @ A_66 @ C_25 ) ) ) ).
thf(fact_2221_dvd__refl,axiom,
! [A_65: nat] : ( dvd_dvd_nat @ A_65 @ A_65 ) ).
thf(fact_2222_dvd__refl,axiom,
! [A_65: int] : ( dvd_dvd_int @ A_65 @ A_65 ) ).
thf(fact_2223_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ M ) ) ) ).
thf(fact_2224_diff__eq__diff__eq,axiom,
! [A_64: rat,B_49: rat,C_24: rat,D_20: rat] :
( ( ( minus_minus_rat @ A_64 @ B_49 )
= ( minus_minus_rat @ C_24 @ D_20 ) )
=> ( ( A_64 = B_49 )
<=> ( C_24 = D_20 ) ) ) ).
thf(fact_2225_diff__eq__diff__eq,axiom,
! [A_64: complex,B_49: complex,C_24: complex,D_20: complex] :
( ( ( minus_minus_complex @ A_64 @ B_49 )
= ( minus_minus_complex @ C_24 @ D_20 ) )
=> ( ( A_64 = B_49 )
<=> ( C_24 = D_20 ) ) ) ).
thf(fact_2226_diff__eq__diff__eq,axiom,
! [A_64: real,B_49: real,C_24: real,D_20: real] :
( ( ( minus_minus_real @ A_64 @ B_49 )
= ( minus_minus_real @ C_24 @ D_20 ) )
=> ( ( A_64 = B_49 )
<=> ( C_24 = D_20 ) ) ) ).
thf(fact_2227_diff__eq__diff__eq,axiom,
! [A_64: int,B_49: int,C_24: int,D_20: int] :
( ( ( minus_minus_int @ A_64 @ B_49 )
= ( minus_minus_int @ C_24 @ D_20 ) )
=> ( ( A_64 = B_49 )
<=> ( C_24 = D_20 ) ) ) ).
thf(fact_2228_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_2229_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_2230_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_2231_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_2232_inv__not__p__minus__1__aux,axiom,
! [A_1: int,P_3: int] :
( ( zcong @ ( times_times_int @ A_1 @ ( minus_minus_int @ P_3 @ one_one_int ) ) @ one_one_int @ P_3 )
<=> ( zcong @ A_1 @ ( minus_minus_int @ P_3 @ one_one_int ) @ P_3 ) ) ).
thf(fact_2233_inf__period_I4_J,axiom,
! [T_4: int,D_19: int,D_18: int] :
( ( dvd_dvd_int @ D_19 @ D_18 )
=> ! [X_1: int,K_1: int] :
( ~ ( dvd_dvd_int @ D_19 @ ( plus_plus_int @ X_1 @ T_4 ) )
<=> ~ ( dvd_dvd_int @ D_19 @ ( plus_plus_int @ ( minus_minus_int @ X_1 @ ( times_times_int @ K_1 @ D_18 ) ) @ T_4 ) ) ) ) ).
thf(fact_2234_inf__period_I4_J,axiom,
! [T_4: rat,D_19: rat,D_18: rat] :
( ( dvd_dvd_rat @ D_19 @ D_18 )
=> ! [X_1: rat,K_1: rat] :
( ~ ( dvd_dvd_rat @ D_19 @ ( plus_plus_rat @ X_1 @ T_4 ) )
<=> ~ ( dvd_dvd_rat @ D_19 @ ( plus_plus_rat @ ( minus_minus_rat @ X_1 @ ( times_times_rat @ K_1 @ D_18 ) ) @ T_4 ) ) ) ) ).
thf(fact_2235_inf__period_I4_J,axiom,
! [T_4: complex,D_19: complex,D_18: complex] :
( ( dvd_dvd_complex @ D_19 @ D_18 )
=> ! [X_1: complex,K_1: complex] :
( ~ ( dvd_dvd_complex @ D_19 @ ( plus_plus_complex @ X_1 @ T_4 ) )
<=> ~ ( dvd_dvd_complex @ D_19 @ ( plus_plus_complex @ ( minus_minus_complex @ X_1 @ ( times_times_complex @ K_1 @ D_18 ) ) @ T_4 ) ) ) ) ).
thf(fact_2236_inf__period_I4_J,axiom,
! [T_4: real,D_19: real,D_18: real] :
( ( dvd_dvd_real @ D_19 @ D_18 )
=> ! [X_1: real,K_1: real] :
( ~ ( dvd_dvd_real @ D_19 @ ( plus_plus_real @ X_1 @ T_4 ) )
<=> ~ ( dvd_dvd_real @ D_19 @ ( plus_plus_real @ ( minus_minus_real @ X_1 @ ( times_times_real @ K_1 @ D_18 ) ) @ T_4 ) ) ) ) ).
thf(fact_2237_inf__period_I3_J,axiom,
! [T_3: int,D_17: int,D_16: int] :
( ( dvd_dvd_int @ D_17 @ D_16 )
=> ! [X_1: int,K_1: int] :
( ( dvd_dvd_int @ D_17 @ ( plus_plus_int @ X_1 @ T_3 ) )
<=> ( dvd_dvd_int @ D_17 @ ( plus_plus_int @ ( minus_minus_int @ X_1 @ ( times_times_int @ K_1 @ D_16 ) ) @ T_3 ) ) ) ) ).
thf(fact_2238_inf__period_I3_J,axiom,
! [T_3: rat,D_17: rat,D_16: rat] :
( ( dvd_dvd_rat @ D_17 @ D_16 )
=> ! [X_1: rat,K_1: rat] :
( ( dvd_dvd_rat @ D_17 @ ( plus_plus_rat @ X_1 @ T_3 ) )
<=> ( dvd_dvd_rat @ D_17 @ ( plus_plus_rat @ ( minus_minus_rat @ X_1 @ ( times_times_rat @ K_1 @ D_16 ) ) @ T_3 ) ) ) ) ).
thf(fact_2239_inf__period_I3_J,axiom,
! [T_3: complex,D_17: complex,D_16: complex] :
( ( dvd_dvd_complex @ D_17 @ D_16 )
=> ! [X_1: complex,K_1: complex] :
( ( dvd_dvd_complex @ D_17 @ ( plus_plus_complex @ X_1 @ T_3 ) )
<=> ( dvd_dvd_complex @ D_17 @ ( plus_plus_complex @ ( minus_minus_complex @ X_1 @ ( times_times_complex @ K_1 @ D_16 ) ) @ T_3 ) ) ) ) ).
thf(fact_2240_inf__period_I3_J,axiom,
! [T_3: real,D_17: real,D_16: real] :
( ( dvd_dvd_real @ D_17 @ D_16 )
=> ! [X_1: real,K_1: real] :
( ( dvd_dvd_real @ D_17 @ ( plus_plus_real @ X_1 @ T_3 ) )
<=> ( dvd_dvd_real @ D_17 @ ( plus_plus_real @ ( minus_minus_real @ X_1 @ ( times_times_real @ K_1 @ D_16 ) ) @ T_3 ) ) ) ) ).
thf(fact_2241_of__nat__diff,axiom,
! [N_16: nat,M_13: nat] :
( ( ord_less_eq_nat @ N_16 @ M_13 )
=> ( ( semiri151668891at_rat @ ( minus_minus_nat @ M_13 @ N_16 ) )
= ( minus_minus_rat @ ( semiri151668891at_rat @ M_13 ) @ ( semiri151668891at_rat @ N_16 ) ) ) ) ).
thf(fact_2242_of__nat__diff,axiom,
! [N_16: nat,M_13: nat] :
( ( ord_less_eq_nat @ N_16 @ M_13 )
=> ( ( semiri2020571505omplex @ ( minus_minus_nat @ M_13 @ N_16 ) )
= ( minus_minus_complex @ ( semiri2020571505omplex @ M_13 ) @ ( semiri2020571505omplex @ N_16 ) ) ) ) ).
thf(fact_2243_of__nat__diff,axiom,
! [N_16: nat,M_13: nat] :
( ( ord_less_eq_nat @ N_16 @ M_13 )
=> ( ( semiri132038758t_real @ ( minus_minus_nat @ M_13 @ N_16 ) )
= ( minus_minus_real @ ( semiri132038758t_real @ M_13 ) @ ( semiri132038758t_real @ N_16 ) ) ) ) ).
thf(fact_2244_of__nat__diff,axiom,
! [N_16: nat,M_13: nat] :
( ( ord_less_eq_nat @ N_16 @ M_13 )
=> ( ( semiri1621563631at_int @ ( minus_minus_nat @ M_13 @ N_16 ) )
= ( minus_minus_int @ ( semiri1621563631at_int @ M_13 ) @ ( semiri1621563631at_int @ N_16 ) ) ) ) ).
thf(fact_2245_dvd__0__left,axiom,
! [A_63: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A_63 )
=> ( A_63 = zero_zero_nat ) ) ).
thf(fact_2246_dvd__0__left,axiom,
! [A_63: int] :
( ( dvd_dvd_int @ zero_zero_int @ A_63 )
=> ( A_63 = zero_zero_int ) ) ).
thf(fact_2247_dvd__0__left,axiom,
! [A_63: rat] :
( ( dvd_dvd_rat @ zero_zero_rat @ A_63 )
=> ( A_63 = zero_zero_rat ) ) ).
thf(fact_2248_dvd__0__left,axiom,
! [A_63: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ zero_z891286103de_int @ A_63 )
=> ( A_63 = zero_z891286103de_int ) ) ).
thf(fact_2249_dvd__0__left,axiom,
! [A_63: complex] :
( ( dvd_dvd_complex @ zero_zero_complex @ A_63 )
=> ( A_63 = zero_zero_complex ) ) ).
thf(fact_2250_dvd__0__left,axiom,
! [A_63: code_code_numeral] :
( ( dvd_dv174992974umeral @ zero_z126310315umeral @ A_63 )
=> ( A_63 = zero_z126310315umeral ) ) ).
thf(fact_2251_dvd__0__left,axiom,
! [A_63: real] :
( ( dvd_dvd_real @ zero_zero_real @ A_63 )
=> ( A_63 = zero_zero_real ) ) ).
thf(fact_2252_dvd__mult__right,axiom,
! [A_62: nat,B_48: nat,C_23: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A_62 @ B_48 ) @ C_23 )
=> ( dvd_dvd_nat @ B_48 @ C_23 ) ) ).
thf(fact_2253_dvd__mult__right,axiom,
! [A_62: int,B_48: int,C_23: int] :
( ( dvd_dvd_int @ ( times_times_int @ A_62 @ B_48 ) @ C_23 )
=> ( dvd_dvd_int @ B_48 @ C_23 ) ) ).
thf(fact_2254_dvd__mult__right,axiom,
! [A_62: rat,B_48: rat,C_23: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A_62 @ B_48 ) @ C_23 )
=> ( dvd_dvd_rat @ B_48 @ C_23 ) ) ).
thf(fact_2255_dvd__mult__right,axiom,
! [A_62: quickcheck_code_int,B_48: quickcheck_code_int,C_23: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ ( times_123202395de_int @ A_62 @ B_48 ) @ C_23 )
=> ( dvd_dv1760642554de_int @ B_48 @ C_23 ) ) ).
thf(fact_2256_dvd__mult__right,axiom,
! [A_62: complex,B_48: complex,C_23: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A_62 @ B_48 ) @ C_23 )
=> ( dvd_dvd_complex @ B_48 @ C_23 ) ) ).
thf(fact_2257_dvd__mult__right,axiom,
! [A_62: real,B_48: real,C_23: real] :
( ( dvd_dvd_real @ ( times_times_real @ A_62 @ B_48 ) @ C_23 )
=> ( dvd_dvd_real @ B_48 @ C_23 ) ) ).
thf(fact_2258_dvd__mult__left,axiom,
! [A_61: nat,B_47: nat,C_22: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A_61 @ B_47 ) @ C_22 )
=> ( dvd_dvd_nat @ A_61 @ C_22 ) ) ).
thf(fact_2259_dvd__mult__left,axiom,
! [A_61: int,B_47: int,C_22: int] :
( ( dvd_dvd_int @ ( times_times_int @ A_61 @ B_47 ) @ C_22 )
=> ( dvd_dvd_int @ A_61 @ C_22 ) ) ).
thf(fact_2260_dvd__mult__left,axiom,
! [A_61: rat,B_47: rat,C_22: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A_61 @ B_47 ) @ C_22 )
=> ( dvd_dvd_rat @ A_61 @ C_22 ) ) ).
thf(fact_2261_dvd__mult__left,axiom,
! [A_61: quickcheck_code_int,B_47: quickcheck_code_int,C_22: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ ( times_123202395de_int @ A_61 @ B_47 ) @ C_22 )
=> ( dvd_dv1760642554de_int @ A_61 @ C_22 ) ) ).
thf(fact_2262_dvd__mult__left,axiom,
! [A_61: complex,B_47: complex,C_22: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A_61 @ B_47 ) @ C_22 )
=> ( dvd_dvd_complex @ A_61 @ C_22 ) ) ).
thf(fact_2263_dvd__mult__left,axiom,
! [A_61: real,B_47: real,C_22: real] :
( ( dvd_dvd_real @ ( times_times_real @ A_61 @ B_47 ) @ C_22 )
=> ( dvd_dvd_real @ A_61 @ C_22 ) ) ).
thf(fact_2264_dvdI,axiom,
! [A_60: nat,B_46: nat,K_7: nat] :
( ( A_60
= ( times_times_nat @ B_46 @ K_7 ) )
=> ( dvd_dvd_nat @ B_46 @ A_60 ) ) ).
thf(fact_2265_dvdI,axiom,
! [A_60: int,B_46: int,K_7: int] :
( ( A_60
= ( times_times_int @ B_46 @ K_7 ) )
=> ( dvd_dvd_int @ B_46 @ A_60 ) ) ).
thf(fact_2266_dvdI,axiom,
! [A_60: rat,B_46: rat,K_7: rat] :
( ( A_60
= ( times_times_rat @ B_46 @ K_7 ) )
=> ( dvd_dvd_rat @ B_46 @ A_60 ) ) ).
thf(fact_2267_dvdI,axiom,
! [A_60: quickcheck_code_int,B_46: quickcheck_code_int,K_7: quickcheck_code_int] :
( ( A_60
= ( times_123202395de_int @ B_46 @ K_7 ) )
=> ( dvd_dv1760642554de_int @ B_46 @ A_60 ) ) ).
thf(fact_2268_dvdI,axiom,
! [A_60: complex,B_46: complex,K_7: complex] :
( ( A_60
= ( times_times_complex @ B_46 @ K_7 ) )
=> ( dvd_dvd_complex @ B_46 @ A_60 ) ) ).
thf(fact_2269_dvdI,axiom,
! [A_60: real,B_46: real,K_7: real] :
( ( A_60
= ( times_times_real @ B_46 @ K_7 ) )
=> ( dvd_dvd_real @ B_46 @ A_60 ) ) ).
thf(fact_2270_mult__dvd__mono,axiom,
! [C_21: nat,D_15: nat,A_59: nat,B_45: nat] :
( ( dvd_dvd_nat @ A_59 @ B_45 )
=> ( ( dvd_dvd_nat @ C_21 @ D_15 )
=> ( dvd_dvd_nat @ ( times_times_nat @ A_59 @ C_21 ) @ ( times_times_nat @ B_45 @ D_15 ) ) ) ) ).
thf(fact_2271_mult__dvd__mono,axiom,
! [C_21: int,D_15: int,A_59: int,B_45: int] :
( ( dvd_dvd_int @ A_59 @ B_45 )
=> ( ( dvd_dvd_int @ C_21 @ D_15 )
=> ( dvd_dvd_int @ ( times_times_int @ A_59 @ C_21 ) @ ( times_times_int @ B_45 @ D_15 ) ) ) ) ).
thf(fact_2272_mult__dvd__mono,axiom,
! [C_21: rat,D_15: rat,A_59: rat,B_45: rat] :
( ( dvd_dvd_rat @ A_59 @ B_45 )
=> ( ( dvd_dvd_rat @ C_21 @ D_15 )
=> ( dvd_dvd_rat @ ( times_times_rat @ A_59 @ C_21 ) @ ( times_times_rat @ B_45 @ D_15 ) ) ) ) ).
thf(fact_2273_mult__dvd__mono,axiom,
! [C_21: quickcheck_code_int,D_15: quickcheck_code_int,A_59: quickcheck_code_int,B_45: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ A_59 @ B_45 )
=> ( ( dvd_dv1760642554de_int @ C_21 @ D_15 )
=> ( dvd_dv1760642554de_int @ ( times_123202395de_int @ A_59 @ C_21 ) @ ( times_123202395de_int @ B_45 @ D_15 ) ) ) ) ).
thf(fact_2274_mult__dvd__mono,axiom,
! [C_21: complex,D_15: complex,A_59: complex,B_45: complex] :
( ( dvd_dvd_complex @ A_59 @ B_45 )
=> ( ( dvd_dvd_complex @ C_21 @ D_15 )
=> ( dvd_dvd_complex @ ( times_times_complex @ A_59 @ C_21 ) @ ( times_times_complex @ B_45 @ D_15 ) ) ) ) ).
thf(fact_2275_mult__dvd__mono,axiom,
! [C_21: real,D_15: real,A_59: real,B_45: real] :
( ( dvd_dvd_real @ A_59 @ B_45 )
=> ( ( dvd_dvd_real @ C_21 @ D_15 )
=> ( dvd_dvd_real @ ( times_times_real @ A_59 @ C_21 ) @ ( times_times_real @ B_45 @ D_15 ) ) ) ) ).
thf(fact_2276_dvd__mult,axiom,
! [B_44: nat,A_58: nat,C_20: nat] :
( ( dvd_dvd_nat @ A_58 @ C_20 )
=> ( dvd_dvd_nat @ A_58 @ ( times_times_nat @ B_44 @ C_20 ) ) ) ).
thf(fact_2277_dvd__mult,axiom,
! [B_44: int,A_58: int,C_20: int] :
( ( dvd_dvd_int @ A_58 @ C_20 )
=> ( dvd_dvd_int @ A_58 @ ( times_times_int @ B_44 @ C_20 ) ) ) ).
thf(fact_2278_dvd__mult,axiom,
! [B_44: rat,A_58: rat,C_20: rat] :
( ( dvd_dvd_rat @ A_58 @ C_20 )
=> ( dvd_dvd_rat @ A_58 @ ( times_times_rat @ B_44 @ C_20 ) ) ) ).
thf(fact_2279_dvd__mult,axiom,
! [B_44: quickcheck_code_int,A_58: quickcheck_code_int,C_20: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ A_58 @ C_20 )
=> ( dvd_dv1760642554de_int @ A_58 @ ( times_123202395de_int @ B_44 @ C_20 ) ) ) ).
thf(fact_2280_dvd__mult,axiom,
! [B_44: complex,A_58: complex,C_20: complex] :
( ( dvd_dvd_complex @ A_58 @ C_20 )
=> ( dvd_dvd_complex @ A_58 @ ( times_times_complex @ B_44 @ C_20 ) ) ) ).
thf(fact_2281_dvd__mult,axiom,
! [B_44: real,A_58: real,C_20: real] :
( ( dvd_dvd_real @ A_58 @ C_20 )
=> ( dvd_dvd_real @ A_58 @ ( times_times_real @ B_44 @ C_20 ) ) ) ).
thf(fact_2282_dvd__mult2,axiom,
! [C_19: nat,A_57: nat,B_43: nat] :
( ( dvd_dvd_nat @ A_57 @ B_43 )
=> ( dvd_dvd_nat @ A_57 @ ( times_times_nat @ B_43 @ C_19 ) ) ) ).
thf(fact_2283_dvd__mult2,axiom,
! [C_19: int,A_57: int,B_43: int] :
( ( dvd_dvd_int @ A_57 @ B_43 )
=> ( dvd_dvd_int @ A_57 @ ( times_times_int @ B_43 @ C_19 ) ) ) ).
thf(fact_2284_dvd__mult2,axiom,
! [C_19: rat,A_57: rat,B_43: rat] :
( ( dvd_dvd_rat @ A_57 @ B_43 )
=> ( dvd_dvd_rat @ A_57 @ ( times_times_rat @ B_43 @ C_19 ) ) ) ).
thf(fact_2285_dvd__mult2,axiom,
! [C_19: quickcheck_code_int,A_57: quickcheck_code_int,B_43: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ A_57 @ B_43 )
=> ( dvd_dv1760642554de_int @ A_57 @ ( times_123202395de_int @ B_43 @ C_19 ) ) ) ).
thf(fact_2286_dvd__mult2,axiom,
! [C_19: complex,A_57: complex,B_43: complex] :
( ( dvd_dvd_complex @ A_57 @ B_43 )
=> ( dvd_dvd_complex @ A_57 @ ( times_times_complex @ B_43 @ C_19 ) ) ) ).
thf(fact_2287_dvd__mult2,axiom,
! [C_19: real,A_57: real,B_43: real] :
( ( dvd_dvd_real @ A_57 @ B_43 )
=> ( dvd_dvd_real @ A_57 @ ( times_times_real @ B_43 @ C_19 ) ) ) ).
thf(fact_2288_dvd__triv__right,axiom,
! [A_56: nat,B_42: nat] : ( dvd_dvd_nat @ A_56 @ ( times_times_nat @ B_42 @ A_56 ) ) ).
thf(fact_2289_dvd__triv__right,axiom,
! [A_56: int,B_42: int] : ( dvd_dvd_int @ A_56 @ ( times_times_int @ B_42 @ A_56 ) ) ).
thf(fact_2290_dvd__triv__right,axiom,
! [A_56: rat,B_42: rat] : ( dvd_dvd_rat @ A_56 @ ( times_times_rat @ B_42 @ A_56 ) ) ).
thf(fact_2291_dvd__triv__right,axiom,
! [A_56: quickcheck_code_int,B_42: quickcheck_code_int] : ( dvd_dv1760642554de_int @ A_56 @ ( times_123202395de_int @ B_42 @ A_56 ) ) ).
thf(fact_2292_dvd__triv__right,axiom,
! [A_56: complex,B_42: complex] : ( dvd_dvd_complex @ A_56 @ ( times_times_complex @ B_42 @ A_56 ) ) ).
thf(fact_2293_dvd__triv__right,axiom,
! [A_56: real,B_42: real] : ( dvd_dvd_real @ A_56 @ ( times_times_real @ B_42 @ A_56 ) ) ).
thf(fact_2294_dvd__triv__left,axiom,
! [A_55: nat,B_41: nat] : ( dvd_dvd_nat @ A_55 @ ( times_times_nat @ A_55 @ B_41 ) ) ).
thf(fact_2295_dvd__triv__left,axiom,
! [A_55: int,B_41: int] : ( dvd_dvd_int @ A_55 @ ( times_times_int @ A_55 @ B_41 ) ) ).
thf(fact_2296_dvd__triv__left,axiom,
! [A_55: rat,B_41: rat] : ( dvd_dvd_rat @ A_55 @ ( times_times_rat @ A_55 @ B_41 ) ) ).
thf(fact_2297_dvd__triv__left,axiom,
! [A_55: quickcheck_code_int,B_41: quickcheck_code_int] : ( dvd_dv1760642554de_int @ A_55 @ ( times_123202395de_int @ A_55 @ B_41 ) ) ).
thf(fact_2298_dvd__triv__left,axiom,
! [A_55: complex,B_41: complex] : ( dvd_dvd_complex @ A_55 @ ( times_times_complex @ A_55 @ B_41 ) ) ).
thf(fact_2299_dvd__triv__left,axiom,
! [A_55: real,B_41: real] : ( dvd_dvd_real @ A_55 @ ( times_times_real @ A_55 @ B_41 ) ) ).
thf(fact_2300_dvd__add,axiom,
! [C_18: nat,A_54: nat,B_40: nat] :
( ( dvd_dvd_nat @ A_54 @ B_40 )
=> ( ( dvd_dvd_nat @ A_54 @ C_18 )
=> ( dvd_dvd_nat @ A_54 @ ( plus_plus_nat @ B_40 @ C_18 ) ) ) ) ).
thf(fact_2301_dvd__add,axiom,
! [C_18: int,A_54: int,B_40: int] :
( ( dvd_dvd_int @ A_54 @ B_40 )
=> ( ( dvd_dvd_int @ A_54 @ C_18 )
=> ( dvd_dvd_int @ A_54 @ ( plus_plus_int @ B_40 @ C_18 ) ) ) ) ).
thf(fact_2302_dvd__add,axiom,
! [C_18: rat,A_54: rat,B_40: rat] :
( ( dvd_dvd_rat @ A_54 @ B_40 )
=> ( ( dvd_dvd_rat @ A_54 @ C_18 )
=> ( dvd_dvd_rat @ A_54 @ ( plus_plus_rat @ B_40 @ C_18 ) ) ) ) ).
thf(fact_2303_dvd__add,axiom,
! [C_18: quickcheck_code_int,A_54: quickcheck_code_int,B_40: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ A_54 @ B_40 )
=> ( ( dvd_dv1760642554de_int @ A_54 @ C_18 )
=> ( dvd_dv1760642554de_int @ A_54 @ ( plus_p1446045655de_int @ B_40 @ C_18 ) ) ) ) ).
thf(fact_2304_dvd__add,axiom,
! [C_18: complex,A_54: complex,B_40: complex] :
( ( dvd_dvd_complex @ A_54 @ B_40 )
=> ( ( dvd_dvd_complex @ A_54 @ C_18 )
=> ( dvd_dvd_complex @ A_54 @ ( plus_plus_complex @ B_40 @ C_18 ) ) ) ) ).
thf(fact_2305_dvd__add,axiom,
! [C_18: real,A_54: real,B_40: real] :
( ( dvd_dvd_real @ A_54 @ B_40 )
=> ( ( dvd_dvd_real @ A_54 @ C_18 )
=> ( dvd_dvd_real @ A_54 @ ( plus_plus_real @ B_40 @ C_18 ) ) ) ) ).
thf(fact_2306_one__dvd,axiom,
! [A_53: nat] : ( dvd_dvd_nat @ one_one_nat @ A_53 ) ).
thf(fact_2307_one__dvd,axiom,
! [A_53: int] : ( dvd_dvd_int @ one_one_int @ A_53 ) ).
thf(fact_2308_one__dvd,axiom,
! [A_53: rat] : ( dvd_dvd_rat @ one_one_rat @ A_53 ) ).
thf(fact_2309_one__dvd,axiom,
! [A_53: quickcheck_code_int] : ( dvd_dv1760642554de_int @ one_on1684967323de_int @ A_53 ) ).
thf(fact_2310_one__dvd,axiom,
! [A_53: complex] : ( dvd_dvd_complex @ one_one_complex @ A_53 ) ).
thf(fact_2311_one__dvd,axiom,
! [A_53: code_code_numeral] : ( dvd_dv174992974umeral @ one_on1645066479umeral @ A_53 ) ).
thf(fact_2312_one__dvd,axiom,
! [A_53: real] : ( dvd_dvd_real @ one_one_real @ A_53 ) ).
thf(fact_2313_dvd__power__same,axiom,
! [N_15: nat,X_13: nat,Y_9: nat] :
( ( dvd_dvd_nat @ X_13 @ Y_9 )
=> ( dvd_dvd_nat @ ( power_power_nat @ X_13 @ N_15 ) @ ( power_power_nat @ Y_9 @ N_15 ) ) ) ).
thf(fact_2314_dvd__power__same,axiom,
! [N_15: nat,X_13: int,Y_9: int] :
( ( dvd_dvd_int @ X_13 @ Y_9 )
=> ( dvd_dvd_int @ ( power_power_int @ X_13 @ N_15 ) @ ( power_power_int @ Y_9 @ N_15 ) ) ) ).
thf(fact_2315_dvd__power__same,axiom,
! [N_15: nat,X_13: complex,Y_9: complex] :
( ( dvd_dvd_complex @ X_13 @ Y_9 )
=> ( dvd_dvd_complex @ ( power_power_complex @ X_13 @ N_15 ) @ ( power_power_complex @ Y_9 @ N_15 ) ) ) ).
thf(fact_2316_dvd__power__same,axiom,
! [N_15: nat,X_13: real,Y_9: real] :
( ( dvd_dvd_real @ X_13 @ Y_9 )
=> ( dvd_dvd_real @ ( power_power_real @ X_13 @ N_15 ) @ ( power_power_real @ Y_9 @ N_15 ) ) ) ).
thf(fact_2317_right__minus__eq,axiom,
! [A_52: rat,B_39: rat] :
( ( ( minus_minus_rat @ A_52 @ B_39 )
= zero_zero_rat )
<=> ( A_52 = B_39 ) ) ).
thf(fact_2318_right__minus__eq,axiom,
! [A_52: complex,B_39: complex] :
( ( ( minus_minus_complex @ A_52 @ B_39 )
= zero_zero_complex )
<=> ( A_52 = B_39 ) ) ).
thf(fact_2319_right__minus__eq,axiom,
! [A_52: real,B_39: real] :
( ( ( minus_minus_real @ A_52 @ B_39 )
= zero_zero_real )
<=> ( A_52 = B_39 ) ) ).
thf(fact_2320_right__minus__eq,axiom,
! [A_52: int,B_39: int] :
( ( ( minus_minus_int @ A_52 @ B_39 )
= zero_zero_int )
<=> ( A_52 = B_39 ) ) ).
thf(fact_2321_eq__iff__diff__eq__0,axiom,
! [A_51: rat,B_38: rat] :
( ( A_51 = B_38 )
<=> ( ( minus_minus_rat @ A_51 @ B_38 )
= zero_zero_rat ) ) ).
thf(fact_2322_eq__iff__diff__eq__0,axiom,
! [A_51: complex,B_38: complex] :
( ( A_51 = B_38 )
<=> ( ( minus_minus_complex @ A_51 @ B_38 )
= zero_zero_complex ) ) ).
thf(fact_2323_eq__iff__diff__eq__0,axiom,
! [A_51: real,B_38: real] :
( ( A_51 = B_38 )
<=> ( ( minus_minus_real @ A_51 @ B_38 )
= zero_zero_real ) ) ).
thf(fact_2324_eq__iff__diff__eq__0,axiom,
! [A_51: int,B_38: int] :
( ( A_51 = B_38 )
<=> ( ( minus_minus_int @ A_51 @ B_38 )
= zero_zero_int ) ) ).
thf(fact_2325_diff__self,axiom,
! [A_50: rat] :
( ( minus_minus_rat @ A_50 @ A_50 )
= zero_zero_rat ) ).
thf(fact_2326_diff__self,axiom,
! [A_50: complex] :
( ( minus_minus_complex @ A_50 @ A_50 )
= zero_zero_complex ) ).
thf(fact_2327_diff__self,axiom,
! [A_50: real] :
( ( minus_minus_real @ A_50 @ A_50 )
= zero_zero_real ) ).
thf(fact_2328_diff__self,axiom,
! [A_50: int] :
( ( minus_minus_int @ A_50 @ A_50 )
= zero_zero_int ) ).
thf(fact_2329_diff__0__right,axiom,
! [A_49: rat] :
( ( minus_minus_rat @ A_49 @ zero_zero_rat )
= A_49 ) ).
thf(fact_2330_diff__0__right,axiom,
! [A_49: complex] :
( ( minus_minus_complex @ A_49 @ zero_zero_complex )
= A_49 ) ).
thf(fact_2331_diff__0__right,axiom,
! [A_49: real] :
( ( minus_minus_real @ A_49 @ zero_zero_real )
= A_49 ) ).
thf(fact_2332_diff__0__right,axiom,
! [A_49: int] :
( ( minus_minus_int @ A_49 @ zero_zero_int )
= A_49 ) ).
thf(fact_2333_diff__eq__diff__less__eq,axiom,
! [A_48: rat,B_37: rat,C_17: rat,D_14: rat] :
( ( ( minus_minus_rat @ A_48 @ B_37 )
= ( minus_minus_rat @ C_17 @ D_14 ) )
=> ( ( ord_less_eq_rat @ A_48 @ B_37 )
<=> ( ord_less_eq_rat @ C_17 @ D_14 ) ) ) ).
thf(fact_2334_diff__eq__diff__less__eq,axiom,
! [A_48: real,B_37: real,C_17: real,D_14: real] :
( ( ( minus_minus_real @ A_48 @ B_37 )
= ( minus_minus_real @ C_17 @ D_14 ) )
=> ( ( ord_less_eq_real @ A_48 @ B_37 )
<=> ( ord_less_eq_real @ C_17 @ D_14 ) ) ) ).
thf(fact_2335_diff__eq__diff__less__eq,axiom,
! [A_48: int,B_37: int,C_17: int,D_14: int] :
( ( ( minus_minus_int @ A_48 @ B_37 )
= ( minus_minus_int @ C_17 @ D_14 ) )
=> ( ( ord_less_eq_int @ A_48 @ B_37 )
<=> ( ord_less_eq_int @ C_17 @ D_14 ) ) ) ).
thf(fact_2336_diff__eq__diff__less,axiom,
! [A_47: rat,B_36: rat,C_16: rat,D_13: rat] :
( ( ( minus_minus_rat @ A_47 @ B_36 )
= ( minus_minus_rat @ C_16 @ D_13 ) )
=> ( ( ord_less_rat @ A_47 @ B_36 )
<=> ( ord_less_rat @ C_16 @ D_13 ) ) ) ).
thf(fact_2337_diff__eq__diff__less,axiom,
! [A_47: real,B_36: real,C_16: real,D_13: real] :
( ( ( minus_minus_real @ A_47 @ B_36 )
= ( minus_minus_real @ C_16 @ D_13 ) )
=> ( ( ord_less_real @ A_47 @ B_36 )
<=> ( ord_less_real @ C_16 @ D_13 ) ) ) ).
thf(fact_2338_diff__eq__diff__less,axiom,
! [A_47: int,B_36: int,C_16: int,D_13: int] :
( ( ( minus_minus_int @ A_47 @ B_36 )
= ( minus_minus_int @ C_16 @ D_13 ) )
=> ( ( ord_less_int @ A_47 @ B_36 )
<=> ( ord_less_int @ C_16 @ D_13 ) ) ) ).
thf(fact_2339_add__diff__cancel,axiom,
! [A_46: rat,B_35: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A_46 @ B_35 ) @ B_35 )
= A_46 ) ).
thf(fact_2340_add__diff__cancel,axiom,
! [A_46: complex,B_35: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A_46 @ B_35 ) @ B_35 )
= A_46 ) ).
thf(fact_2341_add__diff__cancel,axiom,
! [A_46: real,B_35: real] :
( ( minus_minus_real @ ( plus_plus_real @ A_46 @ B_35 ) @ B_35 )
= A_46 ) ).
thf(fact_2342_add__diff__cancel,axiom,
! [A_46: int,B_35: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_46 @ B_35 ) @ B_35 )
= A_46 ) ).
thf(fact_2343_diff__add__cancel,axiom,
! [A_45: rat,B_34: rat] :
( ( plus_plus_rat @ ( minus_minus_rat @ A_45 @ B_34 ) @ B_34 )
= A_45 ) ).
thf(fact_2344_diff__add__cancel,axiom,
! [A_45: complex,B_34: complex] :
( ( plus_plus_complex @ ( minus_minus_complex @ A_45 @ B_34 ) @ B_34 )
= A_45 ) ).
thf(fact_2345_diff__add__cancel,axiom,
! [A_45: real,B_34: real] :
( ( plus_plus_real @ ( minus_minus_real @ A_45 @ B_34 ) @ B_34 )
= A_45 ) ).
thf(fact_2346_diff__add__cancel,axiom,
! [A_45: int,B_34: int] :
( ( plus_plus_int @ ( minus_minus_int @ A_45 @ B_34 ) @ B_34 )
= A_45 ) ).
thf(fact_2347_add__diff__add,axiom,
! [A_44: rat,C_15: rat,B_33: rat,D_12: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A_44 @ C_15 ) @ ( plus_plus_rat @ B_33 @ D_12 ) )
= ( plus_plus_rat @ ( minus_minus_rat @ A_44 @ B_33 ) @ ( minus_minus_rat @ C_15 @ D_12 ) ) ) ).
thf(fact_2348_add__diff__add,axiom,
! [A_44: complex,C_15: complex,B_33: complex,D_12: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A_44 @ C_15 ) @ ( plus_plus_complex @ B_33 @ D_12 ) )
= ( plus_plus_complex @ ( minus_minus_complex @ A_44 @ B_33 ) @ ( minus_minus_complex @ C_15 @ D_12 ) ) ) ).
thf(fact_2349_add__diff__add,axiom,
! [A_44: real,C_15: real,B_33: real,D_12: real] :
( ( minus_minus_real @ ( plus_plus_real @ A_44 @ C_15 ) @ ( plus_plus_real @ B_33 @ D_12 ) )
= ( plus_plus_real @ ( minus_minus_real @ A_44 @ B_33 ) @ ( minus_minus_real @ C_15 @ D_12 ) ) ) ).
thf(fact_2350_add__diff__add,axiom,
! [A_44: int,C_15: int,B_33: int,D_12: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_44 @ C_15 ) @ ( plus_plus_int @ B_33 @ D_12 ) )
= ( plus_plus_int @ ( minus_minus_int @ A_44 @ B_33 ) @ ( minus_minus_int @ C_15 @ D_12 ) ) ) ).
thf(fact_2351_diff__bin__simps_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ pls )
= K ) ).
thf(fact_2352_diff__bin__simps_I7_J,axiom,
! [K: int,L: int] :
( ( minus_minus_int @ ( bit0 @ K ) @ ( bit0 @ L ) )
= ( bit0 @ ( minus_minus_int @ K @ L ) ) ) ).
thf(fact_2353_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_2354_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_2355_dvd__mult__cancel__right,axiom,
! [A_43: int,C_14: int,B_32: int] :
( ( dvd_dvd_int @ ( times_times_int @ A_43 @ C_14 ) @ ( times_times_int @ B_32 @ C_14 ) )
<=> ( ( C_14 = zero_zero_int )
| ( dvd_dvd_int @ A_43 @ B_32 ) ) ) ).
thf(fact_2356_dvd__mult__cancel__right,axiom,
! [A_43: rat,C_14: rat,B_32: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A_43 @ C_14 ) @ ( times_times_rat @ B_32 @ C_14 ) )
<=> ( ( C_14 = zero_zero_rat )
| ( dvd_dvd_rat @ A_43 @ B_32 ) ) ) ).
thf(fact_2357_dvd__mult__cancel__right,axiom,
! [A_43: complex,C_14: complex,B_32: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A_43 @ C_14 ) @ ( times_times_complex @ B_32 @ C_14 ) )
<=> ( ( C_14 = zero_zero_complex )
| ( dvd_dvd_complex @ A_43 @ B_32 ) ) ) ).
thf(fact_2358_dvd__mult__cancel__right,axiom,
! [A_43: real,C_14: real,B_32: real] :
( ( dvd_dvd_real @ ( times_times_real @ A_43 @ C_14 ) @ ( times_times_real @ B_32 @ C_14 ) )
<=> ( ( C_14 = zero_zero_real )
| ( dvd_dvd_real @ A_43 @ B_32 ) ) ) ).
thf(fact_2359_dvd__mult__cancel__left,axiom,
! [C_13: int,A_42: int,B_31: int] :
( ( dvd_dvd_int @ ( times_times_int @ C_13 @ A_42 ) @ ( times_times_int @ C_13 @ B_31 ) )
<=> ( ( C_13 = zero_zero_int )
| ( dvd_dvd_int @ A_42 @ B_31 ) ) ) ).
thf(fact_2360_dvd__mult__cancel__left,axiom,
! [C_13: rat,A_42: rat,B_31: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ C_13 @ A_42 ) @ ( times_times_rat @ C_13 @ B_31 ) )
<=> ( ( C_13 = zero_zero_rat )
| ( dvd_dvd_rat @ A_42 @ B_31 ) ) ) ).
thf(fact_2361_dvd__mult__cancel__left,axiom,
! [C_13: complex,A_42: complex,B_31: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ C_13 @ A_42 ) @ ( times_times_complex @ C_13 @ B_31 ) )
<=> ( ( C_13 = zero_zero_complex )
| ( dvd_dvd_complex @ A_42 @ B_31 ) ) ) ).
thf(fact_2362_dvd__mult__cancel__left,axiom,
! [C_13: real,A_42: real,B_31: real] :
( ( dvd_dvd_real @ ( times_times_real @ C_13 @ A_42 ) @ ( times_times_real @ C_13 @ B_31 ) )
<=> ( ( C_13 = zero_zero_real )
| ( dvd_dvd_real @ A_42 @ B_31 ) ) ) ).
thf(fact_2363_power__le__dvd,axiom,
! [M_12: nat,A_41: nat,N_14: nat,B_30: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A_41 @ N_14 ) @ B_30 )
=> ( ( ord_less_eq_nat @ M_12 @ N_14 )
=> ( dvd_dvd_nat @ ( power_power_nat @ A_41 @ M_12 ) @ B_30 ) ) ) ).
thf(fact_2364_power__le__dvd,axiom,
! [M_12: nat,A_41: int,N_14: nat,B_30: int] :
( ( dvd_dvd_int @ ( power_power_int @ A_41 @ N_14 ) @ B_30 )
=> ( ( ord_less_eq_nat @ M_12 @ N_14 )
=> ( dvd_dvd_int @ ( power_power_int @ A_41 @ M_12 ) @ B_30 ) ) ) ).
thf(fact_2365_power__le__dvd,axiom,
! [M_12: nat,A_41: complex,N_14: nat,B_30: complex] :
( ( dvd_dvd_complex @ ( power_power_complex @ A_41 @ N_14 ) @ B_30 )
=> ( ( ord_less_eq_nat @ M_12 @ N_14 )
=> ( dvd_dvd_complex @ ( power_power_complex @ A_41 @ M_12 ) @ B_30 ) ) ) ).
thf(fact_2366_power__le__dvd,axiom,
! [M_12: nat,A_41: real,N_14: nat,B_30: real] :
( ( dvd_dvd_real @ ( power_power_real @ A_41 @ N_14 ) @ B_30 )
=> ( ( ord_less_eq_nat @ M_12 @ N_14 )
=> ( dvd_dvd_real @ ( power_power_real @ A_41 @ M_12 ) @ B_30 ) ) ) ).
thf(fact_2367_dvd__power__le,axiom,
! [N_13: nat,M_11: nat,X_12: nat,Y_8: nat] :
( ( dvd_dvd_nat @ X_12 @ Y_8 )
=> ( ( ord_less_eq_nat @ N_13 @ M_11 )
=> ( dvd_dvd_nat @ ( power_power_nat @ X_12 @ N_13 ) @ ( power_power_nat @ Y_8 @ M_11 ) ) ) ) ).
thf(fact_2368_dvd__power__le,axiom,
! [N_13: nat,M_11: nat,X_12: int,Y_8: int] :
( ( dvd_dvd_int @ X_12 @ Y_8 )
=> ( ( ord_less_eq_nat @ N_13 @ M_11 )
=> ( dvd_dvd_int @ ( power_power_int @ X_12 @ N_13 ) @ ( power_power_int @ Y_8 @ M_11 ) ) ) ) ).
thf(fact_2369_dvd__power__le,axiom,
! [N_13: nat,M_11: nat,X_12: complex,Y_8: complex] :
( ( dvd_dvd_complex @ X_12 @ Y_8 )
=> ( ( ord_less_eq_nat @ N_13 @ M_11 )
=> ( dvd_dvd_complex @ ( power_power_complex @ X_12 @ N_13 ) @ ( power_power_complex @ Y_8 @ M_11 ) ) ) ) ).
thf(fact_2370_dvd__power__le,axiom,
! [N_13: nat,M_11: nat,X_12: real,Y_8: real] :
( ( dvd_dvd_real @ X_12 @ Y_8 )
=> ( ( ord_less_eq_nat @ N_13 @ M_11 )
=> ( dvd_dvd_real @ ( power_power_real @ X_12 @ N_13 ) @ ( power_power_real @ Y_8 @ M_11 ) ) ) ) ).
thf(fact_2371_le__imp__power__dvd,axiom,
! [A_40: nat,M_10: nat,N_12: nat] :
( ( ord_less_eq_nat @ M_10 @ N_12 )
=> ( dvd_dvd_nat @ ( power_power_nat @ A_40 @ M_10 ) @ ( power_power_nat @ A_40 @ N_12 ) ) ) ).
thf(fact_2372_le__imp__power__dvd,axiom,
! [A_40: int,M_10: nat,N_12: nat] :
( ( ord_less_eq_nat @ M_10 @ N_12 )
=> ( dvd_dvd_int @ ( power_power_int @ A_40 @ M_10 ) @ ( power_power_int @ A_40 @ N_12 ) ) ) ).
thf(fact_2373_le__imp__power__dvd,axiom,
! [A_40: complex,M_10: nat,N_12: nat] :
( ( ord_less_eq_nat @ M_10 @ N_12 )
=> ( dvd_dvd_complex @ ( power_power_complex @ A_40 @ M_10 ) @ ( power_power_complex @ A_40 @ N_12 ) ) ) ).
thf(fact_2374_le__imp__power__dvd,axiom,
! [A_40: real,M_10: nat,N_12: nat] :
( ( ord_less_eq_nat @ M_10 @ N_12 )
=> ( dvd_dvd_real @ ( power_power_real @ A_40 @ M_10 ) @ ( power_power_real @ A_40 @ N_12 ) ) ) ).
thf(fact_2375_le__iff__diff__le__0,axiom,
! [A_39: rat,B_29: rat] :
( ( ord_less_eq_rat @ A_39 @ B_29 )
<=> ( ord_less_eq_rat @ ( minus_minus_rat @ A_39 @ B_29 ) @ zero_zero_rat ) ) ).
thf(fact_2376_le__iff__diff__le__0,axiom,
! [A_39: real,B_29: real] :
( ( ord_less_eq_real @ A_39 @ B_29 )
<=> ( ord_less_eq_real @ ( minus_minus_real @ A_39 @ B_29 ) @ zero_zero_real ) ) ).
thf(fact_2377_le__iff__diff__le__0,axiom,
! [A_39: int,B_29: int] :
( ( ord_less_eq_int @ A_39 @ B_29 )
<=> ( ord_less_eq_int @ ( minus_minus_int @ A_39 @ B_29 ) @ zero_zero_int ) ) ).
thf(fact_2378_less__iff__diff__less__0,axiom,
! [A_38: rat,B_28: rat] :
( ( ord_less_rat @ A_38 @ B_28 )
<=> ( ord_less_rat @ ( minus_minus_rat @ A_38 @ B_28 ) @ zero_zero_rat ) ) ).
thf(fact_2379_less__iff__diff__less__0,axiom,
! [A_38: real,B_28: real] :
( ( ord_less_real @ A_38 @ B_28 )
<=> ( ord_less_real @ ( minus_minus_real @ A_38 @ B_28 ) @ zero_zero_real ) ) ).
thf(fact_2380_less__iff__diff__less__0,axiom,
! [A_38: int,B_28: int] :
( ( ord_less_int @ A_38 @ B_28 )
<=> ( ord_less_int @ ( minus_minus_int @ A_38 @ B_28 ) @ zero_zero_int ) ) ).
thf(fact_2381_eq__add__iff1,axiom,
! [A_37: rat,E_7: rat,C_12: rat,B_27: rat,D_11: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A_37 @ E_7 ) @ C_12 )
= ( plus_plus_rat @ ( times_times_rat @ B_27 @ E_7 ) @ D_11 ) )
<=> ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A_37 @ B_27 ) @ E_7 ) @ C_12 )
= D_11 ) ) ).
thf(fact_2382_eq__add__iff1,axiom,
! [A_37: complex,E_7: complex,C_12: complex,B_27: complex,D_11: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ A_37 @ E_7 ) @ C_12 )
= ( plus_plus_complex @ ( times_times_complex @ B_27 @ E_7 ) @ D_11 ) )
<=> ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A_37 @ B_27 ) @ E_7 ) @ C_12 )
= D_11 ) ) ).
thf(fact_2383_eq__add__iff1,axiom,
! [A_37: real,E_7: real,C_12: real,B_27: real,D_11: real] :
( ( ( plus_plus_real @ ( times_times_real @ A_37 @ E_7 ) @ C_12 )
= ( plus_plus_real @ ( times_times_real @ B_27 @ E_7 ) @ D_11 ) )
<=> ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A_37 @ B_27 ) @ E_7 ) @ C_12 )
= D_11 ) ) ).
thf(fact_2384_eq__add__iff1,axiom,
! [A_37: int,E_7: int,C_12: int,B_27: int,D_11: int] :
( ( ( plus_plus_int @ ( times_times_int @ A_37 @ E_7 ) @ C_12 )
= ( plus_plus_int @ ( times_times_int @ B_27 @ E_7 ) @ D_11 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_37 @ B_27 ) @ E_7 ) @ C_12 )
= D_11 ) ) ).
thf(fact_2385_eq__add__iff2,axiom,
! [A_36: rat,E_6: rat,C_11: rat,B_26: rat,D_10: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A_36 @ E_6 ) @ C_11 )
= ( plus_plus_rat @ ( times_times_rat @ B_26 @ E_6 ) @ D_10 ) )
<=> ( C_11
= ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B_26 @ A_36 ) @ E_6 ) @ D_10 ) ) ) ).
thf(fact_2386_eq__add__iff2,axiom,
! [A_36: complex,E_6: complex,C_11: complex,B_26: complex,D_10: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ A_36 @ E_6 ) @ C_11 )
= ( plus_plus_complex @ ( times_times_complex @ B_26 @ E_6 ) @ D_10 ) )
<=> ( C_11
= ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B_26 @ A_36 ) @ E_6 ) @ D_10 ) ) ) ).
thf(fact_2387_eq__add__iff2,axiom,
! [A_36: real,E_6: real,C_11: real,B_26: real,D_10: real] :
( ( ( plus_plus_real @ ( times_times_real @ A_36 @ E_6 ) @ C_11 )
= ( plus_plus_real @ ( times_times_real @ B_26 @ E_6 ) @ D_10 ) )
<=> ( C_11
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B_26 @ A_36 ) @ E_6 ) @ D_10 ) ) ) ).
thf(fact_2388_eq__add__iff2,axiom,
! [A_36: int,E_6: int,C_11: int,B_26: int,D_10: int] :
( ( ( plus_plus_int @ ( times_times_int @ A_36 @ E_6 ) @ C_11 )
= ( plus_plus_int @ ( times_times_int @ B_26 @ E_6 ) @ D_10 ) )
<=> ( C_11
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_26 @ A_36 ) @ E_6 ) @ D_10 ) ) ) ).
thf(fact_2389_mult__diff__mult,axiom,
! [X_11: rat,Y_7: rat,A_35: rat,B_25: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X_11 @ Y_7 ) @ ( times_times_rat @ A_35 @ B_25 ) )
= ( plus_plus_rat @ ( times_times_rat @ X_11 @ ( minus_minus_rat @ Y_7 @ B_25 ) ) @ ( times_times_rat @ ( minus_minus_rat @ X_11 @ A_35 ) @ B_25 ) ) ) ).
thf(fact_2390_mult__diff__mult,axiom,
! [X_11: complex,Y_7: complex,A_35: complex,B_25: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X_11 @ Y_7 ) @ ( times_times_complex @ A_35 @ B_25 ) )
= ( plus_plus_complex @ ( times_times_complex @ X_11 @ ( minus_minus_complex @ Y_7 @ B_25 ) ) @ ( times_times_complex @ ( minus_minus_complex @ X_11 @ A_35 ) @ B_25 ) ) ) ).
thf(fact_2391_mult__diff__mult,axiom,
! [X_11: real,Y_7: real,A_35: real,B_25: real] :
( ( minus_minus_real @ ( times_times_real @ X_11 @ Y_7 ) @ ( times_times_real @ A_35 @ B_25 ) )
= ( plus_plus_real @ ( times_times_real @ X_11 @ ( minus_minus_real @ Y_7 @ B_25 ) ) @ ( times_times_real @ ( minus_minus_real @ X_11 @ A_35 ) @ B_25 ) ) ) ).
thf(fact_2392_mult__diff__mult,axiom,
! [X_11: int,Y_7: int,A_35: int,B_25: int] :
( ( minus_minus_int @ ( times_times_int @ X_11 @ Y_7 ) @ ( times_times_int @ A_35 @ B_25 ) )
= ( plus_plus_int @ ( times_times_int @ X_11 @ ( minus_minus_int @ Y_7 @ B_25 ) ) @ ( times_times_int @ ( minus_minus_int @ X_11 @ A_35 ) @ B_25 ) ) ) ).
thf(fact_2393_right__diff__distrib__number__of,axiom,
! [V_5: int,B_24: rat,C_10: rat] :
( ( times_times_rat @ ( number_number_of_rat @ V_5 ) @ ( minus_minus_rat @ B_24 @ C_10 ) )
= ( minus_minus_rat @ ( times_times_rat @ ( number_number_of_rat @ V_5 ) @ B_24 ) @ ( times_times_rat @ ( number_number_of_rat @ V_5 ) @ C_10 ) ) ) ).
thf(fact_2394_right__diff__distrib__number__of,axiom,
! [V_5: int,B_24: complex,C_10: complex] :
( ( times_times_complex @ ( number528085621omplex @ V_5 ) @ ( minus_minus_complex @ B_24 @ C_10 ) )
= ( minus_minus_complex @ ( times_times_complex @ ( number528085621omplex @ V_5 ) @ B_24 ) @ ( times_times_complex @ ( number528085621omplex @ V_5 ) @ C_10 ) ) ) ).
thf(fact_2395_right__diff__distrib__number__of,axiom,
! [V_5: int,B_24: real,C_10: real] :
( ( times_times_real @ ( number267125858f_real @ V_5 ) @ ( minus_minus_real @ B_24 @ C_10 ) )
= ( minus_minus_real @ ( times_times_real @ ( number267125858f_real @ V_5 ) @ B_24 ) @ ( times_times_real @ ( number267125858f_real @ V_5 ) @ C_10 ) ) ) ).
thf(fact_2396_right__diff__distrib__number__of,axiom,
! [V_5: int,B_24: int,C_10: int] :
( ( times_times_int @ ( number_number_of_int @ V_5 ) @ ( minus_minus_int @ B_24 @ C_10 ) )
= ( minus_minus_int @ ( times_times_int @ ( number_number_of_int @ V_5 ) @ B_24 ) @ ( times_times_int @ ( number_number_of_int @ V_5 ) @ C_10 ) ) ) ).
thf(fact_2397_left__diff__distrib__number__of,axiom,
! [A_34: rat,B_23: rat,V_4: int] :
( ( times_times_rat @ ( minus_minus_rat @ A_34 @ B_23 ) @ ( number_number_of_rat @ V_4 ) )
= ( minus_minus_rat @ ( times_times_rat @ A_34 @ ( number_number_of_rat @ V_4 ) ) @ ( times_times_rat @ B_23 @ ( number_number_of_rat @ V_4 ) ) ) ) ).
thf(fact_2398_left__diff__distrib__number__of,axiom,
! [A_34: complex,B_23: complex,V_4: int] :
( ( times_times_complex @ ( minus_minus_complex @ A_34 @ B_23 ) @ ( number528085621omplex @ V_4 ) )
= ( minus_minus_complex @ ( times_times_complex @ A_34 @ ( number528085621omplex @ V_4 ) ) @ ( times_times_complex @ B_23 @ ( number528085621omplex @ V_4 ) ) ) ) ).
thf(fact_2399_left__diff__distrib__number__of,axiom,
! [A_34: real,B_23: real,V_4: int] :
( ( times_times_real @ ( minus_minus_real @ A_34 @ B_23 ) @ ( number267125858f_real @ V_4 ) )
= ( minus_minus_real @ ( times_times_real @ A_34 @ ( number267125858f_real @ V_4 ) ) @ ( times_times_real @ B_23 @ ( number267125858f_real @ V_4 ) ) ) ) ).
thf(fact_2400_left__diff__distrib__number__of,axiom,
! [A_34: int,B_23: int,V_4: int] :
( ( times_times_int @ ( minus_minus_int @ A_34 @ B_23 ) @ ( number_number_of_int @ V_4 ) )
= ( minus_minus_int @ ( times_times_int @ A_34 @ ( number_number_of_int @ V_4 ) ) @ ( times_times_int @ B_23 @ ( number_number_of_int @ V_4 ) ) ) ) ).
thf(fact_2401_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_2402_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
thf(fact_2403_zdvd__mono,axiom,
! [M: int,T: int,K: int] :
( ( K != zero_zero_int )
=> ( ( dvd_dvd_int @ M @ T )
<=> ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
thf(fact_2404_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_2405_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_2406_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
<=> ( dvd_dvd_int @ K @ N ) ) ).
thf(fact_2407_zdvd__period,axiom,
! [C: int,X: int,T: int,A_1: int,D_1: int] :
( ( dvd_dvd_int @ A_1 @ D_1 )
=> ( ( dvd_dvd_int @ A_1 @ ( plus_plus_int @ X @ T ) )
<=> ( dvd_dvd_int @ A_1 @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D_1 ) ) @ T ) ) ) ) ).
thf(fact_2408_diff__bin__simps_I10_J,axiom,
! [K: int,L: int] :
( ( minus_minus_int @ ( bit1 @ K ) @ ( bit1 @ L ) )
= ( bit0 @ ( minus_minus_int @ K @ L ) ) ) ).
thf(fact_2409_diff__bin__simps_I9_J,axiom,
! [K: int,L: int] :
( ( minus_minus_int @ ( bit1 @ K ) @ ( bit0 @ L ) )
= ( bit1 @ ( minus_minus_int @ K @ L ) ) ) ).
thf(fact_2410_diff__bin__simps_I3_J,axiom,
! [L: int] :
( ( minus_minus_int @ pls @ ( bit0 @ L ) )
= ( bit0 @ ( minus_minus_int @ pls @ L ) ) ) ).
thf(fact_2411_less__bin__lemma,axiom,
! [K: int,L: int] :
( ( ord_less_int @ K @ L )
<=> ( ord_less_int @ ( minus_minus_int @ K @ L ) @ zero_zero_int ) ) ).
thf(fact_2412_zprime__zdvd__power,axiom,
! [A_1: int,N: nat,P_3: int] :
( ( zprime @ P_3 )
=> ( ( dvd_dvd_int @ P_3 @ ( power_power_int @ A_1 @ N ) )
=> ( dvd_dvd_int @ P_3 @ A_1 ) ) ) ).
thf(fact_2413_diff__bin__simps_I2_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ min )
= ( succ @ K ) ) ).
thf(fact_2414_unity__coeff__ex,axiom,
! [P_5: nat > $o,L_3: nat] :
( ? [X_1: nat] : ( P_5 @ ( times_times_nat @ L_3 @ X_1 ) )
<=> ? [X_1: nat] :
( ( dvd_dvd_nat @ L_3 @ ( plus_plus_nat @ X_1 @ zero_zero_nat ) )
& ( P_5 @ X_1 ) ) ) ).
thf(fact_2415_unity__coeff__ex,axiom,
! [P_5: int > $o,L_3: int] :
( ? [X_1: int] : ( P_5 @ ( times_times_int @ L_3 @ X_1 ) )
<=> ? [X_1: int] :
( ( dvd_dvd_int @ L_3 @ ( plus_plus_int @ X_1 @ zero_zero_int ) )
& ( P_5 @ X_1 ) ) ) ).
thf(fact_2416_unity__coeff__ex,axiom,
! [P_5: rat > $o,L_3: rat] :
( ? [X_1: rat] : ( P_5 @ ( times_times_rat @ L_3 @ X_1 ) )
<=> ? [X_1: rat] :
( ( dvd_dvd_rat @ L_3 @ ( plus_plus_rat @ X_1 @ zero_zero_rat ) )
& ( P_5 @ X_1 ) ) ) ).
