TPTP Problem File: NUM923^4.p
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%------------------------------------------------------------------------------
% File : NUM923^4 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 23, 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_l23 [Bla11]
% Status : Theorem
% Rating : 1.00 v5.3.0
% Syntax : Number of formulae : 5639 (2036 unt; 427 typ; 0 def)
% Number of atoms : 12059 (4121 equ; 67 cnn)
% Maximal formula atoms : 14 ( 2 avg)
% Number of connectives : 48273 (1060 ~; 212 |; 748 &;40799 @)
% (1004 <=>;4422 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 1526 (1526 >; 0 *; 0 +; 0 <<)
% Number of symbols : 416 ( 407 usr; 31 con; 0-8 aty)
% ( 10 !!; 18 ??; 0 @@+; 0 @@-)
% Number of variables : 11262 ( 263 ^;10732 !; 267 ?;11262 :)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:58:10
%------------------------------------------------------------------------------
%----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 (428)
thf(sy_c_Archimedean__Field_Oceiling_000tc__Rat__Orat,type,
archim1419379755ng_rat: rat > int ).
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_Oin__rel_000_062_Itc__Int__Oint_M_Eo_J_000_062_Itc__Int__Oint_M_Eo_J,type,
in_rel_int_o_int_o: ( produc975137661_int_o > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_FunDef_Oin__rel_000tc__Code____Numeral__Ocode____numeral_000tc__Code____Num,type,
in_rel1251239042umeral: ( produc1359518119umeral > $o ) > code_code_numeral > code_code_numeral > $o ).
thf(sy_c_FunDef_Oin__rel_000tc__Int__Oint_000tc__Int__Oint,type,
in_rel_int_int: ( product_prod_int_int > $o ) > int > int > $o ).
thf(sy_c_FunDef_Oin__rel_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_,type,
in_rel1239871883nt_int: ( produc393999548nt_int > $o ) > int > product_prod_int_int > $o ).
thf(sy_c_FunDef_Oin__rel_000tc__Nat__Onat_000tc__Nat__Onat,type,
in_rel_nat_nat: ( product_prod_nat_nat > $o ) > nat > nat > $o ).
thf(sy_c_FunDef_Oin__rel_000tc__Quickcheck____Narrowing__Ocode____int_000tc__Quickch,type,
in_rel1707483778de_int: ( produc167071911de_int > $o ) > quickcheck_code_int > quickcheck_code_int > $o ).
thf(sy_c_FunDef_Oin__rel_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
in_rel_real_real: ( produc914805421l_real > $o ) > real > real > $o ).
thf(sy_c_FunDef_Oin__rel_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod_Itc,type,
in_rel682537726nt_int: ( produc1137372701nt_int > $o ) > product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_FunDef_Oin__rel_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__prod_Itc,type,
in_rel293241086at_nat: ( produc1322466333at_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $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_000_062_Itc__Int__Oint_M_Eo_J,type,
uminus_uminus_int_o: ( int > $o ) > int > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_000_062_Itc__Nat__Onat_M_Eo_J,type,
uminus_uminus_nat_o: ( nat > $o ) > nat > $o ).
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__Complex__Ocomplex,type,
ring_1_Ints_complex: complex > $o ).
thf(sy_c_Int_Oring__1__class_OInts_000tc__Int__Oint,type,
ring_1_Ints_int: int > $o ).
thf(sy_c_Int_Oring__1__class_OInts_000tc__Rat__Orat,type,
ring_1_Ints_rat: rat > $o ).
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__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__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__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__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_Ocurry_000_062_Itc__Int__Oint_M_Eo_J_000_062_Itc__Int__Oint_M,type,
produc144226034nt_o_o: ( produc975137661_int_o > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_Product__Type_Ocurry_000tc__Code____Numeral__Ocode____numeral_000tc__Code__,type,
produc350732110eral_o: ( produc1359518119umeral > $o ) > code_code_numeral > code_code_numeral > $o ).
thf(sy_c_Product__Type_Ocurry_000tc__Int__Oint_000tc__Int__Oint_000_Eo,type,
produc176579150_int_o: ( product_prod_int_int > $o ) > int > int > $o ).
thf(sy_c_Product__Type_Ocurry_000tc__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__,type,
produc122891093_int_o: ( produc393999548nt_int > $o ) > int > product_prod_int_int > $o ).
thf(sy_c_Product__Type_Ocurry_000tc__Nat__Onat_000tc__Nat__Onat_000_Eo,type,
produc764619086_nat_o: ( product_prod_nat_nat > $o ) > nat > nat > $o ).
thf(sy_c_Product__Type_Ocurry_000tc__Quickcheck____Narrowing__Ocode____int_000tc__Qu,type,
produc1585697102_int_o: ( produc167071911de_int > $o ) > quickcheck_code_int > quickcheck_code_int > $o ).
thf(sy_c_Product__Type_Ocurry_000tc__RealDef__Oreal_000tc__RealDef__Oreal_000_Eo,type,
produc1509576130real_o: ( produc914805421l_real > $o ) > real > real > $o ).
thf(sy_c_Product__Type_Ocurry_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__pro,type,
produc1384121810_int_o: ( produc1137372701nt_int > $o ) > product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_Product__Type_Ocurry_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__pro,type,
produc435992530_nat_o: ( produc1322466333at_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Product__Type_Ofst_000_062_Itc__Int__Oint_M_Eo_J_000_062_Itc__Int__Oint_M_E,type,
produc2019076742_int_o: produc975137661_int_o > int > $o ).
thf(sy_c_Product__Type_Ofst_000tc__Code____Numeral__Ocode____numeral_000tc__Code____,type,
produc310214954umeral: produc1359518119umeral > code_code_numeral ).
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__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oi,type,
produc2110120675nt_int: produc393999548nt_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__Quickcheck____Narrowing__Ocode____int_000tc__Quic,type,
produc1968959274de_int: produc167071911de_int > quickcheck_code_int ).
thf(sy_c_Product__Type_Ofst_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
produc1935615926l_real: produc914805421l_real > real ).
thf(sy_c_Product__Type_Ofst_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod_,type,
produc1096438694nt_int: produc1137372701nt_int > product_prod_int_int ).
thf(sy_c_Product__Type_Ofst_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__prod_,type,
produc707142054at_nat: produc1322466333at_nat > product_prod_nat_nat ).
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_000_062_Itc__Int__Oint_M_Eo_J_000_062_Itc__Int__Oint_M_E,type,
produc883676360_int_o: produc975137661_int_o > int > $o ).
thf(sy_c_Product__Type_Osnd_000tc__Code____Numeral__Ocode____numeral_000tc__Code____,type,
produc1857364076umeral: produc1359518119umeral > code_code_numeral ).
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__Int__Oint_000tc__prod_Itc__Int__Oint_Mtc__Int__Oi,type,
produc8341793nt_int: produc393999548nt_int > product_prod_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__Quickcheck____Narrowing__Ocode____int_000tc__Quic,type,
produc221809260de_int: produc167071911de_int > quickcheck_code_int ).
thf(sy_c_Product__Type_Osnd_000tc__RealDef__Oreal_000tc__RealDef__Oreal,type,
produc556554744l_real: produc914805421l_real > real ).
thf(sy_c_Product__Type_Osnd_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__prod_,type,
produc1011150824nt_int: produc1137372701nt_int > product_prod_int_int ).
thf(sy_c_Product__Type_Osnd_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J_000tc__prod_,type,
produc621854184at_nat: produc1322466333at_nat > product_prod_nat_nat ).
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__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_SEQ_OBseq_000tc__RealDef__Oreal,type,
bseq_real: ( nat > real ) > $o ).
thf(sy_c_SEQ_OCauchy_000tc__Complex__Ocomplex,type,
cauchy_complex: ( nat > complex ) > $o ).
thf(sy_c_SEQ_OCauchy_000tc__RealDef__Oreal,type,
cauchy_real: ( nat > real ) > $o ).
thf(sy_c_SEQ_Omonoseq_000tc__RealDef__Oreal,type,
monoseq_real: ( nat > real ) > $o ).
thf(sy_c_SMT_Oz3div,type,
z3div: int > int > int ).
thf(sy_c_SMT_Oz3mod,type,
z3mod: int > int > int ).
thf(sy_c_Series_Osuminf_000tc__Complex__Ocomplex,type,
suminf_complex: ( nat > complex ) > complex ).
thf(sy_c_Series_Osuminf_000tc__RealDef__Oreal,type,
suminf_real: ( nat > real ) > real ).
thf(sy_c_Series_Osummable_000tc__Complex__Ocomplex,type,
summable_complex: ( nat > complex ) > $o ).
thf(sy_c_Series_Osummable_000tc__RealDef__Oreal,type,
summable_real: ( nat > real ) > $o ).
thf(sy_c_Series_Osums_000tc__Complex__Ocomplex,type,
sums_complex: ( nat > complex ) > complex > $o ).
thf(sy_c_Series_Osums_000tc__RealDef__Oreal,type,
sums_real: ( nat > real ) > real > $o ).
thf(sy_c_SetInterval_Oord__class_OatLeastAtMost_000tc__Int__Oint,type,
ord_at875362053st_int: int > int > int > $o ).
thf(sy_c_SetInterval_Oord__class_OatLeastAtMost_000tc__Nat__Onat,type,
ord_at238088361st_nat: nat > nat > nat > $o ).
thf(sy_c_SetInterval_Oord__class_OatLeastAtMost_000tc__RealDef__Oreal,type,
ord_at1589558736t_real: real > real > real > $o ).
thf(sy_c_SetInterval_Oord__class_OatLeastLessThan_000tc__Int__Oint,type,
ord_at641636577an_int: int > int > int > $o ).
thf(sy_c_SetInterval_Oord__class_OatLeastLessThan_000tc__Nat__Onat,type,
ord_at4362885an_nat: nat > nat > nat > $o ).
thf(sy_c_SetInterval_Oord__class_OatLeastLessThan_000tc__RealDef__Oreal,type,
ord_at1496968948n_real: real > real > real > $o ).
thf(sy_c_SetInterval_Oord__class_OatMost_000tc__Nat__Onat,type,
ord_atMost_nat: nat > nat > $o ).
thf(sy_c_SetInterval_Oord__class_OgreaterThanLessThan_000tc__Int__Oint,type,
ord_gr1297742076an_int: int > int > int > $o ).
thf(sy_c_SetInterval_Oord__class_OgreaterThanLessThan_000tc__Nat__Onat,type,
ord_gr660468384an_nat: nat > nat > nat > $o ).
thf(sy_c_SetInterval_Oord__class_OgreaterThanLessThan_000tc__RealDef__Oreal,type,
ord_gr788844697n_real: real > real > real > $o ).
thf(sy_c_SetInterval_Oord__class_OlessThan_000tc__Nat__Onat,type,
ord_lessThan_nat: nat > nat > $o ).
thf(sy_c_SetInterval_Oord__class_OlessThan_000tc__RealDef__Oreal,type,
ord_lessThan_real: real > real > $o ).
thf(sy_c_Set_OCollect_000tc__Int__Oint,type,
collect_int: ( int > $o ) > int > $o ).
thf(sy_c_Set_OCollect_000tc__Nat__Onat,type,
collect_nat: ( nat > $o ) > nat > $o ).
thf(sy_c_Set_OCollect_000tc__RealDef__Oreal,type,
collect_real: ( real > $o ) > real > $o ).
thf(sy_c_Set_OCollect_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
collec1347809874nt_int: ( product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Set_OCollect_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
collec1979865426at_nat: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_Set_OCollect_000tc__prod_Itc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_Mtc__pro,type,
collec50511176nt_int: ( produc1137372701nt_int > $o ) > produc1137372701nt_int > $o ).
thf(sy_c_Set_Oimage_000tc__Int__Oint_000_062_Itc__Int__Oint_M_Eo_J,type,
image_int_int_o: ( int > int > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_Set_Oimage_000tc__Int__Oint_000tc__Int__Oint,type,
image_int_int: ( int > int ) > ( int > $o ) > int > $o ).
thf(sy_c_Set_Oimage_000tc__Int__Oint_000tc__Nat__Onat,type,
image_int_nat: ( int > nat ) > ( int > $o ) > nat > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Int__Oint,type,
image_nat_int: ( nat > int ) > ( nat > $o ) > int > $o ).
thf(sy_c_Set_Oimage_000tc__Nat__Onat_000tc__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Transcendental_Oarccos,type,
arccos: real > real ).
thf(sy_c_Transcendental_Oarcsin,type,
arcsin: real > real ).
thf(sy_c_Transcendental_Oarctan,type,
arctan: real > real ).
thf(sy_c_Transcendental_Ocos,type,
cos: real > real ).
thf(sy_c_Transcendental_Ocos__coeff,type,
cos_coeff: nat > real ).
thf(sy_c_Transcendental_Odiffs_000tc__RealDef__Oreal,type,
diffs_real: ( nat > real ) > nat > real ).
thf(sy_c_Transcendental_Oexp_000tc__RealDef__Oreal,type,
exp_real: real > real ).
thf(sy_c_Transcendental_Oln,type,
ln: real > real ).
thf(sy_c_Transcendental_Opi,type,
pi: real ).
thf(sy_c_Transcendental_Osin,type,
sin: real > real ).
thf(sy_c_Transcendental_Osin__coeff,type,
sin_coeff: nat > real ).
thf(sy_c_Transcendental_Otan,type,
tan: real > real ).
thf(sy_c_TwoSquares__Mirabelle__xzcihllart_Ois__sum2sq,type,
twoSqu1152398899sum2sq: int > $o ).
thf(sy_c_TwoSquares__Mirabelle__xzcihllart_Osum2sq,type,
twoSqu2072599593sum2sq: product_prod_int_int > int ).
thf(sy_c_Wellfounded_Oaccp_000tc__Int__Oint,type,
accp_int: ( int > int > $o ) > int > $o ).
thf(sy_c_Wellfounded_Oaccp_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
accp_P2006205492nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Wellfounded_Oaccp_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
accp_P490777396at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_nat: product_prod_nat_nat > $o ).
thf(sy_c_WilsonRuss_Oinv,type,
inv: int > int > int ).
thf(sy_c_WilsonRuss_Owset,type,
wset: int > int > int > $o ).
thf(sy_c_member_000_062_Itc__Int__Oint_M_Eo_J,type,
member_int_o: ( int > $o ) > ( ( int > $o ) > $o ) > $o ).
thf(sy_c_member_000tc__Complex__Ocomplex,type,
member_complex: complex > ( complex > $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__Rat__Orat,type,
member_rat: rat > ( rat > $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__Code____Numeral__Ocode____numeral_Mtc__Code____Nume,type,
member1246959636umeral: produc1359518119umeral > ( produc1359518119umeral > $o ) > $o ).
thf(sy_c_member_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
member1747683604nt_int: product_prod_int_int > ( product_prod_int_int > $o ) > $o ).
thf(sy_c_member_000tc__prod_Itc__Int__Oint_Mtc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
member791087631nt_int: produc393999548nt_int > ( produc393999548nt_int > $o ) > $o ).
thf(sy_c_member_000tc__prod_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
member232255508at_nat: product_prod_nat_nat > ( product_prod_nat_nat > $o ) > $o ).
thf(sy_c_member_000tc__prod_Itc__Quickcheck____Narrowing__Ocode____int_Mtc__Quickche,type,
member2092545300de_int: produc167071911de_int > ( produc167071911de_int > $o ) > $o ).
thf(sy_c_member_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J,type,
member1344826522l_real: produc914805421l_real > ( produc914805421l_real > $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_a,type,
a: int ).
thf(sy_v_b,type,
b: int ).
thf(sy_v_p,type,
p: int ).
thf(sy_v_q,type,
q: int ).
%----Relevant facts (5196)
thf(fact_0_xzgcda__linear__aux1,axiom,
! [A: int,R_1: int,B: int,M: int,C: int,D: int,N: int] :
( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ ( times_times_int @ R_1 @ B ) ) @ M ) @ ( times_times_int @ ( minus_minus_int @ C @ ( times_times_int @ R_1 @ D ) ) @ N ) )
= ( minus_minus_int @ ( plus_plus_int @ ( times_times_int @ A @ M ) @ ( times_times_int @ C @ N ) ) @ ( times_times_int @ R_1 @ ( plus_plus_int @ ( times_times_int @ B @ M ) @ ( times_times_int @ D @ N ) ) ) ) ) ).
thf(fact_1_mult__diff__mult,axiom,
! [X_4: rat,Y_3: rat,A_4: rat,B_3: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X_4 @ Y_3 ) @ ( times_times_rat @ A_4 @ B_3 ) )
= ( plus_plus_rat @ ( times_times_rat @ X_4 @ ( minus_minus_rat @ Y_3 @ B_3 ) ) @ ( times_times_rat @ ( minus_minus_rat @ X_4 @ A_4 ) @ B_3 ) ) ) ).
thf(fact_2_mult__diff__mult,axiom,
! [X_4: complex,Y_3: complex,A_4: complex,B_3: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X_4 @ Y_3 ) @ ( times_times_complex @ A_4 @ B_3 ) )
= ( plus_plus_complex @ ( times_times_complex @ X_4 @ ( minus_minus_complex @ Y_3 @ B_3 ) ) @ ( times_times_complex @ ( minus_minus_complex @ X_4 @ A_4 ) @ B_3 ) ) ) ).
thf(fact_3_mult__diff__mult,axiom,
! [X_4: real,Y_3: real,A_4: real,B_3: real] :
( ( minus_minus_real @ ( times_times_real @ X_4 @ Y_3 ) @ ( times_times_real @ A_4 @ B_3 ) )
= ( plus_plus_real @ ( times_times_real @ X_4 @ ( minus_minus_real @ Y_3 @ B_3 ) ) @ ( times_times_real @ ( minus_minus_real @ X_4 @ A_4 ) @ B_3 ) ) ) ).
thf(fact_4_mult__diff__mult,axiom,
! [X_4: int,Y_3: int,A_4: int,B_3: int] :
( ( minus_minus_int @ ( times_times_int @ X_4 @ Y_3 ) @ ( times_times_int @ A_4 @ B_3 ) )
= ( plus_plus_int @ ( times_times_int @ X_4 @ ( minus_minus_int @ Y_3 @ B_3 ) ) @ ( times_times_int @ ( minus_minus_int @ X_4 @ A_4 ) @ B_3 ) ) ) ).
thf(fact_5_mult_Oprod__diff__prod,axiom,
! [X_138: complex,Y_93: complex,A_327: complex,B_244: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X_138 @ Y_93 ) @ ( times_times_complex @ A_327 @ B_244 ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ X_138 @ A_327 ) @ ( minus_minus_complex @ Y_93 @ B_244 ) ) @ ( times_times_complex @ ( minus_minus_complex @ X_138 @ A_327 ) @ B_244 ) ) @ ( times_times_complex @ A_327 @ ( minus_minus_complex @ Y_93 @ B_244 ) ) ) ) ).
thf(fact_6_mult_Oprod__diff__prod,axiom,
! [X_138: real,Y_93: real,A_327: real,B_244: real] :
( ( minus_minus_real @ ( times_times_real @ X_138 @ Y_93 ) @ ( times_times_real @ A_327 @ B_244 ) )
= ( plus_plus_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ X_138 @ A_327 ) @ ( minus_minus_real @ Y_93 @ B_244 ) ) @ ( times_times_real @ ( minus_minus_real @ X_138 @ A_327 ) @ B_244 ) ) @ ( times_times_real @ A_327 @ ( minus_minus_real @ Y_93 @ B_244 ) ) ) ) ).
thf(fact_7_eq__add__iff2,axiom,
! [A_326: rat,E_8: rat,C_122: rat,B_243: rat,D_30: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A_326 @ E_8 ) @ C_122 )
= ( plus_plus_rat @ ( times_times_rat @ B_243 @ E_8 ) @ D_30 ) )
<=> ( C_122
= ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B_243 @ A_326 ) @ E_8 ) @ D_30 ) ) ) ).
thf(fact_8_eq__add__iff2,axiom,
! [A_326: complex,E_8: complex,C_122: complex,B_243: complex,D_30: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ A_326 @ E_8 ) @ C_122 )
= ( plus_plus_complex @ ( times_times_complex @ B_243 @ E_8 ) @ D_30 ) )
<=> ( C_122
= ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B_243 @ A_326 ) @ E_8 ) @ D_30 ) ) ) ).
thf(fact_9_eq__add__iff2,axiom,
! [A_326: real,E_8: real,C_122: real,B_243: real,D_30: real] :
( ( ( plus_plus_real @ ( times_times_real @ A_326 @ E_8 ) @ C_122 )
= ( plus_plus_real @ ( times_times_real @ B_243 @ E_8 ) @ D_30 ) )
<=> ( C_122
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B_243 @ A_326 ) @ E_8 ) @ D_30 ) ) ) ).
thf(fact_10_eq__add__iff2,axiom,
! [A_326: int,E_8: int,C_122: int,B_243: int,D_30: int] :
( ( ( plus_plus_int @ ( times_times_int @ A_326 @ E_8 ) @ C_122 )
= ( plus_plus_int @ ( times_times_int @ B_243 @ E_8 ) @ D_30 ) )
<=> ( C_122
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_243 @ A_326 ) @ E_8 ) @ D_30 ) ) ) ).
thf(fact_11_eq__add__iff1,axiom,
! [A_325: rat,E_7: rat,C_121: rat,B_242: rat,D_29: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A_325 @ E_7 ) @ C_121 )
= ( plus_plus_rat @ ( times_times_rat @ B_242 @ E_7 ) @ D_29 ) )
<=> ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A_325 @ B_242 ) @ E_7 ) @ C_121 )
= D_29 ) ) ).
thf(fact_12_eq__add__iff1,axiom,
! [A_325: complex,E_7: complex,C_121: complex,B_242: complex,D_29: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ A_325 @ E_7 ) @ C_121 )
= ( plus_plus_complex @ ( times_times_complex @ B_242 @ E_7 ) @ D_29 ) )
<=> ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A_325 @ B_242 ) @ E_7 ) @ C_121 )
= D_29 ) ) ).
thf(fact_13_eq__add__iff1,axiom,
! [A_325: real,E_7: real,C_121: real,B_242: real,D_29: real] :
( ( ( plus_plus_real @ ( times_times_real @ A_325 @ E_7 ) @ C_121 )
= ( plus_plus_real @ ( times_times_real @ B_242 @ E_7 ) @ D_29 ) )
<=> ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A_325 @ B_242 ) @ E_7 ) @ C_121 )
= D_29 ) ) ).
thf(fact_14_eq__add__iff1,axiom,
! [A_325: int,E_7: int,C_121: int,B_242: int,D_29: int] :
( ( ( plus_plus_int @ ( times_times_int @ A_325 @ E_7 ) @ C_121 )
= ( plus_plus_int @ ( times_times_int @ B_242 @ E_7 ) @ D_29 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_325 @ B_242 ) @ E_7 ) @ C_121 )
= D_29 ) ) ).
thf(fact_15_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_16_Int2_Oaux1,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
=> ( A
= ( plus_plus_int @ C @ B ) ) ) ).
thf(fact_17_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_18_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_19_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_20_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_21_diff__add__cancel,axiom,
! [A_324: rat,B_241: rat] :
( ( plus_plus_rat @ ( minus_minus_rat @ A_324 @ B_241 ) @ B_241 )
= A_324 ) ).
thf(fact_22_diff__add__cancel,axiom,
! [A_324: complex,B_241: complex] :
( ( plus_plus_complex @ ( minus_minus_complex @ A_324 @ B_241 ) @ B_241 )
= A_324 ) ).
thf(fact_23_diff__add__cancel,axiom,
! [A_324: real,B_241: real] :
( ( plus_plus_real @ ( minus_minus_real @ A_324 @ B_241 ) @ B_241 )
= A_324 ) ).
thf(fact_24_diff__add__cancel,axiom,
! [A_324: int,B_241: int] :
( ( plus_plus_int @ ( minus_minus_int @ A_324 @ B_241 ) @ B_241 )
= A_324 ) ).
thf(fact_25_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_323: rat,B_240: rat,C_120: rat] :
( ( times_times_rat @ ( times_times_rat @ A_323 @ B_240 ) @ C_120 )
= ( times_times_rat @ A_323 @ ( times_times_rat @ B_240 @ C_120 ) ) ) ).
thf(fact_26_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_323: quickcheck_code_int,B_240: quickcheck_code_int,C_120: quickcheck_code_int] :
( ( times_123202395de_int @ ( times_123202395de_int @ A_323 @ B_240 ) @ C_120 )
= ( times_123202395de_int @ A_323 @ ( times_123202395de_int @ B_240 @ C_120 ) ) ) ).
thf(fact_27_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_323: complex,B_240: complex,C_120: complex] :
( ( times_times_complex @ ( times_times_complex @ A_323 @ B_240 ) @ C_120 )
= ( times_times_complex @ A_323 @ ( times_times_complex @ B_240 @ C_120 ) ) ) ).
thf(fact_28_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_323: nat,B_240: nat,C_120: nat] :
( ( times_times_nat @ ( times_times_nat @ A_323 @ B_240 ) @ C_120 )
= ( times_times_nat @ A_323 @ ( times_times_nat @ B_240 @ C_120 ) ) ) ).
thf(fact_29_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_323: real,B_240: real,C_120: real] :
( ( times_times_real @ ( times_times_real @ A_323 @ B_240 ) @ C_120 )
= ( times_times_real @ A_323 @ ( times_times_real @ B_240 @ C_120 ) ) ) ).
thf(fact_30_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_323: int,B_240: int,C_120: int] :
( ( times_times_int @ ( times_times_int @ A_323 @ B_240 ) @ C_120 )
= ( times_times_int @ A_323 @ ( times_times_int @ B_240 @ C_120 ) ) ) ).
thf(fact_31_add__right__imp__eq,axiom,
! [B_239: rat,A_322: rat,C_119: rat] :
( ( ( plus_plus_rat @ B_239 @ A_322 )
= ( plus_plus_rat @ C_119 @ A_322 ) )
=> ( B_239 = C_119 ) ) ).
thf(fact_32_add__right__imp__eq,axiom,
! [B_239: quickcheck_code_int,A_322: quickcheck_code_int,C_119: quickcheck_code_int] :
( ( ( plus_p1446045655de_int @ B_239 @ A_322 )
= ( plus_p1446045655de_int @ C_119 @ A_322 ) )
=> ( B_239 = C_119 ) ) ).
thf(fact_33_add__right__imp__eq,axiom,
! [B_239: complex,A_322: complex,C_119: complex] :
( ( ( plus_plus_complex @ B_239 @ A_322 )
= ( plus_plus_complex @ C_119 @ A_322 ) )
=> ( B_239 = C_119 ) ) ).
thf(fact_34_add__right__imp__eq,axiom,
! [B_239: nat,A_322: nat,C_119: nat] :
( ( ( plus_plus_nat @ B_239 @ A_322 )
= ( plus_plus_nat @ C_119 @ A_322 ) )
=> ( B_239 = C_119 ) ) ).
thf(fact_35_add__right__imp__eq,axiom,
! [B_239: real,A_322: real,C_119: real] :
( ( ( plus_plus_real @ B_239 @ A_322 )
= ( plus_plus_real @ C_119 @ A_322 ) )
=> ( B_239 = C_119 ) ) ).
thf(fact_36_add__right__imp__eq,axiom,
! [B_239: int,A_322: int,C_119: int] :
( ( ( plus_plus_int @ B_239 @ A_322 )
= ( plus_plus_int @ C_119 @ A_322 ) )
=> ( B_239 = C_119 ) ) ).
thf(fact_37_add__imp__eq,axiom,
! [A_321: rat,B_238: rat,C_118: rat] :
( ( ( plus_plus_rat @ A_321 @ B_238 )
= ( plus_plus_rat @ A_321 @ C_118 ) )
=> ( B_238 = C_118 ) ) ).
thf(fact_38_add__imp__eq,axiom,
! [A_321: quickcheck_code_int,B_238: quickcheck_code_int,C_118: quickcheck_code_int] :
( ( ( plus_p1446045655de_int @ A_321 @ B_238 )
= ( plus_p1446045655de_int @ A_321 @ C_118 ) )
=> ( B_238 = C_118 ) ) ).
thf(fact_39_add__imp__eq,axiom,
! [A_321: complex,B_238: complex,C_118: complex] :
( ( ( plus_plus_complex @ A_321 @ B_238 )
= ( plus_plus_complex @ A_321 @ C_118 ) )
=> ( B_238 = C_118 ) ) ).
thf(fact_40_add__imp__eq,axiom,
! [A_321: nat,B_238: nat,C_118: nat] :
( ( ( plus_plus_nat @ A_321 @ B_238 )
= ( plus_plus_nat @ A_321 @ C_118 ) )
=> ( B_238 = C_118 ) ) ).
thf(fact_41_add__imp__eq,axiom,
! [A_321: real,B_238: real,C_118: real] :
( ( ( plus_plus_real @ A_321 @ B_238 )
= ( plus_plus_real @ A_321 @ C_118 ) )
=> ( B_238 = C_118 ) ) ).
thf(fact_42_add__imp__eq,axiom,
! [A_321: int,B_238: int,C_118: int] :
( ( ( plus_plus_int @ A_321 @ B_238 )
= ( plus_plus_int @ A_321 @ C_118 ) )
=> ( B_238 = C_118 ) ) ).
thf(fact_43_add__left__imp__eq,axiom,
! [A_320: rat,B_237: rat,C_117: rat] :
( ( ( plus_plus_rat @ A_320 @ B_237 )
= ( plus_plus_rat @ A_320 @ C_117 ) )
=> ( B_237 = C_117 ) ) ).
thf(fact_44_add__left__imp__eq,axiom,
! [A_320: quickcheck_code_int,B_237: quickcheck_code_int,C_117: quickcheck_code_int] :
( ( ( plus_p1446045655de_int @ A_320 @ B_237 )
= ( plus_p1446045655de_int @ A_320 @ C_117 ) )
=> ( B_237 = C_117 ) ) ).
thf(fact_45_add__left__imp__eq,axiom,
! [A_320: complex,B_237: complex,C_117: complex] :
( ( ( plus_plus_complex @ A_320 @ B_237 )
= ( plus_plus_complex @ A_320 @ C_117 ) )
=> ( B_237 = C_117 ) ) ).
thf(fact_46_add__left__imp__eq,axiom,
! [A_320: nat,B_237: nat,C_117: nat] :
( ( ( plus_plus_nat @ A_320 @ B_237 )
= ( plus_plus_nat @ A_320 @ C_117 ) )
=> ( B_237 = C_117 ) ) ).
thf(fact_47_add__left__imp__eq,axiom,
! [A_320: real,B_237: real,C_117: real] :
( ( ( plus_plus_real @ A_320 @ B_237 )
= ( plus_plus_real @ A_320 @ C_117 ) )
=> ( B_237 = C_117 ) ) ).
thf(fact_48_add__left__imp__eq,axiom,
! [A_320: int,B_237: int,C_117: int] :
( ( ( plus_plus_int @ A_320 @ B_237 )
= ( plus_plus_int @ A_320 @ C_117 ) )
=> ( B_237 = C_117 ) ) ).
thf(fact_49_add__right__cancel,axiom,
! [B_236: rat,A_319: rat,C_116: rat] :
( ( ( plus_plus_rat @ B_236 @ A_319 )
= ( plus_plus_rat @ C_116 @ A_319 ) )
<=> ( B_236 = C_116 ) ) ).
thf(fact_50_add__right__cancel,axiom,
! [B_236: quickcheck_code_int,A_319: quickcheck_code_int,C_116: quickcheck_code_int] :
( ( ( plus_p1446045655de_int @ B_236 @ A_319 )
= ( plus_p1446045655de_int @ C_116 @ A_319 ) )
<=> ( B_236 = C_116 ) ) ).
thf(fact_51_add__right__cancel,axiom,
! [B_236: complex,A_319: complex,C_116: complex] :
( ( ( plus_plus_complex @ B_236 @ A_319 )
= ( plus_plus_complex @ C_116 @ A_319 ) )
<=> ( B_236 = C_116 ) ) ).
thf(fact_52_add__right__cancel,axiom,
! [B_236: nat,A_319: nat,C_116: nat] :
( ( ( plus_plus_nat @ B_236 @ A_319 )
= ( plus_plus_nat @ C_116 @ A_319 ) )
<=> ( B_236 = C_116 ) ) ).
thf(fact_53_add__right__cancel,axiom,
! [B_236: real,A_319: real,C_116: real] :
( ( ( plus_plus_real @ B_236 @ A_319 )
= ( plus_plus_real @ C_116 @ A_319 ) )
<=> ( B_236 = C_116 ) ) ).
thf(fact_54_add__right__cancel,axiom,
! [B_236: int,A_319: int,C_116: int] :
( ( ( plus_plus_int @ B_236 @ A_319 )
= ( plus_plus_int @ C_116 @ A_319 ) )
<=> ( B_236 = C_116 ) ) ).
thf(fact_55_add__left__cancel,axiom,
! [A_318: rat,B_235: rat,C_115: rat] :
( ( ( plus_plus_rat @ A_318 @ B_235 )
= ( plus_plus_rat @ A_318 @ C_115 ) )
<=> ( B_235 = C_115 ) ) ).
thf(fact_56_add__left__cancel,axiom,
! [A_318: quickcheck_code_int,B_235: quickcheck_code_int,C_115: quickcheck_code_int] :
( ( ( plus_p1446045655de_int @ A_318 @ B_235 )
= ( plus_p1446045655de_int @ A_318 @ C_115 ) )
<=> ( B_235 = C_115 ) ) ).
thf(fact_57_add__left__cancel,axiom,
! [A_318: complex,B_235: complex,C_115: complex] :
( ( ( plus_plus_complex @ A_318 @ B_235 )
= ( plus_plus_complex @ A_318 @ C_115 ) )
<=> ( B_235 = C_115 ) ) ).
thf(fact_58_add__left__cancel,axiom,
! [A_318: nat,B_235: nat,C_115: nat] :
( ( ( plus_plus_nat @ A_318 @ B_235 )
= ( plus_plus_nat @ A_318 @ C_115 ) )
<=> ( B_235 = C_115 ) ) ).
thf(fact_59_add__left__cancel,axiom,
! [A_318: real,B_235: real,C_115: real] :
( ( ( plus_plus_real @ A_318 @ B_235 )
= ( plus_plus_real @ A_318 @ C_115 ) )
<=> ( B_235 = C_115 ) ) ).
thf(fact_60_add__left__cancel,axiom,
! [A_318: int,B_235: int,C_115: int] :
( ( ( plus_plus_int @ A_318 @ B_235 )
= ( plus_plus_int @ A_318 @ C_115 ) )
<=> ( B_235 = C_115 ) ) ).
thf(fact_61_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_317: rat,B_234: rat,C_114: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A_317 @ B_234 ) @ C_114 )
= ( plus_plus_rat @ A_317 @ ( plus_plus_rat @ B_234 @ C_114 ) ) ) ).
thf(fact_62_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_317: quickcheck_code_int,B_234: quickcheck_code_int,C_114: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_317 @ B_234 ) @ C_114 )
= ( plus_p1446045655de_int @ A_317 @ ( plus_p1446045655de_int @ B_234 @ C_114 ) ) ) ).
thf(fact_63_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_317: complex,B_234: complex,C_114: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A_317 @ B_234 ) @ C_114 )
= ( plus_plus_complex @ A_317 @ ( plus_plus_complex @ B_234 @ C_114 ) ) ) ).
thf(fact_64_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_317: nat,B_234: nat,C_114: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_317 @ B_234 ) @ C_114 )
= ( plus_plus_nat @ A_317 @ ( plus_plus_nat @ B_234 @ C_114 ) ) ) ).
thf(fact_65_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_317: real,B_234: real,C_114: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_317 @ B_234 ) @ C_114 )
= ( plus_plus_real @ A_317 @ ( plus_plus_real @ B_234 @ C_114 ) ) ) ).
thf(fact_66_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_317: int,B_234: int,C_114: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_317 @ B_234 ) @ C_114 )
= ( plus_plus_int @ A_317 @ ( plus_plus_int @ B_234 @ C_114 ) ) ) ).
thf(fact_67_diff__eq__diff__eq,axiom,
! [A_316: rat,B_233: rat,C_113: rat,D_28: rat] :
( ( ( minus_minus_rat @ A_316 @ B_233 )
= ( minus_minus_rat @ C_113 @ D_28 ) )
=> ( ( A_316 = B_233 )
<=> ( C_113 = D_28 ) ) ) ).
thf(fact_68_diff__eq__diff__eq,axiom,
! [A_316: complex,B_233: complex,C_113: complex,D_28: complex] :
( ( ( minus_minus_complex @ A_316 @ B_233 )
= ( minus_minus_complex @ C_113 @ D_28 ) )
=> ( ( A_316 = B_233 )
<=> ( C_113 = D_28 ) ) ) ).
thf(fact_69_diff__eq__diff__eq,axiom,
! [A_316: real,B_233: real,C_113: real,D_28: real] :
( ( ( minus_minus_real @ A_316 @ B_233 )
= ( minus_minus_real @ C_113 @ D_28 ) )
=> ( ( A_316 = B_233 )
<=> ( C_113 = D_28 ) ) ) ).
thf(fact_70_diff__eq__diff__eq,axiom,
! [A_316: int,B_233: int,C_113: int,D_28: int] :
( ( ( minus_minus_int @ A_316 @ B_233 )
= ( minus_minus_int @ C_113 @ D_28 ) )
=> ( ( A_316 = B_233 )
<=> ( C_113 = D_28 ) ) ) ).
thf(fact_71_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_72_zmult__commute,axiom,
! [Z_1: int,W: int] :
( ( times_times_int @ Z_1 @ W )
= ( times_times_int @ W @ Z_1 ) ) ).
thf(fact_73_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_74_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_75_zadd__commute,axiom,
! [Z_1: int,W: int] :
( ( plus_plus_int @ Z_1 @ W )
= ( plus_plus_int @ W @ Z_1 ) ) ).
thf(fact_76_combine__common__factor,axiom,
! [A_315: rat,E_6: rat,B_232: rat,C_112: rat] :
( ( plus_plus_rat @ ( times_times_rat @ A_315 @ E_6 ) @ ( plus_plus_rat @ ( times_times_rat @ B_232 @ E_6 ) @ C_112 ) )
= ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A_315 @ B_232 ) @ E_6 ) @ C_112 ) ) ).
thf(fact_77_combine__common__factor,axiom,
! [A_315: quickcheck_code_int,E_6: quickcheck_code_int,B_232: quickcheck_code_int,C_112: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( times_123202395de_int @ A_315 @ E_6 ) @ ( plus_p1446045655de_int @ ( times_123202395de_int @ B_232 @ E_6 ) @ C_112 ) )
= ( plus_p1446045655de_int @ ( times_123202395de_int @ ( plus_p1446045655de_int @ A_315 @ B_232 ) @ E_6 ) @ C_112 ) ) ).
thf(fact_78_combine__common__factor,axiom,
! [A_315: complex,E_6: complex,B_232: complex,C_112: complex] :
( ( plus_plus_complex @ ( times_times_complex @ A_315 @ E_6 ) @ ( plus_plus_complex @ ( times_times_complex @ B_232 @ E_6 ) @ C_112 ) )
= ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A_315 @ B_232 ) @ E_6 ) @ C_112 ) ) ).
thf(fact_79_combine__common__factor,axiom,
! [A_315: nat,E_6: nat,B_232: nat,C_112: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_315 @ E_6 ) @ ( plus_plus_nat @ ( times_times_nat @ B_232 @ E_6 ) @ C_112 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A_315 @ B_232 ) @ E_6 ) @ C_112 ) ) ).
thf(fact_80_combine__common__factor,axiom,
! [A_315: real,E_6: real,B_232: real,C_112: real] :
( ( plus_plus_real @ ( times_times_real @ A_315 @ E_6 ) @ ( plus_plus_real @ ( times_times_real @ B_232 @ E_6 ) @ C_112 ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A_315 @ B_232 ) @ E_6 ) @ C_112 ) ) ).
thf(fact_81_combine__common__factor,axiom,
! [A_315: int,E_6: int,B_232: int,C_112: int] :
( ( plus_plus_int @ ( times_times_int @ A_315 @ E_6 ) @ ( plus_plus_int @ ( times_times_int @ B_232 @ E_6 ) @ C_112 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A_315 @ B_232 ) @ E_6 ) @ C_112 ) ) ).
thf(fact_82_mult__left_Oadd,axiom,
! [X_137: complex,Y_92: complex,Ya_4: complex] :
( ( times_times_complex @ ( plus_plus_complex @ X_137 @ Y_92 ) @ Ya_4 )
= ( plus_plus_complex @ ( times_times_complex @ X_137 @ Ya_4 ) @ ( times_times_complex @ Y_92 @ Ya_4 ) ) ) ).
thf(fact_83_mult__left_Oadd,axiom,
! [X_137: real,Y_92: real,Ya_4: real] :
( ( times_times_real @ ( plus_plus_real @ X_137 @ Y_92 ) @ Ya_4 )
= ( plus_plus_real @ ( times_times_real @ X_137 @ Ya_4 ) @ ( times_times_real @ Y_92 @ Ya_4 ) ) ) ).
thf(fact_84_mult_Oadd__left,axiom,
! [A_314: complex,A_313: complex,B_231: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A_314 @ A_313 ) @ B_231 )
= ( plus_plus_complex @ ( times_times_complex @ A_314 @ B_231 ) @ ( times_times_complex @ A_313 @ B_231 ) ) ) ).
thf(fact_85_mult_Oadd__left,axiom,
! [A_314: real,A_313: real,B_231: real] :
( ( times_times_real @ ( plus_plus_real @ A_314 @ A_313 ) @ B_231 )
= ( plus_plus_real @ ( times_times_real @ A_314 @ B_231 ) @ ( times_times_real @ A_313 @ B_231 ) ) ) ).
thf(fact_86_comm__semiring__class_Odistrib,axiom,
! [A_312: rat,B_230: rat,C_111: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A_312 @ B_230 ) @ C_111 )
= ( plus_plus_rat @ ( times_times_rat @ A_312 @ C_111 ) @ ( times_times_rat @ B_230 @ C_111 ) ) ) ).
thf(fact_87_comm__semiring__class_Odistrib,axiom,
! [A_312: quickcheck_code_int,B_230: quickcheck_code_int,C_111: quickcheck_code_int] :
( ( times_123202395de_int @ ( plus_p1446045655de_int @ A_312 @ B_230 ) @ C_111 )
= ( plus_p1446045655de_int @ ( times_123202395de_int @ A_312 @ C_111 ) @ ( times_123202395de_int @ B_230 @ C_111 ) ) ) ).
thf(fact_88_comm__semiring__class_Odistrib,axiom,
! [A_312: complex,B_230: complex,C_111: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A_312 @ B_230 ) @ C_111 )
= ( plus_plus_complex @ ( times_times_complex @ A_312 @ C_111 ) @ ( times_times_complex @ B_230 @ C_111 ) ) ) ).
thf(fact_89_comm__semiring__class_Odistrib,axiom,
! [A_312: nat,B_230: nat,C_111: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_312 @ B_230 ) @ C_111 )
= ( plus_plus_nat @ ( times_times_nat @ A_312 @ C_111 ) @ ( times_times_nat @ B_230 @ C_111 ) ) ) ).
thf(fact_90_comm__semiring__class_Odistrib,axiom,
! [A_312: real,B_230: real,C_111: real] :
( ( times_times_real @ ( plus_plus_real @ A_312 @ B_230 ) @ C_111 )
= ( plus_plus_real @ ( times_times_real @ A_312 @ C_111 ) @ ( times_times_real @ B_230 @ C_111 ) ) ) ).
thf(fact_91_comm__semiring__class_Odistrib,axiom,
! [A_312: int,B_230: int,C_111: int] :
( ( times_times_int @ ( plus_plus_int @ A_312 @ B_230 ) @ C_111 )
= ( plus_plus_int @ ( times_times_int @ A_312 @ C_111 ) @ ( times_times_int @ B_230 @ C_111 ) ) ) ).
thf(fact_92_mult__right_Oadd,axiom,
! [Xa_4: complex,X_136: complex,Y_91: complex] :
( ( times_times_complex @ Xa_4 @ ( plus_plus_complex @ X_136 @ Y_91 ) )
= ( plus_plus_complex @ ( times_times_complex @ Xa_4 @ X_136 ) @ ( times_times_complex @ Xa_4 @ Y_91 ) ) ) ).
thf(fact_93_mult__right_Oadd,axiom,
! [Xa_4: real,X_136: real,Y_91: real] :
( ( times_times_real @ Xa_4 @ ( plus_plus_real @ X_136 @ Y_91 ) )
= ( plus_plus_real @ ( times_times_real @ Xa_4 @ X_136 ) @ ( times_times_real @ Xa_4 @ Y_91 ) ) ) ).
thf(fact_94_mult_Oadd__right,axiom,
! [A_311: complex,B_229: complex,B_228: complex] :
( ( times_times_complex @ A_311 @ ( plus_plus_complex @ B_229 @ B_228 ) )
= ( plus_plus_complex @ ( times_times_complex @ A_311 @ B_229 ) @ ( times_times_complex @ A_311 @ B_228 ) ) ) ).
thf(fact_95_mult_Oadd__right,axiom,
! [A_311: real,B_229: real,B_228: real] :
( ( times_times_real @ A_311 @ ( plus_plus_real @ B_229 @ B_228 ) )
= ( plus_plus_real @ ( times_times_real @ A_311 @ B_229 ) @ ( times_times_real @ A_311 @ B_228 ) ) ) ).
thf(fact_96_mult__left_Odiff,axiom,
! [X_135: complex,Y_90: complex,Ya_3: complex] :
( ( times_times_complex @ ( minus_minus_complex @ X_135 @ Y_90 ) @ Ya_3 )
= ( minus_minus_complex @ ( times_times_complex @ X_135 @ Ya_3 ) @ ( times_times_complex @ Y_90 @ Ya_3 ) ) ) ).
thf(fact_97_mult__left_Odiff,axiom,
! [X_135: real,Y_90: real,Ya_3: real] :
( ( times_times_real @ ( minus_minus_real @ X_135 @ Y_90 ) @ Ya_3 )
= ( minus_minus_real @ ( times_times_real @ X_135 @ Ya_3 ) @ ( times_times_real @ Y_90 @ Ya_3 ) ) ) ).
thf(fact_98_mult_Odiff__left,axiom,
! [A_310: complex,A_309: complex,B_227: complex] :
( ( times_times_complex @ ( minus_minus_complex @ A_310 @ A_309 ) @ B_227 )
= ( minus_minus_complex @ ( times_times_complex @ A_310 @ B_227 ) @ ( times_times_complex @ A_309 @ B_227 ) ) ) ).
thf(fact_99_mult_Odiff__left,axiom,
! [A_310: real,A_309: real,B_227: real] :
( ( times_times_real @ ( minus_minus_real @ A_310 @ A_309 ) @ B_227 )
= ( minus_minus_real @ ( times_times_real @ A_310 @ B_227 ) @ ( times_times_real @ A_309 @ B_227 ) ) ) ).
thf(fact_100_mult__right_Odiff,axiom,
! [Xa_3: complex,X_134: complex,Y_89: complex] :
( ( times_times_complex @ Xa_3 @ ( minus_minus_complex @ X_134 @ Y_89 ) )
= ( minus_minus_complex @ ( times_times_complex @ Xa_3 @ X_134 ) @ ( times_times_complex @ Xa_3 @ Y_89 ) ) ) ).
thf(fact_101_mult__right_Odiff,axiom,
! [Xa_3: real,X_134: real,Y_89: real] :
( ( times_times_real @ Xa_3 @ ( minus_minus_real @ X_134 @ Y_89 ) )
= ( minus_minus_real @ ( times_times_real @ Xa_3 @ X_134 ) @ ( times_times_real @ Xa_3 @ Y_89 ) ) ) ).
thf(fact_102_mult_Odiff__right,axiom,
! [A_308: complex,B_226: complex,B_225: complex] :
( ( times_times_complex @ A_308 @ ( minus_minus_complex @ B_226 @ B_225 ) )
= ( minus_minus_complex @ ( times_times_complex @ A_308 @ B_226 ) @ ( times_times_complex @ A_308 @ B_225 ) ) ) ).
thf(fact_103_mult_Odiff__right,axiom,
! [A_308: real,B_226: real,B_225: real] :
( ( times_times_real @ A_308 @ ( minus_minus_real @ B_226 @ B_225 ) )
= ( minus_minus_real @ ( times_times_real @ A_308 @ B_226 ) @ ( times_times_real @ A_308 @ B_225 ) ) ) ).
thf(fact_104_add__diff__add,axiom,
! [A_307: rat,C_110: rat,B_224: rat,D_27: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A_307 @ C_110 ) @ ( plus_plus_rat @ B_224 @ D_27 ) )
= ( plus_plus_rat @ ( minus_minus_rat @ A_307 @ B_224 ) @ ( minus_minus_rat @ C_110 @ D_27 ) ) ) ).
thf(fact_105_add__diff__add,axiom,
! [A_307: complex,C_110: complex,B_224: complex,D_27: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A_307 @ C_110 ) @ ( plus_plus_complex @ B_224 @ D_27 ) )
= ( plus_plus_complex @ ( minus_minus_complex @ A_307 @ B_224 ) @ ( minus_minus_complex @ C_110 @ D_27 ) ) ) ).
thf(fact_106_add__diff__add,axiom,
! [A_307: real,C_110: real,B_224: real,D_27: real] :
( ( minus_minus_real @ ( plus_plus_real @ A_307 @ C_110 ) @ ( plus_plus_real @ B_224 @ D_27 ) )
= ( plus_plus_real @ ( minus_minus_real @ A_307 @ B_224 ) @ ( minus_minus_real @ C_110 @ D_27 ) ) ) ).
thf(fact_107_add__diff__add,axiom,
! [A_307: int,C_110: int,B_224: int,D_27: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_307 @ C_110 ) @ ( plus_plus_int @ B_224 @ D_27 ) )
= ( plus_plus_int @ ( minus_minus_int @ A_307 @ B_224 ) @ ( minus_minus_int @ C_110 @ D_27 ) ) ) ).
thf(fact_108_add__diff__cancel,axiom,
! [A_306: rat,B_223: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A_306 @ B_223 ) @ B_223 )
= A_306 ) ).
thf(fact_109_add__diff__cancel,axiom,
! [A_306: complex,B_223: complex] :
( ( minus_minus_complex @ ( plus_plus_complex @ A_306 @ B_223 ) @ B_223 )
= A_306 ) ).
thf(fact_110_add__diff__cancel,axiom,
! [A_306: real,B_223: real] :
( ( minus_minus_real @ ( plus_plus_real @ A_306 @ B_223 ) @ B_223 )
= A_306 ) ).
thf(fact_111_add__diff__cancel,axiom,
! [A_306: int,B_223: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_306 @ B_223 ) @ B_223 )
= A_306 ) ).
thf(fact_112_crossproduct__eq,axiom,
! [W_20: rat,Y_88: rat,X_133: rat,Z_22: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ W_20 @ Y_88 ) @ ( times_times_rat @ X_133 @ Z_22 ) )
= ( plus_plus_rat @ ( times_times_rat @ W_20 @ Z_22 ) @ ( times_times_rat @ X_133 @ Y_88 ) ) )
<=> ( ( W_20 = X_133 )
| ( Y_88 = Z_22 ) ) ) ).
thf(fact_113_crossproduct__eq,axiom,
! [W_20: complex,Y_88: complex,X_133: complex,Z_22: complex] :
( ( ( plus_plus_complex @ ( times_times_complex @ W_20 @ Y_88 ) @ ( times_times_complex @ X_133 @ Z_22 ) )
= ( plus_plus_complex @ ( times_times_complex @ W_20 @ Z_22 ) @ ( times_times_complex @ X_133 @ Y_88 ) ) )
<=> ( ( W_20 = X_133 )
| ( Y_88 = Z_22 ) ) ) ).
thf(fact_114_crossproduct__eq,axiom,
! [W_20: nat,Y_88: nat,X_133: nat,Z_22: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W_20 @ Y_88 ) @ ( times_times_nat @ X_133 @ Z_22 ) )
= ( plus_plus_nat @ ( times_times_nat @ W_20 @ Z_22 ) @ ( times_times_nat @ X_133 @ Y_88 ) ) )
<=> ( ( W_20 = X_133 )
| ( Y_88 = Z_22 ) ) ) ).
thf(fact_115_crossproduct__eq,axiom,
! [W_20: real,Y_88: real,X_133: real,Z_22: real] :
( ( ( plus_plus_real @ ( times_times_real @ W_20 @ Y_88 ) @ ( times_times_real @ X_133 @ Z_22 ) )
= ( plus_plus_real @ ( times_times_real @ W_20 @ Z_22 ) @ ( times_times_real @ X_133 @ Y_88 ) ) )
<=> ( ( W_20 = X_133 )
| ( Y_88 = Z_22 ) ) ) ).
thf(fact_116_crossproduct__eq,axiom,
! [W_20: int,Y_88: int,X_133: int,Z_22: int] :
( ( ( plus_plus_int @ ( times_times_int @ W_20 @ Y_88 ) @ ( times_times_int @ X_133 @ Z_22 ) )
= ( plus_plus_int @ ( times_times_int @ W_20 @ Z_22 ) @ ( times_times_int @ X_133 @ Y_88 ) ) )
<=> ( ( W_20 = X_133 )
| ( Y_88 = Z_22 ) ) ) ).
thf(fact_117_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_305: rat,M_11: rat,B_222: rat] :
( ( plus_plus_rat @ ( times_times_rat @ A_305 @ M_11 ) @ ( times_times_rat @ B_222 @ M_11 ) )
= ( times_times_rat @ ( plus_plus_rat @ A_305 @ B_222 ) @ M_11 ) ) ).
thf(fact_118_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_305: quickcheck_code_int,M_11: quickcheck_code_int,B_222: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( times_123202395de_int @ A_305 @ M_11 ) @ ( times_123202395de_int @ B_222 @ M_11 ) )
= ( times_123202395de_int @ ( plus_p1446045655de_int @ A_305 @ B_222 ) @ M_11 ) ) ).
thf(fact_119_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_305: complex,M_11: complex,B_222: complex] :
( ( plus_plus_complex @ ( times_times_complex @ A_305 @ M_11 ) @ ( times_times_complex @ B_222 @ M_11 ) )
= ( times_times_complex @ ( plus_plus_complex @ A_305 @ B_222 ) @ M_11 ) ) ).
thf(fact_120_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_305: nat,M_11: nat,B_222: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_305 @ M_11 ) @ ( times_times_nat @ B_222 @ M_11 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_305 @ B_222 ) @ M_11 ) ) ).
thf(fact_121_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_305: real,M_11: real,B_222: real] :
( ( plus_plus_real @ ( times_times_real @ A_305 @ M_11 ) @ ( times_times_real @ B_222 @ M_11 ) )
= ( times_times_real @ ( plus_plus_real @ A_305 @ B_222 ) @ M_11 ) ) ).
thf(fact_122_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_305: int,M_11: int,B_222: int] :
( ( plus_plus_int @ ( times_times_int @ A_305 @ M_11 ) @ ( times_times_int @ B_222 @ M_11 ) )
= ( times_times_int @ ( plus_plus_int @ A_305 @ B_222 ) @ M_11 ) ) ).
thf(fact_123_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_304: rat,B_221: rat,C_109: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A_304 @ B_221 ) @ C_109 )
= ( plus_plus_rat @ ( times_times_rat @ A_304 @ C_109 ) @ ( times_times_rat @ B_221 @ C_109 ) ) ) ).
thf(fact_124_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_304: quickcheck_code_int,B_221: quickcheck_code_int,C_109: quickcheck_code_int] :
( ( times_123202395de_int @ ( plus_p1446045655de_int @ A_304 @ B_221 ) @ C_109 )
= ( plus_p1446045655de_int @ ( times_123202395de_int @ A_304 @ C_109 ) @ ( times_123202395de_int @ B_221 @ C_109 ) ) ) ).
thf(fact_125_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_304: complex,B_221: complex,C_109: complex] :
( ( times_times_complex @ ( plus_plus_complex @ A_304 @ B_221 ) @ C_109 )
= ( plus_plus_complex @ ( times_times_complex @ A_304 @ C_109 ) @ ( times_times_complex @ B_221 @ C_109 ) ) ) ).
thf(fact_126_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_304: nat,B_221: nat,C_109: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A_304 @ B_221 ) @ C_109 )
= ( plus_plus_nat @ ( times_times_nat @ A_304 @ C_109 ) @ ( times_times_nat @ B_221 @ C_109 ) ) ) ).
thf(fact_127_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_304: real,B_221: real,C_109: real] :
( ( times_times_real @ ( plus_plus_real @ A_304 @ B_221 ) @ C_109 )
= ( plus_plus_real @ ( times_times_real @ A_304 @ C_109 ) @ ( times_times_real @ B_221 @ C_109 ) ) ) ).
thf(fact_128_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_304: int,B_221: int,C_109: int] :
( ( times_times_int @ ( plus_plus_int @ A_304 @ B_221 ) @ C_109 )
= ( plus_plus_int @ ( times_times_int @ A_304 @ C_109 ) @ ( times_times_int @ B_221 @ C_109 ) ) ) ).
thf(fact_129_crossproduct__noteq,axiom,
! [C_108: rat,D_26: rat,A_303: rat,B_220: rat] :
( ( ( A_303 != B_220 )
& ( C_108 != D_26 ) )
<=> ( ( plus_plus_rat @ ( times_times_rat @ A_303 @ C_108 ) @ ( times_times_rat @ B_220 @ D_26 ) )
!= ( plus_plus_rat @ ( times_times_rat @ A_303 @ D_26 ) @ ( times_times_rat @ B_220 @ C_108 ) ) ) ) ).
thf(fact_130_crossproduct__noteq,axiom,
! [C_108: complex,D_26: complex,A_303: complex,B_220: complex] :
( ( ( A_303 != B_220 )
& ( C_108 != D_26 ) )
<=> ( ( plus_plus_complex @ ( times_times_complex @ A_303 @ C_108 ) @ ( times_times_complex @ B_220 @ D_26 ) )
!= ( plus_plus_complex @ ( times_times_complex @ A_303 @ D_26 ) @ ( times_times_complex @ B_220 @ C_108 ) ) ) ) ).
thf(fact_131_crossproduct__noteq,axiom,
! [C_108: nat,D_26: nat,A_303: nat,B_220: nat] :
( ( ( A_303 != B_220 )
& ( C_108 != D_26 ) )
<=> ( ( plus_plus_nat @ ( times_times_nat @ A_303 @ C_108 ) @ ( times_times_nat @ B_220 @ D_26 ) )
!= ( plus_plus_nat @ ( times_times_nat @ A_303 @ D_26 ) @ ( times_times_nat @ B_220 @ C_108 ) ) ) ) ).
thf(fact_132_crossproduct__noteq,axiom,
! [C_108: real,D_26: real,A_303: real,B_220: real] :
( ( ( A_303 != B_220 )
& ( C_108 != D_26 ) )
<=> ( ( plus_plus_real @ ( times_times_real @ A_303 @ C_108 ) @ ( times_times_real @ B_220 @ D_26 ) )
!= ( plus_plus_real @ ( times_times_real @ A_303 @ D_26 ) @ ( times_times_real @ B_220 @ C_108 ) ) ) ) ).
thf(fact_133_crossproduct__noteq,axiom,
! [C_108: int,D_26: int,A_303: int,B_220: int] :
( ( ( A_303 != B_220 )
& ( C_108 != D_26 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ A_303 @ C_108 ) @ ( times_times_int @ B_220 @ D_26 ) )
!= ( plus_plus_int @ ( times_times_int @ A_303 @ D_26 ) @ ( times_times_int @ B_220 @ C_108 ) ) ) ) ).
thf(fact_134_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_132: rat,Y_87: rat,Z_21: rat] :
( ( times_times_rat @ X_132 @ ( plus_plus_rat @ Y_87 @ Z_21 ) )
= ( plus_plus_rat @ ( times_times_rat @ X_132 @ Y_87 ) @ ( times_times_rat @ X_132 @ Z_21 ) ) ) ).
thf(fact_135_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_132: quickcheck_code_int,Y_87: quickcheck_code_int,Z_21: quickcheck_code_int] :
( ( times_123202395de_int @ X_132 @ ( plus_p1446045655de_int @ Y_87 @ Z_21 ) )
= ( plus_p1446045655de_int @ ( times_123202395de_int @ X_132 @ Y_87 ) @ ( times_123202395de_int @ X_132 @ Z_21 ) ) ) ).
thf(fact_136_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_132: complex,Y_87: complex,Z_21: complex] :
( ( times_times_complex @ X_132 @ ( plus_plus_complex @ Y_87 @ Z_21 ) )
= ( plus_plus_complex @ ( times_times_complex @ X_132 @ Y_87 ) @ ( times_times_complex @ X_132 @ Z_21 ) ) ) ).
thf(fact_137_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_132: nat,Y_87: nat,Z_21: nat] :
( ( times_times_nat @ X_132 @ ( plus_plus_nat @ Y_87 @ Z_21 ) )
= ( plus_plus_nat @ ( times_times_nat @ X_132 @ Y_87 ) @ ( times_times_nat @ X_132 @ Z_21 ) ) ) ).
thf(fact_138_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_132: real,Y_87: real,Z_21: real] :
( ( times_times_real @ X_132 @ ( plus_plus_real @ Y_87 @ Z_21 ) )
= ( plus_plus_real @ ( times_times_real @ X_132 @ Y_87 ) @ ( times_times_real @ X_132 @ Z_21 ) ) ) ).
thf(fact_139_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_132: int,Y_87: int,Z_21: int] :
( ( times_times_int @ X_132 @ ( plus_plus_int @ Y_87 @ Z_21 ) )
= ( plus_plus_int @ ( times_times_int @ X_132 @ Y_87 ) @ ( times_times_int @ X_132 @ Z_21 ) ) ) ).
thf(fact_140_Pair__inject,axiom,
! [A_302: int > $o,B_219: int > $o,A_301: int > $o,B_218: int > $o] :
( ( ( produc398918003_int_o @ A_302 @ B_219 )
= ( produc398918003_int_o @ A_301 @ B_218 ) )
=> ~ ( ( A_302 = A_301 )
=> ( B_219 != B_218 ) ) ) ).
thf(fact_141_Pair__inject,axiom,
! [A_302: product_prod_int_int,B_219: product_prod_int_int,A_301: product_prod_int_int,B_218: product_prod_int_int] :
( ( ( produc883642259nt_int @ A_302 @ B_219 )
= ( produc883642259nt_int @ A_301 @ B_218 ) )
=> ~ ( ( A_302 = A_301 )
=> ( B_219 != B_218 ) ) ) ).
thf(fact_142_Pair__inject,axiom,
! [A_302: quickcheck_code_int,B_219: quickcheck_code_int,A_301: quickcheck_code_int,B_218: quickcheck_code_int] :
( ( ( produc1318306967de_int @ A_302 @ B_219 )
= ( produc1318306967de_int @ A_301 @ B_218 ) )
=> ~ ( ( A_302 = A_301 )
=> ( B_219 != B_218 ) ) ) ).
thf(fact_143_Pair__inject,axiom,
! [A_302: code_code_numeral,B_219: code_code_numeral,A_301: code_code_numeral,B_218: code_code_numeral] :
( ( ( produc2136830103umeral @ A_302 @ B_219 )
= ( produc2136830103umeral @ A_301 @ B_218 ) )
=> ~ ( ( A_302 = A_301 )
=> ( B_219 != B_218 ) ) ) ).
thf(fact_144_Pair__inject,axiom,
! [A_302: real,B_219: real,A_301: real,B_218: real] :
( ( ( produc865579683l_real @ A_302 @ B_219 )
= ( produc865579683l_real @ A_301 @ B_218 ) )
=> ~ ( ( A_302 = A_301 )
=> ( B_219 != B_218 ) ) ) ).
thf(fact_145_Pair__inject,axiom,
! [A_302: product_prod_nat_nat,B_219: product_prod_nat_nat,A_301: product_prod_nat_nat,B_218: product_prod_nat_nat] :
( ( ( produc494345619at_nat @ A_302 @ B_219 )
= ( produc494345619at_nat @ A_301 @ B_218 ) )
=> ~ ( ( A_302 = A_301 )
=> ( B_219 != B_218 ) ) ) ).
thf(fact_146_Pair__inject,axiom,
! [A_302: nat,B_219: nat,A_301: nat,B_218: nat] :
( ( ( product_Pair_nat_nat @ A_302 @ B_219 )
= ( product_Pair_nat_nat @ A_301 @ B_218 ) )
=> ~ ( ( A_302 = A_301 )
=> ( B_219 != B_218 ) ) ) ).
thf(fact_147_Pair__inject,axiom,
! [A_302: int,B_219: product_prod_int_int,A_301: int,B_218: product_prod_int_int] :
( ( ( produc282740534nt_int @ A_302 @ B_219 )
= ( produc282740534nt_int @ A_301 @ B_218 ) )
=> ~ ( ( A_302 = A_301 )
=> ( B_219 != B_218 ) ) ) ).
thf(fact_148_Pair__inject,axiom,
! [A_302: int,B_219: int,A_301: int,B_218: int] :
( ( ( product_Pair_int_int @ A_302 @ B_219 )
= ( product_Pair_int_int @ A_301 @ B_218 ) )
=> ~ ( ( A_302 = A_301 )
=> ( B_219 != B_218 ) ) ) ).
thf(fact_149_Pair__eq,axiom,
! [A_300: int > $o,B_217: int > $o,A_299: int > $o,B_216: int > $o] :
( ( ( produc398918003_int_o @ A_300 @ B_217 )
= ( produc398918003_int_o @ A_299 @ B_216 ) )
<=> ( ( A_300 = A_299 )
& ( B_217 = B_216 ) ) ) ).
thf(fact_150_Pair__eq,axiom,
! [A_300: product_prod_int_int,B_217: product_prod_int_int,A_299: product_prod_int_int,B_216: product_prod_int_int] :
( ( ( produc883642259nt_int @ A_300 @ B_217 )
= ( produc883642259nt_int @ A_299 @ B_216 ) )
<=> ( ( A_300 = A_299 )
& ( B_217 = B_216 ) ) ) ).
thf(fact_151_Pair__eq,axiom,
! [A_300: quickcheck_code_int,B_217: quickcheck_code_int,A_299: quickcheck_code_int,B_216: quickcheck_code_int] :
( ( ( produc1318306967de_int @ A_300 @ B_217 )
= ( produc1318306967de_int @ A_299 @ B_216 ) )
<=> ( ( A_300 = A_299 )
& ( B_217 = B_216 ) ) ) ).
thf(fact_152_Pair__eq,axiom,
! [A_300: code_code_numeral,B_217: code_code_numeral,A_299: code_code_numeral,B_216: code_code_numeral] :
( ( ( produc2136830103umeral @ A_300 @ B_217 )
= ( produc2136830103umeral @ A_299 @ B_216 ) )
<=> ( ( A_300 = A_299 )
& ( B_217 = B_216 ) ) ) ).
thf(fact_153_Pair__eq,axiom,
! [A_300: real,B_217: real,A_299: real,B_216: real] :
( ( ( produc865579683l_real @ A_300 @ B_217 )
= ( produc865579683l_real @ A_299 @ B_216 ) )
<=> ( ( A_300 = A_299 )
& ( B_217 = B_216 ) ) ) ).
thf(fact_154_Pair__eq,axiom,
! [A_300: product_prod_nat_nat,B_217: product_prod_nat_nat,A_299: product_prod_nat_nat,B_216: product_prod_nat_nat] :
( ( ( produc494345619at_nat @ A_300 @ B_217 )
= ( produc494345619at_nat @ A_299 @ B_216 ) )
<=> ( ( A_300 = A_299 )
& ( B_217 = B_216 ) ) ) ).
thf(fact_155_Pair__eq,axiom,
! [A_300: nat,B_217: nat,A_299: nat,B_216: nat] :
( ( ( product_Pair_nat_nat @ A_300 @ B_217 )
= ( product_Pair_nat_nat @ A_299 @ B_216 ) )
<=> ( ( A_300 = A_299 )
& ( B_217 = B_216 ) ) ) ).
thf(fact_156_Pair__eq,axiom,
! [A_300: int,B_217: product_prod_int_int,A_299: int,B_216: product_prod_int_int] :
( ( ( produc282740534nt_int @ A_300 @ B_217 )
= ( produc282740534nt_int @ A_299 @ B_216 ) )
<=> ( ( A_300 = A_299 )
& ( B_217 = B_216 ) ) ) ).
thf(fact_157_Pair__eq,axiom,
! [A_300: int,B_217: int,A_299: int,B_216: int] :
( ( ( product_Pair_int_int @ A_300 @ B_217 )
= ( product_Pair_int_int @ A_299 @ B_216 ) )
<=> ( ( A_300 = A_299 )
& ( B_217 = B_216 ) ) ) ).
thf(fact_158_split__paired__All,axiom,
! [P_12: produc975137661_int_o > $o] :
( ( !! @ produc975137661_int_o @ P_12 )
<=> ! [A_2: int > $o,B_4: int > $o] : ( P_12 @ ( produc398918003_int_o @ A_2 @ B_4 ) ) ) ).
thf(fact_159_split__paired__All,axiom,
! [P_12: produc1137372701nt_int > $o] :
( ( !! @ produc1137372701nt_int @ P_12 )
<=> ! [A_2: product_prod_int_int,B_4: product_prod_int_int] : ( P_12 @ ( produc883642259nt_int @ A_2 @ B_4 ) ) ) ).
thf(fact_160_split__paired__All,axiom,
! [P_12: produc167071911de_int > $o] :
( ( !! @ produc167071911de_int @ P_12 )
<=> ! [A_2: quickcheck_code_int,B_4: quickcheck_code_int] : ( P_12 @ ( produc1318306967de_int @ A_2 @ B_4 ) ) ) ).
thf(fact_161_split__paired__All,axiom,
! [P_12: produc1359518119umeral > $o] :
( ( !! @ produc1359518119umeral @ P_12 )
<=> ! [A_2: code_code_numeral,B_4: code_code_numeral] : ( P_12 @ ( produc2136830103umeral @ A_2 @ B_4 ) ) ) ).
thf(fact_162_split__paired__All,axiom,
! [P_12: produc914805421l_real > $o] :
( ( !! @ produc914805421l_real @ P_12 )
<=> ! [A_2: real,B_4: real] : ( P_12 @ ( produc865579683l_real @ A_2 @ B_4 ) ) ) ).
thf(fact_163_split__paired__All,axiom,
! [P_12: produc1322466333at_nat > $o] :
( ( !! @ produc1322466333at_nat @ P_12 )
<=> ! [A_2: product_prod_nat_nat,B_4: product_prod_nat_nat] : ( P_12 @ ( produc494345619at_nat @ A_2 @ B_4 ) ) ) ).
thf(fact_164_split__paired__All,axiom,
! [P_12: product_prod_nat_nat > $o] :
( ( !! @ product_prod_nat_nat @ P_12 )
<=> ! [A_2: nat,B_4: nat] : ( P_12 @ ( product_Pair_nat_nat @ A_2 @ B_4 ) ) ) ).
thf(fact_165_split__paired__All,axiom,
! [P_12: produc393999548nt_int > $o] :
( ( !! @ produc393999548nt_int @ P_12 )
<=> ! [A_2: int,B_4: product_prod_int_int] : ( P_12 @ ( produc282740534nt_int @ A_2 @ B_4 ) ) ) ).
thf(fact_166_split__paired__All,axiom,
! [P_12: product_prod_int_int > $o] :
( ( !! @ product_prod_int_int @ P_12 )
<=> ! [A_2: int,B_4: int] : ( P_12 @ ( product_Pair_int_int @ A_2 @ B_4 ) ) ) ).
thf(fact_167_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_298: rat,B_215: rat,C_107: rat,D_25: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A_298 @ B_215 ) @ ( plus_plus_rat @ C_107 @ D_25 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ A_298 @ C_107 ) @ ( plus_plus_rat @ B_215 @ D_25 ) ) ) ).
thf(fact_168_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_298: quickcheck_code_int,B_215: quickcheck_code_int,C_107: quickcheck_code_int,D_25: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_298 @ B_215 ) @ ( plus_p1446045655de_int @ C_107 @ D_25 ) )
= ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_298 @ C_107 ) @ ( plus_p1446045655de_int @ B_215 @ D_25 ) ) ) ).
thf(fact_169_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_298: complex,B_215: complex,C_107: complex,D_25: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A_298 @ B_215 ) @ ( plus_plus_complex @ C_107 @ D_25 ) )
= ( plus_plus_complex @ ( plus_plus_complex @ A_298 @ C_107 ) @ ( plus_plus_complex @ B_215 @ D_25 ) ) ) ).
thf(fact_170_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_298: nat,B_215: nat,C_107: nat,D_25: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_298 @ B_215 ) @ ( plus_plus_nat @ C_107 @ D_25 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_298 @ C_107 ) @ ( plus_plus_nat @ B_215 @ D_25 ) ) ) ).
thf(fact_171_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_298: real,B_215: real,C_107: real,D_25: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_298 @ B_215 ) @ ( plus_plus_real @ C_107 @ D_25 ) )
= ( plus_plus_real @ ( plus_plus_real @ A_298 @ C_107 ) @ ( plus_plus_real @ B_215 @ D_25 ) ) ) ).
thf(fact_172_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_298: int,B_215: int,C_107: int,D_25: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_298 @ B_215 ) @ ( plus_plus_int @ C_107 @ D_25 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_298 @ C_107 ) @ ( plus_plus_int @ B_215 @ D_25 ) ) ) ).
thf(fact_173_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_297: rat,B_214: rat,C_106: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A_297 @ B_214 ) @ C_106 )
= ( plus_plus_rat @ ( plus_plus_rat @ A_297 @ C_106 ) @ B_214 ) ) ).
thf(fact_174_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_297: quickcheck_code_int,B_214: quickcheck_code_int,C_106: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_297 @ B_214 ) @ C_106 )
= ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_297 @ C_106 ) @ B_214 ) ) ).
thf(fact_175_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_297: complex,B_214: complex,C_106: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A_297 @ B_214 ) @ C_106 )
= ( plus_plus_complex @ ( plus_plus_complex @ A_297 @ C_106 ) @ B_214 ) ) ).
thf(fact_176_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_297: nat,B_214: nat,C_106: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_297 @ B_214 ) @ C_106 )
= ( plus_plus_nat @ ( plus_plus_nat @ A_297 @ C_106 ) @ B_214 ) ) ).
thf(fact_177_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_297: real,B_214: real,C_106: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_297 @ B_214 ) @ C_106 )
= ( plus_plus_real @ ( plus_plus_real @ A_297 @ C_106 ) @ B_214 ) ) ).
thf(fact_178_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_297: int,B_214: int,C_106: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_297 @ B_214 ) @ C_106 )
= ( plus_plus_int @ ( plus_plus_int @ A_297 @ C_106 ) @ B_214 ) ) ).