thf(fact_2417_unity__coeff__ex,axiom,
! [P_5: quickcheck_code_int > $o,L_3: quickcheck_code_int] :
( ? [X_1: quickcheck_code_int] : ( P_5 @ ( times_123202395de_int @ L_3 @ X_1 ) )
<=> ? [X_1: quickcheck_code_int] :
( ( dvd_dv1760642554de_int @ L_3 @ ( plus_p1446045655de_int @ X_1 @ zero_z891286103de_int ) )
& ( P_5 @ X_1 ) ) ) ).
thf(fact_2418_unity__coeff__ex,axiom,
! [P_5: complex > $o,L_3: complex] :
( ? [X_1: complex] : ( P_5 @ ( times_times_complex @ L_3 @ X_1 ) )
<=> ? [X_1: complex] :
( ( dvd_dvd_complex @ L_3 @ ( plus_plus_complex @ X_1 @ zero_zero_complex ) )
& ( P_5 @ X_1 ) ) ) ).
thf(fact_2419_unity__coeff__ex,axiom,
! [P_5: real > $o,L_3: real] :
( ? [X_1: real] : ( P_5 @ ( times_times_real @ L_3 @ X_1 ) )
<=> ? [X_1: real] :
( ( dvd_dvd_real @ L_3 @ ( plus_plus_real @ X_1 @ zero_zero_real ) )
& ( P_5 @ X_1 ) ) ) ).
thf(fact_2420_unity__coeff__ex,axiom,
! [P_5: code_code_numeral > $o,L_3: code_code_numeral] :
( ? [X_1: code_code_numeral] : ( P_5 @ ( times_1655362735umeral @ L_3 @ X_1 ) )
<=> ? [X_1: code_code_numeral] :
( ( dvd_dv174992974umeral @ L_3 @ ( plus_p1627245867umeral @ X_1 @ zero_z126310315umeral ) )
& ( P_5 @ X_1 ) ) ) ).
thf(fact_2421_le__add__iff2,axiom,
! [A_33: rat,E_5: rat,C_9: rat,B_22: rat,D_9: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A_33 @ E_5 ) @ C_9 ) @ ( plus_plus_rat @ ( times_times_rat @ B_22 @ E_5 ) @ D_9 ) )
<=> ( ord_less_eq_rat @ C_9 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B_22 @ A_33 ) @ E_5 ) @ D_9 ) ) ) ).
thf(fact_2422_le__add__iff2,axiom,
! [A_33: real,E_5: real,C_9: real,B_22: real,D_9: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A_33 @ E_5 ) @ C_9 ) @ ( plus_plus_real @ ( times_times_real @ B_22 @ E_5 ) @ D_9 ) )
<=> ( ord_less_eq_real @ C_9 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B_22 @ A_33 ) @ E_5 ) @ D_9 ) ) ) ).
thf(fact_2423_le__add__iff2,axiom,
! [A_33: int,E_5: int,C_9: int,B_22: int,D_9: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A_33 @ E_5 ) @ C_9 ) @ ( plus_plus_int @ ( times_times_int @ B_22 @ E_5 ) @ D_9 ) )
<=> ( ord_less_eq_int @ C_9 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_22 @ A_33 ) @ E_5 ) @ D_9 ) ) ) ).
thf(fact_2424_le__add__iff1,axiom,
! [A_32: rat,E_4: rat,C_8: rat,B_21: rat,D_8: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A_32 @ E_4 ) @ C_8 ) @ ( plus_plus_rat @ ( times_times_rat @ B_21 @ E_4 ) @ D_8 ) )
<=> ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A_32 @ B_21 ) @ E_4 ) @ C_8 ) @ D_8 ) ) ).
thf(fact_2425_le__add__iff1,axiom,
! [A_32: real,E_4: real,C_8: real,B_21: real,D_8: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A_32 @ E_4 ) @ C_8 ) @ ( plus_plus_real @ ( times_times_real @ B_21 @ E_4 ) @ D_8 ) )
<=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A_32 @ B_21 ) @ E_4 ) @ C_8 ) @ D_8 ) ) ).
thf(fact_2426_le__add__iff1,axiom,
! [A_32: int,E_4: int,C_8: int,B_21: int,D_8: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A_32 @ E_4 ) @ C_8 ) @ ( plus_plus_int @ ( times_times_int @ B_21 @ E_4 ) @ D_8 ) )
<=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_32 @ B_21 ) @ E_4 ) @ C_8 ) @ D_8 ) ) ).
thf(fact_2427_less__add__iff2,axiom,
! [A_31: rat,E_3: rat,C_7: rat,B_20: rat,D_7: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A_31 @ E_3 ) @ C_7 ) @ ( plus_plus_rat @ ( times_times_rat @ B_20 @ E_3 ) @ D_7 ) )
<=> ( ord_less_rat @ C_7 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B_20 @ A_31 ) @ E_3 ) @ D_7 ) ) ) ).
thf(fact_2428_less__add__iff2,axiom,
! [A_31: real,E_3: real,C_7: real,B_20: real,D_7: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A_31 @ E_3 ) @ C_7 ) @ ( plus_plus_real @ ( times_times_real @ B_20 @ E_3 ) @ D_7 ) )
<=> ( ord_less_real @ C_7 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B_20 @ A_31 ) @ E_3 ) @ D_7 ) ) ) ).
thf(fact_2429_less__add__iff2,axiom,
! [A_31: int,E_3: int,C_7: int,B_20: int,D_7: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A_31 @ E_3 ) @ C_7 ) @ ( plus_plus_int @ ( times_times_int @ B_20 @ E_3 ) @ D_7 ) )
<=> ( ord_less_int @ C_7 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_20 @ A_31 ) @ E_3 ) @ D_7 ) ) ) ).
thf(fact_2430_less__add__iff1,axiom,
! [A_30: rat,E_2: rat,C_6: rat,B_19: rat,D_6: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A_30 @ E_2 ) @ C_6 ) @ ( plus_plus_rat @ ( times_times_rat @ B_19 @ E_2 ) @ D_6 ) )
<=> ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A_30 @ B_19 ) @ E_2 ) @ C_6 ) @ D_6 ) ) ).
thf(fact_2431_less__add__iff1,axiom,
! [A_30: real,E_2: real,C_6: real,B_19: real,D_6: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A_30 @ E_2 ) @ C_6 ) @ ( plus_plus_real @ ( times_times_real @ B_19 @ E_2 ) @ D_6 ) )
<=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A_30 @ B_19 ) @ E_2 ) @ C_6 ) @ D_6 ) ) ).
thf(fact_2432_less__add__iff1,axiom,
! [A_30: int,E_2: int,C_6: int,B_19: int,D_6: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A_30 @ E_2 ) @ C_6 ) @ ( plus_plus_int @ ( times_times_int @ B_19 @ E_2 ) @ D_6 ) )
<=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_30 @ B_19 ) @ E_2 ) @ C_6 ) @ D_6 ) ) ).
thf(fact_2433_real__squared__diff__one__factored,axiom,
! [X_10: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X_10 @ X_10 ) @ one_one_rat )
= ( times_times_rat @ ( plus_plus_rat @ X_10 @ one_one_rat ) @ ( minus_minus_rat @ X_10 @ one_one_rat ) ) ) ).
thf(fact_2434_real__squared__diff__one__factored,axiom,
! [X_10: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X_10 @ X_10 ) @ one_one_complex )
= ( times_times_complex @ ( plus_plus_complex @ X_10 @ one_one_complex ) @ ( minus_minus_complex @ X_10 @ one_one_complex ) ) ) ).
thf(fact_2435_real__squared__diff__one__factored,axiom,
! [X_10: real] :
( ( minus_minus_real @ ( times_times_real @ X_10 @ X_10 ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X_10 @ one_one_real ) @ ( minus_minus_real @ X_10 @ one_one_real ) ) ) ).
thf(fact_2436_real__squared__diff__one__factored,axiom,
! [X_10: int] :
( ( minus_minus_int @ ( times_times_int @ X_10 @ X_10 ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X_10 @ one_one_int ) @ ( minus_minus_int @ X_10 @ one_one_int ) ) ) ).
thf(fact_2437_add__number__of__diff1,axiom,
! [V_3: int,W_2: int,C_5: rat] :
( ( plus_plus_rat @ ( number_number_of_rat @ V_3 ) @ ( minus_minus_rat @ ( number_number_of_rat @ W_2 ) @ C_5 ) )
= ( minus_minus_rat @ ( number_number_of_rat @ ( plus_plus_int @ V_3 @ W_2 ) ) @ C_5 ) ) ).
thf(fact_2438_add__number__of__diff1,axiom,
! [V_3: int,W_2: int,C_5: complex] :
( ( plus_plus_complex @ ( number528085621omplex @ V_3 ) @ ( minus_minus_complex @ ( number528085621omplex @ W_2 ) @ C_5 ) )
= ( minus_minus_complex @ ( number528085621omplex @ ( plus_plus_int @ V_3 @ W_2 ) ) @ C_5 ) ) ).
thf(fact_2439_add__number__of__diff1,axiom,
! [V_3: int,W_2: int,C_5: real] :
( ( plus_plus_real @ ( number267125858f_real @ V_3 ) @ ( minus_minus_real @ ( number267125858f_real @ W_2 ) @ C_5 ) )
= ( minus_minus_real @ ( number267125858f_real @ ( plus_plus_int @ V_3 @ W_2 ) ) @ C_5 ) ) ).
thf(fact_2440_add__number__of__diff1,axiom,
! [V_3: int,W_2: int,C_5: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_3 ) @ ( minus_minus_int @ ( number_number_of_int @ W_2 ) @ C_5 ) )
= ( minus_minus_int @ ( number_number_of_int @ ( plus_plus_int @ V_3 @ W_2 ) ) @ C_5 ) ) ).
thf(fact_2441_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_2442_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_2443_diff__bin__simps_I4_J,axiom,
! [L: int] :
( ( minus_minus_int @ pls @ ( bit1 @ L ) )
= ( bit1 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_2444_diff__bin__simps_I6_J,axiom,
! [L: int] :
( ( minus_minus_int @ min @ ( bit1 @ L ) )
= ( bit0 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_2445_diff__bin__simps_I5_J,axiom,
! [L: int] :
( ( minus_minus_int @ min @ ( bit0 @ L ) )
= ( bit1 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_2446_zprime__power__zdvd__cancel__left,axiom,
! [N: nat,B_1: int,A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ~ ( dvd_dvd_int @ P_3 @ A_1 )
=> ( ( dvd_dvd_int @ ( power_power_int @ P_3 @ N ) @ ( times_times_int @ A_1 @ B_1 ) )
=> ( dvd_dvd_int @ ( power_power_int @ P_3 @ N ) @ B_1 ) ) ) ) ).
thf(fact_2447_zprime__power__zdvd__cancel__right,axiom,
! [N: nat,A_1: int,B_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ~ ( dvd_dvd_int @ P_3 @ B_1 )
=> ( ( dvd_dvd_int @ ( power_power_int @ P_3 @ N ) @ ( times_times_int @ A_1 @ B_1 ) )
=> ( dvd_dvd_int @ ( power_power_int @ P_3 @ N ) @ A_1 ) ) ) ) ).
thf(fact_2448_dvd__power,axiom,
! [X_9: nat,N_11: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N_11 )
| ( X_9 = one_one_nat ) )
=> ( dvd_dvd_nat @ X_9 @ ( power_power_nat @ X_9 @ N_11 ) ) ) ).
thf(fact_2449_dvd__power,axiom,
! [X_9: int,N_11: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N_11 )
| ( X_9 = one_one_int ) )
=> ( dvd_dvd_int @ X_9 @ ( power_power_int @ X_9 @ N_11 ) ) ) ).
thf(fact_2450_dvd__power,axiom,
! [X_9: rat,N_11: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N_11 )
| ( X_9 = one_one_rat ) )
=> ( dvd_dvd_rat @ X_9 @ ( power_power_rat @ X_9 @ N_11 ) ) ) ).
thf(fact_2451_dvd__power,axiom,
! [X_9: quickcheck_code_int,N_11: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N_11 )
| ( X_9 = one_on1684967323de_int ) )
=> ( dvd_dv1760642554de_int @ X_9 @ ( power_881366806de_int @ X_9 @ N_11 ) ) ) ).
thf(fact_2452_dvd__power,axiom,
! [X_9: complex,N_11: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N_11 )
| ( X_9 = one_one_complex ) )
=> ( dvd_dvd_complex @ X_9 @ ( power_power_complex @ X_9 @ N_11 ) ) ) ).
thf(fact_2453_dvd__power,axiom,
! [X_9: code_code_numeral,N_11: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N_11 )
| ( X_9 = one_on1645066479umeral ) )
=> ( dvd_dv174992974umeral @ X_9 @ ( power_2100829034umeral @ X_9 @ N_11 ) ) ) ).
thf(fact_2454_dvd__power,axiom,
! [X_9: real,N_11: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N_11 )
| ( X_9 = one_one_real ) )
=> ( dvd_dvd_real @ X_9 @ ( power_power_real @ X_9 @ N_11 ) ) ) ).
thf(fact_2455_zspecial__product,axiom,
! [A_1: int,B_1: int] :
( ( times_times_int @ ( plus_plus_int @ A_1 @ B_1 ) @ ( minus_minus_int @ A_1 @ B_1 ) )
= ( minus_minus_int @ ( power_power_int @ A_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ B_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_2456_power2__diff,axiom,
! [X_8: rat,Y_6: rat] :
( ( power_power_rat @ ( minus_minus_rat @ X_8 @ Y_6 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X_8 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_rat @ Y_6 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( number_number_of_rat @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_8 ) @ Y_6 ) ) ) ).
thf(fact_2457_power2__diff,axiom,
! [X_8: complex,Y_6: complex] :
( ( power_power_complex @ ( minus_minus_complex @ X_8 @ Y_6 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X_8 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_complex @ Y_6 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( number528085621omplex @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_8 ) @ Y_6 ) ) ) ).
thf(fact_2458_power2__diff,axiom,
! [X_8: real,Y_6: real] :
( ( power_power_real @ ( minus_minus_real @ X_8 @ Y_6 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X_8 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ Y_6 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_8 ) @ Y_6 ) ) ) ).
thf(fact_2459_power2__diff,axiom,
! [X_8: int,Y_6: int] :
( ( power_power_int @ ( minus_minus_int @ X_8 @ Y_6 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X_8 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ Y_6 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ X_8 ) @ Y_6 ) ) ) ).
thf(fact_2460_zdiff__power3,axiom,
! [A_1: int,B_1: int] :
( ( power_power_int @ ( minus_minus_int @ A_1 @ B_1 ) @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( minus_minus_int @ ( plus_plus_int @ ( minus_minus_int @ ( power_power_int @ A_1 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) @ ( power_power_int @ A_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ B_1 ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) @ A_1 ) @ ( power_power_int @ B_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( power_power_int @ B_1 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_2461_zdiff__power2,axiom,
! [A_1: int,B_1: int] :
( ( power_power_int @ ( minus_minus_int @ A_1 @ B_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ ( minus_minus_int @ ( power_power_int @ A_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A_1 ) @ B_1 ) ) @ ( power_power_int @ B_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_2462_neg__one__power__eq__mod__m,axiom,
! [J: nat,K: 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 ) @ M )
=> ( ( power_power_int @ ( number_number_of_int @ min ) @ J )
= ( power_power_int @ ( number_number_of_int @ min ) @ K ) ) ) ) ).
thf(fact_2463_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_2464_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_2465_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_2466_Legendre__def,axiom,
! [A_1: int,P_3: int] :
( ( ( zcong @ A_1 @ zero_zero_int @ P_3 )
=> ( ( legendre @ A_1 @ P_3 )
= zero_zero_int ) )
& ( ~ ( zcong @ A_1 @ zero_zero_int @ P_3 )
=> ( ( ( quadRes @ P_3 @ A_1 )
=> ( ( legendre @ A_1 @ P_3 )
= one_one_int ) )
& ( ~ ( quadRes @ P_3 @ A_1 )
=> ( ( legendre @ A_1 @ P_3 )
= ( number_number_of_int @ min ) ) ) ) ) ) ).
thf(fact_2467_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_2468_zcong__square__zless,axiom,
! [A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ zero_zero_int @ A_1 )
=> ( ( ord_less_int @ A_1 @ P_3 )
=> ( ( zcong @ ( times_times_int @ A_1 @ A_1 ) @ one_one_int @ P_3 )
=> ( ( A_1 = one_one_int )
| ( A_1
= ( minus_minus_int @ P_3 @ one_one_int ) ) ) ) ) ) ) ).
thf(fact_2469_zcong__square,axiom,
! [A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ zero_zero_int @ A_1 )
=> ( ( zcong @ ( times_times_int @ A_1 @ A_1 ) @ one_one_int @ P_3 )
=> ( ( zcong @ A_1 @ one_one_int @ P_3 )
| ( zcong @ A_1 @ ( minus_minus_int @ P_3 @ one_one_int ) @ P_3 ) ) ) ) ) ).
thf(fact_2470_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_2471_tn,axiom,
( tn
= ( minus_minus_nat @ ( nat_1 @ t ) @ one_one_nat ) ) ).
thf(fact_2472_dvd__diff__nat,axiom,
! [N: nat,K: nat,M: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_2473_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
thf(fact_2474_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ M ) ) ) ) ).
thf(fact_2475_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ N ) ) ) ) ).
thf(fact_2476_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_2477_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
thf(fact_2478_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_2479_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_2480_less__imp__diff__less,axiom,
! [N: nat,J: nat,K: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
thf(fact_2481_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_2482_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
thf(fact_2483_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
thf(fact_2484_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_2485_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_2486_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_2487_le__diff__iff,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ) ) ).
thf(fact_2488_Nat_Odiff__diff__eq,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_2489_eq__diff__iff,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
<=> ( M = N ) ) ) ) ).
thf(fact_2490_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_2491_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_2492_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_2493_Nat_Odiff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
thf(fact_2494_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
thf(fact_2495_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
thf(fact_2496_dvd__reduce,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( plus_plus_nat @ N @ K ) )
<=> ( dvd_dvd_nat @ K @ N ) ) ).
thf(fact_2497_divides__add__revr,axiom,
! [B_1: nat,D_1: nat,A_1: nat] :
( ( dvd_dvd_nat @ D_1 @ A_1 )
=> ( ( dvd_dvd_nat @ D_1 @ ( plus_plus_nat @ A_1 @ B_1 ) )
=> ( dvd_dvd_nat @ D_1 @ B_1 ) ) ) ).
thf(fact_2498_divides__mul__l,axiom,
! [C: nat,A_1: nat,B_1: nat] :
( ( dvd_dvd_nat @ A_1 @ B_1 )
=> ( dvd_dvd_nat @ ( times_times_nat @ C @ A_1 ) @ ( times_times_nat @ C @ B_1 ) ) ) ).
thf(fact_2499_divides__mul__r,axiom,
! [C: nat,A_1: nat,B_1: nat] :
( ( dvd_dvd_nat @ A_1 @ B_1 )
=> ( dvd_dvd_nat @ ( times_times_nat @ A_1 @ C ) @ ( times_times_nat @ B_1 @ C ) ) ) ).
thf(fact_2500_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
<=> ( M = one_one_nat ) ) ).
thf(fact_2501_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_2502_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_2503_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_2504_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_2505_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
thf(fact_2506_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_2507_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_2508_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_2509_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
thf(fact_2510_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_2511_divides__ge,axiom,
! [A_1: nat,B_1: nat] :
( ( dvd_dvd_nat @ A_1 @ B_1 )
=> ( ( B_1 = zero_zero_nat )
| ( ord_less_eq_nat @ A_1 @ B_1 ) ) ) ).
thf(fact_2512_diff__less__mono,axiom,
! [C: nat,A_1: nat,B_1: nat] :
( ( ord_less_nat @ A_1 @ B_1 )
=> ( ( ord_less_eq_nat @ C @ A_1 )
=> ( ord_less_nat @ ( minus_minus_nat @ A_1 @ C ) @ ( minus_minus_nat @ B_1 @ C ) ) ) ) ).
thf(fact_2513_less__diff__iff,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
<=> ( ord_less_nat @ M @ N ) ) ) ) ).
thf(fact_2514_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
thf(fact_2515_nat__mult__dvd__cancel__disj_H,axiom,
! [M: nat,K: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
<=> ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
thf(fact_2516_diff__diff__right,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
thf(fact_2517_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
<=> ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
thf(fact_2518_le__add__diff,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_eq_nat @ M @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K ) ) ) ).
thf(fact_2519_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_2520_add__diff__assoc,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
thf(fact_2521_le__diff__conv2,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
thf(fact_2522_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_2523_le__imp__diff__is__add,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
<=> ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
thf(fact_2524_diff__add__assoc,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
thf(fact_2525_add__diff__assoc2,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
thf(fact_2526_diff__add__assoc2,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
thf(fact_2527_divides__rev,axiom,
! [A_1: nat,N: nat,B_1: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A_1 @ N ) @ ( power_power_nat @ B_1 @ N ) )
=> ( ( N != zero_zero_nat )
=> ( dvd_dvd_nat @ A_1 @ B_1 ) ) ) ).
thf(fact_2528_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_2529_zdvd__int,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
<=> ( dvd_dvd_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) ) ) ).
thf(fact_2530_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_2531_nat__diff__split__asm,axiom,
! [P: nat > $o,A_1: nat,B_1: nat] :
( ( P @ ( minus_minus_nat @ A_1 @ B_1 ) )
<=> ~ ( ( ( ord_less_nat @ A_1 @ B_1 )
& ~ ( P @ zero_zero_nat ) )
| ? [D_2: nat] :
( ( A_1
= ( plus_plus_nat @ B_1 @ D_2 ) )
& ~ ( P @ D_2 ) ) ) ) ).
thf(fact_2532_nat__diff__split,axiom,
! [P: nat > $o,A_1: nat,B_1: nat] :
( ( P @ ( minus_minus_nat @ A_1 @ B_1 ) )
<=> ( ( ( ord_less_nat @ A_1 @ B_1 )
=> ( P @ zero_zero_nat ) )
& ! [D_2: nat] :
( ( A_1
= ( plus_plus_nat @ B_1 @ D_2 ) )
=> ( P @ D_2 ) ) ) ) ).
thf(fact_2533_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ) ).
thf(fact_2534_nat__mult__dvd__cancel1,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
<=> ( dvd_dvd_nat @ M @ N ) ) ) ).
thf(fact_2535_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
thf(fact_2536_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_2537_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_2538_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_2539_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_2540_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_2541_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_2542_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_2543_zcong__trans,axiom,
! [C: int,A_1: int,B_1: int,M: int] :
( ( zcong @ A_1 @ B_1 @ M )
=> ( ( zcong @ B_1 @ C @ M )
=> ( zcong @ A_1 @ C @ M ) ) ) ).
thf(fact_2544_zcong__eq__trans,axiom,
! [D_1: int,C: int,A_1: int,B_1: int,M: int] :
( ( zcong @ A_1 @ B_1 @ M )
=> ( ( B_1 = C )
=> ( ( zcong @ C @ D_1 @ M )
=> ( zcong @ A_1 @ D_1 @ M ) ) ) ) ).
thf(fact_2545_zcong__refl,axiom,
! [K: int,M: int] : ( zcong @ K @ K @ M ) ).
thf(fact_2546_zcong__sym,axiom,
! [A_1: int,B_1: int,M: int] :
( ( zcong @ A_1 @ B_1 @ M )
<=> ( zcong @ B_1 @ A_1 @ M ) ) ).
thf(fact_2547_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_2548_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_2549_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_2550_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_2551_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_2552_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_2553_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_2554_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_2555_realpow__minus__mult,axiom,
! [X_7: complex,N_10: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_10 )
=> ( ( times_times_complex @ ( power_power_complex @ X_7 @ ( minus_minus_nat @ N_10 @ one_one_nat ) ) @ X_7 )
= ( power_power_complex @ X_7 @ N_10 ) ) ) ).
thf(fact_2556_realpow__minus__mult,axiom,
! [X_7: real,N_10: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_10 )
=> ( ( times_times_real @ ( power_power_real @ X_7 @ ( minus_minus_nat @ N_10 @ one_one_nat ) ) @ X_7 )
= ( power_power_real @ X_7 @ N_10 ) ) ) ).
thf(fact_2557_realpow__minus__mult,axiom,
! [X_7: nat,N_10: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_10 )
=> ( ( times_times_nat @ ( power_power_nat @ X_7 @ ( minus_minus_nat @ N_10 @ one_one_nat ) ) @ X_7 )
= ( power_power_nat @ X_7 @ N_10 ) ) ) ).
thf(fact_2558_realpow__minus__mult,axiom,
! [X_7: int,N_10: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_10 )
=> ( ( times_times_int @ ( power_power_int @ X_7 @ ( minus_minus_nat @ N_10 @ one_one_nat ) ) @ X_7 )
= ( power_power_int @ X_7 @ N_10 ) ) ) ).
thf(fact_2559_realpow__minus__mult,axiom,
! [X_7: rat,N_10: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_10 )
=> ( ( times_times_rat @ ( power_power_rat @ X_7 @ ( minus_minus_nat @ N_10 @ one_one_nat ) ) @ X_7 )
= ( power_power_rat @ X_7 @ N_10 ) ) ) ).
thf(fact_2560_realpow__minus__mult,axiom,
! [X_7: quickcheck_code_int,N_10: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_10 )
=> ( ( times_123202395de_int @ ( power_881366806de_int @ X_7 @ ( minus_minus_nat @ N_10 @ one_one_nat ) ) @ X_7 )
= ( power_881366806de_int @ X_7 @ N_10 ) ) ) ).
thf(fact_2561_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_2562_order__le__neq__implies__less,axiom,
! [X_6: rat,Y_5: rat] :
( ( ord_less_eq_rat @ X_6 @ Y_5 )
=> ( ( X_6 != Y_5 )
=> ( ord_less_rat @ X_6 @ Y_5 ) ) ) ).
thf(fact_2563_order__le__neq__implies__less,axiom,
! [X_6: int > $o,Y_5: int > $o] :
( ( ord_less_eq_int_o @ X_6 @ Y_5 )
=> ( ( X_6 != Y_5 )
=> ( ord_less_int_o @ X_6 @ Y_5 ) ) ) ).
thf(fact_2564_order__le__neq__implies__less,axiom,
! [X_6: nat > $o,Y_5: nat > $o] :
( ( ord_less_eq_nat_o @ X_6 @ Y_5 )
=> ( ( X_6 != Y_5 )
=> ( ord_less_nat_o @ X_6 @ Y_5 ) ) ) ).
thf(fact_2565_order__le__neq__implies__less,axiom,
! [X_6: quickcheck_code_int,Y_5: quickcheck_code_int] :
( ( ord_le258702272de_int @ X_6 @ Y_5 )
=> ( ( X_6 != Y_5 )
=> ( ord_le1860547276de_int @ X_6 @ Y_5 ) ) ) ).
thf(fact_2566_order__le__neq__implies__less,axiom,
! [X_6: real,Y_5: real] :
( ( ord_less_eq_real @ X_6 @ Y_5 )
=> ( ( X_6 != Y_5 )
=> ( ord_less_real @ X_6 @ Y_5 ) ) ) ).
thf(fact_2567_order__le__neq__implies__less,axiom,
! [X_6: nat,Y_5: nat] :
( ( ord_less_eq_nat @ X_6 @ Y_5 )
=> ( ( X_6 != Y_5 )
=> ( ord_less_nat @ X_6 @ Y_5 ) ) ) ).
thf(fact_2568_order__le__neq__implies__less,axiom,
! [X_6: int,Y_5: int] :
( ( ord_less_eq_int @ X_6 @ Y_5 )
=> ( ( X_6 != Y_5 )
=> ( ord_less_int @ X_6 @ Y_5 ) ) ) ).
thf(fact_2569_Euler_Oaux2,axiom,
! [B_1: int,A_1: int,C: int] :
( ( ord_less_int @ A_1 @ C )
=> ( ( ord_less_int @ B_1 @ C )
=> ( ( ord_less_eq_int @ A_1 @ B_1 )
| ( ord_less_eq_int @ B_1 @ A_1 ) ) ) ) ).
thf(fact_2570_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_2571_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_2572_zcong__id,axiom,
! [M: int] : ( zcong @ M @ zero_zero_int @ M ) ).
thf(fact_2573_IntPrimes_Ozcong__zero,axiom,
! [A_1: int,B_1: int] :
( ( zcong @ A_1 @ B_1 @ zero_zero_int )
<=> ( A_1 = B_1 ) ) ).
thf(fact_2574_Int2_Oaux1,axiom,
! [A_1: int,B_1: int,C: int] :
( ( ( minus_minus_int @ A_1 @ B_1 )
= C )
=> ( A_1
= ( plus_plus_int @ C @ B_1 ) ) ) ).
thf(fact_2575_zcong__1,axiom,
! [A_1: int,B_1: int] : ( zcong @ A_1 @ B_1 @ one_one_int ) ).
thf(fact_2576_zcong__zmult,axiom,
! [C: int,D_1: int,A_1: int,B_1: int,M: int] :
( ( zcong @ A_1 @ B_1 @ M )
=> ( ( zcong @ C @ D_1 @ M )
=> ( zcong @ ( times_times_int @ A_1 @ C ) @ ( times_times_int @ B_1 @ D_1 ) @ M ) ) ) ).
thf(fact_2577_zcong__scalar2,axiom,
! [K: int,A_1: int,B_1: int,M: int] :
( ( zcong @ A_1 @ B_1 @ M )
=> ( zcong @ ( times_times_int @ K @ A_1 ) @ ( times_times_int @ K @ B_1 ) @ M ) ) ).
thf(fact_2578_zcong__scalar,axiom,
! [K: int,A_1: int,B_1: int,M: int] :
( ( zcong @ A_1 @ B_1 @ M )
=> ( zcong @ ( times_times_int @ A_1 @ K ) @ ( times_times_int @ B_1 @ K ) @ M ) ) ).
thf(fact_2579_zcong__zmult__prop2,axiom,
! [C: int,D_1: int,A_1: int,B_1: int,M: int] :
( ( zcong @ A_1 @ B_1 @ M )
=> ( ( zcong @ C @ ( times_times_int @ D_1 @ A_1 ) @ M )
<=> ( zcong @ C @ ( times_times_int @ D_1 @ B_1 ) @ M ) ) ) ).
thf(fact_2580_zcong__zmult__prop1,axiom,
! [C: int,D_1: int,A_1: int,B_1: int,M: int] :
( ( zcong @ A_1 @ B_1 @ M )
=> ( ( zcong @ C @ ( times_times_int @ A_1 @ D_1 ) @ M )
<=> ( zcong @ C @ ( times_times_int @ B_1 @ D_1 ) @ M ) ) ) ).
thf(fact_2581_zcong__zmult__self,axiom,
! [A_1: int,M: int,B_1: int] : ( zcong @ ( times_times_int @ A_1 @ M ) @ ( times_times_int @ B_1 @ M ) @ M ) ).
thf(fact_2582_zcong__zadd,axiom,
! [C: int,D_1: int,A_1: int,B_1: int,M: int] :
( ( zcong @ A_1 @ B_1 @ M )
=> ( ( zcong @ C @ D_1 @ M )
=> ( zcong @ ( plus_plus_int @ A_1 @ C ) @ ( plus_plus_int @ B_1 @ D_1 ) @ M ) ) ) ).
thf(fact_2583_zcong__shift,axiom,
! [C: int,A_1: int,B_1: int,M: int] :
( ( zcong @ A_1 @ B_1 @ M )
=> ( zcong @ ( plus_plus_int @ A_1 @ C ) @ ( plus_plus_int @ B_1 @ C ) @ M ) ) ).
thf(fact_2584_zcong__zdiff,axiom,
! [C: int,D_1: int,A_1: int,B_1: int,M: int] :
( ( zcong @ A_1 @ B_1 @ M )
=> ( ( zcong @ C @ D_1 @ M )
=> ( zcong @ ( minus_minus_int @ A_1 @ C ) @ ( minus_minus_int @ B_1 @ D_1 ) @ M ) ) ) ).
thf(fact_2585_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_2586_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_2587_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_2588_zcong__not,axiom,
! [B_1: int,M: int,A_1: int] :
( ( ord_less_int @ zero_zero_int @ A_1 )
=> ( ( ord_less_int @ A_1 @ M )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_int @ B_1 @ A_1 )
=> ~ ( zcong @ A_1 @ B_1 @ M ) ) ) ) ) ).
thf(fact_2589_xzgcda__linear__aux1,axiom,
! [A_1: int,R_1: int,B_1: int,M: int,C: int,D_1: int,N: int] :
( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_1 @ ( times_times_int @ R_1 @ B_1 ) ) @ M ) @ ( times_times_int @ ( minus_minus_int @ C @ ( times_times_int @ R_1 @ D_1 ) ) @ N ) )
= ( minus_minus_int @ ( plus_plus_int @ ( times_times_int @ A_1 @ M ) @ ( times_times_int @ C @ N ) ) @ ( times_times_int @ R_1 @ ( plus_plus_int @ ( times_times_int @ B_1 @ M ) @ ( times_times_int @ D_1 @ N ) ) ) ) ) ).
thf(fact_2590_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_2591_zcong__iff__lin,axiom,
! [A_1: int,B_1: int,M: int] :
( ( zcong @ A_1 @ B_1 @ M )
<=> ? [K_1: int] :
( B_1
= ( plus_plus_int @ A_1 @ ( times_times_int @ M @ K_1 ) ) ) ) ).
thf(fact_2592_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_2593_zcong__zero__equiv__div,axiom,
! [A_1: int,M: int] :
( ( zcong @ A_1 @ zero_zero_int @ M )
<=> ( dvd_dvd_int @ M @ A_1 ) ) ).
thf(fact_2594_zcong__def,axiom,
! [A_1: int,B_1: int,M: int] :
( ( zcong @ A_1 @ B_1 @ M )
<=> ( dvd_dvd_int @ M @ ( minus_minus_int @ A_1 @ B_1 ) ) ) ).
thf(fact_2595_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_2596_Euler_Oaux1,axiom,
! [A_1: int,X: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ A_1 )
=> ( ( X
!= ( minus_minus_int @ A_1 @ one_one_int ) )
=> ( ord_less_int @ X @ ( minus_minus_int @ A_1 @ one_one_int ) ) ) ) ) ).
thf(fact_2597_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_2598_zcong__zless__0,axiom,
! [M: int,A_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_1 )
=> ( ( ord_less_int @ A_1 @ M )
=> ( ( zcong @ A_1 @ zero_zero_int @ M )
=> ( A_1 = zero_zero_int ) ) ) ) ).
thf(fact_2599_zcong__zless__imp__eq,axiom,
! [B_1: int,M: int,A_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_1 )
=> ( ( ord_less_int @ A_1 @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_int @ B_1 @ M )
=> ( ( zcong @ A_1 @ B_1 @ M )
=> ( A_1 = B_1 ) ) ) ) ) ) ).
thf(fact_2600_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_2601_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_2602_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_2603_zcong__zprime__prod__zero__contra,axiom,
! [B_1: int,A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ zero_zero_int @ A_1 )
=> ( ( ~ ( zcong @ A_1 @ zero_zero_int @ P_3 )
& ~ ( zcong @ B_1 @ zero_zero_int @ P_3 ) )
=> ~ ( zcong @ ( times_times_int @ A_1 @ B_1 ) @ zero_zero_int @ P_3 ) ) ) ) ).
thf(fact_2604_zcong__zprime__prod__zero,axiom,
! [B_1: int,A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ zero_zero_int @ A_1 )
=> ( ( zcong @ ( times_times_int @ A_1 @ B_1 ) @ zero_zero_int @ P_3 )
=> ( ( zcong @ A_1 @ zero_zero_int @ P_3 )
| ( zcong @ B_1 @ zero_zero_int @ P_3 ) ) ) ) ) ).
thf(fact_2605_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_2606_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_2607_realpow__num__eq__if,axiom,
! [M_9: rat,N_9: nat] :
( ( ( N_9 = zero_zero_nat )
=> ( ( power_power_rat @ M_9 @ N_9 )
= one_one_rat ) )
& ( ( N_9 != zero_zero_nat )
=> ( ( power_power_rat @ M_9 @ N_9 )
= ( times_times_rat @ M_9 @ ( power_power_rat @ M_9 @ ( minus_minus_nat @ N_9 @ one_one_nat ) ) ) ) ) ) ).
thf(fact_2608_realpow__num__eq__if,axiom,
! [M_9: quickcheck_code_int,N_9: nat] :
( ( ( N_9 = zero_zero_nat )
=> ( ( power_881366806de_int @ M_9 @ N_9 )
= one_on1684967323de_int ) )
& ( ( N_9 != zero_zero_nat )
=> ( ( power_881366806de_int @ M_9 @ N_9 )
= ( times_123202395de_int @ M_9 @ ( power_881366806de_int @ M_9 @ ( minus_minus_nat @ N_9 @ one_one_nat ) ) ) ) ) ) ).
thf(fact_2609_realpow__num__eq__if,axiom,
! [M_9: complex,N_9: nat] :
( ( ( N_9 = zero_zero_nat )
=> ( ( power_power_complex @ M_9 @ N_9 )
= one_one_complex ) )
& ( ( N_9 != zero_zero_nat )
=> ( ( power_power_complex @ M_9 @ N_9 )
= ( times_times_complex @ M_9 @ ( power_power_complex @ M_9 @ ( minus_minus_nat @ N_9 @ one_one_nat ) ) ) ) ) ) ).
thf(fact_2610_realpow__num__eq__if,axiom,
! [M_9: code_code_numeral,N_9: nat] :
( ( ( N_9 = zero_zero_nat )
=> ( ( power_2100829034umeral @ M_9 @ N_9 )
= one_on1645066479umeral ) )
& ( ( N_9 != zero_zero_nat )
=> ( ( power_2100829034umeral @ M_9 @ N_9 )
= ( times_1655362735umeral @ M_9 @ ( power_2100829034umeral @ M_9 @ ( minus_minus_nat @ N_9 @ one_one_nat ) ) ) ) ) ) ).
thf(fact_2611_realpow__num__eq__if,axiom,
! [M_9: real,N_9: nat] :
( ( ( N_9 = zero_zero_nat )
=> ( ( power_power_real @ M_9 @ N_9 )
= one_one_real ) )
& ( ( N_9 != zero_zero_nat )
=> ( ( power_power_real @ M_9 @ N_9 )
= ( times_times_real @ M_9 @ ( power_power_real @ M_9 @ ( minus_minus_nat @ N_9 @ one_one_nat ) ) ) ) ) ) ).
thf(fact_2612_realpow__num__eq__if,axiom,
! [M_9: nat,N_9: nat] :
( ( ( N_9 = zero_zero_nat )
=> ( ( power_power_nat @ M_9 @ N_9 )
= one_one_nat ) )
& ( ( N_9 != zero_zero_nat )
=> ( ( power_power_nat @ M_9 @ N_9 )
= ( times_times_nat @ M_9 @ ( power_power_nat @ M_9 @ ( minus_minus_nat @ N_9 @ one_one_nat ) ) ) ) ) ) ).
thf(fact_2613_realpow__num__eq__if,axiom,
! [M_9: int,N_9: nat] :
( ( ( N_9 = zero_zero_nat )
=> ( ( power_power_int @ M_9 @ N_9 )
= one_one_int ) )
& ( ( N_9 != zero_zero_nat )
=> ( ( power_power_int @ M_9 @ N_9 )
= ( times_times_int @ M_9 @ ( power_power_int @ M_9 @ ( minus_minus_nat @ N_9 @ one_one_nat ) ) ) ) ) ) ).
thf(fact_2614__096_B_Bthesis_O_A_I_B_Br_Av_O_Av_A_061_Ax_A_N_Ar_A_K_A_I1_A_L_Aint_An,axiom,
~ ! [R: int,V_2: int] :
~ ( ( V_2
= ( minus_minus_int @ x @ ( times_times_int @ R @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) )
& ( ord_less_eq_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( abs_abs_int @ V_2 ) ) @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) ).
thf(fact_2615__096_B_Bthesis_O_A_I_B_Bs_Aw_O_Aw_A_061_Ay_A_N_As_A_K_A_I1_A_L_Aint_An,axiom,
~ ! [S_2: int,W_1: int] :
~ ( ( W_1
= ( minus_minus_int @ y @ ( times_times_int @ S_2 @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) )
& ( ord_less_eq_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( abs_abs_int @ W_1 ) ) @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) ).
thf(fact_2616_decr__mult__lemma,axiom,
! [K: int,P: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( minus_minus_int @ X_1 @ D_1 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( minus_minus_int @ X_1 @ ( times_times_int @ K @ D_1 ) ) ) ) ) ) ) ).
thf(fact_2617_zprime__factor__exists,axiom,
! [A_1: int] :
( ( ord_less_int @ one_one_int @ A_1 )
=> ? [P_4: int] :
( ( zprime @ P_4 )
& ( dvd_dvd_int @ P_4 @ A_1 ) ) ) ).
thf(fact_2618_zcong__zless__unique,axiom,
! [A_1: 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_1 @ X_1 @ M )
& ! [Y_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Y_1 )
& ( ord_less_int @ Y_1 @ M )
& ( zcong @ A_1 @ Y_1 @ M ) )
=> ( Y_1 = X_1 ) ) ) ) ).
thf(fact_2619__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_2620_norR__mem__unique__aux,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_eq_int @ A_1 @ ( minus_minus_int @ B_1 @ one_one_int ) )
=> ( ord_less_int @ A_1 @ B_1 ) ) ).
thf(fact_2621_dvd_Oorder__refl,axiom,
! [X: nat] : ( dvd_dvd_nat @ X @ X ) ).
thf(fact_2622__096_B_Bthesis_O_A_I_B_Btn_O_A_091_124_Atn_A_061_Anat_At_A_N_A1_059_A0,axiom,
ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( nat_1 @ t ) @ one_one_nat ) ).
thf(fact_2623_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_2624_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_2625_dvd_Oless__asym_H,axiom,
! [A_1: nat,B_1: nat] :
( ( ( dvd_dvd_nat @ A_1 @ B_1 )
& ~ ( dvd_dvd_nat @ B_1 @ A_1 ) )
=> ~ ( ( dvd_dvd_nat @ B_1 @ A_1 )
& ~ ( dvd_dvd_nat @ A_1 @ B_1 ) ) ) ).
thf(fact_2626_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_2627_dvd_Oord__less__eq__trans,axiom,
! [C: nat,A_1: nat,B_1: nat] :
( ( ( dvd_dvd_nat @ A_1 @ B_1 )
& ~ ( dvd_dvd_nat @ B_1 @ A_1 ) )
=> ( ( B_1 = C )
=> ( ( dvd_dvd_nat @ A_1 @ C )
& ~ ( dvd_dvd_nat @ C @ A_1 ) ) ) ) ).
thf(fact_2628_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_2629_dvd_Oless__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( Y != X ) ) ).
thf(fact_2630_dvd_Oless__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( X != Y ) ) ).
thf(fact_2631_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_2632_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_2633_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_2634_dvd_Oless__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( X != Y ) ) ).
thf(fact_2635_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_2636_dvd_Oord__eq__less__trans,axiom,
! [C: nat,A_1: nat,B_1: nat] :
( ( A_1 = B_1 )
=> ( ( ( dvd_dvd_nat @ B_1 @ C )
& ~ ( dvd_dvd_nat @ C @ B_1 ) )
=> ( ( dvd_dvd_nat @ A_1 @ C )
& ~ ( dvd_dvd_nat @ C @ A_1 ) ) ) ) ).
thf(fact_2637_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_2638_dvd_Oantisym,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( dvd_dvd_nat @ Y @ X )
=> ( X = Y ) ) ) ).
thf(fact_2639_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
thf(fact_2640_dvd_Oord__le__eq__trans,axiom,
! [C: nat,A_1: nat,B_1: nat] :
( ( dvd_dvd_nat @ A_1 @ B_1 )
=> ( ( B_1 = C )
=> ( dvd_dvd_nat @ A_1 @ C ) ) ) ).
thf(fact_2641_dvd_Oord__eq__le__trans,axiom,
! [C: nat,A_1: nat,B_1: nat] :
( ( A_1 = B_1 )
=> ( ( dvd_dvd_nat @ B_1 @ C )
=> ( dvd_dvd_nat @ A_1 @ C ) ) ) ).
thf(fact_2642_dvd_Ole__neq__trans,axiom,
! [A_1: nat,B_1: nat] :
( ( dvd_dvd_nat @ A_1 @ B_1 )
=> ( ( A_1 != B_1 )
=> ( ( dvd_dvd_nat @ A_1 @ B_1 )
& ~ ( dvd_dvd_nat @ B_1 @ A_1 ) ) ) ) ).
thf(fact_2643_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_2644_dvd_Oantisym__conv,axiom,
! [Y: nat,X: nat] :
( ( dvd_dvd_nat @ Y @ X )
=> ( ( dvd_dvd_nat @ X @ Y )
<=> ( X = Y ) ) ) ).
thf(fact_2645_dvd_Oeq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( dvd_dvd_nat @ X @ Y ) ) ).
thf(fact_2646_dvd_Oneq__le__trans,axiom,
! [A_1: nat,B_1: nat] :
( ( A_1 != B_1 )
=> ( ( dvd_dvd_nat @ A_1 @ B_1 )
=> ( ( dvd_dvd_nat @ A_1 @ B_1 )
& ~ ( dvd_dvd_nat @ B_1 @ A_1 ) ) ) ) ).
thf(fact_2647_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_2648_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_2649_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_2650_dvd_Oeq__iff,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
<=> ( ( dvd_dvd_nat @ X @ Y )
& ( dvd_dvd_nat @ Y @ X ) ) ) ).
thf(fact_2651_dvd_Oless__irrefl,axiom,
! [X: nat] :
~ ( ( dvd_dvd_nat @ X @ X )
& ~ ( dvd_dvd_nat @ X @ X ) ) ).
thf(fact_2652_divides__antisym,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ( dvd_dvd_nat @ Y @ X ) )
<=> ( X = Y ) ) ).
thf(fact_2653_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_2654_abs__idempotent,axiom,
! [A_29: rat] :
( ( abs_abs_rat @ ( abs_abs_rat @ A_29 ) )
= ( abs_abs_rat @ A_29 ) ) ).
thf(fact_2655_abs__idempotent,axiom,
! [A_29: real] :
( ( abs_abs_real @ ( abs_abs_real @ A_29 ) )
= ( abs_abs_real @ A_29 ) ) ).
thf(fact_2656_abs__idempotent,axiom,
! [A_29: int] :
( ( abs_abs_int @ ( abs_abs_int @ A_29 ) )
= ( abs_abs_int @ A_29 ) ) ).
thf(fact_2657_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_2658_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_2659_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_2660_abs__eq__0,axiom,
! [A_28: rat] :
( ( ( abs_abs_rat @ A_28 )
= zero_zero_rat )
<=> ( A_28 = zero_zero_rat ) ) ).
thf(fact_2661_abs__eq__0,axiom,
! [A_28: real] :
( ( ( abs_abs_real @ A_28 )
= zero_zero_real )
<=> ( A_28 = zero_zero_real ) ) ).
thf(fact_2662_abs__eq__0,axiom,
! [A_28: int] :
( ( ( abs_abs_int @ A_28 )
= zero_zero_int )
<=> ( A_28 = zero_zero_int ) ) ).
thf(fact_2663_abs__zero,axiom,
( ( abs_abs_rat @ zero_zero_rat )
= zero_zero_rat ) ).
thf(fact_2664_abs__zero,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_2665_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_2666_abs__ge__self,axiom,
! [A_27: rat] : ( ord_less_eq_rat @ A_27 @ ( abs_abs_rat @ A_27 ) ) ).
thf(fact_2667_abs__ge__self,axiom,
! [A_27: real] : ( ord_less_eq_real @ A_27 @ ( abs_abs_real @ A_27 ) ) ).
thf(fact_2668_abs__ge__self,axiom,
! [A_27: int] : ( ord_less_eq_int @ A_27 @ ( abs_abs_int @ A_27 ) ) ).
thf(fact_2669_abs__le__D1,axiom,
! [A_26: rat,B_18: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A_26 ) @ B_18 )
=> ( ord_less_eq_rat @ A_26 @ B_18 ) ) ).
thf(fact_2670_abs__le__D1,axiom,
! [A_26: real,B_18: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A_26 ) @ B_18 )
=> ( ord_less_eq_real @ A_26 @ B_18 ) ) ).
thf(fact_2671_abs__le__D1,axiom,
! [A_26: int,B_18: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A_26 ) @ B_18 )
=> ( ord_less_eq_int @ A_26 @ B_18 ) ) ).
thf(fact_2672_abs__mult__self,axiom,
! [A_25: rat] :
( ( times_times_rat @ ( abs_abs_rat @ A_25 ) @ ( abs_abs_rat @ A_25 ) )
= ( times_times_rat @ A_25 @ A_25 ) ) ).
thf(fact_2673_abs__mult__self,axiom,
! [A_25: real] :
( ( times_times_real @ ( abs_abs_real @ A_25 ) @ ( abs_abs_real @ A_25 ) )
= ( times_times_real @ A_25 @ A_25 ) ) ).
thf(fact_2674_abs__mult__self,axiom,
! [A_25: int] :
( ( times_times_int @ ( abs_abs_int @ A_25 ) @ ( abs_abs_int @ A_25 ) )
= ( times_times_int @ A_25 @ A_25 ) ) ).
thf(fact_2675_abs__mult,axiom,
! [A_24: rat,B_17: rat] :
( ( abs_abs_rat @ ( times_times_rat @ A_24 @ B_17 ) )
= ( times_times_rat @ ( abs_abs_rat @ A_24 ) @ ( abs_abs_rat @ B_17 ) ) ) ).
thf(fact_2676_abs__mult,axiom,
! [A_24: real,B_17: real] :
( ( abs_abs_real @ ( times_times_real @ A_24 @ B_17 ) )
= ( times_times_real @ ( abs_abs_real @ A_24 ) @ ( abs_abs_real @ B_17 ) ) ) ).
thf(fact_2677_abs__mult,axiom,
! [A_24: int,B_17: int] :
( ( abs_abs_int @ ( times_times_int @ A_24 @ B_17 ) )
= ( times_times_int @ ( abs_abs_int @ A_24 ) @ ( abs_abs_int @ B_17 ) ) ) ).
thf(fact_2678_abs__add__abs,axiom,
! [A_23: rat,B_16: rat] :
( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A_23 ) @ ( abs_abs_rat @ B_16 ) ) )
= ( plus_plus_rat @ ( abs_abs_rat @ A_23 ) @ ( abs_abs_rat @ B_16 ) ) ) ).
thf(fact_2679_abs__add__abs,axiom,
! [A_23: real,B_16: real] :
( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A_23 ) @ ( abs_abs_real @ B_16 ) ) )
= ( plus_plus_real @ ( abs_abs_real @ A_23 ) @ ( abs_abs_real @ B_16 ) ) ) ).
thf(fact_2680_abs__add__abs,axiom,
! [A_23: int,B_16: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A_23 ) @ ( abs_abs_int @ B_16 ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A_23 ) @ ( abs_abs_int @ B_16 ) ) ) ).
thf(fact_2681_abs__one,axiom,
( ( abs_abs_rat @ one_one_rat )
= one_one_rat ) ).
thf(fact_2682_abs__one,axiom,
( ( abs_abs_real @ one_one_real )
= one_one_real ) ).
thf(fact_2683_abs__one,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
thf(fact_2684_abs__minus__commute,axiom,
! [A_22: rat,B_15: rat] :
( ( abs_abs_rat @ ( minus_minus_rat @ A_22 @ B_15 ) )
= ( abs_abs_rat @ ( minus_minus_rat @ B_15 @ A_22 ) ) ) ).
thf(fact_2685_abs__minus__commute,axiom,
! [A_22: real,B_15: real] :
( ( abs_abs_real @ ( minus_minus_real @ A_22 @ B_15 ) )
= ( abs_abs_real @ ( minus_minus_real @ B_15 @ A_22 ) ) ) ).
thf(fact_2686_abs__minus__commute,axiom,
! [A_22: int,B_15: int] :
( ( abs_abs_int @ ( minus_minus_int @ A_22 @ B_15 ) )
= ( abs_abs_int @ ( minus_minus_int @ B_15 @ A_22 ) ) ) ).
thf(fact_2687_power__abs,axiom,
! [A_21: rat,N_8: nat] :
( ( abs_abs_rat @ ( power_power_rat @ A_21 @ N_8 ) )
= ( power_power_rat @ ( abs_abs_rat @ A_21 ) @ N_8 ) ) ).
thf(fact_2688_power__abs,axiom,
! [A_21: real,N_8: nat] :
( ( abs_abs_real @ ( power_power_real @ A_21 @ N_8 ) )
= ( power_power_real @ ( abs_abs_real @ A_21 ) @ N_8 ) ) ).
thf(fact_2689_power__abs,axiom,
! [A_21: int,N_8: nat] :
( ( abs_abs_int @ ( power_power_int @ A_21 @ N_8 ) )
= ( power_power_int @ ( abs_abs_int @ A_21 ) @ N_8 ) ) ).
thf(fact_2690_dvd__if__abs__eq,axiom,
! [L_2: int,K_6: int] :
( ( ( abs_abs_int @ L_2 )
= ( abs_abs_int @ K_6 ) )
=> ( dvd_dvd_int @ L_2 @ K_6 ) ) ).
thf(fact_2691_dvd__if__abs__eq,axiom,
! [L_2: rat,K_6: rat] :
( ( ( abs_abs_rat @ L_2 )
= ( abs_abs_rat @ K_6 ) )
=> ( dvd_dvd_rat @ L_2 @ K_6 ) ) ).
thf(fact_2692_dvd__if__abs__eq,axiom,
! [L_2: real,K_6: real] :
( ( ( abs_abs_real @ L_2 )
= ( abs_abs_real @ K_6 ) )
=> ( dvd_dvd_real @ L_2 @ K_6 ) ) ).
thf(fact_2693_abs__dvd__iff,axiom,
! [M_8: int,K_5: int] :
( ( dvd_dvd_int @ ( abs_abs_int @ M_8 ) @ K_5 )
<=> ( dvd_dvd_int @ M_8 @ K_5 ) ) ).
thf(fact_2694_abs__dvd__iff,axiom,
! [M_8: rat,K_5: rat] :
( ( dvd_dvd_rat @ ( abs_abs_rat @ M_8 ) @ K_5 )
<=> ( dvd_dvd_rat @ M_8 @ K_5 ) ) ).
thf(fact_2695_abs__dvd__iff,axiom,
! [M_8: real,K_5: real] :
( ( dvd_dvd_real @ ( abs_abs_real @ M_8 ) @ K_5 )
<=> ( dvd_dvd_real @ M_8 @ K_5 ) ) ).
thf(fact_2696_dvd__abs__iff,axiom,
! [M_7: int,K_4: int] :
( ( dvd_dvd_int @ M_7 @ ( abs_abs_int @ K_4 ) )
<=> ( dvd_dvd_int @ M_7 @ K_4 ) ) ).
thf(fact_2697_dvd__abs__iff,axiom,
! [M_7: rat,K_4: rat] :
( ( dvd_dvd_rat @ M_7 @ ( abs_abs_rat @ K_4 ) )
<=> ( dvd_dvd_rat @ M_7 @ K_4 ) ) ).
thf(fact_2698_dvd__abs__iff,axiom,
! [M_7: real,K_4: real] :
( ( dvd_dvd_real @ M_7 @ ( abs_abs_real @ K_4 ) )
<=> ( dvd_dvd_real @ M_7 @ K_4 ) ) ).
thf(fact_2699_abs__of__nat,axiom,
! [N_7: nat] :
( ( abs_abs_rat @ ( semiri151668891at_rat @ N_7 ) )
= ( semiri151668891at_rat @ N_7 ) ) ).
thf(fact_2700_abs__of__nat,axiom,
! [N_7: nat] :
( ( abs_abs_real @ ( semiri132038758t_real @ N_7 ) )
= ( semiri132038758t_real @ N_7 ) ) ).
thf(fact_2701_abs__of__nat,axiom,
! [N_7: nat] :
( ( abs_abs_int @ ( semiri1621563631at_int @ N_7 ) )
= ( semiri1621563631at_int @ N_7 ) ) ).
thf(fact_2702_abs__int__eq,axiom,
! [M: nat] :
( ( abs_abs_int @ ( semiri1621563631at_int @ M ) )
= ( semiri1621563631at_int @ M ) ) ).
thf(fact_2703_nat__int,axiom,
! [N: nat] :
( ( nat_1 @ ( semiri1621563631at_int @ N ) )
= N ) ).
thf(fact_2704_zdvd__antisym__abs,axiom,
! [A_1: int,B_1: int] :
( ( dvd_dvd_int @ A_1 @ B_1 )
=> ( ( dvd_dvd_int @ B_1 @ A_1 )
=> ( ( abs_abs_int @ A_1 )
= ( abs_abs_int @ B_1 ) ) ) ) ).
thf(fact_2705_abs__ge__zero,axiom,
! [A_20: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A_20 ) ) ).
thf(fact_2706_abs__ge__zero,axiom,
! [A_20: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A_20 ) ) ).
thf(fact_2707_abs__ge__zero,axiom,
! [A_20: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A_20 ) ) ).
thf(fact_2708_abs__le__zero__iff,axiom,
! [A_19: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A_19 ) @ zero_zero_rat )
<=> ( A_19 = zero_zero_rat ) ) ).
thf(fact_2709_abs__le__zero__iff,axiom,
! [A_19: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A_19 ) @ zero_zero_real )
<=> ( A_19 = zero_zero_real ) ) ).
thf(fact_2710_abs__le__zero__iff,axiom,
! [A_19: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A_19 ) @ zero_zero_int )
<=> ( A_19 = zero_zero_int ) ) ).
thf(fact_2711_abs__of__nonneg,axiom,
! [A_18: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_18 )
=> ( ( abs_abs_rat @ A_18 )
= A_18 ) ) ).
thf(fact_2712_abs__of__nonneg,axiom,
! [A_18: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_18 )
=> ( ( abs_abs_real @ A_18 )
= A_18 ) ) ).
thf(fact_2713_abs__of__nonneg,axiom,
! [A_18: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_18 )
=> ( ( abs_abs_int @ A_18 )
= A_18 ) ) ).
thf(fact_2714_abs__of__pos,axiom,
! [A_17: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_17 )
=> ( ( abs_abs_rat @ A_17 )
= A_17 ) ) ).
thf(fact_2715_abs__of__pos,axiom,
! [A_17: real] :
( ( ord_less_real @ zero_zero_real @ A_17 )
=> ( ( abs_abs_real @ A_17 )
= A_17 ) ) ).
thf(fact_2716_abs__of__pos,axiom,
! [A_17: int] :
( ( ord_less_int @ zero_zero_int @ A_17 )
=> ( ( abs_abs_int @ A_17 )
= A_17 ) ) ).
thf(fact_2717_zero__less__abs__iff,axiom,
! [A_16: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A_16 ) )
<=> ( A_16 != zero_zero_rat ) ) ).
thf(fact_2718_zero__less__abs__iff,axiom,
! [A_16: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A_16 ) )
<=> ( A_16 != zero_zero_real ) ) ).
thf(fact_2719_zero__less__abs__iff,axiom,
! [A_16: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A_16 ) )
<=> ( A_16 != zero_zero_int ) ) ).