thf(fact_179_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_296: rat,B_213: rat] :
( ( times_times_rat @ A_296 @ B_213 )
= ( times_times_rat @ B_213 @ A_296 ) ) ).
thf(fact_180_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_296: quickcheck_code_int,B_213: quickcheck_code_int] :
( ( times_123202395de_int @ A_296 @ B_213 )
= ( times_123202395de_int @ B_213 @ A_296 ) ) ).
thf(fact_181_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_296: complex,B_213: complex] :
( ( times_times_complex @ A_296 @ B_213 )
= ( times_times_complex @ B_213 @ A_296 ) ) ).
thf(fact_182_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_296: nat,B_213: nat] :
( ( times_times_nat @ A_296 @ B_213 )
= ( times_times_nat @ B_213 @ A_296 ) ) ).
thf(fact_183_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_296: real,B_213: real] :
( ( times_times_real @ A_296 @ B_213 )
= ( times_times_real @ B_213 @ A_296 ) ) ).
thf(fact_184_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_296: int,B_213: int] :
( ( times_times_int @ A_296 @ B_213 )
= ( times_times_int @ B_213 @ A_296 ) ) ).
thf(fact_185_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_186_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_187_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_188_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_189_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_190_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_191_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_192_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_193_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_194_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_195_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_196_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_197_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_198_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_199_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_200_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_201_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_202_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_203_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_204_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_205_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_206_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_207_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_208_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_209_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_210_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_211_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_212_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_213_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_214_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_215_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_216_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_217_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_218_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_219_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_220_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_221_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_222_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_223_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_224_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_225_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_226_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_227_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_295: rat,C_105: rat] :
( ( plus_plus_rat @ A_295 @ C_105 )
= ( plus_plus_rat @ C_105 @ A_295 ) ) ).
thf(fact_228_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_295: quickcheck_code_int,C_105: quickcheck_code_int] :
( ( plus_p1446045655de_int @ A_295 @ C_105 )
= ( plus_p1446045655de_int @ C_105 @ A_295 ) ) ).
thf(fact_229_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_295: complex,C_105: complex] :
( ( plus_plus_complex @ A_295 @ C_105 )
= ( plus_plus_complex @ C_105 @ A_295 ) ) ).
thf(fact_230_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_295: nat,C_105: nat] :
( ( plus_plus_nat @ A_295 @ C_105 )
= ( plus_plus_nat @ C_105 @ A_295 ) ) ).
thf(fact_231_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_295: real,C_105: real] :
( ( plus_plus_real @ A_295 @ C_105 )
= ( plus_plus_real @ C_105 @ A_295 ) ) ).
thf(fact_232_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_295: int,C_105: int] :
( ( plus_plus_int @ A_295 @ C_105 )
= ( plus_plus_int @ C_105 @ A_295 ) ) ).
thf(fact_233_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_294: rat,C_104: rat,D_24: rat] :
( ( plus_plus_rat @ A_294 @ ( plus_plus_rat @ C_104 @ D_24 ) )
= ( plus_plus_rat @ C_104 @ ( plus_plus_rat @ A_294 @ D_24 ) ) ) ).
thf(fact_234_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_294: quickcheck_code_int,C_104: quickcheck_code_int,D_24: quickcheck_code_int] :
( ( plus_p1446045655de_int @ A_294 @ ( plus_p1446045655de_int @ C_104 @ D_24 ) )
= ( plus_p1446045655de_int @ C_104 @ ( plus_p1446045655de_int @ A_294 @ D_24 ) ) ) ).
thf(fact_235_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_294: complex,C_104: complex,D_24: complex] :
( ( plus_plus_complex @ A_294 @ ( plus_plus_complex @ C_104 @ D_24 ) )
= ( plus_plus_complex @ C_104 @ ( plus_plus_complex @ A_294 @ D_24 ) ) ) ).
thf(fact_236_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_294: nat,C_104: nat,D_24: nat] :
( ( plus_plus_nat @ A_294 @ ( plus_plus_nat @ C_104 @ D_24 ) )
= ( plus_plus_nat @ C_104 @ ( plus_plus_nat @ A_294 @ D_24 ) ) ) ).
thf(fact_237_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_294: real,C_104: real,D_24: real] :
( ( plus_plus_real @ A_294 @ ( plus_plus_real @ C_104 @ D_24 ) )
= ( plus_plus_real @ C_104 @ ( plus_plus_real @ A_294 @ D_24 ) ) ) ).
thf(fact_238_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_294: int,C_104: int,D_24: int] :
( ( plus_plus_int @ A_294 @ ( plus_plus_int @ C_104 @ D_24 ) )
= ( plus_plus_int @ C_104 @ ( plus_plus_int @ A_294 @ D_24 ) ) ) ).
thf(fact_239_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_293: rat,C_103: rat,D_23: rat] :
( ( plus_plus_rat @ A_293 @ ( plus_plus_rat @ C_103 @ D_23 ) )
= ( plus_plus_rat @ ( plus_plus_rat @ A_293 @ C_103 ) @ D_23 ) ) ).
thf(fact_240_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_293: quickcheck_code_int,C_103: quickcheck_code_int,D_23: quickcheck_code_int] :
( ( plus_p1446045655de_int @ A_293 @ ( plus_p1446045655de_int @ C_103 @ D_23 ) )
= ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_293 @ C_103 ) @ D_23 ) ) ).
thf(fact_241_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_293: complex,C_103: complex,D_23: complex] :
( ( plus_plus_complex @ A_293 @ ( plus_plus_complex @ C_103 @ D_23 ) )
= ( plus_plus_complex @ ( plus_plus_complex @ A_293 @ C_103 ) @ D_23 ) ) ).
thf(fact_242_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_293: nat,C_103: nat,D_23: nat] :
( ( plus_plus_nat @ A_293 @ ( plus_plus_nat @ C_103 @ D_23 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ A_293 @ C_103 ) @ D_23 ) ) ).
thf(fact_243_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_293: real,C_103: real,D_23: real] :
( ( plus_plus_real @ A_293 @ ( plus_plus_real @ C_103 @ D_23 ) )
= ( plus_plus_real @ ( plus_plus_real @ A_293 @ C_103 ) @ D_23 ) ) ).
thf(fact_244_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_293: int,C_103: int,D_23: int] :
( ( plus_plus_int @ A_293 @ ( plus_plus_int @ C_103 @ D_23 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_293 @ C_103 ) @ D_23 ) ) ).
thf(fact_245_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_292: rat,B_212: rat,C_102: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A_292 @ B_212 ) @ C_102 )
= ( plus_plus_rat @ A_292 @ ( plus_plus_rat @ B_212 @ C_102 ) ) ) ).
thf(fact_246_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_292: quickcheck_code_int,B_212: quickcheck_code_int,C_102: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ A_292 @ B_212 ) @ C_102 )
= ( plus_p1446045655de_int @ A_292 @ ( plus_p1446045655de_int @ B_212 @ C_102 ) ) ) ).
thf(fact_247_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_292: complex,B_212: complex,C_102: complex] :
( ( plus_plus_complex @ ( plus_plus_complex @ A_292 @ B_212 ) @ C_102 )
= ( plus_plus_complex @ A_292 @ ( plus_plus_complex @ B_212 @ C_102 ) ) ) ).
thf(fact_248_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_292: nat,B_212: nat,C_102: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A_292 @ B_212 ) @ C_102 )
= ( plus_plus_nat @ A_292 @ ( plus_plus_nat @ B_212 @ C_102 ) ) ) ).
thf(fact_249_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_292: real,B_212: real,C_102: real] :
( ( plus_plus_real @ ( plus_plus_real @ A_292 @ B_212 ) @ C_102 )
= ( plus_plus_real @ A_292 @ ( plus_plus_real @ B_212 @ C_102 ) ) ) ).
thf(fact_250_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_292: int,B_212: int,C_102: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_292 @ B_212 ) @ C_102 )
= ( plus_plus_int @ A_292 @ ( plus_plus_int @ B_212 @ C_102 ) ) ) ).
thf(fact_251_split__paired__Ex,axiom,
! [P_11: produc975137661_int_o > $o] :
( ( ?? @ produc975137661_int_o @ P_11 )
<=> ? [A_2: int > $o,B_4: int > $o] : ( P_11 @ ( produc398918003_int_o @ A_2 @ B_4 ) ) ) ).
thf(fact_252_split__paired__Ex,axiom,
! [P_11: produc1137372701nt_int > $o] :
( ( ?? @ produc1137372701nt_int @ P_11 )
<=> ? [A_2: product_prod_int_int,B_4: product_prod_int_int] : ( P_11 @ ( produc883642259nt_int @ A_2 @ B_4 ) ) ) ).
thf(fact_253_split__paired__Ex,axiom,
! [P_11: produc167071911de_int > $o] :
( ( ?? @ produc167071911de_int @ P_11 )
<=> ? [A_2: quickcheck_code_int,B_4: quickcheck_code_int] : ( P_11 @ ( produc1318306967de_int @ A_2 @ B_4 ) ) ) ).
thf(fact_254_split__paired__Ex,axiom,
! [P_11: produc1359518119umeral > $o] :
( ( ?? @ produc1359518119umeral @ P_11 )
<=> ? [A_2: code_code_numeral,B_4: code_code_numeral] : ( P_11 @ ( produc2136830103umeral @ A_2 @ B_4 ) ) ) ).
thf(fact_255_split__paired__Ex,axiom,
! [P_11: produc914805421l_real > $o] :
( ( ?? @ produc914805421l_real @ P_11 )
<=> ? [A_2: real,B_4: real] : ( P_11 @ ( produc865579683l_real @ A_2 @ B_4 ) ) ) ).
thf(fact_256_split__paired__Ex,axiom,
! [P_11: produc1322466333at_nat > $o] :
( ( ?? @ produc1322466333at_nat @ P_11 )
<=> ? [A_2: product_prod_nat_nat,B_4: product_prod_nat_nat] : ( P_11 @ ( produc494345619at_nat @ A_2 @ B_4 ) ) ) ).
thf(fact_257_split__paired__Ex,axiom,
! [P_11: product_prod_nat_nat > $o] :
( ( ?? @ product_prod_nat_nat @ P_11 )
<=> ? [A_2: nat,B_4: nat] : ( P_11 @ ( product_Pair_nat_nat @ A_2 @ B_4 ) ) ) ).
thf(fact_258_split__paired__Ex,axiom,
! [P_11: produc393999548nt_int > $o] :
( ( ?? @ produc393999548nt_int @ P_11 )
<=> ? [A_2: int,B_4: product_prod_int_int] : ( P_11 @ ( produc282740534nt_int @ A_2 @ B_4 ) ) ) ).
thf(fact_259_split__paired__Ex,axiom,
! [P_11: product_prod_int_int > $o] :
( ( ?? @ product_prod_int_int @ P_11 )
<=> ? [A_2: int,B_4: int] : ( P_11 @ ( product_Pair_int_int @ A_2 @ B_4 ) ) ) ).
thf(fact_260_prod__induct3,axiom,
! [X_131: produc1137372701nt_int,P_10: produc1137372701nt_int > $o] :
( ! [A_2: product_prod_int_int,B_4: int,C_2: int] : ( P_10 @ ( produc883642259nt_int @ A_2 @ ( product_Pair_int_int @ B_4 @ C_2 ) ) )
=> ( P_10 @ X_131 ) ) ).
thf(fact_261_prod__induct3,axiom,
! [X_131: produc1322466333at_nat,P_10: produc1322466333at_nat > $o] :
( ! [A_2: product_prod_nat_nat,B_4: nat,C_2: nat] : ( P_10 @ ( produc494345619at_nat @ A_2 @ ( product_Pair_nat_nat @ B_4 @ C_2 ) ) )
=> ( P_10 @ X_131 ) ) ).
thf(fact_262_prod__induct3,axiom,
! [X_131: produc393999548nt_int,P_10: produc393999548nt_int > $o] :
( ! [A_2: int,B_4: int,C_2: int] : ( P_10 @ ( produc282740534nt_int @ A_2 @ ( product_Pair_int_int @ B_4 @ C_2 ) ) )
=> ( P_10 @ X_131 ) ) ).
thf(fact_263_prod__cases3,axiom,
! [Y_86: produc1137372701nt_int] :
~ ! [A_2: product_prod_int_int,B_4: int,C_2: int] :
( Y_86
!= ( produc883642259nt_int @ A_2 @ ( product_Pair_int_int @ B_4 @ C_2 ) ) ) ).
thf(fact_264_prod__cases3,axiom,
! [Y_86: produc1322466333at_nat] :
~ ! [A_2: product_prod_nat_nat,B_4: nat,C_2: nat] :
( Y_86
!= ( produc494345619at_nat @ A_2 @ ( product_Pair_nat_nat @ B_4 @ C_2 ) ) ) ).
thf(fact_265_prod__cases3,axiom,
! [Y_86: produc393999548nt_int] :
~ ! [A_2: int,B_4: int,C_2: int] :
( Y_86
!= ( produc282740534nt_int @ A_2 @ ( product_Pair_int_int @ B_4 @ C_2 ) ) ) ).
thf(fact_266_PairE,axiom,
! [P_9: produc975137661_int_o] :
~ ! [X_1: int > $o,Y_1: int > $o] :
( P_9
!= ( produc398918003_int_o @ X_1 @ Y_1 ) ) ).
thf(fact_267_PairE,axiom,
! [P_9: produc1137372701nt_int] :
~ ! [X_1: product_prod_int_int,Y_1: product_prod_int_int] :
( P_9
!= ( produc883642259nt_int @ X_1 @ Y_1 ) ) ).
thf(fact_268_PairE,axiom,
! [P_9: produc167071911de_int] :
~ ! [X_1: quickcheck_code_int,Y_1: quickcheck_code_int] :
( P_9
!= ( produc1318306967de_int @ X_1 @ Y_1 ) ) ).
thf(fact_269_PairE,axiom,
! [P_9: produc1359518119umeral] :
~ ! [X_1: code_code_numeral,Y_1: code_code_numeral] :
( P_9
!= ( produc2136830103umeral @ X_1 @ Y_1 ) ) ).
thf(fact_270_PairE,axiom,
! [P_9: produc914805421l_real] :
~ ! [X_1: real,Y_1: real] :
( P_9
!= ( produc865579683l_real @ X_1 @ Y_1 ) ) ).
thf(fact_271_PairE,axiom,
! [P_9: produc1322466333at_nat] :
~ ! [X_1: product_prod_nat_nat,Y_1: product_prod_nat_nat] :
( P_9
!= ( produc494345619at_nat @ X_1 @ Y_1 ) ) ).
thf(fact_272_PairE,axiom,
! [P_9: product_prod_nat_nat] :
~ ! [X_1: nat,Y_1: nat] :
( P_9
!= ( product_Pair_nat_nat @ X_1 @ Y_1 ) ) ).
thf(fact_273_PairE,axiom,
! [P_9: produc393999548nt_int] :
~ ! [X_1: int,Y_1: product_prod_int_int] :
( P_9
!= ( produc282740534nt_int @ X_1 @ Y_1 ) ) ).
thf(fact_274_PairE,axiom,
! [P_9: product_prod_int_int] :
~ ! [X_1: int,Y_1: int] :
( P_9
!= ( product_Pair_int_int @ X_1 @ Y_1 ) ) ).
thf(fact_275_mem__def,axiom,
! [X_130: produc975137661_int_o,A_291: produc975137661_int_o > $o] :
( ( member1329254762_int_o @ X_130 @ A_291 )
<=> ( A_291 @ X_130 ) ) ).
thf(fact_276_mem__def,axiom,
! [X_130: produc1137372701nt_int,A_291: produc1137372701nt_int > $o] :
( ( member2143287562nt_int @ X_130 @ A_291 )
<=> ( A_291 @ X_130 ) ) ).
thf(fact_277_mem__def,axiom,
! [X_130: nat,A_291: nat > $o] :
( ( member_nat @ X_130 @ A_291 )
<=> ( A_291 @ X_130 ) ) ).
thf(fact_278_mem__def,axiom,
! [X_130: int > $o,A_291: ( int > $o ) > $o] :
( ( member_int_o @ X_130 @ A_291 )
<=> ( A_291 @ X_130 ) ) ).
thf(fact_279_mem__def,axiom,
! [X_130: produc1322466333at_nat,A_291: produc1322466333at_nat > $o] :
( ( member180897546at_nat @ X_130 @ A_291 )
<=> ( A_291 @ X_130 ) ) ).
thf(fact_280_mem__def,axiom,
! [X_130: real,A_291: real > $o] :
( ( member_real @ X_130 @ A_291 )
<=> ( A_291 @ X_130 ) ) ).
thf(fact_281_mem__def,axiom,
! [X_130: int,A_291: int > $o] :
( ( member_int @ X_130 @ A_291 )
<=> ( A_291 @ X_130 ) ) ).
thf(fact_282_Collect__def,axiom,
! [P_8: product_prod_nat_nat > $o] :
( ( collec1979865426at_nat @ P_8 )
= P_8 ) ).
thf(fact_283_Collect__def,axiom,
! [P_8: real > $o] :
( ( collect_real @ P_8 )
= P_8 ) ).
thf(fact_284_Collect__def,axiom,
! [P_8: product_prod_int_int > $o] :
( ( collec1347809874nt_int @ P_8 )
= P_8 ) ).
thf(fact_285_Collect__def,axiom,
! [P_8: produc1137372701nt_int > $o] :
( ( collec50511176nt_int @ P_8 )
= P_8 ) ).
thf(fact_286_Collect__def,axiom,
! [P_8: nat > $o] :
( ( collect_nat @ P_8 )
= P_8 ) ).
thf(fact_287_Collect__def,axiom,
! [P_8: int > $o] :
( ( collect_int @ P_8 )
= P_8 ) ).
thf(fact_288_prod_Oexhaust,axiom,
! [Y_85: produc975137661_int_o] :
~ ! [A_2: int > $o,B_4: int > $o] :
( Y_85
!= ( produc398918003_int_o @ A_2 @ B_4 ) ) ).
thf(fact_289_prod_Oexhaust,axiom,
! [Y_85: produc1137372701nt_int] :
~ ! [A_2: product_prod_int_int,B_4: product_prod_int_int] :
( Y_85
!= ( produc883642259nt_int @ A_2 @ B_4 ) ) ).
thf(fact_290_prod_Oexhaust,axiom,
! [Y_85: produc167071911de_int] :
~ ! [A_2: quickcheck_code_int,B_4: quickcheck_code_int] :
( Y_85
!= ( produc1318306967de_int @ A_2 @ B_4 ) ) ).
thf(fact_291_prod_Oexhaust,axiom,
! [Y_85: produc1359518119umeral] :
~ ! [A_2: code_code_numeral,B_4: code_code_numeral] :
( Y_85
!= ( produc2136830103umeral @ A_2 @ B_4 ) ) ).
thf(fact_292_prod_Oexhaust,axiom,
! [Y_85: produc914805421l_real] :
~ ! [A_2: real,B_4: real] :
( Y_85
!= ( produc865579683l_real @ A_2 @ B_4 ) ) ).
thf(fact_293_prod_Oexhaust,axiom,
! [Y_85: produc1322466333at_nat] :
~ ! [A_2: product_prod_nat_nat,B_4: product_prod_nat_nat] :
( Y_85
!= ( produc494345619at_nat @ A_2 @ B_4 ) ) ).
thf(fact_294_prod_Oexhaust,axiom,
! [Y_85: product_prod_nat_nat] :
~ ! [A_2: nat,B_4: nat] :
( Y_85
!= ( product_Pair_nat_nat @ A_2 @ B_4 ) ) ).
thf(fact_295_prod_Oexhaust,axiom,
! [Y_85: produc393999548nt_int] :
~ ! [A_2: int,B_4: product_prod_int_int] :
( Y_85
!= ( produc282740534nt_int @ A_2 @ B_4 ) ) ).
thf(fact_296_prod_Oexhaust,axiom,
! [Y_85: product_prod_int_int] :
~ ! [A_2: int,B_4: int] :
( Y_85
!= ( product_Pair_int_int @ A_2 @ B_4 ) ) ).
thf(fact_297_curryI,axiom,
! [F_10: produc975137661_int_o > $o,A_290: int > $o,B_211: int > $o] :
( ( F_10 @ ( produc398918003_int_o @ A_290 @ B_211 ) )
=> ( produc144226034nt_o_o @ F_10 @ A_290 @ B_211 ) ) ).
thf(fact_298_curryI,axiom,
! [F_10: produc1137372701nt_int > $o,A_290: product_prod_int_int,B_211: product_prod_int_int] :
( ( F_10 @ ( produc883642259nt_int @ A_290 @ B_211 ) )
=> ( produc1384121810_int_o @ F_10 @ A_290 @ B_211 ) ) ).
thf(fact_299_curryI,axiom,
! [F_10: produc167071911de_int > $o,A_290: quickcheck_code_int,B_211: quickcheck_code_int] :
( ( F_10 @ ( produc1318306967de_int @ A_290 @ B_211 ) )
=> ( produc1585697102_int_o @ F_10 @ A_290 @ B_211 ) ) ).
thf(fact_300_curryI,axiom,
! [F_10: produc1359518119umeral > $o,A_290: code_code_numeral,B_211: code_code_numeral] :
( ( F_10 @ ( produc2136830103umeral @ A_290 @ B_211 ) )
=> ( produc350732110eral_o @ F_10 @ A_290 @ B_211 ) ) ).
thf(fact_301_curryI,axiom,
! [F_10: produc914805421l_real > $o,A_290: real,B_211: real] :
( ( F_10 @ ( produc865579683l_real @ A_290 @ B_211 ) )
=> ( produc1509576130real_o @ F_10 @ A_290 @ B_211 ) ) ).
thf(fact_302_curryI,axiom,
! [F_10: produc1322466333at_nat > $o,A_290: product_prod_nat_nat,B_211: product_prod_nat_nat] :
( ( F_10 @ ( produc494345619at_nat @ A_290 @ B_211 ) )
=> ( produc435992530_nat_o @ F_10 @ A_290 @ B_211 ) ) ).
thf(fact_303_curryI,axiom,
! [F_10: product_prod_nat_nat > $o,A_290: nat,B_211: nat] :
( ( F_10 @ ( product_Pair_nat_nat @ A_290 @ B_211 ) )
=> ( produc764619086_nat_o @ F_10 @ A_290 @ B_211 ) ) ).
thf(fact_304_curryI,axiom,
! [F_10: produc393999548nt_int > $o,A_290: int,B_211: product_prod_int_int] :
( ( F_10 @ ( produc282740534nt_int @ A_290 @ B_211 ) )
=> ( produc122891093_int_o @ F_10 @ A_290 @ B_211 ) ) ).
thf(fact_305_curryI,axiom,
! [F_10: product_prod_int_int > $o,A_290: int,B_211: int] :
( ( F_10 @ ( product_Pair_int_int @ A_290 @ B_211 ) )
=> ( produc176579150_int_o @ F_10 @ A_290 @ B_211 ) ) ).
thf(fact_306_le__add__iff1,axiom,
! [A_289: rat,E_5: rat,C_101: rat,B_210: rat,D_22: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A_289 @ E_5 ) @ C_101 ) @ ( plus_plus_rat @ ( times_times_rat @ B_210 @ E_5 ) @ D_22 ) )
<=> ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A_289 @ B_210 ) @ E_5 ) @ C_101 ) @ D_22 ) ) ).
thf(fact_307_le__add__iff1,axiom,
! [A_289: real,E_5: real,C_101: real,B_210: real,D_22: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A_289 @ E_5 ) @ C_101 ) @ ( plus_plus_real @ ( times_times_real @ B_210 @ E_5 ) @ D_22 ) )
<=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A_289 @ B_210 ) @ E_5 ) @ C_101 ) @ D_22 ) ) ).
thf(fact_308_le__add__iff1,axiom,
! [A_289: int,E_5: int,C_101: int,B_210: int,D_22: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A_289 @ E_5 ) @ C_101 ) @ ( plus_plus_int @ ( times_times_int @ B_210 @ E_5 ) @ D_22 ) )
<=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_289 @ B_210 ) @ E_5 ) @ C_101 ) @ D_22 ) ) ).
thf(fact_309_le__add__iff2,axiom,
! [A_288: rat,E_4: rat,C_100: rat,B_209: rat,D_21: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A_288 @ E_4 ) @ C_100 ) @ ( plus_plus_rat @ ( times_times_rat @ B_209 @ E_4 ) @ D_21 ) )
<=> ( ord_less_eq_rat @ C_100 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B_209 @ A_288 ) @ E_4 ) @ D_21 ) ) ) ).
thf(fact_310_le__add__iff2,axiom,
! [A_288: real,E_4: real,C_100: real,B_209: real,D_21: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A_288 @ E_4 ) @ C_100 ) @ ( plus_plus_real @ ( times_times_real @ B_209 @ E_4 ) @ D_21 ) )
<=> ( ord_less_eq_real @ C_100 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B_209 @ A_288 ) @ E_4 ) @ D_21 ) ) ) ).
thf(fact_311_le__add__iff2,axiom,
! [A_288: int,E_4: int,C_100: int,B_209: int,D_21: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A_288 @ E_4 ) @ C_100 ) @ ( plus_plus_int @ ( times_times_int @ B_209 @ E_4 ) @ D_21 ) )
<=> ( ord_less_eq_int @ C_100 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_209 @ A_288 ) @ E_4 ) @ D_21 ) ) ) ).
thf(fact_312_real__squared__diff__one__factored,axiom,
! [X_129: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X_129 @ X_129 ) @ one_one_rat )
= ( times_times_rat @ ( plus_plus_rat @ X_129 @ one_one_rat ) @ ( minus_minus_rat @ X_129 @ one_one_rat ) ) ) ).
thf(fact_313_real__squared__diff__one__factored,axiom,
! [X_129: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X_129 @ X_129 ) @ one_one_complex )
= ( times_times_complex @ ( plus_plus_complex @ X_129 @ one_one_complex ) @ ( minus_minus_complex @ X_129 @ one_one_complex ) ) ) ).
thf(fact_314_real__squared__diff__one__factored,axiom,
! [X_129: real] :
( ( minus_minus_real @ ( times_times_real @ X_129 @ X_129 ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X_129 @ one_one_real ) @ ( minus_minus_real @ X_129 @ one_one_real ) ) ) ).
thf(fact_315_real__squared__diff__one__factored,axiom,
! [X_129: int] :
( ( minus_minus_int @ ( times_times_int @ X_129 @ X_129 ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X_129 @ one_one_int ) @ ( minus_minus_int @ X_129 @ one_one_int ) ) ) ).
thf(fact_316_less__add__iff1,axiom,
! [A_287: rat,E_3: rat,C_99: rat,B_208: rat,D_20: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A_287 @ E_3 ) @ C_99 ) @ ( plus_plus_rat @ ( times_times_rat @ B_208 @ E_3 ) @ D_20 ) )
<=> ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A_287 @ B_208 ) @ E_3 ) @ C_99 ) @ D_20 ) ) ).
thf(fact_317_less__add__iff1,axiom,
! [A_287: real,E_3: real,C_99: real,B_208: real,D_20: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A_287 @ E_3 ) @ C_99 ) @ ( plus_plus_real @ ( times_times_real @ B_208 @ E_3 ) @ D_20 ) )
<=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A_287 @ B_208 ) @ E_3 ) @ C_99 ) @ D_20 ) ) ).
thf(fact_318_less__add__iff1,axiom,
! [A_287: int,E_3: int,C_99: int,B_208: int,D_20: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A_287 @ E_3 ) @ C_99 ) @ ( plus_plus_int @ ( times_times_int @ B_208 @ E_3 ) @ D_20 ) )
<=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_287 @ B_208 ) @ E_3 ) @ C_99 ) @ D_20 ) ) ).
thf(fact_319_less__add__iff2,axiom,
! [A_286: rat,E_2: rat,C_98: rat,B_207: rat,D_19: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A_286 @ E_2 ) @ C_98 ) @ ( plus_plus_rat @ ( times_times_rat @ B_207 @ E_2 ) @ D_19 ) )
<=> ( ord_less_rat @ C_98 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B_207 @ A_286 ) @ E_2 ) @ D_19 ) ) ) ).
thf(fact_320_less__add__iff2,axiom,
! [A_286: real,E_2: real,C_98: real,B_207: real,D_19: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A_286 @ E_2 ) @ C_98 ) @ ( plus_plus_real @ ( times_times_real @ B_207 @ E_2 ) @ D_19 ) )
<=> ( ord_less_real @ C_98 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B_207 @ A_286 ) @ E_2 ) @ D_19 ) ) ) ).
thf(fact_321_less__add__iff2,axiom,
! [A_286: int,E_2: int,C_98: int,B_207: int,D_19: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A_286 @ E_2 ) @ C_98 ) @ ( plus_plus_int @ ( times_times_int @ B_207 @ E_2 ) @ D_19 ) )
<=> ( ord_less_int @ C_98 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_207 @ A_286 ) @ E_2 ) @ D_19 ) ) ) ).
thf(fact_322_add__number__of__diff1,axiom,
! [V_32: int,W_19: int,C_97: rat] :
( ( plus_plus_rat @ ( number_number_of_rat @ V_32 ) @ ( minus_minus_rat @ ( number_number_of_rat @ W_19 ) @ C_97 ) )
= ( minus_minus_rat @ ( number_number_of_rat @ ( plus_plus_int @ V_32 @ W_19 ) ) @ C_97 ) ) ).
thf(fact_323_add__number__of__diff1,axiom,
! [V_32: int,W_19: int,C_97: complex] :
( ( plus_plus_complex @ ( number528085621omplex @ V_32 ) @ ( minus_minus_complex @ ( number528085621omplex @ W_19 ) @ C_97 ) )
= ( minus_minus_complex @ ( number528085621omplex @ ( plus_plus_int @ V_32 @ W_19 ) ) @ C_97 ) ) ).
thf(fact_324_add__number__of__diff1,axiom,
! [V_32: int,W_19: int,C_97: real] :
( ( plus_plus_real @ ( number267125858f_real @ V_32 ) @ ( minus_minus_real @ ( number267125858f_real @ W_19 ) @ C_97 ) )
= ( minus_minus_real @ ( number267125858f_real @ ( plus_plus_int @ V_32 @ W_19 ) ) @ C_97 ) ) ).
thf(fact_325_add__number__of__diff1,axiom,
! [V_32: int,W_19: int,C_97: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_32 ) @ ( minus_minus_int @ ( number_number_of_int @ W_19 ) @ C_97 ) )
= ( minus_minus_int @ ( number_number_of_int @ ( plus_plus_int @ V_32 @ W_19 ) ) @ C_97 ) ) ).
thf(fact_326_in__rel__def,axiom,
! [R_7: produc975137661_int_o > $o,X_128: int > $o,Y_84: int > $o] :
( ( in_rel_int_o_int_o @ R_7 @ X_128 @ Y_84 )
<=> ( member1329254762_int_o @ ( produc398918003_int_o @ X_128 @ Y_84 ) @ R_7 ) ) ).
thf(fact_327_in__rel__def,axiom,
! [R_7: produc1137372701nt_int > $o,X_128: product_prod_int_int,Y_84: product_prod_int_int] :
( ( in_rel682537726nt_int @ R_7 @ X_128 @ Y_84 )
<=> ( member2143287562nt_int @ ( produc883642259nt_int @ X_128 @ Y_84 ) @ R_7 ) ) ).
thf(fact_328_in__rel__def,axiom,
! [R_7: produc1322466333at_nat > $o,X_128: product_prod_nat_nat,Y_84: product_prod_nat_nat] :
( ( in_rel293241086at_nat @ R_7 @ X_128 @ Y_84 )
<=> ( member180897546at_nat @ ( produc494345619at_nat @ X_128 @ Y_84 ) @ R_7 ) ) ).
thf(fact_329_in__rel__def,axiom,
! [R_7: produc167071911de_int > $o,X_128: quickcheck_code_int,Y_84: quickcheck_code_int] :
( ( in_rel1707483778de_int @ R_7 @ X_128 @ Y_84 )
<=> ( member2092545300de_int @ ( produc1318306967de_int @ X_128 @ Y_84 ) @ R_7 ) ) ).
thf(fact_330_in__rel__def,axiom,
! [R_7: produc1359518119umeral > $o,X_128: code_code_numeral,Y_84: code_code_numeral] :
( ( in_rel1251239042umeral @ R_7 @ X_128 @ Y_84 )
<=> ( member1246959636umeral @ ( produc2136830103umeral @ X_128 @ Y_84 ) @ R_7 ) ) ).
thf(fact_331_in__rel__def,axiom,
! [R_7: produc914805421l_real > $o,X_128: real,Y_84: real] :
( ( in_rel_real_real @ R_7 @ X_128 @ Y_84 )
<=> ( member1344826522l_real @ ( produc865579683l_real @ X_128 @ Y_84 ) @ R_7 ) ) ).
thf(fact_332_in__rel__def,axiom,
! [R_7: product_prod_nat_nat > $o,X_128: nat,Y_84: nat] :
( ( in_rel_nat_nat @ R_7 @ X_128 @ Y_84 )
<=> ( member232255508at_nat @ ( product_Pair_nat_nat @ X_128 @ Y_84 ) @ R_7 ) ) ).
thf(fact_333_in__rel__def,axiom,
! [R_7: produc393999548nt_int > $o,X_128: int,Y_84: product_prod_int_int] :
( ( in_rel1239871883nt_int @ R_7 @ X_128 @ Y_84 )
<=> ( member791087631nt_int @ ( produc282740534nt_int @ X_128 @ Y_84 ) @ R_7 ) ) ).
thf(fact_334_in__rel__def,axiom,
! [R_7: product_prod_int_int > $o,X_128: int,Y_84: int] :
( ( in_rel_int_int @ R_7 @ X_128 @ Y_84 )
<=> ( member1747683604nt_int @ ( product_Pair_int_int @ X_128 @ Y_84 ) @ R_7 ) ) ).
thf(fact_335_curryE,axiom,
! [F_9: produc975137661_int_o > $o,A_285: int > $o,B_206: int > $o] :
( ( produc144226034nt_o_o @ F_9 @ A_285 @ B_206 )
=> ( F_9 @ ( produc398918003_int_o @ A_285 @ B_206 ) ) ) ).
thf(fact_336_curryE,axiom,
! [F_9: produc1137372701nt_int > $o,A_285: product_prod_int_int,B_206: product_prod_int_int] :
( ( produc1384121810_int_o @ F_9 @ A_285 @ B_206 )
=> ( F_9 @ ( produc883642259nt_int @ A_285 @ B_206 ) ) ) ).
thf(fact_337_curryE,axiom,
! [F_9: produc167071911de_int > $o,A_285: quickcheck_code_int,B_206: quickcheck_code_int] :
( ( produc1585697102_int_o @ F_9 @ A_285 @ B_206 )
=> ( F_9 @ ( produc1318306967de_int @ A_285 @ B_206 ) ) ) ).
thf(fact_338_curryE,axiom,
! [F_9: produc1359518119umeral > $o,A_285: code_code_numeral,B_206: code_code_numeral] :
( ( produc350732110eral_o @ F_9 @ A_285 @ B_206 )
=> ( F_9 @ ( produc2136830103umeral @ A_285 @ B_206 ) ) ) ).
thf(fact_339_curryE,axiom,
! [F_9: produc914805421l_real > $o,A_285: real,B_206: real] :
( ( produc1509576130real_o @ F_9 @ A_285 @ B_206 )
=> ( F_9 @ ( produc865579683l_real @ A_285 @ B_206 ) ) ) ).
thf(fact_340_curryE,axiom,
! [F_9: produc1322466333at_nat > $o,A_285: product_prod_nat_nat,B_206: product_prod_nat_nat] :
( ( produc435992530_nat_o @ F_9 @ A_285 @ B_206 )
=> ( F_9 @ ( produc494345619at_nat @ A_285 @ B_206 ) ) ) ).
thf(fact_341_curryE,axiom,
! [F_9: product_prod_nat_nat > $o,A_285: nat,B_206: nat] :
( ( produc764619086_nat_o @ F_9 @ A_285 @ B_206 )
=> ( F_9 @ ( product_Pair_nat_nat @ A_285 @ B_206 ) ) ) ).
thf(fact_342_curryE,axiom,
! [F_9: produc393999548nt_int > $o,A_285: int,B_206: product_prod_int_int] :
( ( produc122891093_int_o @ F_9 @ A_285 @ B_206 )
=> ( F_9 @ ( produc282740534nt_int @ A_285 @ B_206 ) ) ) ).
thf(fact_343_curryE,axiom,
! [F_9: product_prod_int_int > $o,A_285: int,B_206: int] :
( ( produc176579150_int_o @ F_9 @ A_285 @ B_206 )
=> ( F_9 @ ( product_Pair_int_int @ A_285 @ B_206 ) ) ) ).
thf(fact_344_one__reorient,axiom,
! [X_127: rat] :
( ( one_one_rat = X_127 )
<=> ( X_127 = one_one_rat ) ) ).
thf(fact_345_one__reorient,axiom,
! [X_127: quickcheck_code_int] :
( ( one_on1684967323de_int = X_127 )
<=> ( X_127 = one_on1684967323de_int ) ) ).
thf(fact_346_one__reorient,axiom,
! [X_127: complex] :
( ( one_one_complex = X_127 )
<=> ( X_127 = one_one_complex ) ) ).
thf(fact_347_one__reorient,axiom,
! [X_127: code_code_numeral] :
( ( one_on1645066479umeral = X_127 )
<=> ( X_127 = one_on1645066479umeral ) ) ).