thf(fact_2720_abs__not__less__zero,axiom,
! [A_15: rat] :
~ ( ord_less_rat @ ( abs_abs_rat @ A_15 ) @ zero_zero_rat ) ).
thf(fact_2721_abs__not__less__zero,axiom,
! [A_15: real] :
~ ( ord_less_real @ ( abs_abs_real @ A_15 ) @ zero_zero_real ) ).
thf(fact_2722_abs__not__less__zero,axiom,
! [A_15: int] :
~ ( ord_less_int @ ( abs_abs_int @ A_15 ) @ zero_zero_int ) ).
thf(fact_2723_abs__mult__less,axiom,
! [B_14: rat,D_5: rat,A_14: rat,C_4: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ A_14 ) @ C_4 )
=> ( ( ord_less_rat @ ( abs_abs_rat @ B_14 ) @ D_5 )
=> ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A_14 ) @ ( abs_abs_rat @ B_14 ) ) @ ( times_times_rat @ C_4 @ D_5 ) ) ) ) ).
thf(fact_2724_abs__mult__less,axiom,
! [B_14: real,D_5: real,A_14: real,C_4: real] :
( ( ord_less_real @ ( abs_abs_real @ A_14 ) @ C_4 )
=> ( ( ord_less_real @ ( abs_abs_real @ B_14 ) @ D_5 )
=> ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A_14 ) @ ( abs_abs_real @ B_14 ) ) @ ( times_times_real @ C_4 @ D_5 ) ) ) ) ).
thf(fact_2725_abs__mult__less,axiom,
! [B_14: int,D_5: int,A_14: int,C_4: int] :
( ( ord_less_int @ ( abs_abs_int @ A_14 ) @ C_4 )
=> ( ( ord_less_int @ ( abs_abs_int @ B_14 ) @ D_5 )
=> ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A_14 ) @ ( abs_abs_int @ B_14 ) ) @ ( times_times_int @ C_4 @ D_5 ) ) ) ) ).
thf(fact_2726_abs__triangle__ineq,axiom,
! [A_13: rat,B_13: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A_13 @ B_13 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A_13 ) @ ( abs_abs_rat @ B_13 ) ) ) ).
thf(fact_2727_abs__triangle__ineq,axiom,
! [A_13: real,B_13: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A_13 @ B_13 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A_13 ) @ ( abs_abs_real @ B_13 ) ) ) ).
thf(fact_2728_abs__triangle__ineq,axiom,
! [A_13: int,B_13: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A_13 @ B_13 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A_13 ) @ ( abs_abs_int @ B_13 ) ) ) ).
thf(fact_2729_abs__triangle__ineq2__sym,axiom,
! [A_12: rat,B_12: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A_12 ) @ ( abs_abs_rat @ B_12 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B_12 @ A_12 ) ) ) ).
thf(fact_2730_abs__triangle__ineq2__sym,axiom,
! [A_12: real,B_12: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A_12 ) @ ( abs_abs_real @ B_12 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B_12 @ A_12 ) ) ) ).
thf(fact_2731_abs__triangle__ineq2__sym,axiom,
! [A_12: int,B_12: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A_12 ) @ ( abs_abs_int @ B_12 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B_12 @ A_12 ) ) ) ).
thf(fact_2732_abs__triangle__ineq2,axiom,
! [A_11: rat,B_11: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A_11 ) @ ( abs_abs_rat @ B_11 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A_11 @ B_11 ) ) ) ).
thf(fact_2733_abs__triangle__ineq2,axiom,
! [A_11: real,B_11: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A_11 ) @ ( abs_abs_real @ B_11 ) ) @ ( abs_abs_real @ ( minus_minus_real @ A_11 @ B_11 ) ) ) ).
thf(fact_2734_abs__triangle__ineq2,axiom,
! [A_11: int,B_11: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A_11 ) @ ( abs_abs_int @ B_11 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A_11 @ B_11 ) ) ) ).
thf(fact_2735_abs__triangle__ineq3,axiom,
! [A_10: rat,B_10: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A_10 ) @ ( abs_abs_rat @ B_10 ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A_10 @ B_10 ) ) ) ).
thf(fact_2736_abs__triangle__ineq3,axiom,
! [A_10: real,B_10: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A_10 ) @ ( abs_abs_real @ B_10 ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A_10 @ B_10 ) ) ) ).
thf(fact_2737_abs__triangle__ineq3,axiom,
! [A_10: int,B_10: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A_10 ) @ ( abs_abs_int @ B_10 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A_10 @ B_10 ) ) ) ).
thf(fact_2738_nat__0,axiom,
( ( nat_1 @ zero_zero_int )
= zero_zero_nat ) ).
thf(fact_2739_transfer__nat__int__numerals_I1_J,axiom,
( zero_zero_nat
= ( nat_1 @ zero_zero_int ) ) ).
thf(fact_2740_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_2741_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_2742_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_2743_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_2744_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_2745_nat__number__of__def,axiom,
! [V: int] :
( ( number_number_of_nat @ V )
= ( nat_1 @ ( number_number_of_int @ V ) ) ) ).
thf(fact_2746_nat__number__of,axiom,
! [W: int] :
( ( nat_1 @ ( number_number_of_int @ W ) )
= ( number_number_of_nat @ W ) ) ).
thf(fact_2747_transfer__nat__int__numerals_I2_J,axiom,
( one_one_nat
= ( nat_1 @ one_one_int ) ) ).
thf(fact_2748_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_2749_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
<=> ( ( abs_abs_int @ X )
= one_one_int ) ) ).
thf(fact_2750_mult__sum2sq,axiom,
! [A_1: int,B_1: int,P_3: int,Q: int] :
( ( times_times_int @ ( twoSqu2072599593sum2sq @ ( product_Pair_int_int @ A_1 @ B_1 ) ) @ ( twoSqu2072599593sum2sq @ ( product_Pair_int_int @ P_3 @ Q ) ) )
= ( twoSqu2072599593sum2sq @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A_1 @ P_3 ) @ ( times_times_int @ B_1 @ Q ) ) @ ( minus_minus_int @ ( times_times_int @ A_1 @ Q ) @ ( times_times_int @ B_1 @ P_3 ) ) ) ) ) ).
thf(fact_2751_abs__mult__pos,axiom,
! [Y_4: rat,X_5: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X_5 )
=> ( ( times_times_rat @ ( abs_abs_rat @ Y_4 ) @ X_5 )
= ( abs_abs_rat @ ( times_times_rat @ Y_4 @ X_5 ) ) ) ) ).
thf(fact_2752_abs__mult__pos,axiom,
! [Y_4: real,X_5: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_5 )
=> ( ( times_times_real @ ( abs_abs_real @ Y_4 ) @ X_5 )
= ( abs_abs_real @ ( times_times_real @ Y_4 @ X_5 ) ) ) ) ).
thf(fact_2753_abs__mult__pos,axiom,
! [Y_4: int,X_5: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_5 )
=> ( ( times_times_int @ ( abs_abs_int @ Y_4 ) @ X_5 )
= ( abs_abs_int @ ( times_times_int @ Y_4 @ X_5 ) ) ) ) ).
thf(fact_2754_abs__eq__mult,axiom,
! [B_9: rat,A_9: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A_9 )
| ( ord_less_eq_rat @ A_9 @ zero_zero_rat ) )
& ( ( ord_less_eq_rat @ zero_zero_rat @ B_9 )
| ( ord_less_eq_rat @ B_9 @ zero_zero_rat ) ) )
=> ( ( abs_abs_rat @ ( times_times_rat @ A_9 @ B_9 ) )
= ( times_times_rat @ ( abs_abs_rat @ A_9 ) @ ( abs_abs_rat @ B_9 ) ) ) ) ).
thf(fact_2755_abs__eq__mult,axiom,
! [B_9: real,A_9: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A_9 )
| ( ord_less_eq_real @ A_9 @ zero_zero_real ) )
& ( ( ord_less_eq_real @ zero_zero_real @ B_9 )
| ( ord_less_eq_real @ B_9 @ zero_zero_real ) ) )
=> ( ( abs_abs_real @ ( times_times_real @ A_9 @ B_9 ) )
= ( times_times_real @ ( abs_abs_real @ A_9 ) @ ( abs_abs_real @ B_9 ) ) ) ) ).
thf(fact_2756_abs__eq__mult,axiom,
! [B_9: int,A_9: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A_9 )
| ( ord_less_eq_int @ A_9 @ zero_zero_int ) )
& ( ( ord_less_eq_int @ zero_zero_int @ B_9 )
| ( ord_less_eq_int @ B_9 @ zero_zero_int ) ) )
=> ( ( abs_abs_int @ ( times_times_int @ A_9 @ B_9 ) )
= ( times_times_int @ ( abs_abs_int @ A_9 ) @ ( abs_abs_int @ B_9 ) ) ) ) ).
thf(fact_2757_zero__le__power__abs,axiom,
! [A_8: rat,N_6: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A_8 ) @ N_6 ) ) ).
thf(fact_2758_zero__le__power__abs,axiom,
! [A_8: real,N_6: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A_8 ) @ N_6 ) ) ).
thf(fact_2759_zero__le__power__abs,axiom,
! [A_8: int,N_6: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A_8 ) @ N_6 ) ) ).
thf(fact_2760_abs__triangle__ineq4,axiom,
! [A_7: rat,B_8: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A_7 @ B_8 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A_7 ) @ ( abs_abs_rat @ B_8 ) ) ) ).
thf(fact_2761_abs__triangle__ineq4,axiom,
! [A_7: real,B_8: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A_7 @ B_8 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A_7 ) @ ( abs_abs_real @ B_8 ) ) ) ).
thf(fact_2762_abs__triangle__ineq4,axiom,
! [A_7: int,B_8: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A_7 @ B_8 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A_7 ) @ ( abs_abs_int @ B_8 ) ) ) ).
thf(fact_2763_abs__diff__triangle__ineq,axiom,
! [A_6: rat,B_7: rat,C_3: rat,D_4: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A_6 @ B_7 ) @ ( plus_plus_rat @ C_3 @ D_4 ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A_6 @ C_3 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B_7 @ D_4 ) ) ) ) ).
thf(fact_2764_abs__diff__triangle__ineq,axiom,
! [A_6: real,B_7: real,C_3: real,D_4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A_6 @ B_7 ) @ ( plus_plus_real @ C_3 @ D_4 ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A_6 @ C_3 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B_7 @ D_4 ) ) ) ) ).
thf(fact_2765_abs__diff__triangle__ineq,axiom,
! [A_6: int,B_7: int,C_3: int,D_4: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A_6 @ B_7 ) @ ( plus_plus_int @ C_3 @ D_4 ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A_6 @ C_3 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B_7 @ D_4 ) ) ) ) ).
thf(fact_2766_abs__minus__one,axiom,
( ( abs_abs_rat @ ( number_number_of_rat @ min ) )
= one_one_rat ) ).
thf(fact_2767_abs__minus__one,axiom,
( ( abs_abs_real @ ( number267125858f_real @ min ) )
= one_one_real ) ).
thf(fact_2768_abs__minus__one,axiom,
( ( abs_abs_int @ ( number_number_of_int @ min ) )
= one_one_int ) ).
thf(fact_2769_nat__0__iff,axiom,
! [I: int] :
( ( ( nat_1 @ I )
= zero_zero_nat )
<=> ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
thf(fact_2770_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_2771_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_2772_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_2773_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_2774_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_2775_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_2776_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_2777_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_2778_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_2779_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_2780_dvd__imp__le__int,axiom,
! [D_1: int,I: int] :
( ( I != zero_zero_int )
=> ( ( dvd_dvd_int @ D_1 @ I )
=> ( ord_less_eq_int @ ( abs_abs_int @ D_1 ) @ ( abs_abs_int @ I ) ) ) ) ).
thf(fact_2781_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_2782_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_2783_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_2784_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_2785_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_2786_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_2787_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_2788_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_2789_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_2790_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_2791_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_2792_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_2793_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_2794_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_2795_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_2796_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_2797_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_2798_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_2799_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_2800_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_2801_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_2802_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_2803_power2__eq__iff__abs__eq,axiom,
! [A_1: int,B_1: int] :
( ( ( power_power_int @ A_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_int @ B_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( ( abs_abs_int @ A_1 )
= ( abs_abs_int @ B_1 ) ) ) ).
thf(fact_2804_abs__power2__distrib,axiom,
! [A_1: int] :
( ( abs_abs_int @ ( power_power_int @ A_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( power_power_int @ ( abs_abs_int @ A_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_2805_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_2806_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_2807_power2__eq1__iff,axiom,
! [A_1: int] :
( ( ( power_power_int @ A_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_int )
=> ( ( abs_abs_int @ A_1 )
= one_one_int ) ) ).
thf(fact_2808_decr__lemma,axiom,
! [X: int,Z_1: int,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( 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_1 ) ) @ Z_1 ) ) ).
thf(fact_2809_incr__lemma,axiom,
! [Z_1: int,X: int,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( 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_1 ) ) ) ) ).
thf(fact_2810_Euler_Oaux__1,axiom,
! [A_1: int,P_3: int] :
( ( ord_less_int @ zero_zero_int @ P_3 )
=> ( ( power_power_int @ A_1 @ ( nat_1 @ P_3 ) )
= ( times_times_int @ A_1 @ ( power_power_int @ A_1 @ ( minus_minus_nat @ ( nat_1 @ P_3 ) @ one_one_nat ) ) ) ) ) ).
thf(fact_2811_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_2812_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_2813_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_2814_nat0__intermed__int__val,axiom,
! [K: 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 )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I_1: nat] :
( ( ord_less_eq_nat @ I_1 @ N )
& ( ( F @ I_1 )
= K ) ) ) ) ) ).
thf(fact_2815_int__val__lemma,axiom,
! [K: 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 )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I_1: nat] :
( ( ord_less_eq_nat @ I_1 @ N )
& ( ( F @ I_1 )
= K ) ) ) ) ) ).
thf(fact_2816_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_2817_nat__intermed__int__val,axiom,
! [K: 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 )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I_1: nat] :
( ( ord_less_eq_nat @ M @ I_1 )
& ( ord_less_eq_nat @ I_1 @ N )
& ( ( F @ I_1 )
= K ) ) ) ) ) ) ).
thf(fact_2818_nat__aux__def,axiom,
! [I: int,N: nat] :
( ( nat_aux @ I @ N )
= ( plus_plus_nat @ ( nat_1 @ I ) @ N ) ) ).
thf(fact_2819_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_2820_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_2821_zdiv__zero,axiom,
! [B_1: int] :
( ( div_div_int @ zero_zero_int @ B_1 )
= zero_zero_int ) ).
thf(fact_2822_zdiv__self,axiom,
! [A_1: int] :
( ( A_1 != zero_zero_int )
=> ( ( div_div_int @ A_1 @ A_1 )
= one_one_int ) ) ).
thf(fact_2823_pos__imp__zdiv__neg__iff,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_int @ ( div_div_int @ A_1 @ B_1 ) @ zero_zero_int )
<=> ( ord_less_int @ A_1 @ zero_zero_int ) ) ) ).
thf(fact_2824_neg__imp__zdiv__neg__iff,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ B_1 @ zero_zero_int )
=> ( ( ord_less_int @ ( div_div_int @ A_1 @ B_1 ) @ zero_zero_int )
<=> ( ord_less_int @ zero_zero_int @ A_1 ) ) ) ).
thf(fact_2825_div__neg__pos__less0,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_int @ A_1 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ord_less_int @ ( div_div_int @ A_1 @ B_1 ) @ zero_zero_int ) ) ) ).
thf(fact_2826_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_2827_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_2828_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_2829_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_2830_int__div__less__self,axiom,
! [K: int,X: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( div_div_int @ X @ K ) @ X ) ) ) ).
thf(fact_2831_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_2832_zdiv__zmult2__eq,axiom,
! [A_1: int,B_1: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( div_div_int @ A_1 @ ( times_times_int @ B_1 @ C ) )
= ( div_div_int @ ( div_div_int @ A_1 @ B_1 ) @ C ) ) ) ).
thf(fact_2833_zdiv__mono1__neg,axiom,
! [B_1: int,A_1: int,A_5: int] :
( ( ord_less_eq_int @ A_1 @ A_5 )
=> ( ( ord_less_int @ B_1 @ zero_zero_int )
=> ( ord_less_eq_int @ ( div_div_int @ A_5 @ B_1 ) @ ( div_div_int @ A_1 @ B_1 ) ) ) ) ).
thf(fact_2834_zdiv__mono1,axiom,
! [B_1: int,A_1: int,A_5: int] :
( ( ord_less_eq_int @ A_1 @ A_5 )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ord_less_eq_int @ ( div_div_int @ A_1 @ B_1 ) @ ( div_div_int @ A_5 @ B_1 ) ) ) ) ).
thf(fact_2835_div__neg__neg__trivial,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_eq_int @ A_1 @ zero_zero_int )
=> ( ( ord_less_int @ B_1 @ A_1 )
=> ( ( div_div_int @ A_1 @ B_1 )
= zero_zero_int ) ) ) ).
thf(fact_2836_zdiv__mono2__neg,axiom,
! [B_1: int,B_5: int,A_1: int] :
( ( ord_less_int @ A_1 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B_5 )
=> ( ( ord_less_eq_int @ B_5 @ B_1 )
=> ( ord_less_eq_int @ ( div_div_int @ A_1 @ B_5 ) @ ( div_div_int @ A_1 @ B_1 ) ) ) ) ) ).
thf(fact_2837_div__nonpos__pos__le0,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_eq_int @ A_1 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ord_less_eq_int @ ( div_div_int @ A_1 @ B_1 ) @ zero_zero_int ) ) ) ).
thf(fact_2838_neg__imp__zdiv__nonneg__iff,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ B_1 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( div_div_int @ A_1 @ B_1 ) )
<=> ( ord_less_eq_int @ A_1 @ zero_zero_int ) ) ) ).
thf(fact_2839_div__pos__pos__trivial,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_1 )
=> ( ( ord_less_int @ A_1 @ B_1 )
=> ( ( div_div_int @ A_1 @ B_1 )
= zero_zero_int ) ) ) ).
thf(fact_2840_div__nonneg__neg__le0,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_1 )
=> ( ( ord_less_int @ B_1 @ zero_zero_int )
=> ( ord_less_eq_int @ ( div_div_int @ A_1 @ B_1 ) @ zero_zero_int ) ) ) ).
thf(fact_2841_zdiv__mono2,axiom,
! [B_1: int,B_5: int,A_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_1 )
=> ( ( ord_less_int @ zero_zero_int @ B_5 )
=> ( ( ord_less_eq_int @ B_5 @ B_1 )
=> ( ord_less_eq_int @ ( div_div_int @ A_1 @ B_1 ) @ ( div_div_int @ A_1 @ B_5 ) ) ) ) ) ).
thf(fact_2842_nonneg1__imp__zdiv__pos__iff,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_1 )
=> ( ( ord_less_int @ zero_zero_int @ ( div_div_int @ A_1 @ B_1 ) )
<=> ( ( ord_less_eq_int @ B_1 @ A_1 )
& ( ord_less_int @ zero_zero_int @ B_1 ) ) ) ) ).
thf(fact_2843_pos__imp__zdiv__pos__iff,axiom,
! [I: int,K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( div_div_int @ I @ K ) )
<=> ( ord_less_eq_int @ K @ I ) ) ) ).
thf(fact_2844_pos__imp__zdiv__nonneg__iff,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( div_div_int @ A_1 @ B_1 ) )
<=> ( ord_less_eq_int @ zero_zero_int @ A_1 ) ) ) ).
thf(fact_2845_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( div_div_int @ I @ K )
= zero_zero_int )
<=> ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
thf(fact_2846_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_2847_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_2848_div__eq__minus1,axiom,
! [B_1: int] :
( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( div_div_int @ ( number_number_of_int @ min ) @ B_1 )
= ( number_number_of_int @ min ) ) ) ).
thf(fact_2849_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_2850_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_2851_divmod__int__rel__div__eq,axiom,
! [A_4: int,B_3: int,Y: int,R_4: int] :
( ( A_4
= ( plus_plus_int @ ( times_times_int @ B_3 @ Y ) @ R_4 ) )
=> ( ( ( ( ord_less_int @ zero_zero_int @ B_3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_4 )
& ( ord_less_int @ R_4 @ B_3 ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ B_3 )
=> ( ( ord_less_int @ B_3 @ R_4 )
& ( ord_less_eq_int @ R_4 @ zero_zero_int ) ) ) )
=> ( ( B_3 != zero_zero_int )
=> ( ( div_div_int @ A_4 @ B_3 )
= Y ) ) ) ) ).
thf(fact_2852_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( div_div_int @ N @ K ) )
<=> ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I_1: int,J_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J_1 )
& ( ord_less_int @ J_1 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I_1 ) @ J_1 ) ) )
=> ( P @ I_1 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I_1: int,J_1: int] :
( ( ( ord_less_int @ K @ J_1 )
& ( ord_less_eq_int @ J_1 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I_1 ) @ J_1 ) ) )
=> ( P @ I_1 ) ) ) ) ) ).
thf(fact_2853_div__pos__neg__trivial,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_int @ zero_zero_int @ A_1 )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ A_1 @ B_1 ) @ zero_zero_int )
=> ( ( div_div_int @ A_1 @ B_1 )
= ( number_number_of_int @ min ) ) ) ) ).
thf(fact_2854_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_2855_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_2856_pos__zdiv__mult__2,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_1 )
=> ( ( div_div_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B_1 ) ) @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A_1 ) )
= ( div_div_int @ B_1 @ A_1 ) ) ) ).
thf(fact_2857_neg__zdiv__mult__2,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_eq_int @ A_1 @ zero_zero_int )
=> ( ( div_div_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B_1 ) ) @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A_1 ) )
= ( div_div_int @ ( plus_plus_int @ B_1 @ one_one_int ) @ A_1 ) ) ) ).
thf(fact_2858_zfact__prop,axiom,
! [A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A_1 @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A_1 )
=> ( zcong @ ( zfact @ ( minus_minus_int @ P_3 @ one_one_int ) ) @ ( power_power_int @ A_1 @ ( nat_1 @ ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ P_3 ) ) ) ) ) ).
thf(fact_2859_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_2860_Euler__part2,axiom,
! [A_1: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ( zcong @ A_1 @ zero_zero_int @ P_3 )
=> ( zcong @ zero_zero_int @ ( power_power_int @ A_1 @ ( nat_1 @ ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ P_3 ) ) ) ) ).
thf(fact_2861_Euler__Criterion,axiom,
! [A_1: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( zcong @ ( legendre @ A_1 @ P_3 ) @ ( power_power_int @ A_1 @ ( nat_1 @ ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ P_3 ) ) ) ).
thf(fact_2862_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_2863_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_2864_rv,axiom,
( ( v
= ( minus_minus_int @ x @ ( times_times_int @ r @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) )
& ( ord_less_eq_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( abs_abs_int @ v ) ) @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) ).
thf(fact_2865_sw,axiom,
( ( w
= ( minus_minus_int @ y @ ( times_times_int @ sa @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) )
& ( ord_less_eq_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( abs_abs_int @ w ) ) @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) ) ).
thf(fact_2866_inv__inv,axiom,
! [A_1: 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_1 )
=> ( ( ord_less_int @ A_1 @ P_3 )
=> ( ( inv @ P_3 @ ( inv @ P_3 @ A_1 ) )
= A_1 ) ) ) ) ) ).
thf(fact_2867_MultInvPair__distinct,axiom,
! [J: int,A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A_1 @ zero_zero_int @ P_3 )
=> ( ~ ( zcong @ J @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A_1 )
=> ~ ( zcong @ J @ ( times_times_int @ A_1 @ ( multInv @ P_3 @ J ) ) @ P_3 ) ) ) ) ) ) ).
thf(fact_2868_int__le__induct,axiom,
! [P: int > $o,I: int,K: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ I_1 @ K )
=> ( ( P @ I_1 )
=> ( P @ ( minus_minus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_2869_m1,axiom,
( ( plus_plus_int @ ( power_power_int @ v @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ w @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ m1 ) ) ).
thf(fact_2870__0961_A_L_Aint_An_Advd_Av_A_094_A2_A_L_Aw_A_094_A2_096,axiom,
dvd_dvd_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( plus_plus_int @ ( power_power_int @ v @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ w @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_2871__096_Ix_A_094_A2_A_L_Ay_A_094_A2_J_A_K_A_Iv_A_094_A2_A_L_Aw_A_094_A2_J,axiom,
( ( times_times_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 ) ) ) ) ) @ ( plus_plus_int @ ( power_power_int @ v @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ w @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
= ( plus_plus_int @ ( power_power_int @ ( plus_plus_int @ ( times_times_int @ x @ v ) @ ( times_times_int @ y @ w ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ ( minus_minus_int @ ( times_times_int @ x @ w ) @ ( times_times_int @ y @ v ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_2872__096_I_Ir_A_K_A_I1_A_L_Aint_An_J_A_L_Av_J_A_K_Av_A_L_A_Is_A_K_A_I1_A_L,axiom,
( ( plus_plus_int @ ( power_power_int @ ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ r @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) @ v ) @ v ) @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ sa @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) @ w ) @ w ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ ( minus_minus_int @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ r @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) @ v ) @ w ) @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ sa @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) @ w ) @ v ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( plus_plus_int @ ( power_power_int @ ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( times_times_int @ r @ v ) ) @ ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( times_times_int @ sa @ w ) ) ) @ ( plus_plus_int @ ( power_power_int @ v @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ w @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ ( minus_minus_int @ ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( times_times_int @ r @ w ) ) @ ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( times_times_int @ sa @ v ) ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_2873__096_I_I1_A_L_Aint_An_J_A_K_A_Ir_A_K_Av_J_A_L_A_I1_A_L_Aint_An_J_A_K_A,axiom,
( ( plus_plus_int @ ( power_power_int @ ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( times_times_int @ r @ v ) ) @ ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( times_times_int @ sa @ w ) ) ) @ ( plus_plus_int @ ( power_power_int @ v @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ w @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ ( minus_minus_int @ ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( times_times_int @ r @ w ) ) @ ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( times_times_int @ sa @ v ) ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( plus_plus_int @ ( power_power_int @ ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( times_times_int @ r @ v ) ) @ ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( times_times_int @ sa @ w ) ) ) @ ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ m1 ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ ( minus_minus_int @ ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( times_times_int @ r @ w ) ) @ ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( times_times_int @ sa @ v ) ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_2874__096_I4_A_K_Am_A_L_A1_J_A_K_A_I1_A_L_Aint_An_J_A_K_A_Iv_A_094_A2_A_L_A,axiom,
( ( times_times_int @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) @ ( plus_plus_int @ ( power_power_int @ v @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ w @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
= ( times_times_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 ) ) ) ) ) @ ( plus_plus_int @ ( power_power_int @ v @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ w @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_2875__096_Ix_A_K_Av_A_L_Ay_A_K_Aw_J_A_094_A2_A_L_A_Ix_A_K_Aw_A_N_Ay_A_K_Av_,axiom,
( ( plus_plus_int @ ( power_power_int @ ( plus_plus_int @ ( times_times_int @ x @ v ) @ ( times_times_int @ y @ w ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ ( minus_minus_int @ ( times_times_int @ x @ w ) @ ( times_times_int @ y @ v ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( plus_plus_int @ ( power_power_int @ ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ r @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) @ v ) @ v ) @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ sa @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) @ w ) @ w ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ ( minus_minus_int @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ r @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) @ v ) @ w ) @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ sa @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) @ w ) @ v ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_2876__096_I4_A_K_Am_A_L_A1_J_A_K_A_I1_A_L_Aint_An_J_A_K_A_Iv_A_094_A2_A_L_A,axiom,
( ( times_times_int @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) @ ( plus_plus_int @ ( power_power_int @ v @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ w @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
= ( plus_plus_int @ ( power_power_int @ ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( times_times_int @ r @ v ) ) @ ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( times_times_int @ sa @ w ) ) ) @ ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ m1 ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ ( minus_minus_int @ ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( times_times_int @ r @ w ) ) @ ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( times_times_int @ sa @ v ) ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_2877__096_I4_A_K_Am_A_L_A1_J_A_K_A_I1_A_L_Aint_An_J_A_K_A_Iv_A_094_A2_A_L_A,axiom,
( ( times_times_int @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) ) @ ( plus_plus_int @ ( power_power_int @ v @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ w @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ r @ v ) @ ( times_times_int @ sa @ w ) ) @ m1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ ( minus_minus_int @ ( times_times_int @ r @ w ) @ ( times_times_int @ sa @ v ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_2878__096_I1_A_L_Aint_An_J_A_094_A2_A_K_A_I_I4_A_K_Am_A_L_A1_J_A_K_Am1_J_A_,axiom,
( ( times_times_int @ ( power_power_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ m1 ) )
= ( times_times_int @ ( power_power_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( plus_plus_int @ ( power_power_int @ ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ r @ v ) @ ( times_times_int @ sa @ w ) ) @ m1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ ( minus_minus_int @ ( times_times_int @ r @ w ) @ ( times_times_int @ sa @ v ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_2879_calculation,axiom,
( ( times_times_int @ ( power_power_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( minus_minus_int @ ( minus_minus_int @ ( times_times_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ m ) @ one_one_int ) @ m1 ) @ ( power_power_int @ ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ r @ v ) @ ( times_times_int @ sa @ w ) ) @ m1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( power_power_int @ ( minus_minus_int @ ( times_times_int @ r @ w ) @ ( times_times_int @ sa @ v ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
= zero_zero_int ) ).
thf(fact_2880__096_B_Bthesis_O_A_I_B_Bm1_O_Av_A_094_A2_A_L_Aw_A_094_A2_A_061_A_I1_A_,axiom,
~ ! [M1: int] :
( ( plus_plus_int @ ( power_power_int @ v @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ w @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
!= ( times_times_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ M1 ) ) ).
thf(fact_2881_div__le__mono,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( div_div_nat @ M @ K ) @ ( div_div_nat @ N @ K ) ) ) ).
thf(fact_2882_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( div_div_nat @ M @ N ) @ M ) ).
thf(fact_2883_div__mult2__eq,axiom,
! [A_1: nat,B_1: nat,C: nat] :
( ( div_div_nat @ A_1 @ ( times_times_nat @ B_1 @ C ) )
= ( div_div_nat @ ( div_div_nat @ A_1 @ B_1 ) @ C ) ) ).
thf(fact_2884_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( div_div_nat @ M @ N )
= zero_zero_nat ) ) ).
thf(fact_2885_nat__mult__div__cancel__disj,axiom,
! [M: nat,N: nat,K: nat] :
( ( ( K = zero_zero_nat )
=> ( ( div_div_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( div_div_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( div_div_nat @ M @ N ) ) ) ) ).
thf(fact_2886_zdiv__int,axiom,
! [A_1: nat,B_1: nat] :
( ( semiri1621563631at_int @ ( div_div_nat @ A_1 @ B_1 ) )
= ( div_div_int @ ( semiri1621563631at_int @ A_1 ) @ ( semiri1621563631at_int @ B_1 ) ) ) ).
thf(fact_2887_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_2888_aux______3,axiom,
! [J: int,K: int,A_1: int,P_3: int] :
( ( zcong @ ( times_times_int @ J @ K ) @ A_1 @ P_3 )
=> ( zcong @ ( times_times_int @ ( times_times_int @ ( multInv @ P_3 @ J ) @ J ) @ K ) @ ( times_times_int @ ( multInv @ P_3 @ J ) @ A_1 ) @ P_3 ) ) ).
thf(fact_2889_aux______1,axiom,
! [J: int,A_1: int,P_3: int,K: int] :
( ( zcong @ J @ ( times_times_int @ A_1 @ ( multInv @ P_3 @ K ) ) @ P_3 )
=> ( zcong @ ( times_times_int @ J @ K ) @ ( times_times_int @ ( times_times_int @ A_1 @ ( multInv @ P_3 @ K ) ) @ K ) @ P_3 ) ) ).
thf(fact_2890_div__le__mono2,axiom,
! [K: 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 @ N ) @ ( div_div_nat @ K @ M ) ) ) ) ).
thf(fact_2891_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_2892_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_2893_nat__mult__div__cancel1,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( div_div_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( div_div_nat @ M @ N ) ) ) ).
thf(fact_2894_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_2895_split__div,axiom,
! [P: nat > $o,N: nat,K: nat] :
( ( P @ ( div_div_nat @ N @ K ) )
<=> ( ( ( K = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ! [I_1: nat,J_1: nat] :
( ( ord_less_nat @ J_1 @ K )
=> ( ( N
= ( plus_plus_nat @ ( times_times_nat @ K @ I_1 ) @ J_1 ) )
=> ( P @ I_1 ) ) ) ) ) ) ).
thf(fact_2896_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_2897_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_2898_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_2899_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_2900_inv__less__p__minus__1,axiom,
! [A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ one_one_int @ A_1 )
=> ( ( ord_less_int @ A_1 @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ord_less_int @ ( inv @ P_3 @ A_1 ) @ ( minus_minus_int @ P_3 @ one_one_int ) ) ) ) ) ).
thf(fact_2901_inv__g__1,axiom,
! [A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ one_one_int @ A_1 )
=> ( ( ord_less_int @ A_1 @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ord_less_int @ one_one_int @ ( inv @ P_3 @ A_1 ) ) ) ) ) ).
thf(fact_2902_inv__not__p__minus__1,axiom,
! [A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ one_one_int @ A_1 )
=> ( ( ord_less_int @ A_1 @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( inv @ P_3 @ A_1 )
!= ( minus_minus_int @ P_3 @ one_one_int ) ) ) ) ) ).
thf(fact_2903_inv__not__1,axiom,
! [A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ one_one_int @ A_1 )
=> ( ( ord_less_int @ A_1 @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( inv @ P_3 @ A_1 )
!= one_one_int ) ) ) ) ).
thf(fact_2904_inv__distinct,axiom,
! [A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ one_one_int @ A_1 )
=> ( ( ord_less_int @ A_1 @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( A_1
!= ( inv @ P_3 @ A_1 ) ) ) ) ) ).
thf(fact_2905_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_2906_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_2907_inv__not__0,axiom,
! [A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ one_one_int @ A_1 )
=> ( ( ord_less_int @ A_1 @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( inv @ P_3 @ A_1 )
!= zero_zero_int ) ) ) ) ).
thf(fact_2908_MultInv__zcong__prop1,axiom,
! [A_1: int,J: int,K: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zcong @ J @ K @ P_3 )
=> ( zcong @ ( times_times_int @ A_1 @ ( multInv @ P_3 @ J ) ) @ ( times_times_int @ A_1 @ ( multInv @ P_3 @ K ) ) @ P_3 ) ) ) ).
thf(fact_2909_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_2910_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_2911_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_2912_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_2913_inv__is__inv,axiom,
! [A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ zero_zero_int @ A_1 )
=> ( ( ord_less_int @ A_1 @ P_3 )
=> ( zcong @ ( times_times_int @ A_1 @ ( inv @ P_3 @ A_1 ) ) @ one_one_int @ P_3 ) ) ) ) ).
thf(fact_2914_aux______4,axiom,
! [K: int,A_1: 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 ) @ ( times_times_int @ ( multInv @ P_3 @ J ) @ A_1 ) @ P_3 )
=> ( zcong @ K @ ( times_times_int @ A_1 @ ( multInv @ P_3 @ J ) ) @ P_3 ) ) ) ) ) ).
thf(fact_2915_aux______2,axiom,
! [J: int,A_1: int,K: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ K @ zero_zero_int @ P_3 )
=> ( ( zcong @ ( times_times_int @ J @ K ) @ ( times_times_int @ ( times_times_int @ A_1 @ ( multInv @ P_3 @ K ) ) @ K ) @ P_3 )
=> ( zcong @ ( times_times_int @ J @ K ) @ A_1 @ P_3 ) ) ) ) ) ).
thf(fact_2916_MultInv__zcong__prop2,axiom,
! [A_1: int,J: int,K: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ K @ zero_zero_int @ P_3 )
=> ( ~ ( zcong @ J @ zero_zero_int @ P_3 )
=> ( ( zcong @ J @ ( times_times_int @ A_1 @ ( multInv @ P_3 @ K ) ) @ P_3 )
=> ( zcong @ K @ ( times_times_int @ A_1 @ ( multInv @ P_3 @ J ) ) @ P_3 ) ) ) ) ) ) ).
thf(fact_2917_MultInv__zcong__prop3,axiom,
! [J: int,K: int,A_1: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ A_1 @ zero_zero_int @ P_3 )
=> ( ~ ( zcong @ K @ zero_zero_int @ P_3 )
=> ( ~ ( zcong @ J @ zero_zero_int @ P_3 )
=> ( ( zcong @ ( times_times_int @ A_1 @ ( multInv @ P_3 @ J ) ) @ ( times_times_int @ A_1 @ ( multInv @ P_3 @ K ) ) @ P_3 )
=> ( zcong @ J @ K @ P_3 ) ) ) ) ) ) ) ).
thf(fact_2918_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_2919_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_2920_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_2921_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_2922_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_2923_wset__mem__inv__mem,axiom,
! [B_1: int,A_1: 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_1 @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( member_int @ B_1 @ ( wset @ A_1 @ P_3 ) )
=> ( member_int @ ( inv @ P_3 @ B_1 ) @ ( wset @ A_1 @ P_3 ) ) ) ) ) ) ).
thf(fact_2924_wset__inv__mem__mem,axiom,
! [B_1: int,A_1: 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_1 @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( ord_less_int @ one_one_int @ B_1 )
=> ( ( ord_less_int @ B_1 @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( member_int @ ( inv @ P_3 @ B_1 ) @ ( wset @ A_1 @ P_3 ) )
=> ( member_int @ B_1 @ ( wset @ A_1 @ P_3 ) ) ) ) ) ) ) ) ).
thf(fact_2925_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_2926_int__less__induct,axiom,
! [P: int > $o,I: int,K: int] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I_1: int] :
( ( ord_less_int @ I_1 @ K )
=> ( ( P @ I_1 )
=> ( P @ ( minus_minus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_2927_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_2928_wset__mem__mem,axiom,
! [P_3: int,A_1: int] :
( ( ord_less_int @ one_one_int @ A_1 )
=> ( member_int @ A_1 @ ( wset @ A_1 @ P_3 ) ) ) ).
thf(fact_2929_wset__subset,axiom,
! [B_1: int,P_3: int,A_1: int] :
( ( ord_less_int @ one_one_int @ A_1 )
=> ( ( member_int @ B_1 @ ( wset @ ( minus_minus_int @ A_1 @ one_one_int ) @ P_3 ) )
=> ( member_int @ B_1 @ ( wset @ A_1 @ P_3 ) ) ) ) ).
thf(fact_2930_wset__g__1,axiom,
! [B_1: int,A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ A_1 @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( member_int @ B_1 @ ( wset @ A_1 @ P_3 ) )
=> ( ord_less_int @ one_one_int @ B_1 ) ) ) ) ).
thf(fact_2931_wset__less,axiom,
! [B_1: int,A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ A_1 @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( member_int @ B_1 @ ( wset @ A_1 @ P_3 ) )
=> ( ord_less_int @ B_1 @ ( minus_minus_int @ P_3 @ one_one_int ) ) ) ) ) ).
thf(fact_2932_wset__mem__imp__or,axiom,
! [B_1: int,P_3: int,A_1: int] :
( ( ord_less_int @ one_one_int @ A_1 )
=> ( ~ ( member_int @ B_1 @ ( wset @ ( minus_minus_int @ A_1 @ one_one_int ) @ P_3 ) )
=> ( ( member_int @ B_1 @ ( wset @ A_1 @ P_3 ) )
=> ( ( B_1 = A_1 )
| ( B_1
= ( inv @ P_3 @ A_1 ) ) ) ) ) ) ).
thf(fact_2933_le__Suc__ex__iff,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
<=> ? [N_1: nat] :
( L
= ( plus_plus_nat @ K @ N_1 ) ) ) ).
thf(fact_2934_wset__mem,axiom,
! [B_1: int,A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ A_1 @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( ord_less_int @ one_one_int @ B_1 )
=> ( ( ord_less_eq_int @ B_1 @ A_1 )
=> ( member_int @ B_1 @ ( wset @ A_1 @ P_3 ) ) ) ) ) ) ).
thf(fact_2935_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_2936_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_2937_phi__prime,axiom,
! [P_3: int] :
( ( zprime @ P_3 )
=> ( ( phi @ P_3 )
= ( nat_1 @ ( minus_minus_int @ P_3 @ one_one_int ) ) ) ) ).
thf(fact_2938_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_2939_int__nat__abs,axiom,
! [X: int] :
( ( semiri1621563631at_int @ ( nat_1 @ ( abs_abs_int @ X ) ) )
= ( abs_abs_int @ X ) ) ).
thf(fact_2940_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_2941_odd__mult__odd__prop,axiom,
! [X: int,Y: int] :
( ( member_int @ ( times_times_int @ X @ Y ) @ zOdd )
=> ( member_int @ X @ zOdd ) ) ).
thf(fact_2942_d22set__le__swap,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ A_1 @ B_1 )
=> ~ ( member_int @ B_1 @ ( d22set @ A_1 ) ) ) ).
thf(fact_2943_d22set__le,axiom,
! [B_1: int,A_1: int] :
( ( member_int @ B_1 @ ( d22set @ A_1 ) )
=> ( ord_less_eq_int @ B_1 @ A_1 ) ) ).
thf(fact_2944_d22set__g__1,axiom,
! [B_1: int,A_1: int] :
( ( member_int @ B_1 @ ( d22set @ A_1 ) )
=> ( ord_less_int @ one_one_int @ B_1 ) ) ).
thf(fact_2945_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_2946_d22set__mem,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ one_one_int @ B_1 )
=> ( ( ord_less_eq_int @ B_1 @ A_1 )
=> ( member_int @ B_1 @ ( d22set @ A_1 ) ) ) ) ).
thf(fact_2947_zdvd__self__abs1,axiom,
! [D_1: int] : ( dvd_dvd_int @ D_1 @ ( abs_abs_int @ D_1 ) ) ).
thf(fact_2948_zdvd__self__abs2,axiom,
! [D_1: int] : ( dvd_dvd_int @ ( abs_abs_int @ D_1 ) @ D_1 ) ).
thf(fact_2949_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_2950_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_2951_zOddI,axiom,
! [X: int,K: int] :
( ( X
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ K ) @ one_one_int ) )
=> ( member_int @ X @ zOdd ) ) ).
thf(fact_2952_divides__le,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( ord_less_eq_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
thf(fact_2953_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_2954_zOddE,axiom,
! [X: int] :
( ( member_int @ X @ zOdd )
=> ~ ! [K_1: int] :
( X
!= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_1 ) @ one_one_int ) ) ) ).
thf(fact_2955_pow__divides__eq__int,axiom,
! [A_1: int,B_1: int,N: nat] :
( ( N != zero_zero_nat )
=> ( ( dvd_dvd_int @ ( power_power_int @ A_1 @ N ) @ ( power_power_int @ B_1 @ N ) )
<=> ( dvd_dvd_int @ A_1 @ B_1 ) ) ) ).
thf(fact_2956_pow__divides__pow__int,axiom,
! [A_1: int,N: nat,B_1: int] :
( ( dvd_dvd_int @ ( power_power_int @ A_1 @ N ) @ ( power_power_int @ B_1 @ N ) )
=> ( ( N != zero_zero_nat )
=> ( dvd_dvd_int @ A_1 @ B_1 ) ) ) ).
thf(fact_2957_pow__divides__eq__nat,axiom,
! [A_1: nat,B_1: nat,N: nat] :
( ( N != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A_1 @ N ) @ ( power_power_nat @ B_1 @ N ) )
<=> ( dvd_dvd_nat @ A_1 @ B_1 ) ) ) ).
thf(fact_2958_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_2959_gcd__lcm__complete__lattice__nat_Otop__le,axiom,
! [A_1: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A_1 )
=> ( A_1 = zero_zero_nat ) ) ).
thf(fact_2960_gcd__lcm__complete__lattice__nat_Otop__unique,axiom,
! [A_1: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A_1 )
<=> ( A_1 = zero_zero_nat ) ) ).
thf(fact_2961_gcd__lcm__complete__lattice__nat_Oless__top,axiom,
! [A_1: nat] :
( ( A_1 != zero_zero_nat )
<=> ( ( dvd_dvd_nat @ A_1 @ zero_zero_nat )
& ~ ( dvd_dvd_nat @ zero_zero_nat @ A_1 ) ) ) ).
thf(fact_2962_gcd__lcm__complete__lattice__nat_Otop__greatest,axiom,
! [A_1: nat] : ( dvd_dvd_nat @ A_1 @ zero_zero_nat ) ).
thf(fact_2963_gcd__lcm__complete__lattice__nat_Onot__top__less,axiom,
! [A_1: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A_1 )
& ~ ( dvd_dvd_nat @ A_1 @ zero_zero_nat ) ) ).
thf(fact_2964_gcd__lcm__complete__lattice__nat_Onot__less__bot,axiom,
! [A_1: nat] :
~ ( ( dvd_dvd_nat @ A_1 @ one_one_nat )
& ~ ( dvd_dvd_nat @ one_one_nat @ A_1 ) ) ).
thf(fact_2965_gcd__lcm__complete__lattice__nat_Obot__least,axiom,
! [A_1: nat] : ( dvd_dvd_nat @ one_one_nat @ A_1 ) ).
thf(fact_2966_gcd__lcm__complete__lattice__nat_Obot__less,axiom,
! [A_1: nat] :
( ( A_1 != one_one_nat )
<=> ( ( dvd_dvd_nat @ one_one_nat @ A_1 )
& ~ ( dvd_dvd_nat @ A_1 @ one_one_nat ) ) ) ).
thf(fact_2967_gcd__lcm__complete__lattice__nat_Obot__unique,axiom,
! [A_1: nat] :
( ( dvd_dvd_nat @ A_1 @ one_one_nat )
<=> ( A_1 = one_one_nat ) ) ).
thf(fact_2968_gcd__lcm__complete__lattice__nat_Ole__bot,axiom,
! [A_1: nat] :
( ( dvd_dvd_nat @ A_1 @ one_one_nat )
=> ( A_1 = one_one_nat ) ) ).
thf(fact_2969_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_2970_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_2971_neg__zmod__mult__2,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_eq_int @ A_1 @ zero_zero_int )
=> ( ( div_mod_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B_1 ) ) @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A_1 ) )
= ( minus_minus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( div_mod_int @ ( plus_plus_int @ B_1 @ one_one_int ) @ A_1 ) ) @ one_one_int ) ) ) ).
thf(fact_2972_transfer__morphism__nat__int,axiom,
nat_tr876908586nt_nat @ nat_1 @ ( ord_less_eq_int @ zero_zero_int ) ).
thf(fact_2973_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_2974_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_2975_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_2976_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
thf(fact_2977_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
thf(fact_2978_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
thf(fact_2979_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
thf(fact_2980_StandardRes__SR__prop,axiom,
! [X: int,P_3: int] :
( ( member_int @ X @ ( sr @ P_3 ) )
=> ( ( standardRes @ P_3 @ X )
= X ) ) ).
thf(fact_2981_StandardRes__def,axiom,
! [M: int,X: int] :
( ( standardRes @ M @ X )
= ( div_mod_int @ X @ M ) ) ).
thf(fact_2982_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
thf(fact_2983_nat_Oinject,axiom,
! [Nat: nat,Nat_3: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat_3 ) )
<=> ( Nat = Nat_3 ) ) ).
thf(fact_2984_Suc__n__not__n,axiom,
! [N: nat] :
( ( suc @ N )
!= N ) ).
thf(fact_2985_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
thf(fact_2986_zmod__self,axiom,
! [A_1: int] :
( ( div_mod_int @ A_1 @ A_1 )
= zero_zero_int ) ).
thf(fact_2987_zmod__zero,axiom,
! [B_1: int] :
( ( div_mod_int @ zero_zero_int @ B_1 )
= zero_zero_int ) ).
thf(fact_2988_zmod__zmult1__eq,axiom,
! [A_1: int,B_1: int,C: int] :
( ( div_mod_int @ ( times_times_int @ A_1 @ B_1 ) @ C )
= ( div_mod_int @ ( times_times_int @ A_1 @ ( div_mod_int @ B_1 @ C ) ) @ C ) ) ).
thf(fact_2989_zmod__simps_I3_J,axiom,
! [A_1: int,B_1: int,C: int] :
( ( div_mod_int @ ( times_times_int @ A_1 @ ( div_mod_int @ B_1 @ C ) ) @ C )
= ( div_mod_int @ ( times_times_int @ A_1 @ B_1 ) @ C ) ) ).
thf(fact_2990_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_2991_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_2992_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_2993_nat_Osimps_I2_J,axiom,
! [Nat_3: nat] :
( zero_zero_nat
!= ( suc @ Nat_3 ) ) ).
thf(fact_2994_Suc__not__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_2995_nat_Osimps_I3_J,axiom,
! [Nat_2: nat] :
( ( suc @ Nat_2 )
!= zero_zero_nat ) ).
thf(fact_2996_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_2997_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_2998_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_2999_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_3000_mod__mod__is__mod,axiom,
! [X: int,M: int] : ( zcong @ X @ ( div_mod_int @ X @ M ) @ M ) ).
thf(fact_3001_zcong__zmod,axiom,
! [A_1: int,B_1: int,M: int] :
( ( zcong @ A_1 @ B_1 @ M )
<=> ( zcong @ ( div_mod_int @ A_1 @ M ) @ ( div_mod_int @ B_1 @ M ) @ M ) ) ).
thf(fact_3002_zdvd__zmod__imp__zdvd,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ ( div_mod_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ M ) ) ) ).
thf(fact_3003_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_3004_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_3005_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_3006_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
thf(fact_3007_less__trans__Suc,axiom,
! [K: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
thf(fact_3008_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
thf(fact_3009_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_3010_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
thf(fact_3011_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_3012_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_3013_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
<=> ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
thf(fact_3014_not__less__eq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
<=> ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
thf(fact_3015_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_3016_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
thf(fact_3017_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
thf(fact_3018_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
thf(fact_3019_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_eq_nat @ M @ N )
<=> ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
thf(fact_3020_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_3021_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
<=> ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_3022_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_3023_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_3024_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_3025_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
<=> ( M = N ) ) ).
thf(fact_3026_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_3027_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
thf(fact_3028_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_3029_StandardRes__prop1,axiom,
! [X: int,M: int] : ( zcong @ X @ ( standardRes @ M @ X ) @ M ) ).
thf(fact_3030_neg__mod__bound,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ B_1 @ zero_zero_int )
=> ( ord_less_int @ B_1 @ ( div_mod_int @ A_1 @ B_1 ) ) ) ).
thf(fact_3031_pos__mod__bound,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ord_less_int @ ( div_mod_int @ A_1 @ B_1 ) @ B_1 ) ) ).
thf(fact_3032_zmod__eq__0__iff,axiom,
! [M: int,D_1: int] :
( ( ( div_mod_int @ M @ D_1 )
= zero_zero_int )
<=> ? [Q_2: int] :
( M
= ( times_times_int @ D_1 @ Q_2 ) ) ) ).
thf(fact_3033_zmod__le__nonneg__dividend,axiom,
! [K: int,M: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ord_less_eq_int @ ( div_mod_int @ M @ K ) @ M ) ) ).
thf(fact_3034_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_3035_zmod__zdiv__trivial,axiom,
! [A_1: int,B_1: int] :
( ( div_div_int @ ( div_mod_int @ A_1 @ B_1 ) @ B_1 )
= zero_zero_int ) ).
thf(fact_3036_DIVISION__BY__ZERO,axiom,
! [A_1: int] :
( ( ( div_div_int @ A_1 @ zero_zero_int )
= zero_zero_int )
& ( ( div_mod_int @ A_1 @ zero_zero_int )
= A_1 ) ) ).
thf(fact_3037_zdiv__zadd1__eq,axiom,
! [A_1: int,B_1: int,C: int] :
( ( div_div_int @ ( plus_plus_int @ A_1 @ B_1 ) @ C )
= ( plus_plus_int @ ( plus_plus_int @ ( div_div_int @ A_1 @ C ) @ ( div_div_int @ B_1 @ C ) ) @ ( div_div_int @ ( plus_plus_int @ ( div_mod_int @ A_1 @ C ) @ ( div_mod_int @ B_1 @ C ) ) @ C ) ) ) ).
thf(fact_3038_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_3039_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_3040_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_3041_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
<=> ? [M_2: nat] :
( N
= ( suc @ M_2 ) ) ) ).
thf(fact_3042_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
<=> ( M
= ( suc @ zero_zero_nat ) ) ) ).
thf(fact_3043_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_3044_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_3045_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_3046_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_3047_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
thf(fact_3048_less__iff__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
<=> ? [K_1: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K_1 ) ) ) ) ).
thf(fact_3049_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
thf(fact_3050_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
thf(fact_3051_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_3052_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_3053_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
thf(fact_3054_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_3055_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_3056_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_3057_less__eq__Suc__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
<=> ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
thf(fact_3058_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_3059_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
thf(fact_3060_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
thf(fact_3061_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_3062_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_3063_Suc__eq__plus1,axiom,
! [N: nat] :
( ( suc @ N )
= ( plus_plus_nat @ N @ one_one_nat ) ) ).
thf(fact_3064_Suc__eq__plus1__left,axiom,
! [N: nat] :
( ( suc @ N )
= ( plus_plus_nat @ one_one_nat @ N ) ) ).
thf(fact_3065_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_3066_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
thf(fact_3067_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_3068_div__1,axiom,
! [M: nat] :
( ( div_div_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
thf(fact_3069_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
thf(fact_3070_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_3071_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_3072_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_3073_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_3074_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_3075_StandardRes__prop3,axiom,
! [X: int,P_3: int] :
( ~ ( zcong @ X @ zero_zero_int @ P_3 )
<=> ( ( standardRes @ P_3 @ X )
!= zero_zero_int ) ) ).
thf(fact_3076_StandardRes__eq__zcong,axiom,
! [M: int,X: int] :
( ( ( standardRes @ M @ X )
= zero_zero_int )
<=> ( zcong @ X @ zero_zero_int @ M ) ) ).
thf(fact_3077_pos__mod__sign,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ord_less_eq_int @ zero_zero_int @ ( div_mod_int @ A_1 @ B_1 ) ) ) ).
thf(fact_3078_pos__mod__conj,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( div_mod_int @ A_1 @ B_1 ) )
& ( ord_less_int @ ( div_mod_int @ A_1 @ B_1 ) @ B_1 ) ) ) ).
thf(fact_3079_mod__pos__pos__trivial,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_1 )
=> ( ( ord_less_int @ A_1 @ B_1 )
=> ( ( div_mod_int @ A_1 @ B_1 )
= A_1 ) ) ) ).
thf(fact_3080_neg__mod__sign,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ B_1 @ zero_zero_int )
=> ( ord_less_eq_int @ ( div_mod_int @ A_1 @ B_1 ) @ zero_zero_int ) ) ).
thf(fact_3081_neg__mod__conj,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ B_1 @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( div_mod_int @ A_1 @ B_1 ) @ zero_zero_int )
& ( ord_less_int @ B_1 @ ( div_mod_int @ A_1 @ B_1 ) ) ) ) ).
thf(fact_3082_mod__neg__neg__trivial,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_eq_int @ A_1 @ zero_zero_int )
=> ( ( ord_less_int @ B_1 @ A_1 )
=> ( ( div_mod_int @ A_1 @ B_1 )
= A_1 ) ) ) ).
thf(fact_3083_zcong__zmod__eq,axiom,
! [A_1: int,B_1: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( zcong @ A_1 @ B_1 @ M )
<=> ( ( div_mod_int @ A_1 @ M )
= ( div_mod_int @ B_1 @ M ) ) ) ) ).
thf(fact_3084_zmod__zdvd__zmod,axiom,
! [A_1: int,B_1: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( dvd_dvd_int @ M @ B_1 )
=> ( ( div_mod_int @ ( div_mod_int @ A_1 @ B_1 ) @ M )
= ( div_mod_int @ A_1 @ M ) ) ) ) ).
thf(fact_3085_zmod__minus1__right,axiom,
! [A_1: int] :
( ( div_mod_int @ A_1 @ ( number_number_of_int @ min ) )
= zero_zero_int ) ).
thf(fact_3086_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_3087_zmod__zdiv__equality,axiom,
! [A_1: int,B_1: int] :
( A_1
= ( plus_plus_int @ ( times_times_int @ B_1 @ ( div_div_int @ A_1 @ B_1 ) ) @ ( div_mod_int @ A_1 @ B_1 ) ) ) ).
thf(fact_3088_zdiv__zmult1__eq,axiom,
! [A_1: int,B_1: int,C: int] :
( ( div_div_int @ ( times_times_int @ A_1 @ B_1 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A_1 @ ( div_div_int @ B_1 @ C ) ) @ ( div_div_int @ ( times_times_int @ A_1 @ ( div_mod_int @ B_1 @ C ) ) @ C ) ) ) ).
thf(fact_3089_zdiv__zmod__equality,axiom,
! [B_1: int,A_1: int,K: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ ( div_div_int @ A_1 @ B_1 ) ) @ ( div_mod_int @ A_1 @ B_1 ) ) @ K )
= ( plus_plus_int @ A_1 @ K ) ) ).
thf(fact_3090_zdiv__zmod__equality2,axiom,
! [A_1: int,B_1: int,K: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( div_div_int @ A_1 @ B_1 ) @ B_1 ) @ ( div_mod_int @ A_1 @ B_1 ) ) @ K )
= ( plus_plus_int @ A_1 @ K ) ) ).
thf(fact_3091_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_3092_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_3093_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_3094_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_3095_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_3096_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_3097_prod__mn__less__k,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ( ( times_times_nat @ M @ N )
= K )
=> ( ord_less_nat @ N @ K ) ) ) ) ) ).
thf(fact_3098_one__less__k,axiom,
! [M: nat,K: nat] :
( ( M
!= ( times_times_nat @ M @ K ) )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ K ) )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ K ) ) ) ).
thf(fact_3099_one__less__m,axiom,
! [M: nat,K: nat] :
( ( M
!= ( times_times_nat @ M @ K ) )
=> ( ( M
!= ( suc @ zero_zero_nat ) )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M ) ) ) ).
thf(fact_3100_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_3101_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
thf(fact_3102_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_3103_int__Suc0__eq__1,axiom,
( ( semiri1621563631at_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
thf(fact_3104_nat__1,axiom,
( ( nat_1 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
thf(fact_3105_int__Suc,axiom,
! [M: nat] :
( ( semiri1621563631at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ M ) ) ) ).
thf(fact_3106_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_3107_diff__Suc__diff__eq2,axiom,
! [M: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ M )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ M ) ) ) ) ).
thf(fact_3108_diff__Suc__diff__eq1,axiom,
! [M: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ M @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( suc @ J ) ) ) ) ).
thf(fact_3109_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_3110_mod__pos__neg__trivial,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_int @ zero_zero_int @ A_1 )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ A_1 @ B_1 ) @ zero_zero_int )
=> ( ( div_mod_int @ A_1 @ B_1 )
= ( plus_plus_int @ A_1 @ B_1 ) ) ) ) ).
thf(fact_3111_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_3112_zcong__zmod__aux,axiom,
! [A_1: int,B_1: int,M: int] :
( ( minus_minus_int @ A_1 @ B_1 )
= ( plus_plus_int @ ( times_times_int @ M @ ( minus_minus_int @ ( div_div_int @ A_1 @ M ) @ ( div_div_int @ B_1 @ M ) ) ) @ ( minus_minus_int @ ( div_mod_int @ A_1 @ M ) @ ( div_mod_int @ B_1 @ M ) ) ) ) ).
thf(fact_3113_numeral__3__eq__3,axiom,
( ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) )
= ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
thf(fact_3114_numeral__1__eq__Suc__0,axiom,
( ( number_number_of_nat @ ( bit1 @ pls ) )
= ( suc @ zero_zero_nat ) ) ).
thf(fact_3115_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) @ N ) ) ).
thf(fact_3116_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
thf(fact_3117_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
thf(fact_3118_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_3119_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_3120_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_3121_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_3122_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_3123_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_3124_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_3125_split__zmod,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( div_mod_int @ N @ K ) )
<=> ( ( ( K = zero_zero_int )
=> ( P @ N ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I_1: int,J_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J_1 )
& ( ord_less_int @ J_1 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I_1 ) @ J_1 ) ) )
=> ( P @ J_1 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I_1: int,J_1: int] :
( ( ( ord_less_int @ K @ J_1 )
& ( ord_less_eq_int @ J_1 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I_1 ) @ J_1 ) ) )
=> ( P @ J_1 ) ) ) ) ) ).
thf(fact_3126_zmult2__lemma__aux3,axiom,
! [Q: int,B_1: 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_1 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ ( div_mod_int @ Q @ C ) ) @ R_1 ) ) ) ) ) ).
thf(fact_3127_zmult2__lemma__aux4,axiom,
! [Q: int,B_1: 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_1 )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ ( div_mod_int @ Q @ C ) ) @ R_1 ) @ ( times_times_int @ B_1 @ C ) ) ) ) ) ).
thf(fact_3128_zmult2__lemma__aux1,axiom,
! [Q: int,B_1: int,R_1: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ B_1 @ R_1 )
=> ( ( ord_less_eq_int @ R_1 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B_1 @ C ) @ ( plus_plus_int @ ( times_times_int @ B_1 @ ( div_mod_int @ Q @ C ) ) @ R_1 ) ) ) ) ) ).
thf(fact_3129_zmult2__lemma__aux2,axiom,
! [Q: int,B_1: int,R_1: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ B_1 @ R_1 )
=> ( ( ord_less_eq_int @ R_1 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B_1 @ ( div_mod_int @ Q @ C ) ) @ R_1 ) @ zero_zero_int ) ) ) ) ).
thf(fact_3130_divmod__int__rel__mod__eq,axiom,
! [A_4: int,B_3: int,Q_4: int,Y: int] :
( ( A_4
= ( plus_plus_int @ ( times_times_int @ B_3 @ Q_4 ) @ 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_3131_zmod__zmult2__eq,axiom,
! [A_1: int,B_1: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( div_mod_int @ A_1 @ ( times_times_int @ B_1 @ C ) )
= ( plus_plus_int @ ( times_times_int @ B_1 @ ( div_mod_int @ ( div_div_int @ A_1 @ B_1 ) @ C ) ) @ ( div_mod_int @ A_1 @ B_1 ) ) ) ) ).
thf(fact_3132_tsub__eq,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( nat_tsub @ X @ Y )
= ( minus_minus_int @ X @ Y ) ) ) ).
thf(fact_3133_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_3134_numeral__2__eq__2,axiom,
( ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_3135_semiring__norm_I115_J,axiom,
( ( suc @ ( suc @ zero_zero_nat ) )
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3136_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
thf(fact_3137_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( suc @ ( suc @ N ) ) ) ).
thf(fact_3138_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_3139_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_3140_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_3141_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_3142_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_3143_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_3144_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_3145_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_3146_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_3147_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_3148_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_3149_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_3150_split__pos__lemma,axiom,
! [P: int > int > $o,N: int,K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( P @ ( div_div_int @ N @ K ) @ ( div_mod_int @ N @ K ) )
<=> ! [I_1: int,J_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J_1 )
& ( ord_less_int @ J_1 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I_1 ) @ J_1 ) ) )
=> ( P @ I_1 @ J_1 ) ) ) ) ).
thf(fact_3151_split__neg__lemma,axiom,
! [P: int > int > $o,N: int,K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ( ( P @ ( div_div_int @ N @ K ) @ ( div_mod_int @ N @ K ) )
<=> ! [I_1: int,J_1: int] :
( ( ( ord_less_int @ K @ J_1 )
& ( ord_less_eq_int @ J_1 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I_1 ) @ J_1 ) ) )
=> ( P @ I_1 @ J_1 ) ) ) ) ).
thf(fact_3152_zmod__minus1,axiom,
! [B_1: int] :
( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( div_mod_int @ ( number_number_of_int @ min ) @ B_1 )
= ( minus_minus_int @ B_1 @ one_one_int ) ) ) ).
thf(fact_3153_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_3154_nat__2,axiom,
( ( nat_1 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_3155_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_3156_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_3157_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_3158_int__power__div__base,axiom,
! [K: int,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ( div_div_int @ ( power_power_int @ K @ M ) @ K )
= ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
thf(fact_3159_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_3160_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_3161_WilsonRuss_Oinv__def,axiom,
! [P_3: int,A_1: int] :
( ( inv @ P_3 @ A_1 )
= ( div_mod_int @ ( power_power_int @ A_1 @ ( nat_1 @ ( minus_minus_int @ P_3 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ P_3 ) ) ).
thf(fact_3162_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_3163_pos__zmod__mult__2,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_1 )
=> ( ( div_mod_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B_1 ) ) @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A_1 ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( div_mod_int @ B_1 @ A_1 ) ) ) ) ) ).
thf(fact_3164_SRStar__mult__prop2,axiom,
! [X: int,A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A_1 @ zero_zero_int @ P_3 )
=> ( ( member_int @ X @ ( sRStar @ P_3 ) )
=> ( member_int @ ( standardRes @ P_3 @ ( times_times_int @ A_1 @ ( multInv @ P_3 @ X ) ) ) @ ( sRStar @ P_3 ) ) ) ) ) ) ).
thf(fact_3165_realpow__pos__nth2,axiom,
! [N: nat,A_1: real] :
( ( ord_less_real @ zero_zero_real @ A_1 )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ ( suc @ N ) )
= A_1 ) ) ) ).
thf(fact_3166_zmod__eq__0D,axiom,
! [M_6: int,D_3: int] :
( ( ( div_mod_int @ M_6 @ D_3 )
= zero_zero_int )
=> ? [Q_2: int] :
( M_6
= ( times_times_int @ D_3 @ Q_2 ) ) ) ).
thf(fact_3167_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_3168_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_3169_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K_1: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K_1 ) ) ) ) ).
thf(fact_3170_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_3171_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( div_mod_nat @ M @ N )
= M ) ) ).
thf(fact_3172_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( div_mod_nat @ M @ N ) @ M ) ).
thf(fact_3173_mod__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( div_mod_nat @ M @ N ) )
= ( div_mod_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
thf(fact_3174_mod__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( div_mod_nat @ M @ N ) @ K )
= ( div_mod_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
thf(fact_3175_StandardRes__SRStar__prop3,axiom,
! [X: int,P_3: int] :
( ( member_int @ X @ ( sRStar @ P_3 ) )
=> ( ( standardRes @ P_3 @ X )
= X ) ) ).
thf(fact_3176_mod__1,axiom,
! [M: nat] :
( ( div_mod_nat @ M @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
thf(fact_3177_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_3178_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_3179_mod__eq__0__iff,axiom,
! [M: nat,D_1: nat] :
( ( ( div_mod_nat @ M @ D_1 )
= zero_zero_nat )
<=> ? [Q_2: nat] :
( M
= ( times_times_nat @ D_1 @ Q_2 ) ) ) ).
thf(fact_3180_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_3181_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_3182_mod__mult__self3,axiom,
! [K: nat,N: nat,M: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) @ N )
= ( div_mod_nat @ M @ N ) ) ).
thf(fact_3183_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_3184_div__add1__eq,axiom,
! [A_1: nat,B_1: nat,C: nat] :
( ( div_div_nat @ ( plus_plus_nat @ A_1 @ B_1 ) @ C )
= ( plus_plus_nat @ ( plus_plus_nat @ ( div_div_nat @ A_1 @ C ) @ ( div_div_nat @ B_1 @ C ) ) @ ( div_div_nat @ ( plus_plus_nat @ ( div_mod_nat @ A_1 @ C ) @ ( div_mod_nat @ B_1 @ C ) ) @ C ) ) ) ).
thf(fact_3185_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_3186_zmod__int,axiom,
! [A_1: nat,B_1: nat] :
( ( semiri1621563631at_int @ ( div_mod_nat @ A_1 @ B_1 ) )
= ( div_mod_int @ ( semiri1621563631at_int @ A_1 ) @ ( semiri1621563631at_int @ B_1 ) ) ) ).
thf(fact_3187_SRStar__SR__prop,axiom,
! [X: int,P_3: int] :
( ( member_int @ X @ ( sRStar @ P_3 ) )
=> ( member_int @ X @ ( sr @ P_3 ) ) ) ).
thf(fact_3188_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_3189_mod__mult__self4,axiom,
! [K: nat,N: nat,M: nat] :
( ( div_mod_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
= ( div_mod_nat @ ( suc @ M ) @ N ) ) ).
thf(fact_3190_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_3191_mod__mult2__eq,axiom,
! [A_1: nat,B_1: nat,C: nat] :
( ( div_mod_nat @ A_1 @ ( times_times_nat @ B_1 @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ B_1 @ ( div_mod_nat @ ( div_div_nat @ A_1 @ B_1 ) @ C ) ) @ ( div_mod_nat @ A_1 @ B_1 ) ) ) ).
thf(fact_3192_div__mult1__eq,axiom,
! [A_1: nat,B_1: nat,C: nat] :
( ( div_div_nat @ ( times_times_nat @ A_1 @ B_1 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A_1 @ ( div_div_nat @ B_1 @ C ) ) @ ( div_div_nat @ ( times_times_nat @ A_1 @ ( div_mod_nat @ B_1 @ C ) ) @ C ) ) ) ).
thf(fact_3193_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_3194_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_3195_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_3196_StandardRes__SRStar__prop1a,axiom,
! [X: int,P_3: int] :
( ( member_int @ X @ ( sRStar @ P_3 ) )
=> ~ ( zcong @ X @ zero_zero_int @ P_3 ) ) ).
thf(fact_3197_split__mod,axiom,
! [P: nat > $o,N: nat,K: nat] :
( ( P @ ( div_mod_nat @ N @ K ) )
<=> ( ( ( K = zero_zero_nat )
=> ( P @ N ) )
& ( ( K != zero_zero_nat )
=> ! [I_1: nat,J_1: nat] :
( ( ord_less_nat @ J_1 @ K )
=> ( ( N
= ( plus_plus_nat @ ( times_times_nat @ K @ I_1 ) @ J_1 ) )
=> ( P @ J_1 ) ) ) ) ) ) ).
thf(fact_3198_mod__lemma,axiom,
! [Q: nat,R_1: nat,B_1: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ( ord_less_nat @ R_1 @ B_1 )
=> ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ B_1 @ ( div_mod_nat @ Q @ C ) ) @ R_1 ) @ ( times_times_nat @ B_1 @ C ) ) ) ) ).
thf(fact_3199_Suc__times__mod__eq,axiom,
! [M: nat,K: nat] :
( ( ord_less_nat @ one_one_nat @ K )
=> ( ( div_mod_nat @ ( suc @ ( times_times_nat @ K @ M ) ) @ K )
= one_one_nat ) ) ).
thf(fact_3200_div__Suc,axiom,
! [A_1: nat,C: nat] :
( ( div_div_nat @ ( suc @ A_1 ) @ C )
= ( plus_plus_nat @ ( plus_plus_nat @ ( div_div_nat @ A_1 @ C ) @ ( div_div_nat @ ( suc @ zero_zero_nat ) @ C ) ) @ ( div_div_nat @ ( plus_plus_nat @ ( div_mod_nat @ A_1 @ C ) @ ( div_mod_nat @ ( suc @ zero_zero_nat ) @ C ) ) @ C ) ) ) ).
thf(fact_3201_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_3202_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_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_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_3205_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_3206_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_3207_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_3208_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_3209_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_3210_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_3211_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_3212_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_3213_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_3214_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_3215_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_3216_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_3217_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M_2: nat] :
( N
= ( suc @ M_2 ) ) ) ).
thf(fact_3218_mod__eqD,axiom,
! [M: nat,D_1: nat,R_1: nat] :
( ( ( div_mod_nat @ M @ D_1 )
= R_1 )
=> ? [Q_2: nat] :
( M
= ( plus_plus_nat @ R_1 @ ( times_times_nat @ Q_2 @ D_1 ) ) ) ) ).
thf(fact_3219_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_3220_mod__eq__0D,axiom,
! [M_6: nat,D_3: nat] :
( ( ( div_mod_nat @ M_6 @ D_3 )
= zero_zero_nat )
=> ? [Q_2: nat] :
( M_6
= ( times_times_nat @ D_3 @ Q_2 ) ) ) ).
thf(fact_3221_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_3222_pdivmod__def,axiom,
! [K: int,L: int] :
( ( pdivmod @ K @ L )
= ( product_Pair_int_int @ ( div_div_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) @ ( div_mod_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
thf(fact_3223_zmult2__lemma,axiom,
! [C: int,A_1: int,B_1: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A_1 @ B_1 @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( B_1 != zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( divmod_int_rel @ A_1 @ ( times_times_int @ B_1 @ C ) @ ( product_Pair_int_int @ ( div_div_int @ Q @ C ) @ ( plus_plus_int @ ( times_times_int @ B_1 @ ( div_mod_int @ Q @ C ) ) @ R_1 ) ) ) ) ) ) ).
thf(fact_3224_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_3225_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_3226_lemma__interval,axiom,
! [B_1: real,A_1: real,X: real] :
( ( ord_less_real @ A_1 @ X )
=> ( ( ord_less_real @ X @ B_1 )
=> ? [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_1 @ Y_1 )
& ( ord_less_eq_real @ Y_1 @ B_1 ) ) ) ) ) ) ).
thf(fact_3227_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_3228_not__neg__0,axiom,
~ ( nat_neg @ zero_zero_int ) ).
thf(fact_3229_even__zero__int,axiom,
even_odd_even_int @ zero_zero_int ).
thf(fact_3230_not__neg__1,axiom,
~ ( nat_neg @ one_one_int ) ).
thf(fact_3231_odd__one__int,axiom,
~ ( even_odd_even_int @ one_one_int ) ).
thf(fact_3232_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_3233_anything__times__even,axiom,
! [X: int,Y: int] :
( ( even_odd_even_int @ Y )
=> ( even_odd_even_int @ ( times_times_int @ X @ Y ) ) ) ).
thf(fact_3234_even__times__anything,axiom,
! [Y: int,X: int] :
( ( even_odd_even_int @ X )
=> ( even_odd_even_int @ ( times_times_int @ X @ Y ) ) ) ).
thf(fact_3235_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_3236_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_3237_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_3238_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_3239_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_3240_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_3241_negDivAlg__correct,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_int @ A_1 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( divmod_int_rel @ A_1 @ B_1 @ ( negDivAlg @ A_1 @ B_1 ) ) ) ) ).
thf(fact_3242_not__neg__int,axiom,
! [N: nat] :
~ ( nat_neg @ ( semiri1621563631at_int @ N ) ) ).
thf(fact_3243_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_3244_odd__pow,axiom,
! [N: nat,X: int] :
( ~ ( even_odd_even_int @ X )
=> ~ ( even_odd_even_int @ ( power_power_int @ X @ N ) ) ) ).
thf(fact_3245_neg__def,axiom,
! [Z_4: int] :
( ( nat_neg @ Z_4 )
<=> ( ord_less_int @ Z_4 @ zero_zero_int ) ) ).
thf(fact_3246_neg__number__of__Bit1,axiom,
! [W: int] :
( ( nat_neg @ ( number_number_of_int @ ( bit1 @ W ) ) )
<=> ( nat_neg @ ( number_number_of_int @ W ) ) ) ).
thf(fact_3247_not__neg__eq__ge__0,axiom,
! [X: int] :
( ~ ( nat_neg @ X )
<=> ( ord_less_eq_int @ zero_zero_int @ X ) ) ).
thf(fact_3248_not__neg__number__of__Pls,axiom,
~ ( nat_neg @ ( number_number_of_int @ pls ) ) ).
thf(fact_3249_neg__number__of__Bit0,axiom,
! [W: int] :
( ( nat_neg @ ( number_number_of_int @ ( bit0 @ W ) ) )
<=> ( nat_neg @ ( number_number_of_int @ W ) ) ) ).
thf(fact_3250_neg__nat,axiom,
! [Z_1: int] :
( ( nat_neg @ Z_1 )
=> ( ( nat_1 @ Z_1 )
= zero_zero_nat ) ) ).
thf(fact_3251_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_3252_neg__number__of__Min,axiom,
nat_neg @ ( number_number_of_int @ min ) ).
thf(fact_3253_not__neg__nat,axiom,
! [Z_1: int] :
( ~ ( nat_neg @ Z_1 )
=> ( ( semiri1621563631at_int @ ( nat_1 @ Z_1 ) )
= Z_1 ) ) ).
thf(fact_3254_unique__remainder,axiom,
! [Q_3: int,R_3: int,A_1: int,B_1: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A_1 @ B_1 @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( divmod_int_rel @ A_1 @ B_1 @ ( product_Pair_int_int @ Q_3 @ R_3 ) )
=> ( ( B_1 != zero_zero_int )
=> ( R_1 = R_3 ) ) ) ) ).
thf(fact_3255_unique__quotient,axiom,
! [Q_3: int,R_3: int,A_1: int,B_1: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A_1 @ B_1 @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( divmod_int_rel @ A_1 @ B_1 @ ( product_Pair_int_int @ Q_3 @ R_3 ) )
=> ( ( B_1 != zero_zero_int )
=> ( Q = Q_3 ) ) ) ) ).
thf(fact_3256_self__remainder,axiom,
! [A_1: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A_1 @ A_1 @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( A_1 != zero_zero_int )
=> ( R_1 = zero_zero_int ) ) ) ).
thf(fact_3257_divmod__int__rel__0,axiom,
! [B_1: int] :
( ( B_1 != zero_zero_int )
=> ( divmod_int_rel @ zero_zero_int @ B_1 @ ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) ) ).
thf(fact_3258_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_3259_self__quotient,axiom,
! [A_1: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A_1 @ A_1 @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( A_1 != zero_zero_int )
=> ( Q = one_one_int ) ) ) ).
thf(fact_3260_divmod__int__rel__div,axiom,
! [A_1: int,B_1: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A_1 @ B_1 @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( B_1 != zero_zero_int )
=> ( ( div_div_int @ A_1 @ B_1 )
= Q ) ) ) ).
thf(fact_3261_divmod__int__rel__mod,axiom,
! [A_1: int,B_1: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A_1 @ B_1 @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( B_1 != zero_zero_int )
=> ( ( div_mod_int @ A_1 @ B_1 )
= R_1 ) ) ) ).
thf(fact_3262_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_3263_nat__number__of__add__left,axiom,
! [V_1: int,K: 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 ) )
= ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K ) ) )
& ( ~ ( 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 ) )
= ( plus_plus_nat @ ( number_number_of_nat @ V ) @ K ) ) )
& ( ~ ( 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 ) )
= ( plus_plus_nat @ ( number_number_of_nat @ ( plus_plus_int @ V @ V_1 ) ) @ K ) ) ) ) ) ) ).
thf(fact_3264_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_3265_divmod__int__rel__div__mod,axiom,
! [A_1: int,B_1: int] :
( ( B_1 != zero_zero_int )
=> ( divmod_int_rel @ A_1 @ B_1 @ ( product_Pair_int_int @ ( div_div_int @ A_1 @ B_1 ) @ ( div_mod_int @ A_1 @ B_1 ) ) ) ) ).
thf(fact_3266_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_3267_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_3268_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_3269_zadd1__lemma,axiom,
! [B_1: int,Bq: int,Br: int,A_1: int,C: int,Aq: int,Ar: int] :
( ( divmod_int_rel @ A_1 @ C @ ( product_Pair_int_int @ Aq @ Ar ) )
=> ( ( divmod_int_rel @ B_1 @ C @ ( product_Pair_int_int @ Bq @ Br ) )
=> ( ( C != zero_zero_int )
=> ( divmod_int_rel @ ( plus_plus_int @ A_1 @ B_1 ) @ 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_3270_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_3271_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_3272_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_3273_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_3274_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_3275_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_3276_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_3277_negDivAlg__minus1,axiom,
! [B_1: int] :
( ( negDivAlg @ ( number_number_of_int @ min ) @ B_1 )
= ( product_Pair_int_int @ ( number_number_of_int @ min ) @ ( minus_minus_int @ B_1 @ one_one_int ) ) ) ).
thf(fact_3278_negDivAlg__div__mod,axiom,
! [L: int,K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ L )
=> ( ( negDivAlg @ K @ L )
= ( product_Pair_int_int @ ( div_div_int @ K @ L ) @ ( div_mod_int @ K @ L ) ) ) ) ) ).
thf(fact_3279_negDivAlg_Osimps,axiom,
! [A_1: int,B_1: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A_1 @ B_1 ) )
| ( ord_less_eq_int @ B_1 @ zero_zero_int ) )
=> ( ( negDivAlg @ A_1 @ B_1 )
= ( product_Pair_int_int @ ( number_number_of_int @ min ) @ ( plus_plus_int @ A_1 @ B_1 ) ) ) )
& ( ~ ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A_1 @ B_1 ) )
| ( ord_less_eq_int @ B_1 @ zero_zero_int ) )
=> ( ( negDivAlg @ A_1 @ B_1 )
= ( adjust @ B_1 @ ( negDivAlg @ A_1 @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B_1 ) ) ) ) ) ) ).
thf(fact_3280_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_3281_divmod__int__relI,axiom,
! [A_1: int,B_1: int,Q: int,R_1: int] :
( ( A_1
= ( plus_plus_int @ ( times_times_int @ B_1 @ Q ) @ R_1 ) )
=> ( ( ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
& ( ord_less_int @ R_1 @ B_1 ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_int @ B_1 @ R_1 )
& ( ord_less_eq_int @ R_1 @ zero_zero_int ) ) ) )
=> ( divmod_int_rel @ A_1 @ B_1 @ ( product_Pair_int_int @ Q @ R_1 ) ) ) ) ).
thf(fact_3282_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_3283_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_3284_zmult1__lemma,axiom,
! [A_1: int,B_1: int,C: int,Q: int,R_1: int] :
( ( divmod_int_rel @ B_1 @ C @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( C != zero_zero_int )
=> ( divmod_int_rel @ ( times_times_int @ A_1 @ B_1 ) @ C @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A_1 @ Q ) @ ( div_div_int @ ( times_times_int @ A_1 @ R_1 ) @ C ) ) @ ( div_mod_int @ ( times_times_int @ A_1 @ R_1 ) @ C ) ) ) ) ) ).
thf(fact_3285_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_3286_negDivAlg__eqn,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A_1 @ B_1 ) )
=> ( ( negDivAlg @ A_1 @ B_1 )
= ( product_Pair_int_int @ ( number_number_of_int @ min ) @ ( plus_plus_int @ A_1 @ B_1 ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A_1 @ B_1 ) )
=> ( ( negDivAlg @ A_1 @ B_1 )
= ( adjust @ B_1 @ ( negDivAlg @ A_1 @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B_1 ) ) ) ) ) ) ) ).
thf(fact_3287_lemma__interval__lt,axiom,
! [B_1: real,A_1: real,X: real] :
( ( ord_less_real @ A_1 @ X )
=> ( ( ord_less_real @ X @ B_1 )
=> ? [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_1 @ Y_1 )
& ( ord_less_real @ Y_1 @ B_1 ) ) ) ) ) ) ).
thf(fact_3288_posDivAlg_Osimps,axiom,
! [A_1: int,B_1: int] :
( ( ( ( ord_less_int @ A_1 @ B_1 )
| ( ord_less_eq_int @ B_1 @ zero_zero_int ) )
=> ( ( posDivAlg @ A_1 @ B_1 )
= ( product_Pair_int_int @ zero_zero_int @ A_1 ) ) )
& ( ~ ( ( ord_less_int @ A_1 @ B_1 )
| ( ord_less_eq_int @ B_1 @ zero_zero_int ) )
=> ( ( posDivAlg @ A_1 @ B_1 )
= ( adjust @ B_1 @ ( posDivAlg @ A_1 @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B_1 ) ) ) ) ) ) ).
thf(fact_3289_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_3290_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_3291_posDivAlg__eqn,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ( ord_less_int @ A_1 @ B_1 )
=> ( ( posDivAlg @ A_1 @ B_1 )
= ( product_Pair_int_int @ zero_zero_int @ A_1 ) ) )
& ( ~ ( ord_less_int @ A_1 @ B_1 )
=> ( ( posDivAlg @ A_1 @ B_1 )
= ( adjust @ B_1 @ ( posDivAlg @ A_1 @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B_1 ) ) ) ) ) ) ) ).
thf(fact_3292_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_3293_posDivAlg__0,axiom,
! [B_1: int] :
( ( posDivAlg @ zero_zero_int @ B_1 )
= ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) ).
thf(fact_3294_posDivAlg__correct,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_1 )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( divmod_int_rel @ A_1 @ B_1 @ ( posDivAlg @ A_1 @ B_1 ) ) ) ) ).
thf(fact_3295_pdivmod__posDivAlg,axiom,
! [K: int,L: int] :
( ( ( L = zero_zero_int )
=> ( ( pdivmod @ K @ L )
= ( product_Pair_int_int @ zero_zero_int @ ( abs_abs_int @ K ) ) ) )
& ( ( L != zero_zero_int )
=> ( ( pdivmod @ K @ L )
= ( posDivAlg @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ) ).
thf(fact_3296_posDivAlg__div__mod,axiom,
! [L: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( posDivAlg @ K @ L )
= ( product_Pair_int_int @ ( div_div_int @ K @ L ) @ ( div_mod_int @ K @ L ) ) ) ) ) ).
thf(fact_3297_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_3298_negDivAlg_Opsimps,axiom,
! [A_1: int,B_1: int] :
( ( accp_P2006205492nt_int @ negDivAlg_rel @ ( product_Pair_int_int @ A_1 @ B_1 ) )
=> ( ( ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A_1 @ B_1 ) )
| ( ord_less_eq_int @ B_1 @ zero_zero_int ) )
=> ( ( negDivAlg @ A_1 @ B_1 )
= ( product_Pair_int_int @ ( number_number_of_int @ min ) @ ( plus_plus_int @ A_1 @ B_1 ) ) ) )
& ( ~ ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A_1 @ B_1 ) )
| ( ord_less_eq_int @ B_1 @ zero_zero_int ) )
=> ( ( negDivAlg @ A_1 @ B_1 )
= ( adjust @ B_1 @ ( negDivAlg @ A_1 @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B_1 ) ) ) ) ) ) ) ).
thf(fact_3299_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_3300_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_3301_posDivAlg_Opsimps,axiom,
! [A_1: int,B_1: int] :
( ( accp_P2006205492nt_int @ posDivAlg_rel @ ( product_Pair_int_int @ A_1 @ B_1 ) )
=> ( ( ( ( ord_less_int @ A_1 @ B_1 )
| ( ord_less_eq_int @ B_1 @ zero_zero_int ) )
=> ( ( posDivAlg @ A_1 @ B_1 )
= ( product_Pair_int_int @ zero_zero_int @ A_1 ) ) )
& ( ~ ( ( ord_less_int @ A_1 @ B_1 )
| ( ord_less_eq_int @ B_1 @ zero_zero_int ) )
=> ( ( posDivAlg @ A_1 @ B_1 )
= ( adjust @ B_1 @ ( posDivAlg @ A_1 @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B_1 ) ) ) ) ) ) ) ).
thf(fact_3302_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_3303_natfloor__zero,axiom,
( ( natfloor @ zero_zero_real )
= zero_zero_nat ) ).
thf(fact_3304_zero__le__natfloor,axiom,
! [X: real] : ( ord_less_eq_nat @ zero_zero_nat @ ( natfloor @ X ) ) ).
thf(fact_3305_natfloor__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( natfloor @ X ) @ ( natfloor @ Y ) ) ) ).
thf(fact_3306_natfloor__number__of__eq,axiom,
! [N: int] :
( ( natfloor @ ( number267125858f_real @ N ) )
= ( number_number_of_nat @ N ) ) ).
thf(fact_3307_natfloor__one,axiom,
( ( natfloor @ one_one_real )
= one_one_nat ) ).
thf(fact_3308_natceiling__zero,axiom,
( ( natceiling @ zero_zero_real )
= zero_zero_nat ) ).
thf(fact_3309_zero__le__natceiling,axiom,
! [X: real] : ( ord_less_eq_nat @ zero_zero_nat @ ( natceiling @ X ) ) ).
thf(fact_3310_natceiling__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( natceiling @ X ) @ ( natceiling @ Y ) ) ) ).
thf(fact_3311_natceiling__number__of__eq,axiom,
! [N: int] :
( ( natceiling @ ( number267125858f_real @ N ) )
= ( number_number_of_nat @ N ) ) ).
thf(fact_3312_natceiling__one,axiom,
( ( natceiling @ one_one_real )
= one_one_nat ) ).
thf(fact_3313_natfloor__neg,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( natfloor @ X )
= zero_zero_nat ) ) ).
thf(fact_3314_natceiling__neg,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( natceiling @ X )
= zero_zero_nat ) ) ).
thf(fact_3315_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_3316_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_3317_le__mult__natfloor,axiom,
! [B_1: real,A_1: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_1 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_1 )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( natfloor @ A_1 ) @ ( natfloor @ B_1 ) ) @ ( natfloor @ ( times_times_real @ A_1 @ B_1 ) ) ) ) ) ).
thf(fact_3318_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_3319_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_3320_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_3321_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_3322_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_3323_MultInvPair__card__two,axiom,
! [J: int,A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A_1 @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A_1 )
=> ( ~ ( zcong @ J @ zero_zero_int @ P_3 )
=> ( ( finite_card_int @ ( multInvPair @ A_1 @ P_3 @ J ) )
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_3324_SetS__elems__card,axiom,
! [A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A_1 @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A_1 )
=> ! [X_1: int > $o] :
( ( member_int_o @ X_1 @ ( setS @ A_1 @ P_3 ) )
=> ( ( finite_card_int @ X_1 )
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_3325_Code__Numeral_Oint__of__code,axiom,
! [K: code_code_numeral] :
( ( ( K = zero_z126310315umeral )
=> ( ( code_int_of @ K )
= zero_zero_int ) )
& ( ( K != zero_z126310315umeral )
=> ( ( ( ( div_mo1740067990umeral @ K @ ( number1443263063umeral @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_z126310315umeral )
=> ( ( code_int_of @ K )
= ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( code_int_of @ ( div_di1218280263umeral @ K @ ( number1443263063umeral @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) )
& ( ( ( div_mo1740067990umeral @ K @ ( number1443263063umeral @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= zero_z126310315umeral )
=> ( ( code_int_of @ K )
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( code_int_of @ ( div_di1218280263umeral @ K @ ( number1443263063umeral @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
thf(fact_3326_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_3327_code__numeral__zero__minus__one,axiom,
( ( minus_1690775515umeral @ zero_z126310315umeral @ one_on1645066479umeral )
= zero_z126310315umeral ) ).
thf(fact_3328_zero__code__numeral__code,axiom,
( zero_z126310315umeral
= ( number1443263063umeral @ pls ) ) ).
thf(fact_3329_one__code__numeral__code,axiom,
( one_on1645066479umeral
= ( number1443263063umeral @ ( bit1 @ pls ) ) ) ).
thf(fact_3330_SetS__card,axiom,
! [A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A_1 @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A_1 )
=> ( ( semiri1621563631at_int @ ( finite_card_int_o @ ( setS @ A_1 @ P_3 ) ) )
= ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_3331_MultInvPair__prop2,axiom,
! [A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A_1 @ zero_zero_int @ P_3 )
=> ( ( comple1092985777_int_o @ ( setS @ A_1 @ P_3 ) )
= ( sRStar @ P_3 ) ) ) ) ) ).
thf(fact_3332_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_3333_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N_1: nat] :
( L
= ( plus_plus_nat @ K @ N_1 ) ) ) ).
thf(fact_3334_zero__induct__lemma,axiom,
! [I: nat,P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N_1: nat] :
( ( P @ ( suc @ N_1 ) )
=> ( P @ N_1 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
thf(fact_3335_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_3336_one__not__even,axiom,
~ ( member_int @ one_one_int @ zEven ) ).
thf(fact_3337_even__times__either,axiom,
! [Y: int,X: int] :
( ( member_int @ X @ zEven )
=> ( member_int @ ( times_times_int @ X @ Y ) @ zEven ) ) ).
thf(fact_3338_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_3339_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_3340_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_3341_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_3342_not__odd__impl__even,axiom,
! [X: int] :
( ~ ( member_int @ X @ zOdd )
=> ( member_int @ X @ zEven ) ) ).
thf(fact_3343_even__odd__disj,axiom,
! [X: int] :
( ( member_int @ X @ zOdd )
| ( member_int @ X @ zEven ) ) ).
thf(fact_3344_odd__iff__not__even,axiom,
! [X: int] :
( ( member_int @ X @ zOdd )
<=> ~ ( member_int @ X @ zEven ) ) ).
thf(fact_3345_even__odd__conj,axiom,
! [X: int] :
~ ( ( member_int @ X @ zOdd )
& ( member_int @ X @ zEven ) ) ).
thf(fact_3346_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_3347_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_3348_ln__less__self,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ ( ln @ X ) @ X ) ) ).
thf(fact_3349_ln__one,axiom,
( ( ln @ one_one_real )
= zero_zero_real ) ).
thf(fact_3350_even__plus__odd__prop2,axiom,
! [A_1: int,B_1: int] :
( ( member_int @ ( plus_plus_int @ A_1 @ B_1 ) @ zOdd )
=> ( ( member_int @ A_1 @ zEven )
=> ( member_int @ B_1 @ zOdd ) ) ) ).
thf(fact_3351_even__plus__odd__prop1,axiom,
! [A_1: int,B_1: int] :
( ( member_int @ ( plus_plus_int @ A_1 @ B_1 ) @ zOdd )
=> ( ( member_int @ A_1 @ zOdd )
=> ( member_int @ B_1 @ zEven ) ) ) ).
thf(fact_3352_IntNatAux_Oeven__plus__odd,axiom,
! [B_1: int,A_1: int] :
( ( member_int @ A_1 @ zEven )
=> ( ( member_int @ B_1 @ zOdd )
=> ( member_int @ ( plus_plus_int @ A_1 @ B_1 ) @ zOdd ) ) ) ).
thf(fact_3353_IntNatAux_Oodd__plus__odd,axiom,
! [B_1: int,A_1: int] :
( ( member_int @ A_1 @ zOdd )
=> ( ( member_int @ B_1 @ zOdd )
=> ( member_int @ ( plus_plus_int @ A_1 @ B_1 ) @ zEven ) ) ) ).
thf(fact_3354_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_3355_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_3356_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_3357_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_3358_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_3359_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_3360_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_3361_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_3362_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_3363_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_3364_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_3365_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( ln @ X ) ) ) ).
thf(fact_3366_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_3367_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_3368_odd__minus__one__even,axiom,
! [X: int] :
( ( member_int @ X @ zOdd )
=> ( member_int @ ( minus_minus_int @ X @ one_one_int ) @ zEven ) ) ).
thf(fact_3369_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_3370_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_3371_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_3372_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_3373_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_3374_zEvenI,axiom,
! [X: int,K: int] :
( ( X
= ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ K ) )
=> ( member_int @ X @ zEven ) ) ).
thf(fact_3375_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_3376_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_3377_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_3378_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_3379_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_3380_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_3381_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_3382_zEvenE,axiom,
! [X: int] :
( ( member_int @ X @ zEven )
=> ~ ! [K_1: int] :
( X
!= ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_1 ) ) ) ).
thf(fact_3383_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_3384_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_3385_ln__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( ln @ X ) @ X ) ) ).
thf(fact_3386_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_3387_exp__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
thf(fact_3388_exp__inj__iff,axiom,
! [X: real,Y: real] :
( ( ( exp_real @ X )
= ( exp_real @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_3389_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_3390_exp__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
thf(fact_3391_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_3392_real__divide__square__eq,axiom,
! [R_1: real,A_1: real] :
( ( inverse_divide_real @ ( times_times_real @ R_1 @ A_1 ) @ ( times_times_real @ R_1 @ R_1 ) )
= ( inverse_divide_real @ A_1 @ R_1 ) ) ).
thf(fact_3393_abs__exp__cancel,axiom,
! [X: real] :
( ( abs_abs_real @ ( exp_real @ X ) )
= ( exp_real @ X ) ) ).
thf(fact_3394_ln__exp,axiom,
! [X: real] :
( ( ln @ ( exp_real @ X ) )
= X ) ).
thf(fact_3395_exp__ln__eq,axiom,
! [U: real,X: real] :
( ( ( exp_real @ U )
= X )
=> ( ( ln @ X )
= U ) ) ).
thf(fact_3396_ln__unique,axiom,
! [Y: real,X: real] :
( ( ( exp_real @ Y )
= X )
=> ( ( ln @ X )
= Y ) ) ).
thf(fact_3397_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_3398_not__exp__le__zero,axiom,
! [X: real] :
~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).
thf(fact_3399_exp__ge__zero,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).
thf(fact_3400_not__exp__less__zero,axiom,
! [X: real] :
~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).
thf(fact_3401_exp__gt__zero,axiom,
! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).
thf(fact_3402_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_3403_exp__eq__one__iff,axiom,
! [X: real] :
( ( ( exp_real @ X )
= one_one_real )
<=> ( X = zero_zero_real ) ) ).
thf(fact_3404_lemma__MVT,axiom,
! [F: real > real,A_1: real,B_1: real] :
( ( minus_minus_real @ ( F @ A_1 ) @ ( times_times_real @ ( inverse_divide_real @ ( minus_minus_real @ ( F @ B_1 ) @ ( F @ A_1 ) ) @ ( minus_minus_real @ B_1 @ A_1 ) ) @ A_1 ) )
= ( minus_minus_real @ ( F @ B_1 ) @ ( times_times_real @ ( inverse_divide_real @ ( minus_minus_real @ ( F @ B_1 ) @ ( F @ A_1 ) ) @ ( minus_minus_real @ B_1 @ A_1 ) ) @ B_1 ) ) ) ).
thf(fact_3405_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_3406_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_3407_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_3408_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_3409_exp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
thf(fact_3410_exp__ge__add__one__self,axiom,
! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).
thf(fact_3411_exp__ln,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( exp_real @ ( ln @ X ) )
= X ) ) ).
thf(fact_3412_exp__ln__iff,axiom,
! [X: real] :
( ( ( exp_real @ ( ln @ X ) )
= X )
<=> ( ord_less_real @ zero_zero_real @ X ) ) ).
thf(fact_3413_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_3414_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_3415_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_3416_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_3417_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_3418_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_3419_real__average__minus__first,axiom,
! [A_1: real,B_1: real] :
( ( minus_minus_real @ ( inverse_divide_real @ ( plus_plus_real @ A_1 @ B_1 ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A_1 )
= ( inverse_divide_real @ ( minus_minus_real @ B_1 @ A_1 ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3420_real__average__minus__second,axiom,
! [B_1: real,A_1: real] :
( ( minus_minus_real @ ( inverse_divide_real @ ( plus_plus_real @ B_1 @ A_1 ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A_1 )
= ( inverse_divide_real @ ( minus_minus_real @ B_1 @ A_1 ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3421_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_3422_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_3423_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_3424_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_3425_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_3426_real__norm__def,axiom,
! [R_1: real] :
( ( norm_norm_real @ R_1 )
= ( abs_abs_real @ R_1 ) ) ).
thf(fact_3427_complex__mod__triangle__ineq2,axiom,
! [B_1: complex,A_1: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( norm_norm_complex @ ( plus_plus_complex @ B_1 @ A_1 ) ) @ ( norm_norm_complex @ B_1 ) ) @ ( norm_norm_complex @ A_1 ) ) ).
thf(fact_3428_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_3429_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_3430_real__add__minus__iff,axiom,
! [X: real,A_1: real] :
( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A_1 ) )
= zero_zero_real )
<=> ( X = A_1 ) ) ).
thf(fact_3431_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_3432_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_3433_minus__real__def,axiom,
! [X: real,Y: real] :
( ( minus_minus_real @ X @ Y )
= ( plus_plus_real @ X @ ( uminus_uminus_real @ Y ) ) ) ).
thf(fact_3434_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_3435_log__one,axiom,
! [A_1: real] :
( ( log @ A_1 @ one_one_real )
= zero_zero_real ) ).
thf(fact_3436_Log_Olog__def,axiom,
! [A_1: real,X: real] :
( ( log @ A_1 @ X )
= ( inverse_divide_real @ ( ln @ X ) @ ( ln @ A_1 ) ) ) ).
thf(fact_3437_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_3438_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_3439_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_3440_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_3441_abs__real__def,axiom,
! [A_1: real] :
( ( ( ord_less_real @ A_1 @ zero_zero_real )
=> ( ( abs_abs_real @ A_1 )
= ( uminus_uminus_real @ A_1 ) ) )
& ( ~ ( ord_less_real @ A_1 @ zero_zero_real )
=> ( ( abs_abs_real @ A_1 )
= A_1 ) ) ) ).
thf(fact_3442_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_3443_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_3444_log__eq__one,axiom,
! [A_1: real] :
( ( ord_less_real @ zero_zero_real @ A_1 )
=> ( ( A_1 != one_one_real )
=> ( ( log @ A_1 @ A_1 )
= one_one_real ) ) ) ).
thf(fact_3445_log__less__cancel__iff,axiom,
! [Y: real,X: real,A_1: real] :
( ( ord_less_real @ one_one_real @ A_1 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( log @ A_1 @ X ) @ ( log @ A_1 @ Y ) )
<=> ( ord_less_real @ X @ Y ) ) ) ) ) ).
thf(fact_3446_log__ln,axiom,
! [X: real] :
( ( ln @ X )
= ( log @ ( exp_real @ one_one_real ) @ X ) ) ).
thf(fact_3447_log__le__cancel__iff,axiom,
! [Y: real,X: real,A_1: real] :
( ( ord_less_real @ one_one_real @ A_1 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( log @ A_1 @ X ) @ ( log @ A_1 @ Y ) )
<=> ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).
thf(fact_3448_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_3449_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_3450_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_3451_log__mult,axiom,
! [Y: real,X: real,A_1: real] :
( ( ord_less_real @ zero_zero_real @ A_1 )
=> ( ( A_1 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( log @ A_1 @ ( times_times_real @ X @ Y ) )
= ( plus_plus_real @ ( log @ A_1 @ X ) @ ( log @ A_1 @ Y ) ) ) ) ) ) ) ).
thf(fact_3452_log__divide,axiom,
! [Y: real,X: real,A_1: real] :
( ( ord_less_real @ zero_zero_real @ A_1 )
=> ( ( A_1 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( log @ A_1 @ ( inverse_divide_real @ X @ Y ) )
= ( minus_minus_real @ ( log @ A_1 @ X ) @ ( log @ A_1 @ Y ) ) ) ) ) ) ) ).
thf(fact_3453_log__eq__div__ln__mult__log,axiom,
! [X: real,B_1: real,A_1: real] :
( ( ord_less_real @ zero_zero_real @ A_1 )
=> ( ( A_1 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ B_1 )
=> ( ( B_1 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ A_1 @ X )
= ( times_times_real @ ( inverse_divide_real @ ( ln @ B_1 ) @ ( ln @ A_1 ) ) @ ( log @ B_1 @ X ) ) ) ) ) ) ) ) ).
thf(fact_3454_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_3455_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_3456_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1621563631at_int @ N ) ) @ ( semiri1621563631at_int @ M ) ) ).
thf(fact_3457_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1621563631at_int @ ( suc @ N ) ) ) @ ( semiri1621563631at_int @ M ) ) ).
thf(fact_3458_complex__diff__def,axiom,
! [X: complex,Y: complex] :
( ( minus_minus_complex @ X @ Y )
= ( plus_plus_complex @ X @ ( uminus473333897omplex @ Y ) ) ) ).
thf(fact_3459_zminus__zminus,axiom,
! [Z_1: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ Z_1 ) )
= Z_1 ) ).
thf(fact_3460_minus__Pls,axiom,
( ( uminus_uminus_int @ pls )
= pls ) ).
thf(fact_3461_minus__Bit0,axiom,
! [K: int] :
( ( uminus_uminus_int @ ( bit0 @ K ) )
= ( bit0 @ ( uminus_uminus_int @ K ) ) ) ).
thf(fact_3462_zminus__0,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_3463_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_3464_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_3465_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_3466_zcong__zminus,axiom,
! [A_1: int,B_1: int,M: int] :
( ( zcong @ A_1 @ B_1 @ ( uminus_uminus_int @ M ) )
<=> ( zcong @ A_1 @ B_1 @ M ) ) ).
thf(fact_3467_uminus__dvd__conv_I1_J,axiom,
! [D_1: int,T: int] :
( ( dvd_dvd_int @ D_1 @ T )
<=> ( dvd_dvd_int @ ( uminus_uminus_int @ D_1 ) @ T ) ) ).
thf(fact_3468_uminus__dvd__conv_I2_J,axiom,
! [D_1: int,T: int] :
( ( dvd_dvd_int @ D_1 @ T )
<=> ( dvd_dvd_int @ D_1 @ ( uminus_uminus_int @ T ) ) ) ).
thf(fact_3469_zdiv__zminus__zminus,axiom,
! [A_1: int,B_1: int] :
( ( div_div_int @ ( uminus_uminus_int @ A_1 ) @ ( uminus_uminus_int @ B_1 ) )
= ( div_div_int @ A_1 @ B_1 ) ) ).
thf(fact_3470_zdiv__zminus2,axiom,
! [A_1: int,B_1: int] :
( ( div_div_int @ A_1 @ ( uminus_uminus_int @ B_1 ) )
= ( div_div_int @ ( uminus_uminus_int @ A_1 ) @ B_1 ) ) ).
thf(fact_3471_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_3472_zmod__zminus__zminus,axiom,
! [A_1: int,B_1: int] :
( ( div_mod_int @ ( uminus_uminus_int @ A_1 ) @ ( uminus_uminus_int @ B_1 ) )
= ( uminus_uminus_int @ ( div_mod_int @ A_1 @ B_1 ) ) ) ).
thf(fact_3473_zmod__zminus2,axiom,
! [A_1: int,B_1: int] :
( ( div_mod_int @ A_1 @ ( uminus_uminus_int @ B_1 ) )
= ( uminus_uminus_int @ ( div_mod_int @ ( uminus_uminus_int @ A_1 ) @ B_1 ) ) ) ).
thf(fact_3474_even__neg,axiom,
! [X: int] :
( ( even_odd_even_int @ ( uminus_uminus_int @ X ) )
<=> ( even_odd_even_int @ X ) ) ).
thf(fact_3475_zadd__zminus__inverse2,axiom,
! [Z_1: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ Z_1 ) @ Z_1 )
= zero_zero_int ) ).
thf(fact_3476_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_3477_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1621563631at_int @ N ) @ ( uminus_uminus_int @ ( semiri1621563631at_int @ M ) ) ) ).
thf(fact_3478_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_3479_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_3480_Int_OMin__def,axiom,
( min
= ( uminus_uminus_int @ one_one_int ) ) ).
thf(fact_3481_mult__Min,axiom,
! [K: int] :
( ( times_times_int @ min @ K )
= ( uminus_uminus_int @ K ) ) ).
thf(fact_3482_zmod__zminus1__not__zero,axiom,
! [K: int,L: int] :
( ( ( div_mod_int @ ( uminus_uminus_int @ K ) @ L )
!= zero_zero_int )
=> ( ( div_mod_int @ K @ L )
!= zero_zero_int ) ) ).
thf(fact_3483_zmod__zminus2__not__zero,axiom,
! [K: int,L: int] :
( ( ( div_mod_int @ K @ ( uminus_uminus_int @ L ) )
!= zero_zero_int )
=> ( ( div_mod_int @ K @ L )
!= zero_zero_int ) ) ).
thf(fact_3484_minus__Min,axiom,
( ( uminus_uminus_int @ min )
= ( bit1 @ pls ) ) ).
thf(fact_3485_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_3486_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1621563631at_int @ N ) ) @ zero_zero_int ) ).
thf(fact_3487_nat__zminus__int,axiom,
! [N: nat] :
( ( nat_1 @ ( uminus_uminus_int @ ( semiri1621563631at_int @ N ) ) )
= zero_zero_nat ) ).
thf(fact_3488_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_3489_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_3490_zmod__zminus1__eq__if,axiom,
! [A_1: int,B_1: int] :
( ( ( ( div_mod_int @ A_1 @ B_1 )
= zero_zero_int )
=> ( ( div_mod_int @ ( uminus_uminus_int @ A_1 ) @ B_1 )
= zero_zero_int ) )
& ( ( ( div_mod_int @ A_1 @ B_1 )
!= zero_zero_int )
=> ( ( div_mod_int @ ( uminus_uminus_int @ A_1 ) @ B_1 )
= ( minus_minus_int @ B_1 @ ( div_mod_int @ A_1 @ B_1 ) ) ) ) ) ).
thf(fact_3491_zmod__zminus2__eq__if,axiom,
! [A_1: int,B_1: int] :
( ( ( ( div_mod_int @ A_1 @ B_1 )
= zero_zero_int )
=> ( ( div_mod_int @ A_1 @ ( uminus_uminus_int @ B_1 ) )
= zero_zero_int ) )
& ( ( ( div_mod_int @ A_1 @ B_1 )
!= zero_zero_int )
=> ( ( div_mod_int @ A_1 @ ( uminus_uminus_int @ B_1 ) )
= ( minus_minus_int @ ( div_mod_int @ A_1 @ B_1 ) @ B_1 ) ) ) ) ).
thf(fact_3492_zdiv__minus1__right,axiom,
! [A_1: int] :
( ( div_div_int @ A_1 @ ( number_number_of_int @ min ) )
= ( uminus_uminus_int @ A_1 ) ) ).
thf(fact_3493_neg__zminus__int,axiom,
! [N: nat] : ( nat_neg @ ( uminus_uminus_int @ ( semiri1621563631at_int @ ( suc @ N ) ) ) ) ).
thf(fact_3494_power3__minus,axiom,
! [A_1: int] :
( ( power_power_int @ ( uminus_uminus_int @ A_1 ) @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( uminus_uminus_int @ ( power_power_int @ A_1 @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3495_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1621563631at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
thf(fact_3496_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1621563631at_int @ ( suc @ N ) ) ) ) ).
thf(fact_3497_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_3498_zdiv__zminus2__eq__if,axiom,
! [A_1: int,B_1: int] :
( ( B_1 != zero_zero_int )
=> ( ( ( ( div_mod_int @ A_1 @ B_1 )
= zero_zero_int )
=> ( ( div_div_int @ A_1 @ ( uminus_uminus_int @ B_1 ) )
= ( uminus_uminus_int @ ( div_div_int @ A_1 @ B_1 ) ) ) )
& ( ( ( div_mod_int @ A_1 @ B_1 )
!= zero_zero_int )
=> ( ( div_div_int @ A_1 @ ( uminus_uminus_int @ B_1 ) )
= ( minus_minus_int @ ( uminus_uminus_int @ ( div_div_int @ A_1 @ B_1 ) ) @ one_one_int ) ) ) ) ) ).
thf(fact_3499_zdiv__zminus1__eq__if,axiom,
! [A_1: int,B_1: int] :
( ( B_1 != zero_zero_int )
=> ( ( ( ( div_mod_int @ A_1 @ B_1 )
= zero_zero_int )
=> ( ( div_div_int @ ( uminus_uminus_int @ A_1 ) @ B_1 )
= ( uminus_uminus_int @ ( div_div_int @ A_1 @ B_1 ) ) ) )
& ( ( ( div_mod_int @ A_1 @ B_1 )
!= zero_zero_int )
=> ( ( div_div_int @ ( uminus_uminus_int @ A_1 ) @ B_1 )
= ( minus_minus_int @ ( uminus_uminus_int @ ( div_div_int @ A_1 @ B_1 ) ) @ one_one_int ) ) ) ) ) ).
thf(fact_3500_neg__even__power,axiom,
! [A_1: int,X: int] :
( ( member_int @ X @ zEven )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( power_power_int @ ( uminus_uminus_int @ A_1 ) @ ( nat_1 @ X ) )
= ( power_power_int @ A_1 @ ( nat_1 @ X ) ) ) ) ) ).
thf(fact_3501_neg__odd__power,axiom,
! [A_1: int,X: int] :
( ( member_int @ X @ zOdd )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( power_power_int @ ( uminus_uminus_int @ A_1 ) @ ( nat_1 @ X ) )
= ( uminus_uminus_int @ ( power_power_int @ A_1 @ ( nat_1 @ X ) ) ) ) ) ) ).
thf(fact_3502_zminus1__lemma,axiom,
! [A_1: int,B_1: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A_1 @ B_1 @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( divmod_int_rel @ ( uminus_uminus_int @ A_1 ) @ B_1 @ ( 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_1 @ R_1 ) ) ) ) ) ).
thf(fact_3503_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_3504_exp__total,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ? [X_1: real] :
( ( exp_real @ X_1 )
= Y ) ) ).
thf(fact_3505_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_3506_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_3507_real__sqrt__divide,axiom,
! [X: real,Y: real] :
( ( sqrt @ ( inverse_divide_real @ X @ Y ) )
= ( inverse_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
thf(fact_3508_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_3509_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_3510_real__sqrt__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( sqrt @ X )
= ( sqrt @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_3511_real__sqrt__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
thf(fact_3512_real__sqrt__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
<=> ( ord_less_real @ X @ Y ) ) ).
thf(fact_3513_real__sqrt__eq__0__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= zero_zero_real )
<=> ( X = zero_zero_real ) ) ).
thf(fact_3514_real__sqrt__zero,axiom,
( ( sqrt @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_3515_real__sqrt__one,axiom,
( ( sqrt @ one_one_real )
= one_one_real ) ).
thf(fact_3516_real__sqrt__eq__1__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= one_one_real )
<=> ( X = one_one_real ) ) ).
thf(fact_3517_real__sqrt__power,axiom,
! [X: real,K: nat] :
( ( sqrt @ ( power_power_real @ X @ K ) )
= ( power_power_real @ ( sqrt @ X ) @ K ) ) ).
thf(fact_3518_real__sqrt__mult,axiom,
! [X: real,Y: real] :
( ( sqrt @ ( times_times_real @ X @ Y ) )
= ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
thf(fact_3519_real__sqrt__minus,axiom,
! [X: real] :
( ( sqrt @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).
thf(fact_3520_arctan__monotone_H,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
thf(fact_3521_arctan__zero__zero,axiom,
( ( arctan @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_3522_arctan__monotone,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
thf(fact_3523_arctan__minus,axiom,
! [X: real] :
( ( arctan @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( arctan @ X ) ) ) ).
thf(fact_3524_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_3525_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_3526_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_3527_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_3528_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_3529_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_3530_real__sqrt__not__eq__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( sqrt @ X )
!= zero_zero_real ) ) ).
thf(fact_3531_real__sqrt__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
thf(fact_3532_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_3533_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_3534_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_3535_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_3536_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_3537_real__sqrt__abs2,axiom,
! [X: real] :
( ( sqrt @ ( times_times_real @ X @ X ) )
= ( abs_abs_real @ X ) ) ).
thf(fact_3538_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_3539_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_3540_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_3541_real__sqrt__two__ge__zero,axiom,
ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3542_real__sqrt__two__gt__zero,axiom,
ord_less_real @ zero_zero_real @ ( sqrt @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3543_real__sqrt__two__gt__one,axiom,
ord_less_real @ one_one_real @ ( sqrt @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3544_real__sqrt__abs,axiom,
! [X: real] :
( ( sqrt @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( abs_abs_real @ X ) ) ).
thf(fact_3545_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_3546_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_3547_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_3548_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_3549_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_3550_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_3551_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_3552_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A_1: real,C: real,B_1: real,D_1: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A_1 @ C ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( plus_plus_real @ B_1 @ D_1 ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ B_1 @ ( 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_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ).
thf(fact_3553_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_3554_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_3555_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_3556_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_3557_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_3558_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_3559_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_3560_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_3561_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_3562_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_3563_z3mod__def,axiom,
! [K: int,L: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( z3mod @ K @ L )
= ( div_mod_int @ K @ L ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( z3mod @ K @ L )
= ( div_mod_int @ K @ ( uminus_uminus_int @ L ) ) ) ) ) ).
thf(fact_3564_z3div__def,axiom,
! [K: int,L: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( z3div @ K @ L )
= ( div_div_int @ K @ L ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( z3div @ K @ L )
= ( uminus_uminus_int @ ( div_div_int @ K @ ( uminus_uminus_int @ L ) ) ) ) ) ) ).
thf(fact_3565_pi__neq__zero,axiom,
pi != zero_zero_real ).
thf(fact_3566_pi__ge__zero,axiom,
ord_less_eq_real @ zero_zero_real @ pi ).
thf(fact_3567_pi__gt__zero,axiom,
ord_less_real @ zero_zero_real @ pi ).
thf(fact_3568_pi__not__less__zero,axiom,
~ ( ord_less_real @ pi @ zero_zero_real ) ).
thf(fact_3569_pi__ge__two,axiom,
ord_less_eq_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ).
thf(fact_3570_pi__less__4,axiom,
ord_less_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3571_pi__half__neq__two,axiom,
( ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3572_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_3573_pi__half__neq__zero,axiom,
( ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= zero_zero_real ) ).
thf(fact_3574_pi__half__less__two,axiom,
ord_less_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_3575_pi__half__ge__zero,axiom,
ord_less_eq_real @ zero_zero_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3576_pi__half__gt__zero,axiom,
ord_less_real @ zero_zero_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3577_m2pi__less__pi,axiom,
ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ) ) @ pi ).
thf(fact_3578_arctan__ubound,axiom,
! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3579_arctan1__eq__pi4,axiom,
( ( arctan @ one_one_real )
= ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3580_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_3581_arctan__lbound,axiom,
! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( arctan @ Y ) ) ).
thf(fact_3582_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_3583_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_3584_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_3585_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_3586_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_3587_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_3588_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_3589_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_3590_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_3591_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_3592_sin__minus,axiom,
! [X: real] :
( ( sin @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( sin @ X ) ) ) ).
thf(fact_3593_cos__minus,axiom,
! [X: real] :
( ( cos @ ( uminus_uminus_real @ X ) )
= ( cos @ X ) ) ).
thf(fact_3594_sin__zero,axiom,
( ( sin @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_3595_cos__one__sin__zero,axiom,
! [X: real] :
( ( ( cos @ X )
= one_one_real )
=> ( ( sin @ X )
= zero_zero_real ) ) ).
thf(fact_3596_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_3597_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_3598_sin__zero__abs__cos__one,axiom,
! [X: real] :
( ( ( sin @ X )
= zero_zero_real )
=> ( ( abs_abs_real @ ( cos @ X ) )
= one_one_real ) ) ).
thf(fact_3599_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_3600_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_3601_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_3602_sin__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( sin @ X ) @ one_one_real ) ).
thf(fact_3603_cos__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( cos @ X ) @ one_one_real ) ).
thf(fact_3604_cos__zero,axiom,
( ( cos @ zero_zero_real )
= one_one_real ) ).
thf(fact_3605_sin__pi,axiom,
( ( sin @ pi )
= zero_zero_real ) ).