thf(fact_348_one__reorient,axiom,
! [X_127: nat] :
( ( one_one_nat = X_127 )
<=> ( X_127 = one_one_nat ) ) ).
thf(fact_349_one__reorient,axiom,
! [X_127: real] :
( ( one_one_real = X_127 )
<=> ( X_127 = one_one_real ) ) ).
thf(fact_350_one__reorient,axiom,
! [X_127: int] :
( ( one_one_int = X_127 )
<=> ( X_127 = one_one_int ) ) ).
thf(fact_351_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_352_number__of__reorient,axiom,
! [W_18: int,X_126: rat] :
( ( ( number_number_of_rat @ W_18 )
= X_126 )
<=> ( X_126
= ( number_number_of_rat @ W_18 ) ) ) ).
thf(fact_353_number__of__reorient,axiom,
! [W_18: int,X_126: quickcheck_code_int] :
( ( ( number1226105091de_int @ W_18 )
= X_126 )
<=> ( X_126
= ( number1226105091de_int @ W_18 ) ) ) ).
thf(fact_354_number__of__reorient,axiom,
! [W_18: int,X_126: complex] :
( ( ( number528085621omplex @ W_18 )
= X_126 )
<=> ( X_126
= ( number528085621omplex @ W_18 ) ) ) ).
thf(fact_355_number__of__reorient,axiom,
! [W_18: int,X_126: code_code_numeral] :
( ( ( number1443263063umeral @ W_18 )
= X_126 )
<=> ( X_126
= ( number1443263063umeral @ W_18 ) ) ) ).
thf(fact_356_number__of__reorient,axiom,
! [W_18: int,X_126: nat] :
( ( ( number_number_of_nat @ W_18 )
= X_126 )
<=> ( X_126
= ( number_number_of_nat @ W_18 ) ) ) ).
thf(fact_357_number__of__reorient,axiom,
! [W_18: int,X_126: real] :
( ( ( number267125858f_real @ W_18 )
= X_126 )
<=> ( X_126
= ( number267125858f_real @ W_18 ) ) ) ).
thf(fact_358_number__of__reorient,axiom,
! [W_18: int,X_126: int] :
( ( ( number_number_of_int @ W_18 )
= X_126 )
<=> ( X_126
= ( number_number_of_int @ W_18 ) ) ) ).
thf(fact_359_le__number__of__eq__not__less,axiom,
! [V_31: int,W_17: int] :
( ( ord_less_eq_rat @ ( number_number_of_rat @ V_31 ) @ ( number_number_of_rat @ W_17 ) )
<=> ~ ( ord_less_rat @ ( number_number_of_rat @ W_17 ) @ ( number_number_of_rat @ V_31 ) ) ) ).
thf(fact_360_le__number__of__eq__not__less,axiom,
! [V_31: int,W_17: int] :
( ( ord_le258702272de_int @ ( number1226105091de_int @ V_31 ) @ ( number1226105091de_int @ W_17 ) )
<=> ~ ( ord_le1860547276de_int @ ( number1226105091de_int @ W_17 ) @ ( number1226105091de_int @ V_31 ) ) ) ).
thf(fact_361_le__number__of__eq__not__less,axiom,
! [V_31: int,W_17: int] :
( ( ord_less_eq_nat @ ( number_number_of_nat @ V_31 ) @ ( number_number_of_nat @ W_17 ) )
<=> ~ ( ord_less_nat @ ( number_number_of_nat @ W_17 ) @ ( number_number_of_nat @ V_31 ) ) ) ).
thf(fact_362_le__number__of__eq__not__less,axiom,
! [V_31: int,W_17: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ V_31 ) @ ( number267125858f_real @ W_17 ) )
<=> ~ ( ord_less_real @ ( number267125858f_real @ W_17 ) @ ( number267125858f_real @ V_31 ) ) ) ).
thf(fact_363_le__number__of__eq__not__less,axiom,
! [V_31: int,W_17: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ V_31 ) @ ( number_number_of_int @ W_17 ) )
<=> ~ ( ord_less_int @ ( number_number_of_int @ W_17 ) @ ( number_number_of_int @ V_31 ) ) ) ).
thf(fact_364_le__number__of__eq__not__less,axiom,
! [V_31: int,W_17: int] :
( ( ord_le565307924umeral @ ( number1443263063umeral @ V_31 ) @ ( number1443263063umeral @ W_17 ) )
<=> ~ ( ord_le1304079648umeral @ ( number1443263063umeral @ W_17 ) @ ( number1443263063umeral @ V_31 ) ) ) ).
thf(fact_365_eq__number__of,axiom,
! [X_125: int,Y_83: int] :
( ( ( number_number_of_rat @ X_125 )
= ( number_number_of_rat @ Y_83 ) )
<=> ( X_125 = Y_83 ) ) ).
thf(fact_366_eq__number__of,axiom,
! [X_125: int,Y_83: int] :
( ( ( number528085621omplex @ X_125 )
= ( number528085621omplex @ Y_83 ) )
<=> ( X_125 = Y_83 ) ) ).
thf(fact_367_eq__number__of,axiom,
! [X_125: int,Y_83: int] :
( ( ( number267125858f_real @ X_125 )
= ( number267125858f_real @ Y_83 ) )
<=> ( X_125 = Y_83 ) ) ).
thf(fact_368_eq__number__of,axiom,
! [X_125: int,Y_83: int] :
( ( ( number_number_of_int @ X_125 )
= ( number_number_of_int @ Y_83 ) )
<=> ( X_125 = Y_83 ) ) ).
thf(fact_369_less__number__of,axiom,
! [X_124: int,Y_82: int] :
( ( ord_less_rat @ ( number_number_of_rat @ X_124 ) @ ( number_number_of_rat @ Y_82 ) )
<=> ( ord_less_int @ X_124 @ Y_82 ) ) ).
thf(fact_370_less__number__of,axiom,
! [X_124: int,Y_82: int] :
( ( ord_less_real @ ( number267125858f_real @ X_124 ) @ ( number267125858f_real @ Y_82 ) )
<=> ( ord_less_int @ X_124 @ Y_82 ) ) ).
thf(fact_371_less__number__of,axiom,
! [X_124: int,Y_82: int] :
( ( ord_less_int @ ( number_number_of_int @ X_124 ) @ ( number_number_of_int @ Y_82 ) )
<=> ( ord_less_int @ X_124 @ Y_82 ) ) ).
thf(fact_372_le__number__of,axiom,
! [X_123: int,Y_81: int] :
( ( ord_less_eq_rat @ ( number_number_of_rat @ X_123 ) @ ( number_number_of_rat @ Y_81 ) )
<=> ( ord_less_eq_int @ X_123 @ Y_81 ) ) ).
thf(fact_373_le__number__of,axiom,
! [X_123: int,Y_81: int] :
( ( ord_less_eq_real @ ( number267125858f_real @ X_123 ) @ ( number267125858f_real @ Y_81 ) )
<=> ( ord_less_eq_int @ X_123 @ Y_81 ) ) ).
thf(fact_374_le__number__of,axiom,
! [X_123: int,Y_81: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ X_123 ) @ ( number_number_of_int @ Y_81 ) )
<=> ( ord_less_eq_int @ X_123 @ Y_81 ) ) ).
thf(fact_375_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_376_linorder__neqE__linordered__idom,axiom,
! [X_122: rat,Y_80: rat] :
( ( X_122 != Y_80 )
=> ( ~ ( ord_less_rat @ X_122 @ Y_80 )
=> ( ord_less_rat @ Y_80 @ X_122 ) ) ) ).
thf(fact_377_linorder__neqE__linordered__idom,axiom,
! [X_122: real,Y_80: real] :
( ( X_122 != Y_80 )
=> ( ~ ( ord_less_real @ X_122 @ Y_80 )
=> ( ord_less_real @ Y_80 @ X_122 ) ) ) ).
thf(fact_378_linorder__neqE__linordered__idom,axiom,
! [X_122: int,Y_80: int] :
( ( X_122 != Y_80 )
=> ( ~ ( ord_less_int @ X_122 @ Y_80 )
=> ( ord_less_int @ Y_80 @ X_122 ) ) ) ).
thf(fact_379_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_380_order__le__neq__implies__less,axiom,
! [X_121: rat,Y_79: rat] :
( ( ord_less_eq_rat @ X_121 @ Y_79 )
=> ( ( X_121 != Y_79 )
=> ( ord_less_rat @ X_121 @ Y_79 ) ) ) ).
thf(fact_381_order__le__neq__implies__less,axiom,
! [X_121: int > $o,Y_79: int > $o] :
( ( ord_less_eq_int_o @ X_121 @ Y_79 )
=> ( ( X_121 != Y_79 )
=> ( ord_less_int_o @ X_121 @ Y_79 ) ) ) ).
thf(fact_382_order__le__neq__implies__less,axiom,
! [X_121: nat > $o,Y_79: nat > $o] :
( ( ord_less_eq_nat_o @ X_121 @ Y_79 )
=> ( ( X_121 != Y_79 )
=> ( ord_less_nat_o @ X_121 @ Y_79 ) ) ) ).
thf(fact_383_order__le__neq__implies__less,axiom,
! [X_121: quickcheck_code_int,Y_79: quickcheck_code_int] :
( ( ord_le258702272de_int @ X_121 @ Y_79 )
=> ( ( X_121 != Y_79 )
=> ( ord_le1860547276de_int @ X_121 @ Y_79 ) ) ) ).
thf(fact_384_order__le__neq__implies__less,axiom,
! [X_121: nat,Y_79: nat] :
( ( ord_less_eq_nat @ X_121 @ Y_79 )
=> ( ( X_121 != Y_79 )
=> ( ord_less_nat @ X_121 @ Y_79 ) ) ) ).
thf(fact_385_order__le__neq__implies__less,axiom,
! [X_121: real,Y_79: real] :
( ( ord_less_eq_real @ X_121 @ Y_79 )
=> ( ( X_121 != Y_79 )
=> ( ord_less_real @ X_121 @ Y_79 ) ) ) ).
thf(fact_386_order__le__neq__implies__less,axiom,
! [X_121: int,Y_79: int] :
( ( ord_less_eq_int @ X_121 @ Y_79 )
=> ( ( X_121 != Y_79 )
=> ( ord_less_int @ X_121 @ Y_79 ) ) ) ).
thf(fact_387_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_388_less__1__mult,axiom,
! [N_6: rat,M_10: rat] :
( ( ord_less_rat @ one_one_rat @ M_10 )
=> ( ( ord_less_rat @ one_one_rat @ N_6 )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M_10 @ N_6 ) ) ) ) ).
thf(fact_389_less__1__mult,axiom,
! [N_6: quickcheck_code_int,M_10: quickcheck_code_int] :
( ( ord_le1860547276de_int @ one_on1684967323de_int @ M_10 )
=> ( ( ord_le1860547276de_int @ one_on1684967323de_int @ N_6 )
=> ( ord_le1860547276de_int @ one_on1684967323de_int @ ( times_123202395de_int @ M_10 @ N_6 ) ) ) ) ).
thf(fact_390_less__1__mult,axiom,
! [N_6: code_code_numeral,M_10: code_code_numeral] :
( ( ord_le1304079648umeral @ one_on1645066479umeral @ M_10 )
=> ( ( ord_le1304079648umeral @ one_on1645066479umeral @ N_6 )
=> ( ord_le1304079648umeral @ one_on1645066479umeral @ ( times_1655362735umeral @ M_10 @ N_6 ) ) ) ) ).
thf(fact_391_less__1__mult,axiom,
! [N_6: nat,M_10: nat] :
( ( ord_less_nat @ one_one_nat @ M_10 )
=> ( ( ord_less_nat @ one_one_nat @ N_6 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M_10 @ N_6 ) ) ) ) ).
thf(fact_392_less__1__mult,axiom,
! [N_6: real,M_10: real] :
( ( ord_less_real @ one_one_real @ M_10 )
=> ( ( ord_less_real @ one_one_real @ N_6 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M_10 @ N_6 ) ) ) ) ).
thf(fact_393_less__1__mult,axiom,
! [N_6: int,M_10: int] :
( ( ord_less_int @ one_one_int @ M_10 )
=> ( ( ord_less_int @ one_one_int @ N_6 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M_10 @ N_6 ) ) ) ) ).
thf(fact_394_less__add__one,axiom,
! [A_284: rat] : ( ord_less_rat @ A_284 @ ( plus_plus_rat @ A_284 @ one_one_rat ) ) ).
thf(fact_395_less__add__one,axiom,
! [A_284: quickcheck_code_int] : ( ord_le1860547276de_int @ A_284 @ ( plus_p1446045655de_int @ A_284 @ one_on1684967323de_int ) ) ).
thf(fact_396_less__add__one,axiom,
! [A_284: code_code_numeral] : ( ord_le1304079648umeral @ A_284 @ ( plus_p1627245867umeral @ A_284 @ one_on1645066479umeral ) ) ).
thf(fact_397_less__add__one,axiom,
! [A_284: nat] : ( ord_less_nat @ A_284 @ ( plus_plus_nat @ A_284 @ one_one_nat ) ) ).
thf(fact_398_less__add__one,axiom,
! [A_284: real] : ( ord_less_real @ A_284 @ ( plus_plus_real @ A_284 @ one_one_real ) ) ).
thf(fact_399_less__add__one,axiom,
! [A_284: int] : ( ord_less_int @ A_284 @ ( plus_plus_int @ A_284 @ one_one_int ) ) ).
thf(fact_400_add__le__less__mono,axiom,
! [C_96: rat,D_18: rat,A_283: rat,B_205: rat] :
( ( ord_less_eq_rat @ A_283 @ B_205 )
=> ( ( ord_less_rat @ C_96 @ D_18 )
=> ( ord_less_rat @ ( plus_plus_rat @ A_283 @ C_96 ) @ ( plus_plus_rat @ B_205 @ D_18 ) ) ) ) ).
thf(fact_401_add__le__less__mono,axiom,
! [C_96: quickcheck_code_int,D_18: quickcheck_code_int,A_283: quickcheck_code_int,B_205: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_283 @ B_205 )
=> ( ( ord_le1860547276de_int @ C_96 @ D_18 )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_283 @ C_96 ) @ ( plus_p1446045655de_int @ B_205 @ D_18 ) ) ) ) ).
thf(fact_402_add__le__less__mono,axiom,
! [C_96: nat,D_18: nat,A_283: nat,B_205: nat] :
( ( ord_less_eq_nat @ A_283 @ B_205 )
=> ( ( ord_less_nat @ C_96 @ D_18 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_283 @ C_96 ) @ ( plus_plus_nat @ B_205 @ D_18 ) ) ) ) ).
thf(fact_403_add__le__less__mono,axiom,
! [C_96: real,D_18: real,A_283: real,B_205: real] :
( ( ord_less_eq_real @ A_283 @ B_205 )
=> ( ( ord_less_real @ C_96 @ D_18 )
=> ( ord_less_real @ ( plus_plus_real @ A_283 @ C_96 ) @ ( plus_plus_real @ B_205 @ D_18 ) ) ) ) ).
thf(fact_404_add__le__less__mono,axiom,
! [C_96: int,D_18: int,A_283: int,B_205: int] :
( ( ord_less_eq_int @ A_283 @ B_205 )
=> ( ( ord_less_int @ C_96 @ D_18 )
=> ( ord_less_int @ ( plus_plus_int @ A_283 @ C_96 ) @ ( plus_plus_int @ B_205 @ D_18 ) ) ) ) ).
thf(fact_405_add__less__le__mono,axiom,
! [C_95: rat,D_17: rat,A_282: rat,B_204: rat] :
( ( ord_less_rat @ A_282 @ B_204 )
=> ( ( ord_less_eq_rat @ C_95 @ D_17 )
=> ( ord_less_rat @ ( plus_plus_rat @ A_282 @ C_95 ) @ ( plus_plus_rat @ B_204 @ D_17 ) ) ) ) ).
thf(fact_406_add__less__le__mono,axiom,
! [C_95: quickcheck_code_int,D_17: quickcheck_code_int,A_282: quickcheck_code_int,B_204: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_282 @ B_204 )
=> ( ( ord_le258702272de_int @ C_95 @ D_17 )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_282 @ C_95 ) @ ( plus_p1446045655de_int @ B_204 @ D_17 ) ) ) ) ).
thf(fact_407_add__less__le__mono,axiom,
! [C_95: nat,D_17: nat,A_282: nat,B_204: nat] :
( ( ord_less_nat @ A_282 @ B_204 )
=> ( ( ord_less_eq_nat @ C_95 @ D_17 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_282 @ C_95 ) @ ( plus_plus_nat @ B_204 @ D_17 ) ) ) ) ).
thf(fact_408_add__less__le__mono,axiom,
! [C_95: real,D_17: real,A_282: real,B_204: real] :
( ( ord_less_real @ A_282 @ B_204 )
=> ( ( ord_less_eq_real @ C_95 @ D_17 )
=> ( ord_less_real @ ( plus_plus_real @ A_282 @ C_95 ) @ ( plus_plus_real @ B_204 @ D_17 ) ) ) ) ).
thf(fact_409_add__less__le__mono,axiom,
! [C_95: int,D_17: int,A_282: int,B_204: int] :
( ( ord_less_int @ A_282 @ B_204 )
=> ( ( ord_less_eq_int @ C_95 @ D_17 )
=> ( ord_less_int @ ( plus_plus_int @ A_282 @ C_95 ) @ ( plus_plus_int @ B_204 @ D_17 ) ) ) ) ).
thf(fact_410_zadd__zless__mono,axiom,
! [Z_3: int,Z_1: int,W_16: int,W: int] :
( ( ord_less_int @ W_16 @ W )
=> ( ( ord_less_eq_int @ Z_3 @ Z_1 )
=> ( ord_less_int @ ( plus_plus_int @ W_16 @ Z_3 ) @ ( plus_plus_int @ W @ Z_1 ) ) ) ) ).
thf(fact_411_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_412_add__less__cancel__right,axiom,
! [A_281: rat,C_94: rat,B_203: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A_281 @ C_94 ) @ ( plus_plus_rat @ B_203 @ C_94 ) )
<=> ( ord_less_rat @ A_281 @ B_203 ) ) ).
thf(fact_413_add__less__cancel__right,axiom,
! [A_281: quickcheck_code_int,C_94: quickcheck_code_int,B_203: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_281 @ C_94 ) @ ( plus_p1446045655de_int @ B_203 @ C_94 ) )
<=> ( ord_le1860547276de_int @ A_281 @ B_203 ) ) ).
thf(fact_414_add__less__cancel__right,axiom,
! [A_281: nat,C_94: nat,B_203: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_281 @ C_94 ) @ ( plus_plus_nat @ B_203 @ C_94 ) )
<=> ( ord_less_nat @ A_281 @ B_203 ) ) ).
thf(fact_415_add__less__cancel__right,axiom,
! [A_281: real,C_94: real,B_203: real] :
( ( ord_less_real @ ( plus_plus_real @ A_281 @ C_94 ) @ ( plus_plus_real @ B_203 @ C_94 ) )
<=> ( ord_less_real @ A_281 @ B_203 ) ) ).
thf(fact_416_add__less__cancel__right,axiom,
! [A_281: int,C_94: int,B_203: int] :
( ( ord_less_int @ ( plus_plus_int @ A_281 @ C_94 ) @ ( plus_plus_int @ B_203 @ C_94 ) )
<=> ( ord_less_int @ A_281 @ B_203 ) ) ).
thf(fact_417_add__less__cancel__left,axiom,
! [C_93: rat,A_280: rat,B_202: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C_93 @ A_280 ) @ ( plus_plus_rat @ C_93 @ B_202 ) )
<=> ( ord_less_rat @ A_280 @ B_202 ) ) ).
thf(fact_418_add__less__cancel__left,axiom,
! [C_93: quickcheck_code_int,A_280: quickcheck_code_int,B_202: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ C_93 @ A_280 ) @ ( plus_p1446045655de_int @ C_93 @ B_202 ) )
<=> ( ord_le1860547276de_int @ A_280 @ B_202 ) ) ).
thf(fact_419_add__less__cancel__left,axiom,
! [C_93: nat,A_280: nat,B_202: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_93 @ A_280 ) @ ( plus_plus_nat @ C_93 @ B_202 ) )
<=> ( ord_less_nat @ A_280 @ B_202 ) ) ).
thf(fact_420_add__less__cancel__left,axiom,
! [C_93: real,A_280: real,B_202: real] :
( ( ord_less_real @ ( plus_plus_real @ C_93 @ A_280 ) @ ( plus_plus_real @ C_93 @ B_202 ) )
<=> ( ord_less_real @ A_280 @ B_202 ) ) ).
thf(fact_421_add__less__cancel__left,axiom,
! [C_93: int,A_280: int,B_202: int] :
( ( ord_less_int @ ( plus_plus_int @ C_93 @ A_280 ) @ ( plus_plus_int @ C_93 @ B_202 ) )
<=> ( ord_less_int @ A_280 @ B_202 ) ) ).
thf(fact_422_add__strict__right__mono,axiom,
! [C_92: rat,A_279: rat,B_201: rat] :
( ( ord_less_rat @ A_279 @ B_201 )
=> ( ord_less_rat @ ( plus_plus_rat @ A_279 @ C_92 ) @ ( plus_plus_rat @ B_201 @ C_92 ) ) ) ).
thf(fact_423_add__strict__right__mono,axiom,
! [C_92: quickcheck_code_int,A_279: quickcheck_code_int,B_201: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_279 @ B_201 )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_279 @ C_92 ) @ ( plus_p1446045655de_int @ B_201 @ C_92 ) ) ) ).
thf(fact_424_add__strict__right__mono,axiom,
! [C_92: nat,A_279: nat,B_201: nat] :
( ( ord_less_nat @ A_279 @ B_201 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_279 @ C_92 ) @ ( plus_plus_nat @ B_201 @ C_92 ) ) ) ).
thf(fact_425_add__strict__right__mono,axiom,
! [C_92: real,A_279: real,B_201: real] :
( ( ord_less_real @ A_279 @ B_201 )
=> ( ord_less_real @ ( plus_plus_real @ A_279 @ C_92 ) @ ( plus_plus_real @ B_201 @ C_92 ) ) ) ).
thf(fact_426_add__strict__right__mono,axiom,
! [C_92: int,A_279: int,B_201: int] :
( ( ord_less_int @ A_279 @ B_201 )
=> ( ord_less_int @ ( plus_plus_int @ A_279 @ C_92 ) @ ( plus_plus_int @ B_201 @ C_92 ) ) ) ).
thf(fact_427_add__strict__left__mono,axiom,
! [C_91: rat,A_278: rat,B_200: rat] :
( ( ord_less_rat @ A_278 @ B_200 )
=> ( ord_less_rat @ ( plus_plus_rat @ C_91 @ A_278 ) @ ( plus_plus_rat @ C_91 @ B_200 ) ) ) ).
thf(fact_428_add__strict__left__mono,axiom,
! [C_91: quickcheck_code_int,A_278: quickcheck_code_int,B_200: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_278 @ B_200 )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ C_91 @ A_278 ) @ ( plus_p1446045655de_int @ C_91 @ B_200 ) ) ) ).
thf(fact_429_add__strict__left__mono,axiom,
! [C_91: nat,A_278: nat,B_200: nat] :
( ( ord_less_nat @ A_278 @ B_200 )
=> ( ord_less_nat @ ( plus_plus_nat @ C_91 @ A_278 ) @ ( plus_plus_nat @ C_91 @ B_200 ) ) ) ).
thf(fact_430_add__strict__left__mono,axiom,
! [C_91: real,A_278: real,B_200: real] :
( ( ord_less_real @ A_278 @ B_200 )
=> ( ord_less_real @ ( plus_plus_real @ C_91 @ A_278 ) @ ( plus_plus_real @ C_91 @ B_200 ) ) ) ).
thf(fact_431_add__strict__left__mono,axiom,
! [C_91: int,A_278: int,B_200: int] :
( ( ord_less_int @ A_278 @ B_200 )
=> ( ord_less_int @ ( plus_plus_int @ C_91 @ A_278 ) @ ( plus_plus_int @ C_91 @ B_200 ) ) ) ).
thf(fact_432_add__strict__mono,axiom,
! [C_90: rat,D_16: rat,A_277: rat,B_199: rat] :
( ( ord_less_rat @ A_277 @ B_199 )
=> ( ( ord_less_rat @ C_90 @ D_16 )
=> ( ord_less_rat @ ( plus_plus_rat @ A_277 @ C_90 ) @ ( plus_plus_rat @ B_199 @ D_16 ) ) ) ) ).
thf(fact_433_add__strict__mono,axiom,
! [C_90: quickcheck_code_int,D_16: quickcheck_code_int,A_277: quickcheck_code_int,B_199: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_277 @ B_199 )
=> ( ( ord_le1860547276de_int @ C_90 @ D_16 )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_277 @ C_90 ) @ ( plus_p1446045655de_int @ B_199 @ D_16 ) ) ) ) ).
thf(fact_434_add__strict__mono,axiom,
! [C_90: nat,D_16: nat,A_277: nat,B_199: nat] :
( ( ord_less_nat @ A_277 @ B_199 )
=> ( ( ord_less_nat @ C_90 @ D_16 )
=> ( ord_less_nat @ ( plus_plus_nat @ A_277 @ C_90 ) @ ( plus_plus_nat @ B_199 @ D_16 ) ) ) ) ).
thf(fact_435_add__strict__mono,axiom,
! [C_90: real,D_16: real,A_277: real,B_199: real] :
( ( ord_less_real @ A_277 @ B_199 )
=> ( ( ord_less_real @ C_90 @ D_16 )
=> ( ord_less_real @ ( plus_plus_real @ A_277 @ C_90 ) @ ( plus_plus_real @ B_199 @ D_16 ) ) ) ) ).
thf(fact_436_add__strict__mono,axiom,
! [C_90: int,D_16: int,A_277: int,B_199: int] :
( ( ord_less_int @ A_277 @ B_199 )
=> ( ( ord_less_int @ C_90 @ D_16 )
=> ( ord_less_int @ ( plus_plus_int @ A_277 @ C_90 ) @ ( plus_plus_int @ B_199 @ D_16 ) ) ) ) ).
thf(fact_437_add__less__imp__less__right,axiom,
! [A_276: rat,C_89: rat,B_198: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A_276 @ C_89 ) @ ( plus_plus_rat @ B_198 @ C_89 ) )
=> ( ord_less_rat @ A_276 @ B_198 ) ) ).
thf(fact_438_add__less__imp__less__right,axiom,
! [A_276: quickcheck_code_int,C_89: quickcheck_code_int,B_198: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_276 @ C_89 ) @ ( plus_p1446045655de_int @ B_198 @ C_89 ) )
=> ( ord_le1860547276de_int @ A_276 @ B_198 ) ) ).
thf(fact_439_add__less__imp__less__right,axiom,
! [A_276: nat,C_89: nat,B_198: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A_276 @ C_89 ) @ ( plus_plus_nat @ B_198 @ C_89 ) )
=> ( ord_less_nat @ A_276 @ B_198 ) ) ).
thf(fact_440_add__less__imp__less__right,axiom,
! [A_276: real,C_89: real,B_198: real] :
( ( ord_less_real @ ( plus_plus_real @ A_276 @ C_89 ) @ ( plus_plus_real @ B_198 @ C_89 ) )
=> ( ord_less_real @ A_276 @ B_198 ) ) ).
thf(fact_441_add__less__imp__less__right,axiom,
! [A_276: int,C_89: int,B_198: int] :
( ( ord_less_int @ ( plus_plus_int @ A_276 @ C_89 ) @ ( plus_plus_int @ B_198 @ C_89 ) )
=> ( ord_less_int @ A_276 @ B_198 ) ) ).
thf(fact_442_add__less__imp__less__left,axiom,
! [C_88: rat,A_275: rat,B_197: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C_88 @ A_275 ) @ ( plus_plus_rat @ C_88 @ B_197 ) )
=> ( ord_less_rat @ A_275 @ B_197 ) ) ).
thf(fact_443_add__less__imp__less__left,axiom,
! [C_88: quickcheck_code_int,A_275: quickcheck_code_int,B_197: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ C_88 @ A_275 ) @ ( plus_p1446045655de_int @ C_88 @ B_197 ) )
=> ( ord_le1860547276de_int @ A_275 @ B_197 ) ) ).
thf(fact_444_add__less__imp__less__left,axiom,
! [C_88: nat,A_275: nat,B_197: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C_88 @ A_275 ) @ ( plus_plus_nat @ C_88 @ B_197 ) )
=> ( ord_less_nat @ A_275 @ B_197 ) ) ).
thf(fact_445_add__less__imp__less__left,axiom,
! [C_88: real,A_275: real,B_197: real] :
( ( ord_less_real @ ( plus_plus_real @ C_88 @ A_275 ) @ ( plus_plus_real @ C_88 @ B_197 ) )
=> ( ord_less_real @ A_275 @ B_197 ) ) ).
thf(fact_446_add__less__imp__less__left,axiom,
! [C_88: int,A_275: int,B_197: int] :
( ( ord_less_int @ ( plus_plus_int @ C_88 @ A_275 ) @ ( plus_plus_int @ C_88 @ B_197 ) )
=> ( ord_less_int @ A_275 @ B_197 ) ) ).
thf(fact_447_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_274: rat] :
( ( times_times_rat @ one_one_rat @ A_274 )
= A_274 ) ).
thf(fact_448_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_274: quickcheck_code_int] :
( ( times_123202395de_int @ one_on1684967323de_int @ A_274 )
= A_274 ) ).
thf(fact_449_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_274: complex] :
( ( times_times_complex @ one_one_complex @ A_274 )
= A_274 ) ).
thf(fact_450_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_274: code_code_numeral] :
( ( times_1655362735umeral @ one_on1645066479umeral @ A_274 )
= A_274 ) ).
thf(fact_451_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_274: nat] :
( ( times_times_nat @ one_one_nat @ A_274 )
= A_274 ) ).
thf(fact_452_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_274: real] :
( ( times_times_real @ one_one_real @ A_274 )
= A_274 ) ).
thf(fact_453_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_274: int] :
( ( times_times_int @ one_one_int @ A_274 )
= A_274 ) ).
thf(fact_454_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_273: rat] :
( ( times_times_rat @ A_273 @ one_one_rat )
= A_273 ) ).
thf(fact_455_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_273: quickcheck_code_int] :
( ( times_123202395de_int @ A_273 @ one_on1684967323de_int )
= A_273 ) ).
thf(fact_456_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_273: complex] :
( ( times_times_complex @ A_273 @ one_one_complex )
= A_273 ) ).
thf(fact_457_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_273: code_code_numeral] :
( ( times_1655362735umeral @ A_273 @ one_on1645066479umeral )
= A_273 ) ).
thf(fact_458_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_273: nat] :
( ( times_times_nat @ A_273 @ one_one_nat )
= A_273 ) ).
thf(fact_459_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_273: real] :
( ( times_times_real @ A_273 @ one_one_real )
= A_273 ) ).
thf(fact_460_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_273: int] :
( ( times_times_int @ A_273 @ one_one_int )
= A_273 ) ).
thf(fact_461_mult__1__left,axiom,
! [A_272: rat] :
( ( times_times_rat @ one_one_rat @ A_272 )
= A_272 ) ).
thf(fact_462_mult__1__left,axiom,
! [A_272: quickcheck_code_int] :
( ( times_123202395de_int @ one_on1684967323de_int @ A_272 )
= A_272 ) ).
thf(fact_463_mult__1__left,axiom,
! [A_272: complex] :
( ( times_times_complex @ one_one_complex @ A_272 )
= A_272 ) ).
thf(fact_464_mult__1__left,axiom,
! [A_272: code_code_numeral] :
( ( times_1655362735umeral @ one_on1645066479umeral @ A_272 )
= A_272 ) ).
thf(fact_465_mult__1__left,axiom,
! [A_272: nat] :
( ( times_times_nat @ one_one_nat @ A_272 )
= A_272 ) ).
thf(fact_466_mult__1__left,axiom,
! [A_272: real] :
( ( times_times_real @ one_one_real @ A_272 )
= A_272 ) ).
thf(fact_467_mult__1__left,axiom,
! [A_272: int] :
( ( times_times_int @ one_one_int @ A_272 )
= A_272 ) ).
thf(fact_468_mult__1,axiom,
! [A_271: rat] :
( ( times_times_rat @ one_one_rat @ A_271 )
= A_271 ) ).
thf(fact_469_mult__1,axiom,
! [A_271: quickcheck_code_int] :
( ( times_123202395de_int @ one_on1684967323de_int @ A_271 )
= A_271 ) ).
thf(fact_470_mult__1,axiom,
! [A_271: complex] :
( ( times_times_complex @ one_one_complex @ A_271 )
= A_271 ) ).
thf(fact_471_mult__1,axiom,
! [A_271: code_code_numeral] :
( ( times_1655362735umeral @ one_on1645066479umeral @ A_271 )
= A_271 ) ).
thf(fact_472_mult__1,axiom,
! [A_271: nat] :
( ( times_times_nat @ one_one_nat @ A_271 )
= A_271 ) ).
thf(fact_473_mult__1,axiom,
! [A_271: real] :
( ( times_times_real @ one_one_real @ A_271 )
= A_271 ) ).
thf(fact_474_mult__1,axiom,
! [A_271: int] :
( ( times_times_int @ one_one_int @ A_271 )
= A_271 ) ).
thf(fact_475_mult__1__right,axiom,
! [A_270: rat] :
( ( times_times_rat @ A_270 @ one_one_rat )
= A_270 ) ).
thf(fact_476_mult__1__right,axiom,
! [A_270: quickcheck_code_int] :
( ( times_123202395de_int @ A_270 @ one_on1684967323de_int )
= A_270 ) ).
thf(fact_477_mult__1__right,axiom,
! [A_270: complex] :
( ( times_times_complex @ A_270 @ one_one_complex )
= A_270 ) ).
thf(fact_478_mult__1__right,axiom,
! [A_270: code_code_numeral] :
( ( times_1655362735umeral @ A_270 @ one_on1645066479umeral )
= A_270 ) ).
thf(fact_479_mult__1__right,axiom,
! [A_270: nat] :
( ( times_times_nat @ A_270 @ one_one_nat )
= A_270 ) ).
thf(fact_480_mult__1__right,axiom,
! [A_270: real] :
( ( times_times_real @ A_270 @ one_one_real )
= A_270 ) ).
thf(fact_481_mult__1__right,axiom,
! [A_270: int] :
( ( times_times_int @ A_270 @ one_one_int )
= A_270 ) ).
thf(fact_482_mult_Ocomm__neutral,axiom,
! [A_269: rat] :
( ( times_times_rat @ A_269 @ one_one_rat )
= A_269 ) ).
thf(fact_483_mult_Ocomm__neutral,axiom,
! [A_269: quickcheck_code_int] :
( ( times_123202395de_int @ A_269 @ one_on1684967323de_int )
= A_269 ) ).
thf(fact_484_mult_Ocomm__neutral,axiom,
! [A_269: complex] :
( ( times_times_complex @ A_269 @ one_one_complex )
= A_269 ) ).
thf(fact_485_mult_Ocomm__neutral,axiom,
! [A_269: code_code_numeral] :
( ( times_1655362735umeral @ A_269 @ one_on1645066479umeral )
= A_269 ) ).
thf(fact_486_mult_Ocomm__neutral,axiom,
! [A_269: nat] :
( ( times_times_nat @ A_269 @ one_one_nat )
= A_269 ) ).
thf(fact_487_mult_Ocomm__neutral,axiom,
! [A_269: real] :
( ( times_times_real @ A_269 @ one_one_real )
= A_269 ) ).
thf(fact_488_mult_Ocomm__neutral,axiom,
! [A_269: int] :
( ( times_times_int @ A_269 @ one_one_int )
= A_269 ) ).
thf(fact_489_diff__eq__diff__less,axiom,
! [A_268: rat,B_196: rat,C_87: rat,D_15: rat] :
( ( ( minus_minus_rat @ A_268 @ B_196 )
= ( minus_minus_rat @ C_87 @ D_15 ) )
=> ( ( ord_less_rat @ A_268 @ B_196 )
<=> ( ord_less_rat @ C_87 @ D_15 ) ) ) ).
thf(fact_490_diff__eq__diff__less,axiom,
! [A_268: real,B_196: real,C_87: real,D_15: real] :
( ( ( minus_minus_real @ A_268 @ B_196 )
= ( minus_minus_real @ C_87 @ D_15 ) )
=> ( ( ord_less_real @ A_268 @ B_196 )
<=> ( ord_less_real @ C_87 @ D_15 ) ) ) ).
thf(fact_491_diff__eq__diff__less,axiom,
! [A_268: int,B_196: int,C_87: int,D_15: int] :
( ( ( minus_minus_int @ A_268 @ B_196 )
= ( minus_minus_int @ C_87 @ D_15 ) )
=> ( ( ord_less_int @ A_268 @ B_196 )
<=> ( ord_less_int @ C_87 @ D_15 ) ) ) ).
thf(fact_492_add__le__cancel__right,axiom,
! [A_267: rat,C_86: rat,B_195: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A_267 @ C_86 ) @ ( plus_plus_rat @ B_195 @ C_86 ) )
<=> ( ord_less_eq_rat @ A_267 @ B_195 ) ) ).
thf(fact_493_add__le__cancel__right,axiom,
! [A_267: quickcheck_code_int,C_86: quickcheck_code_int,B_195: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( plus_p1446045655de_int @ A_267 @ C_86 ) @ ( plus_p1446045655de_int @ B_195 @ C_86 ) )
<=> ( ord_le258702272de_int @ A_267 @ B_195 ) ) ).