thf(fact_3606_cos__arctan__not__zero,axiom,
! [X: real] :
( ( cos @ ( arctan @ X ) )
!= zero_zero_real ) ).
thf(fact_3607_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_3608_real__sgn__eq,axiom,
! [X: real] :
( ( sgn_sgn_real @ X )
= ( inverse_divide_real @ X @ ( abs_abs_real @ X ) ) ) ).
thf(fact_3609_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_3610_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_3611_sin__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( number267125858f_real @ min ) @ ( sin @ X ) ) ).
thf(fact_3612_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_3613_abs__sin__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin @ X ) ) @ one_one_real ) ).
thf(fact_3614_cos__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( number267125858f_real @ min ) @ ( cos @ X ) ) ).
thf(fact_3615_abs__cos__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos @ X ) ) @ one_one_real ) ).
thf(fact_3616_sin__periodic__pi2,axiom,
! [X: real] :
( ( sin @ ( plus_plus_real @ pi @ X ) )
= ( uminus_uminus_real @ ( sin @ X ) ) ) ).
thf(fact_3617_sin__periodic__pi,axiom,
! [X: real] :
( ( sin @ ( plus_plus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( sin @ X ) ) ) ).
thf(fact_3618_cos__pi,axiom,
( ( cos @ pi )
= ( number267125858f_real @ min ) ) ).
thf(fact_3619_cos__periodic__pi,axiom,
! [X: real] :
( ( cos @ ( plus_plus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( cos @ X ) ) ) ).
thf(fact_3620_sin__cos__squared__add2__mult,axiom,
! [R_1: real,A_1: real] :
( ( plus_plus_real @ ( power_power_real @ ( times_times_real @ R_1 @ ( cos @ A_1 ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( times_times_real @ R_1 @ ( sin @ A_1 ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( power_power_real @ R_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3621_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_3622_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_3623_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_3624_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_3625_cos__sin__eq,axiom,
! [X: real] :
( ( cos @ X )
= ( sin @ ( minus_minus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ X ) ) ) ).
thf(fact_3626_sin__cos__eq,axiom,
! [X: real] :
( ( sin @ X )
= ( cos @ ( minus_minus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ X ) ) ) ).
thf(fact_3627_real__sgn__pos,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( sgn_sgn_real @ X )
= one_one_real ) ) ).
thf(fact_3628_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_3629_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_3630_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_3631_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_3632_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_3633_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_3634_real__sgn__neg,axiom,
! [X: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( sgn_sgn_real @ X )
= ( number267125858f_real @ min ) ) ) ).
thf(fact_3635_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_3636_sgn__real__def,axiom,
! [A_1: real] :
( ( ( A_1 = zero_zero_real )
=> ( ( sgn_sgn_real @ A_1 )
= zero_zero_real ) )
& ( ( A_1 != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ A_1 )
=> ( ( sgn_sgn_real @ A_1 )
= one_one_real ) )
& ( ~ ( ord_less_real @ zero_zero_real @ A_1 )
=> ( ( sgn_sgn_real @ A_1 )
= ( uminus_uminus_real @ one_one_real ) ) ) ) ) ) ).
thf(fact_3637_cos__two__neq__zero,axiom,
( ( cos @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= zero_zero_real ) ).
thf(fact_3638_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_3639_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_3640_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_3641_cos__two__le__zero,axiom,
ord_less_eq_real @ ( cos @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_real ).
thf(fact_3642_cos__two__less__zero,axiom,
ord_less_real @ ( cos @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_real ).
thf(fact_3643_sin__two__pi,axiom,
( ( sin @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ) )
= zero_zero_real ) ).
thf(fact_3644_sin__periodic,axiom,
! [X: real] :
( ( sin @ ( plus_plus_real @ X @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ) ) )
= ( sin @ X ) ) ).
thf(fact_3645_sin__pi__half,axiom,
( ( sin @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= one_one_real ) ).
thf(fact_3646_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_3647_cos__pi__half,axiom,
( ( cos @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_real ) ).
thf(fact_3648_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_3649_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_3650_cos__two__pi,axiom,
( ( cos @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ) )
= one_one_real ) ).
thf(fact_3651_cos__periodic,axiom,
! [X: real] :
( ( cos @ ( plus_plus_real @ X @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ) ) )
= ( cos @ X ) ) ).
thf(fact_3652_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_3653_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_3654_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_3655_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_3656_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_3657_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_3658_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_3659_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_3660_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_3661_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_3662_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_3663_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_3664_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_3665_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_3666_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_3667_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_3668_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_3669_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_3670_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_3671_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_3672_cos__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( cos @ ( arccos @ Y ) )
= Y ) ) ).
thf(fact_3673_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_3674_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_3675_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_3676_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_3677_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_3678_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_3679_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_3680_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_3681_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_3682_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_3683_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_3684_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_3685_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_3686_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_3687_real__of__nat__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( real_nat @ N ) ) ).
thf(fact_3688_tan__minus,axiom,
! [X: real] :
( ( tan @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( tan @ X ) ) ) ).
thf(fact_3689_natceiling__real__of__nat,axiom,
! [N: nat] :
( ( natceiling @ ( real_nat @ N ) )
= N ) ).
thf(fact_3690_real__of__nat__inject,axiom,
! [N: nat,M: nat] :
( ( ( real_nat @ N )
= ( real_nat @ M ) )
<=> ( N = M ) ) ).
thf(fact_3691_tan__zero,axiom,
( ( tan @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_3692_abs__real__of__nat__cancel,axiom,
! [X: nat] :
( ( abs_abs_real @ ( real_nat @ X ) )
= ( real_nat @ X ) ) ).
thf(fact_3693_natfloor__real__of__nat,axiom,
! [N: nat] :
( ( natfloor @ ( real_nat @ N ) )
= N ) ).
thf(fact_3694_tan__arctan,axiom,
! [Y: real] :
( ( tan @ ( arctan @ Y ) )
= Y ) ).
thf(fact_3695_tan__npi,axiom,
! [N: nat] :
( ( tan @ ( times_times_real @ ( real_nat @ N ) @ pi ) )
= zero_zero_real ) ).
thf(fact_3696_tan__periodic__nat,axiom,
! [X: real,N: nat] :
( ( tan @ ( plus_plus_real @ X @ ( times_times_real @ ( real_nat @ N ) @ pi ) ) )
= ( tan @ X ) ) ).
thf(fact_3697_real__of__nat__def,axiom,
real_nat = semiri132038758t_real ).
thf(fact_3698_real__eq__of__nat,axiom,
real_nat = semiri132038758t_real ).
thf(fact_3699_real__of__nat__zero,axiom,
( ( real_nat @ zero_zero_nat )
= zero_zero_real ) ).
thf(fact_3700_real__of__nat__zero__iff,axiom,
! [N: nat] :
( ( ( real_nat @ N )
= zero_zero_real )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_3701_not__real__of__nat__less__zero,axiom,
! [N: nat] :
~ ( ord_less_real @ ( real_nat @ N ) @ zero_zero_real ) ).
thf(fact_3702_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_3703_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_3704_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_3705_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_3706_real__of__nat__1,axiom,
( ( real_nat @ one_one_nat )
= one_one_real ) ).
thf(fact_3707_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_3708_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_3709_real__natceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( real_nat @ ( natceiling @ X ) ) ) ).
thf(fact_3710_tan__pi,axiom,
( ( tan @ pi )
= zero_zero_real ) ).
thf(fact_3711_tan__periodic__pi,axiom,
! [X: real] :
( ( tan @ ( plus_plus_real @ X @ pi ) )
= ( tan @ X ) ) ).
thf(fact_3712_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_3713_real__of__nat__Suc__gt__zero,axiom,
! [N: nat] : ( ord_less_real @ zero_zero_real @ ( real_nat @ ( suc @ N ) ) ) ).
thf(fact_3714_real__of__nat__one,axiom,
( ( real_nat @ ( suc @ zero_zero_nat ) )
= one_one_real ) ).
thf(fact_3715_real__of__nat__Suc,axiom,
! [N: nat] :
( ( real_nat @ ( suc @ N ) )
= ( plus_plus_real @ ( real_nat @ N ) @ one_one_real ) ) ).
thf(fact_3716_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_3717_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_3718_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_3719_le__natfloor,axiom,
! [X: nat,A_1: real] :
( ( ord_less_eq_real @ ( real_nat @ X ) @ A_1 )
=> ( ord_less_eq_nat @ X @ ( natfloor @ A_1 ) ) ) ).
thf(fact_3720_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_3721_natceiling__le,axiom,
! [X: real,A_1: nat] :
( ( ord_less_eq_real @ X @ ( real_nat @ A_1 ) )
=> ( ord_less_eq_nat @ ( natceiling @ X ) @ A_1 ) ) ).
thf(fact_3722_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_3723_tan__def,axiom,
! [X: real] :
( ( tan @ X )
= ( inverse_divide_real @ ( sin @ X ) @ ( cos @ X ) ) ) ).
thf(fact_3724_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_3725_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_3726_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_3727_sin__npi,axiom,
! [N: nat] :
( ( sin @ ( times_times_real @ ( real_nat @ N ) @ pi ) )
= zero_zero_real ) ).
thf(fact_3728_sin__npi2,axiom,
! [N: nat] :
( ( sin @ ( times_times_real @ pi @ ( real_nat @ N ) ) )
= zero_zero_real ) ).
thf(fact_3729_le__natfloor__eq,axiom,
! [A_1: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_nat @ A_1 @ ( natfloor @ X ) )
<=> ( ord_less_eq_real @ ( real_nat @ A_1 ) @ X ) ) ) ).
thf(fact_3730_natceiling__le__eq,axiom,
! [A_1: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_nat @ ( natceiling @ X ) @ A_1 )
<=> ( ord_less_eq_real @ X @ ( real_nat @ A_1 ) ) ) ) ).
thf(fact_3731_real__natfloor__add__one__gt,axiom,
! [X: real] : ( ord_less_real @ X @ ( plus_plus_real @ ( real_nat @ ( natfloor @ X ) ) @ one_one_real ) ) ).
thf(fact_3732_natfloor__subtract,axiom,
! [A_1: nat,X: real] :
( ( ord_less_eq_real @ ( real_nat @ A_1 ) @ X )
=> ( ( natfloor @ ( minus_minus_real @ X @ ( real_nat @ A_1 ) ) )
= ( minus_minus_nat @ ( natfloor @ X ) @ A_1 ) ) ) ).
thf(fact_3733_real__natfloor__gt__diff__one,axiom,
! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ one_one_real ) @ ( real_nat @ ( natfloor @ X ) ) ) ).
thf(fact_3734_natceiling__subtract,axiom,
! [A_1: nat,X: real] :
( ( ord_less_eq_real @ ( real_nat @ A_1 ) @ X )
=> ( ( natceiling @ ( minus_minus_real @ X @ ( real_nat @ A_1 ) ) )
= ( minus_minus_nat @ ( natceiling @ X ) @ A_1 ) ) ) ).
thf(fact_3735_tan__periodic__n,axiom,
! [X: real,N: int] :
( ( tan @ ( plus_plus_real @ X @ ( times_times_real @ ( number267125858f_real @ N ) @ pi ) ) )
= ( tan @ X ) ) ).
thf(fact_3736_real__of__nat__div,axiom,
! [N: nat,D_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ D_1 )
=> ( ( dvd_dvd_nat @ D_1 @ N )
=> ( ( real_nat @ ( div_div_nat @ N @ D_1 ) )
= ( inverse_divide_real @ ( real_nat @ N ) @ ( real_nat @ D_1 ) ) ) ) ) ).
thf(fact_3737_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_3738_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_3739_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_3740_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_3741_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_3742_natfloor__add,axiom,
! [A_1: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( natfloor @ ( plus_plus_real @ X @ ( real_nat @ A_1 ) ) )
= ( plus_plus_nat @ ( natfloor @ X ) @ A_1 ) ) ) ).
thf(fact_3743_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_3744_log__nat__power,axiom,
! [B_1: real,N: nat,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ B_1 @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( real_nat @ N ) @ ( log @ B_1 @ X ) ) ) ) ).
thf(fact_3745_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_3746_natceiling__add,axiom,
! [A_1: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( natceiling @ ( plus_plus_real @ X @ ( real_nat @ A_1 ) ) )
= ( plus_plus_nat @ ( natceiling @ X ) @ A_1 ) ) ) ).
thf(fact_3747_two__realpow__gt,axiom,
! [N: nat] : ( ord_less_real @ ( real_nat @ N ) @ ( power_power_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) ).
thf(fact_3748_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_3749_real__of__nat__div__aux,axiom,
! [X: nat,D_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ D_1 )
=> ( ( inverse_divide_real @ ( real_nat @ X ) @ ( real_nat @ D_1 ) )
= ( plus_plus_real @ ( real_nat @ ( div_div_nat @ X @ D_1 ) ) @ ( inverse_divide_real @ ( real_nat @ ( div_mod_nat @ X @ D_1 ) ) @ ( real_nat @ D_1 ) ) ) ) ) ).
thf(fact_3750_cos__npi,axiom,
! [N: nat] :
( ( cos @ ( times_times_real @ ( real_nat @ N ) @ pi ) )
= ( power_power_real @ ( number267125858f_real @ min ) @ N ) ) ).
thf(fact_3751_cos__npi2,axiom,
! [N: nat] :
( ( cos @ ( times_times_real @ pi @ ( real_nat @ N ) ) )
= ( power_power_real @ ( number267125858f_real @ min ) @ N ) ) ).
thf(fact_3752_tan__60,axiom,
( ( tan @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit1 @ ( bit1 @ pls ) ) ) ) )
= ( sqrt @ ( number267125858f_real @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3753_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_3754_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_3755_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_3756_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_3757_tan__periodic,axiom,
! [X: real] :
( ( tan @ ( plus_plus_real @ X @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ) ) )
= ( tan @ X ) ) ).
thf(fact_3758_tan__45,axiom,
( ( tan @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
= one_one_real ) ).
thf(fact_3759_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_3760_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_3761_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_3762_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_3763_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_3764_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_3765_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_3766_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_3767_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_3768_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_3769_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_3770_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_3771_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_3772_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_3773_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_3774_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_3775_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_3776_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_3777_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_3778_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_3779_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_3780_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_3781_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_3782_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_3783_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_3784_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_3785_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_3786_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_3787_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_3788_even__zero__nat,axiom,
even_odd_even_nat @ zero_zero_nat ).
thf(fact_3789_even__Suc,axiom,
! [X: nat] :
( ( even_odd_even_nat @ ( suc @ X ) )
<=> ~ ( even_odd_even_nat @ X ) ) ).
thf(fact_3790_odd__1__nat,axiom,
~ ( even_odd_even_nat @ one_one_nat ) ).
thf(fact_3791_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_3792_real__of__nat__fact__not__zero,axiom,
! [N: nat] :
( ( real_nat @ ( fact_fact_nat @ N ) )
!= zero_zero_real ) ).
thf(fact_3793_odd__pos,axiom,
! [N: nat] :
( ~ ( even_odd_even_nat @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_3794_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_3795_even__nat__def,axiom,
! [X: nat] :
( ( even_odd_even_nat @ X )
<=> ( even_odd_even_int @ ( semiri1621563631at_int @ X ) ) ) ).
thf(fact_3796_Parity_Otransfer__int__nat__relations,axiom,
! [X: nat] :
( ( even_odd_even_int @ ( semiri1621563631at_int @ X ) )
<=> ( even_odd_even_nat @ X ) ) ).
thf(fact_3797_real__of__nat__fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( real_nat @ ( fact_fact_nat @ N ) ) ) ).
thf(fact_3798_real__of__nat__fact__gt__zero,axiom,
! [N: nat] : ( ord_less_real @ zero_zero_real @ ( real_nat @ ( fact_fact_nat @ N ) ) ) ).
thf(fact_3799_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_3800_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_3801_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_3802_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_3803_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_3804_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_3805_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_3806_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_3807_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_3808_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_3809_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_3810_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_3811_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_3812_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_3813_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_3814_even__dvd,axiom,
! [N: nat] :
( ( even_odd_even_nat @ N )
<=> ( dvd_dvd_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) ).
thf(fact_3815_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_3816_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_3817_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_3818_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_3819_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_3820_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_3821_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_3822_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_3823_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_3824_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_3825_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_3826_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_3827_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_3828_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_3829_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_3830_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_3831_fact__nonzero__nat,axiom,
! [N: nat] :
( ( fact_fact_nat @ N )
!= zero_zero_nat ) ).
thf(fact_3832_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_3833_fact__1__nat,axiom,
( ( fact_fact_nat @ one_one_nat )
= one_one_nat ) ).
thf(fact_3834_Fact_Ofact__0__nat,axiom,
( ( fact_fact_nat @ zero_zero_nat )
= ( suc @ zero_zero_nat ) ) ).
thf(fact_3835_fact__Suc__0__nat,axiom,
( ( fact_fact_nat @ ( suc @ zero_zero_nat ) )
= ( suc @ zero_zero_nat ) ) ).
thf(fact_3836_fact__gt__zero__nat,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( fact_fact_nat @ N ) ) ).
thf(fact_3837_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_3838_Fact_Ofact__Suc,axiom,
! [X: nat] :
( ( fact_fact_nat @ ( suc @ X ) )
= ( times_times_nat @ ( suc @ X ) @ ( fact_fact_nat @ X ) ) ) ).
thf(fact_3839_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_3840_fact__ge__one__nat,axiom,
! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( fact_fact_nat @ N ) ) ).
thf(fact_3841_fact__ge__Suc__0__nat,axiom,
! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( fact_fact_nat @ N ) ) ).
thf(fact_3842_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_3843_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_3844_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_3845_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_3846_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_3847_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_3848_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_3849_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_3850_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_3851_real__sqrt__inverse,axiom,
! [X: real] :
( ( sqrt @ ( inverse_inverse_real @ X ) )
= ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).
thf(fact_3852_INVERSE__ZERO,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_3853_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_3854_divide__real__def,axiom,
! [X: real,Y: real] :
( ( inverse_divide_real @ X @ Y )
= ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ).
thf(fact_3855_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_3856_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_3857_real__mult__inverse__left,axiom,
! [X: real] :
( ( X != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ X ) @ X )
= one_one_real ) ) ).
thf(fact_3858_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_3859_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_3860_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_3861_ln__inverse,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln @ ( inverse_inverse_real @ X ) )
= ( uminus_uminus_real @ ( ln @ X ) ) ) ) ).
thf(fact_3862_log__inverse,axiom,
! [X: real,A_1: real] :
( ( ord_less_real @ zero_zero_real @ A_1 )
=> ( ( A_1 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ A_1 @ ( inverse_inverse_real @ X ) )
= ( uminus_uminus_real @ ( log @ A_1 @ X ) ) ) ) ) ) ).
thf(fact_3863_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_3864_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_3865_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_3866_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_3867_DERIV__sin,axiom,
! [X: real] : ( deriv_real @ sin @ X @ ( cos @ X ) ) ).
thf(fact_3868_DERIV__cos,axiom,
! [X: real] : ( deriv_real @ cos @ X @ ( uminus_uminus_real @ ( sin @ X ) ) ) ).
thf(fact_3869_DERIV__ln,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( deriv_real @ ln @ X @ ( inverse_inverse_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_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_3872_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_3873_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_3874_DERIV__real__sqrt__generic,axiom,
! [D: real,X: real] :
( ( X != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ X )
=> ( D
= ( inverse_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
=> ( ( ( ord_less_real @ X @ zero_zero_real )
=> ( D
= ( inverse_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
=> ( deriv_real @ sqrt @ X @ D ) ) ) ) ).
thf(fact_3875_DERIV__const__average,axiom,
! [V: real > real,K: real,A_1: real,B_1: real] :
( ( A_1 != B_1 )
=> ( ! [X_1: real] : ( deriv_real @ V @ X_1 @ K )
=> ( ( V @ ( inverse_divide_real @ ( plus_plus_real @ A_1 @ B_1 ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( inverse_divide_real @ ( plus_plus_real @ ( V @ A_1 ) @ ( V @ B_1 ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3876_DERIV__local__max,axiom,
! [D_1: real,F: real > real,X: real,L: real] :
( ( deriv_real @ F @ X @ L )
=> ( ( ord_less_real @ zero_zero_real @ D_1 )
=> ( ! [Y_1: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y_1 ) ) @ D_1 )
=> ( ord_less_eq_real @ ( F @ Y_1 ) @ ( F @ X ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
thf(fact_3877_DERIV__local__min,axiom,
! [D_1: real,F: real > real,X: real,L: real] :
( ( deriv_real @ F @ X @ L )
=> ( ( ord_less_real @ zero_zero_real @ D_1 )
=> ( ! [Y_1: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y_1 ) ) @ D_1 )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y_1 ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
thf(fact_3878_DERIV__local__const,axiom,
! [D_1: real,F: real > real,X: real,L: real] :
( ( deriv_real @ F @ X @ L )
=> ( ( ord_less_real @ zero_zero_real @ D_1 )
=> ( ! [Y_1: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y_1 ) ) @ D_1 )
=> ( ( F @ X )
= ( F @ Y_1 ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
thf(fact_3879_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_3880_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_3881_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_3882_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_3883_DERIV__const__ratio__const2,axiom,
! [F: real > real,K: real,A_1: real,B_1: real] :
( ( A_1 != B_1 )
=> ( ! [X_1: real] : ( deriv_real @ F @ X_1 @ K )
=> ( ( inverse_divide_real @ ( minus_minus_real @ ( F @ B_1 ) @ ( F @ A_1 ) ) @ ( minus_minus_real @ B_1 @ A_1 ) )
= K ) ) ) ).
thf(fact_3884_DERIV__const__ratio__const,axiom,
! [F: real > real,K: real,A_1: real,B_1: real] :
( ( A_1 != B_1 )
=> ( ! [X_1: real] : ( deriv_real @ F @ X_1 @ K )
=> ( ( minus_minus_real @ ( F @ B_1 ) @ ( F @ A_1 ) )
= ( times_times_real @ ( minus_minus_real @ B_1 @ A_1 ) @ K ) ) ) ) ).
thf(fact_3885_DERIV__real__root__generic,axiom,
! [D: 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
= ( 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
= ( 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
= ( 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 ) ) ) ) ) ) ).
thf(fact_3886_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_3887_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_3888_MVT2,axiom,
! [F: real > real,F_1: real > real,A_1: real,B_1: real] :
( ( ord_less_real @ A_1 @ B_1 )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A_1 @ X_1 )
& ( ord_less_eq_real @ X_1 @ B_1 ) )
=> ( deriv_real @ F @ X_1 @ ( F_1 @ X_1 ) ) )
=> ? [Z: real] :
( ( ord_less_real @ A_1 @ Z )
& ( ord_less_real @ Z @ B_1 )
& ( ( minus_minus_real @ ( F @ B_1 ) @ ( F @ A_1 ) )
= ( times_times_real @ ( minus_minus_real @ B_1 @ A_1 ) @ ( F_1 @ Z ) ) ) ) ) ) ).
thf(fact_3889_real__root__zero,axiom,
! [N: nat] :
( ( root @ N @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_3890_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ zero_zero_nat ) @ X )
= X ) ).
thf(fact_3891_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_3892_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_3893_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_3894_odd__real__root__pow,axiom,
! [X: real,N: nat] :
( ~ ( even_odd_even_nat @ N )
=> ( ( power_power_real @ ( root @ N @ X ) @ N )
= X ) ) ).
thf(fact_3895_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_3896_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_3897_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_3898_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_3899_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_3900_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_3901_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_3902_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_3903_real__root__power,axiom,
! [X: real,K: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( power_power_real @ X @ K ) )
= ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).
thf(fact_3904_real__root__one,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ one_one_real )
= one_one_real ) ) ).
thf(fact_3905_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_3906_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_3907_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_3908_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_3909_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_3910_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_3911_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_3912_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_3913_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_3914_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_3915_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_3916_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_3917_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_3918_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_3919_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_3920_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_3921_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_3922_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_3923_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_3924_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_3925_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_3926_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_3927_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_3928_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_3929_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_3930_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_3931_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_3932_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_3933_sqrt__def,axiom,
( sqrt
= ( root @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3934_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_3935_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_3936_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_3937_reals__Archimedean2,axiom,
! [X: real] :
? [N_1: nat] : ( ord_less_real @ X @ ( real_nat @ N_1 ) ) ).
thf(fact_3938_Union__SetS__finite,axiom,
! [A_1: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( finite_finite_int @ ( comple1092985777_int_o @ ( setS @ A_1 @ P_3 ) ) ) ) ).
thf(fact_3939_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_3940_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_3941_wset__fin,axiom,
! [A_1: int,P_3: int] : ( finite_finite_int @ ( wset @ A_1 @ P_3 ) ) ).
thf(fact_3942_d22set__fin,axiom,
! [A_1: int] : ( finite_finite_int @ ( d22set @ A_1 ) ) ).
thf(fact_3943_SetS__elems__finite,axiom,
! [A_1: int,P_3: int,X_1: int > $o] :
( ( member_int_o @ X_1 @ ( setS @ A_1 @ P_3 ) )
=> ( finite_finite_int @ X_1 ) ) ).
thf(fact_3944_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_3945_SetS__finite,axiom,
! [A_1: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( finite_finite_int_o @ ( setS @ A_1 @ P_3 ) ) ) ).
thf(fact_3946_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( K
!= ( suc @ J_1 ) ) ) ) ).
thf(fact_3947_lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( K
!= ( suc @ J_1 ) ) ) ) ) ).
thf(fact_3948_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_3949_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_3950_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_3951_Complex__eq__0,axiom,
! [A_1: real,B_1: real] :
( ( ( complex_1 @ A_1 @ B_1 )
= zero_zero_complex )
<=> ( ( A_1 = zero_zero_real )
& ( B_1 = zero_zero_real ) ) ) ).
thf(fact_3952_complex__zero__def,axiom,
( zero_zero_complex
= ( complex_1 @ zero_zero_real @ zero_zero_real ) ) ).
thf(fact_3953_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_3954_Complex__eq__number__of,axiom,
! [A_1: real,B_1: real,W: int] :
( ( ( complex_1 @ A_1 @ B_1 )
= ( number528085621omplex @ W ) )
<=> ( ( A_1
= ( number267125858f_real @ W ) )
& ( B_1 = zero_zero_real ) ) ) ).
thf(fact_3955_Complex__eq__1,axiom,
! [A_1: real,B_1: real] :
( ( ( complex_1 @ A_1 @ B_1 )
= one_one_complex )
<=> ( ( A_1 = one_one_real )
& ( B_1 = zero_zero_real ) ) ) ).
thf(fact_3956_complex__one__def,axiom,
( one_one_complex
= ( complex_1 @ one_one_real @ zero_zero_real ) ) ).
thf(fact_3957_complex__add,axiom,
! [A_1: real,B_1: real,C: real,D_1: real] :
( ( plus_plus_complex @ ( complex_1 @ A_1 @ B_1 ) @ ( complex_1 @ C @ D_1 ) )
= ( complex_1 @ ( plus_plus_real @ A_1 @ C ) @ ( plus_plus_real @ B_1 @ D_1 ) ) ) ).
thf(fact_3958_complex__minus,axiom,
! [A_1: real,B_1: real] :
( ( uminus473333897omplex @ ( complex_1 @ A_1 @ B_1 ) )
= ( complex_1 @ ( uminus_uminus_real @ A_1 ) @ ( uminus_uminus_real @ B_1 ) ) ) ).
thf(fact_3959_complex__diff,axiom,
! [A_1: real,B_1: real,C: real,D_1: real] :
( ( minus_minus_complex @ ( complex_1 @ A_1 @ B_1 ) @ ( complex_1 @ C @ D_1 ) )
= ( complex_1 @ ( minus_minus_real @ A_1 @ C ) @ ( minus_minus_real @ B_1 @ D_1 ) ) ) ).
thf(fact_3960_complex__mult,axiom,
! [A_1: real,B_1: real,C: real,D_1: real] :
( ( times_times_complex @ ( complex_1 @ A_1 @ B_1 ) @ ( complex_1 @ C @ D_1 ) )
= ( complex_1 @ ( minus_minus_real @ ( times_times_real @ A_1 @ C ) @ ( times_times_real @ B_1 @ D_1 ) ) @ ( plus_plus_real @ ( times_times_real @ A_1 @ D_1 ) @ ( times_times_real @ B_1 @ C ) ) ) ) ).
thf(fact_3961_cmod__unit__one,axiom,
! [A_1: real] :
( ( norm_norm_complex @ ( complex_1 @ ( cos @ A_1 ) @ ( sin @ A_1 ) ) )
= one_one_real ) ).
thf(fact_3962_complex__inverse,axiom,
! [A_1: real,B_1: real] :
( ( invers1449016382omplex @ ( complex_1 @ A_1 @ B_1 ) )
= ( complex_1 @ ( inverse_divide_real @ A_1 @ ( plus_plus_real @ ( power_power_real @ A_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ B_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( inverse_divide_real @ ( uminus_uminus_real @ B_1 ) @ ( plus_plus_real @ ( power_power_real @ A_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ B_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ).
thf(fact_3963_complex_Osize_I1_J,axiom,
! [Real1: real,Real2: real] :
( ( complex_size @ ( complex_1 @ Real1 @ Real2 ) )
= zero_zero_nat ) ).
thf(fact_3964_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_3965_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_3966_complex__divide__def,axiom,
! [X: complex,Y: complex] :
( ( invers1025623611omplex @ X @ Y )
= ( times_times_complex @ X @ ( invers1449016382omplex @ Y ) ) ) ).
thf(fact_3967_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_3968_complex_Osize_I2_J,axiom,
! [Real1: real,Real2: real] :
( ( size_size_complex @ ( complex_1 @ Real1 @ Real2 ) )
= zero_zero_nat ) ).
thf(fact_3969_power2__i,axiom,
( ( power_power_complex @ ii @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( number528085621omplex @ min ) ) ).
thf(fact_3970_complex__i__not__zero,axiom,
ii != zero_zero_complex ).
thf(fact_3971_complex__i__not__one,axiom,
ii != one_one_complex ).
thf(fact_3972_complex__i__not__number__of,axiom,
! [W: int] :
( ii
!= ( number528085621omplex @ W ) ) ).
thf(fact_3973_complex__i__mult__minus,axiom,
! [X: complex] :
( ( times_times_complex @ ii @ ( times_times_complex @ ii @ X ) )
= ( uminus473333897omplex @ X ) ) ).
thf(fact_3974_inverse__i,axiom,
( ( invers1449016382omplex @ ii )
= ( uminus473333897omplex @ ii ) ) ).
thf(fact_3975_i__def,axiom,
( ii
= ( complex_1 @ zero_zero_real @ one_one_real ) ) ).
thf(fact_3976_Complex__eq__i,axiom,
! [X: real,Y: real] :
( ( ( complex_1 @ X @ Y )
= ii )
<=> ( ( X = zero_zero_real )
& ( Y = one_one_real ) ) ) ).
thf(fact_3977_Complex__mult__i,axiom,
! [A_1: real,B_1: real] :
( ( times_times_complex @ ( complex_1 @ A_1 @ B_1 ) @ ii )
= ( complex_1 @ ( uminus_uminus_real @ B_1 ) @ A_1 ) ) ).
thf(fact_3978_i__mult__Complex,axiom,
! [A_1: real,B_1: real] :
( ( times_times_complex @ ii @ ( complex_1 @ A_1 @ B_1 ) )
= ( complex_1 @ ( uminus_uminus_real @ B_1 ) @ A_1 ) ) ).
thf(fact_3979_i__squared,axiom,
( ( times_times_complex @ ii @ ii )
= ( number528085621omplex @ min ) ) ).
thf(fact_3980_i__mult__eq2,axiom,
( ( times_times_complex @ ii @ ii )
= ( uminus473333897omplex @ one_one_complex ) ) ).
thf(fact_3981_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_3982_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M_2: nat] :
( N
= ( suc @ M_2 ) ) ) ).
thf(fact_3983_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_3984_complex__of__real__def,axiom,
! [R_1: real] :
( ( of_real_complex @ R_1 )
= ( complex_1 @ R_1 @ zero_zero_real ) ) ).
thf(fact_3985_sgn__eq,axiom,
! [Z_1: complex] :
( ( sgn_sgn_complex @ Z_1 )
= ( invers1025623611omplex @ Z_1 @ ( of_real_complex @ ( norm_norm_complex @ Z_1 ) ) ) ) ).
thf(fact_3986_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_3987_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_3988_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_3989_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_3990_i__mult__eq,axiom,
( ( times_times_complex @ ii @ ii )
= ( of_real_complex @ ( number267125858f_real @ min ) ) ) ).
thf(fact_3991_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_3992_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_3993_complex__of__real__minus__one,axiom,
( ( of_real_complex @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus473333897omplex @ one_one_complex ) ) ).
thf(fact_3994_cmod__complex__polar,axiom,
! [R_1: real,A_1: real] :
( ( norm_norm_complex @ ( times_times_complex @ ( of_real_complex @ R_1 ) @ ( complex_1 @ ( cos @ A_1 ) @ ( sin @ A_1 ) ) ) )
= ( abs_abs_real @ R_1 ) ) ).
thf(fact_3995_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_3996_expi__add,axiom,
! [A_1: complex,B_1: complex] :
( ( expi @ ( plus_plus_complex @ A_1 @ B_1 ) )
= ( times_times_complex @ ( expi @ A_1 ) @ ( expi @ B_1 ) ) ) ).
thf(fact_3997_expi__zero,axiom,
( ( expi @ zero_zero_complex )
= one_one_complex ) ).
thf(fact_3998_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_3999_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_4000_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat_1: nat] :
( Y
!= ( suc @ Nat_1 ) ) ) ).
thf(fact_4001_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N_1: nat] :
( ( P @ ( suc @ N_1 ) )
=> ( P @ N_1 ) )
=> ( P @ zero_zero_nat ) ) ) ).
thf(fact_4002_DERIV__neg__imp__decreasing,axiom,
! [F: real > real,A_1: real,B_1: real] :
( ( ord_less_real @ A_1 @ B_1 )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A_1 @ X_1 )
& ( ord_less_eq_real @ X_1 @ B_1 ) )
=> ? [Y_1: real] :
( ( deriv_real @ F @ X_1 @ Y_1 )
& ( ord_less_real @ Y_1 @ zero_zero_real ) ) )
=> ( ord_less_real @ ( F @ B_1 ) @ ( F @ A_1 ) ) ) ) ).
thf(fact_4003_DERIV__pos__imp__increasing,axiom,
! [F: real > real,A_1: real,B_1: real] :
( ( ord_less_real @ A_1 @ B_1 )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A_1 @ X_1 )
& ( ord_less_eq_real @ X_1 @ B_1 ) )
=> ? [Y_1: real] :
( ( deriv_real @ F @ X_1 @ Y_1 )
& ( ord_less_real @ zero_zero_real @ Y_1 ) ) )
=> ( ord_less_real @ ( F @ A_1 ) @ ( F @ B_1 ) ) ) ) ).
thf(fact_4004_int__induct,axiom,
! [I: int,P: int > $o,K: int] :
( ( P @ K )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ K @ I_1 )
=> ( ( P @ I_1 )
=> ( P @ ( plus_plus_int @ I_1 @ one_one_int ) ) ) )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ I_1 @ K )
=> ( ( P @ I_1 )
=> ( P @ ( minus_minus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_4005_minusinfinity,axiom,
! [P: int > $o,P1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ! [X_1: int,K_1: int] :
( ( P1 @ X_1 )
<=> ( P1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K_1 @ D_1 ) ) ) )
=> ( ? [Z: int] :
! [X_1: int] :
( ( ord_less_int @ X_1 @ Z )
=> ( ( P @ X_1 )
<=> ( P1 @ X_1 ) ) )
=> ( ( ?? @ int @ P1 )
=> ( ?? @ int @ P ) ) ) ) ) ).
thf(fact_4006_plusinfinity,axiom,
! [P: int > $o,P_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ! [X_1: int,K_1: int] :
( ( P_1 @ X_1 )
<=> ( P_1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K_1 @ D_1 ) ) ) )
=> ( ? [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_4007_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_4008_coprime__exp,axiom,
! [N: nat,D_1: nat,A_1: nat] :
( ( coprime @ D_1 @ A_1 )
=> ( coprime @ D_1 @ ( power_power_nat @ A_1 @ N ) ) ) ).
thf(fact_4009_coprime__exp__imp,axiom,
! [N: nat,A_1: nat,B_1: nat] :
( ( coprime @ A_1 @ B_1 )
=> ( coprime @ ( power_power_nat @ A_1 @ N ) @ ( power_power_nat @ B_1 @ N ) ) ) ).
thf(fact_4010_coprime__commute,axiom,
! [A_1: nat,B_1: nat] :
( ( coprime @ A_1 @ B_1 )
<=> ( coprime @ B_1 @ A_1 ) ) ).
thf(fact_4011_coprime__divisors,axiom,
! [E_1: nat,B_1: nat,D_1: nat,A_1: nat] :
( ( dvd_dvd_nat @ D_1 @ A_1 )
=> ( ( dvd_dvd_nat @ E_1 @ B_1 )
=> ( ( coprime @ A_1 @ B_1 )
=> ( coprime @ D_1 @ E_1 ) ) ) ) ).
thf(fact_4012_coprime__lmul2,axiom,
! [D_1: nat,A_1: nat,B_1: nat] :
( ( coprime @ D_1 @ ( times_times_nat @ A_1 @ B_1 ) )
=> ( coprime @ D_1 @ B_1 ) ) ).
thf(fact_4013_coprime__rmul2,axiom,
! [D_1: nat,A_1: nat,B_1: nat] :
( ( coprime @ D_1 @ ( times_times_nat @ A_1 @ B_1 ) )
=> ( coprime @ D_1 @ A_1 ) ) ).
thf(fact_4014_coprime__mul,axiom,
! [B_1: nat,D_1: nat,A_1: nat] :
( ( coprime @ D_1 @ A_1 )
=> ( ( coprime @ D_1 @ B_1 )
=> ( coprime @ D_1 @ ( times_times_nat @ A_1 @ B_1 ) ) ) ) ).
thf(fact_4015_coprime__mul__eq,axiom,
! [D_1: nat,A_1: nat,B_1: nat] :
( ( coprime @ D_1 @ ( times_times_nat @ A_1 @ B_1 ) )
<=> ( ( coprime @ D_1 @ A_1 )
& ( coprime @ D_1 @ B_1 ) ) ) ).
thf(fact_4016_coprime__1_H,axiom,
! [A_1: nat] : ( coprime @ one_one_nat @ A_1 ) ).
thf(fact_4017_coprime__1,axiom,
! [A_1: nat] : ( coprime @ A_1 @ one_one_nat ) ).
thf(fact_4018_coprime__refl,axiom,
! [N: nat] :
( ( coprime @ N @ N )
<=> ( N = one_one_nat ) ) ).
thf(fact_4019_floor__real__of__nat,axiom,
! [N: nat] :
( ( archim1246769320r_real @ ( real_nat @ N ) )
= ( semiri1621563631at_int @ N ) ) ).
thf(fact_4020_natfloor__def,axiom,
! [X: real] :
( ( natfloor @ X )
= ( nat_1 @ ( archim1246769320r_real @ X ) ) ) ).
thf(fact_4021_coprime__Suc0,axiom,
! [A_1: nat] : ( coprime @ A_1 @ ( suc @ zero_zero_nat ) ) ).
thf(fact_4022_coprime__Suc0_H,axiom,
! [A_1: nat] : ( coprime @ ( suc @ zero_zero_nat ) @ A_1 ) ).
thf(fact_4023_coprime__0,axiom,
! [D_1: nat] :
( ( coprime @ D_1 @ zero_zero_nat )
<=> ( D_1 = one_one_nat ) ) ).
thf(fact_4024_coprime__0_H,axiom,
! [D_1: nat] :
( ( coprime @ zero_zero_nat @ D_1 )
<=> ( D_1 = one_one_nat ) ) ).
thf(fact_4025_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_4026_coprime__divprod,axiom,
! [D_1: nat,A_1: nat,B_1: nat] :
( ( dvd_dvd_nat @ D_1 @ ( times_times_nat @ A_1 @ B_1 ) )
=> ( ( coprime @ D_1 @ A_1 )
=> ( dvd_dvd_nat @ D_1 @ B_1 ) ) ) ).
thf(fact_4027_coprime,axiom,
! [A_1: nat,B_1: nat] :
( ( coprime @ A_1 @ B_1 )
<=> ! [D_2: nat] :
( ( ( dvd_dvd_nat @ D_2 @ A_1 )
& ( dvd_dvd_nat @ D_2 @ B_1 ) )
<=> ( D_2 = one_one_nat ) ) ) ).
thf(fact_4028_coprime__plus1,axiom,
! [N: nat] : ( coprime @ ( plus_plus_nat @ N @ one_one_nat ) @ N ) ).
thf(fact_4029_coprime__exp2,axiom,
! [A_1: nat,N: nat,B_1: nat] :
( ( coprime @ ( power_power_nat @ A_1 @ ( suc @ N ) ) @ ( power_power_nat @ B_1 @ ( suc @ N ) ) )
<=> ( coprime @ A_1 @ B_1 ) ) ).
thf(fact_4030_coprime__minus1,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( coprime @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
thf(fact_4031_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_4032_floor__minus__real__of__nat,axiom,
! [N: nat] :
( ( archim1246769320r_real @ ( uminus_uminus_real @ ( real_nat @ N ) ) )
= ( uminus_uminus_int @ ( semiri1621563631at_int @ N ) ) ) ).
thf(fact_4033_le__mult__floor,axiom,
! [B_1: real,A_1: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_1 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_1 )
=> ( ord_less_eq_int @ ( times_times_int @ ( archim1246769320r_real @ A_1 ) @ ( archim1246769320r_real @ B_1 ) ) @ ( archim1246769320r_real @ ( times_times_real @ A_1 @ B_1 ) ) ) ) ) ).
thf(fact_4034_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_4035_coprime__bezout,axiom,
! [A_1: nat,B_1: nat] :
( ( coprime @ A_1 @ B_1 )
<=> ? [X_1: nat,Y_1: nat] :
( ( ( minus_minus_nat @ ( times_times_nat @ A_1 @ X_1 ) @ ( times_times_nat @ B_1 @ Y_1 ) )
= one_one_nat )
| ( ( minus_minus_nat @ ( times_times_nat @ B_1 @ X_1 ) @ ( times_times_nat @ A_1 @ Y_1 ) )
= one_one_nat ) ) ) ).
thf(fact_4036_coprime__bezout__strong,axiom,
! [A_1: nat,B_1: nat] :
( ( coprime @ A_1 @ B_1 )
=> ( ( B_1 != one_one_nat )
=> ? [X_1: nat,Y_1: nat] :
( ( times_times_nat @ A_1 @ X_1 )
= ( plus_plus_nat @ ( times_times_nat @ B_1 @ Y_1 ) @ one_one_nat ) ) ) ) ).
thf(fact_4037_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_4038_succ__pred,axiom,
! [X: int] :
( ( succ @ ( pred @ X ) )
= X ) ).
thf(fact_4039_pred__Bit0,axiom,
! [K: int] :
( ( pred @ ( bit0 @ K ) )
= ( bit1 @ ( pred @ K ) ) ) ).
thf(fact_4040_pred__Bit1,axiom,
! [K: int] :
( ( pred @ ( bit1 @ K ) )
= ( bit0 @ K ) ) ).
thf(fact_4041_le__iff__pred__less,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ L )
<=> ( ord_less_int @ ( pred @ K ) @ L ) ) ).
thf(fact_4042_pred__Pls,axiom,
( ( pred @ pls )
= min ) ).
thf(fact_4043_pred__Min,axiom,
( ( pred @ min )
= ( bit0 @ min ) ) ).
thf(fact_4044_pred__def,axiom,
! [K: int] :
( ( pred @ K )
= ( minus_minus_int @ K @ one_one_int ) ) ).
thf(fact_4045_minus__Bit1,axiom,
! [K: int] :
( ( uminus_uminus_int @ ( bit1 @ K ) )
= ( bit1 @ ( pred @ ( uminus_uminus_int @ K ) ) ) ) ).
thf(fact_4046_add__Min__right,axiom,
! [K: int] :
( ( plus_plus_int @ K @ min )
= ( pred @ K ) ) ).
thf(fact_4047_add__Min,axiom,
! [K: int] :
( ( plus_plus_int @ min @ K )
= ( pred @ K ) ) ).
thf(fact_4048_diff__bin__simps_I8_J,axiom,
! [K: int,L: int] :
( ( minus_minus_int @ ( bit0 @ K ) @ ( bit1 @ L ) )
= ( bit1 @ ( minus_minus_int @ ( pred @ K ) @ L ) ) ) ).
thf(fact_4049_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_4050_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_4051_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_4052_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_4053_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_4054_ResSet__SRStar__prop,axiom,
! [P_3: int] : ( resSet @ P_3 @ ( sRStar @ P_3 ) ) ).
thf(fact_4055_nat_Osize_I2_J,axiom,
! [Nat: nat] :
( ( nat_size @ ( suc @ Nat ) )
= ( plus_plus_nat @ ( nat_size @ Nat ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_4056_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_4057_code__numeral_Osize_I1_J,axiom,
( ( code_c271388182l_size @ zero_z126310315umeral )
= zero_zero_nat ) ).
thf(fact_4058_nat__size,axiom,
! [N: nat] :
( ( size_size_nat @ N )
= N ) ).
thf(fact_4059_nat_Osize_I3_J,axiom,
( ( size_size_nat @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_4060_nat_Osize_I1_J,axiom,
( ( nat_size @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_4061_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_4062_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_4063_complex__cnj__complex__of__real,axiom,
! [X: real] :
( ( cnj @ ( of_real_complex @ X ) )
= ( of_real_complex @ X ) ) ).
thf(fact_4064_complex__cnj__divide,axiom,
! [X: complex,Y: complex] :
( ( cnj @ ( invers1025623611omplex @ X @ Y ) )
= ( invers1025623611omplex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
thf(fact_4065_complex__cnj__power,axiom,
! [X: complex,N: nat] :
( ( cnj @ ( power_power_complex @ X @ N ) )
= ( power_power_complex @ ( cnj @ X ) @ N ) ) ).
thf(fact_4066_cnj_Oadd,axiom,
! [X: complex,Y: complex] :
( ( cnj @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
thf(fact_4067_cnj_Ominus,axiom,
! [X: complex] :
( ( cnj @ ( uminus473333897omplex @ X ) )
= ( uminus473333897omplex @ ( cnj @ X ) ) ) ).
thf(fact_4068_complex__cnj__cnj,axiom,
! [Z_1: complex] :
( ( cnj @ ( cnj @ Z_1 ) )
= Z_1 ) ).
thf(fact_4069_complex__cnj__cancel__iff,axiom,
! [X: complex,Y: complex] :
( ( ( cnj @ X )
= ( cnj @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_4070_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_4071_complex__cnj__of__nat,axiom,
! [N: nat] :
( ( cnj @ ( semiri2020571505omplex @ N ) )
= ( semiri2020571505omplex @ N ) ) ).
thf(fact_4072_complex__mod__cnj,axiom,
! [Z_1: complex] :
( ( norm_norm_complex @ ( cnj @ Z_1 ) )
= ( norm_norm_complex @ Z_1 ) ) ).
thf(fact_4073_cnj_Odiff,axiom,
! [X: complex,Y: complex] :
( ( cnj @ ( minus_minus_complex @ X @ Y ) )
= ( minus_minus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
thf(fact_4074_complex__cnj__inverse,axiom,
! [X: complex] :
( ( cnj @ ( invers1449016382omplex @ X ) )
= ( invers1449016382omplex @ ( cnj @ X ) ) ) ).
thf(fact_4075_cnj_Ozero,axiom,
( ( cnj @ zero_zero_complex )
= zero_zero_complex ) ).
thf(fact_4076_complex__cnj__zero__iff,axiom,
! [Z_1: complex] :
( ( ( cnj @ Z_1 )
= zero_zero_complex )
<=> ( Z_1 = zero_zero_complex ) ) ).
thf(fact_4077_complex__cnj__number__of,axiom,
! [W: int] :
( ( cnj @ ( number528085621omplex @ W ) )
= ( number528085621omplex @ W ) ) ).
thf(fact_4078_complex__cnj__mult,axiom,
! [X: complex,Y: complex] :
( ( cnj @ ( times_times_complex @ X @ Y ) )
= ( times_times_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
thf(fact_4079_complex__cnj__one,axiom,
( ( cnj @ one_one_complex )
= one_one_complex ) ).
thf(fact_4080_code__numeral_Osimps_I3_J,axiom,
! [Code_numeral_3: code_code_numeral] :
( ( code_S1047413653umeral @ Code_numeral_3 )
!= zero_z126310315umeral ) ).
thf(fact_4081_code__numeral_Osimps_I2_J,axiom,
! [Code_numeral_2: code_code_numeral] :
( zero_z126310315umeral
!= ( code_S1047413653umeral @ Code_numeral_2 ) ) ).
thf(fact_4082_complex__cnj,axiom,
! [A_1: real,B_1: real] :
( ( cnj @ ( complex_1 @ A_1 @ B_1 ) )
= ( complex_1 @ A_1 @ ( uminus_uminus_real @ B_1 ) ) ) ).
thf(fact_4083_complex__cnj__i,axiom,
( ( cnj @ ii )
= ( uminus473333897omplex @ ii ) ) ).
thf(fact_4084_Suc__code__numeral__minus__one,axiom,
! [N: code_code_numeral] :
( ( minus_1690775515umeral @ ( code_S1047413653umeral @ N ) @ one_on1645066479umeral )
= N ) ).
thf(fact_4085_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_4086_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_4087_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_4088_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_4089_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_4090_code__numeral_Oexhaust,axiom,
! [Y: code_code_numeral] :
( ( Y != zero_z126310315umeral )
=> ~ ! [Code_numeral: code_code_numeral] :
( Y
!= ( code_S1047413653umeral @ Code_numeral ) ) ) ).
thf(fact_4091_Im,axiom,
! [X: real,Y: real] :
( ( im @ ( complex_1 @ X @ Y ) )
= Y ) ).
thf(fact_4092_complex__Im__cnj,axiom,
! [X: complex] :
( ( im @ ( cnj @ X ) )
= ( uminus_uminus_real @ ( im @ X ) ) ) ).
thf(fact_4093_Im__complex__of__real,axiom,
! [Z_1: real] :
( ( im @ ( of_real_complex @ Z_1 ) )
= zero_zero_real ) ).
thf(fact_4094_complex__Im__i,axiom,
( ( im @ ii )
= one_one_real ) ).
thf(fact_4095_Im_Ozero,axiom,
( ( im @ zero_zero_complex )
= zero_zero_real ) ).
thf(fact_4096_complex__Im__one,axiom,
( ( im @ one_one_complex )
= zero_zero_real ) ).
thf(fact_4097_Im_Oadd,axiom,
! [X: complex,Y: complex] :
( ( im @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).
thf(fact_4098_complex__Im__number__of,axiom,
! [V: int] :
( ( im @ ( number528085621omplex @ V ) )
= zero_zero_real ) ).
thf(fact_4099_Im_Ominus,axiom,
! [X: complex] :
( ( im @ ( uminus473333897omplex @ X ) )
= ( uminus_uminus_real @ ( im @ X ) ) ) ).
thf(fact_4100_Im_Odiff,axiom,
! [X: complex,Y: complex] :
( ( im @ ( minus_minus_complex @ X @ Y ) )
= ( minus_minus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).
thf(fact_4101_complex__Im__of__nat,axiom,
! [N: nat] :
( ( im @ ( semiri2020571505omplex @ N ) )
= zero_zero_real ) ).
thf(fact_4102_abs__Im__le__cmod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( norm_norm_complex @ X ) ) ).
thf(fact_4103_complex__In__mult__cnj__zero,axiom,
! [Z_1: complex] :
( ( im @ ( times_times_complex @ Z_1 @ ( cnj @ Z_1 ) ) )
= zero_zero_real ) ).
thf(fact_4104_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_4105_code__numeral_Osize_I3_J,axiom,
( ( size_s945831648umeral @ zero_z126310315umeral )
= zero_zero_nat ) ).
thf(fact_4106_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_4107_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_4108_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_4109_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_4110_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_4111_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_4112_Re__complex__of__real,axiom,
! [Z_1: real] :
( ( re @ ( of_real_complex @ Z_1 ) )
= Z_1 ) ).
thf(fact_4113_Re,axiom,
! [X: real,Y: real] :
( ( re @ ( complex_1 @ X @ Y ) )
= X ) ).
thf(fact_4114_complex__Re__cnj,axiom,
! [X: complex] :
( ( re @ ( cnj @ X ) )
= ( re @ X ) ) ).
thf(fact_4115_complex__eqI,axiom,
! [X: complex,Y: complex] :
( ( ( re @ X )
= ( re @ Y ) )
=> ( ( ( im @ X )
= ( im @ Y ) )
=> ( X = Y ) ) ) ).
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__Re__le__cmod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( re @ X ) @ ( norm_norm_complex @ X ) ) ).
thf(fact_4118_complex__Re__i,axiom,
( ( re @ ii )
= zero_zero_real ) ).
thf(fact_4119_complex__surj,axiom,
! [Z_1: complex] :
( ( complex_1 @ ( re @ Z_1 ) @ ( im @ Z_1 ) )
= Z_1 ) ).
thf(fact_4120_Re_Ozero,axiom,
( ( re @ zero_zero_complex )
= zero_zero_real ) ).
thf(fact_4121_Re_Oadd,axiom,
! [X: complex,Y: complex] :
( ( re @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).
thf(fact_4122_complex__Re__one,axiom,
( ( re @ one_one_complex )
= one_one_real ) ).
thf(fact_4123_Re_Ominus,axiom,
! [X: complex] :
( ( re @ ( uminus473333897omplex @ X ) )
= ( uminus_uminus_real @ ( re @ X ) ) ) ).
thf(fact_4124_complex__Re__number__of,axiom,
! [V: int] :
( ( re @ ( number528085621omplex @ V ) )
= ( number267125858f_real @ V ) ) ).
thf(fact_4125_Re_Odiff,axiom,
! [X: complex,Y: complex] :
( ( re @ ( minus_minus_complex @ X @ Y ) )
= ( minus_minus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).
thf(fact_4126_complex__Re__of__nat,axiom,
! [N: nat] :
( ( re @ ( semiri2020571505omplex @ N ) )
= ( semiri132038758t_real @ N ) ) ).
thf(fact_4127_abs__Re__le__cmod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( norm_norm_complex @ X ) ) ).
thf(fact_4128_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_4129_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_4130_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_4131_cnj__def,axiom,
! [Z_1: complex] :
( ( cnj @ Z_1 )
= ( complex_1 @ ( re @ Z_1 ) @ ( uminus_uminus_real @ ( im @ Z_1 ) ) ) ) ).
thf(fact_4132_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_4133_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_4134_complex__minus__def,axiom,
! [X: complex] :
( ( uminus473333897omplex @ X )
= ( complex_1 @ ( uminus_uminus_real @ ( re @ X ) ) @ ( uminus_uminus_real @ ( im @ X ) ) ) ) ).
thf(fact_4135_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_4136_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_4137_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_4138_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_4139_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_4140_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_4141_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_4142_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_4143_sin__n__Im__cis__pow__n,axiom,
! [N: nat,A_1: real] :
( ( sin @ ( times_times_real @ ( real_nat @ N ) @ A_1 ) )
= ( im @ ( power_power_complex @ ( cis @ A_1 ) @ N ) ) ) ).
thf(fact_4144_cos__n__Re__cis__pow__n,axiom,
! [N: nat,A_1: real] :
( ( cos @ ( times_times_real @ ( real_nat @ N ) @ A_1 ) )
= ( re @ ( power_power_complex @ ( cis @ A_1 ) @ N ) ) ) ).
thf(fact_4145_Re__cis,axiom,
! [A_1: real] :
( ( re @ ( cis @ A_1 ) )
= ( cos @ A_1 ) ) ).
thf(fact_4146_Im__cis,axiom,
! [A_1: real] :
( ( im @ ( cis @ A_1 ) )
= ( sin @ A_1 ) ) ).
thf(fact_4147_cis__mult,axiom,
! [A_1: real,B_1: real] :
( ( times_times_complex @ ( cis @ A_1 ) @ ( cis @ B_1 ) )
= ( cis @ ( plus_plus_real @ A_1 @ B_1 ) ) ) ).
thf(fact_4148_cis__zero,axiom,
( ( cis @ zero_zero_real )
= one_one_complex ) ).
thf(fact_4149_cis__inverse,axiom,
! [A_1: real] :
( ( invers1449016382omplex @ ( cis @ A_1 ) )
= ( cis @ ( uminus_uminus_real @ A_1 ) ) ) ).
thf(fact_4150_cis__divide,axiom,
! [A_1: real,B_1: real] :
( ( invers1025623611omplex @ ( cis @ A_1 ) @ ( cis @ B_1 ) )
= ( cis @ ( minus_minus_real @ A_1 @ B_1 ) ) ) ).
thf(fact_4151_cis__def,axiom,
! [A_1: real] :
( ( cis @ A_1 )
= ( complex_1 @ ( cos @ A_1 ) @ ( sin @ A_1 ) ) ) ).
thf(fact_4152_DeMoivre,axiom,
! [A_1: real,N: nat] :
( ( power_power_complex @ ( cis @ A_1 ) @ N )
= ( cis @ ( times_times_real @ ( real_nat @ N ) @ A_1 ) ) ) ).
thf(fact_4153_cis__real__of__nat__Suc__mult,axiom,
! [N: nat,A_1: real] :
( ( cis @ ( times_times_real @ ( real_nat @ ( suc @ N ) ) @ A_1 ) )
= ( times_times_complex @ ( cis @ A_1 ) @ ( cis @ ( times_times_real @ ( real_nat @ N ) @ A_1 ) ) ) ) ).
thf(fact_4154_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
thf(fact_4155_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_4156_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_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_I5_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
thf(fact_4159_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_4160_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_4161_divmod__int__def,axiom,
! [B_1: int,A_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ A_1 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ B_1 )
=> ( ( divmod_int @ A_1 @ B_1 )
= ( posDivAlg @ A_1 @ B_1 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ B_1 )
=> ( ( ( A_1 = zero_zero_int )
=> ( ( divmod_int @ A_1 @ B_1 )
= ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) )
& ( ( A_1 != zero_zero_int )
=> ( ( divmod_int @ A_1 @ B_1 )
= ( negateSnd @ ( negDivAlg @ ( uminus_uminus_int @ A_1 ) @ ( uminus_uminus_int @ B_1 ) ) ) ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ A_1 )
=> ( ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( divmod_int @ A_1 @ B_1 )
= ( negDivAlg @ A_1 @ B_1 ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( divmod_int @ A_1 @ B_1 )
= ( negateSnd @ ( posDivAlg @ ( uminus_uminus_int @ A_1 ) @ ( uminus_uminus_int @ B_1 ) ) ) ) ) ) ) ) ).
thf(fact_4162_primes__eq,axiom,
! [Q: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( prime @ Q )
=> ( ( dvd_dvd_nat @ P_3 @ Q )
=> ( P_3 = Q ) ) ) ) ).
thf(fact_4163_prime__1,axiom,
~ ( prime @ one_one_nat ) ).
thf(fact_4164_prime__0,axiom,
~ ( prime @ zero_zero_nat ) ).
thf(fact_4165_distinct__prime__coprime,axiom,
! [Q: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( prime @ Q )
=> ( ( P_3 != Q )
=> ( coprime @ P_3 @ Q ) ) ) ) ).
thf(fact_4166_prime__Suc0,axiom,
~ ( prime @ ( suc @ zero_zero_nat ) ) ).
thf(fact_4167_prime__g__zero,axiom,
! [P_3: nat] :
( ( prime @ P_3 )
=> ( ord_less_nat @ zero_zero_nat @ P_3 ) ) ).
thf(fact_4168_prime__divprod__eq,axiom,
! [A_1: nat,B_1: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( dvd_dvd_nat @ P_3 @ ( times_times_nat @ A_1 @ B_1 ) )
<=> ( ( dvd_dvd_nat @ P_3 @ A_1 )
| ( dvd_dvd_nat @ P_3 @ B_1 ) ) ) ) ).
thf(fact_4169_prime__divprod,axiom,
! [A_1: nat,B_1: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( dvd_dvd_nat @ P_3 @ ( times_times_nat @ A_1 @ B_1 ) )
=> ( ( dvd_dvd_nat @ P_3 @ A_1 )
| ( dvd_dvd_nat @ P_3 @ B_1 ) ) ) ) ).
thf(fact_4170_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_4171_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_4172_prime__dvd__power,axiom,
! [A_1: nat,N: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( dvd_dvd_nat @ P_3 @ ( power_power_nat @ A_1 @ N ) )
=> ( dvd_dvd_nat @ P_3 @ A_1 ) ) ) ).