thf(fact_494_add__le__cancel__right,axiom,
! [A_267: nat,C_86: nat,B_195: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_267 @ C_86 ) @ ( plus_plus_nat @ B_195 @ C_86 ) )
<=> ( ord_less_eq_nat @ A_267 @ B_195 ) ) ).
thf(fact_495_add__le__cancel__right,axiom,
! [A_267: real,C_86: real,B_195: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A_267 @ C_86 ) @ ( plus_plus_real @ B_195 @ C_86 ) )
<=> ( ord_less_eq_real @ A_267 @ B_195 ) ) ).
thf(fact_496_add__le__cancel__right,axiom,
! [A_267: int,C_86: int,B_195: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_267 @ C_86 ) @ ( plus_plus_int @ B_195 @ C_86 ) )
<=> ( ord_less_eq_int @ A_267 @ B_195 ) ) ).
thf(fact_497_add__le__cancel__left,axiom,
! [C_85: rat,A_266: rat,B_194: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C_85 @ A_266 ) @ ( plus_plus_rat @ C_85 @ B_194 ) )
<=> ( ord_less_eq_rat @ A_266 @ B_194 ) ) ).
thf(fact_498_add__le__cancel__left,axiom,
! [C_85: quickcheck_code_int,A_266: quickcheck_code_int,B_194: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( plus_p1446045655de_int @ C_85 @ A_266 ) @ ( plus_p1446045655de_int @ C_85 @ B_194 ) )
<=> ( ord_le258702272de_int @ A_266 @ B_194 ) ) ).
thf(fact_499_add__le__cancel__left,axiom,
! [C_85: nat,A_266: nat,B_194: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_85 @ A_266 ) @ ( plus_plus_nat @ C_85 @ B_194 ) )
<=> ( ord_less_eq_nat @ A_266 @ B_194 ) ) ).
thf(fact_500_add__le__cancel__left,axiom,
! [C_85: real,A_266: real,B_194: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C_85 @ A_266 ) @ ( plus_plus_real @ C_85 @ B_194 ) )
<=> ( ord_less_eq_real @ A_266 @ B_194 ) ) ).
thf(fact_501_add__le__cancel__left,axiom,
! [C_85: int,A_266: int,B_194: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_85 @ A_266 ) @ ( plus_plus_int @ C_85 @ B_194 ) )
<=> ( ord_less_eq_int @ A_266 @ B_194 ) ) ).
thf(fact_502_add__right__mono,axiom,
! [C_84: rat,A_265: rat,B_193: rat] :
( ( ord_less_eq_rat @ A_265 @ B_193 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A_265 @ C_84 ) @ ( plus_plus_rat @ B_193 @ C_84 ) ) ) ).
thf(fact_503_add__right__mono,axiom,
! [C_84: quickcheck_code_int,A_265: quickcheck_code_int,B_193: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_265 @ B_193 )
=> ( ord_le258702272de_int @ ( plus_p1446045655de_int @ A_265 @ C_84 ) @ ( plus_p1446045655de_int @ B_193 @ C_84 ) ) ) ).
thf(fact_504_add__right__mono,axiom,
! [C_84: nat,A_265: nat,B_193: nat] :
( ( ord_less_eq_nat @ A_265 @ B_193 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_265 @ C_84 ) @ ( plus_plus_nat @ B_193 @ C_84 ) ) ) ).
thf(fact_505_add__right__mono,axiom,
! [C_84: real,A_265: real,B_193: real] :
( ( ord_less_eq_real @ A_265 @ B_193 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A_265 @ C_84 ) @ ( plus_plus_real @ B_193 @ C_84 ) ) ) ).
thf(fact_506_add__right__mono,axiom,
! [C_84: int,A_265: int,B_193: int] :
( ( ord_less_eq_int @ A_265 @ B_193 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_265 @ C_84 ) @ ( plus_plus_int @ B_193 @ C_84 ) ) ) ).
thf(fact_507_add__left__mono,axiom,
! [C_83: rat,A_264: rat,B_192: rat] :
( ( ord_less_eq_rat @ A_264 @ B_192 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ C_83 @ A_264 ) @ ( plus_plus_rat @ C_83 @ B_192 ) ) ) ).
thf(fact_508_add__left__mono,axiom,
! [C_83: quickcheck_code_int,A_264: quickcheck_code_int,B_192: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_264 @ B_192 )
=> ( ord_le258702272de_int @ ( plus_p1446045655de_int @ C_83 @ A_264 ) @ ( plus_p1446045655de_int @ C_83 @ B_192 ) ) ) ).
thf(fact_509_add__left__mono,axiom,
! [C_83: nat,A_264: nat,B_192: nat] :
( ( ord_less_eq_nat @ A_264 @ B_192 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C_83 @ A_264 ) @ ( plus_plus_nat @ C_83 @ B_192 ) ) ) ).
thf(fact_510_add__left__mono,axiom,
! [C_83: real,A_264: real,B_192: real] :
( ( ord_less_eq_real @ A_264 @ B_192 )
=> ( ord_less_eq_real @ ( plus_plus_real @ C_83 @ A_264 ) @ ( plus_plus_real @ C_83 @ B_192 ) ) ) ).
thf(fact_511_add__left__mono,axiom,
! [C_83: int,A_264: int,B_192: int] :
( ( ord_less_eq_int @ A_264 @ B_192 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C_83 @ A_264 ) @ ( plus_plus_int @ C_83 @ B_192 ) ) ) ).
thf(fact_512_add__mono,axiom,
! [C_82: rat,D_14: rat,A_263: rat,B_191: rat] :
( ( ord_less_eq_rat @ A_263 @ B_191 )
=> ( ( ord_less_eq_rat @ C_82 @ D_14 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A_263 @ C_82 ) @ ( plus_plus_rat @ B_191 @ D_14 ) ) ) ) ).
thf(fact_513_add__mono,axiom,
! [C_82: quickcheck_code_int,D_14: quickcheck_code_int,A_263: quickcheck_code_int,B_191: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_263 @ B_191 )
=> ( ( ord_le258702272de_int @ C_82 @ D_14 )
=> ( ord_le258702272de_int @ ( plus_p1446045655de_int @ A_263 @ C_82 ) @ ( plus_p1446045655de_int @ B_191 @ D_14 ) ) ) ) ).
thf(fact_514_add__mono,axiom,
! [C_82: nat,D_14: nat,A_263: nat,B_191: nat] :
( ( ord_less_eq_nat @ A_263 @ B_191 )
=> ( ( ord_less_eq_nat @ C_82 @ D_14 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_263 @ C_82 ) @ ( plus_plus_nat @ B_191 @ D_14 ) ) ) ) ).
thf(fact_515_add__mono,axiom,
! [C_82: real,D_14: real,A_263: real,B_191: real] :
( ( ord_less_eq_real @ A_263 @ B_191 )
=> ( ( ord_less_eq_real @ C_82 @ D_14 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A_263 @ C_82 ) @ ( plus_plus_real @ B_191 @ D_14 ) ) ) ) ).
thf(fact_516_add__mono,axiom,
! [C_82: int,D_14: int,A_263: int,B_191: int] :
( ( ord_less_eq_int @ A_263 @ B_191 )
=> ( ( ord_less_eq_int @ C_82 @ D_14 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_263 @ C_82 ) @ ( plus_plus_int @ B_191 @ D_14 ) ) ) ) ).
thf(fact_517_add__le__imp__le__right,axiom,
! [A_262: rat,C_81: rat,B_190: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A_262 @ C_81 ) @ ( plus_plus_rat @ B_190 @ C_81 ) )
=> ( ord_less_eq_rat @ A_262 @ B_190 ) ) ).
thf(fact_518_add__le__imp__le__right,axiom,
! [A_262: quickcheck_code_int,C_81: quickcheck_code_int,B_190: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( plus_p1446045655de_int @ A_262 @ C_81 ) @ ( plus_p1446045655de_int @ B_190 @ C_81 ) )
=> ( ord_le258702272de_int @ A_262 @ B_190 ) ) ).
thf(fact_519_add__le__imp__le__right,axiom,
! [A_262: nat,C_81: nat,B_190: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A_262 @ C_81 ) @ ( plus_plus_nat @ B_190 @ C_81 ) )
=> ( ord_less_eq_nat @ A_262 @ B_190 ) ) ).
thf(fact_520_add__le__imp__le__right,axiom,
! [A_262: real,C_81: real,B_190: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A_262 @ C_81 ) @ ( plus_plus_real @ B_190 @ C_81 ) )
=> ( ord_less_eq_real @ A_262 @ B_190 ) ) ).
thf(fact_521_add__le__imp__le__right,axiom,
! [A_262: int,C_81: int,B_190: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_262 @ C_81 ) @ ( plus_plus_int @ B_190 @ C_81 ) )
=> ( ord_less_eq_int @ A_262 @ B_190 ) ) ).
thf(fact_522_add__le__imp__le__left,axiom,
! [C_80: rat,A_261: rat,B_189: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C_80 @ A_261 ) @ ( plus_plus_rat @ C_80 @ B_189 ) )
=> ( ord_less_eq_rat @ A_261 @ B_189 ) ) ).
thf(fact_523_add__le__imp__le__left,axiom,
! [C_80: quickcheck_code_int,A_261: quickcheck_code_int,B_189: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( plus_p1446045655de_int @ C_80 @ A_261 ) @ ( plus_p1446045655de_int @ C_80 @ B_189 ) )
=> ( ord_le258702272de_int @ A_261 @ B_189 ) ) ).
thf(fact_524_add__le__imp__le__left,axiom,
! [C_80: nat,A_261: nat,B_189: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C_80 @ A_261 ) @ ( plus_plus_nat @ C_80 @ B_189 ) )
=> ( ord_less_eq_nat @ A_261 @ B_189 ) ) ).
thf(fact_525_add__le__imp__le__left,axiom,
! [C_80: real,A_261: real,B_189: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C_80 @ A_261 ) @ ( plus_plus_real @ C_80 @ B_189 ) )
=> ( ord_less_eq_real @ A_261 @ B_189 ) ) ).
thf(fact_526_add__le__imp__le__left,axiom,
! [C_80: int,A_261: int,B_189: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_80 @ A_261 ) @ ( plus_plus_int @ C_80 @ B_189 ) )
=> ( ord_less_eq_int @ A_261 @ B_189 ) ) ).
thf(fact_527_diff__eq__diff__less__eq,axiom,
! [A_260: rat,B_188: rat,C_79: rat,D_13: rat] :
( ( ( minus_minus_rat @ A_260 @ B_188 )
= ( minus_minus_rat @ C_79 @ D_13 ) )
=> ( ( ord_less_eq_rat @ A_260 @ B_188 )
<=> ( ord_less_eq_rat @ C_79 @ D_13 ) ) ) ).
thf(fact_528_diff__eq__diff__less__eq,axiom,
! [A_260: real,B_188: real,C_79: real,D_13: real] :
( ( ( minus_minus_real @ A_260 @ B_188 )
= ( minus_minus_real @ C_79 @ D_13 ) )
=> ( ( ord_less_eq_real @ A_260 @ B_188 )
<=> ( ord_less_eq_real @ C_79 @ D_13 ) ) ) ).
thf(fact_529_diff__eq__diff__less__eq,axiom,
! [A_260: int,B_188: int,C_79: int,D_13: int] :
( ( ( minus_minus_int @ A_260 @ B_188 )
= ( minus_minus_int @ C_79 @ D_13 ) )
=> ( ( ord_less_eq_int @ A_260 @ B_188 )
<=> ( ord_less_eq_int @ C_79 @ D_13 ) ) ) ).
thf(fact_530_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_531_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_532_zadd__strict__right__mono,axiom,
! [K_1: int,I: int,J: int] :
( ( ord_less_int @ I @ J )
=> ( ord_less_int @ ( plus_plus_int @ I @ K_1 ) @ ( plus_plus_int @ J @ K_1 ) ) ) ).
thf(fact_533_zadd__left__mono,axiom,
! [K_1: int,I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ord_less_eq_int @ ( plus_plus_int @ K_1 @ I ) @ ( plus_plus_int @ K_1 @ J ) ) ) ).
thf(fact_534_zmult__1,axiom,
! [Z_1: int] :
( ( times_times_int @ one_one_int @ Z_1 )
= Z_1 ) ).
thf(fact_535_zmult__1__right,axiom,
! [Z_1: int] :
( ( times_times_int @ Z_1 @ one_one_int )
= Z_1 ) ).
thf(fact_536_right__distrib__number__of,axiom,
! [V_30: int,B_187: rat,C_78: rat] :
( ( times_times_rat @ ( number_number_of_rat @ V_30 ) @ ( plus_plus_rat @ B_187 @ C_78 ) )
= ( plus_plus_rat @ ( times_times_rat @ ( number_number_of_rat @ V_30 ) @ B_187 ) @ ( times_times_rat @ ( number_number_of_rat @ V_30 ) @ C_78 ) ) ) ).
thf(fact_537_right__distrib__number__of,axiom,
! [V_30: int,B_187: quickcheck_code_int,C_78: quickcheck_code_int] :
( ( times_123202395de_int @ ( number1226105091de_int @ V_30 ) @ ( plus_p1446045655de_int @ B_187 @ C_78 ) )
= ( plus_p1446045655de_int @ ( times_123202395de_int @ ( number1226105091de_int @ V_30 ) @ B_187 ) @ ( times_123202395de_int @ ( number1226105091de_int @ V_30 ) @ C_78 ) ) ) ).
thf(fact_538_right__distrib__number__of,axiom,
! [V_30: int,B_187: complex,C_78: complex] :
( ( times_times_complex @ ( number528085621omplex @ V_30 ) @ ( plus_plus_complex @ B_187 @ C_78 ) )
= ( plus_plus_complex @ ( times_times_complex @ ( number528085621omplex @ V_30 ) @ B_187 ) @ ( times_times_complex @ ( number528085621omplex @ V_30 ) @ C_78 ) ) ) ).
thf(fact_539_right__distrib__number__of,axiom,
! [V_30: int,B_187: nat,C_78: nat] :
( ( times_times_nat @ ( number_number_of_nat @ V_30 ) @ ( plus_plus_nat @ B_187 @ C_78 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ V_30 ) @ B_187 ) @ ( times_times_nat @ ( number_number_of_nat @ V_30 ) @ C_78 ) ) ) ).
thf(fact_540_right__distrib__number__of,axiom,
! [V_30: int,B_187: real,C_78: real] :
( ( times_times_real @ ( number267125858f_real @ V_30 ) @ ( plus_plus_real @ B_187 @ C_78 ) )
= ( plus_plus_real @ ( times_times_real @ ( number267125858f_real @ V_30 ) @ B_187 ) @ ( times_times_real @ ( number267125858f_real @ V_30 ) @ C_78 ) ) ) ).
thf(fact_541_right__distrib__number__of,axiom,
! [V_30: int,B_187: int,C_78: int] :
( ( times_times_int @ ( number_number_of_int @ V_30 ) @ ( plus_plus_int @ B_187 @ C_78 ) )
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ V_30 ) @ B_187 ) @ ( times_times_int @ ( number_number_of_int @ V_30 ) @ C_78 ) ) ) ).
thf(fact_542_right__distrib__number__of,axiom,
! [V_30: int,B_187: code_code_numeral,C_78: code_code_numeral] :
( ( times_1655362735umeral @ ( number1443263063umeral @ V_30 ) @ ( plus_p1627245867umeral @ B_187 @ C_78 ) )
= ( plus_p1627245867umeral @ ( times_1655362735umeral @ ( number1443263063umeral @ V_30 ) @ B_187 ) @ ( times_1655362735umeral @ ( number1443263063umeral @ V_30 ) @ C_78 ) ) ) ).
thf(fact_543_left__distrib__number__of,axiom,
! [A_259: rat,B_186: rat,V_29: int] :
( ( times_times_rat @ ( plus_plus_rat @ A_259 @ B_186 ) @ ( number_number_of_rat @ V_29 ) )
= ( plus_plus_rat @ ( times_times_rat @ A_259 @ ( number_number_of_rat @ V_29 ) ) @ ( times_times_rat @ B_186 @ ( number_number_of_rat @ V_29 ) ) ) ) ).
thf(fact_544_left__distrib__number__of,axiom,
! [A_259: quickcheck_code_int,B_186: quickcheck_code_int,V_29: int] :
( ( times_123202395de_int @ ( plus_p1446045655de_int @ A_259 @ B_186 ) @ ( number1226105091de_int @ V_29 ) )
= ( plus_p1446045655de_int @ ( times_123202395de_int @ A_259 @ ( number1226105091de_int @ V_29 ) ) @ ( times_123202395de_int @ B_186 @ ( number1226105091de_int @ V_29 ) ) ) ) ).
thf(fact_545_left__distrib__number__of,axiom,
! [A_259: complex,B_186: complex,V_29: int] :
( ( times_times_complex @ ( plus_plus_complex @ A_259 @ B_186 ) @ ( number528085621omplex @ V_29 ) )
= ( plus_plus_complex @ ( times_times_complex @ A_259 @ ( number528085621omplex @ V_29 ) ) @ ( times_times_complex @ B_186 @ ( number528085621omplex @ V_29 ) ) ) ) ).
thf(fact_546_left__distrib__number__of,axiom,
! [A_259: nat,B_186: nat,V_29: int] :
( ( times_times_nat @ ( plus_plus_nat @ A_259 @ B_186 ) @ ( number_number_of_nat @ V_29 ) )
= ( plus_plus_nat @ ( times_times_nat @ A_259 @ ( number_number_of_nat @ V_29 ) ) @ ( times_times_nat @ B_186 @ ( number_number_of_nat @ V_29 ) ) ) ) ).
thf(fact_547_left__distrib__number__of,axiom,
! [A_259: real,B_186: real,V_29: int] :
( ( times_times_real @ ( plus_plus_real @ A_259 @ B_186 ) @ ( number267125858f_real @ V_29 ) )
= ( plus_plus_real @ ( times_times_real @ A_259 @ ( number267125858f_real @ V_29 ) ) @ ( times_times_real @ B_186 @ ( number267125858f_real @ V_29 ) ) ) ) ).
thf(fact_548_left__distrib__number__of,axiom,
! [A_259: int,B_186: int,V_29: int] :
( ( times_times_int @ ( plus_plus_int @ A_259 @ B_186 ) @ ( number_number_of_int @ V_29 ) )
= ( plus_plus_int @ ( times_times_int @ A_259 @ ( number_number_of_int @ V_29 ) ) @ ( times_times_int @ B_186 @ ( number_number_of_int @ V_29 ) ) ) ) ).
thf(fact_549_left__distrib__number__of,axiom,
! [A_259: code_code_numeral,B_186: code_code_numeral,V_29: int] :
( ( times_1655362735umeral @ ( plus_p1627245867umeral @ A_259 @ B_186 ) @ ( number1443263063umeral @ V_29 ) )
= ( plus_p1627245867umeral @ ( times_1655362735umeral @ A_259 @ ( number1443263063umeral @ V_29 ) ) @ ( times_1655362735umeral @ B_186 @ ( number1443263063umeral @ V_29 ) ) ) ) ).
thf(fact_550_right__diff__distrib__number__of,axiom,
! [V_28: int,B_185: rat,C_77: rat] :
( ( times_times_rat @ ( number_number_of_rat @ V_28 ) @ ( minus_minus_rat @ B_185 @ C_77 ) )
= ( minus_minus_rat @ ( times_times_rat @ ( number_number_of_rat @ V_28 ) @ B_185 ) @ ( times_times_rat @ ( number_number_of_rat @ V_28 ) @ C_77 ) ) ) ).
thf(fact_551_right__diff__distrib__number__of,axiom,
! [V_28: int,B_185: complex,C_77: complex] :
( ( times_times_complex @ ( number528085621omplex @ V_28 ) @ ( minus_minus_complex @ B_185 @ C_77 ) )
= ( minus_minus_complex @ ( times_times_complex @ ( number528085621omplex @ V_28 ) @ B_185 ) @ ( times_times_complex @ ( number528085621omplex @ V_28 ) @ C_77 ) ) ) ).
thf(fact_552_right__diff__distrib__number__of,axiom,
! [V_28: int,B_185: real,C_77: real] :
( ( times_times_real @ ( number267125858f_real @ V_28 ) @ ( minus_minus_real @ B_185 @ C_77 ) )
= ( minus_minus_real @ ( times_times_real @ ( number267125858f_real @ V_28 ) @ B_185 ) @ ( times_times_real @ ( number267125858f_real @ V_28 ) @ C_77 ) ) ) ).
thf(fact_553_right__diff__distrib__number__of,axiom,
! [V_28: int,B_185: int,C_77: int] :
( ( times_times_int @ ( number_number_of_int @ V_28 ) @ ( minus_minus_int @ B_185 @ C_77 ) )
= ( minus_minus_int @ ( times_times_int @ ( number_number_of_int @ V_28 ) @ B_185 ) @ ( times_times_int @ ( number_number_of_int @ V_28 ) @ C_77 ) ) ) ).
thf(fact_554_left__diff__distrib__number__of,axiom,
! [A_258: rat,B_184: rat,V_27: int] :
( ( times_times_rat @ ( minus_minus_rat @ A_258 @ B_184 ) @ ( number_number_of_rat @ V_27 ) )
= ( minus_minus_rat @ ( times_times_rat @ A_258 @ ( number_number_of_rat @ V_27 ) ) @ ( times_times_rat @ B_184 @ ( number_number_of_rat @ V_27 ) ) ) ) ).
thf(fact_555_left__diff__distrib__number__of,axiom,
! [A_258: complex,B_184: complex,V_27: int] :
( ( times_times_complex @ ( minus_minus_complex @ A_258 @ B_184 ) @ ( number528085621omplex @ V_27 ) )
= ( minus_minus_complex @ ( times_times_complex @ A_258 @ ( number528085621omplex @ V_27 ) ) @ ( times_times_complex @ B_184 @ ( number528085621omplex @ V_27 ) ) ) ) ).
thf(fact_556_left__diff__distrib__number__of,axiom,
! [A_258: real,B_184: real,V_27: int] :
( ( times_times_real @ ( minus_minus_real @ A_258 @ B_184 ) @ ( number267125858f_real @ V_27 ) )
= ( minus_minus_real @ ( times_times_real @ A_258 @ ( number267125858f_real @ V_27 ) ) @ ( times_times_real @ B_184 @ ( number267125858f_real @ V_27 ) ) ) ) ).
thf(fact_557_left__diff__distrib__number__of,axiom,
! [A_258: int,B_184: int,V_27: int] :
( ( times_times_int @ ( minus_minus_int @ A_258 @ B_184 ) @ ( number_number_of_int @ V_27 ) )
= ( minus_minus_int @ ( times_times_int @ A_258 @ ( number_number_of_int @ V_27 ) ) @ ( times_times_int @ B_184 @ ( number_number_of_int @ V_27 ) ) ) ) ).
thf(fact_558_number__of__mult,axiom,
! [V_26: int,W_15: int] :
( ( number_number_of_rat @ ( times_times_int @ V_26 @ W_15 ) )
= ( times_times_rat @ ( number_number_of_rat @ V_26 ) @ ( number_number_of_rat @ W_15 ) ) ) ).
thf(fact_559_number__of__mult,axiom,
! [V_26: int,W_15: int] :
( ( number528085621omplex @ ( times_times_int @ V_26 @ W_15 ) )
= ( times_times_complex @ ( number528085621omplex @ V_26 ) @ ( number528085621omplex @ W_15 ) ) ) ).
thf(fact_560_number__of__mult,axiom,
! [V_26: int,W_15: int] :
( ( number267125858f_real @ ( times_times_int @ V_26 @ W_15 ) )
= ( times_times_real @ ( number267125858f_real @ V_26 ) @ ( number267125858f_real @ W_15 ) ) ) ).
thf(fact_561_number__of__mult,axiom,
! [V_26: int,W_15: int] :
( ( number_number_of_int @ ( times_times_int @ V_26 @ W_15 ) )
= ( times_times_int @ ( number_number_of_int @ V_26 ) @ ( number_number_of_int @ W_15 ) ) ) ).
thf(fact_562_arith__simps_I32_J,axiom,
! [V_25: int,W_14: int] :
( ( times_times_rat @ ( number_number_of_rat @ V_25 ) @ ( number_number_of_rat @ W_14 ) )
= ( number_number_of_rat @ ( times_times_int @ V_25 @ W_14 ) ) ) ).
thf(fact_563_arith__simps_I32_J,axiom,
! [V_25: int,W_14: int] :
( ( times_times_complex @ ( number528085621omplex @ V_25 ) @ ( number528085621omplex @ W_14 ) )
= ( number528085621omplex @ ( times_times_int @ V_25 @ W_14 ) ) ) ).
thf(fact_564_arith__simps_I32_J,axiom,
! [V_25: int,W_14: int] :
( ( times_times_real @ ( number267125858f_real @ V_25 ) @ ( number267125858f_real @ W_14 ) )
= ( number267125858f_real @ ( times_times_int @ V_25 @ W_14 ) ) ) ).
thf(fact_565_arith__simps_I32_J,axiom,
! [V_25: int,W_14: int] :
( ( times_times_int @ ( number_number_of_int @ V_25 ) @ ( number_number_of_int @ W_14 ) )
= ( number_number_of_int @ ( times_times_int @ V_25 @ W_14 ) ) ) ).
thf(fact_566_mult__number__of__left,axiom,
! [V_24: int,W_13: int,Z_20: rat] :
( ( times_times_rat @ ( number_number_of_rat @ V_24 ) @ ( times_times_rat @ ( number_number_of_rat @ W_13 ) @ Z_20 ) )
= ( times_times_rat @ ( number_number_of_rat @ ( times_times_int @ V_24 @ W_13 ) ) @ Z_20 ) ) ).
thf(fact_567_mult__number__of__left,axiom,
! [V_24: int,W_13: int,Z_20: complex] :
( ( times_times_complex @ ( number528085621omplex @ V_24 ) @ ( times_times_complex @ ( number528085621omplex @ W_13 ) @ Z_20 ) )
= ( times_times_complex @ ( number528085621omplex @ ( times_times_int @ V_24 @ W_13 ) ) @ Z_20 ) ) ).
thf(fact_568_mult__number__of__left,axiom,
! [V_24: int,W_13: int,Z_20: real] :
( ( times_times_real @ ( number267125858f_real @ V_24 ) @ ( times_times_real @ ( number267125858f_real @ W_13 ) @ Z_20 ) )
= ( times_times_real @ ( number267125858f_real @ ( times_times_int @ V_24 @ W_13 ) ) @ Z_20 ) ) ).
thf(fact_569_mult__number__of__left,axiom,
! [V_24: int,W_13: int,Z_20: int] :
( ( times_times_int @ ( number_number_of_int @ V_24 ) @ ( times_times_int @ ( number_number_of_int @ W_13 ) @ Z_20 ) )
= ( times_times_int @ ( number_number_of_int @ ( times_times_int @ V_24 @ W_13 ) ) @ Z_20 ) ) ).
thf(fact_570_number__of__add,axiom,
! [V_23: int,W_12: int] :
( ( number_number_of_rat @ ( plus_plus_int @ V_23 @ W_12 ) )
= ( plus_plus_rat @ ( number_number_of_rat @ V_23 ) @ ( number_number_of_rat @ W_12 ) ) ) ).
thf(fact_571_number__of__add,axiom,
! [V_23: int,W_12: int] :
( ( number528085621omplex @ ( plus_plus_int @ V_23 @ W_12 ) )
= ( plus_plus_complex @ ( number528085621omplex @ V_23 ) @ ( number528085621omplex @ W_12 ) ) ) ).
thf(fact_572_number__of__add,axiom,
! [V_23: int,W_12: int] :
( ( number267125858f_real @ ( plus_plus_int @ V_23 @ W_12 ) )
= ( plus_plus_real @ ( number267125858f_real @ V_23 ) @ ( number267125858f_real @ W_12 ) ) ) ).
thf(fact_573_number__of__add,axiom,
! [V_23: int,W_12: int] :
( ( number_number_of_int @ ( plus_plus_int @ V_23 @ W_12 ) )
= ( plus_plus_int @ ( number_number_of_int @ V_23 ) @ ( number_number_of_int @ W_12 ) ) ) ).
thf(fact_574_add__number__of__eq,axiom,
! [V_22: int,W_11: int] :
( ( plus_plus_rat @ ( number_number_of_rat @ V_22 ) @ ( number_number_of_rat @ W_11 ) )
= ( number_number_of_rat @ ( plus_plus_int @ V_22 @ W_11 ) ) ) ).
thf(fact_575_add__number__of__eq,axiom,
! [V_22: int,W_11: int] :
( ( plus_plus_complex @ ( number528085621omplex @ V_22 ) @ ( number528085621omplex @ W_11 ) )
= ( number528085621omplex @ ( plus_plus_int @ V_22 @ W_11 ) ) ) ).
thf(fact_576_add__number__of__eq,axiom,
! [V_22: int,W_11: int] :
( ( plus_plus_real @ ( number267125858f_real @ V_22 ) @ ( number267125858f_real @ W_11 ) )
= ( number267125858f_real @ ( plus_plus_int @ V_22 @ W_11 ) ) ) ).
thf(fact_577_add__number__of__eq,axiom,
! [V_22: int,W_11: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_22 ) @ ( number_number_of_int @ W_11 ) )
= ( number_number_of_int @ ( plus_plus_int @ V_22 @ W_11 ) ) ) ).
thf(fact_578_add__number__of__left,axiom,
! [V_21: int,W_10: int,Z_19: rat] :
( ( plus_plus_rat @ ( number_number_of_rat @ V_21 ) @ ( plus_plus_rat @ ( number_number_of_rat @ W_10 ) @ Z_19 ) )
= ( plus_plus_rat @ ( number_number_of_rat @ ( plus_plus_int @ V_21 @ W_10 ) ) @ Z_19 ) ) ).
thf(fact_579_add__number__of__left,axiom,
! [V_21: int,W_10: int,Z_19: complex] :
( ( plus_plus_complex @ ( number528085621omplex @ V_21 ) @ ( plus_plus_complex @ ( number528085621omplex @ W_10 ) @ Z_19 ) )
= ( plus_plus_complex @ ( number528085621omplex @ ( plus_plus_int @ V_21 @ W_10 ) ) @ Z_19 ) ) ).
thf(fact_580_add__number__of__left,axiom,
! [V_21: int,W_10: int,Z_19: real] :
( ( plus_plus_real @ ( number267125858f_real @ V_21 ) @ ( plus_plus_real @ ( number267125858f_real @ W_10 ) @ Z_19 ) )
= ( plus_plus_real @ ( number267125858f_real @ ( plus_plus_int @ V_21 @ W_10 ) ) @ Z_19 ) ) ).
thf(fact_581_add__number__of__left,axiom,
! [V_21: int,W_10: int,Z_19: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_21 ) @ ( plus_plus_int @ ( number_number_of_int @ W_10 ) @ Z_19 ) )
= ( plus_plus_int @ ( number_number_of_int @ ( plus_plus_int @ V_21 @ W_10 ) ) @ Z_19 ) ) ).
thf(fact_582_number__of__diff,axiom,
! [V_20: int,W_9: int] :
( ( number_number_of_rat @ ( minus_minus_int @ V_20 @ W_9 ) )
= ( minus_minus_rat @ ( number_number_of_rat @ V_20 ) @ ( number_number_of_rat @ W_9 ) ) ) ).
thf(fact_583_number__of__diff,axiom,
! [V_20: int,W_9: int] :
( ( number528085621omplex @ ( minus_minus_int @ V_20 @ W_9 ) )
= ( minus_minus_complex @ ( number528085621omplex @ V_20 ) @ ( number528085621omplex @ W_9 ) ) ) ).
thf(fact_584_number__of__diff,axiom,
! [V_20: int,W_9: int] :
( ( number267125858f_real @ ( minus_minus_int @ V_20 @ W_9 ) )
= ( minus_minus_real @ ( number267125858f_real @ V_20 ) @ ( number267125858f_real @ W_9 ) ) ) ).
thf(fact_585_number__of__diff,axiom,
! [V_20: int,W_9: int] :
( ( number_number_of_int @ ( minus_minus_int @ V_20 @ W_9 ) )
= ( minus_minus_int @ ( number_number_of_int @ V_20 ) @ ( number_number_of_int @ W_9 ) ) ) ).
thf(fact_586_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_9: rat] :
( ( plus_plus_rat @ M_9 @ M_9 )
= ( times_times_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) @ M_9 ) ) ).
thf(fact_587_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_9: quickcheck_code_int] :
( ( plus_p1446045655de_int @ M_9 @ M_9 )
= ( times_123202395de_int @ ( plus_p1446045655de_int @ one_on1684967323de_int @ one_on1684967323de_int ) @ M_9 ) ) ).
thf(fact_588_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_9: complex] :
( ( plus_plus_complex @ M_9 @ M_9 )
= ( times_times_complex @ ( plus_plus_complex @ one_one_complex @ one_one_complex ) @ M_9 ) ) ).
thf(fact_589_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_9: code_code_numeral] :
( ( plus_p1627245867umeral @ M_9 @ M_9 )
= ( times_1655362735umeral @ ( plus_p1627245867umeral @ one_on1645066479umeral @ one_on1645066479umeral ) @ M_9 ) ) ).
thf(fact_590_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_9: nat] :
( ( plus_plus_nat @ M_9 @ M_9 )
= ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) @ M_9 ) ) ).
thf(fact_591_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_9: real] :
( ( plus_plus_real @ M_9 @ M_9 )
= ( times_times_real @ ( plus_plus_real @ one_one_real @ one_one_real ) @ M_9 ) ) ).
thf(fact_592_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_9: int] :
( ( plus_plus_int @ M_9 @ M_9 )
= ( times_times_int @ ( plus_plus_int @ one_one_int @ one_one_int ) @ M_9 ) ) ).
thf(fact_593_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_8: rat,A_257: rat] :
( ( plus_plus_rat @ M_8 @ ( times_times_rat @ A_257 @ M_8 ) )
= ( times_times_rat @ ( plus_plus_rat @ A_257 @ one_one_rat ) @ M_8 ) ) ).
thf(fact_594_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_8: quickcheck_code_int,A_257: quickcheck_code_int] :
( ( plus_p1446045655de_int @ M_8 @ ( times_123202395de_int @ A_257 @ M_8 ) )
= ( times_123202395de_int @ ( plus_p1446045655de_int @ A_257 @ one_on1684967323de_int ) @ M_8 ) ) ).
thf(fact_595_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_8: complex,A_257: complex] :
( ( plus_plus_complex @ M_8 @ ( times_times_complex @ A_257 @ M_8 ) )
= ( times_times_complex @ ( plus_plus_complex @ A_257 @ one_one_complex ) @ M_8 ) ) ).
thf(fact_596_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_8: code_code_numeral,A_257: code_code_numeral] :
( ( plus_p1627245867umeral @ M_8 @ ( times_1655362735umeral @ A_257 @ M_8 ) )
= ( times_1655362735umeral @ ( plus_p1627245867umeral @ A_257 @ one_on1645066479umeral ) @ M_8 ) ) ).
thf(fact_597_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_8: nat,A_257: nat] :
( ( plus_plus_nat @ M_8 @ ( times_times_nat @ A_257 @ M_8 ) )
= ( times_times_nat @ ( plus_plus_nat @ A_257 @ one_one_nat ) @ M_8 ) ) ).
thf(fact_598_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_8: real,A_257: real] :
( ( plus_plus_real @ M_8 @ ( times_times_real @ A_257 @ M_8 ) )
= ( times_times_real @ ( plus_plus_real @ A_257 @ one_one_real ) @ M_8 ) ) ).
thf(fact_599_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_8: int,A_257: int] :
( ( plus_plus_int @ M_8 @ ( times_times_int @ A_257 @ M_8 ) )
= ( times_times_int @ ( plus_plus_int @ A_257 @ one_one_int ) @ M_8 ) ) ).
thf(fact_600_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_256: rat,M_7: rat] :
( ( plus_plus_rat @ ( times_times_rat @ A_256 @ M_7 ) @ M_7 )
= ( times_times_rat @ ( plus_plus_rat @ A_256 @ one_one_rat ) @ M_7 ) ) ).
thf(fact_601_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_256: quickcheck_code_int,M_7: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( times_123202395de_int @ A_256 @ M_7 ) @ M_7 )
= ( times_123202395de_int @ ( plus_p1446045655de_int @ A_256 @ one_on1684967323de_int ) @ M_7 ) ) ).
thf(fact_602_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_256: complex,M_7: complex] :
( ( plus_plus_complex @ ( times_times_complex @ A_256 @ M_7 ) @ M_7 )
= ( times_times_complex @ ( plus_plus_complex @ A_256 @ one_one_complex ) @ M_7 ) ) ).
thf(fact_603_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_256: code_code_numeral,M_7: code_code_numeral] :
( ( plus_p1627245867umeral @ ( times_1655362735umeral @ A_256 @ M_7 ) @ M_7 )
= ( times_1655362735umeral @ ( plus_p1627245867umeral @ A_256 @ one_on1645066479umeral ) @ M_7 ) ) ).
thf(fact_604_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_256: nat,M_7: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A_256 @ M_7 ) @ M_7 )
= ( times_times_nat @ ( plus_plus_nat @ A_256 @ one_one_nat ) @ M_7 ) ) ).
thf(fact_605_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_256: real,M_7: real] :
( ( plus_plus_real @ ( times_times_real @ A_256 @ M_7 ) @ M_7 )
= ( times_times_real @ ( plus_plus_real @ A_256 @ one_one_real ) @ M_7 ) ) ).