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__exp,axiom,
! [P_3: nat,N: nat] :
( ( prime @ ( power_power_nat @ P_3 @ N ) )
<=> ( ( prime @ P_3 )
& ( N = one_one_nat ) ) ) ).
thf(fact_4176_prime__impl__zprime__int,axiom,
! [A_1: nat] :
( ( prime @ A_1 )
=> ( zprime @ ( semiri1621563631at_int @ A_1 ) ) ) ).
thf(fact_4177_coprime__prime,axiom,
! [P_3: nat,A_1: nat,B_1: nat] :
( ( coprime @ A_1 @ B_1 )
=> ~ ( ( prime @ P_3 )
& ( dvd_dvd_nat @ P_3 @ A_1 )
& ( dvd_dvd_nat @ P_3 @ B_1 ) ) ) ).
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_1: nat,B_1: nat] :
( ( coprime @ A_1 @ B_1 )
<=> ! [P_4: nat] :
~ ( ( prime @ P_4 )
& ( dvd_dvd_nat @ P_4 @ A_1 )
& ( dvd_dvd_nat @ P_4 @ B_1 ) ) ) ).
thf(fact_4180_prime__g__one,axiom,
! [P_3: nat] :
( ( prime @ P_3 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ P_3 ) ) ).
thf(fact_4181_prime__nd__one,axiom,
! [P_3: nat] :
( ( prime @ P_3 )
=> ~ ( dvd_dvd_nat @ P_3 @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_4182_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_4183_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_4184_divides__primepow,axiom,
! [D_1: nat,K: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( dvd_dvd_nat @ D_1 @ ( power_power_nat @ P_3 @ K ) )
<=> ? [I_1: nat] :
( ( ord_less_eq_nat @ I_1 @ K )
& ( D_1
= ( power_power_nat @ P_3 @ I_1 ) ) ) ) ) ).
thf(fact_4185_prime__power__dvd__cancel__right,axiom,
! [N: nat,A_1: nat,B_1: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ~ ( dvd_dvd_nat @ P_3 @ B_1 )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ P_3 @ N ) @ ( times_times_nat @ A_1 @ B_1 ) )
=> ( dvd_dvd_nat @ ( power_power_nat @ P_3 @ N ) @ A_1 ) ) ) ) ).
thf(fact_4186_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_4187_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_4188_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_4189_divmod__int__rel__neg,axiom,
! [A_1: int,B_1: int,Qr: product_prod_int_int] :
( ( divmod_int_rel @ ( uminus_uminus_int @ A_1 ) @ ( uminus_uminus_int @ B_1 ) @ Qr )
=> ( divmod_int_rel @ A_1 @ B_1 @ ( negateSnd @ Qr ) ) ) ).
thf(fact_4190_two__is__prime,axiom,
prime @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_4191_divmod__int__correct,axiom,
! [A_1: int,B_1: int] :
( ( B_1 != zero_zero_int )
=> ( divmod_int_rel @ A_1 @ B_1 @ ( divmod_int @ A_1 @ B_1 ) ) ) ).
thf(fact_4192_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_4193_prime__divprod__pow,axiom,
! [N: nat,A_1: nat,B_1: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( coprime @ A_1 @ B_1 )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ P_3 @ N ) @ ( times_times_nat @ A_1 @ B_1 ) )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ P_3 @ N ) @ A_1 )
| ( dvd_dvd_nat @ ( power_power_nat @ P_3 @ N ) @ B_1 ) ) ) ) ) ).
thf(fact_4194_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_4195_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_4196_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_4197_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_4198_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_4199_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_4200_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_4201_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_4202_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_4203_prime__power__exp,axiom,
! [X: nat,K: nat,N: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( N != zero_zero_nat )
=> ( ( ( power_power_nat @ X @ N )
= ( power_power_nat @ P_3 @ K ) )
=> ? [I_1: nat] :
( X
= ( power_power_nat @ P_3 @ I_1 ) ) ) ) ) ).
thf(fact_4204_prime__factor,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ? [P_4: nat] :
( ( prime @ P_4 )
& ( dvd_dvd_nat @ P_4 @ N ) ) ) ).
thf(fact_4205_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_4206_bezout__prime,axiom,
! [A_1: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ~ ( dvd_dvd_nat @ P_3 @ A_1 )
=> ? [X_1: nat,Y_1: nat] :
( ( times_times_nat @ A_1 @ X_1 )
= ( plus_plus_nat @ ( times_times_nat @ P_3 @ Y_1 ) @ one_one_nat ) ) ) ) ).
thf(fact_4207_not__prime__ex__mk,axiom,
! [N: nat] :
( ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
& ~ ( prime @ N ) )
=> ? [M_2: nat,K_1: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M_2 )
& ( ord_less_nat @ ( suc @ zero_zero_nat ) @ K_1 )
& ( ord_less_nat @ M_2 @ N )
& ( ord_less_nat @ K_1 @ N )
& ( N
= ( times_times_nat @ M_2 @ K_1 ) ) ) ) ).
thf(fact_4208_divmod__nat__zero,axiom,
! [M: nat] :
( ( divmod_nat @ M @ zero_zero_nat )
= ( product_Pair_nat_nat @ zero_zero_nat @ M ) ) ).
thf(fact_4209_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_4210_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_4211_divmod__nat__rel__mult2__eq,axiom,
! [C: nat,A_1: nat,B_1: nat,Q: nat,R_1: nat] :
( ( divmod_nat_rel @ A_1 @ B_1 @ ( product_Pair_nat_nat @ Q @ R_1 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ B_1 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( divmod_nat_rel @ A_1 @ ( times_times_nat @ B_1 @ C ) @ ( product_Pair_nat_nat @ ( div_div_nat @ Q @ C ) @ ( plus_plus_nat @ ( times_times_nat @ B_1 @ ( div_mod_nat @ Q @ C ) ) @ R_1 ) ) ) ) ) ) ).
thf(fact_4212_divmod__nat__rel__mult1__eq,axiom,
! [A_1: nat,B_1: nat,C: nat,Q: nat,R_1: nat] :
( ( divmod_nat_rel @ B_1 @ C @ ( product_Pair_nat_nat @ Q @ R_1 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( divmod_nat_rel @ ( times_times_nat @ A_1 @ B_1 ) @ C @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ A_1 @ Q ) @ ( div_div_nat @ ( times_times_nat @ A_1 @ R_1 ) @ C ) ) @ ( div_mod_nat @ ( times_times_nat @ A_1 @ R_1 ) @ C ) ) ) ) ) ).
thf(fact_4213_divmod__nat__rel__add1__eq,axiom,
! [B_1: nat,Bq: nat,Br: nat,A_1: nat,C: nat,Aq: nat,Ar: nat] :
( ( divmod_nat_rel @ A_1 @ C @ ( product_Pair_nat_nat @ Aq @ Ar ) )
=> ( ( divmod_nat_rel @ B_1 @ C @ ( product_Pair_nat_nat @ Bq @ Br ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( divmod_nat_rel @ ( plus_plus_nat @ A_1 @ B_1 ) @ 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_4214_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_4215_divmod__nat__rel__divmod__nat,axiom,
! [M: nat,N: nat] : ( divmod_nat_rel @ M @ N @ ( divmod_nat @ M @ N ) ) ).
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_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_4218_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_4219_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_4220_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_4221_euclid,axiom,
! [N: nat] :
? [P_4: nat] :
( ( prime @ P_4 )
& ( ord_less_nat @ N @ P_4 ) ) ).
thf(fact_4222_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_4223_div__int__def,axiom,
! [A_1: int,B_1: int] :
( ( div_div_int @ A_1 @ B_1 )
= ( product_fst_int_int @ ( divmod_int @ A_1 @ B_1 ) ) ) ).
thf(fact_4224_div__neg__pos,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_int @ A_1 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( div_div_int @ A_1 @ B_1 )
= ( product_fst_int_int @ ( negDivAlg @ A_1 @ B_1 ) ) ) ) ) ).
thf(fact_4225_div__pos__pos,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_int @ zero_zero_int @ A_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_1 )
=> ( ( div_div_int @ A_1 @ B_1 )
= ( product_fst_int_int @ ( posDivAlg @ A_1 @ B_1 ) ) ) ) ) ).
thf(fact_4226_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_4227_div__pos__neg,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_int @ zero_zero_int @ A_1 )
=> ( ( ord_less_int @ B_1 @ zero_zero_int )
=> ( ( div_div_int @ A_1 @ B_1 )
= ( product_fst_int_int @ ( negateSnd @ ( negDivAlg @ ( uminus_uminus_int @ A_1 ) @ ( uminus_uminus_int @ B_1 ) ) ) ) ) ) ) ).
thf(fact_4228_div__neg__neg,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_int @ A_1 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B_1 @ zero_zero_int )
=> ( ( div_div_int @ A_1 @ B_1 )
= ( product_fst_int_int @ ( negateSnd @ ( posDivAlg @ ( uminus_uminus_int @ A_1 ) @ ( uminus_uminus_int @ B_1 ) ) ) ) ) ) ) ).
thf(fact_4229_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_4230_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_4231_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_4232_mod__int__def,axiom,
! [A_1: int,B_1: int] :
( ( div_mod_int @ A_1 @ B_1 )
= ( product_snd_int_int @ ( divmod_int @ A_1 @ B_1 ) ) ) ).
thf(fact_4233_mod__neg__pos,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_int @ A_1 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( div_mod_int @ A_1 @ B_1 )
= ( product_snd_int_int @ ( negDivAlg @ A_1 @ B_1 ) ) ) ) ) ).
thf(fact_4234_mod__pos__pos,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_int @ zero_zero_int @ A_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_1 )
=> ( ( div_mod_int @ A_1 @ B_1 )
= ( product_snd_int_int @ ( posDivAlg @ A_1 @ B_1 ) ) ) ) ) ).
thf(fact_4235_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_4236_mod__pos__neg,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_int @ zero_zero_int @ A_1 )
=> ( ( ord_less_int @ B_1 @ zero_zero_int )
=> ( ( div_mod_int @ A_1 @ B_1 )
= ( product_snd_int_int @ ( negateSnd @ ( negDivAlg @ ( uminus_uminus_int @ A_1 ) @ ( uminus_uminus_int @ B_1 ) ) ) ) ) ) ) ).
thf(fact_4237_mod__neg__neg,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_int @ A_1 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B_1 @ zero_zero_int )
=> ( ( div_mod_int @ A_1 @ B_1 )
= ( product_snd_int_int @ ( negateSnd @ ( posDivAlg @ ( uminus_uminus_int @ A_1 ) @ ( uminus_uminus_int @ B_1 ) ) ) ) ) ) ) ).
thf(fact_4238_div__nat__def,axiom,
! [M: nat,N: nat] :
( ( div_div_nat @ M @ N )
= ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) ) ) ).
thf(fact_4239_mod__nat__def,axiom,
! [M: nat,N: nat] :
( ( div_mod_nat @ M @ N )
= ( product_snd_nat_nat @ ( divmod_nat @ M @ N ) ) ) ).
thf(fact_4240_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_4241_Bolzano__bisect__diff,axiom,
! [P: produc914805421l_real > $o,N: nat,A_1: real,B_1: real] :
( ( ord_less_eq_real @ A_1 @ B_1 )
=> ( ( minus_minus_real @ ( produc556554744l_real @ ( bolzano_bisect @ P @ A_1 @ B_1 @ N ) ) @ ( produc1935615926l_real @ ( bolzano_bisect @ P @ A_1 @ B_1 @ N ) ) )
= ( inverse_divide_real @ ( minus_minus_real @ B_1 @ A_1 ) @ ( power_power_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) ) ) ).
thf(fact_4242_Bolzano__bisect__le,axiom,
! [P: produc914805421l_real > $o,A_1: real,B_1: real] :
( ( ord_less_eq_real @ A_1 @ B_1 )
=> ! [N_1: nat] : ( ord_less_eq_real @ ( produc1935615926l_real @ ( bolzano_bisect @ P @ A_1 @ B_1 @ N_1 ) ) @ ( produc556554744l_real @ ( bolzano_bisect @ P @ A_1 @ B_1 @ N_1 ) ) ) ) ).
thf(fact_4243_Bolzano__bisect_Osimps_I1_J,axiom,
! [P: produc914805421l_real > $o,A_1: real,B_1: real] :
( ( bolzano_bisect @ P @ A_1 @ B_1 @ zero_zero_nat )
= ( produc865579683l_real @ A_1 @ B_1 ) ) ).
thf(fact_4244_Bolzano__bisect__fst__le__Suc,axiom,
! [P: produc914805421l_real > $o,A_1: real,B_1: real] :
( ( ord_less_eq_real @ A_1 @ B_1 )
=> ! [N_1: nat] : ( ord_less_eq_real @ ( produc1935615926l_real @ ( bolzano_bisect @ P @ A_1 @ B_1 @ N_1 ) ) @ ( produc1935615926l_real @ ( bolzano_bisect @ P @ A_1 @ B_1 @ ( suc @ N_1 ) ) ) ) ) ).
thf(fact_4245_Bolzano__bisect__Suc__le__snd,axiom,
! [P: produc914805421l_real > $o,A_1: real,B_1: real] :
( ( ord_less_eq_real @ A_1 @ B_1 )
=> ! [N_1: nat] : ( ord_less_eq_real @ ( produc556554744l_real @ ( bolzano_bisect @ P @ A_1 @ B_1 @ ( suc @ N_1 ) ) ) @ ( produc556554744l_real @ ( bolzano_bisect @ P @ A_1 @ B_1 @ N_1 ) ) ) ) ).
thf(fact_4246_not__P__Bolzano__bisect,axiom,
! [N: nat,A_1: real,B_1: 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_1 @ B_1 ) )
=> ( ( ord_less_eq_real @ A_1 @ B_1 )
=> ~ ( P @ ( produc865579683l_real @ ( produc1935615926l_real @ ( bolzano_bisect @ P @ A_1 @ B_1 @ N ) ) @ ( produc556554744l_real @ ( bolzano_bisect @ P @ A_1 @ B_1 @ N ) ) ) ) ) ) ) ).
thf(fact_4247_not__P__Bolzano__bisect_H,axiom,
! [A_1: real,B_1: 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_1 @ B_1 ) )
=> ( ( ord_less_eq_real @ A_1 @ B_1 )
=> ! [N_1: nat] :
~ ( P @ ( produc865579683l_real @ ( produc1935615926l_real @ ( bolzano_bisect @ P @ A_1 @ B_1 @ N_1 ) ) @ ( produc556554744l_real @ ( bolzano_bisect @ P @ A_1 @ B_1 @ N_1 ) ) ) ) ) ) ) ).
thf(fact_4248_one__code__int__code,axiom,
( one_on1684967323de_int
= ( number1226105091de_int @ ( bit1 @ pls ) ) ) ).
thf(fact_4249_Quickcheck__Narrowing_Oint__of__code,axiom,
! [K: quickcheck_code_int] :
( ( ( K = zero_z891286103de_int )
=> ( ( quickcheck_int_of @ K )
= zero_zero_int ) )
& ( ( K != zero_z891286103de_int )
=> ( ( ( ( div_mo231679042de_int @ K @ ( number1226105091de_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_z891286103de_int )
=> ( ( quickcheck_int_of @ K )
= ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( quickcheck_int_of @ ( div_di1430059507de_int @ K @ ( number1226105091de_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) )
& ( ( ( div_mo231679042de_int @ K @ ( number1226105091de_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= zero_z891286103de_int )
=> ( ( quickcheck_int_of @ K )
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( quickcheck_int_of @ ( div_di1430059507de_int @ K @ ( number1226105091de_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
thf(fact_4250_DeMoivre2,axiom,
! [R_1: real,A_1: real,N: nat] :
( ( power_power_complex @ ( rcis @ R_1 @ A_1 ) @ N )
= ( rcis @ ( power_power_real @ R_1 @ N ) @ ( times_times_real @ ( real_nat @ N ) @ A_1 ) ) ) ).
thf(fact_4251_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_4252_int__of__number,axiom,
! [K: int] :
( ( quickcheck_int_of @ ( number1226105091de_int @ K ) )
= ( number_number_of_int @ K ) ) ).
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_Quickcheck__Narrowing_Oint__of__inject,axiom,
! [K: quickcheck_code_int,L: quickcheck_code_int] :
( ( ( quickcheck_int_of @ K )
= ( quickcheck_int_of @ L ) )
<=> ( K = L ) ) ).
thf(fact_4255_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_4256_zero__code__int__code,axiom,
( zero_z891286103de_int
= ( number1226105091de_int @ pls ) ) ).
thf(fact_4257_rcis__zero__arg,axiom,
! [R_1: real] :
( ( rcis @ R_1 @ zero_zero_real )
= ( of_real_complex @ R_1 ) ) ).
thf(fact_4258_complex__mod__rcis,axiom,
! [R_1: real,A_1: real] :
( ( norm_norm_complex @ ( rcis @ R_1 @ A_1 ) )
= ( abs_abs_real @ R_1 ) ) ).
thf(fact_4259_cis__rcis__eq,axiom,
! [A_1: real] :
( ( cis @ A_1 )
= ( rcis @ one_one_real @ A_1 ) ) ).
thf(fact_4260_rcis__zero__mod,axiom,
! [A_1: real] :
( ( rcis @ zero_zero_real @ A_1 )
= zero_zero_complex ) ).
thf(fact_4261_Re__rcis,axiom,
! [R_1: real,A_1: real] :
( ( re @ ( rcis @ R_1 @ A_1 ) )
= ( times_times_real @ R_1 @ ( cos @ A_1 ) ) ) ).
thf(fact_4262_Im__rcis,axiom,
! [R_1: real,A_1: real] :
( ( im @ ( rcis @ R_1 @ A_1 ) )
= ( times_times_real @ R_1 @ ( sin @ A_1 ) ) ) ).
thf(fact_4263_rcis__mult,axiom,
! [R1: real,A_1: real,R2: real,B_1: real] :
( ( times_times_complex @ ( rcis @ R1 @ A_1 ) @ ( rcis @ R2 @ B_1 ) )
= ( rcis @ ( times_times_real @ R1 @ R2 ) @ ( plus_plus_real @ A_1 @ B_1 ) ) ) ).
thf(fact_4264_rcis__divide,axiom,
! [R1: real,A_1: real,R2: real,B_1: real] :
( ( invers1025623611omplex @ ( rcis @ R1 @ A_1 ) @ ( rcis @ R2 @ B_1 ) )
= ( rcis @ ( inverse_divide_real @ R1 @ R2 ) @ ( minus_minus_real @ A_1 @ B_1 ) ) ) ).
thf(fact_4265_rcis__def,axiom,
! [R_1: real,A_1: real] :
( ( rcis @ R_1 @ A_1 )
= ( times_times_complex @ ( of_real_complex @ R_1 ) @ ( cis @ A_1 ) ) ) ).
thf(fact_4266_rcis__inverse,axiom,
! [R_1: real,A_1: real] :
( ( invers1449016382omplex @ ( rcis @ R_1 @ A_1 ) )
= ( rcis @ ( inverse_divide_real @ one_one_real @ R_1 ) @ ( uminus_uminus_real @ A_1 ) ) ) ).
thf(fact_4267_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_4268_nat__of__def,axiom,
! [I: quickcheck_code_int] :
( ( quickcheck_nat_of @ I )
= ( nat_1 @ ( quickcheck_int_of @ I ) ) ) ).
thf(fact_4269_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_4270_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_4271_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_4272_fact__1__int,axiom,
( ( fact_fact_int @ one_one_int )
= one_one_int ) ).
thf(fact_4273_fact__0__int,axiom,
( ( fact_fact_int @ zero_zero_int )
= one_one_int ) ).
thf(fact_4274_fact__neg__int,axiom,
! [M: int] :
( ( ord_less_int @ M @ zero_zero_int )
=> ( ( fact_fact_int @ M )
= zero_zero_int ) ) ).
thf(fact_4275_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_4276_fact__ge__zero__int,axiom,
! [M: int] : ( ord_less_eq_int @ zero_zero_int @ ( fact_fact_int @ M ) ) ).
thf(fact_4277_fact__nonzero__int,axiom,
! [N: int] :
( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( fact_fact_int @ N )
!= zero_zero_int ) ) ).
thf(fact_4278_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_4279_transfer__int__nat__factorial,axiom,
! [X: nat] :
( ( fact_fact_int @ ( semiri1621563631at_int @ X ) )
= ( semiri1621563631at_int @ ( fact_fact_nat @ X ) ) ) ).
thf(fact_4280_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_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__mono__int__aux,axiom,
! [M: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ord_less_eq_int @ ( fact_fact_int @ M ) @ ( fact_fact_int @ ( plus_plus_int @ M @ K ) ) ) ) ).
thf(fact_4283_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_4284_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_4285_fact__less__mono__int__aux,axiom,
! [M: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( 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 ) ) ) ) ) ).
thf(fact_4286_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_4287_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_4288_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_4289_Bnor__prime,axiom,
! [A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ A_1 @ P_3 )
=> ( ( finite_card_int @ ( bnorRset @ A_1 @ P_3 ) )
= ( nat_1 @ A_1 ) ) ) ) ).
thf(fact_4290_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_4291_Bnor__mem__zle,axiom,
! [B_1: int,A_1: int,M: int] :
( ( member_int @ B_1 @ ( bnorRset @ A_1 @ M ) )
=> ( ord_less_eq_int @ B_1 @ A_1 ) ) ).
thf(fact_4292_Bnor__mem__zle__swap,axiom,
! [M: int,A_1: int,B_1: int] :
( ( ord_less_int @ A_1 @ B_1 )
=> ~ ( member_int @ B_1 @ ( bnorRset @ A_1 @ M ) ) ) ).
thf(fact_4293_Bnor__fin,axiom,
! [A_1: int,M: int] : ( finite_finite_int @ ( bnorRset @ A_1 @ M ) ) ).
thf(fact_4294_Bnor__mem__zg,axiom,
! [B_1: int,A_1: int,M: int] :
( ( member_int @ B_1 @ ( bnorRset @ A_1 @ M ) )
=> ( ord_less_int @ zero_zero_int @ B_1 ) ) ).
thf(fact_4295_lemma__BOLZANO,axiom,
! [A_1: real,B_1: 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_1 @ B_1 )
=> ( P @ ( produc865579683l_real @ A_1 @ B_1 ) ) ) ) ) ).
thf(fact_4296_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_4297_gcd__dvd1__nat,axiom,
! [M: nat,N: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ M @ N ) @ M ) ).
thf(fact_4298_gcd__dvd2__nat,axiom,
! [M: nat,N: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ M @ N ) @ N ) ).
thf(fact_4299_gcd__dvd__prod__nat,axiom,
! [M: nat,N: nat,K: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ M @ N ) @ ( times_times_nat @ K @ N ) ) ).
thf(fact_4300_coprime__exp2__nat,axiom,
! [N: nat,M: nat,A_1: nat,B_1: nat] :
( ( ( gcd_gcd_nat @ A_1 @ B_1 )
= one_one_nat )
=> ( ( gcd_gcd_nat @ ( power_power_nat @ A_1 @ N ) @ ( power_power_nat @ B_1 @ M ) )
= one_one_nat ) ) ).
thf(fact_4301_gcd__mult__distrib__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
= ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
thf(fact_4302_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_4303_gcd__assoc__nat,axiom,
! [A_1: nat,B_1: nat,C: nat] :
( ( gcd_gcd_nat @ ( gcd_gcd_nat @ A_1 @ B_1 ) @ C )
= ( gcd_gcd_nat @ A_1 @ ( gcd_gcd_nat @ B_1 @ C ) ) ) ).
thf(fact_4304_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_4305_gcd__left__commute__nat,axiom,
! [B_1: nat,A_1: nat,C: nat] :
( ( gcd_gcd_nat @ B_1 @ ( gcd_gcd_nat @ A_1 @ C ) )
= ( gcd_gcd_nat @ A_1 @ ( gcd_gcd_nat @ B_1 @ C ) ) ) ).
thf(fact_4306_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_4307_gcd__semilattice__nat_Oinf_Oleft__idem,axiom,
! [A_1: nat,B_1: nat] :
( ( gcd_gcd_nat @ A_1 @ ( gcd_gcd_nat @ A_1 @ B_1 ) )
= ( gcd_gcd_nat @ A_1 @ B_1 ) ) ).
thf(fact_4308_gcd__semilattice__nat_Oinf__commute,axiom,
! [X: nat,Y: nat] :
( ( gcd_gcd_nat @ X @ Y )
= ( gcd_gcd_nat @ Y @ X ) ) ).
thf(fact_4309_gcd__commute__nat,axiom,
! [A_1: nat,B_1: nat] :
( ( gcd_gcd_nat @ A_1 @ B_1 )
= ( gcd_gcd_nat @ B_1 @ A_1 ) ) ).
thf(fact_4310_gcd__idem__nat,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ X @ X )
= X ) ).
thf(fact_4311_gcd__semilattice__nat_Oinf_Oidem,axiom,
! [A_1: nat] :
( ( gcd_gcd_nat @ A_1 @ A_1 )
= A_1 ) ).
thf(fact_4312_gcd__semilattice__nat_Oinf__le1,axiom,
! [X: nat,Y: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ X @ Y ) @ X ) ).
thf(fact_4313_gcd__semilattice__nat_Oinf__le2,axiom,
! [X: nat,Y: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ X @ Y ) @ Y ) ).
thf(fact_4314_gcd__semilattice__nat_Ole__iff__inf,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
<=> ( ( gcd_gcd_nat @ X @ Y )
= X ) ) ).
thf(fact_4315_gcd__greatest__iff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
<=> ( ( dvd_dvd_nat @ K @ M )
& ( dvd_dvd_nat @ K @ N ) ) ) ).
thf(fact_4316_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_4317_gcd__unique__nat,axiom,
! [B_1: nat,D_1: nat,A_1: nat] :
( ( ( dvd_dvd_nat @ D_1 @ A_1 )
& ( dvd_dvd_nat @ D_1 @ B_1 )
& ! [E: nat] :
( ( ( dvd_dvd_nat @ E @ A_1 )
& ( dvd_dvd_nat @ E @ B_1 ) )
=> ( dvd_dvd_nat @ E @ D_1 ) ) )
<=> ( D_1
= ( gcd_gcd_nat @ A_1 @ B_1 ) ) ) ).
thf(fact_4318_gcd__semilattice__nat_Ole__infI1,axiom,
! [B_1: nat,A_1: nat,X: nat] :
( ( dvd_dvd_nat @ A_1 @ X )
=> ( dvd_dvd_nat @ ( gcd_gcd_nat @ A_1 @ B_1 ) @ X ) ) ).
thf(fact_4319_gcd__semilattice__nat_Ole__infI2,axiom,
! [A_1: nat,B_1: nat,X: nat] :
( ( dvd_dvd_nat @ B_1 @ X )
=> ( dvd_dvd_nat @ ( gcd_gcd_nat @ A_1 @ B_1 ) @ X ) ) ).
thf(fact_4320_gcd__proj1__if__dvd__nat,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( gcd_gcd_nat @ X @ Y )
= X ) ) ).
thf(fact_4321_gcd__proj2__if__dvd__nat,axiom,
! [Y: nat,X: nat] :
( ( dvd_dvd_nat @ Y @ X )
=> ( ( gcd_gcd_nat @ X @ Y )
= Y ) ) ).
thf(fact_4322_gcd__greatest__nat,axiom,
! [N: nat,K: nat,M: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( gcd_gcd_nat @ M @ N ) ) ) ) ).
thf(fact_4323_gcd__semilattice__nat_Ole__infI,axiom,
! [B_1: nat,X: nat,A_1: nat] :
( ( dvd_dvd_nat @ X @ A_1 )
=> ( ( dvd_dvd_nat @ X @ B_1 )
=> ( dvd_dvd_nat @ X @ ( gcd_gcd_nat @ A_1 @ B_1 ) ) ) ) ).
thf(fact_4324_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_4325_gcd__semilattice__nat_Oinf__mono,axiom,
! [B_1: nat,D_1: nat,A_1: nat,C: nat] :
( ( dvd_dvd_nat @ A_1 @ C )
=> ( ( dvd_dvd_nat @ B_1 @ D_1 )
=> ( dvd_dvd_nat @ ( gcd_gcd_nat @ A_1 @ B_1 ) @ ( gcd_gcd_nat @ C @ D_1 ) ) ) ) ).
thf(fact_4326_dvd__gcd__D1__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
=> ( dvd_dvd_nat @ K @ M ) ) ).
thf(fact_4327_dvd__gcd__D2__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
=> ( dvd_dvd_nat @ K @ N ) ) ).
thf(fact_4328_gcd__semilattice__nat_Ole__infE,axiom,
! [X: nat,A_1: nat,B_1: nat] :
( ( dvd_dvd_nat @ X @ ( gcd_gcd_nat @ A_1 @ B_1 ) )
=> ~ ( ( dvd_dvd_nat @ X @ A_1 )
=> ~ ( dvd_dvd_nat @ X @ B_1 ) ) ) ).
thf(fact_4329_gcd__semilattice__nat_Oless__infI1,axiom,
! [B_1: nat,A_1: nat,X: nat] :
( ( ( dvd_dvd_nat @ A_1 @ X )
& ~ ( dvd_dvd_nat @ X @ A_1 ) )
=> ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A_1 @ B_1 ) @ X )
& ~ ( dvd_dvd_nat @ X @ ( gcd_gcd_nat @ A_1 @ B_1 ) ) ) ) ).
thf(fact_4330_gcd__semilattice__nat_Oless__infI2,axiom,
! [A_1: nat,B_1: nat,X: nat] :
( ( ( dvd_dvd_nat @ B_1 @ X )
& ~ ( dvd_dvd_nat @ X @ B_1 ) )
=> ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A_1 @ B_1 ) @ X )
& ~ ( dvd_dvd_nat @ X @ ( gcd_gcd_nat @ A_1 @ B_1 ) ) ) ) ).
thf(fact_4331_gcd__add2__nat,axiom,
! [M: nat,N: nat] :
( ( gcd_gcd_nat @ M @ ( plus_plus_nat @ M @ N ) )
= ( gcd_gcd_nat @ M @ N ) ) ).
thf(fact_4332_gcd__add1__nat,axiom,
! [M: nat,N: nat] :
( ( gcd_gcd_nat @ ( plus_plus_nat @ M @ N ) @ N )
= ( gcd_gcd_nat @ M @ N ) ) ).
thf(fact_4333_gcd__exp__nat,axiom,
! [A_1: nat,N: nat,B_1: nat] :
( ( gcd_gcd_nat @ ( power_power_nat @ A_1 @ N ) @ ( power_power_nat @ B_1 @ N ) )
= ( power_power_nat @ ( gcd_gcd_nat @ A_1 @ B_1 ) @ N ) ) ).
thf(fact_4334_gcd__red__nat,axiom,
! [X: nat,Y: nat] :
( ( gcd_gcd_nat @ X @ Y )
= ( gcd_gcd_nat @ Y @ ( div_mod_nat @ X @ Y ) ) ) ).
thf(fact_4335_gcd__lcm__complete__lattice__nat_Oinf__bot__right,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ X @ one_one_nat )
= 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__left,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ one_one_nat @ X )
= 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__Suc__0,axiom,
! [M: nat] :
( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( suc @ zero_zero_nat ) ) ).
thf(fact_4343_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_4344_gcd__le2__nat,axiom,
! [A_1: nat,B_1: nat] :
( ( B_1 != zero_zero_nat )
=> ( ord_less_eq_nat @ ( gcd_gcd_nat @ A_1 @ B_1 ) @ B_1 ) ) ).
thf(fact_4345_gcd__le1__nat,axiom,
! [B_1: nat,A_1: nat] :
( ( A_1 != zero_zero_nat )
=> ( ord_less_eq_nat @ ( gcd_gcd_nat @ A_1 @ B_1 ) @ A_1 ) ) ).
thf(fact_4346_coprime__Suc__nat,axiom,
! [N: nat] :
( ( gcd_gcd_nat @ ( suc @ N ) @ N )
= one_one_nat ) ).
thf(fact_4347_gcd__add__mult__nat,axiom,
! [M: nat,K: nat,N: nat] :
( ( gcd_gcd_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ N ) )
= ( gcd_gcd_nat @ M @ N ) ) ).
thf(fact_4348_coprime__common__divisor__nat,axiom,
! [X: nat,A_1: nat,B_1: nat] :
( ( ( gcd_gcd_nat @ A_1 @ B_1 )
= one_one_nat )
=> ( ( dvd_dvd_nat @ X @ A_1 )
=> ( ( dvd_dvd_nat @ X @ B_1 )
=> ( X = one_one_nat ) ) ) ) ).
thf(fact_4349_coprime__nat,axiom,
! [A_1: nat,B_1: nat] :
( ( ( gcd_gcd_nat @ A_1 @ B_1 )
= one_one_nat )
<=> ! [D_2: nat] :
( ( ( dvd_dvd_nat @ D_2 @ A_1 )
& ( dvd_dvd_nat @ D_2 @ B_1 ) )
<=> ( D_2 = one_one_nat ) ) ) ).
thf(fact_4350_coprime__plus__one__nat,axiom,
! [N: nat] :
( ( gcd_gcd_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ N )
= one_one_nat ) ).
thf(fact_4351_coprime__mul__eq__nat,axiom,
! [D_1: nat,A_1: nat,B_1: nat] :
( ( ( gcd_gcd_nat @ D_1 @ ( times_times_nat @ A_1 @ B_1 ) )
= one_one_nat )
<=> ( ( ( gcd_gcd_nat @ D_1 @ A_1 )
= one_one_nat )
& ( ( gcd_gcd_nat @ D_1 @ B_1 )
= one_one_nat ) ) ) ).
thf(fact_4352_gcd__mult__cancel__nat,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( gcd_gcd_nat @ K @ N )
= one_one_nat )
=> ( ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ N )
= ( gcd_gcd_nat @ M @ N ) ) ) ).
thf(fact_4353_coprime__mult__nat,axiom,
! [B_1: nat,D_1: nat,A_1: nat] :
( ( ( gcd_gcd_nat @ D_1 @ A_1 )
= one_one_nat )
=> ( ( ( gcd_gcd_nat @ D_1 @ B_1 )
= one_one_nat )
=> ( ( gcd_gcd_nat @ D_1 @ ( times_times_nat @ A_1 @ B_1 ) )
= one_one_nat ) ) ) ).
thf(fact_4354_coprime__crossproduct__nat,axiom,
! [B_1: nat,C: nat,A_1: nat,D_1: nat] :
( ( ( gcd_gcd_nat @ A_1 @ D_1 )
= one_one_nat )
=> ( ( ( gcd_gcd_nat @ B_1 @ C )
= one_one_nat )
=> ( ( ( times_times_nat @ A_1 @ C )
= ( times_times_nat @ B_1 @ D_1 ) )
<=> ( ( A_1 = B_1 )
& ( C = D_1 ) ) ) ) ) ).
thf(fact_4355_coprime__lmult__nat,axiom,
! [D_1: nat,A_1: nat,B_1: nat] :
( ( ( gcd_gcd_nat @ D_1 @ ( times_times_nat @ A_1 @ B_1 ) )
= one_one_nat )
=> ( ( gcd_gcd_nat @ D_1 @ A_1 )
= one_one_nat ) ) ).
thf(fact_4356_coprime__rmult__nat,axiom,
! [D_1: nat,A_1: nat,B_1: nat] :
( ( ( gcd_gcd_nat @ D_1 @ ( times_times_nat @ A_1 @ B_1 ) )
= one_one_nat )
=> ( ( gcd_gcd_nat @ D_1 @ B_1 )
= one_one_nat ) ) ).
thf(fact_4357_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_4358_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_4359_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_4360_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_4361_coprime__exp__nat,axiom,
! [N: nat,D_1: nat,A_1: nat] :
( ( ( gcd_gcd_nat @ D_1 @ A_1 )
= one_one_nat )
=> ( ( gcd_gcd_nat @ D_1 @ ( power_power_nat @ A_1 @ N ) )
= one_one_nat ) ) ).
thf(fact_4362_ceiling__real__of__nat,axiom,
! [N: nat] :
( ( archim856651990g_real @ ( real_nat @ N ) )
= ( semiri1621563631at_int @ N ) ) ).
thf(fact_4363_natceiling__def,axiom,
! [X: real] :
( ( natceiling @ X )
= ( nat_1 @ ( archim856651990g_real @ X ) ) ) ).
thf(fact_4364_gcd__coprime__nat,axiom,
! [B_5: nat,A_5: nat,A_1: nat,B_1: nat] :
( ( ( gcd_gcd_nat @ A_1 @ B_1 )
!= zero_zero_nat )
=> ( ( A_1
= ( times_times_nat @ A_5 @ ( gcd_gcd_nat @ A_1 @ B_1 ) ) )
=> ( ( B_1
= ( times_times_nat @ B_5 @ ( gcd_gcd_nat @ A_1 @ B_1 ) ) )
=> ( ( gcd_gcd_nat @ A_5 @ B_5 )
= one_one_nat ) ) ) ) ).
thf(fact_4365_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_4366_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_4367_coprime__dvd__mult__iff__nat,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( gcd_gcd_nat @ K @ N )
= one_one_nat )
=> ( ( dvd_dvd_nat @ K @ ( times_times_nat @ M @ N ) )
<=> ( dvd_dvd_nat @ K @ M ) ) ) ).
thf(fact_4368_coprime__dvd__mult__nat,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( gcd_gcd_nat @ K @ N )
= one_one_nat )
=> ( ( dvd_dvd_nat @ K @ ( times_times_nat @ M @ N ) )
=> ( dvd_dvd_nat @ K @ M ) ) ) ).
thf(fact_4369_div__gcd__coprime__nat,axiom,
! [B_1: nat,A_1: nat] :
( ( ( A_1 != zero_zero_nat )
| ( B_1 != zero_zero_nat ) )
=> ( ( gcd_gcd_nat @ ( div_div_nat @ A_1 @ ( gcd_gcd_nat @ A_1 @ B_1 ) ) @ ( div_div_nat @ B_1 @ ( gcd_gcd_nat @ A_1 @ B_1 ) ) )
= one_one_nat ) ) ).
thf(fact_4370_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_4371_coprime__Suc__0__nat,axiom,
! [A_1: nat,B_1: nat] :
( ( ( gcd_gcd_nat @ A_1 @ B_1 )
= one_one_nat )
<=> ! [D_2: nat] :
( ( ( dvd_dvd_nat @ D_2 @ A_1 )
& ( dvd_dvd_nat @ D_2 @ B_1 ) )
<=> ( D_2
= ( suc @ zero_zero_nat ) ) ) ) ).
thf(fact_4372_gcd__coprime__exists__nat,axiom,
! [A_1: nat,B_1: nat] :
( ( ( gcd_gcd_nat @ A_1 @ B_1 )
!= zero_zero_nat )
=> ? [A_3: nat,B_2: nat] :
( ( A_1
= ( times_times_nat @ A_3 @ ( gcd_gcd_nat @ A_1 @ B_1 ) ) )
& ( B_1
= ( times_times_nat @ B_2 @ ( gcd_gcd_nat @ A_1 @ B_1 ) ) )
& ( ( gcd_gcd_nat @ A_3 @ B_2 )
= one_one_nat ) ) ) ).
thf(fact_4373_bezout__nat,axiom,
! [B_1: nat,A_1: nat] :
( ( A_1 != zero_zero_nat )
=> ? [X_1: nat,Y_1: nat] :
( ( times_times_nat @ A_1 @ X_1 )
= ( plus_plus_nat @ ( times_times_nat @ B_1 @ Y_1 ) @ ( gcd_gcd_nat @ A_1 @ B_1 ) ) ) ) ).
thf(fact_4374_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_4375_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_4376_ln__powr__bound,axiom,
! [A_1: real,X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ A_1 )
=> ( ord_less_eq_real @ ( ln @ X ) @ ( inverse_divide_real @ ( powr @ X @ A_1 ) @ A_1 ) ) ) ) ).
thf(fact_4377_real__number__of,axiom,
! [V: int] :
( ( real_int @ ( number_number_of_int @ V ) )
= ( number267125858f_real @ V ) ) ).
thf(fact_4378_real__of__int__zero__cancel,axiom,
! [X: int] :
( ( ( real_int @ X )
= zero_zero_real )
<=> ( X = zero_zero_int ) ) ).
thf(fact_4379_real__of__int__zero,axiom,
( ( real_int @ zero_zero_int )
= zero_zero_real ) ).
thf(fact_4380_powr__not__zero,axiom,
! [X: real,A_1: real] :
( ( powr @ X @ A_1 )
!= zero_zero_real ) ).
thf(fact_4381_powr__one__eq__one,axiom,
! [A_1: real] :
( ( powr @ one_one_real @ A_1 )
= one_one_real ) ).
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_floor__real__of__int,axiom,
! [N: int] :
( ( archim1246769320r_real @ ( real_int @ N ) )
= N ) ).
thf(fact_4384_real__of__int__inject,axiom,
! [X: int,Y: int] :
( ( ( real_int @ X )
= ( real_int @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_4385_powr__powr__swap,axiom,
! [X: real,A_1: real,B_1: real] :
( ( powr @ ( powr @ X @ A_1 ) @ B_1 )
= ( powr @ ( powr @ X @ B_1 ) @ A_1 ) ) ).
thf(fact_4386_powr__powr,axiom,
! [X: real,A_1: real,B_1: real] :
( ( powr @ ( powr @ X @ A_1 ) @ B_1 )
= ( powr @ X @ ( times_times_real @ A_1 @ B_1 ) ) ) ).
thf(fact_4387_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_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__one,axiom,
( ( real_int @ one_one_int )
= one_one_real ) ).
thf(fact_4390_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_4391_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_4392_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_4393_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_4394_real__of__int__of__nat__eq,axiom,
! [N: nat] :
( ( real_int @ ( semiri1621563631at_int @ N ) )
= ( real_nat @ N ) ) ).
thf(fact_4395_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_4396_real__of__int__ceiling__cancel,axiom,
! [X: real] :
( ( ( real_int @ ( archim856651990g_real @ X ) )
= X )
<=> ? [N_1: int] :
( X
= ( real_int @ N_1 ) ) ) ).
thf(fact_4397_ceiling__real__of__int,axiom,
! [N: int] :
( ( archim856651990g_real @ ( real_int @ N ) )
= N ) ).
thf(fact_4398_real__of__int__minus,axiom,
! [X: int] :
( ( real_int @ ( uminus_uminus_int @ X ) )
= ( uminus_uminus_real @ ( real_int @ X ) ) ) ).
thf(fact_4399_real__of__int__abs,axiom,
! [X: int] :
( ( real_int @ ( abs_abs_int @ X ) )
= ( abs_abs_real @ ( real_int @ X ) ) ) ).
thf(fact_4400_real__of__int__floor__le,axiom,
! [R_1: real] : ( ord_less_eq_real @ ( real_int @ ( archim1246769320r_real @ R_1 ) ) @ R_1 ) ).
thf(fact_4401_real__of__int__ceiling__ge,axiom,
! [R_1: real] : ( ord_less_eq_real @ R_1 @ ( real_int @ ( archim856651990g_real @ R_1 ) ) ) ).
thf(fact_4402_powr__ge__pzero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr @ X @ Y ) ) ).
thf(fact_4403_powr__gt__zero,axiom,
! [X: real,A_1: real] : ( ord_less_real @ zero_zero_real @ ( powr @ X @ A_1 ) ) ).
thf(fact_4404_powr__less__mono2,axiom,
! [Y: real,X: real,A_1: real] :
( ( ord_less_real @ zero_zero_real @ A_1 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( powr @ X @ A_1 ) @ ( powr @ Y @ A_1 ) ) ) ) ) ).
thf(fact_4405_powr__less__mono2__neg,axiom,
! [Y: real,X: real,A_1: real] :
( ( ord_less_real @ A_1 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( powr @ Y @ A_1 ) @ ( powr @ X @ A_1 ) ) ) ) ) ).
thf(fact_4406_powr__mono,axiom,
! [X: real,A_1: real,B_1: real] :
( ( ord_less_eq_real @ A_1 @ B_1 )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( powr @ X @ A_1 ) @ ( powr @ X @ B_1 ) ) ) ) ).
thf(fact_4407_powr__zero__eq__one,axiom,
! [X: real] :
( ( powr @ X @ zero_zero_real )
= one_one_real ) ).
thf(fact_4408_powr__less__cancel,axiom,
! [X: real,A_1: real,B_1: real] :
( ( ord_less_real @ ( powr @ X @ A_1 ) @ ( powr @ X @ B_1 ) )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ A_1 @ B_1 ) ) ) ).
thf(fact_4409_powr__less__mono,axiom,
! [X: real,A_1: real,B_1: real] :
( ( ord_less_real @ A_1 @ B_1 )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( powr @ X @ A_1 ) @ ( powr @ X @ B_1 ) ) ) ) ).
thf(fact_4410_powr__less__cancel__iff,axiom,
! [A_1: real,B_1: real,X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ ( powr @ X @ A_1 ) @ ( powr @ X @ B_1 ) )
<=> ( ord_less_real @ A_1 @ B_1 ) ) ) ).
thf(fact_4411_powr__add,axiom,
! [X: real,A_1: real,B_1: real] :
( ( powr @ X @ ( plus_plus_real @ A_1 @ B_1 ) )
= ( times_times_real @ ( powr @ X @ A_1 ) @ ( powr @ X @ B_1 ) ) ) ).
thf(fact_4412_powr__divide2,axiom,
! [X: real,A_1: real,B_1: real] :
( ( inverse_divide_real @ ( powr @ X @ A_1 ) @ ( powr @ X @ B_1 ) )
= ( powr @ X @ ( minus_minus_real @ A_1 @ B_1 ) ) ) ).
thf(fact_4413_powr__minus,axiom,
! [X: real,A_1: real] :
( ( powr @ X @ ( uminus_uminus_real @ A_1 ) )
= ( inverse_inverse_real @ ( powr @ X @ A_1 ) ) ) ).
thf(fact_4414_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_4415_floor__less__eq,axiom,
! [X: real,A_1: int] :
( ( ord_less_int @ ( archim1246769320r_real @ X ) @ A_1 )
<=> ( ord_less_real @ X @ ( real_int @ A_1 ) ) ) ).
thf(fact_4416_le__floor,axiom,
! [A_1: int,X: real] :
( ( ord_less_eq_real @ ( real_int @ A_1 ) @ X )
=> ( ord_less_eq_int @ A_1 @ ( archim1246769320r_real @ X ) ) ) ).
thf(fact_4417_real__le__floor,axiom,
! [A_1: int,X: real] :
( ( ord_less_eq_int @ A_1 @ ( archim1246769320r_real @ X ) )
=> ( ord_less_eq_real @ ( real_int @ A_1 ) @ X ) ) ).
thf(fact_4418_le__floor__eq,axiom,
! [A_1: int,X: real] :
( ( ord_less_eq_int @ A_1 @ ( archim1246769320r_real @ X ) )
<=> ( ord_less_eq_real @ ( real_int @ A_1 ) @ X ) ) ).
thf(fact_4419_floor__add,axiom,
! [X: real,A_1: int] :
( ( archim1246769320r_real @ ( plus_plus_real @ X @ ( real_int @ A_1 ) ) )
= ( plus_plus_int @ ( archim1246769320r_real @ X ) @ A_1 ) ) ).
thf(fact_4420_less__ceiling__eq,axiom,
! [A_1: int,X: real] :
( ( ord_less_int @ A_1 @ ( archim856651990g_real @ X ) )
<=> ( ord_less_real @ ( real_int @ A_1 ) @ X ) ) ).
thf(fact_4421_ceiling__le__eq,axiom,
! [X: real,A_1: int] :
( ( ord_less_eq_int @ ( archim856651990g_real @ X ) @ A_1 )
<=> ( ord_less_eq_real @ X @ ( real_int @ A_1 ) ) ) ).
thf(fact_4422_ceiling__le,axiom,
! [X: real,A_1: int] :
( ( ord_less_eq_real @ X @ ( real_int @ A_1 ) )
=> ( ord_less_eq_int @ ( archim856651990g_real @ X ) @ A_1 ) ) ).
thf(fact_4423_ceiling__le__real,axiom,
! [X: real,A_1: int] :
( ( ord_less_eq_int @ ( archim856651990g_real @ X ) @ A_1 )
=> ( ord_less_eq_real @ X @ ( real_int @ A_1 ) ) ) ).
thf(fact_4424_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_4425_floor__subtract,axiom,
! [X: real,A_1: int] :
( ( archim1246769320r_real @ ( minus_minus_real @ X @ ( real_int @ A_1 ) ) )
= ( minus_minus_int @ ( archim1246769320r_real @ X ) @ A_1 ) ) ).
thf(fact_4426_ceiling__add,axiom,
! [X: real,A_1: int] :
( ( archim856651990g_real @ ( plus_plus_real @ X @ ( real_int @ A_1 ) ) )
= ( plus_plus_int @ ( archim856651990g_real @ X ) @ A_1 ) ) ).
thf(fact_4427_floor__minus__real__of__int,axiom,
! [N: int] :
( ( archim1246769320r_real @ ( uminus_uminus_real @ ( real_int @ N ) ) )
= ( uminus_uminus_int @ N ) ) ).
thf(fact_4428_ceiling__subtract,axiom,
! [X: real,A_1: int] :
( ( archim856651990g_real @ ( minus_minus_real @ X @ ( real_int @ A_1 ) ) )
= ( minus_minus_int @ ( archim856651990g_real @ X ) @ A_1 ) ) ).
thf(fact_4429_powr__mono2,axiom,
! [Y: real,X: real,A_1: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_1 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( powr @ X @ A_1 ) @ ( powr @ Y @ A_1 ) ) ) ) ) ).
thf(fact_4430_powr__mult,axiom,
! [A_1: 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_1 )
= ( times_times_real @ ( powr @ X @ A_1 ) @ ( powr @ Y @ A_1 ) ) ) ) ) ).
thf(fact_4431_ge__one__powr__ge__zero,axiom,
! [A_1: real,X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_1 )
=> ( ord_less_eq_real @ one_one_real @ ( powr @ X @ A_1 ) ) ) ) ).
thf(fact_4432_powr__le__cancel__iff,axiom,
! [A_1: real,B_1: real,X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ ( powr @ X @ A_1 ) @ ( powr @ X @ B_1 ) )
<=> ( ord_less_eq_real @ A_1 @ B_1 ) ) ) ).
thf(fact_4433_powr__one__gt__zero__iff,axiom,
! [X: real] :
( ( ( powr @ X @ one_one_real )
= X )
<=> ( ord_less_real @ zero_zero_real @ X ) ) ).
thf(fact_4434_powr__divide,axiom,
! [A_1: 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_1 )
= ( inverse_divide_real @ ( powr @ X @ A_1 ) @ ( powr @ Y @ A_1 ) ) ) ) ) ).
thf(fact_4435_powr__minus__divide,axiom,
! [X: real,A_1: real] :
( ( powr @ X @ ( uminus_uminus_real @ A_1 ) )
= ( inverse_divide_real @ one_one_real @ ( powr @ X @ A_1 ) ) ) ).
thf(fact_4436_powr__def,axiom,
! [X: real,A_1: real] :
( ( powr @ X @ A_1 )
= ( exp_real @ ( times_times_real @ A_1 @ ( ln @ X ) ) ) ) ).
thf(fact_4437_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_4438_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_4439_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_4440_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_4441_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_4442_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_4443_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_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_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_4446_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_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_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_4449_real__of__int__div,axiom,
! [N: int,D_1: int] :
( ( D_1 != zero_zero_int )
=> ( ( dvd_dvd_int @ D_1 @ N )
=> ( ( real_int @ ( div_div_int @ N @ D_1 ) )
= ( inverse_divide_real @ ( real_int @ N ) @ ( real_int @ D_1 ) ) ) ) ) ).
thf(fact_4450_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_4451_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_4452_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_4453_tan__periodic__int,axiom,
! [X: real,I: int] :
( ( tan @ ( plus_plus_real @ X @ ( times_times_real @ ( real_int @ I ) @ pi ) ) )
= ( tan @ X ) ) ).
thf(fact_4454_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_4455_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_4456_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_4457_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_4458_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_4459_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_4460_powr__log__cancel,axiom,
! [X: real,A_1: real] :
( ( ord_less_real @ zero_zero_real @ A_1 )
=> ( ( A_1 != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr @ A_1 @ ( log @ A_1 @ X ) )
= X ) ) ) ) ).
thf(fact_4461_log__powr__cancel,axiom,
! [Y: real,A_1: real] :
( ( ord_less_real @ zero_zero_real @ A_1 )
=> ( ( A_1 != one_one_real )
=> ( ( log @ A_1 @ ( powr @ A_1 @ Y ) )
= Y ) ) ) ).
thf(fact_4462_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_4463_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_4464_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_4465_real__of__int__div__aux,axiom,
! [X: int,D_1: int] :
( ( D_1 != zero_zero_int )
=> ( ( inverse_divide_real @ ( real_int @ X ) @ ( real_int @ D_1 ) )
= ( plus_plus_real @ ( real_int @ ( div_div_int @ X @ D_1 ) ) @ ( inverse_divide_real @ ( real_int @ ( div_mod_int @ X @ D_1 ) ) @ ( real_int @ D_1 ) ) ) ) ) ).
thf(fact_4466_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_4467_less__floor__eq,axiom,
! [A_1: int,X: real] :
( ( ord_less_int @ A_1 @ ( archim1246769320r_real @ X ) )
<=> ( ord_less_eq_real @ ( plus_plus_real @ ( real_int @ A_1 ) @ one_one_real ) @ X ) ) ).
thf(fact_4468_floor__le__eq,axiom,
! [X: real,A_1: int] :
( ( ord_less_eq_int @ ( archim1246769320r_real @ X ) @ A_1 )
<=> ( ord_less_real @ X @ ( plus_plus_real @ ( real_int @ A_1 ) @ one_one_real ) ) ) ).
thf(fact_4469_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_4470_ceiling__less__eq,axiom,
! [X: real,A_1: int] :
( ( ord_less_int @ ( archim856651990g_real @ X ) @ A_1 )
<=> ( ord_less_eq_real @ X @ ( minus_minus_real @ ( real_int @ A_1 ) @ one_one_real ) ) ) ).
thf(fact_4471_le__ceiling__eq,axiom,
! [A_1: int,X: real] :
( ( ord_less_eq_int @ A_1 @ ( archim856651990g_real @ X ) )
<=> ( ord_less_real @ ( minus_minus_real @ ( real_int @ A_1 ) @ one_one_real ) @ X ) ) ).
thf(fact_4472_log__powr,axiom,
! [B_1: real,Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( log @ B_1 @ ( powr @ X @ Y ) )
= ( times_times_real @ Y @ ( log @ B_1 @ X ) ) ) ) ) ).
thf(fact_4473_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_4474_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_4475_ln__powr__bound2,axiom,
! [A_1: real,X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ A_1 )
=> ( ord_less_eq_real @ ( powr @ ( ln @ X ) @ A_1 ) @ ( times_times_real @ ( powr @ A_1 @ A_1 ) @ X ) ) ) ) ).
thf(fact_4476_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_4477_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_4478_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_4479_coprime__pow,axiom,
! [C: nat,N: nat,A_1: nat,B_1: nat] :
( ( coprime @ A_1 @ B_1 )
=> ( ( ( times_times_nat @ A_1 @ B_1 )
= ( power_power_nat @ C @ N ) )
=> ? [R: nat,S_2: nat] :
( ( A_1
= ( power_power_nat @ R @ N ) )
& ( B_1
= ( power_power_nat @ S_2 @ N ) ) ) ) ) ).
thf(fact_4480_prime__power__mult,axiom,
! [X: nat,Y: nat,K: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( ( times_times_nat @ X @ Y )
= ( power_power_nat @ P_3 @ K ) )
=> ? [I_1: nat,J_1: nat] :
( ( X
= ( power_power_nat @ P_3 @ I_1 ) )
& ( Y
= ( power_power_nat @ P_3 @ J_1 ) ) ) ) ) ).
thf(fact_4481_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_4482_zgcd__zdvd2,axiom,
! [I: int,J: int] : ( dvd_dvd_int @ ( legacy_zgcd @ I @ J ) @ J ) ).
thf(fact_4483_zgcd__zdvd1,axiom,
! [I: int,J: int] : ( dvd_dvd_int @ ( legacy_zgcd @ I @ J ) @ I ) ).
thf(fact_4484_zgcd__zdvd__zgcd__zmult,axiom,
! [M: int,N: int,K: int] : ( dvd_dvd_int @ ( legacy_zgcd @ M @ N ) @ ( legacy_zgcd @ ( times_times_int @ K @ M ) @ N ) ) ).
thf(fact_4485_zgcd__zmult__cancel,axiom,
! [M: int,K: int,N: int] :
( ( ( legacy_zgcd @ K @ N )
= one_one_int )
=> ( ( legacy_zgcd @ ( times_times_int @ K @ M ) @ N )
= ( legacy_zgcd @ M @ N ) ) ) ).
thf(fact_4486_zgcd__zgcd__zmult,axiom,
! [N: int,K: int,M: int] :
( ( ( legacy_zgcd @ K @ M )
= one_one_int )
=> ( ( ( legacy_zgcd @ N @ M )
= one_one_int )
=> ( ( legacy_zgcd @ ( times_times_int @ K @ N ) @ M )
= one_one_int ) ) ) ).
thf(fact_4487_zgcd__1,axiom,
! [M: int] :
( ( legacy_zgcd @ M @ one_one_int )
= one_one_int ) ).
thf(fact_4488_zgcd__1__left,axiom,
! [M: int] :
( ( legacy_zgcd @ one_one_int @ M )
= one_one_int ) ).
thf(fact_4489_zgcd0,axiom,
! [I: int,J: int] :
( ( ( legacy_zgcd @ I @ J )
= zero_zero_int )
<=> ( ( I = zero_zero_int )
& ( J = zero_zero_int ) ) ) ).
thf(fact_4490_zgcd__eq,axiom,
! [M: int,N: int] :
( ( legacy_zgcd @ M @ N )
= ( legacy_zgcd @ N @ ( div_mod_int @ M @ N ) ) ) ).
thf(fact_4491_zgcd__power__distrib,axiom,
! [A_1: int,B_1: int,N: nat] :
( ( power_power_int @ ( legacy_zgcd @ A_1 @ B_1 ) @ N )
= ( legacy_zgcd @ ( power_power_int @ A_1 @ N ) @ ( power_power_int @ B_1 @ N ) ) ) ).
thf(fact_4492_zgcd__greatest,axiom,
! [N: int,K: int,M: int] :
( ( dvd_dvd_int @ K @ M )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ ( legacy_zgcd @ M @ N ) ) ) ) ).
thf(fact_4493_zgcd__greatest__iff,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ ( legacy_zgcd @ M @ N ) )
<=> ( ( dvd_dvd_int @ K @ M )
& ( dvd_dvd_int @ K @ N ) ) ) ).
thf(fact_4494_zgcd__commute,axiom,
! [I: int,J: int] :
( ( legacy_zgcd @ I @ J )
= ( legacy_zgcd @ J @ I ) ) ).
thf(fact_4495_zgcd__left__commute,axiom,
! [K: int,M: int,N: int] :
( ( legacy_zgcd @ K @ ( legacy_zgcd @ M @ N ) )
= ( legacy_zgcd @ M @ ( legacy_zgcd @ K @ N ) ) ) ).
thf(fact_4496_zgcd__assoc,axiom,
! [K: int,M: int,N: int] :
( ( legacy_zgcd @ ( legacy_zgcd @ K @ M ) @ N )
= ( legacy_zgcd @ K @ ( legacy_zgcd @ M @ 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__zadd__zmult,axiom,
! [M: int,N: int,K: int] :
( ( legacy_zgcd @ ( plus_plus_int @ M @ ( times_times_int @ N @ K ) ) @ N )
= ( legacy_zgcd @ M @ N ) ) ).