thf(fact_606_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_256: int,M_7: int] :
( ( plus_plus_int @ ( times_times_int @ A_256 @ M_7 ) @ M_7 )
= ( times_times_int @ ( plus_plus_int @ A_256 @ one_one_int ) @ M_7 ) ) ).
thf(fact_607_curryD,axiom,
! [F_8: produc975137661_int_o > $o,A_255: int > $o,B_183: int > $o] :
( ( produc144226034nt_o_o @ F_8 @ A_255 @ B_183 )
=> ( F_8 @ ( produc398918003_int_o @ A_255 @ B_183 ) ) ) ).
thf(fact_608_curryD,axiom,
! [F_8: produc1137372701nt_int > $o,A_255: product_prod_int_int,B_183: product_prod_int_int] :
( ( produc1384121810_int_o @ F_8 @ A_255 @ B_183 )
=> ( F_8 @ ( produc883642259nt_int @ A_255 @ B_183 ) ) ) ).
thf(fact_609_curryD,axiom,
! [F_8: produc167071911de_int > $o,A_255: quickcheck_code_int,B_183: quickcheck_code_int] :
( ( produc1585697102_int_o @ F_8 @ A_255 @ B_183 )
=> ( F_8 @ ( produc1318306967de_int @ A_255 @ B_183 ) ) ) ).
thf(fact_610_curryD,axiom,
! [F_8: produc1359518119umeral > $o,A_255: code_code_numeral,B_183: code_code_numeral] :
( ( produc350732110eral_o @ F_8 @ A_255 @ B_183 )
=> ( F_8 @ ( produc2136830103umeral @ A_255 @ B_183 ) ) ) ).
thf(fact_611_curryD,axiom,
! [F_8: produc914805421l_real > $o,A_255: real,B_183: real] :
( ( produc1509576130real_o @ F_8 @ A_255 @ B_183 )
=> ( F_8 @ ( produc865579683l_real @ A_255 @ B_183 ) ) ) ).
thf(fact_612_curryD,axiom,
! [F_8: produc1322466333at_nat > $o,A_255: product_prod_nat_nat,B_183: product_prod_nat_nat] :
( ( produc435992530_nat_o @ F_8 @ A_255 @ B_183 )
=> ( F_8 @ ( produc494345619at_nat @ A_255 @ B_183 ) ) ) ).
thf(fact_613_curryD,axiom,
! [F_8: product_prod_nat_nat > $o,A_255: nat,B_183: nat] :
( ( produc764619086_nat_o @ F_8 @ A_255 @ B_183 )
=> ( F_8 @ ( product_Pair_nat_nat @ A_255 @ B_183 ) ) ) ).
thf(fact_614_curryD,axiom,
! [F_8: produc393999548nt_int > $o,A_255: int,B_183: product_prod_int_int] :
( ( produc122891093_int_o @ F_8 @ A_255 @ B_183 )
=> ( F_8 @ ( produc282740534nt_int @ A_255 @ B_183 ) ) ) ).
thf(fact_615_curryD,axiom,
! [F_8: product_prod_int_int > $o,A_255: int,B_183: int] :
( ( produc176579150_int_o @ F_8 @ A_255 @ B_183 )
=> ( F_8 @ ( product_Pair_int_int @ A_255 @ B_183 ) ) ) ).
thf(fact_616_order__refl,axiom,
! [X_120: rat] : ( ord_less_eq_rat @ X_120 @ X_120 ) ).
thf(fact_617_order__refl,axiom,
! [X_120: int > $o] : ( ord_less_eq_int_o @ X_120 @ X_120 ) ).
thf(fact_618_order__refl,axiom,
! [X_120: nat > $o] : ( ord_less_eq_nat_o @ X_120 @ X_120 ) ).
thf(fact_619_order__refl,axiom,
! [X_120: quickcheck_code_int] : ( ord_le258702272de_int @ X_120 @ X_120 ) ).
thf(fact_620_order__refl,axiom,
! [X_120: nat] : ( ord_less_eq_nat @ X_120 @ X_120 ) ).
thf(fact_621_order__refl,axiom,
! [X_120: real] : ( ord_less_eq_real @ X_120 @ X_120 ) ).
thf(fact_622_order__refl,axiom,
! [X_120: int] : ( ord_less_eq_int @ X_120 @ X_120 ) ).
thf(fact_623_xt1_I8_J,axiom,
! [Z_18: rat,Y_78: rat,X_119: rat] :
( ( ord_less_eq_rat @ Y_78 @ X_119 )
=> ( ( ord_less_rat @ Z_18 @ Y_78 )
=> ( ord_less_rat @ Z_18 @ X_119 ) ) ) ).
thf(fact_624_xt1_I8_J,axiom,
! [Z_18: int > $o,Y_78: int > $o,X_119: int > $o] :
( ( ord_less_eq_int_o @ Y_78 @ X_119 )
=> ( ( ord_less_int_o @ Z_18 @ Y_78 )
=> ( ord_less_int_o @ Z_18 @ X_119 ) ) ) ).
thf(fact_625_xt1_I8_J,axiom,
! [Z_18: nat > $o,Y_78: nat > $o,X_119: nat > $o] :
( ( ord_less_eq_nat_o @ Y_78 @ X_119 )
=> ( ( ord_less_nat_o @ Z_18 @ Y_78 )
=> ( ord_less_nat_o @ Z_18 @ X_119 ) ) ) ).
thf(fact_626_xt1_I8_J,axiom,
! [Z_18: quickcheck_code_int,Y_78: quickcheck_code_int,X_119: quickcheck_code_int] :
( ( ord_le258702272de_int @ Y_78 @ X_119 )
=> ( ( ord_le1860547276de_int @ Z_18 @ Y_78 )
=> ( ord_le1860547276de_int @ Z_18 @ X_119 ) ) ) ).
thf(fact_627_xt1_I8_J,axiom,
! [Z_18: nat,Y_78: nat,X_119: nat] :
( ( ord_less_eq_nat @ Y_78 @ X_119 )
=> ( ( ord_less_nat @ Z_18 @ Y_78 )
=> ( ord_less_nat @ Z_18 @ X_119 ) ) ) ).
thf(fact_628_xt1_I8_J,axiom,
! [Z_18: real,Y_78: real,X_119: real] :
( ( ord_less_eq_real @ Y_78 @ X_119 )
=> ( ( ord_less_real @ Z_18 @ Y_78 )
=> ( ord_less_real @ Z_18 @ X_119 ) ) ) ).
thf(fact_629_xt1_I8_J,axiom,
! [Z_18: int,Y_78: int,X_119: int] :
( ( ord_less_eq_int @ Y_78 @ X_119 )
=> ( ( ord_less_int @ Z_18 @ Y_78 )
=> ( ord_less_int @ Z_18 @ X_119 ) ) ) ).
thf(fact_630_order__le__less__trans,axiom,
! [Z_17: rat,X_118: rat,Y_77: rat] :
( ( ord_less_eq_rat @ X_118 @ Y_77 )
=> ( ( ord_less_rat @ Y_77 @ Z_17 )
=> ( ord_less_rat @ X_118 @ Z_17 ) ) ) ).
thf(fact_631_order__le__less__trans,axiom,
! [Z_17: int > $o,X_118: int > $o,Y_77: int > $o] :
( ( ord_less_eq_int_o @ X_118 @ Y_77 )
=> ( ( ord_less_int_o @ Y_77 @ Z_17 )
=> ( ord_less_int_o @ X_118 @ Z_17 ) ) ) ).
thf(fact_632_order__le__less__trans,axiom,
! [Z_17: nat > $o,X_118: nat > $o,Y_77: nat > $o] :
( ( ord_less_eq_nat_o @ X_118 @ Y_77 )
=> ( ( ord_less_nat_o @ Y_77 @ Z_17 )
=> ( ord_less_nat_o @ X_118 @ Z_17 ) ) ) ).
thf(fact_633_order__le__less__trans,axiom,
! [Z_17: quickcheck_code_int,X_118: quickcheck_code_int,Y_77: quickcheck_code_int] :
( ( ord_le258702272de_int @ X_118 @ Y_77 )
=> ( ( ord_le1860547276de_int @ Y_77 @ Z_17 )
=> ( ord_le1860547276de_int @ X_118 @ Z_17 ) ) ) ).
thf(fact_634_order__le__less__trans,axiom,
! [Z_17: nat,X_118: nat,Y_77: nat] :
( ( ord_less_eq_nat @ X_118 @ Y_77 )
=> ( ( ord_less_nat @ Y_77 @ Z_17 )
=> ( ord_less_nat @ X_118 @ Z_17 ) ) ) ).
thf(fact_635_order__le__less__trans,axiom,
! [Z_17: real,X_118: real,Y_77: real] :
( ( ord_less_eq_real @ X_118 @ Y_77 )
=> ( ( ord_less_real @ Y_77 @ Z_17 )
=> ( ord_less_real @ X_118 @ Z_17 ) ) ) ).
thf(fact_636_order__le__less__trans,axiom,
! [Z_17: int,X_118: int,Y_77: int] :
( ( ord_less_eq_int @ X_118 @ Y_77 )
=> ( ( ord_less_int @ Y_77 @ Z_17 )
=> ( ord_less_int @ X_118 @ Z_17 ) ) ) ).
thf(fact_637_xt1_I7_J,axiom,
! [Z_16: rat,Y_76: rat,X_117: rat] :
( ( ord_less_rat @ Y_76 @ X_117 )
=> ( ( ord_less_eq_rat @ Z_16 @ Y_76 )
=> ( ord_less_rat @ Z_16 @ X_117 ) ) ) ).
thf(fact_638_xt1_I7_J,axiom,
! [Z_16: int > $o,Y_76: int > $o,X_117: int > $o] :
( ( ord_less_int_o @ Y_76 @ X_117 )
=> ( ( ord_less_eq_int_o @ Z_16 @ Y_76 )
=> ( ord_less_int_o @ Z_16 @ X_117 ) ) ) ).
thf(fact_639_xt1_I7_J,axiom,
! [Z_16: nat > $o,Y_76: nat > $o,X_117: nat > $o] :
( ( ord_less_nat_o @ Y_76 @ X_117 )
=> ( ( ord_less_eq_nat_o @ Z_16 @ Y_76 )
=> ( ord_less_nat_o @ Z_16 @ X_117 ) ) ) ).
thf(fact_640_xt1_I7_J,axiom,
! [Z_16: quickcheck_code_int,Y_76: quickcheck_code_int,X_117: quickcheck_code_int] :
( ( ord_le1860547276de_int @ Y_76 @ X_117 )
=> ( ( ord_le258702272de_int @ Z_16 @ Y_76 )
=> ( ord_le1860547276de_int @ Z_16 @ X_117 ) ) ) ).
thf(fact_641_xt1_I7_J,axiom,
! [Z_16: nat,Y_76: nat,X_117: nat] :
( ( ord_less_nat @ Y_76 @ X_117 )
=> ( ( ord_less_eq_nat @ Z_16 @ Y_76 )
=> ( ord_less_nat @ Z_16 @ X_117 ) ) ) ).
thf(fact_642_xt1_I7_J,axiom,
! [Z_16: real,Y_76: real,X_117: real] :
( ( ord_less_real @ Y_76 @ X_117 )
=> ( ( ord_less_eq_real @ Z_16 @ Y_76 )
=> ( ord_less_real @ Z_16 @ X_117 ) ) ) ).
thf(fact_643_xt1_I7_J,axiom,
! [Z_16: int,Y_76: int,X_117: int] :
( ( ord_less_int @ Y_76 @ X_117 )
=> ( ( ord_less_eq_int @ Z_16 @ Y_76 )
=> ( ord_less_int @ Z_16 @ X_117 ) ) ) ).
thf(fact_644_order__less__le__trans,axiom,
! [Z_15: rat,X_116: rat,Y_75: rat] :
( ( ord_less_rat @ X_116 @ Y_75 )
=> ( ( ord_less_eq_rat @ Y_75 @ Z_15 )
=> ( ord_less_rat @ X_116 @ Z_15 ) ) ) ).
thf(fact_645_order__less__le__trans,axiom,
! [Z_15: int > $o,X_116: int > $o,Y_75: int > $o] :
( ( ord_less_int_o @ X_116 @ Y_75 )
=> ( ( ord_less_eq_int_o @ Y_75 @ Z_15 )
=> ( ord_less_int_o @ X_116 @ Z_15 ) ) ) ).
thf(fact_646_order__less__le__trans,axiom,
! [Z_15: nat > $o,X_116: nat > $o,Y_75: nat > $o] :
( ( ord_less_nat_o @ X_116 @ Y_75 )
=> ( ( ord_less_eq_nat_o @ Y_75 @ Z_15 )
=> ( ord_less_nat_o @ X_116 @ Z_15 ) ) ) ).
thf(fact_647_order__less__le__trans,axiom,
! [Z_15: quickcheck_code_int,X_116: quickcheck_code_int,Y_75: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_116 @ Y_75 )
=> ( ( ord_le258702272de_int @ Y_75 @ Z_15 )
=> ( ord_le1860547276de_int @ X_116 @ Z_15 ) ) ) ).
thf(fact_648_order__less__le__trans,axiom,
! [Z_15: nat,X_116: nat,Y_75: nat] :
( ( ord_less_nat @ X_116 @ Y_75 )
=> ( ( ord_less_eq_nat @ Y_75 @ Z_15 )
=> ( ord_less_nat @ X_116 @ Z_15 ) ) ) ).
thf(fact_649_order__less__le__trans,axiom,
! [Z_15: real,X_116: real,Y_75: real] :
( ( ord_less_real @ X_116 @ Y_75 )
=> ( ( ord_less_eq_real @ Y_75 @ Z_15 )
=> ( ord_less_real @ X_116 @ Z_15 ) ) ) ).
thf(fact_650_order__less__le__trans,axiom,
! [Z_15: int,X_116: int,Y_75: int] :
( ( ord_less_int @ X_116 @ Y_75 )
=> ( ( ord_less_eq_int @ Y_75 @ Z_15 )
=> ( ord_less_int @ X_116 @ Z_15 ) ) ) ).
thf(fact_651_xt1_I11_J,axiom,
! [B_182: rat,A_254: rat] :
( ( ord_less_eq_rat @ B_182 @ A_254 )
=> ( ( A_254 != B_182 )
=> ( ord_less_rat @ B_182 @ A_254 ) ) ) ).
thf(fact_652_xt1_I11_J,axiom,
! [B_182: int > $o,A_254: int > $o] :
( ( ord_less_eq_int_o @ B_182 @ A_254 )
=> ( ( A_254 != B_182 )
=> ( ord_less_int_o @ B_182 @ A_254 ) ) ) ).
thf(fact_653_xt1_I11_J,axiom,
! [B_182: nat > $o,A_254: nat > $o] :
( ( ord_less_eq_nat_o @ B_182 @ A_254 )
=> ( ( A_254 != B_182 )
=> ( ord_less_nat_o @ B_182 @ A_254 ) ) ) ).
thf(fact_654_xt1_I11_J,axiom,
! [B_182: quickcheck_code_int,A_254: quickcheck_code_int] :
( ( ord_le258702272de_int @ B_182 @ A_254 )
=> ( ( A_254 != B_182 )
=> ( ord_le1860547276de_int @ B_182 @ A_254 ) ) ) ).
thf(fact_655_xt1_I11_J,axiom,
! [B_182: nat,A_254: nat] :
( ( ord_less_eq_nat @ B_182 @ A_254 )
=> ( ( A_254 != B_182 )
=> ( ord_less_nat @ B_182 @ A_254 ) ) ) ).
thf(fact_656_xt1_I11_J,axiom,
! [B_182: real,A_254: real] :
( ( ord_less_eq_real @ B_182 @ A_254 )
=> ( ( A_254 != B_182 )
=> ( ord_less_real @ B_182 @ A_254 ) ) ) ).
thf(fact_657_xt1_I11_J,axiom,
! [B_182: int,A_254: int] :
( ( ord_less_eq_int @ B_182 @ A_254 )
=> ( ( A_254 != B_182 )
=> ( ord_less_int @ B_182 @ A_254 ) ) ) ).
thf(fact_658_order__le__neq__trans,axiom,
! [A_253: rat,B_181: rat] :
( ( ord_less_eq_rat @ A_253 @ B_181 )
=> ( ( A_253 != B_181 )
=> ( ord_less_rat @ A_253 @ B_181 ) ) ) ).
thf(fact_659_order__le__neq__trans,axiom,
! [A_253: int > $o,B_181: int > $o] :
( ( ord_less_eq_int_o @ A_253 @ B_181 )
=> ( ( A_253 != B_181 )
=> ( ord_less_int_o @ A_253 @ B_181 ) ) ) ).
thf(fact_660_order__le__neq__trans,axiom,
! [A_253: nat > $o,B_181: nat > $o] :
( ( ord_less_eq_nat_o @ A_253 @ B_181 )
=> ( ( A_253 != B_181 )
=> ( ord_less_nat_o @ A_253 @ B_181 ) ) ) ).
thf(fact_661_order__le__neq__trans,axiom,
! [A_253: quickcheck_code_int,B_181: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_253 @ B_181 )
=> ( ( A_253 != B_181 )
=> ( ord_le1860547276de_int @ A_253 @ B_181 ) ) ) ).
thf(fact_662_order__le__neq__trans,axiom,
! [A_253: nat,B_181: nat] :
( ( ord_less_eq_nat @ A_253 @ B_181 )
=> ( ( A_253 != B_181 )
=> ( ord_less_nat @ A_253 @ B_181 ) ) ) ).
thf(fact_663_order__le__neq__trans,axiom,
! [A_253: real,B_181: real] :
( ( ord_less_eq_real @ A_253 @ B_181 )
=> ( ( A_253 != B_181 )
=> ( ord_less_real @ A_253 @ B_181 ) ) ) ).
thf(fact_664_order__le__neq__trans,axiom,
! [A_253: int,B_181: int] :
( ( ord_less_eq_int @ A_253 @ B_181 )
=> ( ( A_253 != B_181 )
=> ( ord_less_int @ A_253 @ B_181 ) ) ) ).
thf(fact_665_order__le__imp__less__or__eq,axiom,
! [X_115: rat,Y_74: rat] :
( ( ord_less_eq_rat @ X_115 @ Y_74 )
=> ( ( ord_less_rat @ X_115 @ Y_74 )
| ( X_115 = Y_74 ) ) ) ).
thf(fact_666_order__le__imp__less__or__eq,axiom,
! [X_115: int > $o,Y_74: int > $o] :
( ( ord_less_eq_int_o @ X_115 @ Y_74 )
=> ( ( ord_less_int_o @ X_115 @ Y_74 )
| ( X_115 = Y_74 ) ) ) ).
thf(fact_667_order__le__imp__less__or__eq,axiom,
! [X_115: nat > $o,Y_74: nat > $o] :
( ( ord_less_eq_nat_o @ X_115 @ Y_74 )
=> ( ( ord_less_nat_o @ X_115 @ Y_74 )
| ( X_115 = Y_74 ) ) ) ).
thf(fact_668_order__le__imp__less__or__eq,axiom,
! [X_115: quickcheck_code_int,Y_74: quickcheck_code_int] :
( ( ord_le258702272de_int @ X_115 @ Y_74 )
=> ( ( ord_le1860547276de_int @ X_115 @ Y_74 )
| ( X_115 = Y_74 ) ) ) ).
thf(fact_669_order__le__imp__less__or__eq,axiom,
! [X_115: nat,Y_74: nat] :
( ( ord_less_eq_nat @ X_115 @ Y_74 )
=> ( ( ord_less_nat @ X_115 @ Y_74 )
| ( X_115 = Y_74 ) ) ) ).
thf(fact_670_order__le__imp__less__or__eq,axiom,
! [X_115: real,Y_74: real] :
( ( ord_less_eq_real @ X_115 @ Y_74 )
=> ( ( ord_less_real @ X_115 @ Y_74 )
| ( X_115 = Y_74 ) ) ) ).
thf(fact_671_order__le__imp__less__or__eq,axiom,
! [X_115: int,Y_74: int] :
( ( ord_less_eq_int @ X_115 @ Y_74 )
=> ( ( ord_less_int @ X_115 @ Y_74 )
| ( X_115 = Y_74 ) ) ) ).
thf(fact_672_linorder__antisym__conv2,axiom,
! [X_114: rat,Y_73: rat] :
( ( ord_less_eq_rat @ X_114 @ Y_73 )
=> ( ~ ( ord_less_rat @ X_114 @ Y_73 )
<=> ( X_114 = Y_73 ) ) ) ).
thf(fact_673_linorder__antisym__conv2,axiom,
! [X_114: quickcheck_code_int,Y_73: quickcheck_code_int] :
( ( ord_le258702272de_int @ X_114 @ Y_73 )
=> ( ~ ( ord_le1860547276de_int @ X_114 @ Y_73 )
<=> ( X_114 = Y_73 ) ) ) ).
thf(fact_674_linorder__antisym__conv2,axiom,
! [X_114: nat,Y_73: nat] :
( ( ord_less_eq_nat @ X_114 @ Y_73 )
=> ( ~ ( ord_less_nat @ X_114 @ Y_73 )
<=> ( X_114 = Y_73 ) ) ) ).
thf(fact_675_linorder__antisym__conv2,axiom,
! [X_114: real,Y_73: real] :
( ( ord_less_eq_real @ X_114 @ Y_73 )
=> ( ~ ( ord_less_real @ X_114 @ Y_73 )
<=> ( X_114 = Y_73 ) ) ) ).
thf(fact_676_linorder__antisym__conv2,axiom,
! [X_114: int,Y_73: int] :
( ( ord_less_eq_int @ X_114 @ Y_73 )
=> ( ~ ( ord_less_int @ X_114 @ Y_73 )
<=> ( X_114 = Y_73 ) ) ) ).
thf(fact_677_order__less__imp__le,axiom,
! [X_113: rat,Y_72: rat] :
( ( ord_less_rat @ X_113 @ Y_72 )
=> ( ord_less_eq_rat @ X_113 @ Y_72 ) ) ).
thf(fact_678_order__less__imp__le,axiom,
! [X_113: int > $o,Y_72: int > $o] :
( ( ord_less_int_o @ X_113 @ Y_72 )
=> ( ord_less_eq_int_o @ X_113 @ Y_72 ) ) ).
thf(fact_679_order__less__imp__le,axiom,
! [X_113: nat > $o,Y_72: nat > $o] :
( ( ord_less_nat_o @ X_113 @ Y_72 )
=> ( ord_less_eq_nat_o @ X_113 @ Y_72 ) ) ).
thf(fact_680_order__less__imp__le,axiom,
! [X_113: quickcheck_code_int,Y_72: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_113 @ Y_72 )
=> ( ord_le258702272de_int @ X_113 @ Y_72 ) ) ).
thf(fact_681_order__less__imp__le,axiom,
! [X_113: nat,Y_72: nat] :
( ( ord_less_nat @ X_113 @ Y_72 )
=> ( ord_less_eq_nat @ X_113 @ Y_72 ) ) ).
thf(fact_682_order__less__imp__le,axiom,
! [X_113: real,Y_72: real] :
( ( ord_less_real @ X_113 @ Y_72 )
=> ( ord_less_eq_real @ X_113 @ Y_72 ) ) ).
thf(fact_683_order__less__imp__le,axiom,
! [X_113: int,Y_72: int] :
( ( ord_less_int @ X_113 @ Y_72 )
=> ( ord_less_eq_int @ X_113 @ Y_72 ) ) ).
thf(fact_684_leD,axiom,
! [Y_71: rat,X_112: rat] :
( ( ord_less_eq_rat @ Y_71 @ X_112 )
=> ~ ( ord_less_rat @ X_112 @ Y_71 ) ) ).
thf(fact_685_leD,axiom,
! [Y_71: quickcheck_code_int,X_112: quickcheck_code_int] :
( ( ord_le258702272de_int @ Y_71 @ X_112 )
=> ~ ( ord_le1860547276de_int @ X_112 @ Y_71 ) ) ).
thf(fact_686_leD,axiom,
! [Y_71: nat,X_112: nat] :
( ( ord_less_eq_nat @ Y_71 @ X_112 )
=> ~ ( ord_less_nat @ X_112 @ Y_71 ) ) ).
thf(fact_687_leD,axiom,
! [Y_71: real,X_112: real] :
( ( ord_less_eq_real @ Y_71 @ X_112 )
=> ~ ( ord_less_real @ X_112 @ Y_71 ) ) ).
thf(fact_688_leD,axiom,
! [Y_71: int,X_112: int] :
( ( ord_less_eq_int @ Y_71 @ X_112 )
=> ~ ( ord_less_int @ X_112 @ Y_71 ) ) ).
thf(fact_689_xt1_I12_J,axiom,
! [A_252: rat,B_180: rat] :
( ( A_252 != B_180 )
=> ( ( ord_less_eq_rat @ B_180 @ A_252 )
=> ( ord_less_rat @ B_180 @ A_252 ) ) ) ).
thf(fact_690_xt1_I12_J,axiom,
! [A_252: int > $o,B_180: int > $o] :
( ( A_252 != B_180 )
=> ( ( ord_less_eq_int_o @ B_180 @ A_252 )
=> ( ord_less_int_o @ B_180 @ A_252 ) ) ) ).
thf(fact_691_xt1_I12_J,axiom,
! [A_252: nat > $o,B_180: nat > $o] :
( ( A_252 != B_180 )
=> ( ( ord_less_eq_nat_o @ B_180 @ A_252 )
=> ( ord_less_nat_o @ B_180 @ A_252 ) ) ) ).
thf(fact_692_xt1_I12_J,axiom,
! [A_252: quickcheck_code_int,B_180: quickcheck_code_int] :
( ( A_252 != B_180 )
=> ( ( ord_le258702272de_int @ B_180 @ A_252 )
=> ( ord_le1860547276de_int @ B_180 @ A_252 ) ) ) ).
thf(fact_693_xt1_I12_J,axiom,
! [A_252: nat,B_180: nat] :
( ( A_252 != B_180 )
=> ( ( ord_less_eq_nat @ B_180 @ A_252 )
=> ( ord_less_nat @ B_180 @ A_252 ) ) ) ).
thf(fact_694_xt1_I12_J,axiom,
! [A_252: real,B_180: real] :
( ( A_252 != B_180 )
=> ( ( ord_less_eq_real @ B_180 @ A_252 )
=> ( ord_less_real @ B_180 @ A_252 ) ) ) ).
thf(fact_695_xt1_I12_J,axiom,
! [A_252: int,B_180: int] :
( ( A_252 != B_180 )
=> ( ( ord_less_eq_int @ B_180 @ A_252 )
=> ( ord_less_int @ B_180 @ A_252 ) ) ) ).
thf(fact_696_zle__refl,axiom,
! [W: int] : ( ord_less_eq_int @ W @ W ) ).
thf(fact_697_number__of__is__id,axiom,
! [K_1: int] :
( ( number_number_of_int @ K_1 )
= K_1 ) ).
thf(fact_698_zle__linear,axiom,
! [Z_1: int,W: int] :
( ( ord_less_eq_int @ Z_1 @ W )
| ( ord_less_eq_int @ W @ Z_1 ) ) ).
thf(fact_699_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_700_zless__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
thf(fact_701_less__number__of__int__code,axiom,
! [K_1: int,L: int] :
( ( ord_less_int @ ( number_number_of_int @ K_1 ) @ ( number_number_of_int @ L ) )
<=> ( ord_less_int @ K_1 @ L ) ) ).
thf(fact_702_less__eq__number__of__int__code,axiom,
! [K_1: int,L: int] :
( ( ord_less_eq_int @ ( number_number_of_int @ K_1 ) @ ( number_number_of_int @ L ) )
<=> ( ord_less_eq_int @ K_1 @ L ) ) ).
thf(fact_703_zle__trans,axiom,
! [K_1: int,I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K_1 )
=> ( ord_less_eq_int @ I @ K_1 ) ) ) ).
thf(fact_704_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_705_le__fun__def,axiom,
! [F_7: int > $o,G_5: int > $o] :
( ( ord_less_eq_int_o @ F_7 @ G_5 )
<=> ! [X_1: int] : ( ord_less_eq_o @ ( F_7 @ X_1 ) @ ( G_5 @ X_1 ) ) ) ).
thf(fact_706_le__fun__def,axiom,
! [F_7: nat > $o,G_5: nat > $o] :
( ( ord_less_eq_nat_o @ F_7 @ G_5 )
<=> ! [X_1: nat] : ( ord_less_eq_o @ ( F_7 @ X_1 ) @ ( G_5 @ X_1 ) ) ) ).
thf(fact_707_linorder__linear,axiom,
! [X_111: rat,Y_70: rat] :
( ( ord_less_eq_rat @ X_111 @ Y_70 )
| ( ord_less_eq_rat @ Y_70 @ X_111 ) ) ).
thf(fact_708_linorder__linear,axiom,
! [X_111: quickcheck_code_int,Y_70: quickcheck_code_int] :
( ( ord_le258702272de_int @ X_111 @ Y_70 )
| ( ord_le258702272de_int @ Y_70 @ X_111 ) ) ).
thf(fact_709_linorder__linear,axiom,
! [X_111: nat,Y_70: nat] :
( ( ord_less_eq_nat @ X_111 @ Y_70 )
| ( ord_less_eq_nat @ Y_70 @ X_111 ) ) ).
thf(fact_710_linorder__linear,axiom,
! [X_111: real,Y_70: real] :
( ( ord_less_eq_real @ X_111 @ Y_70 )
| ( ord_less_eq_real @ Y_70 @ X_111 ) ) ).
thf(fact_711_linorder__linear,axiom,
! [X_111: int,Y_70: int] :
( ( ord_less_eq_int @ X_111 @ Y_70 )
| ( ord_less_eq_int @ Y_70 @ X_111 ) ) ).
thf(fact_712_order__eq__iff,axiom,
! [X_110: rat,Y_69: rat] :
( ( X_110 = Y_69 )
<=> ( ( ord_less_eq_rat @ X_110 @ Y_69 )
& ( ord_less_eq_rat @ Y_69 @ X_110 ) ) ) ).
thf(fact_713_order__eq__iff,axiom,
! [X_110: int > $o,Y_69: int > $o] :
( ( X_110 = Y_69 )
<=> ( ( ord_less_eq_int_o @ X_110 @ Y_69 )
& ( ord_less_eq_int_o @ Y_69 @ X_110 ) ) ) ).
thf(fact_714_order__eq__iff,axiom,
! [X_110: nat > $o,Y_69: nat > $o] :
( ( X_110 = Y_69 )
<=> ( ( ord_less_eq_nat_o @ X_110 @ Y_69 )
& ( ord_less_eq_nat_o @ Y_69 @ X_110 ) ) ) ).
thf(fact_715_order__eq__iff,axiom,
! [X_110: quickcheck_code_int,Y_69: quickcheck_code_int] :
( ( X_110 = Y_69 )
<=> ( ( ord_le258702272de_int @ X_110 @ Y_69 )
& ( ord_le258702272de_int @ Y_69 @ X_110 ) ) ) ).
thf(fact_716_order__eq__iff,axiom,
! [X_110: nat,Y_69: nat] :
( ( X_110 = Y_69 )
<=> ( ( ord_less_eq_nat @ X_110 @ Y_69 )
& ( ord_less_eq_nat @ Y_69 @ X_110 ) ) ) ).
thf(fact_717_order__eq__iff,axiom,
! [X_110: real,Y_69: real] :
( ( X_110 = Y_69 )
<=> ( ( ord_less_eq_real @ X_110 @ Y_69 )
& ( ord_less_eq_real @ Y_69 @ X_110 ) ) ) ).
thf(fact_718_order__eq__iff,axiom,
! [X_110: int,Y_69: int] :
( ( X_110 = Y_69 )
<=> ( ( ord_less_eq_int @ X_110 @ Y_69 )
& ( ord_less_eq_int @ Y_69 @ X_110 ) ) ) ).
thf(fact_719_order__eq__refl,axiom,
! [X_109: rat,Y_68: rat] :
( ( X_109 = Y_68 )
=> ( ord_less_eq_rat @ X_109 @ Y_68 ) ) ).
thf(fact_720_order__eq__refl,axiom,
! [X_109: int > $o,Y_68: int > $o] :
( ( X_109 = Y_68 )
=> ( ord_less_eq_int_o @ X_109 @ Y_68 ) ) ).
thf(fact_721_order__eq__refl,axiom,
! [X_109: nat > $o,Y_68: nat > $o] :
( ( X_109 = Y_68 )
=> ( ord_less_eq_nat_o @ X_109 @ Y_68 ) ) ).
thf(fact_722_order__eq__refl,axiom,
! [X_109: quickcheck_code_int,Y_68: quickcheck_code_int] :
( ( X_109 = Y_68 )
=> ( ord_le258702272de_int @ X_109 @ Y_68 ) ) ).
thf(fact_723_order__eq__refl,axiom,
! [X_109: nat,Y_68: nat] :
( ( X_109 = Y_68 )
=> ( ord_less_eq_nat @ X_109 @ Y_68 ) ) ).
thf(fact_724_order__eq__refl,axiom,
! [X_109: real,Y_68: real] :
( ( X_109 = Y_68 )
=> ( ord_less_eq_real @ X_109 @ Y_68 ) ) ).
thf(fact_725_order__eq__refl,axiom,
! [X_109: int,Y_68: int] :
( ( X_109 = Y_68 )
=> ( ord_less_eq_int @ X_109 @ Y_68 ) ) ).
thf(fact_726_le__funD,axiom,
! [X_108: 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_108 ) @ ( G_4 @ X_108 ) ) ) ).
thf(fact_727_le__funD,axiom,
! [X_108: 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_108 ) @ ( G_4 @ X_108 ) ) ) ).
thf(fact_728_order__antisym__conv,axiom,
! [Y_67: rat,X_107: rat] :
( ( ord_less_eq_rat @ Y_67 @ X_107 )
=> ( ( ord_less_eq_rat @ X_107 @ Y_67 )
<=> ( X_107 = Y_67 ) ) ) ).
thf(fact_729_order__antisym__conv,axiom,
! [Y_67: int > $o,X_107: int > $o] :
( ( ord_less_eq_int_o @ Y_67 @ X_107 )
=> ( ( ord_less_eq_int_o @ X_107 @ Y_67 )
<=> ( X_107 = Y_67 ) ) ) ).
thf(fact_730_order__antisym__conv,axiom,
! [Y_67: nat > $o,X_107: nat > $o] :
( ( ord_less_eq_nat_o @ Y_67 @ X_107 )
=> ( ( ord_less_eq_nat_o @ X_107 @ Y_67 )
<=> ( X_107 = Y_67 ) ) ) ).
thf(fact_731_order__antisym__conv,axiom,
! [Y_67: quickcheck_code_int,X_107: quickcheck_code_int] :
( ( ord_le258702272de_int @ Y_67 @ X_107 )
=> ( ( ord_le258702272de_int @ X_107 @ Y_67 )
<=> ( X_107 = Y_67 ) ) ) ).
thf(fact_732_order__antisym__conv,axiom,
! [Y_67: nat,X_107: nat] :
( ( ord_less_eq_nat @ Y_67 @ X_107 )
=> ( ( ord_less_eq_nat @ X_107 @ Y_67 )
<=> ( X_107 = Y_67 ) ) ) ).
thf(fact_733_order__antisym__conv,axiom,
! [Y_67: real,X_107: real] :
( ( ord_less_eq_real @ Y_67 @ X_107 )
=> ( ( ord_less_eq_real @ X_107 @ Y_67 )
<=> ( X_107 = Y_67 ) ) ) ).
thf(fact_734_order__antisym__conv,axiom,
! [Y_67: int,X_107: int] :
( ( ord_less_eq_int @ Y_67 @ X_107 )
=> ( ( ord_less_eq_int @ X_107 @ Y_67 )
<=> ( X_107 = Y_67 ) ) ) ).
thf(fact_735_ord__eq__le__trans,axiom,
! [C_76: rat,A_251: rat,B_179: rat] :
( ( A_251 = B_179 )
=> ( ( ord_less_eq_rat @ B_179 @ C_76 )
=> ( ord_less_eq_rat @ A_251 @ C_76 ) ) ) ).
thf(fact_736_ord__eq__le__trans,axiom,
! [C_76: int > $o,A_251: int > $o,B_179: int > $o] :
( ( A_251 = B_179 )
=> ( ( ord_less_eq_int_o @ B_179 @ C_76 )
=> ( ord_less_eq_int_o @ A_251 @ C_76 ) ) ) ).
thf(fact_737_ord__eq__le__trans,axiom,
! [C_76: nat > $o,A_251: nat > $o,B_179: nat > $o] :
( ( A_251 = B_179 )
=> ( ( ord_less_eq_nat_o @ B_179 @ C_76 )
=> ( ord_less_eq_nat_o @ A_251 @ C_76 ) ) ) ).
thf(fact_738_ord__eq__le__trans,axiom,
! [C_76: quickcheck_code_int,A_251: quickcheck_code_int,B_179: quickcheck_code_int] :
( ( A_251 = B_179 )
=> ( ( ord_le258702272de_int @ B_179 @ C_76 )
=> ( ord_le258702272de_int @ A_251 @ C_76 ) ) ) ).
thf(fact_739_ord__eq__le__trans,axiom,
! [C_76: nat,A_251: nat,B_179: nat] :
( ( A_251 = B_179 )
=> ( ( ord_less_eq_nat @ B_179 @ C_76 )
=> ( ord_less_eq_nat @ A_251 @ C_76 ) ) ) ).