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__self,axiom,
! [M: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( legacy_zgcd @ M @ M )
= M ) ) ).
thf(fact_4502_zgcd__geq__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( legacy_zgcd @ X @ Y ) ) ).
thf(fact_4503_zgcd__1__power__distrib,axiom,
! [N: nat,A_1: int,B_1: int] :
( ( ( legacy_zgcd @ A_1 @ B_1 )
= one_one_int )
=> ( ( legacy_zgcd @ ( power_power_int @ A_1 @ N ) @ ( power_power_int @ B_1 @ N ) )
= one_one_int ) ) ).
thf(fact_4504_zgcd__1__power__left__distrib,axiom,
! [N: nat,A_1: int,B_1: int] :
( ( ( legacy_zgcd @ A_1 @ B_1 )
= one_one_int )
=> ( ( legacy_zgcd @ ( power_power_int @ A_1 @ N ) @ B_1 )
= one_one_int ) ) ).
thf(fact_4505_zgcd__zmult__distrib2__abs,axiom,
! [K: int,M: int,N: int] :
( ( legacy_zgcd @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
= ( times_times_int @ ( abs_abs_int @ K ) @ ( legacy_zgcd @ M @ N ) ) ) ).
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__zmult__eq__self2,axiom,
! [N: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( legacy_zgcd @ ( times_times_int @ K @ N ) @ K )
= K ) ) ).
thf(fact_4509_zgcd__zmult__distrib2,axiom,
! [M: int,N: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( times_times_int @ K @ ( legacy_zgcd @ M @ N ) )
= ( legacy_zgcd @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ) ).
thf(fact_4510_zgcd__zmult__eq__self,axiom,
! [N: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( legacy_zgcd @ K @ ( times_times_int @ K @ N ) )
= K ) ) ).
thf(fact_4511_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_4512_div__zgcd__relprime,axiom,
! [B_1: int,A_1: int] :
( ( ( A_1 != zero_zero_int )
| ( B_1 != zero_zero_int ) )
=> ( ( legacy_zgcd @ ( div_div_int @ A_1 @ ( legacy_zgcd @ A_1 @ B_1 ) ) @ ( div_div_int @ B_1 @ ( legacy_zgcd @ A_1 @ B_1 ) ) )
= one_one_int ) ) ).
thf(fact_4513_zcong__zgcd__zmult__zmod,axiom,
! [N: int,A_1: int,B_1: int,M: int] :
( ( zcong @ A_1 @ B_1 @ M )
=> ( ( zcong @ A_1 @ B_1 @ N )
=> ( ( ( legacy_zgcd @ M @ N )
= one_one_int )
=> ( zcong @ A_1 @ B_1 @ ( times_times_int @ M @ N ) ) ) ) ) ).
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_zgcd__zmult__zdvd__zgcd,axiom,
! [M: int,K: int,N: int] :
( ( ( legacy_zgcd @ K @ N )
= one_one_int )
=> ( dvd_dvd_int @ ( legacy_zgcd @ ( times_times_int @ K @ M ) @ N ) @ ( legacy_zgcd @ M @ N ) ) ) ).
thf(fact_4516_zrelprime__zdvd__zmult,axiom,
! [M: int,N: int,K: int] :
( ( ( legacy_zgcd @ N @ K )
= one_one_int )
=> ( ( dvd_dvd_int @ K @ ( times_times_int @ M @ N ) )
=> ( dvd_dvd_int @ K @ M ) ) ) ).
thf(fact_4517_zrelprime__dvd__mult,axiom,
! [K: int,I: int,J: int] :
( ( ( legacy_zgcd @ I @ J )
= one_one_int )
=> ( ( dvd_dvd_int @ I @ ( times_times_int @ K @ J ) )
=> ( dvd_dvd_int @ I @ K ) ) ) ).
thf(fact_4518_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_4519_zgcd__code,axiom,
! [K: int,L: int] :
( ( legacy_zgcd @ K @ L )
= ( abs_abs_int @ ( if_int @ ( L = zero_zero_int ) @ K @ ( legacy_zgcd @ L @ ( div_mod_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ) ) ).
thf(fact_4520_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_4521_zgcd1__iff__no__common__primedivisor,axiom,
! [A_1: int,B_1: int] :
( ( ( legacy_zgcd @ A_1 @ B_1 )
= one_one_int )
<=> ~ ? [P_4: int] :
( ( zprime @ P_4 )
& ( dvd_dvd_int @ P_4 @ A_1 )
& ( dvd_dvd_int @ P_4 @ B_1 ) ) ) ).
thf(fact_4522_zgcd__zcong__zgcd,axiom,
! [B_1: int,A_1: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( legacy_zgcd @ A_1 @ M )
= one_one_int )
=> ( ( zcong @ A_1 @ B_1 @ M )
=> ( ( legacy_zgcd @ B_1 @ M )
= one_one_int ) ) ) ) ).
thf(fact_4523_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_4524_zcong__cancel2,axiom,
! [A_1: int,B_1: int,K: int,M: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ( legacy_zgcd @ K @ M )
= one_one_int )
=> ( ( zcong @ ( times_times_int @ K @ A_1 ) @ ( times_times_int @ K @ B_1 ) @ M )
<=> ( zcong @ A_1 @ B_1 @ M ) ) ) ) ).
thf(fact_4525_zcong__cancel,axiom,
! [A_1: int,B_1: int,K: int,M: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ( legacy_zgcd @ K @ M )
= one_one_int )
=> ( ( zcong @ ( times_times_int @ A_1 @ K ) @ ( times_times_int @ B_1 @ K ) @ M )
<=> ( zcong @ A_1 @ B_1 @ M ) ) ) ) ).
thf(fact_4526_zrelprime__zdvd__zmult__aux,axiom,
! [M: int,N: int,K: int] :
( ( ( legacy_zgcd @ N @ K )
= one_one_int )
=> ( ( dvd_dvd_int @ K @ ( times_times_int @ M @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( dvd_dvd_int @ K @ M ) ) ) ) ).
thf(fact_4527_Bnor__mem__if,axiom,
! [A_1: int,B_1: int,M: int] :
( ( ( legacy_zgcd @ B_1 @ M )
= one_one_int )
=> ( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_eq_int @ B_1 @ A_1 )
=> ( member_int @ B_1 @ ( bnorRset @ A_1 @ M ) ) ) ) ) ).
thf(fact_4528_int__relprime__odd__power__divisors,axiom,
! [A_1: int,B_1: int,C: int,X: int] :
( ( ( times_times_int @ A_1 @ B_1 )
= ( power_power_int @ C @ ( nat_1 @ X ) ) )
=> ( ( ord_less_nat @ one_one_nat @ ( nat_1 @ X ) )
=> ( ( member_int @ X @ zOdd )
=> ( ( ( legacy_zgcd @ A_1 @ B_1 )
= one_one_int )
=> ? [K_1: int] :
( A_1
= ( power_power_int @ K_1 @ ( nat_1 @ X ) ) ) ) ) ) ) ).
thf(fact_4529_xzgcd__correct,axiom,
! [M: int,K: int,N: int] :
( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( legacy_zgcd @ M @ N )
= K )
<=> ? [S_2: int,T_1: int] :
( ( xzgcd @ M @ N )
= ( produc282740534nt_int @ K @ ( product_Pair_int_int @ S_2 @ T_1 ) ) ) ) ) ).
thf(fact_4530_int__relprime__power__divisors,axiom,
! [A_1: int,B_1: int,C: int,N: nat] :
( ( ( times_times_int @ A_1 @ B_1 )
= ( power_power_int @ C @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ( legacy_zgcd @ A_1 @ B_1 )
= one_one_int )
=> ? [K_1: int] :
( ( abs_abs_int @ A_1 )
= ( power_power_int @ K_1 @ N ) ) ) ) ) ).
thf(fact_4531_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: int] :
( ? [Sn: int,Tn: int] :
( ( xzgcda @ M @ N @ R_3 @ R_1 @ S_3 @ S_1 @ T_2 @ T )
= ( produc282740534nt_int @ K @ ( product_Pair_int_int @ Sn @ Tn ) ) )
=> ( ( ord_less_int @ zero_zero_int @ R_1 )
=> ( ( legacy_zgcd @ R_3 @ R_1 )
= K ) ) ) ).
thf(fact_4532_zcong__lineq__unique,axiom,
! [B_1: int,A_1: int,N: int] :
( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( legacy_zgcd @ A_1 @ 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_1 @ X_1 ) @ B_1 @ N )
& ! [Y_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Y_1 )
& ( ord_less_int @ Y_1 @ N )
& ( zcong @ ( times_times_int @ A_1 @ Y_1 ) @ B_1 @ N ) )
=> ( Y_1 = X_1 ) ) ) ) ) ).
thf(fact_4533_zcong__lineq__ex,axiom,
! [A_1: int,N: int] :
( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( legacy_zgcd @ A_1 @ N )
= one_one_int )
=> ? [X_1: int] : ( zcong @ ( times_times_int @ A_1 @ X_1 ) @ one_one_int @ N ) ) ) ).
thf(fact_4534_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: int] :
( ( ( legacy_zgcd @ R_3 @ R_1 )
= K )
=> ( ( 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 @ ( product_Pair_int_int @ Sn @ Tn ) ) ) ) ) ).
thf(fact_4535_int__triple__relprime__odd__power__divisors,axiom,
! [A_1: int,B_1: int,C: int,D_1: int,X: int] :
( ( ( times_times_int @ ( times_times_int @ A_1 @ B_1 ) @ C )
= ( power_power_int @ D_1 @ ( nat_1 @ X ) ) )
=> ( ( ord_less_nat @ one_one_nat @ ( nat_1 @ X ) )
=> ( ( member_int @ X @ zOdd )
=> ( ( ( legacy_zgcd @ A_1 @ B_1 )
= one_one_int )
=> ( ( ( legacy_zgcd @ B_1 @ C )
= one_one_int )
=> ( ( ( legacy_zgcd @ C @ A_1 )
= one_one_int )
=> ? [K_1: int,L_1: int,M_2: int] :
( ( A_1
= ( power_power_int @ K_1 @ ( nat_1 @ X ) ) )
& ( B_1
= ( power_power_int @ L_1 @ ( nat_1 @ X ) ) )
& ( C
= ( power_power_int @ M_2 @ ( nat_1 @ X ) ) ) ) ) ) ) ) ) ) ).
thf(fact_4536_zgcd__ex__linear,axiom,
! [M: int,K: int,N: int] :
( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( legacy_zgcd @ M @ N )
= K )
=> ? [S_2: int,T_1: int] :
( K
= ( plus_plus_int @ ( times_times_int @ S_2 @ M ) @ ( times_times_int @ T_1 @ N ) ) ) ) ) ).
thf(fact_4537_int__triple__relprime__power__divisors,axiom,
! [A_1: int,B_1: int,C: int,D_1: int,N: nat] :
( ( ( times_times_int @ ( times_times_int @ A_1 @ B_1 ) @ C )
= ( power_power_int @ D_1 @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ( legacy_zgcd @ A_1 @ B_1 )
= one_one_int )
=> ( ( ( legacy_zgcd @ B_1 @ C )
= one_one_int )
=> ( ( ( legacy_zgcd @ C @ A_1 )
= one_one_int )
=> ? [K_1: int,L_1: int,M_2: int] :
( ( ( abs_abs_int @ A_1 )
= ( power_power_int @ K_1 @ N ) )
& ( ( abs_abs_int @ B_1 )
= ( power_power_int @ L_1 @ N ) )
& ( ( abs_abs_int @ C )
= ( power_power_int @ M_2 @ N ) ) ) ) ) ) ) ) ).
thf(fact_4538_make__zrelprime,axiom,
! [B_1: int,A_1: int] :
( ( ( A_1 != zero_zero_int )
| ( B_1 != zero_zero_int ) )
=> ? [C_2: int,D_2: int] :
( ( A_1
= ( times_times_int @ ( legacy_zgcd @ A_1 @ B_1 ) @ C_2 ) )
& ( B_1
= ( times_times_int @ ( legacy_zgcd @ A_1 @ B_1 ) @ D_2 ) )
& ( ( legacy_zgcd @ C_2 @ D_2 )
= one_one_int ) ) ) ).
thf(fact_4539_pair__lessI2,axiom,
! [S_1: nat,T: nat,A_1: nat,B_1: nat] :
( ( ord_less_eq_nat @ A_1 @ B_1 )
=> ( ( ord_less_nat @ S_1 @ T )
=> ( member180897546at_nat @ ( produc494345619at_nat @ ( product_Pair_nat_nat @ A_1 @ S_1 ) @ ( product_Pair_nat_nat @ B_1 @ T ) ) @ pair_less ) ) ) ).
thf(fact_4540_pair__lessI1,axiom,
! [S_1: nat,T: nat,A_1: nat,B_1: nat] :
( ( ord_less_nat @ A_1 @ B_1 )
=> ( member180897546at_nat @ ( produc494345619at_nat @ ( product_Pair_nat_nat @ A_1 @ S_1 ) @ ( product_Pair_nat_nat @ B_1 @ T ) ) @ pair_less ) ) ).
thf(fact_4541_pair__leqI2,axiom,
! [S_1: nat,T: nat,A_1: nat,B_1: nat] :
( ( ord_less_eq_nat @ A_1 @ B_1 )
=> ( ( ord_less_eq_nat @ S_1 @ T )
=> ( member180897546at_nat @ ( produc494345619at_nat @ ( product_Pair_nat_nat @ A_1 @ S_1 ) @ ( product_Pair_nat_nat @ B_1 @ T ) ) @ pair_leq ) ) ) ).
thf(fact_4542_pair__leqI1,axiom,
! [S_1: nat,T: nat,A_1: nat,B_1: nat] :
( ( ord_less_nat @ A_1 @ B_1 )
=> ( member180897546at_nat @ ( produc494345619at_nat @ ( product_Pair_nat_nat @ A_1 @ S_1 ) @ ( product_Pair_nat_nat @ B_1 @ T ) ) @ pair_leq ) ) ).
thf(fact_4543_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_4544_dist__real__def,axiom,
! [X: real,Y: real] :
( ( dist_dist_real @ X @ Y )
= ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) ) ).
thf(fact_4545_dist__complex__def,axiom,
! [X: complex,Y: complex] :
( ( dist_dist_complex @ X @ Y )
= ( norm_norm_complex @ ( minus_minus_complex @ X @ Y ) ) ) ).
thf(fact_4546_DERIV__nonneg__imp__nonincreasing,axiom,
! [F: real > real,A_1: real,B_1: real] :
( ( ord_less_eq_real @ A_1 @ B_1 )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A_1 @ X_1 )
& ( ord_less_eq_real @ X_1 @ B_1 ) )
=> ? [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_1 ) @ ( F @ B_1 ) ) ) ) ).
thf(fact_4547_DERIV__nonpos__imp__nonincreasing,axiom,
! [F: real > real,A_1: real,B_1: real] :
( ( ord_less_eq_real @ A_1 @ B_1 )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A_1 @ X_1 )
& ( ord_less_eq_real @ X_1 @ B_1 ) )
=> ? [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_1 ) @ ( F @ A_1 ) ) ) ) ).
thf(fact_4548_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_4549_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_4550_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_4551_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_4552_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_4553_number__of__code__int__def,axiom,
number1226105091de_int = quickcheck_of_int ).
thf(fact_4554_int__of__of__int,axiom,
! [N: int] :
( ( quickcheck_int_of @ ( quickcheck_of_int @ N ) )
= N ) ).
thf(fact_4555_of__int__int__of,axiom,
! [K: quickcheck_code_int] :
( ( quickcheck_of_int @ ( quickcheck_int_of @ K ) )
= K ) ).
thf(fact_4556_int__of__inverse,axiom,
! [X: quickcheck_code_int] :
( ( quickcheck_of_int @ ( quickcheck_int_of @ X ) )
= X ) ).
thf(fact_4557_Quickcheck__Narrowing_Oof__int__inject,axiom,
! [N: int,M: int] :
( ( ( quickcheck_of_int @ N )
= ( quickcheck_of_int @ M ) )
<=> ( N = M ) ) ).
thf(fact_4558_fact__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( fact @ M ) @ ( fact @ N ) ) ) ).
thf(fact_4559_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_4560_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_4561_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_4562_zero__code__int__def,axiom,
( zero_z891286103de_int
= ( quickcheck_of_int @ zero_zero_int ) ) ).
thf(fact_4563_fact__lt,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( fact @ N ) ) ).
thf(fact_4564_fact_Osimps_I1_J,axiom,
( ( fact @ zero_zero_nat )
= one_one_nat ) ).
thf(fact_4565_fact_Osimps_I2_J,axiom,
! [N: nat] :
( ( fact @ ( suc @ N ) )
= ( times_times_nat @ ( suc @ N ) @ ( fact @ N ) ) ) ).
thf(fact_4566_fact__le,axiom,
! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( fact @ N ) ) ).
thf(fact_4567_one__code__int__def,axiom,
( one_on1684967323de_int
= ( quickcheck_of_int @ one_one_int ) ) ).
thf(fact_4568_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_4569_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_4570_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_1: nat] :
( ( F @ ( plus_plus_nat @ I_1 @ K_1 ) )
= ( F @ I_1 ) ) ) ).
thf(fact_4571_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_4572_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_4573_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_4574_norRRset__def,axiom,
! [M: int] :
( ( norRRset @ M )
= ( bnorRset @ ( minus_minus_int @ M @ one_one_int ) @ M ) ) ).
thf(fact_4575_number__of__real__def,axiom,
! [X: int] :
( ( number267125858f_real @ X )
= ( ring_1_of_int_real @ X ) ) ).
thf(fact_4576_int__number__of__def,axiom,
! [W: int] :
( ( number_number_of_int @ W )
= ( ring_1_of_int_int @ W ) ) ).
thf(fact_4577_complex__number__of__def,axiom,
! [W: int] :
( ( number528085621omplex @ W )
= ( ring_11397209091omplex @ W ) ) ).
thf(fact_4578_of__int__int__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_int @ ( semiri1621563631at_int @ N ) )
= ( semiri1621563631at_int @ N ) ) ).
thf(fact_4579_complex__cnj__of__int,axiom,
! [Z_1: int] :
( ( cnj @ ( ring_11397209091omplex @ Z_1 ) )
= ( ring_11397209091omplex @ Z_1 ) ) ).
thf(fact_4580_complex__Re__of__int,axiom,
! [Z_1: int] :
( ( re @ ( ring_11397209091omplex @ Z_1 ) )
= ( ring_1_of_int_real @ Z_1 ) ) ).
thf(fact_4581_complex__Im__of__int,axiom,
! [Z_1: int] :
( ( im @ ( ring_11397209091omplex @ Z_1 ) )
= zero_zero_real ) ).
thf(fact_4582_real__eq__of__int,axiom,
real_int = ring_1_of_int_real ).
thf(fact_4583_real__of__int__def,axiom,
real_int = ring_1_of_int_real ).
thf(fact_4584_phi__def,axiom,
! [M: int] :
( ( phi @ M )
= ( finite_card_int @ ( norRRset @ M ) ) ) ).
thf(fact_4585_norR__mem__unique,axiom,
! [A_1: int,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ( legacy_zgcd @ A_1 @ M )
= one_one_int )
=> ? [X_1: int] :
( ( zcong @ A_1 @ X_1 @ M )
& ( member_int @ X_1 @ ( norRRset @ M ) )
& ! [Y_1: int] :
( ( ( zcong @ A_1 @ Y_1 @ M )
& ( member_int @ Y_1 @ ( norRRset @ M ) ) )
=> ( Y_1 = X_1 ) ) ) ) ) ).
thf(fact_4586_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_4587_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_4588_RRset__gcd,axiom,
! [A_1: int,A: int > $o,M: int] :
( ( is_RRset @ A @ M )
=> ( ( member_int @ A_1 @ A )
=> ( ( legacy_zgcd @ A_1 @ M )
= one_one_int ) ) ) ).
thf(fact_4589_RRset__zcong__eq,axiom,
! [A_1: int,B_1: int,A: int > $o,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( is_RRset @ A @ M )
=> ( ( zcong @ A_1 @ B_1 @ M )
=> ( ( member_int @ A_1 @ A )
=> ( ( member_int @ B_1 @ A )
=> ( A_1 = B_1 ) ) ) ) ) ) ).
thf(fact_4590_RRset2norRR__correct,axiom,
! [A_1: int,A: int > $o,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( is_RRset @ A @ M )
=> ( ( member_int @ A_1 @ A )
=> ( ( zcong @ A_1 @ ( rRset2norRR @ A @ M @ A_1 ) @ M )
& ( member_int @ ( rRset2norRR @ A @ M @ A_1 ) @ ( norRRset @ M ) ) ) ) ) ) ).
thf(fact_4591_RRset2norRR__correct2,axiom,
! [A_1: int,A: int > $o,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( is_RRset @ A @ M )
=> ( ( member_int @ A_1 @ A )
=> ( member_int @ ( rRset2norRR @ A @ M @ A_1 ) @ ( norRRset @ M ) ) ) ) ) ).
thf(fact_4592_RRset2norRR__correct1,axiom,
! [A_1: int,A: int > $o,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( is_RRset @ A @ M )
=> ( ( member_int @ A_1 @ A )
=> ( zcong @ A_1 @ ( rRset2norRR @ A @ M @ A_1 ) @ M ) ) ) ) ).
thf(fact_4593_is__RRset__def,axiom,
! [A: int > $o,M: int] :
( ( is_RRset @ A @ M )
<=> ( ( member_int_o @ A @ ( rsetR @ M ) )
& ( ( finite_card_int @ A )
= ( phi @ M ) ) ) ) ).
thf(fact_4594_RRset2norRR__eq__norR,axiom,
! [A: int > $o,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( is_RRset @ A @ M )
=> ( ( image_int_int @ ( rRset2norRR @ A @ M ) @ A )
= ( norRRset @ M ) ) ) ) ).
thf(fact_4595_RsetR__fin,axiom,
! [A: int > $o,M: int] :
( ( member_int_o @ A @ ( rsetR @ M ) )
=> ( finite_finite_int @ A ) ) ).
thf(fact_4596_Bnor__in__RsetR,axiom,
! [A_1: int,M: int] :
( ( ord_less_int @ A_1 @ M )
=> ( member_int_o @ ( bnorRset @ A_1 @ M ) @ ( rsetR @ M ) ) ) ).
thf(fact_4597_ResSet__image,axiom,
! [F: int > int,A: int > $o,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( resSet @ M @ A )
=> ( ! [X_1: int] :
( ( member_int @ X_1 @ A )
=> ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( zcong @ ( F @ X_1 ) @ ( F @ Xa ) @ M )
=> ( X_1 = Xa ) ) ) )
=> ( resSet @ M @ ( image_int_int @ F @ A ) ) ) ) ) ).
thf(fact_4598_transfer__nat__int__set__relations_I1_J,axiom,
! [A: nat > $o] :
( ( finite_finite_nat @ A )
<=> ( finite_finite_int @ ( image_nat_int @ semiri1621563631at_int @ A ) ) ) ).
thf(fact_4599_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J,axiom,
! [A: nat > $o] :
( ( finite_card_nat @ A )
= ( finite_card_int @ ( image_nat_int @ semiri1621563631at_int @ A ) ) ) ).
thf(fact_4600_transfer__nat__int__set__relations_I3_J,axiom,
! [A: nat > $o,B: nat > $o] :
( ( A = B )
<=> ( ( image_nat_int @ semiri1621563631at_int @ A )
= ( image_nat_int @ semiri1621563631at_int @ B ) ) ) ).
thf(fact_4601_transfer__nat__int__set__relations_I2_J,axiom,
! [X: nat,A: nat > $o] :
( ( member_nat @ X @ A )
<=> ( member_int @ ( semiri1621563631at_int @ X ) @ ( image_nat_int @ semiri1621563631at_int @ A ) ) ) ).
thf(fact_4602_transfer__int__nat__set__return__embed,axiom,
! [A: nat > $o] :
( ( image_int_nat @ nat_1 @ ( image_nat_int @ semiri1621563631at_int @ A ) )
= A ) ).
thf(fact_4603_transfer__nat__int__set__relations_I4_J,axiom,
! [A: nat > $o,B: nat > $o] :
( ( ord_less_nat_o @ A @ B )
<=> ( ord_less_int_o @ ( image_nat_int @ semiri1621563631at_int @ A ) @ ( image_nat_int @ semiri1621563631at_int @ B ) ) ) ).
thf(fact_4604_transfer__nat__int__set__relations_I5_J,axiom,
! [A: nat > $o,B: nat > $o] :
( ( ord_less_eq_nat_o @ A @ B )
<=> ( ord_less_eq_int_o @ ( image_nat_int @ semiri1621563631at_int @ A ) @ ( image_nat_int @ semiri1621563631at_int @ B ) ) ) ).
thf(fact_4605_SetS__def,axiom,
! [A_1: int,P_3: int] :
( ( setS @ A_1 @ P_3 )
= ( image_int_int_o @ ( multInvPair @ A_1 @ P_3 ) @ ( sRStar @ P_3 ) ) ) ).
thf(fact_4606_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_4607_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_4608_transfer__int__nat__set__relations_I4_J,axiom,
! [B: int > $o,A: int > $o] :
( ( nat_nat_set @ A )
=> ( ( nat_nat_set @ B )
=> ( ( ord_less_int_o @ A @ B )
<=> ( ord_less_nat_o @ ( image_int_nat @ nat_1 @ A ) @ ( image_int_nat @ nat_1 @ B ) ) ) ) ) ).
thf(fact_4609_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_4610_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_4611_Nat__Transfer_Otransfer__nat__int__set__function__closures_I6_J,axiom,
! [X: int,A: int > $o] :
( ( nat_nat_set @ A )
=> ( ( member_int @ X @ A )
=> ( ord_less_eq_int @ zero_zero_int @ X ) ) ) ).
thf(fact_4612_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_4613_transfer__int__nat__set__relations_I3_J,axiom,
! [B: int > $o,A: int > $o] :
( ( nat_nat_set @ A )
=> ( ( nat_nat_set @ B )
=> ( ( A = B )
<=> ( ( image_int_nat @ nat_1 @ A )
= ( image_int_nat @ nat_1 @ B ) ) ) ) ) ).
thf(fact_4614_transfer__nat__int__set__return__embed,axiom,
! [A: int > $o] :
( ( nat_nat_set @ A )
=> ( ( image_nat_int @ semiri1621563631at_int @ ( image_int_nat @ nat_1 @ A ) )
= A ) ) ).
thf(fact_4615_transfer__int__nat__set__relations_I1_J,axiom,
! [A: int > $o] :
( ( nat_nat_set @ A )
=> ( ( finite_finite_int @ A )
<=> ( finite_finite_nat @ ( image_int_nat @ nat_1 @ A ) ) ) ) ).
thf(fact_4616_Nat__Transfer_Otransfer__int__nat__set__functions_I1_J,axiom,
! [A: int > $o] :
( ( nat_nat_set @ A )
=> ( ( finite_card_int @ A )
= ( finite_card_nat @ ( image_int_nat @ nat_1 @ A ) ) ) ) ).
thf(fact_4617_transfer__int__nat__set__relations_I5_J,axiom,
! [B: int > $o,A: int > $o] :
( ( nat_nat_set @ A )
=> ( ( nat_nat_set @ B )
=> ( ( ord_less_eq_int_o @ A @ B )
<=> ( ord_less_eq_nat_o @ ( image_int_nat @ nat_1 @ A ) @ ( image_int_nat @ nat_1 @ B ) ) ) ) ) ).
thf(fact_4618_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_4619_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_4620_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_4621_finite__greaterThanLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( ord_gr1297742076an_int @ L @ U ) ) ).
thf(fact_4622_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_4623_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_4624_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( ord_gr660468384an_nat @ L @ U ) ) ).
thf(fact_4625_Rats__real__nat,axiom,
! [N: nat] : ( member_real @ ( real_nat @ N ) @ field_1210416355s_real ) ).
thf(fact_4626_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( ord_gr660468384an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
thf(fact_4627_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_4628_DERIV__isconst3,axiom,
! [F: real > real,Y: real,X: real,A_1: real,B_1: real] :
( ( ord_less_real @ A_1 @ B_1 )
=> ( ( member_real @ X @ ( ord_gr788844697n_real @ A_1 @ B_1 ) )
=> ( ( member_real @ Y @ ( ord_gr788844697n_real @ A_1 @ B_1 ) )
=> ( ! [X_1: real] :
( ( member_real @ X_1 @ ( ord_gr788844697n_real @ A_1 @ B_1 ) )
=> ( deriv_real @ F @ X_1 @ zero_zero_real ) )
=> ( ( F @ X )
= ( F @ Y ) ) ) ) ) ) ).
thf(fact_4629_iszero__rat,axiom,
! [K: int] :
( ( iszero_rat @ ( number_number_of_rat @ K ) )
<=> ( iszero_int @ ( number_number_of_int @ K ) ) ) ).
thf(fact_4630_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_4631_Sup__greaterThanLessThan,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ Y @ X )
=> ( ( comple124823625p_real @ ( ord_gr788844697n_real @ Y @ X ) )
= X ) ) ).
thf(fact_4632_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_4633_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_4634_Frct__code__post_I5_J,axiom,
! [K: int] :
( ( frct @ ( product_Pair_int_int @ one_one_int @ ( number_number_of_int @ K ) ) )
= ( inverse_divide_rat @ one_one_rat @ ( number_number_of_rat @ K ) ) ) ).
thf(fact_4635_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_4636_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_4637_Frct__code__post_I2_J,axiom,
! [K: int] :
( ( frct @ ( product_Pair_int_int @ K @ zero_zero_int ) )
= zero_zero_rat ) ).
thf(fact_4638_Frct__code__post_I1_J,axiom,
! [K: int] :
( ( frct @ ( product_Pair_int_int @ zero_zero_int @ K ) )
= zero_zero_rat ) ).
thf(fact_4639_Frct__code__post_I3_J,axiom,
( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
= one_one_rat ) ).
thf(fact_4640_Frct__code__post_I6_J,axiom,
! [K: int,L: int] :
( ( frct @ ( product_Pair_int_int @ ( number_number_of_int @ K ) @ ( number_number_of_int @ L ) ) )
= ( inverse_divide_rat @ ( number_number_of_rat @ K ) @ ( number_number_of_rat @ L ) ) ) ).
thf(fact_4641_Frct__code__post_I4_J,axiom,
! [K: int] :
( ( frct @ ( product_Pair_int_int @ ( number_number_of_int @ K ) @ one_one_int ) )
= ( number_number_of_rat @ K ) ) ).
thf(fact_4642_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_4643_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_4644_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_4645_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_4646_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_4647_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_4648_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_4649_quotient__of__inject__eq,axiom,
! [A_1: rat,B_1: rat] :
( ( ( quotient_of @ A_1 )
= ( quotient_of @ B_1 ) )
<=> ( A_1 = B_1 ) ) ).
thf(fact_4650_quotient__of__inject,axiom,
! [A_1: rat,B_1: rat] :
( ( ( quotient_of @ A_1 )
= ( quotient_of @ B_1 ) )
=> ( A_1 = B_1 ) ) ).
thf(fact_4651_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_4652_rat__one__code,axiom,
( ( quotient_of @ one_one_rat )
= ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
thf(fact_4653_rat__zero__code,axiom,
( ( quotient_of @ zero_zero_rat )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
thf(fact_4654_quotient__of__number_I3_J,axiom,
! [K: int] :
( ( quotient_of @ ( number_number_of_rat @ K ) )
= ( product_Pair_int_int @ ( number_number_of_int @ K ) @ one_one_int ) ) ).
thf(fact_4655_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_4656_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_4657_gcd__eq__nitpick__gcd,axiom,
! [X: nat,Y: nat] :
( ( gcd_gcd_nat @ X @ Y )
= ( nat_gcd @ X @ Y ) ) ).
thf(fact_4658_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_4659_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_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__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_4662_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_4663_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_4664_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_4665_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_4666_Fract__1__number__of,axiom,
! [K: int] :
( ( fract @ one_one_int @ ( number_number_of_int @ K ) )
= ( inverse_divide_rat @ one_one_rat @ ( number_number_of_rat @ K ) ) ) ).
thf(fact_4667_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_4668_quotient__of__eq,axiom,
! [A_1: int,B_1: int,P_3: int,Q: int] :
( ( ( quotient_of @ ( fract @ A_1 @ B_1 ) )
= ( product_Pair_int_int @ P_3 @ Q ) )
=> ( ( fract @ P_3 @ Q )
= ( fract @ A_1 @ B_1 ) ) ) ).
thf(fact_4669_minus__rat,axiom,
! [A_1: int,B_1: int] :
( ( uminus_uminus_rat @ ( fract @ A_1 @ B_1 ) )
= ( fract @ ( uminus_uminus_int @ A_1 ) @ B_1 ) ) ).
thf(fact_4670_eq__rat_I2_J,axiom,
! [A_1: int] :
( ( fract @ A_1 @ zero_zero_int )
= ( fract @ zero_zero_int @ one_one_int ) ) ).
thf(fact_4671_eq__rat_I1_J,axiom,
! [A_1: int,C: int,D_1: int,B_1: int] :
( ( B_1 != zero_zero_int )
=> ( ( D_1 != zero_zero_int )
=> ( ( ( fract @ A_1 @ B_1 )
= ( fract @ C @ D_1 ) )
<=> ( ( times_times_int @ A_1 @ D_1 )
= ( times_times_int @ C @ B_1 ) ) ) ) ) ).
thf(fact_4672_mult__rat__cancel,axiom,
! [A_1: int,B_1: int,C: int] :
( ( C != zero_zero_int )
=> ( ( fract @ ( times_times_int @ C @ A_1 ) @ ( times_times_int @ C @ B_1 ) )
= ( fract @ A_1 @ B_1 ) ) ) ).
thf(fact_4673_eq__rat_I3_J,axiom,
! [A_1: int,C: int] :
( ( fract @ zero_zero_int @ A_1 )
= ( fract @ zero_zero_int @ C ) ) ).
thf(fact_4674_rat__number__collapse_I4_J,axiom,
! [K: int] :
( ( fract @ K @ zero_zero_int )
= zero_zero_rat ) ).
thf(fact_4675_rat__number__collapse_I1_J,axiom,
! [K: int] :
( ( fract @ zero_zero_int @ K )
= zero_zero_rat ) ).
thf(fact_4676_inverse__rat,axiom,
! [A_1: int,B_1: int] :
( ( inverse_inverse_rat @ ( fract @ A_1 @ B_1 ) )
= ( fract @ B_1 @ A_1 ) ) ).
thf(fact_4677_minus__rat__cancel,axiom,
! [A_1: int,B_1: int] :
( ( fract @ ( uminus_uminus_int @ A_1 ) @ ( uminus_uminus_int @ B_1 ) )
= ( fract @ A_1 @ B_1 ) ) ).
thf(fact_4678_divide__rat,axiom,
! [A_1: int,B_1: int,C: int,D_1: int] :
( ( inverse_divide_rat @ ( fract @ A_1 @ B_1 ) @ ( fract @ C @ D_1 ) )
= ( fract @ ( times_times_int @ A_1 @ D_1 ) @ ( times_times_int @ B_1 @ C ) ) ) ).
thf(fact_4679_real__floor__code,axiom,
! [X: rat] :
( ( archim1246769320r_real @ ( ratreal @ X ) )
= ( archim791455193or_rat @ X ) ) ).
thf(fact_4680_floor__Fract,axiom,
! [A_1: int,B_1: int] :
( ( archim791455193or_rat @ ( fract @ A_1 @ B_1 ) )
= ( div_div_int @ A_1 @ B_1 ) ) ).
thf(fact_4681_Fract__of__int__eq,axiom,
! [K: int] :
( ( fract @ K @ one_one_int )
= ( ring_1_of_int_rat @ K ) ) ).
thf(fact_4682_of__int__rat,axiom,
! [K: int] :
( ( ring_1_of_int_rat @ K )
= ( fract @ K @ one_one_int ) ) ).
thf(fact_4683_Fract__of__int__quotient,axiom,
! [K: int,L: int] :
( ( fract @ K @ L )
= ( inverse_divide_rat @ ( ring_1_of_int_rat @ K ) @ ( ring_1_of_int_rat @ L ) ) ) ).
thf(fact_4684_rat__number__collapse_I2_J,axiom,
( ( fract @ one_one_int @ one_one_int )
= one_one_rat ) ).
thf(fact_4685_one__rat,axiom,
( one_one_rat
= ( fract @ one_one_int @ one_one_int ) ) ).
thf(fact_4686_of__nat__rat,axiom,
! [K: nat] :
( ( semiri151668891at_rat @ K )
= ( fract @ ( semiri1621563631at_int @ K ) @ one_one_int ) ) ).
thf(fact_4687_Fract__of__nat__eq,axiom,
! [K: nat] :
( ( fract @ ( semiri1621563631at_int @ K ) @ one_one_int )
= ( semiri151668891at_rat @ K ) ) ).
thf(fact_4688_rat__number__of__def,axiom,
! [W: int] :
( ( number_number_of_rat @ W )
= ( fract @ W @ one_one_int ) ) ).
thf(fact_4689_mult__rat,axiom,
! [A_1: int,B_1: int,C: int,D_1: int] :
( ( times_times_rat @ ( fract @ A_1 @ B_1 ) @ ( fract @ C @ D_1 ) )
= ( fract @ ( times_times_int @ A_1 @ C ) @ ( times_times_int @ B_1 @ D_1 ) ) ) ).
thf(fact_4690_abs__rat,axiom,
! [A_1: int,B_1: int] :
( ( abs_abs_rat @ ( fract @ A_1 @ B_1 ) )
= ( fract @ ( abs_abs_int @ A_1 ) @ ( abs_abs_int @ B_1 ) ) ) ).
thf(fact_4691_normalize__eq,axiom,
! [A_1: int,B_1: int,P_3: int,Q: int] :
( ( ( normalize @ ( product_Pair_int_int @ A_1 @ B_1 ) )
= ( product_Pair_int_int @ P_3 @ Q ) )
=> ( ( fract @ P_3 @ Q )
= ( fract @ A_1 @ B_1 ) ) ) ).
thf(fact_4692_real__less__code,axiom,
! [X: rat,Y: rat] :
( ( ord_less_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
<=> ( ord_less_rat @ X @ Y ) ) ).
thf(fact_4693_Ratreal__number__collapse_I1_J,axiom,
( ( ratreal @ zero_zero_rat )
= zero_zero_real ) ).
thf(fact_4694_zero__real__code,axiom,
( zero_zero_real
= ( ratreal @ zero_zero_rat ) ) ).
thf(fact_4695_number__of__real__code,axiom,
! [K: int] :
( ( number267125858f_real @ K )
= ( ratreal @ ( number_number_of_rat @ K ) ) ) ).
thf(fact_4696_Ratreal__number__collapse_I3_J,axiom,
! [K: int] :
( ( ratreal @ ( number_number_of_rat @ K ) )
= ( number267125858f_real @ K ) ) ).
thf(fact_4697_one__real__code,axiom,
( one_one_real
= ( ratreal @ one_one_rat ) ) ).
thf(fact_4698_Ratreal__number__collapse_I2_J,axiom,
( ( ratreal @ one_one_rat )
= one_one_real ) ).
thf(fact_4699_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_4700_real__plus__code,axiom,
! [X: rat,Y: rat] :
( ( plus_plus_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
= ( ratreal @ ( plus_plus_rat @ X @ Y ) ) ) ).
thf(fact_4701_real__minus__code,axiom,
! [X: rat,Y: rat] :
( ( minus_minus_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
= ( ratreal @ ( minus_minus_rat @ X @ Y ) ) ) ).
thf(fact_4702_real__uminus__code,axiom,
! [X: rat] :
( ( uminus_uminus_real @ ( ratreal @ X ) )
= ( ratreal @ ( uminus_uminus_rat @ X ) ) ) ).
thf(fact_4703_real__divide__code,axiom,
! [X: rat,Y: rat] :
( ( inverse_divide_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
= ( ratreal @ ( inverse_divide_rat @ X @ Y ) ) ) ).
thf(fact_4704_real__times__code,axiom,
! [X: rat,Y: rat] :
( ( times_times_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
= ( ratreal @ ( times_times_rat @ X @ Y ) ) ) ).
thf(fact_4705_real__inverse__code,axiom,
! [X: rat] :
( ( inverse_inverse_real @ ( ratreal @ X ) )
= ( ratreal @ ( inverse_inverse_rat @ X ) ) ) ).
thf(fact_4706_zero__rat,axiom,
( zero_zero_rat
= ( fract @ zero_zero_int @ one_one_int ) ) ).
thf(fact_4707_rat__number__collapse_I3_J,axiom,
! [K: int] :
( ( fract @ ( number_number_of_int @ K ) @ one_one_int )
= ( number_number_of_rat @ K ) ) ).
thf(fact_4708_rat__number__expand_I3_J,axiom,
! [K: int] :
( ( number_number_of_rat @ K )
= ( fract @ ( number_number_of_int @ K ) @ one_one_int ) ) ).
thf(fact_4709_Fract__number__of__quotient,axiom,
! [K: int,L: int] :
( ( fract @ ( number_number_of_int @ K ) @ ( number_number_of_int @ L ) )
= ( inverse_divide_rat @ ( number_number_of_rat @ K ) @ ( number_number_of_rat @ L ) ) ) ).
thf(fact_4710_quotient__of__Fract,axiom,
! [A_1: int,B_1: int] :
( ( quotient_of @ ( fract @ A_1 @ B_1 ) )
= ( normalize @ ( product_Pair_int_int @ A_1 @ B_1 ) ) ) ).
thf(fact_4711_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_4712_less__rat,axiom,
! [A_1: int,C: int,D_1: int,B_1: int] :
( ( B_1 != zero_zero_int )
=> ( ( D_1 != zero_zero_int )
=> ( ( ord_less_rat @ ( fract @ A_1 @ B_1 ) @ ( fract @ C @ D_1 ) )
<=> ( ord_less_int @ ( times_times_int @ ( times_times_int @ A_1 @ D_1 ) @ ( times_times_int @ B_1 @ D_1 ) ) @ ( times_times_int @ ( times_times_int @ C @ B_1 ) @ ( times_times_int @ B_1 @ D_1 ) ) ) ) ) ) ).
thf(fact_4713_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_4714_sgn__rat,axiom,
! [A_1: int,B_1: int] :
( ( sgn_sgn_rat @ ( fract @ A_1 @ B_1 ) )
= ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A_1 ) @ ( sgn_sgn_int @ B_1 ) ) ) ) ).
thf(fact_4715_zero__less__Fract__iff,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A_1 @ B_1 ) )
<=> ( ord_less_int @ zero_zero_int @ A_1 ) ) ) ).
thf(fact_4716_Fract__less__zero__iff,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_rat @ ( fract @ A_1 @ B_1 ) @ zero_zero_rat )
<=> ( ord_less_int @ A_1 @ zero_zero_int ) ) ) ).
thf(fact_4717_le__rat,axiom,
! [A_1: int,C: int,D_1: int,B_1: int] :
( ( B_1 != zero_zero_int )
=> ( ( D_1 != zero_zero_int )
=> ( ( ord_less_eq_rat @ ( fract @ A_1 @ B_1 ) @ ( fract @ C @ D_1 ) )
<=> ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A_1 @ D_1 ) @ ( times_times_int @ B_1 @ D_1 ) ) @ ( times_times_int @ ( times_times_int @ C @ B_1 ) @ ( times_times_int @ B_1 @ D_1 ) ) ) ) ) ) ).
thf(fact_4718_add__rat,axiom,
! [A_1: int,C: int,D_1: int,B_1: int] :
( ( B_1 != zero_zero_int )
=> ( ( D_1 != zero_zero_int )
=> ( ( plus_plus_rat @ ( fract @ A_1 @ B_1 ) @ ( fract @ C @ D_1 ) )
= ( fract @ ( plus_plus_int @ ( times_times_int @ A_1 @ D_1 ) @ ( times_times_int @ C @ B_1 ) ) @ ( times_times_int @ B_1 @ D_1 ) ) ) ) ) ).
thf(fact_4719_diff__rat,axiom,
! [A_1: int,C: int,D_1: int,B_1: int] :
( ( B_1 != zero_zero_int )
=> ( ( D_1 != zero_zero_int )
=> ( ( minus_minus_rat @ ( fract @ A_1 @ B_1 ) @ ( fract @ C @ D_1 ) )
= ( fract @ ( minus_minus_int @ ( times_times_int @ A_1 @ D_1 ) @ ( times_times_int @ C @ B_1 ) ) @ ( times_times_int @ B_1 @ D_1 ) ) ) ) ) ).
thf(fact_4720_one__less__Fract__iff,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_rat @ one_one_rat @ ( fract @ A_1 @ B_1 ) )
<=> ( ord_less_int @ B_1 @ A_1 ) ) ) ).
thf(fact_4721_Fract__less__one__iff,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_rat @ ( fract @ A_1 @ B_1 ) @ one_one_rat )
<=> ( ord_less_int @ A_1 @ B_1 ) ) ) ).
thf(fact_4722_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_4723_zero__le__Fract__iff,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A_1 @ B_1 ) )
<=> ( ord_less_eq_int @ zero_zero_int @ A_1 ) ) ) ).
thf(fact_4724_Fract__le__zero__iff,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_eq_rat @ ( fract @ A_1 @ B_1 ) @ zero_zero_rat )
<=> ( ord_less_eq_int @ A_1 @ zero_zero_int ) ) ) ).
thf(fact_4725_rat__floor__lemma,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( div_div_int @ A_1 @ B_1 ) ) @ ( fract @ A_1 @ B_1 ) )
& ( ord_less_rat @ ( fract @ A_1 @ B_1 ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( div_div_int @ A_1 @ B_1 ) @ one_one_int ) ) ) ) ).
thf(fact_4726_one__le__Fract__iff,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A_1 @ B_1 ) )
<=> ( ord_less_eq_int @ B_1 @ A_1 ) ) ) ).
thf(fact_4727_Fract__le__one__iff,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ( ord_less_eq_rat @ ( fract @ A_1 @ B_1 ) @ one_one_rat )
<=> ( ord_less_eq_int @ A_1 @ B_1 ) ) ) ).
thf(fact_4728_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_4729_Ints__real__of__nat,axiom,
! [N: nat] : ( member_real @ ( real_nat @ N ) @ ring_1_Ints_real ) ).
thf(fact_4730_Ints__real__of__int,axiom,
! [X: int] : ( member_real @ ( real_int @ X ) @ ring_1_Ints_real ) ).
thf(fact_4731_adjust__eq,axiom,
! [B_1: int,Q: int,R_1: int] :
( ( adjust @ B_1 @ ( product_Pair_int_int @ Q @ R_1 ) )
= ( if_Pro1731782967nt_int @ ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ R_1 @ B_1 ) ) @ ( 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_1 ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Q ) @ R_1 ) ) ) ).
thf(fact_4732_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_4733_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N_1: nat] : ( ord_less_nat @ N_1 @ K ) ) ) ).
thf(fact_4734_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N_1: nat] : ( ord_less_eq_nat @ N_1 @ K ) ) ) ).
thf(fact_4735_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_4736_DERIV__cos__add,axiom,
! [K: real,Xa_1: real] :
( deriv_real
@ ^ [X_1: real] : ( cos @ ( plus_plus_real @ X_1 @ K ) )
@ Xa_1
@ ( uminus_uminus_real @ ( sin @ ( plus_plus_real @ Xa_1 @ K ) ) ) ) ).
thf(fact_4737_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_4738_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_4739_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_4740_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_4741_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_4742_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_4743_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_4744_less__eq__Suc__le__raw,axiom,
! [X_1: nat] :
( ( ord_less_nat @ X_1 )
= ( ord_less_eq_nat @ ( suc @ X_1 ) ) ) ).
thf(fact_4745_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_4746_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_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_DERIV__sin__add,axiom,
! [K: real,Xa_1: real] :
( deriv_real
@ ^ [X_1: real] : ( sin @ ( plus_plus_real @ X_1 @ K ) )
@ Xa_1
@ ( cos @ ( plus_plus_real @ Xa_1 @ K ) ) ) ).
thf(fact_4749_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_4750_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_4751_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_4752_primes__infinite,axiom,
~ ( finite_finite_nat @ ( collect_nat @ prime ) ) ).
thf(fact_4753_Re_OCauchy,axiom,
! [X_2: nat > complex] :
( ( cauchy_complex @ X_2 )
=> ( cauchy_real
@ ^ [N_1: nat] : ( re @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_4754_Im_OCauchy,axiom,
! [X_2: nat > complex] :
( ( cauchy_complex @ X_2 )
=> ( cauchy_real
@ ^ [N_1: nat] : ( im @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_4755_cnj_OCauchy,axiom,
! [X_2: nat > complex] :
( ( cauchy_complex @ X_2 )
=> ( cauchy_complex
@ ^ [N_1: nat] : ( cnj @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_4756_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_4757_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_4758_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_4759_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I_1: nat] : ( ord_less_nat @ I_1 @ N ) ) )
= N ) ).
thf(fact_4760_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_4761_bdd__nat__set__l__finite,axiom,
! [X: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [Y_1: nat] : ( ord_less_nat @ Y_1 @ X ) ) ) ).
thf(fact_4762_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_4763_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_4764_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_4765_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( P @ K_1 ) @ ( ord_less_nat @ K_1 @ I ) ) ) ) ).
thf(fact_4766_card__less__Suc2,axiom,
! [I: nat,M_3: nat > $o] :
( ~ ( member_nat @ zero_zero_nat @ M_3 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( member_nat @ ( suc @ K_1 ) @ M_3 ) @ ( ord_less_nat @ K_1 @ I ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( member_nat @ K_1 @ M_3 ) @ ( ord_less_nat @ K_1 @ ( suc @ I ) ) ) ) ) ) ) ).
thf(fact_4767_card__less,axiom,
! [I: nat,M_3: nat > $o] :
( ( member_nat @ zero_zero_nat @ M_3 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( member_nat @ K_1 @ M_3 ) @ ( ord_less_nat @ K_1 @ ( suc @ I ) ) ) ) )
!= zero_zero_nat ) ) ).
thf(fact_4768_card__less__Suc,axiom,
! [I: nat,M_3: nat > $o] :
( ( member_nat @ zero_zero_nat @ M_3 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( member_nat @ ( suc @ K_1 ) @ M_3 ) @ ( ord_less_nat @ K_1 @ I ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K_1: nat] : ( (&) @ ( member_nat @ K_1 @ M_3 ) @ ( ord_less_nat @ K_1 @ ( suc @ I ) ) ) ) ) ) ) ).
thf(fact_4769_transfer__morphism__int__nat,axiom,
( nat_tr160667106at_int @ semiri1621563631at_int
@ ^ [N_1: nat] : $true ) ).
thf(fact_4770_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_4771_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_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_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_4775_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_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_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_4779_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_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__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_4784_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_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: int > $o,M: int] :
( ( member_int_o @ A @ ( 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 )
@ ( 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_1: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( deriv_real @ ( log @ B_1 ) @ X @ ( inverse_divide_real @ one_one_real @ ( times_times_real @ ( ln @ B_1 ) @ 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_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_4796_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_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_norm__frac_Opsimps,axiom,
! [A_1: int,B_1: int] :
( ( accp_P2006205492nt_int @ norm_frac_rel @ ( product_Pair_int_int @ A_1 @ B_1 ) )
=> ( ( ( ord_less_int @ B_1 @ zero_zero_int )
=> ( ( norm_frac @ A_1 @ B_1 )
= ( norm_frac @ ( uminus_uminus_int @ A_1 ) @ ( uminus_uminus_int @ B_1 ) ) ) )
& ( ~ ( ord_less_int @ B_1 @ zero_zero_int )
=> ( ( ( ( A_1 = zero_zero_int )
| ( B_1 = zero_zero_int ) )
=> ( ( norm_frac @ A_1 @ B_1 )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) )
& ( ~ ( ( A_1 = zero_zero_int )
| ( B_1 = zero_zero_int ) )
=> ( ( norm_frac @ A_1 @ B_1 )
= ( product_Pair_int_int @ ( div_div_int @ A_1 @ ( int_gcd @ A_1 @ B_1 ) ) @ ( div_div_int @ B_1 @ ( int_gcd @ A_1 @ B_1 ) ) ) ) ) ) ) ) ) ).
thf(fact_4804_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_4805_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_4806_norm__frac_Osimps,axiom,
! [A_1: int,B_1: int] :
( ( ( ord_less_int @ B_1 @ zero_zero_int )
=> ( ( norm_frac @ A_1 @ B_1 )
= ( norm_frac @ ( uminus_uminus_int @ A_1 ) @ ( uminus_uminus_int @ B_1 ) ) ) )
& ( ~ ( ord_less_int @ B_1 @ zero_zero_int )
=> ( ( ( ( A_1 = zero_zero_int )
| ( B_1 = zero_zero_int ) )
=> ( ( norm_frac @ A_1 @ B_1 )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) )
& ( ~ ( ( A_1 = zero_zero_int )
| ( B_1 = zero_zero_int ) )
=> ( ( norm_frac @ A_1 @ B_1 )
= ( product_Pair_int_int @ ( div_div_int @ A_1 @ ( int_gcd @ A_1 @ B_1 ) ) @ ( div_div_int @ B_1 @ ( int_gcd @ A_1 @ B_1 ) ) ) ) ) ) ) ) ).
thf(fact_4807_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_4808_Union__SetS__setprod__prop1,axiom,
! [A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A_1 @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A_1 )
=> ( zcong
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( comple1092985777_int_o @ ( setS @ A_1 @ P_3 ) ) )
@ ( power_power_int @ A_1 @ ( nat_1 @ ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
@ P_3 ) ) ) ) ) ).
thf(fact_4809_d22set__prod__zfact,axiom,
! [A_1: int] :
( ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( d22set @ A_1 ) )
= ( zfact @ A_1 ) ) ).
thf(fact_4810_Bnor__prod__zgcd,axiom,
! [A_1: int,M: int] :
( ( ord_less_int @ A_1 @ M )
=> ( ( legacy_zgcd
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( bnorRset @ A_1 @ M ) )
@ M )
= one_one_int ) ) ).
thf(fact_4811_Bnor__prod__power,axiom,
! [A_1: int,M: int,X: int] :
( ( X != zero_zero_int )
=> ( ( ord_less_int @ A_1 @ M )
=> ( ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( image_int_int
@ ^ [A_2: int] : ( times_times_int @ A_2 @ X )
@ ( bnorRset @ A_1 @ M ) ) )
= ( times_times_int
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( bnorRset @ A_1 @ M ) )
@ ( power_power_int @ X @ ( finite_card_int @ ( bnorRset @ A_1 @ M ) ) ) ) ) ) ) ).
thf(fact_4812_SetS__setprod__prop,axiom,
! [X: int > $o,A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A_1 @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A_1 )
=> ( ( member_int_o @ X @ ( setS @ A_1 @ P_3 ) )
=> ( zcong
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ X )
@ A_1
@ P_3 ) ) ) ) ) ) ).
thf(fact_4813_wset__zcong__prod__1,axiom,
! [A_1: 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_1 @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( zcong
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( wset @ A_1 @ P_3 ) )
@ one_one_int
@ P_3 ) ) ) ) ).
thf(fact_4814_Union__SetS__setprod__prop2,axiom,
! [A_1: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A_1 @ zero_zero_int @ P_3 )
=> ( ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( comple1092985777_int_o @ ( setS @ A_1 @ P_3 ) ) )
= ( zfact @ ( minus_minus_int @ P_3 @ one_one_int ) ) ) ) ) ) ).
thf(fact_4815_transfer__nat__int__sum__prod__closure_I2_J,axiom,
! [F: int > int,A: int > $o] :
( ( nat_nat_set @ A )
=> ( ! [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 ) ) ) ) ).
thf(fact_4816_arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( arctan @ X )
= ( suminf_real
@ ^ [K_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ K_1 ) @ ( times_times_real @ ( inverse_divide_real @ one_one_real @ ( real_nat @ ( plus_plus_nat @ ( times_times_nat @ K_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
thf(fact_4817_DERIV__arctan__series,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( deriv_real
@ ^ [X_3: real] :
( suminf_real
@ ^ [K_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ K_1 ) @ ( times_times_real @ ( inverse_divide_real @ one_one_real @ ( real_nat @ ( plus_plus_nat @ ( times_times_nat @ K_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X_3 @ ( plus_plus_nat @ ( times_times_nat @ K_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) ) )
@ X
@ ( suminf_real
@ ^ [K_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ K_1 ) @ ( power_power_real @ X @ ( times_times_nat @ K_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_4818_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_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_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_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_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_4823_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_4824_pi__series,axiom,
( ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( suminf_real
@ ^ [K_1: nat] : ( inverse_divide_real @ ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ K_1 ) @ one_one_real ) @ ( real_nat @ ( plus_plus_nat @ ( times_times_nat @ K_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) ) ) ).
thf(fact_4825_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_4826_summable__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( summable_real
@ ^ [K_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ K_1 ) @ ( times_times_real @ ( inverse_divide_real @ one_one_real @ ( real_nat @ ( plus_plus_nat @ ( times_times_nat @ K_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) ) ) ) ).
thf(fact_4827_summable__rabs__cancel,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N_1: nat] : ( abs_abs_real @ ( F @ N_1 ) ) )
=> ( summable_real @ F ) ) ).
thf(fact_4828_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_4829_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_4830_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_4831_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_4832_rat__floor__code,axiom,
! [P_3: rat] :
( ( archim791455193or_rat @ P_3 )
= ( produc1298267108nt_int @ div_div_int @ ( quotient_of @ P_3 ) ) ) ).