thf(fact_740_ord__eq__le__trans,axiom,
! [C_76: real,A_251: real,B_179: real] :
( ( A_251 = B_179 )
=> ( ( ord_less_eq_real @ B_179 @ C_76 )
=> ( ord_less_eq_real @ A_251 @ C_76 ) ) ) ).
thf(fact_741_ord__eq__le__trans,axiom,
! [C_76: int,A_251: int,B_179: int] :
( ( A_251 = B_179 )
=> ( ( ord_less_eq_int @ B_179 @ C_76 )
=> ( ord_less_eq_int @ A_251 @ C_76 ) ) ) ).
thf(fact_742_xt1_I3_J,axiom,
! [C_75: rat,A_250: rat,B_178: rat] :
( ( A_250 = B_178 )
=> ( ( ord_less_eq_rat @ C_75 @ B_178 )
=> ( ord_less_eq_rat @ C_75 @ A_250 ) ) ) ).
thf(fact_743_xt1_I3_J,axiom,
! [C_75: int > $o,A_250: int > $o,B_178: int > $o] :
( ( A_250 = B_178 )
=> ( ( ord_less_eq_int_o @ C_75 @ B_178 )
=> ( ord_less_eq_int_o @ C_75 @ A_250 ) ) ) ).
thf(fact_744_xt1_I3_J,axiom,
! [C_75: nat > $o,A_250: nat > $o,B_178: nat > $o] :
( ( A_250 = B_178 )
=> ( ( ord_less_eq_nat_o @ C_75 @ B_178 )
=> ( ord_less_eq_nat_o @ C_75 @ A_250 ) ) ) ).
thf(fact_745_xt1_I3_J,axiom,
! [C_75: quickcheck_code_int,A_250: quickcheck_code_int,B_178: quickcheck_code_int] :
( ( A_250 = B_178 )
=> ( ( ord_le258702272de_int @ C_75 @ B_178 )
=> ( ord_le258702272de_int @ C_75 @ A_250 ) ) ) ).
thf(fact_746_xt1_I3_J,axiom,
! [C_75: nat,A_250: nat,B_178: nat] :
( ( A_250 = B_178 )
=> ( ( ord_less_eq_nat @ C_75 @ B_178 )
=> ( ord_less_eq_nat @ C_75 @ A_250 ) ) ) ).
thf(fact_747_xt1_I3_J,axiom,
! [C_75: real,A_250: real,B_178: real] :
( ( A_250 = B_178 )
=> ( ( ord_less_eq_real @ C_75 @ B_178 )
=> ( ord_less_eq_real @ C_75 @ A_250 ) ) ) ).
thf(fact_748_xt1_I3_J,axiom,
! [C_75: int,A_250: int,B_178: int] :
( ( A_250 = B_178 )
=> ( ( ord_less_eq_int @ C_75 @ B_178 )
=> ( ord_less_eq_int @ C_75 @ A_250 ) ) ) ).
thf(fact_749_ord__le__eq__trans,axiom,
! [C_74: rat,A_249: rat,B_177: rat] :
( ( ord_less_eq_rat @ A_249 @ B_177 )
=> ( ( B_177 = C_74 )
=> ( ord_less_eq_rat @ A_249 @ C_74 ) ) ) ).
thf(fact_750_ord__le__eq__trans,axiom,
! [C_74: int > $o,A_249: int > $o,B_177: int > $o] :
( ( ord_less_eq_int_o @ A_249 @ B_177 )
=> ( ( B_177 = C_74 )
=> ( ord_less_eq_int_o @ A_249 @ C_74 ) ) ) ).
thf(fact_751_ord__le__eq__trans,axiom,
! [C_74: nat > $o,A_249: nat > $o,B_177: nat > $o] :
( ( ord_less_eq_nat_o @ A_249 @ B_177 )
=> ( ( B_177 = C_74 )
=> ( ord_less_eq_nat_o @ A_249 @ C_74 ) ) ) ).
thf(fact_752_ord__le__eq__trans,axiom,
! [C_74: quickcheck_code_int,A_249: quickcheck_code_int,B_177: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_249 @ B_177 )
=> ( ( B_177 = C_74 )
=> ( ord_le258702272de_int @ A_249 @ C_74 ) ) ) ).
thf(fact_753_ord__le__eq__trans,axiom,
! [C_74: nat,A_249: nat,B_177: nat] :
( ( ord_less_eq_nat @ A_249 @ B_177 )
=> ( ( B_177 = C_74 )
=> ( ord_less_eq_nat @ A_249 @ C_74 ) ) ) ).
thf(fact_754_ord__le__eq__trans,axiom,
! [C_74: real,A_249: real,B_177: real] :
( ( ord_less_eq_real @ A_249 @ B_177 )
=> ( ( B_177 = C_74 )
=> ( ord_less_eq_real @ A_249 @ C_74 ) ) ) ).
thf(fact_755_ord__le__eq__trans,axiom,
! [C_74: int,A_249: int,B_177: int] :
( ( ord_less_eq_int @ A_249 @ B_177 )
=> ( ( B_177 = C_74 )
=> ( ord_less_eq_int @ A_249 @ C_74 ) ) ) ).
thf(fact_756_xt1_I4_J,axiom,
! [C_73: rat,B_176: rat,A_248: rat] :
( ( ord_less_eq_rat @ B_176 @ A_248 )
=> ( ( B_176 = C_73 )
=> ( ord_less_eq_rat @ C_73 @ A_248 ) ) ) ).
thf(fact_757_xt1_I4_J,axiom,
! [C_73: int > $o,B_176: int > $o,A_248: int > $o] :
( ( ord_less_eq_int_o @ B_176 @ A_248 )
=> ( ( B_176 = C_73 )
=> ( ord_less_eq_int_o @ C_73 @ A_248 ) ) ) ).
thf(fact_758_xt1_I4_J,axiom,
! [C_73: nat > $o,B_176: nat > $o,A_248: nat > $o] :
( ( ord_less_eq_nat_o @ B_176 @ A_248 )
=> ( ( B_176 = C_73 )
=> ( ord_less_eq_nat_o @ C_73 @ A_248 ) ) ) ).
thf(fact_759_xt1_I4_J,axiom,
! [C_73: quickcheck_code_int,B_176: quickcheck_code_int,A_248: quickcheck_code_int] :
( ( ord_le258702272de_int @ B_176 @ A_248 )
=> ( ( B_176 = C_73 )
=> ( ord_le258702272de_int @ C_73 @ A_248 ) ) ) ).
thf(fact_760_xt1_I4_J,axiom,
! [C_73: nat,B_176: nat,A_248: nat] :
( ( ord_less_eq_nat @ B_176 @ A_248 )
=> ( ( B_176 = C_73 )
=> ( ord_less_eq_nat @ C_73 @ A_248 ) ) ) ).
thf(fact_761_xt1_I4_J,axiom,
! [C_73: real,B_176: real,A_248: real] :
( ( ord_less_eq_real @ B_176 @ A_248 )
=> ( ( B_176 = C_73 )
=> ( ord_less_eq_real @ C_73 @ A_248 ) ) ) ).
thf(fact_762_xt1_I4_J,axiom,
! [C_73: int,B_176: int,A_248: int] :
( ( ord_less_eq_int @ B_176 @ A_248 )
=> ( ( B_176 = C_73 )
=> ( ord_less_eq_int @ C_73 @ A_248 ) ) ) ).
thf(fact_763_order__antisym,axiom,
! [X_106: rat,Y_66: rat] :
( ( ord_less_eq_rat @ X_106 @ Y_66 )
=> ( ( ord_less_eq_rat @ Y_66 @ X_106 )
=> ( X_106 = Y_66 ) ) ) ).
thf(fact_764_order__antisym,axiom,
! [X_106: int > $o,Y_66: int > $o] :
( ( ord_less_eq_int_o @ X_106 @ Y_66 )
=> ( ( ord_less_eq_int_o @ Y_66 @ X_106 )
=> ( X_106 = Y_66 ) ) ) ).
thf(fact_765_order__antisym,axiom,
! [X_106: nat > $o,Y_66: nat > $o] :
( ( ord_less_eq_nat_o @ X_106 @ Y_66 )
=> ( ( ord_less_eq_nat_o @ Y_66 @ X_106 )
=> ( X_106 = Y_66 ) ) ) ).
thf(fact_766_order__antisym,axiom,
! [X_106: quickcheck_code_int,Y_66: quickcheck_code_int] :
( ( ord_le258702272de_int @ X_106 @ Y_66 )
=> ( ( ord_le258702272de_int @ Y_66 @ X_106 )
=> ( X_106 = Y_66 ) ) ) ).
thf(fact_767_order__antisym,axiom,
! [X_106: nat,Y_66: nat] :
( ( ord_less_eq_nat @ X_106 @ Y_66 )
=> ( ( ord_less_eq_nat @ Y_66 @ X_106 )
=> ( X_106 = Y_66 ) ) ) ).
thf(fact_768_order__antisym,axiom,
! [X_106: real,Y_66: real] :
( ( ord_less_eq_real @ X_106 @ Y_66 )
=> ( ( ord_less_eq_real @ Y_66 @ X_106 )
=> ( X_106 = Y_66 ) ) ) ).
thf(fact_769_order__antisym,axiom,
! [X_106: int,Y_66: int] :
( ( ord_less_eq_int @ X_106 @ Y_66 )
=> ( ( ord_less_eq_int @ Y_66 @ X_106 )
=> ( X_106 = Y_66 ) ) ) ).
thf(fact_770_order__trans,axiom,
! [Z_14: rat,X_105: rat,Y_65: rat] :
( ( ord_less_eq_rat @ X_105 @ Y_65 )
=> ( ( ord_less_eq_rat @ Y_65 @ Z_14 )
=> ( ord_less_eq_rat @ X_105 @ Z_14 ) ) ) ).
thf(fact_771_order__trans,axiom,
! [Z_14: int > $o,X_105: int > $o,Y_65: int > $o] :
( ( ord_less_eq_int_o @ X_105 @ Y_65 )
=> ( ( ord_less_eq_int_o @ Y_65 @ Z_14 )
=> ( ord_less_eq_int_o @ X_105 @ Z_14 ) ) ) ).
thf(fact_772_order__trans,axiom,
! [Z_14: nat > $o,X_105: nat > $o,Y_65: nat > $o] :
( ( ord_less_eq_nat_o @ X_105 @ Y_65 )
=> ( ( ord_less_eq_nat_o @ Y_65 @ Z_14 )
=> ( ord_less_eq_nat_o @ X_105 @ Z_14 ) ) ) ).
thf(fact_773_order__trans,axiom,
! [Z_14: quickcheck_code_int,X_105: quickcheck_code_int,Y_65: quickcheck_code_int] :
( ( ord_le258702272de_int @ X_105 @ Y_65 )
=> ( ( ord_le258702272de_int @ Y_65 @ Z_14 )
=> ( ord_le258702272de_int @ X_105 @ Z_14 ) ) ) ).
thf(fact_774_order__trans,axiom,
! [Z_14: nat,X_105: nat,Y_65: nat] :
( ( ord_less_eq_nat @ X_105 @ Y_65 )
=> ( ( ord_less_eq_nat @ Y_65 @ Z_14 )
=> ( ord_less_eq_nat @ X_105 @ Z_14 ) ) ) ).
thf(fact_775_order__trans,axiom,
! [Z_14: real,X_105: real,Y_65: real] :
( ( ord_less_eq_real @ X_105 @ Y_65 )
=> ( ( ord_less_eq_real @ Y_65 @ Z_14 )
=> ( ord_less_eq_real @ X_105 @ Z_14 ) ) ) ).
thf(fact_776_order__trans,axiom,
! [Z_14: int,X_105: int,Y_65: int] :
( ( ord_less_eq_int @ X_105 @ Y_65 )
=> ( ( ord_less_eq_int @ Y_65 @ Z_14 )
=> ( ord_less_eq_int @ X_105 @ Z_14 ) ) ) ).
thf(fact_777_xt1_I5_J,axiom,
! [Y_64: rat,X_104: rat] :
( ( ord_less_eq_rat @ Y_64 @ X_104 )
=> ( ( ord_less_eq_rat @ X_104 @ Y_64 )
=> ( X_104 = Y_64 ) ) ) ).
thf(fact_778_xt1_I5_J,axiom,
! [Y_64: int > $o,X_104: int > $o] :
( ( ord_less_eq_int_o @ Y_64 @ X_104 )
=> ( ( ord_less_eq_int_o @ X_104 @ Y_64 )
=> ( X_104 = Y_64 ) ) ) ).
thf(fact_779_xt1_I5_J,axiom,
! [Y_64: nat > $o,X_104: nat > $o] :
( ( ord_less_eq_nat_o @ Y_64 @ X_104 )
=> ( ( ord_less_eq_nat_o @ X_104 @ Y_64 )
=> ( X_104 = Y_64 ) ) ) ).
thf(fact_780_xt1_I5_J,axiom,
! [Y_64: quickcheck_code_int,X_104: quickcheck_code_int] :
( ( ord_le258702272de_int @ Y_64 @ X_104 )
=> ( ( ord_le258702272de_int @ X_104 @ Y_64 )
=> ( X_104 = Y_64 ) ) ) ).
thf(fact_781_xt1_I5_J,axiom,
! [Y_64: nat,X_104: nat] :
( ( ord_less_eq_nat @ Y_64 @ X_104 )
=> ( ( ord_less_eq_nat @ X_104 @ Y_64 )
=> ( X_104 = Y_64 ) ) ) ).
thf(fact_782_xt1_I5_J,axiom,
! [Y_64: real,X_104: real] :
( ( ord_less_eq_real @ Y_64 @ X_104 )
=> ( ( ord_less_eq_real @ X_104 @ Y_64 )
=> ( X_104 = Y_64 ) ) ) ).
thf(fact_783_xt1_I5_J,axiom,
! [Y_64: int,X_104: int] :
( ( ord_less_eq_int @ Y_64 @ X_104 )
=> ( ( ord_less_eq_int @ X_104 @ Y_64 )
=> ( X_104 = Y_64 ) ) ) ).
thf(fact_784_xt1_I6_J,axiom,
! [Z_13: rat,Y_63: rat,X_103: rat] :
( ( ord_less_eq_rat @ Y_63 @ X_103 )
=> ( ( ord_less_eq_rat @ Z_13 @ Y_63 )
=> ( ord_less_eq_rat @ Z_13 @ X_103 ) ) ) ).
thf(fact_785_xt1_I6_J,axiom,
! [Z_13: int > $o,Y_63: int > $o,X_103: int > $o] :
( ( ord_less_eq_int_o @ Y_63 @ X_103 )
=> ( ( ord_less_eq_int_o @ Z_13 @ Y_63 )
=> ( ord_less_eq_int_o @ Z_13 @ X_103 ) ) ) ).
thf(fact_786_xt1_I6_J,axiom,
! [Z_13: nat > $o,Y_63: nat > $o,X_103: nat > $o] :
( ( ord_less_eq_nat_o @ Y_63 @ X_103 )
=> ( ( ord_less_eq_nat_o @ Z_13 @ Y_63 )
=> ( ord_less_eq_nat_o @ Z_13 @ X_103 ) ) ) ).
thf(fact_787_xt1_I6_J,axiom,
! [Z_13: quickcheck_code_int,Y_63: quickcheck_code_int,X_103: quickcheck_code_int] :
( ( ord_le258702272de_int @ Y_63 @ X_103 )
=> ( ( ord_le258702272de_int @ Z_13 @ Y_63 )
=> ( ord_le258702272de_int @ Z_13 @ X_103 ) ) ) ).
thf(fact_788_xt1_I6_J,axiom,
! [Z_13: nat,Y_63: nat,X_103: nat] :
( ( ord_less_eq_nat @ Y_63 @ X_103 )
=> ( ( ord_less_eq_nat @ Z_13 @ Y_63 )
=> ( ord_less_eq_nat @ Z_13 @ X_103 ) ) ) ).
thf(fact_789_xt1_I6_J,axiom,
! [Z_13: real,Y_63: real,X_103: real] :
( ( ord_less_eq_real @ Y_63 @ X_103 )
=> ( ( ord_less_eq_real @ Z_13 @ Y_63 )
=> ( ord_less_eq_real @ Z_13 @ X_103 ) ) ) ).
thf(fact_790_xt1_I6_J,axiom,
! [Z_13: int,Y_63: int,X_103: int] :
( ( ord_less_eq_int @ Y_63 @ X_103 )
=> ( ( ord_less_eq_int @ Z_13 @ Y_63 )
=> ( ord_less_eq_int @ Z_13 @ X_103 ) ) ) ).
thf(fact_791_le__funE,axiom,
! [X_102: 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_102 ) @ ( G_3 @ X_102 ) ) ) ).
thf(fact_792_le__funE,axiom,
! [X_102: 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_102 ) @ ( G_3 @ X_102 ) ) ) ).
thf(fact_793_linorder__le__cases,axiom,
! [X_101: rat,Y_62: rat] :
( ~ ( ord_less_eq_rat @ X_101 @ Y_62 )
=> ( ord_less_eq_rat @ Y_62 @ X_101 ) ) ).
thf(fact_794_linorder__le__cases,axiom,
! [X_101: quickcheck_code_int,Y_62: quickcheck_code_int] :
( ~ ( ord_le258702272de_int @ X_101 @ Y_62 )
=> ( ord_le258702272de_int @ Y_62 @ X_101 ) ) ).
thf(fact_795_linorder__le__cases,axiom,
! [X_101: nat,Y_62: nat] :
( ~ ( ord_less_eq_nat @ X_101 @ Y_62 )
=> ( ord_less_eq_nat @ Y_62 @ X_101 ) ) ).
thf(fact_796_linorder__le__cases,axiom,
! [X_101: real,Y_62: real] :
( ~ ( ord_less_eq_real @ X_101 @ Y_62 )
=> ( ord_less_eq_real @ Y_62 @ X_101 ) ) ).
thf(fact_797_linorder__le__cases,axiom,
! [X_101: int,Y_62: int] :
( ~ ( ord_less_eq_int @ X_101 @ Y_62 )
=> ( ord_less_eq_int @ Y_62 @ X_101 ) ) ).
thf(fact_798_order__less__irrefl,axiom,
! [X_100: rat] :
~ ( ord_less_rat @ X_100 @ X_100 ) ).
thf(fact_799_order__less__irrefl,axiom,
! [X_100: int > $o] :
~ ( ord_less_int_o @ X_100 @ X_100 ) ).
thf(fact_800_order__less__irrefl,axiom,
! [X_100: nat > $o] :
~ ( ord_less_nat_o @ X_100 @ X_100 ) ).
thf(fact_801_order__less__irrefl,axiom,
! [X_100: quickcheck_code_int] :
~ ( ord_le1860547276de_int @ X_100 @ X_100 ) ).
thf(fact_802_order__less__irrefl,axiom,
! [X_100: nat] :
~ ( ord_less_nat @ X_100 @ X_100 ) ).
thf(fact_803_order__less__irrefl,axiom,
! [X_100: real] :
~ ( ord_less_real @ X_100 @ X_100 ) ).
thf(fact_804_order__less__irrefl,axiom,
! [X_100: int] :
~ ( ord_less_int @ X_100 @ X_100 ) ).
thf(fact_805_linorder__neq__iff,axiom,
! [X_99: rat,Y_61: rat] :
( ( X_99 != Y_61 )
<=> ( ( ord_less_rat @ X_99 @ Y_61 )
| ( ord_less_rat @ Y_61 @ X_99 ) ) ) ).
thf(fact_806_linorder__neq__iff,axiom,
! [X_99: quickcheck_code_int,Y_61: quickcheck_code_int] :
( ( X_99 != Y_61 )
<=> ( ( ord_le1860547276de_int @ X_99 @ Y_61 )
| ( ord_le1860547276de_int @ Y_61 @ X_99 ) ) ) ).
thf(fact_807_linorder__neq__iff,axiom,
! [X_99: nat,Y_61: nat] :
( ( X_99 != Y_61 )
<=> ( ( ord_less_nat @ X_99 @ Y_61 )
| ( ord_less_nat @ Y_61 @ X_99 ) ) ) ).
thf(fact_808_linorder__neq__iff,axiom,
! [X_99: real,Y_61: real] :
( ( X_99 != Y_61 )
<=> ( ( ord_less_real @ X_99 @ Y_61 )
| ( ord_less_real @ Y_61 @ X_99 ) ) ) ).
thf(fact_809_linorder__neq__iff,axiom,
! [X_99: int,Y_61: int] :
( ( X_99 != Y_61 )
<=> ( ( ord_less_int @ X_99 @ Y_61 )
| ( ord_less_int @ Y_61 @ X_99 ) ) ) ).
thf(fact_810_not__less__iff__gr__or__eq,axiom,
! [X_98: rat,Y_60: rat] :
( ~ ( ord_less_rat @ X_98 @ Y_60 )
<=> ( ( ord_less_rat @ Y_60 @ X_98 )
| ( X_98 = Y_60 ) ) ) ).
thf(fact_811_not__less__iff__gr__or__eq,axiom,
! [X_98: quickcheck_code_int,Y_60: quickcheck_code_int] :
( ~ ( ord_le1860547276de_int @ X_98 @ Y_60 )
<=> ( ( ord_le1860547276de_int @ Y_60 @ X_98 )
| ( X_98 = Y_60 ) ) ) ).
thf(fact_812_not__less__iff__gr__or__eq,axiom,
! [X_98: nat,Y_60: nat] :
( ~ ( ord_less_nat @ X_98 @ Y_60 )
<=> ( ( ord_less_nat @ Y_60 @ X_98 )
| ( X_98 = Y_60 ) ) ) ).
thf(fact_813_not__less__iff__gr__or__eq,axiom,
! [X_98: real,Y_60: real] :
( ~ ( ord_less_real @ X_98 @ Y_60 )
<=> ( ( ord_less_real @ Y_60 @ X_98 )
| ( X_98 = Y_60 ) ) ) ).
thf(fact_814_not__less__iff__gr__or__eq,axiom,
! [X_98: int,Y_60: int] :
( ~ ( ord_less_int @ X_98 @ Y_60 )
<=> ( ( ord_less_int @ Y_60 @ X_98 )
| ( X_98 = Y_60 ) ) ) ).
thf(fact_815_linorder__less__linear,axiom,
! [X_97: rat,Y_59: rat] :
( ( ord_less_rat @ X_97 @ Y_59 )
| ( X_97 = Y_59 )
| ( ord_less_rat @ Y_59 @ X_97 ) ) ).
thf(fact_816_linorder__less__linear,axiom,
! [X_97: quickcheck_code_int,Y_59: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_97 @ Y_59 )
| ( X_97 = Y_59 )
| ( ord_le1860547276de_int @ Y_59 @ X_97 ) ) ).
thf(fact_817_linorder__less__linear,axiom,
! [X_97: nat,Y_59: nat] :
( ( ord_less_nat @ X_97 @ Y_59 )
| ( X_97 = Y_59 )
| ( ord_less_nat @ Y_59 @ X_97 ) ) ).
thf(fact_818_linorder__less__linear,axiom,
! [X_97: real,Y_59: real] :
( ( ord_less_real @ X_97 @ Y_59 )
| ( X_97 = Y_59 )
| ( ord_less_real @ Y_59 @ X_97 ) ) ).
thf(fact_819_linorder__less__linear,axiom,
! [X_97: int,Y_59: int] :
( ( ord_less_int @ X_97 @ Y_59 )
| ( X_97 = Y_59 )
| ( ord_less_int @ Y_59 @ X_97 ) ) ).
thf(fact_820_linorder__antisym__conv3,axiom,
! [Y_58: rat,X_96: rat] :
( ~ ( ord_less_rat @ Y_58 @ X_96 )
=> ( ~ ( ord_less_rat @ X_96 @ Y_58 )
<=> ( X_96 = Y_58 ) ) ) ).
thf(fact_821_linorder__antisym__conv3,axiom,
! [Y_58: quickcheck_code_int,X_96: quickcheck_code_int] :
( ~ ( ord_le1860547276de_int @ Y_58 @ X_96 )
=> ( ~ ( ord_le1860547276de_int @ X_96 @ Y_58 )
<=> ( X_96 = Y_58 ) ) ) ).
thf(fact_822_linorder__antisym__conv3,axiom,
! [Y_58: nat,X_96: nat] :
( ~ ( ord_less_nat @ Y_58 @ X_96 )
=> ( ~ ( ord_less_nat @ X_96 @ Y_58 )
<=> ( X_96 = Y_58 ) ) ) ).
thf(fact_823_linorder__antisym__conv3,axiom,
! [Y_58: real,X_96: real] :
( ~ ( ord_less_real @ Y_58 @ X_96 )
=> ( ~ ( ord_less_real @ X_96 @ Y_58 )
<=> ( X_96 = Y_58 ) ) ) ).
thf(fact_824_linorder__antisym__conv3,axiom,
! [Y_58: int,X_96: int] :
( ~ ( ord_less_int @ Y_58 @ X_96 )
=> ( ~ ( ord_less_int @ X_96 @ Y_58 )
<=> ( X_96 = Y_58 ) ) ) ).
thf(fact_825_linorder__neqE,axiom,
! [X_95: rat,Y_57: rat] :
( ( X_95 != Y_57 )
=> ( ~ ( ord_less_rat @ X_95 @ Y_57 )
=> ( ord_less_rat @ Y_57 @ X_95 ) ) ) ).
thf(fact_826_linorder__neqE,axiom,
! [X_95: quickcheck_code_int,Y_57: quickcheck_code_int] :
( ( X_95 != Y_57 )
=> ( ~ ( ord_le1860547276de_int @ X_95 @ Y_57 )
=> ( ord_le1860547276de_int @ Y_57 @ X_95 ) ) ) ).
thf(fact_827_linorder__neqE,axiom,
! [X_95: nat,Y_57: nat] :
( ( X_95 != Y_57 )
=> ( ~ ( ord_less_nat @ X_95 @ Y_57 )
=> ( ord_less_nat @ Y_57 @ X_95 ) ) ) ).
thf(fact_828_linorder__neqE,axiom,
! [X_95: real,Y_57: real] :
( ( X_95 != Y_57 )
=> ( ~ ( ord_less_real @ X_95 @ Y_57 )
=> ( ord_less_real @ Y_57 @ X_95 ) ) ) ).
thf(fact_829_linorder__neqE,axiom,
! [X_95: int,Y_57: int] :
( ( X_95 != Y_57 )
=> ( ~ ( ord_less_int @ X_95 @ Y_57 )
=> ( ord_less_int @ Y_57 @ X_95 ) ) ) ).
thf(fact_830_less__imp__neq,axiom,
! [X_94: rat,Y_56: rat] :
( ( ord_less_rat @ X_94 @ Y_56 )
=> ( X_94 != Y_56 ) ) ).
thf(fact_831_less__imp__neq,axiom,
! [X_94: int > $o,Y_56: int > $o] :
( ( ord_less_int_o @ X_94 @ Y_56 )
=> ( X_94 != Y_56 ) ) ).
thf(fact_832_less__imp__neq,axiom,
! [X_94: nat > $o,Y_56: nat > $o] :
( ( ord_less_nat_o @ X_94 @ Y_56 )
=> ( X_94 != Y_56 ) ) ).
thf(fact_833_less__imp__neq,axiom,
! [X_94: quickcheck_code_int,Y_56: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_94 @ Y_56 )
=> ( X_94 != Y_56 ) ) ).
thf(fact_834_less__imp__neq,axiom,
! [X_94: nat,Y_56: nat] :
( ( ord_less_nat @ X_94 @ Y_56 )
=> ( X_94 != Y_56 ) ) ).
thf(fact_835_less__imp__neq,axiom,
! [X_94: real,Y_56: real] :
( ( ord_less_real @ X_94 @ Y_56 )
=> ( X_94 != Y_56 ) ) ).
thf(fact_836_less__imp__neq,axiom,
! [X_94: int,Y_56: int] :
( ( ord_less_int @ X_94 @ Y_56 )
=> ( X_94 != Y_56 ) ) ).
thf(fact_837_order__less__not__sym,axiom,
! [X_93: rat,Y_55: rat] :
( ( ord_less_rat @ X_93 @ Y_55 )
=> ~ ( ord_less_rat @ Y_55 @ X_93 ) ) ).
thf(fact_838_order__less__not__sym,axiom,
! [X_93: int > $o,Y_55: int > $o] :
( ( ord_less_int_o @ X_93 @ Y_55 )
=> ~ ( ord_less_int_o @ Y_55 @ X_93 ) ) ).
thf(fact_839_order__less__not__sym,axiom,
! [X_93: nat > $o,Y_55: nat > $o] :
( ( ord_less_nat_o @ X_93 @ Y_55 )
=> ~ ( ord_less_nat_o @ Y_55 @ X_93 ) ) ).
thf(fact_840_order__less__not__sym,axiom,
! [X_93: quickcheck_code_int,Y_55: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_93 @ Y_55 )
=> ~ ( ord_le1860547276de_int @ Y_55 @ X_93 ) ) ).
thf(fact_841_order__less__not__sym,axiom,
! [X_93: nat,Y_55: nat] :
( ( ord_less_nat @ X_93 @ Y_55 )
=> ~ ( ord_less_nat @ Y_55 @ X_93 ) ) ).
thf(fact_842_order__less__not__sym,axiom,
! [X_93: real,Y_55: real] :
( ( ord_less_real @ X_93 @ Y_55 )
=> ~ ( ord_less_real @ Y_55 @ X_93 ) ) ).
thf(fact_843_order__less__not__sym,axiom,
! [X_93: int,Y_55: int] :
( ( ord_less_int @ X_93 @ Y_55 )
=> ~ ( ord_less_int @ Y_55 @ X_93 ) ) ).
thf(fact_844_order__less__imp__not__less,axiom,
! [X_92: rat,Y_54: rat] :
( ( ord_less_rat @ X_92 @ Y_54 )
=> ~ ( ord_less_rat @ Y_54 @ X_92 ) ) ).
thf(fact_845_order__less__imp__not__less,axiom,
! [X_92: int > $o,Y_54: int > $o] :
( ( ord_less_int_o @ X_92 @ Y_54 )
=> ~ ( ord_less_int_o @ Y_54 @ X_92 ) ) ).
thf(fact_846_order__less__imp__not__less,axiom,
! [X_92: nat > $o,Y_54: nat > $o] :
( ( ord_less_nat_o @ X_92 @ Y_54 )
=> ~ ( ord_less_nat_o @ Y_54 @ X_92 ) ) ).
thf(fact_847_order__less__imp__not__less,axiom,
! [X_92: quickcheck_code_int,Y_54: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_92 @ Y_54 )
=> ~ ( ord_le1860547276de_int @ Y_54 @ X_92 ) ) ).
thf(fact_848_order__less__imp__not__less,axiom,
! [X_92: nat,Y_54: nat] :
( ( ord_less_nat @ X_92 @ Y_54 )
=> ~ ( ord_less_nat @ Y_54 @ X_92 ) ) ).
thf(fact_849_order__less__imp__not__less,axiom,
! [X_92: real,Y_54: real] :
( ( ord_less_real @ X_92 @ Y_54 )
=> ~ ( ord_less_real @ Y_54 @ X_92 ) ) ).
thf(fact_850_order__less__imp__not__less,axiom,
! [X_92: int,Y_54: int] :
( ( ord_less_int @ X_92 @ Y_54 )
=> ~ ( ord_less_int @ Y_54 @ X_92 ) ) ).
thf(fact_851_order__less__imp__not__eq,axiom,
! [X_91: rat,Y_53: rat] :
( ( ord_less_rat @ X_91 @ Y_53 )
=> ( X_91 != Y_53 ) ) ).
thf(fact_852_order__less__imp__not__eq,axiom,
! [X_91: int > $o,Y_53: int > $o] :
( ( ord_less_int_o @ X_91 @ Y_53 )
=> ( X_91 != Y_53 ) ) ).
thf(fact_853_order__less__imp__not__eq,axiom,
! [X_91: nat > $o,Y_53: nat > $o] :
( ( ord_less_nat_o @ X_91 @ Y_53 )
=> ( X_91 != Y_53 ) ) ).
thf(fact_854_order__less__imp__not__eq,axiom,
! [X_91: quickcheck_code_int,Y_53: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_91 @ Y_53 )
=> ( X_91 != Y_53 ) ) ).
thf(fact_855_order__less__imp__not__eq,axiom,
! [X_91: nat,Y_53: nat] :
( ( ord_less_nat @ X_91 @ Y_53 )
=> ( X_91 != Y_53 ) ) ).
thf(fact_856_order__less__imp__not__eq,axiom,
! [X_91: real,Y_53: real] :
( ( ord_less_real @ X_91 @ Y_53 )
=> ( X_91 != Y_53 ) ) ).
thf(fact_857_order__less__imp__not__eq,axiom,
! [X_91: int,Y_53: int] :
( ( ord_less_int @ X_91 @ Y_53 )
=> ( X_91 != Y_53 ) ) ).
thf(fact_858_order__less__imp__not__eq2,axiom,
! [X_90: rat,Y_52: rat] :
( ( ord_less_rat @ X_90 @ Y_52 )
=> ( Y_52 != X_90 ) ) ).
thf(fact_859_order__less__imp__not__eq2,axiom,
! [X_90: int > $o,Y_52: int > $o] :
( ( ord_less_int_o @ X_90 @ Y_52 )
=> ( Y_52 != X_90 ) ) ).
thf(fact_860_order__less__imp__not__eq2,axiom,
! [X_90: nat > $o,Y_52: nat > $o] :
( ( ord_less_nat_o @ X_90 @ Y_52 )
=> ( Y_52 != X_90 ) ) ).
thf(fact_861_order__less__imp__not__eq2,axiom,
! [X_90: quickcheck_code_int,Y_52: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_90 @ Y_52 )
=> ( Y_52 != X_90 ) ) ).
thf(fact_862_order__less__imp__not__eq2,axiom,
! [X_90: nat,Y_52: nat] :
( ( ord_less_nat @ X_90 @ Y_52 )
=> ( Y_52 != X_90 ) ) ).
thf(fact_863_order__less__imp__not__eq2,axiom,
! [X_90: real,Y_52: real] :
( ( ord_less_real @ X_90 @ Y_52 )
=> ( Y_52 != X_90 ) ) ).
thf(fact_864_order__less__imp__not__eq2,axiom,
! [X_90: int,Y_52: int] :
( ( ord_less_int @ X_90 @ Y_52 )
=> ( Y_52 != X_90 ) ) ).
thf(fact_865_order__less__imp__triv,axiom,
! [P_7: $o,X_89: rat,Y_51: rat] :
( ( ord_less_rat @ X_89 @ Y_51 )
=> ( ( ord_less_rat @ Y_51 @ X_89 )
=> P_7 ) ) ).
thf(fact_866_order__less__imp__triv,axiom,
! [P_7: $o,X_89: int > $o,Y_51: int > $o] :
( ( ord_less_int_o @ X_89 @ Y_51 )
=> ( ( ord_less_int_o @ Y_51 @ X_89 )
=> P_7 ) ) ).
thf(fact_867_order__less__imp__triv,axiom,
! [P_7: $o,X_89: nat > $o,Y_51: nat > $o] :
( ( ord_less_nat_o @ X_89 @ Y_51 )
=> ( ( ord_less_nat_o @ Y_51 @ X_89 )
=> P_7 ) ) ).
thf(fact_868_order__less__imp__triv,axiom,
! [P_7: $o,X_89: quickcheck_code_int,Y_51: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_89 @ Y_51 )
=> ( ( ord_le1860547276de_int @ Y_51 @ X_89 )
=> P_7 ) ) ).
thf(fact_869_order__less__imp__triv,axiom,
! [P_7: $o,X_89: nat,Y_51: nat] :
( ( ord_less_nat @ X_89 @ Y_51 )
=> ( ( ord_less_nat @ Y_51 @ X_89 )
=> P_7 ) ) ).
thf(fact_870_order__less__imp__triv,axiom,
! [P_7: $o,X_89: real,Y_51: real] :
( ( ord_less_real @ X_89 @ Y_51 )
=> ( ( ord_less_real @ Y_51 @ X_89 )
=> P_7 ) ) ).
thf(fact_871_order__less__imp__triv,axiom,
! [P_7: $o,X_89: int,Y_51: int] :
( ( ord_less_int @ X_89 @ Y_51 )
=> ( ( ord_less_int @ Y_51 @ X_89 )
=> P_7 ) ) ).
thf(fact_872_order__less__asym_H,axiom,
! [A_247: rat,B_175: rat] :
( ( ord_less_rat @ A_247 @ B_175 )
=> ~ ( ord_less_rat @ B_175 @ A_247 ) ) ).
thf(fact_873_order__less__asym_H,axiom,
! [A_247: int > $o,B_175: int > $o] :
( ( ord_less_int_o @ A_247 @ B_175 )
=> ~ ( ord_less_int_o @ B_175 @ A_247 ) ) ).
thf(fact_874_order__less__asym_H,axiom,
! [A_247: nat > $o,B_175: nat > $o] :
( ( ord_less_nat_o @ A_247 @ B_175 )
=> ~ ( ord_less_nat_o @ B_175 @ A_247 ) ) ).
thf(fact_875_order__less__asym_H,axiom,
! [A_247: quickcheck_code_int,B_175: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_247 @ B_175 )
=> ~ ( ord_le1860547276de_int @ B_175 @ A_247 ) ) ).
thf(fact_876_order__less__asym_H,axiom,
! [A_247: nat,B_175: nat] :
( ( ord_less_nat @ A_247 @ B_175 )
=> ~ ( ord_less_nat @ B_175 @ A_247 ) ) ).