thf(fact_4833_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_4834_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_4835_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_4836_divmod__int__rel__def,axiom,
! [A_1: int,B_1: int] :
( ( divmod_int_rel @ A_1 @ B_1 )
= ( produc450523309_int_o
@ ^ [Q_2: int,R: int] :
( (&)
@ ( A_1
= ( plus_plus_int @ ( times_times_int @ B_1 @ Q_2 ) @ R ) )
@ ( (&) @ ( (=>) @ ( ord_less_int @ zero_zero_int @ B_1 ) @ ( (&) @ ( ord_less_eq_int @ zero_zero_int @ R ) @ ( ord_less_int @ R @ B_1 ) ) ) @ ( (=>) @ ( (~) @ ( ord_less_int @ zero_zero_int @ B_1 ) ) @ ( (&) @ ( ord_less_int @ B_1 @ R ) @ ( ord_less_eq_int @ R @ zero_zero_int ) ) ) ) ) ) ) ).
thf(fact_4837_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_4838_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_4839_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_4840_adjust__def,axiom,
! [B_1: int] :
( ( adjust @ B_1 )
= ( produc1518849193nt_int
@ ^ [Q_2: int,R: int] : ( if_Pro1731782967nt_int @ ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ R @ B_1 ) ) @ ( 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_1 ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Q_2 ) @ R ) ) ) ) ).
thf(fact_4841_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_4842_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_4843_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_4844_cnj_Osummable,axiom,
! [X_2: nat > complex] :
( ( summable_complex @ X_2 )
=> ( summable_complex
@ ^ [N_1: nat] : ( cnj @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_4845_Re_Osummable,axiom,
! [X_2: nat > complex] :
( ( summable_complex @ X_2 )
=> ( summable_real
@ ^ [N_1: nat] : ( re @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_4846_Im_Osummable,axiom,
! [X_2: nat > complex] :
( ( summable_complex @ X_2 )
=> ( summable_real
@ ^ [N_1: nat] : ( im @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_4847_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_4848_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_4849_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_4850_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_4851_Bolzano__bisect_Osimps_I2_J,axiom,
! [P: produc914805421l_real > $o,A_1: real,B_1: real,N: nat] :
( ( bolzano_bisect @ P @ A_1 @ B_1 @ ( 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_1 @ B_1 @ N ) ) ) ).
thf(fact_4852_suminf__exist__split,axiom,
! [A_1: nat > real,R_1: real] :
( ( ord_less_real @ zero_zero_real @ R_1 )
=> ( ( summable_real @ A_1 )
=> ? [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_1 @ ( plus_plus_nat @ I_1 @ N_1 ) ) ) )
@ R_1 ) ) ) ) ).
thf(fact_4853_divmod__int__pdivmod,axiom,
! [L: int,K: int] :
( ( ( K = zero_zero_int )
=> ( ( divmod_int @ K @ L )
= ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) )
& ( ( K != zero_zero_int )
=> ( ( ( L = zero_zero_int )
=> ( ( divmod_int @ K @ L )
= ( product_Pair_int_int @ zero_zero_int @ K ) ) )
& ( ( L != zero_zero_int )
=> ( ( divmod_int @ K @ 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 ) ) @ ( (&) @ ( ord_less_int @ L @ zero_zero_int ) @ ( ord_less_int @ K @ zero_zero_int ) ) ) @ ( pdivmod @ K @ 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 @ L ) ) ) ) ) ) ) ) ) ).
thf(fact_4854_negateSnd__def,axiom,
( negateSnd
= ( produc713050258nt_int @ uminus_uminus_int ) ) ).
thf(fact_4855_divmod__int__code,axiom,
! [L: int,K: int] :
( ( ( K = zero_zero_int )
=> ( ( divmod_int @ K @ L )
= ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) )
& ( ( K != zero_zero_int )
=> ( ( ( L = zero_zero_int )
=> ( ( divmod_int @ K @ L )
= ( product_Pair_int_int @ zero_zero_int @ K ) ) )
& ( ( L != zero_zero_int )
=> ( ( divmod_int @ K @ L )
= ( produc713050258nt_int @ ( times_times_int @ ( sgn_sgn_int @ L ) )
@ ( if_Pro1731782967nt_int
@ ( ( sgn_sgn_int @ K )
= ( sgn_sgn_int @ L ) )
@ ( pdivmod @ K @ 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 @ L ) ) ) ) ) ) ) ) ) ).
thf(fact_4856_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_4857_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_4858_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_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_int__ge__less__than2__def,axiom,
! [D_1: int] :
( ( int_ge_less_than2 @ D_1 )
= ( collec1347809874nt_int
@ ( produc450523309_int_o
@ ^ [Z_2: int,Z: int] : ( (&) @ ( ord_less_eq_int @ D_1 @ Z ) @ ( ord_less_int @ Z_2 @ Z ) ) ) ) ) ).
thf(fact_4861_int__ge__less__than__def,axiom,
! [D_1: int] :
( ( int_ge_less_than @ D_1 )
= ( collec1347809874nt_int
@ ( produc450523309_int_o
@ ^ [Z_2: int,Z: int] : ( (&) @ ( ord_less_eq_int @ D_1 @ Z_2 ) @ ( ord_less_int @ Z_2 @ Z ) ) ) ) ) ).
thf(fact_4862_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_4863_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_4864_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_4865_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_4866_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_4867_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_4868_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_4869_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_4870_cnj_Osums,axiom,
! [X_2: nat > complex,A_1: complex] :
( ( sums_complex @ X_2 @ A_1 )
=> ( sums_complex
@ ^ [N_1: nat] : ( cnj @ ( X_2 @ N_1 ) )
@ ( cnj @ A_1 ) ) ) ).
thf(fact_4871_Re_Osums,axiom,
! [X_2: nat > complex,A_1: complex] :
( ( sums_complex @ X_2 @ A_1 )
=> ( sums_real
@ ^ [N_1: nat] : ( re @ ( X_2 @ N_1 ) )
@ ( re @ A_1 ) ) ) ).
thf(fact_4872_Im_Osums,axiom,
! [X_2: nat > complex,A_1: complex] :
( ( sums_complex @ X_2 @ A_1 )
=> ( sums_real
@ ^ [N_1: nat] : ( im @ ( X_2 @ N_1 ) )
@ ( im @ A_1 ) ) ) ).
thf(fact_4873_sin__fdiffs2,axiom,
! [N: nat] :
( ( diffs_real @ sin_coeff @ N )
= ( cos_coeff @ N ) ) ).
thf(fact_4874_sin__fdiffs,axiom,
( ( diffs_real @ sin_coeff )
= cos_coeff ) ).
thf(fact_4875_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_4876_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_4877_cos__fdiffs,axiom,
! [X_1: nat] :
( ( diffs_real @ cos_coeff @ X_1 )
= ( uminus_uminus_real @ ( sin_coeff @ X_1 ) ) ) ).
thf(fact_4878_cos__fdiffs2,axiom,
! [N: nat] :
( ( diffs_real @ cos_coeff @ N )
= ( uminus_uminus_real @ ( sin_coeff @ N ) ) ) ).
thf(fact_4879_diff__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( nat_case_nat @ zero_zero_nat
@ ^ [K_1: nat] : K_1
@ ( minus_minus_nat @ M @ N ) ) ) ).
thf(fact_4880_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_4881_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_4882_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_4883_ln__def,axiom,
! [X: real] :
( ( ln @ X )
= ( the_real
@ ^ [U_1: real] :
( ( exp_real @ U_1 )
= X ) ) ) ).
thf(fact_4884_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_4885_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_4886_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_4887_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_4888_divmod__nat__def,axiom,
! [M: nat,N: nat] :
( ( divmod_nat @ M @ N )
= ( the_Pr588456374at_nat @ ( divmod_nat_rel @ M @ N ) ) ) ).
thf(fact_4889_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_4890_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_4891_RRset2norRR__def,axiom,
! [A_1: int,A: int > $o,M: int] :
( ( ( ( ord_less_int @ one_one_int @ M )
& ( is_RRset @ A @ M )
& ( member_int @ A_1 @ A ) )
=> ( ( rRset2norRR @ A @ M @ A_1 )
= ( hilbert_Eps_int
@ ^ [B_4: int] : ( (&) @ ( zcong @ A_1 @ B_4 @ M ) @ ( member_int @ B_4 @ ( norRRset @ M ) ) ) ) ) )
& ( ~ ( ( ord_less_int @ one_one_int @ M )
& ( is_RRset @ A @ M )
& ( member_int @ A_1 @ A ) )
=> ( ( rRset2norRR @ A @ M @ A_1 )
= zero_zero_int ) ) ) ).
thf(fact_4892_aux__some,axiom,
! [A_1: int,A: int > $o,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( is_RRset @ A @ M )
=> ( ( member_int @ A_1 @ A )
=> ( ( zcong @ A_1
@ ( hilbert_Eps_int
@ ^ [B_4: int] : ( (&) @ ( zcong @ A_1 @ B_4 @ M ) @ ( member_int @ B_4 @ ( norRRset @ M ) ) ) )
@ M )
& ( member_int
@ ( hilbert_Eps_int
@ ^ [B_4: int] : ( (&) @ ( zcong @ A_1 @ B_4 @ M ) @ ( member_int @ B_4 @ ( norRRset @ M ) ) ) )
@ ( norRRset @ M ) ) ) ) ) ) ).
thf(fact_4893_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_4894_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_4895_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( ord_at4362885an_nat @ L @ U ) ) ).
thf(fact_4896_sumr__offset4,axiom,
! [K: 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 ) ) )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( F_2 @ ( plus_plus_nat @ M_2 @ K ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N_1 ) )
@ ( big_co604158596t_real @ F_2 @ ( ord_at4362885an_nat @ zero_zero_nat @ K ) ) ) ) ).
thf(fact_4897_sumr__offset2,axiom,
! [K: nat,N: nat,F_2: nat > real] :
( ( big_co604158596t_real
@ ^ [M_2: nat] : ( F_2 @ ( plus_plus_nat @ M_2 @ K ) )
@ ( 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 ) ) ) @ ( big_co604158596t_real @ F_2 @ ( ord_at4362885an_nat @ zero_zero_nat @ K ) ) ) ) ).
thf(fact_4898_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_4899_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_4900_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_4901_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_4902_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( ord_at4362885an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
thf(fact_4903_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_4904_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( ord_at4362885an_nat @ ( suc @ L ) @ U )
= ( ord_gr660468384an_nat @ L @ U ) ) ).
thf(fact_4905_image__add__atLeastLessThan,axiom,
! [K: nat,I: nat,J: nat] :
( ( image_nat_nat
@ ^ [N_1: nat] : ( plus_plus_nat @ N_1 @ K )
@ ( ord_at4362885an_nat @ I @ J ) )
= ( ord_at4362885an_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_4906_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_4907_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_4908_subset__card__intvl__is__intvl,axiom,
! [A: nat > $o,K: nat] :
( ( ord_less_eq_nat_o @ A @ ( ord_at4362885an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A ) ) ) )
=> ( A
= ( ord_at4362885an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A ) ) ) ) ) ).
thf(fact_4909_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_4910_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_4911_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_4912_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_4913_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_4914_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_4915_finite__atLeastLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( ord_at641636577an_int @ L @ U ) ) ).
thf(fact_4916_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite_int @ ( ord_at641636577an_int @ zero_zero_int @ 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_card__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( finite_card_int @ ( ord_at641636577an_int @ zero_zero_int @ U ) )
= ( nat_1 @ U ) ) ).
thf(fact_4919_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_4920_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_4921_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_4922_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_4923_Maclaurin__lemma2,axiom,
! [B: real,K: 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 ) )
=> ! [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 @ ( 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 @ ( 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_4924_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_4925_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_4926_sumr__pos__lt__pair,axiom,
! [K: nat,F: nat > real] :
( ( summable_real @ F )
=> ( ! [D_2: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D_2 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( 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 ) ) @ ( suminf_real @ F ) ) ) ) ).
thf(fact_4927_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_4928_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_4929_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_4930_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_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__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_4933_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_4934_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_4935_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_4936_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_4937_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_4938_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_4939_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_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_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_4942_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_4943_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_4944_transfer__nat__int__sum__prod__closure_I1_J,axiom,
! [F: int > int,A: int > $o] :
( ( nat_nat_set @ A )
=> ( ! [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 ) ) ) ) ).
thf(fact_4945_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_4946_arith__series__int,axiom,
! [A_1: int,D_1: int,N: nat] :
( ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) )
@ ( big_co1024481617at_int
@ ^ [I_1: nat] : ( plus_plus_int @ A_1 @ ( times_times_int @ ( semiri1621563631at_int @ I_1 ) @ D_1 ) )
@ ( ord_lessThan_nat @ N ) ) )
= ( times_times_int @ ( semiri1621563631at_int @ N ) @ ( plus_plus_int @ A_1 @ ( plus_plus_int @ A_1 @ ( times_times_int @ ( semiri1621563631at_int @ ( minus_minus_nat @ N @ one_one_nat ) ) @ D_1 ) ) ) ) ) ).
thf(fact_4947_BseqI2,axiom,
! [K_2: real,K: real,F: nat > real] :
( ! [N_1: nat] :
( ( ord_less_eq_real @ K @ ( F @ N_1 ) )
& ( ord_less_eq_real @ ( F @ N_1 ) @ K_2 ) )
=> ( bseq_real @ F ) ) ).
thf(fact_4948_finite__lessThan,axiom,
! [K: nat] : ( finite_finite_nat @ ( ord_lessThan_nat @ K ) ) ).
thf(fact_4949_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( ord_lessThan_nat @ U ) )
= U ) ).
thf(fact_4950_finite__nat__iff__bounded,axiom,
! [S: nat > $o] :
( ( finite_finite_nat @ S )
<=> ? [K_1: nat] : ( ord_less_eq_nat_o @ S @ ( ord_lessThan_nat @ K_1 ) ) ) ).
thf(fact_4951_atLeast0LessThan,axiom,
! [N: nat] :
( ( ord_at4362885an_nat @ zero_zero_nat @ N )
= ( ord_lessThan_nat @ N ) ) ).
thf(fact_4952_arith__series__nat,axiom,
! [A_1: nat,D_1: nat,N: nat] :
( ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) )
@ ( big_co387207925at_nat
@ ^ [I_1: nat] : ( plus_plus_nat @ A_1 @ ( times_times_nat @ I_1 @ D_1 ) )
@ ( ord_lessThan_nat @ N ) ) )
= ( times_times_nat @ N @ ( plus_plus_nat @ A_1 @ ( plus_plus_nat @ A_1 @ ( times_times_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ D_1 ) ) ) ) ) ).
thf(fact_4953_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_4954_finite__nat__bounded,axiom,
! [S: nat > $o] :
( ( finite_finite_nat @ S )
=> ? [K_1: nat] : ( ord_less_eq_nat_o @ S @ ( ord_lessThan_nat @ K_1 ) ) ) ).
thf(fact_4955_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_4956_Sup__lessThan,axiom,
! [X: real] :
( ( comple124823625p_real @ ( ord_lessThan_real @ X ) )
= X ) ).
thf(fact_4957_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_4958_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_4959_gcd__dvd2__int,axiom,
! [X: int,Y: int] : ( dvd_dvd_int @ ( gcd_gcd_int @ X @ Y ) @ Y ) ).
thf(fact_4960_gcd__dvd1__int,axiom,
! [X: int,Y: int] : ( dvd_dvd_int @ ( gcd_gcd_int @ X @ Y ) @ X ) ).
thf(fact_4961_coprime__exp2__int,axiom,
! [N: nat,M: nat,A_1: int,B_1: int] :
( ( ( gcd_gcd_int @ A_1 @ B_1 )
= one_one_int )
=> ( ( gcd_gcd_int @ ( power_power_int @ A_1 @ N ) @ ( power_power_int @ B_1 @ M ) )
= one_one_int ) ) ).
thf(fact_4962_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_4963_coprime__dvd__mult__iff__int,axiom,
! [M: int,K: int,N: int] :
( ( ( gcd_gcd_int @ K @ N )
= one_one_int )
=> ( ( dvd_dvd_int @ K @ ( times_times_int @ M @ N ) )
<=> ( dvd_dvd_int @ K @ M ) ) ) ).
thf(fact_4964_coprime__dvd__mult__int,axiom,
! [M: int,K: int,N: int] :
( ( ( gcd_gcd_int @ K @ N )
= one_one_int )
=> ( ( dvd_dvd_int @ K @ ( times_times_int @ M @ N ) )
=> ( dvd_dvd_int @ K @ M ) ) ) ).
thf(fact_4965_div__gcd__coprime__int,axiom,
! [B_1: int,A_1: int] :
( ( ( A_1 != zero_zero_int )
| ( B_1 != zero_zero_int ) )
=> ( ( gcd_gcd_int @ ( div_div_int @ A_1 @ ( gcd_gcd_int @ A_1 @ B_1 ) ) @ ( div_div_int @ B_1 @ ( gcd_gcd_int @ A_1 @ B_1 ) ) )
= one_one_int ) ) ).
thf(fact_4966_Fract__coprime,axiom,
! [A_1: int,B_1: int] :
( ( fract @ ( div_div_int @ A_1 @ ( gcd_gcd_int @ A_1 @ B_1 ) ) @ ( div_div_int @ B_1 @ ( gcd_gcd_int @ A_1 @ B_1 ) ) )
= ( fract @ A_1 @ B_1 ) ) ).
thf(fact_4967_gcd__0__int,axiom,
! [X: int] :
( ( gcd_gcd_int @ X @ zero_zero_int )
= ( abs_abs_int @ X ) ) ).
thf(fact_4968_gcd__0__left__int,axiom,
! [X: int] :
( ( gcd_gcd_int @ zero_zero_int @ X )
= ( abs_abs_int @ X ) ) ).
thf(fact_4969_gcd__add__mult__int,axiom,
! [M: int,K: int,N: int] :
( ( gcd_gcd_int @ M @ ( plus_plus_int @ ( times_times_int @ K @ M ) @ N ) )
= ( gcd_gcd_int @ M @ N ) ) ).
thf(fact_4970_gcd__mult__distrib__int,axiom,
! [K: int,M: int,N: int] :
( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
= ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).
thf(fact_4971_gcd__add1__int,axiom,
! [M: int,N: int] :
( ( gcd_gcd_int @ ( plus_plus_int @ M @ N ) @ N )
= ( gcd_gcd_int @ M @ N ) ) ).
thf(fact_4972_gcd__add2__int,axiom,
! [M: int,N: int] :
( ( gcd_gcd_int @ M @ ( plus_plus_int @ M @ N ) )
= ( gcd_gcd_int @ M @ N ) ) ).
thf(fact_4973_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_4974_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_4975_abs__gcd__int,axiom,
! [X: int,Y: int] :
( ( abs_abs_int @ ( gcd_gcd_int @ X @ Y ) )
= ( gcd_gcd_int @ X @ Y ) ) ).
thf(fact_4976_gcd__idem__int,axiom,
! [X: int] :
( ( gcd_gcd_int @ X @ X )
= ( abs_abs_int @ X ) ) ).
thf(fact_4977_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_4978_gcd__abs2__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ X @ ( abs_abs_int @ Y ) )
= ( gcd_gcd_int @ X @ Y ) ) ).
thf(fact_4979_gcd__abs1__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ ( abs_abs_int @ X ) @ Y )
= ( gcd_gcd_int @ X @ Y ) ) ).
thf(fact_4980_coprime__plus__one__int,axiom,
! [N: int] :
( ( gcd_gcd_int @ ( plus_plus_int @ N @ one_one_int ) @ N )
= one_one_int ) ).
thf(fact_4981_coprime__minus__one__int,axiom,
! [N: int] :
( ( gcd_gcd_int @ ( minus_minus_int @ N @ one_one_int ) @ N )
= one_one_int ) ).
thf(fact_4982_gcd__1__int,axiom,
! [M: int] :
( ( gcd_gcd_int @ M @ one_one_int )
= one_one_int ) ).
thf(fact_4983_coprime__divisors__nat,axiom,
! [E_1: int,B_1: int,D_1: int,A_1: int] :
( ( dvd_dvd_int @ D_1 @ A_1 )
=> ( ( dvd_dvd_int @ E_1 @ B_1 )
=> ( ( ( gcd_gcd_int @ A_1 @ B_1 )
= one_one_int )
=> ( ( gcd_gcd_int @ D_1 @ E_1 )
= one_one_int ) ) ) ) ).
thf(fact_4984_coprime__exp__int,axiom,
! [N: nat,D_1: int,A_1: int] :
( ( ( gcd_gcd_int @ D_1 @ A_1 )
= one_one_int )
=> ( ( gcd_gcd_int @ D_1 @ ( power_power_int @ A_1 @ N ) )
= one_one_int ) ) ).
thf(fact_4985_gcd__red__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ X @ Y )
= ( gcd_gcd_int @ Y @ ( div_mod_int @ X @ Y ) ) ) ).
thf(fact_4986_gcd__exp__int,axiom,
! [A_1: int,N: nat,B_1: int] :
( ( gcd_gcd_int @ ( power_power_int @ A_1 @ N ) @ ( power_power_int @ B_1 @ N ) )
= ( power_power_int @ ( gcd_gcd_int @ A_1 @ B_1 ) @ N ) ) ).
thf(fact_4987_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_4988_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_4989_gcd__greatest__int,axiom,
! [N: int,K: int,M: int] :
( ( dvd_dvd_int @ K @ M )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ ( gcd_gcd_int @ M @ N ) ) ) ) ).
thf(fact_4990_gcd__greatest__iff__int,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ ( gcd_gcd_int @ M @ N ) )
<=> ( ( dvd_dvd_int @ K @ M )
& ( dvd_dvd_int @ K @ N ) ) ) ).
thf(fact_4991_gcd__commute__int,axiom,
! [A_1: int,B_1: int] :
( ( gcd_gcd_int @ A_1 @ B_1 )
= ( gcd_gcd_int @ B_1 @ A_1 ) ) ).
thf(fact_4992_gcd__int_Oleft__commute,axiom,
! [B_1: int,A_1: int,C: int] :
( ( gcd_gcd_int @ B_1 @ ( gcd_gcd_int @ A_1 @ C ) )
= ( gcd_gcd_int @ A_1 @ ( gcd_gcd_int @ B_1 @ C ) ) ) ).
thf(fact_4993_gcd__assoc__int,axiom,
! [A_1: int,B_1: int,C: int] :
( ( gcd_gcd_int @ ( gcd_gcd_int @ A_1 @ B_1 ) @ C )
= ( gcd_gcd_int @ A_1 @ ( gcd_gcd_int @ B_1 @ C ) ) ) ).
thf(fact_4994_gcd__neg2__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) )
= ( gcd_gcd_int @ X @ Y ) ) ).
thf(fact_4995_gcd__neg1__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y )
= ( gcd_gcd_int @ X @ Y ) ) ).
thf(fact_4996_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_4997_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_4998_coprime__mul__eq__int,axiom,
! [D_1: int,A_1: int,B_1: int] :
( ( ( gcd_gcd_int @ D_1 @ ( times_times_int @ A_1 @ B_1 ) )
= one_one_int )
<=> ( ( ( gcd_gcd_int @ D_1 @ A_1 )
= one_one_int )
& ( ( gcd_gcd_int @ D_1 @ B_1 )
= one_one_int ) ) ) ).
thf(fact_4999_gcd__mult__cancel__int,axiom,
! [M: int,K: int,N: int] :
( ( ( gcd_gcd_int @ K @ N )
= one_one_int )
=> ( ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ N )
= ( gcd_gcd_int @ M @ N ) ) ) ).
thf(fact_5000_coprime__mult__int,axiom,
! [B_1: int,D_1: int,A_1: int] :
( ( ( gcd_gcd_int @ D_1 @ A_1 )
= one_one_int )
=> ( ( ( gcd_gcd_int @ D_1 @ B_1 )
= one_one_int )
=> ( ( gcd_gcd_int @ D_1 @ ( times_times_int @ A_1 @ B_1 ) )
= one_one_int ) ) ) ).
thf(fact_5001_coprime__lmult__int,axiom,
! [D_1: int,A_1: int,B_1: int] :
( ( ( gcd_gcd_int @ D_1 @ ( times_times_int @ A_1 @ B_1 ) )
= one_one_int )
=> ( ( gcd_gcd_int @ D_1 @ A_1 )
= one_one_int ) ) ).
thf(fact_5002_coprime__rmult__int,axiom,
! [D_1: int,A_1: int,B_1: int] :
( ( ( gcd_gcd_int @ D_1 @ ( times_times_int @ A_1 @ B_1 ) )
= one_one_int )
=> ( ( gcd_gcd_int @ D_1 @ B_1 )
= one_one_int ) ) ).
thf(fact_5003_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_5004_gcd__coprime__int,axiom,
! [B_5: int,A_5: int,A_1: int,B_1: int] :
( ( ( gcd_gcd_int @ A_1 @ B_1 )
!= zero_zero_int )
=> ( ( A_1
= ( times_times_int @ A_5 @ ( gcd_gcd_int @ A_1 @ B_1 ) ) )
=> ( ( B_1
= ( times_times_int @ B_5 @ ( gcd_gcd_int @ A_1 @ B_1 ) ) )
=> ( ( gcd_gcd_int @ A_5 @ B_5 )
= one_one_int ) ) ) ) ).
thf(fact_5005_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_5006_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_5007_coprime__crossproduct__int,axiom,
! [B_1: int,C: int,A_1: int,D_1: int] :
( ( ( gcd_gcd_int @ A_1 @ D_1 )
= one_one_int )
=> ( ( ( gcd_gcd_int @ B_1 @ C )
= one_one_int )
=> ( ( ( times_times_int @ ( abs_abs_int @ A_1 ) @ ( abs_abs_int @ C ) )
= ( times_times_int @ ( abs_abs_int @ B_1 ) @ ( abs_abs_int @ D_1 ) ) )
<=> ( ( ( abs_abs_int @ A_1 )
= ( abs_abs_int @ B_1 ) )
& ( ( abs_abs_int @ C )
= ( abs_abs_int @ D_1 ) ) ) ) ) ) ).
thf(fact_5008_gcd__unique__int,axiom,
! [B_1: int,A_1: int,D_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ D_1 )
& ( dvd_dvd_int @ D_1 @ A_1 )
& ( dvd_dvd_int @ D_1 @ B_1 )
& ! [E: int] :
( ( ( dvd_dvd_int @ E @ A_1 )
& ( dvd_dvd_int @ E @ B_1 ) )
=> ( dvd_dvd_int @ E @ D_1 ) ) )
<=> ( D_1
= ( gcd_gcd_int @ A_1 @ B_1 ) ) ) ).
thf(fact_5009_gcd__le1__int,axiom,
! [B_1: int,A_1: int] :
( ( ord_less_int @ zero_zero_int @ A_1 )
=> ( ord_less_eq_int @ ( gcd_gcd_int @ A_1 @ B_1 ) @ A_1 ) ) ).
thf(fact_5010_gcd__le2__int,axiom,
! [A_1: int,B_1: int] :
( ( ord_less_int @ zero_zero_int @ B_1 )
=> ( ord_less_eq_int @ ( gcd_gcd_int @ A_1 @ B_1 ) @ B_1 ) ) ).
thf(fact_5011_gcd__ge__0__int,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y ) ) ).
thf(fact_5012_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_5013_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_5014_gcd__code__int,axiom,
! [K: int,L: int] :
( ( gcd_gcd_int @ K @ L )
= ( abs_abs_int @ ( if_int @ ( L = zero_zero_int ) @ K @ ( gcd_gcd_int @ L @ ( div_mod_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ) ) ).
thf(fact_5015_coprime__common__divisor__int,axiom,
! [X: int,A_1: int,B_1: int] :
( ( ( gcd_gcd_int @ A_1 @ B_1 )
= one_one_int )
=> ( ( dvd_dvd_int @ X @ A_1 )
=> ( ( dvd_dvd_int @ X @ B_1 )
=> ( ( abs_abs_int @ X )
= one_one_int ) ) ) ) ).
thf(fact_5016_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_5017_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_5018_coprime__int,axiom,
! [A_1: int,B_1: int] :
( ( ( gcd_gcd_int @ A_1 @ B_1 )
= one_one_int )
<=> ! [D_2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ D_2 )
& ( dvd_dvd_int @ D_2 @ A_1 )
& ( dvd_dvd_int @ D_2 @ B_1 ) )
<=> ( D_2 = one_one_int ) ) ) ).
thf(fact_5019_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_5020_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_5021_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_5022_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_5023_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_5024_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_5025_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_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_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_5028_zOdd__def,axiom,
( zOdd
= ( collect_int
@ ^ [X_1: int] :
? [K_1: int] :
( X_1
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_1 ) @ one_one_int ) ) ) ) ).
thf(fact_5029_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_5030_zEven__def,axiom,
( zEven
= ( collect_int
@ ^ [X_1: int] :
? [K_1: int] :
( X_1
= ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_1 ) ) ) ) ).
thf(fact_5031_summable__Leibniz_I5_J,axiom,
! [A_1: nat > real] :
( ( tendsto_nat_real @ A_1 @ zero_zero_real @ sequentially )
=> ( ( monoseq_real @ A_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_1 @ 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_1 @ I_1 ) ) )
@ sequentially ) ) ) ).
thf(fact_5032_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_5033_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_5034_LIMSEQ__divide__realpow__zero,axiom,
! [A_1: real,X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( tendsto_nat_real
@ ^ [N_1: nat] : ( inverse_divide_real @ A_1 @ ( power_power_real @ X @ N_1 ) )
@ zero_zero_real
@ sequentially ) ) ).
thf(fact_5035_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_5036_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_5037_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_5038_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_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__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_5041_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_5042_monoseq__le,axiom,
! [X: real,A_1: nat > real] :
( ( monoseq_real @ A_1 )
=> ( ( tendsto_nat_real @ A_1 @ X @ sequentially )
=> ( ( ! [N_1: nat] : ( ord_less_eq_real @ ( A_1 @ N_1 ) @ X )
& ! [M_2: nat,N_1: nat] :
( ( ord_less_eq_nat @ M_2 @ N_1 )
=> ( ord_less_eq_real @ ( A_1 @ M_2 ) @ ( A_1 @ N_1 ) ) ) )
| ( ! [N_1: nat] : ( ord_less_eq_real @ X @ ( A_1 @ N_1 ) )
& ! [M_2: nat,N_1: nat] :
( ( ord_less_eq_nat @ M_2 @ N_1 )
=> ( ord_less_eq_real @ ( A_1 @ N_1 ) @ ( A_1 @ M_2 ) ) ) ) ) ) ) ).
thf(fact_5043_LIMSEQ__le__const,axiom,
! [A_1: 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_1 @ ( X_2 @ N_1 ) ) )
=> ( ord_less_eq_real @ A_1 @ X ) ) ) ).
thf(fact_5044_LIMSEQ__le__const2,axiom,
! [A_1: 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_1 ) )
=> ( ord_less_eq_real @ X @ A_1 ) ) ) ).
thf(fact_5045_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_5046_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_5047_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_5048_summable__Leibniz_I1_J,axiom,
! [A_1: nat > real] :
( ( tendsto_nat_real @ A_1 @ zero_zero_real @ sequentially )
=> ( ( monoseq_real @ A_1 )
=> ( summable_real
@ ^ [N_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ N_1 ) @ ( A_1 @ N_1 ) ) ) ) ) ).
thf(fact_5049_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_5050_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_5051_summable__Leibniz_I4_J,axiom,
! [A_1: nat > real] :
( ( tendsto_nat_real @ A_1 @ zero_zero_real @ sequentially )
=> ( ( monoseq_real @ A_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_1 @ 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_1 @ I_1 ) ) )
@ sequentially ) ) ) ).
thf(fact_5052_summable__Leibniz_H_I4_J,axiom,
! [N: nat,A_1: nat > real] :
( ( tendsto_nat_real @ A_1 @ zero_zero_real @ sequentially )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A_1 @ N_1 ) )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( A_1 @ ( suc @ N_1 ) ) @ ( A_1 @ N_1 ) )
=> ( ord_less_eq_real
@ ( suminf_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A_1 @ I_1 ) ) )
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A_1 @ 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_5053_summable__Leibniz_H_I5_J,axiom,
! [A_1: nat > real] :
( ( tendsto_nat_real @ A_1 @ zero_zero_real @ sequentially )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A_1 @ N_1 ) )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( A_1 @ ( suc @ N_1 ) ) @ ( A_1 @ 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_1 @ 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_1 @ I_1 ) ) )
@ sequentially ) ) ) ) ).
thf(fact_5054_cnj_OLIMSEQ,axiom,
! [X_2: nat > complex,A_1: complex] :
( ( tendsto_nat_complex @ X_2 @ A_1 @ sequentially )
=> ( tendsto_nat_complex
@ ^ [N_1: nat] : ( cnj @ ( X_2 @ N_1 ) )
@ ( cnj @ A_1 )
@ sequentially ) ) ).
thf(fact_5055_Re_OLIMSEQ,axiom,
! [X_2: nat > complex,A_1: complex] :
( ( tendsto_nat_complex @ X_2 @ A_1 @ sequentially )
=> ( tendsto_nat_real
@ ^ [N_1: nat] : ( re @ ( X_2 @ N_1 ) )
@ ( re @ A_1 )
@ sequentially ) ) ).
thf(fact_5056_Im_OLIMSEQ,axiom,
! [X_2: nat > complex,A_1: complex] :
( ( tendsto_nat_complex @ X_2 @ A_1 @ sequentially )
=> ( tendsto_nat_real
@ ^ [N_1: nat] : ( im @ ( X_2 @ N_1 ) )
@ ( im @ A_1 )
@ sequentially ) ) ).
thf(fact_5057_LIMSEQ__Complex,axiom,
! [Y_2: nat > real,B_1: real,X_2: nat > real,A_1: real] :
( ( tendsto_nat_real @ X_2 @ A_1 @ sequentially )
=> ( ( tendsto_nat_real @ Y_2 @ B_1 @ sequentially )
=> ( tendsto_nat_complex
@ ^ [N_1: nat] : ( complex_1 @ ( X_2 @ N_1 ) @ ( Y_2 @ N_1 ) )
@ ( complex_1 @ A_1 @ B_1 )
@ sequentially ) ) ) ).
thf(fact_5058_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_5059_summable__Leibniz_H_I3_J,axiom,
! [A_1: nat > real] :
( ( tendsto_nat_real @ A_1 @ zero_zero_real @ sequentially )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A_1 @ N_1 ) )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( A_1 @ ( suc @ N_1 ) ) @ ( A_1 @ 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_1 @ 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_1 @ I_1 ) ) )
@ sequentially ) ) ) ) ).
thf(fact_5060_summable__Leibniz_H_I2_J,axiom,
! [N: nat,A_1: nat > real] :
( ( tendsto_nat_real @ A_1 @ zero_zero_real @ sequentially )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A_1 @ N_1 ) )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( A_1 @ ( suc @ N_1 ) ) @ ( A_1 @ N_1 ) )
=> ( ord_less_eq_real
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A_1 @ 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_1 @ I_1 ) ) ) ) ) ) ) ).
thf(fact_5061_summable__Leibniz_I2_J,axiom,
! [A_1: nat > real] :
( ( tendsto_nat_real @ A_1 @ zero_zero_real @ sequentially )
=> ( ( monoseq_real @ A_1 )
=> ( ( ord_less_real @ zero_zero_real @ ( A_1 @ 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_1 @ I_1 ) ) )
@ ( ord_at1589558736t_real
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A_1 @ 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_1 @ 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_5062_summable__Leibniz_I3_J,axiom,
! [A_1: nat > real] :
( ( tendsto_nat_real @ A_1 @ zero_zero_real @ sequentially )
=> ( ( monoseq_real @ A_1 )
=> ( ( ord_less_real @ ( A_1 @ 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_1 @ I_1 ) ) )
@ ( ord_at1589558736t_real
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A_1 @ 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_1 @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) ) ) ) ) ) ) ) ).
thf(fact_5063_Sup__atLeastAtMost,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( comple124823625p_real @ ( ord_at1589558736t_real @ Y @ X ) )
= X ) ) ).
thf(fact_5064_summable,axiom,
! [A_1: nat > real] :
( ( tendsto_nat_real @ A_1 @ zero_zero_real @ sequentially )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A_1 @ N_1 ) )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( A_1 @ ( suc @ N_1 ) ) @ ( A_1 @ N_1 ) )
=> ( summable_real
@ ^ [N_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ N_1 ) @ ( A_1 @ N_1 ) ) ) ) ) ) ).
thf(fact_5065_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( ord_at238088361st_nat @ L @ U ) ) ).
thf(fact_5066_finite__atLeastAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( ord_at875362053st_int @ L @ U ) ) ).
thf(fact_5067_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_5068_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_5069_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_5070_card__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( ord_at238088361st_nat @ L @ U ) )
= ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
thf(fact_5071_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( ord_at4362885an_nat @ L @ ( suc @ U ) )
= ( ord_at238088361st_nat @ L @ U ) ) ).
thf(fact_5072_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_5073_image__add__atLeastAtMost,axiom,
! [K: nat,I: nat,J: nat] :
( ( image_nat_nat
@ ^ [N_1: nat] : ( plus_plus_nat @ N_1 @ K )
@ ( ord_at238088361st_nat @ I @ J ) )
= ( ord_at238088361st_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
thf(fact_5074_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_5075_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_5076_Re_Ocont,axiom,
! [A_1: complex] : ( tendsto_complex_real @ re @ ( re @ A_1 ) @ ( at_complex @ A_1 ) ) ).
thf(fact_5077_Im_Ocont,axiom,
! [A_1: complex] : ( tendsto_complex_real @ im @ ( im @ A_1 ) @ ( at_complex @ A_1 ) ) ).
thf(fact_5078_cnj_Ocont,axiom,
! [A_1: complex] : ( tendst1507391555omplex @ cnj @ ( cnj @ A_1 ) @ ( at_complex @ A_1 ) ) ).
thf(fact_5079_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_5080_setsum__shift__lb__Suc0__0,axiom,
! [K: nat,F: nat > nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( big_co387207925at_nat @ F @ ( ord_at238088361st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( big_co387207925at_nat @ F @ ( ord_at238088361st_nat @ zero_zero_nat @ K ) ) ) ) ).
thf(fact_5081_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_5082_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_5083_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_5084_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_5085_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_5086_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_5087_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_5088_aset_I8_J,axiom,
! [T: int,A: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A )
=> ( 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 ) ) ) ) ) ).
thf(fact_5089_aset_I6_J,axiom,
! [T: int,A: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ X_1 @ T )
=> ( ord_less_eq_int @ ( plus_plus_int @ X_1 @ D ) @ T ) ) ) ) ) ).
thf(fact_5090_bset_I8_J,axiom,
! [T: int,B: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B )
=> ( 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 ) ) ) ) ) ) ).
thf(fact_5091_bset_I6_J,axiom,
! [T: int,B: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ X_1 @ T )
=> ( ord_less_eq_int @ ( minus_minus_int @ X_1 @ D ) @ T ) ) ) ) ).
thf(fact_5092_bset_I3_J,axiom,
! [T: int,B: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( X_1 = T )
=> ( ( minus_minus_int @ X_1 @ D )
= T ) ) ) ) ) ).
thf(fact_5093_aset_I3_J,axiom,
! [T: int,A: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( X_1 = T )
=> ( ( plus_plus_int @ X_1 @ D )
= T ) ) ) ) ) ).
thf(fact_5094_aset_I7_J,axiom,
! [T: int,A: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ T @ X_1 )
=> ( ord_less_int @ T @ ( plus_plus_int @ X_1 @ D ) ) ) ) ) ).
thf(fact_5095_bset_I5_J,axiom,
! [T: int,B: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ X_1 @ T )
=> ( ord_less_int @ ( minus_minus_int @ X_1 @ D ) @ T ) ) ) ) ).
thf(fact_5096_aset_I5_J,axiom,
! [T: int,A: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ( member_int @ T @ A )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ X_1 @ T )
=> ( ord_less_int @ ( plus_plus_int @ X_1 @ D ) @ T ) ) ) ) ) ).
thf(fact_5097_bset_I7_J,axiom,
! [T: int,B: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ( member_int @ T @ B )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ T @ X_1 )
=> ( ord_less_int @ T @ ( minus_minus_int @ X_1 @ D ) ) ) ) ) ) ).
thf(fact_5098_aset_I4_J,axiom,
! [T: int,A: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ( member_int @ T @ A )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( X_1 != T )
=> ( ( plus_plus_int @ X_1 @ D )
!= T ) ) ) ) ) ).
thf(fact_5099_bset_I4_J,axiom,
! [T: int,B: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ( member_int @ T @ B )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( X_1 != T )
=> ( ( minus_minus_int @ X_1 @ D )
!= T ) ) ) ) ) ).
thf(fact_5100_periodic__finite__ex,axiom,
! [P: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ! [X_1: int,K_1: int] :
( ( P @ X_1 )
<=> ( P @ ( minus_minus_int @ X_1 @ ( times_times_int @ K_1 @ D_1 ) ) ) )
=> ( ( ?? @ int @ P )
<=> ? [X_1: int] :
( ( member_int @ X_1 @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
& ( P @ X_1 ) ) ) ) ) ).
thf(fact_5101_aset_I10_J,axiom,
! [T: int,A: int > $o,D_1: int,D: int] :
( ( dvd_dvd_int @ D_1 @ D )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ~ ( dvd_dvd_int @ D_1 @ ( plus_plus_int @ X_1 @ T ) )
=> ~ ( dvd_dvd_int @ D_1 @ ( plus_plus_int @ ( plus_plus_int @ X_1 @ D ) @ T ) ) ) ) ) ).
thf(fact_5102_bset_I10_J,axiom,
! [T: int,B: int > $o,D_1: int,D: int] :
( ( dvd_dvd_int @ D_1 @ D )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ~ ( dvd_dvd_int @ D_1 @ ( plus_plus_int @ X_1 @ T ) )
=> ~ ( dvd_dvd_int @ D_1 @ ( plus_plus_int @ ( minus_minus_int @ X_1 @ D ) @ T ) ) ) ) ) ).
thf(fact_5103_bset_I9_J,axiom,
! [T: int,B: int > $o,D_1: int,D: int] :
( ( dvd_dvd_int @ D_1 @ D )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( dvd_dvd_int @ D_1 @ ( plus_plus_int @ X_1 @ T ) )
=> ( dvd_dvd_int @ D_1 @ ( plus_plus_int @ ( minus_minus_int @ X_1 @ D ) @ T ) ) ) ) ) ).
thf(fact_5104_aset_I9_J,axiom,
! [T: int,A: int > $o,D_1: int,D: int] :
( ( dvd_dvd_int @ D_1 @ D )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( dvd_dvd_int @ D_1 @ ( plus_plus_int @ X_1 @ T ) )
=> ( dvd_dvd_int @ D_1 @ ( plus_plus_int @ ( plus_plus_int @ X_1 @ D ) @ T ) ) ) ) ) ).
thf(fact_5105_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_5106_Re_OscaleR,axiom,
! [R_1: real,X: complex] :
( ( re @ ( scaleR1652505878omplex @ R_1 @ X ) )
= ( scaleR_scaleR_real @ R_1 @ ( re @ X ) ) ) ).
thf(fact_5107_Im_OscaleR,axiom,
! [R_1: real,X: complex] :
( ( im @ ( scaleR1652505878omplex @ R_1 @ X ) )
= ( scaleR_scaleR_real @ R_1 @ ( im @ X ) ) ) ).
thf(fact_5108_cnj_OscaleR,axiom,
! [R_1: real,X: complex] :
( ( cnj @ ( scaleR1652505878omplex @ R_1 @ X ) )
= ( scaleR1652505878omplex @ R_1 @ ( cnj @ X ) ) ) ).
thf(fact_5109_real__scaleR__def,axiom,
! [A_1: real,X: real] :
( ( scaleR_scaleR_real @ A_1 @ X )
= ( times_times_real @ A_1 @ X ) ) ).
thf(fact_5110_complex__scaleR,axiom,
! [R_1: real,A_1: real,B_1: real] :
( ( scaleR1652505878omplex @ R_1 @ ( complex_1 @ A_1 @ B_1 ) )
= ( complex_1 @ ( times_times_real @ R_1 @ A_1 ) @ ( times_times_real @ R_1 @ B_1 ) ) ) ).
thf(fact_5111_complex__Im__scaleR,axiom,
! [R_1: real,X: complex] :
( ( im @ ( scaleR1652505878omplex @ R_1 @ X ) )
= ( times_times_real @ R_1 @ ( im @ X ) ) ) ).
thf(fact_5112_complex__Re__scaleR,axiom,
! [R_1: real,X: complex] :
( ( re @ ( scaleR1652505878omplex @ R_1 @ X ) )
= ( times_times_real @ R_1 @ ( re @ X ) ) ) ).
thf(fact_5113_complex__sgn__def,axiom,
! [X: complex] :
( ( sgn_sgn_complex @ X )
= ( scaleR1652505878omplex @ ( inverse_inverse_real @ ( norm_norm_complex @ X ) ) @ X ) ) ).
thf(fact_5114_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_5115_trivial__limit__sequentially,axiom,
~ ( trivial_limit_nat @ sequentially ) ).
thf(fact_5116_pred__nat__def,axiom,
( pred_nat
= ( collec1979865426at_nat
@ ( produc1038563245_nat_o
@ ^ [M_2: nat,N_1: nat] :
( N_1
= ( suc @ M_2 ) ) ) ) ) ).
thf(fact_5117_gcd__coprime__exists__int,axiom,
! [A_1: int,B_1: int] :
( ( ( gcd_gcd_int @ A_1 @ B_1 )
!= zero_zero_int )
=> ? [A_3: int,B_2: int] :
( ( A_1
= ( times_times_int @ A_3 @ ( gcd_gcd_int @ A_1 @ B_1 ) ) )
& ( B_1
= ( times_times_int @ B_2 @ ( gcd_gcd_int @ A_1 @ B_1 ) ) )
& ( ( gcd_gcd_int @ A_3 @ B_2 )
= one_one_int ) ) ) ).
thf(fact_5118_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_5119_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_5120_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_5121_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_5122_min__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
thf(fact_5123_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_5124_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_5125_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_5126_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_5127_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_5128_is__nat__def,axiom,
! [X: int] :
( ( nat_is_nat @ X )
<=> ( ord_less_eq_int @ zero_zero_int @ X ) ) ).
thf(fact_5129_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_5130_Nat__Transfer_Otransfer__int__nat__function__closures_I6_J,axiom,
nat_is_nat @ one_one_int ).
thf(fact_5131_Nat__Transfer_Otransfer__int__nat__set__function__closures_I6_J,axiom,
! [X: int,A: int > $o] :
( ( nat_nat_set @ A )
=> ( ( member_int @ X @ A )
=> ( nat_is_nat @ X ) ) ) ).
thf(fact_5132_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_5133_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_5134_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_5135_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_5136_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_5137_Nat__Transfer_Otransfer__int__nat__function__closures_I5_J,axiom,
nat_is_nat @ zero_zero_int ).
thf(fact_5138_Nat__Transfer_Otransfer__int__nat__function__closures_I9_J,axiom,
! [Z_1: nat] : ( nat_is_nat @ ( semiri1621563631at_int @ Z_1 ) ) ).
thf(fact_5139_vanishes__minus,axiom,
! [X_2: nat > rat] :
( ( vanishes @ X_2 )
=> ( vanishes
@ ^ [N_1: nat] : ( uminus_uminus_rat @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_5140_vanishes__const,axiom,
! [C: rat] :
( ( vanishes
@ ^ [N_1: nat] : C )
<=> ( C = zero_zero_rat ) ) ).
thf(fact_5141_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_5142_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_5143_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_5144_Nat__Transfer_Otransfer__int__nat__function__closures_I8_J,axiom,
nat_is_nat @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) ).
thf(fact_5145_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_5146_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_5147_transfer__int__nat__set__relations_I2_J,axiom,
! [A: int > $o,X: int] :
( ( nat_is_nat @ X )
=> ( ( nat_nat_set @ A )
=> ( ( member_int @ X @ A )
<=> ( member_nat @ ( nat_1 @ X ) @ ( image_int_nat @ nat_1 @ A ) ) ) ) ) ).
thf(fact_5148_Nat__Transfer_Otransfer__int__nat__function__closures_I7_J,axiom,
nat_is_nat @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_5149_vanishesD,axiom,
! [R_1: rat,X_2: nat > rat] :
( ( vanishes @ X_2 )
=> ( ( ord_less_rat @ zero_zero_rat @ R_1 )
=> ? [K_1: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ K_1 @ N_1 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X_2 @ N_1 ) ) @ R_1 ) ) ) ) ).
thf(fact_5150_vanishes__def,axiom,
! [X_2: nat > rat] :
( ( vanishes @ X_2 )
<=> ! [R: rat] :
( ( ord_less_rat @ zero_zero_rat @ R )
=> ? [K_1: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ K_1 @ N_1 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X_2 @ N_1 ) ) @ R ) ) ) ) ).
thf(fact_5151_vanishesI,axiom,
! [X_2: nat > rat] :
( ! [R: rat] :
( ( ord_less_rat @ zero_zero_rat @ R )
=> ? [K_1: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ K_1 @ N_1 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X_2 @ N_1 ) ) @ R ) ) )
=> ( vanishes @ X_2 ) ) ).
thf(fact_5152_zcongm__def,axiom,
! [M: int,X_1: int,Xa: int] :
( ( zcongm @ M @ X_1 @ Xa )
<=> ( zcong @ X_1 @ Xa @ M ) ) ).
thf(fact_5153_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_5154_bijzcong__zcong__prod,axiom,
! [A: int > $o,B: int > $o,M: int] :
( ( member1329254762_int_o @ ( produc398918003_int_o @ A @ B ) @ ( bijR_int_int @ ( zcongm @ M ) ) )
=> ( zcong
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ A )
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ B )
@ M ) ) ).
thf(fact_5155_isCont__real__root,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( isCont_real_real @ ( root @ N ) @ X ) ) ).
thf(fact_5156_isCont__ln,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( isCont_real_real @ ln @ X ) ) ).
thf(fact_5157_isCont__tan,axiom,
! [X: real] :
( ( ( cos @ X )
!= zero_zero_real )
=> ( isCont_real_real @ tan @ X ) ) ).
thf(fact_5158_isCont__abs,axiom,
! [A_1: real] : ( isCont_real_real @ abs_abs_real @ A_1 ) ).
thf(fact_5159_isCont__sin,axiom,
! [X: real] : ( isCont_real_real @ sin @ X ) ).
thf(fact_5160_isCont__cos,axiom,
! [X: real] : ( isCont_real_real @ cos @ X ) ).
thf(fact_5161_isCont__real__sqrt,axiom,
! [X: real] : ( isCont_real_real @ sqrt @ X ) ).
thf(fact_5162_isCont__arctan,axiom,
! [X: real] : ( isCont_real_real @ arctan @ X ) ).
thf(fact_5163_cnj_OisCont,axiom,
! [A_1: complex] : ( isCont156215680omplex @ cnj @ A_1 ) ).
thf(fact_5164_Im_OisCont,axiom,
! [A_1: complex] : ( isCont_complex_real @ im @ A_1 ) ).
thf(fact_5165_Re_OisCont,axiom,
! [A_1: complex] : ( isCont_complex_real @ re @ A_1 ) ).
thf(fact_5166_isCont__root__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( isCont_real_real @ ( root @ N ) @ zero_zero_real ) ) ).
thf(fact_5167_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_5168_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_5169_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_5170_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_5171_LIM__less__bound,axiom,
! [F: real > real,B_1: real,X: real] :
( ( ord_less_real @ B_1 @ X )
=> ( ! [X_1: real] :
( ( member_real @ X_1 @ ( ord_gr788844697n_real @ B_1 @ 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_5172_DERIV__inverse__function,axiom,
! [B_1: real,A_1: real,F: real > real,G: real > real,X: real,D: real] :
( ( deriv_real @ F @ ( G @ X ) @ D )
=> ( ( D != zero_zero_real )
=> ( ( ord_less_real @ A_1 @ X )
=> ( ( ord_less_real @ X @ B_1 )
=> ( ! [Y_1: real] :
( ( ( ord_less_real @ A_1 @ Y_1 )
& ( ord_less_real @ Y_1 @ B_1 ) )
=> ( ( F @ ( G @ Y_1 ) )
= Y_1 ) )
=> ( ( isCont_real_real @ G @ X )
=> ( deriv_real @ G @ X @ ( inverse_inverse_real @ D ) ) ) ) ) ) ) ) ).
thf(fact_5173_isCont__inv__fun,axiom,
! [G: real > real,F: real > real,X: real,D_1: real] :
( ( ord_less_real @ zero_zero_real @ D_1 )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D_1 )
=> ( ( G @ ( F @ Z ) )
= Z ) )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D_1 )
=> ( isCont_real_real @ F @ Z ) )
=> ( isCont_real_real @ G @ ( F @ X ) ) ) ) ) ).
thf(fact_5174_DERIV__isconst1,axiom,
! [F: real > real,A_1: real,B_1: real] :
( ( ord_less_real @ A_1 @ B_1 )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A_1 @ X_1 )
& ( ord_less_eq_real @ X_1 @ B_1 ) )
=> ( isCont_real_real @ F @ X_1 ) )
=> ( ! [X_1: real] :
( ( ( ord_less_real @ A_1 @ X_1 )
& ( ord_less_real @ X_1 @ B_1 ) )
=> ( deriv_real @ F @ X_1 @ zero_zero_real ) )
=> ! [X_1: real] :
( ( ( ord_less_eq_real @ A_1 @ X_1 )
& ( ord_less_eq_real @ X_1 @ B_1 ) )
=> ( ( F @ X_1 )
= ( F @ A_1 ) ) ) ) ) ) ).
thf(fact_5175_DERIV__isconst2,axiom,
! [X: real,F: real > real,A_1: real,B_1: real] :
( ( ord_less_real @ A_1 @ B_1 )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A_1 @ X_1 )
& ( ord_less_eq_real @ X_1 @ B_1 ) )
=> ( isCont_real_real @ F @ X_1 ) )
=> ( ! [X_1: real] :
( ( ( ord_less_real @ A_1 @ X_1 )
& ( ord_less_real @ X_1 @ B_1 ) )
=> ( deriv_real @ F @ X_1 @ zero_zero_real ) )
=> ( ( ord_less_eq_real @ A_1 @ X )
=> ( ( ord_less_eq_real @ X @ B_1 )
=> ( ( F @ X )
= ( F @ A_1 ) ) ) ) ) ) ) ).
thf(fact_5176_DERIV__isconst__end,axiom,
! [F: real > real,A_1: real,B_1: real] :
( ( ord_less_real @ A_1 @ B_1 )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A_1 @ X_1 )
& ( ord_less_eq_real @ X_1 @ B_1 ) )
=> ( isCont_real_real @ F @ X_1 ) )
=> ( ! [X_1: real] :
( ( ( ord_less_real @ A_1 @ X_1 )
& ( ord_less_real @ X_1 @ B_1 ) )
=> ( deriv_real @ F @ X_1 @ zero_zero_real ) )
=> ( ( F @ B_1 )
= ( F @ A_1 ) ) ) ) ) ).
thf(fact_5177_isCont__inv__fun__inv,axiom,
! [G: real > real,F: real > real,X: real,D_1: real] :
( ( ord_less_real @ zero_zero_real @ D_1 )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D_1 )
=> ( ( G @ ( F @ Z ) )
= Z ) )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D_1 )
=> ( 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_5178_lemma__isCont__inj,axiom,
! [G: real > real,F: real > real,X: real,D_1: real] :
( ( ord_less_real @ zero_zero_real @ D_1 )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D_1 )
=> ( ( G @ ( F @ Z ) )
= Z ) )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D_1 )
=> ( isCont_real_real @ F @ Z ) )
=> ? [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D_1 )
& ( ord_less_real @ ( F @ X ) @ ( F @ Z ) ) ) ) ) ) ).
thf(fact_5179_lemma__isCont__inj2,axiom,
! [G: real > real,F: real > real,X: real,D_1: real] :
( ( ord_less_real @ zero_zero_real @ D_1 )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D_1 )
=> ( ( G @ ( F @ Z ) )
= Z ) )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D_1 )
=> ( isCont_real_real @ F @ Z ) )
=> ? [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D_1 )
& ( ord_less_real @ ( F @ Z ) @ ( F @ X ) ) ) ) ) ) ).
thf(fact_5180_cpmi,axiom,
! [B: int > $o,P_1: int > $o,P: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ? [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 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X_1 )
=> ( P @ ( minus_minus_int @ X_1 @ D ) ) ) )
=> ( ! [X_1: int,K_1: int] :
( ( P_1 @ X_1 )
<=> ( P_1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K_1 @ D ) ) ) )
=> ( ( ?? @ int @ P )
<=> ( ? [X_1: int] :
( ( member_int @ X_1 @ ( ord_at875362053st_int @ one_one_int @ D ) )
& ( P_1 @ X_1 ) )
| ? [X_1: int] :
( ( member_int @ X_1 @ ( ord_at875362053st_int @ one_one_int @ D ) )
& ? [Xa: int] :
( ( member_int @ Xa @ B )
& ( P @ ( plus_plus_int @ Xa @ X_1 ) ) ) ) ) ) ) ) ) ) ).
thf(fact_5181_cppi,axiom,
! [A: int > $o,P_1: int > $o,P: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ? [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 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X_1 )
=> ( P @ ( plus_plus_int @ X_1 @ D ) ) ) )
=> ( ! [X_1: int,K_1: int] :
( ( P_1 @ X_1 )
<=> ( P_1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K_1 @ D ) ) ) )
=> ( ( ?? @ int @ P )
<=> ( ? [X_1: int] :
( ( member_int @ X_1 @ ( ord_at875362053st_int @ one_one_int @ D ) )
& ( P_1 @ X_1 ) )
| ? [X_1: int] :
( ( member_int @ X_1 @ ( ord_at875362053st_int @ one_one_int @ D ) )
& ? [Xa: int] :
( ( member_int @ Xa @ A )
& ( P @ ( minus_minus_int @ Xa @ X_1 ) ) ) ) ) ) ) ) ) ) ).
thf(fact_5182_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_5183_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( ord_atMost_nat @ K ) ) ).
thf(fact_5184_finite__nat__iff__bounded__le,axiom,
! [S: nat > $o] :
( ( finite_finite_nat @ S )
<=> ? [K_1: nat] : ( ord_less_eq_nat_o @ S @ ( ord_atMost_nat @ K_1 ) ) ) ).
thf(fact_5185_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( ord_lessThan_nat @ ( suc @ K ) )
= ( ord_atMost_nat @ K ) ) ).
thf(fact_5186_card__atMost,axiom,
! [U: nat] :
( ( finite_card_nat @ ( ord_atMost_nat @ U ) )
= ( suc @ U ) ) ).
thf(fact_5187_atLeast0AtMost,axiom,
! [N: nat] :
( ( ord_at238088361st_nat @ zero_zero_nat @ N )
= ( ord_atMost_nat @ N ) ) ).
thf(fact_5188_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_5189_max__Suc__number__of,axiom,
! [V: int,N: nat] :
( ( ord_max_nat @ ( number_number_of_nat @ V ) @ ( suc @ N ) )
= ( if_nat @ ( nat_neg @ ( number_number_of_int @ ( pred @ V ) ) ) @ ( suc @ N ) @ ( suc @ ( ord_max_nat @ ( nat_1 @ ( number_number_of_int @ ( pred @ V ) ) ) @ N ) ) ) ) ).
thf(fact_5190_Sup__atMost,axiom,
! [X: real] :
( ( comple124823625p_real @ ( ord_atMost_real @ X ) )
= X ) ).
thf(fact_5191_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_5192_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_5193_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
thf(fact_5194_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
thf(fact_5195_max__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
%----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,
( ( power_power_int @ ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ n ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= zero_zero_int ) ).
%------------------------------------------------------------------------------