thf(fact_877_order__less__asym_H,axiom,
! [A_247: real,B_175: real] :
( ( ord_less_real @ A_247 @ B_175 )
=> ~ ( ord_less_real @ B_175 @ A_247 ) ) ).
thf(fact_878_order__less__asym_H,axiom,
! [A_247: int,B_175: int] :
( ( ord_less_int @ A_247 @ B_175 )
=> ~ ( ord_less_int @ B_175 @ A_247 ) ) ).
thf(fact_879_xt1_I9_J,axiom,
! [B_174: rat,A_246: rat] :
( ( ord_less_rat @ B_174 @ A_246 )
=> ~ ( ord_less_rat @ A_246 @ B_174 ) ) ).
thf(fact_880_xt1_I9_J,axiom,
! [B_174: int > $o,A_246: int > $o] :
( ( ord_less_int_o @ B_174 @ A_246 )
=> ~ ( ord_less_int_o @ A_246 @ B_174 ) ) ).
thf(fact_881_xt1_I9_J,axiom,
! [B_174: nat > $o,A_246: nat > $o] :
( ( ord_less_nat_o @ B_174 @ A_246 )
=> ~ ( ord_less_nat_o @ A_246 @ B_174 ) ) ).
thf(fact_882_xt1_I9_J,axiom,
! [B_174: quickcheck_code_int,A_246: quickcheck_code_int] :
( ( ord_le1860547276de_int @ B_174 @ A_246 )
=> ~ ( ord_le1860547276de_int @ A_246 @ B_174 ) ) ).
thf(fact_883_xt1_I9_J,axiom,
! [B_174: nat,A_246: nat] :
( ( ord_less_nat @ B_174 @ A_246 )
=> ~ ( ord_less_nat @ A_246 @ B_174 ) ) ).
thf(fact_884_xt1_I9_J,axiom,
! [B_174: real,A_246: real] :
( ( ord_less_real @ B_174 @ A_246 )
=> ~ ( ord_less_real @ A_246 @ B_174 ) ) ).
thf(fact_885_xt1_I9_J,axiom,
! [B_174: int,A_246: int] :
( ( ord_less_int @ B_174 @ A_246 )
=> ~ ( ord_less_int @ A_246 @ B_174 ) ) ).
thf(fact_886_ord__eq__less__trans,axiom,
! [C_72: rat,A_245: rat,B_173: rat] :
( ( A_245 = B_173 )
=> ( ( ord_less_rat @ B_173 @ C_72 )
=> ( ord_less_rat @ A_245 @ C_72 ) ) ) ).
thf(fact_887_ord__eq__less__trans,axiom,
! [C_72: int > $o,A_245: int > $o,B_173: int > $o] :
( ( A_245 = B_173 )
=> ( ( ord_less_int_o @ B_173 @ C_72 )
=> ( ord_less_int_o @ A_245 @ C_72 ) ) ) ).
thf(fact_888_ord__eq__less__trans,axiom,
! [C_72: nat > $o,A_245: nat > $o,B_173: nat > $o] :
( ( A_245 = B_173 )
=> ( ( ord_less_nat_o @ B_173 @ C_72 )
=> ( ord_less_nat_o @ A_245 @ C_72 ) ) ) ).
thf(fact_889_ord__eq__less__trans,axiom,
! [C_72: quickcheck_code_int,A_245: quickcheck_code_int,B_173: quickcheck_code_int] :
( ( A_245 = B_173 )
=> ( ( ord_le1860547276de_int @ B_173 @ C_72 )
=> ( ord_le1860547276de_int @ A_245 @ C_72 ) ) ) ).
thf(fact_890_ord__eq__less__trans,axiom,
! [C_72: nat,A_245: nat,B_173: nat] :
( ( A_245 = B_173 )
=> ( ( ord_less_nat @ B_173 @ C_72 )
=> ( ord_less_nat @ A_245 @ C_72 ) ) ) ).
thf(fact_891_ord__eq__less__trans,axiom,
! [C_72: real,A_245: real,B_173: real] :
( ( A_245 = B_173 )
=> ( ( ord_less_real @ B_173 @ C_72 )
=> ( ord_less_real @ A_245 @ C_72 ) ) ) ).
thf(fact_892_ord__eq__less__trans,axiom,
! [C_72: int,A_245: int,B_173: int] :
( ( A_245 = B_173 )
=> ( ( ord_less_int @ B_173 @ C_72 )
=> ( ord_less_int @ A_245 @ C_72 ) ) ) ).
thf(fact_893_xt1_I1_J,axiom,
! [C_71: rat,A_244: rat,B_172: rat] :
( ( A_244 = B_172 )
=> ( ( ord_less_rat @ C_71 @ B_172 )
=> ( ord_less_rat @ C_71 @ A_244 ) ) ) ).
thf(fact_894_xt1_I1_J,axiom,
! [C_71: int > $o,A_244: int > $o,B_172: int > $o] :
( ( A_244 = B_172 )
=> ( ( ord_less_int_o @ C_71 @ B_172 )
=> ( ord_less_int_o @ C_71 @ A_244 ) ) ) ).
thf(fact_895_xt1_I1_J,axiom,
! [C_71: nat > $o,A_244: nat > $o,B_172: nat > $o] :
( ( A_244 = B_172 )
=> ( ( ord_less_nat_o @ C_71 @ B_172 )
=> ( ord_less_nat_o @ C_71 @ A_244 ) ) ) ).
thf(fact_896_xt1_I1_J,axiom,
! [C_71: quickcheck_code_int,A_244: quickcheck_code_int,B_172: quickcheck_code_int] :
( ( A_244 = B_172 )
=> ( ( ord_le1860547276de_int @ C_71 @ B_172 )
=> ( ord_le1860547276de_int @ C_71 @ A_244 ) ) ) ).
thf(fact_897_xt1_I1_J,axiom,
! [C_71: nat,A_244: nat,B_172: nat] :
( ( A_244 = B_172 )
=> ( ( ord_less_nat @ C_71 @ B_172 )
=> ( ord_less_nat @ C_71 @ A_244 ) ) ) ).
thf(fact_898_xt1_I1_J,axiom,
! [C_71: real,A_244: real,B_172: real] :
( ( A_244 = B_172 )
=> ( ( ord_less_real @ C_71 @ B_172 )
=> ( ord_less_real @ C_71 @ A_244 ) ) ) ).
thf(fact_899_xt1_I1_J,axiom,
! [C_71: int,A_244: int,B_172: int] :
( ( A_244 = B_172 )
=> ( ( ord_less_int @ C_71 @ B_172 )
=> ( ord_less_int @ C_71 @ A_244 ) ) ) ).
thf(fact_900_ord__less__eq__trans,axiom,
! [C_70: rat,A_243: rat,B_171: rat] :
( ( ord_less_rat @ A_243 @ B_171 )
=> ( ( B_171 = C_70 )
=> ( ord_less_rat @ A_243 @ C_70 ) ) ) ).
thf(fact_901_ord__less__eq__trans,axiom,
! [C_70: int > $o,A_243: int > $o,B_171: int > $o] :
( ( ord_less_int_o @ A_243 @ B_171 )
=> ( ( B_171 = C_70 )
=> ( ord_less_int_o @ A_243 @ C_70 ) ) ) ).
thf(fact_902_ord__less__eq__trans,axiom,
! [C_70: nat > $o,A_243: nat > $o,B_171: nat > $o] :
( ( ord_less_nat_o @ A_243 @ B_171 )
=> ( ( B_171 = C_70 )
=> ( ord_less_nat_o @ A_243 @ C_70 ) ) ) ).
thf(fact_903_ord__less__eq__trans,axiom,
! [C_70: quickcheck_code_int,A_243: quickcheck_code_int,B_171: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_243 @ B_171 )
=> ( ( B_171 = C_70 )
=> ( ord_le1860547276de_int @ A_243 @ C_70 ) ) ) ).
thf(fact_904_ord__less__eq__trans,axiom,
! [C_70: nat,A_243: nat,B_171: nat] :
( ( ord_less_nat @ A_243 @ B_171 )
=> ( ( B_171 = C_70 )
=> ( ord_less_nat @ A_243 @ C_70 ) ) ) ).
thf(fact_905_ord__less__eq__trans,axiom,
! [C_70: real,A_243: real,B_171: real] :
( ( ord_less_real @ A_243 @ B_171 )
=> ( ( B_171 = C_70 )
=> ( ord_less_real @ A_243 @ C_70 ) ) ) ).
thf(fact_906_ord__less__eq__trans,axiom,
! [C_70: int,A_243: int,B_171: int] :
( ( ord_less_int @ A_243 @ B_171 )
=> ( ( B_171 = C_70 )
=> ( ord_less_int @ A_243 @ C_70 ) ) ) ).
thf(fact_907_xt1_I2_J,axiom,
! [C_69: rat,B_170: rat,A_242: rat] :
( ( ord_less_rat @ B_170 @ A_242 )
=> ( ( B_170 = C_69 )
=> ( ord_less_rat @ C_69 @ A_242 ) ) ) ).
thf(fact_908_xt1_I2_J,axiom,
! [C_69: int > $o,B_170: int > $o,A_242: int > $o] :
( ( ord_less_int_o @ B_170 @ A_242 )
=> ( ( B_170 = C_69 )
=> ( ord_less_int_o @ C_69 @ A_242 ) ) ) ).
thf(fact_909_xt1_I2_J,axiom,
! [C_69: nat > $o,B_170: nat > $o,A_242: nat > $o] :
( ( ord_less_nat_o @ B_170 @ A_242 )
=> ( ( B_170 = C_69 )
=> ( ord_less_nat_o @ C_69 @ A_242 ) ) ) ).
thf(fact_910_xt1_I2_J,axiom,
! [C_69: quickcheck_code_int,B_170: quickcheck_code_int,A_242: quickcheck_code_int] :
( ( ord_le1860547276de_int @ B_170 @ A_242 )
=> ( ( B_170 = C_69 )
=> ( ord_le1860547276de_int @ C_69 @ A_242 ) ) ) ).
thf(fact_911_xt1_I2_J,axiom,
! [C_69: nat,B_170: nat,A_242: nat] :
( ( ord_less_nat @ B_170 @ A_242 )
=> ( ( B_170 = C_69 )
=> ( ord_less_nat @ C_69 @ A_242 ) ) ) ).
thf(fact_912_xt1_I2_J,axiom,
! [C_69: real,B_170: real,A_242: real] :
( ( ord_less_real @ B_170 @ A_242 )
=> ( ( B_170 = C_69 )
=> ( ord_less_real @ C_69 @ A_242 ) ) ) ).
thf(fact_913_xt1_I2_J,axiom,
! [C_69: int,B_170: int,A_242: int] :
( ( ord_less_int @ B_170 @ A_242 )
=> ( ( B_170 = C_69 )
=> ( ord_less_int @ C_69 @ A_242 ) ) ) ).
thf(fact_914_order__less__trans,axiom,
! [Z_12: rat,X_88: rat,Y_50: rat] :
( ( ord_less_rat @ X_88 @ Y_50 )
=> ( ( ord_less_rat @ Y_50 @ Z_12 )
=> ( ord_less_rat @ X_88 @ Z_12 ) ) ) ).
thf(fact_915_order__less__trans,axiom,
! [Z_12: int > $o,X_88: int > $o,Y_50: int > $o] :
( ( ord_less_int_o @ X_88 @ Y_50 )
=> ( ( ord_less_int_o @ Y_50 @ Z_12 )
=> ( ord_less_int_o @ X_88 @ Z_12 ) ) ) ).
thf(fact_916_order__less__trans,axiom,
! [Z_12: nat > $o,X_88: nat > $o,Y_50: nat > $o] :
( ( ord_less_nat_o @ X_88 @ Y_50 )
=> ( ( ord_less_nat_o @ Y_50 @ Z_12 )
=> ( ord_less_nat_o @ X_88 @ Z_12 ) ) ) ).
thf(fact_917_order__less__trans,axiom,
! [Z_12: quickcheck_code_int,X_88: quickcheck_code_int,Y_50: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_88 @ Y_50 )
=> ( ( ord_le1860547276de_int @ Y_50 @ Z_12 )
=> ( ord_le1860547276de_int @ X_88 @ Z_12 ) ) ) ).
thf(fact_918_order__less__trans,axiom,
! [Z_12: nat,X_88: nat,Y_50: nat] :
( ( ord_less_nat @ X_88 @ Y_50 )
=> ( ( ord_less_nat @ Y_50 @ Z_12 )
=> ( ord_less_nat @ X_88 @ Z_12 ) ) ) ).
thf(fact_919_order__less__trans,axiom,
! [Z_12: real,X_88: real,Y_50: real] :
( ( ord_less_real @ X_88 @ Y_50 )
=> ( ( ord_less_real @ Y_50 @ Z_12 )
=> ( ord_less_real @ X_88 @ Z_12 ) ) ) ).
thf(fact_920_order__less__trans,axiom,
! [Z_12: int,X_88: int,Y_50: int] :
( ( ord_less_int @ X_88 @ Y_50 )
=> ( ( ord_less_int @ Y_50 @ Z_12 )
=> ( ord_less_int @ X_88 @ Z_12 ) ) ) ).
thf(fact_921_xt1_I10_J,axiom,
! [Z_11: rat,Y_49: rat,X_87: rat] :
( ( ord_less_rat @ Y_49 @ X_87 )
=> ( ( ord_less_rat @ Z_11 @ Y_49 )
=> ( ord_less_rat @ Z_11 @ X_87 ) ) ) ).
thf(fact_922_xt1_I10_J,axiom,
! [Z_11: int > $o,Y_49: int > $o,X_87: int > $o] :
( ( ord_less_int_o @ Y_49 @ X_87 )
=> ( ( ord_less_int_o @ Z_11 @ Y_49 )
=> ( ord_less_int_o @ Z_11 @ X_87 ) ) ) ).
thf(fact_923_xt1_I10_J,axiom,
! [Z_11: nat > $o,Y_49: nat > $o,X_87: nat > $o] :
( ( ord_less_nat_o @ Y_49 @ X_87 )
=> ( ( ord_less_nat_o @ Z_11 @ Y_49 )
=> ( ord_less_nat_o @ Z_11 @ X_87 ) ) ) ).
thf(fact_924_xt1_I10_J,axiom,
! [Z_11: quickcheck_code_int,Y_49: quickcheck_code_int,X_87: quickcheck_code_int] :
( ( ord_le1860547276de_int @ Y_49 @ X_87 )
=> ( ( ord_le1860547276de_int @ Z_11 @ Y_49 )
=> ( ord_le1860547276de_int @ Z_11 @ X_87 ) ) ) ).
thf(fact_925_xt1_I10_J,axiom,
! [Z_11: nat,Y_49: nat,X_87: nat] :
( ( ord_less_nat @ Y_49 @ X_87 )
=> ( ( ord_less_nat @ Z_11 @ Y_49 )
=> ( ord_less_nat @ Z_11 @ X_87 ) ) ) ).
thf(fact_926_xt1_I10_J,axiom,
! [Z_11: real,Y_49: real,X_87: real] :
( ( ord_less_real @ Y_49 @ X_87 )
=> ( ( ord_less_real @ Z_11 @ Y_49 )
=> ( ord_less_real @ Z_11 @ X_87 ) ) ) ).
thf(fact_927_xt1_I10_J,axiom,
! [Z_11: int,Y_49: int,X_87: int] :
( ( ord_less_int @ Y_49 @ X_87 )
=> ( ( ord_less_int @ Z_11 @ Y_49 )
=> ( ord_less_int @ Z_11 @ X_87 ) ) ) ).
thf(fact_928_order__less__asym,axiom,
! [X_86: rat,Y_48: rat] :
( ( ord_less_rat @ X_86 @ Y_48 )
=> ~ ( ord_less_rat @ Y_48 @ X_86 ) ) ).
thf(fact_929_order__less__asym,axiom,
! [X_86: int > $o,Y_48: int > $o] :
( ( ord_less_int_o @ X_86 @ Y_48 )
=> ~ ( ord_less_int_o @ Y_48 @ X_86 ) ) ).
thf(fact_930_order__less__asym,axiom,
! [X_86: nat > $o,Y_48: nat > $o] :
( ( ord_less_nat_o @ X_86 @ Y_48 )
=> ~ ( ord_less_nat_o @ Y_48 @ X_86 ) ) ).
thf(fact_931_order__less__asym,axiom,
! [X_86: quickcheck_code_int,Y_48: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_86 @ Y_48 )
=> ~ ( ord_le1860547276de_int @ Y_48 @ X_86 ) ) ).
thf(fact_932_order__less__asym,axiom,
! [X_86: nat,Y_48: nat] :
( ( ord_less_nat @ X_86 @ Y_48 )
=> ~ ( ord_less_nat @ Y_48 @ X_86 ) ) ).
thf(fact_933_order__less__asym,axiom,
! [X_86: real,Y_48: real] :
( ( ord_less_real @ X_86 @ Y_48 )
=> ~ ( ord_less_real @ Y_48 @ X_86 ) ) ).
thf(fact_934_order__less__asym,axiom,
! [X_86: int,Y_48: int] :
( ( ord_less_int @ X_86 @ Y_48 )
=> ~ ( ord_less_int @ Y_48 @ X_86 ) ) ).
thf(fact_935_linorder__cases,axiom,
! [X_85: rat,Y_47: rat] :
( ~ ( ord_less_rat @ X_85 @ Y_47 )
=> ( ( X_85 != Y_47 )
=> ( ord_less_rat @ Y_47 @ X_85 ) ) ) ).
thf(fact_936_linorder__cases,axiom,
! [X_85: quickcheck_code_int,Y_47: quickcheck_code_int] :
( ~ ( ord_le1860547276de_int @ X_85 @ Y_47 )
=> ( ( X_85 != Y_47 )
=> ( ord_le1860547276de_int @ Y_47 @ X_85 ) ) ) ).
thf(fact_937_linorder__cases,axiom,
! [X_85: nat,Y_47: nat] :
( ~ ( ord_less_nat @ X_85 @ Y_47 )
=> ( ( X_85 != Y_47 )
=> ( ord_less_nat @ Y_47 @ X_85 ) ) ) ).
thf(fact_938_linorder__cases,axiom,
! [X_85: real,Y_47: real] :
( ~ ( ord_less_real @ X_85 @ Y_47 )
=> ( ( X_85 != Y_47 )
=> ( ord_less_real @ Y_47 @ X_85 ) ) ) ).
thf(fact_939_linorder__cases,axiom,
! [X_85: int,Y_47: int] :
( ~ ( ord_less_int @ X_85 @ Y_47 )
=> ( ( X_85 != Y_47 )
=> ( ord_less_int @ Y_47 @ X_85 ) ) ) ).
thf(fact_940_linorder__not__less,axiom,
! [X_84: rat,Y_46: rat] :
( ~ ( ord_less_rat @ X_84 @ Y_46 )
<=> ( ord_less_eq_rat @ Y_46 @ X_84 ) ) ).
thf(fact_941_linorder__not__less,axiom,
! [X_84: quickcheck_code_int,Y_46: quickcheck_code_int] :
( ~ ( ord_le1860547276de_int @ X_84 @ Y_46 )
<=> ( ord_le258702272de_int @ Y_46 @ X_84 ) ) ).
thf(fact_942_linorder__not__less,axiom,
! [X_84: nat,Y_46: nat] :
( ~ ( ord_less_nat @ X_84 @ Y_46 )
<=> ( ord_less_eq_nat @ Y_46 @ X_84 ) ) ).
thf(fact_943_linorder__not__less,axiom,
! [X_84: real,Y_46: real] :
( ~ ( ord_less_real @ X_84 @ Y_46 )
<=> ( ord_less_eq_real @ Y_46 @ X_84 ) ) ).
thf(fact_944_linorder__not__less,axiom,
! [X_84: int,Y_46: int] :
( ~ ( ord_less_int @ X_84 @ Y_46 )
<=> ( ord_less_eq_int @ Y_46 @ X_84 ) ) ).
thf(fact_945_linorder__not__le,axiom,
! [X_83: rat,Y_45: rat] :
( ~ ( ord_less_eq_rat @ X_83 @ Y_45 )
<=> ( ord_less_rat @ Y_45 @ X_83 ) ) ).
thf(fact_946_linorder__not__le,axiom,
! [X_83: quickcheck_code_int,Y_45: quickcheck_code_int] :
( ~ ( ord_le258702272de_int @ X_83 @ Y_45 )
<=> ( ord_le1860547276de_int @ Y_45 @ X_83 ) ) ).
thf(fact_947_linorder__not__le,axiom,
! [X_83: nat,Y_45: nat] :
( ~ ( ord_less_eq_nat @ X_83 @ Y_45 )
<=> ( ord_less_nat @ Y_45 @ X_83 ) ) ).
thf(fact_948_linorder__not__le,axiom,
! [X_83: real,Y_45: real] :
( ~ ( ord_less_eq_real @ X_83 @ Y_45 )
<=> ( ord_less_real @ Y_45 @ X_83 ) ) ).
thf(fact_949_linorder__not__le,axiom,
! [X_83: int,Y_45: int] :
( ~ ( ord_less_eq_int @ X_83 @ Y_45 )
<=> ( ord_less_int @ Y_45 @ X_83 ) ) ).
thf(fact_950_linorder__le__less__linear,axiom,
! [X_82: rat,Y_44: rat] :
( ( ord_less_eq_rat @ X_82 @ Y_44 )
| ( ord_less_rat @ Y_44 @ X_82 ) ) ).
thf(fact_951_linorder__le__less__linear,axiom,
! [X_82: quickcheck_code_int,Y_44: quickcheck_code_int] :
( ( ord_le258702272de_int @ X_82 @ Y_44 )
| ( ord_le1860547276de_int @ Y_44 @ X_82 ) ) ).
thf(fact_952_linorder__le__less__linear,axiom,
! [X_82: nat,Y_44: nat] :
( ( ord_less_eq_nat @ X_82 @ Y_44 )
| ( ord_less_nat @ Y_44 @ X_82 ) ) ).
thf(fact_953_linorder__le__less__linear,axiom,
! [X_82: real,Y_44: real] :
( ( ord_less_eq_real @ X_82 @ Y_44 )
| ( ord_less_real @ Y_44 @ X_82 ) ) ).
thf(fact_954_linorder__le__less__linear,axiom,
! [X_82: int,Y_44: int] :
( ( ord_less_eq_int @ X_82 @ Y_44 )
| ( ord_less_int @ Y_44 @ X_82 ) ) ).
thf(fact_955_order__less__le,axiom,
! [X_81: rat,Y_43: rat] :
( ( ord_less_rat @ X_81 @ Y_43 )
<=> ( ( ord_less_eq_rat @ X_81 @ Y_43 )
& ( X_81 != Y_43 ) ) ) ).
thf(fact_956_order__less__le,axiom,
! [X_81: int > $o,Y_43: int > $o] :
( ( ord_less_int_o @ X_81 @ Y_43 )
<=> ( ( ord_less_eq_int_o @ X_81 @ Y_43 )
& ( X_81 != Y_43 ) ) ) ).
thf(fact_957_order__less__le,axiom,
! [X_81: nat > $o,Y_43: nat > $o] :
( ( ord_less_nat_o @ X_81 @ Y_43 )
<=> ( ( ord_less_eq_nat_o @ X_81 @ Y_43 )
& ( X_81 != Y_43 ) ) ) ).
thf(fact_958_order__less__le,axiom,
! [X_81: quickcheck_code_int,Y_43: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_81 @ Y_43 )
<=> ( ( ord_le258702272de_int @ X_81 @ Y_43 )
& ( X_81 != Y_43 ) ) ) ).
thf(fact_959_order__less__le,axiom,
! [X_81: nat,Y_43: nat] :
( ( ord_less_nat @ X_81 @ Y_43 )
<=> ( ( ord_less_eq_nat @ X_81 @ Y_43 )
& ( X_81 != Y_43 ) ) ) ).
thf(fact_960_order__less__le,axiom,
! [X_81: real,Y_43: real] :
( ( ord_less_real @ X_81 @ Y_43 )
<=> ( ( ord_less_eq_real @ X_81 @ Y_43 )
& ( X_81 != Y_43 ) ) ) ).
thf(fact_961_order__less__le,axiom,
! [X_81: int,Y_43: int] :
( ( ord_less_int @ X_81 @ Y_43 )
<=> ( ( ord_less_eq_int @ X_81 @ Y_43 )
& ( X_81 != Y_43 ) ) ) ).
thf(fact_962_less__le__not__le,axiom,
! [X_80: rat,Y_42: rat] :
( ( ord_less_rat @ X_80 @ Y_42 )
<=> ( ( ord_less_eq_rat @ X_80 @ Y_42 )
& ~ ( ord_less_eq_rat @ Y_42 @ X_80 ) ) ) ).
thf(fact_963_less__le__not__le,axiom,
! [X_80: int > $o,Y_42: int > $o] :
( ( ord_less_int_o @ X_80 @ Y_42 )
<=> ( ( ord_less_eq_int_o @ X_80 @ Y_42 )
& ~ ( ord_less_eq_int_o @ Y_42 @ X_80 ) ) ) ).
thf(fact_964_less__le__not__le,axiom,
! [X_80: nat > $o,Y_42: nat > $o] :
( ( ord_less_nat_o @ X_80 @ Y_42 )
<=> ( ( ord_less_eq_nat_o @ X_80 @ Y_42 )
& ~ ( ord_less_eq_nat_o @ Y_42 @ X_80 ) ) ) ).
thf(fact_965_less__le__not__le,axiom,
! [X_80: quickcheck_code_int,Y_42: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_80 @ Y_42 )
<=> ( ( ord_le258702272de_int @ X_80 @ Y_42 )
& ~ ( ord_le258702272de_int @ Y_42 @ X_80 ) ) ) ).
thf(fact_966_less__le__not__le,axiom,
! [X_80: nat,Y_42: nat] :
( ( ord_less_nat @ X_80 @ Y_42 )
<=> ( ( ord_less_eq_nat @ X_80 @ Y_42 )
& ~ ( ord_less_eq_nat @ Y_42 @ X_80 ) ) ) ).
thf(fact_967_less__le__not__le,axiom,
! [X_80: real,Y_42: real] :
( ( ord_less_real @ X_80 @ Y_42 )
<=> ( ( ord_less_eq_real @ X_80 @ Y_42 )
& ~ ( ord_less_eq_real @ Y_42 @ X_80 ) ) ) ).
thf(fact_968_less__le__not__le,axiom,
! [X_80: int,Y_42: int] :
( ( ord_less_int @ X_80 @ Y_42 )
<=> ( ( ord_less_eq_int @ X_80 @ Y_42 )
& ~ ( ord_less_eq_int @ Y_42 @ X_80 ) ) ) ).
thf(fact_969_order__le__less,axiom,
! [X_79: rat,Y_41: rat] :
( ( ord_less_eq_rat @ X_79 @ Y_41 )
<=> ( ( ord_less_rat @ X_79 @ Y_41 )
| ( X_79 = Y_41 ) ) ) ).
thf(fact_970_order__le__less,axiom,
! [X_79: int > $o,Y_41: int > $o] :
( ( ord_less_eq_int_o @ X_79 @ Y_41 )
<=> ( ( ord_less_int_o @ X_79 @ Y_41 )
| ( X_79 = Y_41 ) ) ) ).
thf(fact_971_order__le__less,axiom,
! [X_79: nat > $o,Y_41: nat > $o] :
( ( ord_less_eq_nat_o @ X_79 @ Y_41 )
<=> ( ( ord_less_nat_o @ X_79 @ Y_41 )
| ( X_79 = Y_41 ) ) ) ).
thf(fact_972_order__le__less,axiom,
! [X_79: quickcheck_code_int,Y_41: quickcheck_code_int] :
( ( ord_le258702272de_int @ X_79 @ Y_41 )
<=> ( ( ord_le1860547276de_int @ X_79 @ Y_41 )
| ( X_79 = Y_41 ) ) ) ).
thf(fact_973_order__le__less,axiom,
! [X_79: nat,Y_41: nat] :
( ( ord_less_eq_nat @ X_79 @ Y_41 )
<=> ( ( ord_less_nat @ X_79 @ Y_41 )
| ( X_79 = Y_41 ) ) ) ).
thf(fact_974_order__le__less,axiom,
! [X_79: real,Y_41: real] :
( ( ord_less_eq_real @ X_79 @ Y_41 )
<=> ( ( ord_less_real @ X_79 @ Y_41 )
| ( X_79 = Y_41 ) ) ) ).
thf(fact_975_order__le__less,axiom,
! [X_79: int,Y_41: int] :
( ( ord_less_eq_int @ X_79 @ Y_41 )
<=> ( ( ord_less_int @ X_79 @ Y_41 )
| ( X_79 = Y_41 ) ) ) ).
thf(fact_976_leI,axiom,
! [X_78: rat,Y_40: rat] :
( ~ ( ord_less_rat @ X_78 @ Y_40 )
=> ( ord_less_eq_rat @ Y_40 @ X_78 ) ) ).
thf(fact_977_leI,axiom,
! [X_78: quickcheck_code_int,Y_40: quickcheck_code_int] :
( ~ ( ord_le1860547276de_int @ X_78 @ Y_40 )
=> ( ord_le258702272de_int @ Y_40 @ X_78 ) ) ).
thf(fact_978_leI,axiom,
! [X_78: nat,Y_40: nat] :
( ~ ( ord_less_nat @ X_78 @ Y_40 )
=> ( ord_less_eq_nat @ Y_40 @ X_78 ) ) ).
thf(fact_979_leI,axiom,
! [X_78: real,Y_40: real] :
( ~ ( ord_less_real @ X_78 @ Y_40 )
=> ( ord_less_eq_real @ Y_40 @ X_78 ) ) ).
thf(fact_980_leI,axiom,
! [X_78: int,Y_40: int] :
( ~ ( ord_less_int @ X_78 @ Y_40 )
=> ( ord_less_eq_int @ Y_40 @ X_78 ) ) ).
thf(fact_981_not__leE,axiom,
! [Y_39: rat,X_77: rat] :
( ~ ( ord_less_eq_rat @ Y_39 @ X_77 )
=> ( ord_less_rat @ X_77 @ Y_39 ) ) ).
thf(fact_982_not__leE,axiom,
! [Y_39: quickcheck_code_int,X_77: quickcheck_code_int] :
( ~ ( ord_le258702272de_int @ Y_39 @ X_77 )
=> ( ord_le1860547276de_int @ X_77 @ Y_39 ) ) ).
thf(fact_983_not__leE,axiom,
! [Y_39: nat,X_77: nat] :
( ~ ( ord_less_eq_nat @ Y_39 @ X_77 )
=> ( ord_less_nat @ X_77 @ Y_39 ) ) ).
thf(fact_984_not__leE,axiom,
! [Y_39: real,X_77: real] :
( ~ ( ord_less_eq_real @ Y_39 @ X_77 )
=> ( ord_less_real @ X_77 @ Y_39 ) ) ).
thf(fact_985_not__leE,axiom,
! [Y_39: int,X_77: int] :
( ~ ( ord_less_eq_int @ Y_39 @ X_77 )
=> ( ord_less_int @ X_77 @ Y_39 ) ) ).
thf(fact_986_linorder__antisym__conv1,axiom,
! [X_76: rat,Y_38: rat] :
( ~ ( ord_less_rat @ X_76 @ Y_38 )
=> ( ( ord_less_eq_rat @ X_76 @ Y_38 )
<=> ( X_76 = Y_38 ) ) ) ).
thf(fact_987_linorder__antisym__conv1,axiom,
! [X_76: quickcheck_code_int,Y_38: quickcheck_code_int] :
( ~ ( ord_le1860547276de_int @ X_76 @ Y_38 )
=> ( ( ord_le258702272de_int @ X_76 @ Y_38 )
<=> ( X_76 = Y_38 ) ) ) ).
thf(fact_988_linorder__antisym__conv1,axiom,
! [X_76: nat,Y_38: nat] :
( ~ ( ord_less_nat @ X_76 @ Y_38 )
=> ( ( ord_less_eq_nat @ X_76 @ Y_38 )
<=> ( X_76 = Y_38 ) ) ) ).
thf(fact_989_linorder__antisym__conv1,axiom,
! [X_76: real,Y_38: real] :
( ~ ( ord_less_real @ X_76 @ Y_38 )
=> ( ( ord_less_eq_real @ X_76 @ Y_38 )
<=> ( X_76 = Y_38 ) ) ) ).
thf(fact_990_linorder__antisym__conv1,axiom,
! [X_76: int,Y_38: int] :
( ~ ( ord_less_int @ X_76 @ Y_38 )
=> ( ( ord_less_eq_int @ X_76 @ Y_38 )
<=> ( X_76 = Y_38 ) ) ) ).
thf(fact_991_order__neq__le__trans,axiom,
! [A_241: rat,B_169: rat] :
( ( A_241 != B_169 )
=> ( ( ord_less_eq_rat @ A_241 @ B_169 )
=> ( ord_less_rat @ A_241 @ B_169 ) ) ) ).
thf(fact_992_order__neq__le__trans,axiom,
! [A_241: int > $o,B_169: int > $o] :
( ( A_241 != B_169 )
=> ( ( ord_less_eq_int_o @ A_241 @ B_169 )
=> ( ord_less_int_o @ A_241 @ B_169 ) ) ) ).
thf(fact_993_order__neq__le__trans,axiom,
! [A_241: nat > $o,B_169: nat > $o] :
( ( A_241 != B_169 )
=> ( ( ord_less_eq_nat_o @ A_241 @ B_169 )
=> ( ord_less_nat_o @ A_241 @ B_169 ) ) ) ).
thf(fact_994_order__neq__le__trans,axiom,
! [A_241: quickcheck_code_int,B_169: quickcheck_code_int] :
( ( A_241 != B_169 )
=> ( ( ord_le258702272de_int @ A_241 @ B_169 )
=> ( ord_le1860547276de_int @ A_241 @ B_169 ) ) ) ).
thf(fact_995_order__neq__le__trans,axiom,
! [A_241: nat,B_169: nat] :
( ( A_241 != B_169 )
=> ( ( ord_less_eq_nat @ A_241 @ B_169 )
=> ( ord_less_nat @ A_241 @ B_169 ) ) ) ).
thf(fact_996_order__neq__le__trans,axiom,
! [A_241: real,B_169: real] :
( ( A_241 != B_169 )
=> ( ( ord_less_eq_real @ A_241 @ B_169 )
=> ( ord_less_real @ A_241 @ B_169 ) ) ) ).
thf(fact_997_order__neq__le__trans,axiom,
! [A_241: int,B_169: int] :
( ( A_241 != B_169 )
=> ( ( ord_less_eq_int @ A_241 @ B_169 )
=> ( ord_less_int @ A_241 @ B_169 ) ) ) ).
thf(fact_998_norR__mem__unique__aux,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ B @ one_one_int ) )
=> ( ord_less_int @ A @ B ) ) ).
thf(fact_999_int__le__induct,axiom,
! [P: int > $o,I: int,K_1: int] :
( ( ord_less_eq_int @ I @ K_1 )
=> ( ( P @ K_1 )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ I_1 @ K_1 )
=> ( ( P @ I_1 )
=> ( P @ ( minus_minus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_1000_int__less__induct,axiom,
! [P: int > $o,I: int,K_1: int] :
( ( ord_less_int @ I @ K_1 )
=> ( ( P @ ( minus_minus_int @ K_1 @ one_one_int ) )
=> ( ! [I_1: int] :
( ( ord_less_int @ I_1 @ K_1 )
=> ( ( P @ I_1 )
=> ( P @ ( minus_minus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_1001_int__ge__induct,axiom,
! [P: int > $o,K_1: int,I: int] :
( ( ord_less_eq_int @ K_1 @ I )
=> ( ( P @ K_1 )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ K_1 @ I_1 )
=> ( ( P @ I_1 )
=> ( P @ ( plus_plus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_1002_int__gr__induct,axiom,
! [P: int > $o,K_1: int,I: int] :
( ( ord_less_int @ K_1 @ I )
=> ( ( P @ ( plus_plus_int @ K_1 @ one_one_int ) )
=> ( ! [I_1: int] :
( ( ord_less_int @ K_1 @ I_1 )
=> ( ( P @ I_1 )
=> ( P @ ( plus_plus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_1003_dense__le,axiom,
! [Z_10: rat,Y_37: rat] :
( ! [X_1: rat] :
( ( ord_less_rat @ X_1 @ Y_37 )
=> ( ord_less_eq_rat @ X_1 @ Z_10 ) )
=> ( ord_less_eq_rat @ Y_37 @ Z_10 ) ) ).
thf(fact_1004_dense__le,axiom,
! [Z_10: real,Y_37: real] :
( ! [X_1: real] :
( ( ord_less_real @ X_1 @ Y_37 )
=> ( ord_less_eq_real @ X_1 @ Z_10 ) )
=> ( ord_less_eq_real @ Y_37 @ Z_10 ) ) ).
thf(fact_1005_dense__le__bounded,axiom,
! [Z_9: rat,X_75: rat,Y_36: rat] :
( ( ord_less_rat @ X_75 @ Y_36 )
=> ( ! [W_8: rat] :
( ( ord_less_rat @ X_75 @ W_8 )
=> ( ( ord_less_rat @ W_8 @ Y_36 )
=> ( ord_less_eq_rat @ W_8 @ Z_9 ) ) )
=> ( ord_less_eq_rat @ Y_36 @ Z_9 ) ) ) ).
thf(fact_1006_dense__le__bounded,axiom,
! [Z_9: real,X_75: real,Y_36: real] :
( ( ord_less_real @ X_75 @ Y_36 )
=> ( ! [W_8: real] :
( ( ord_less_real @ X_75 @ W_8 )
=> ( ( ord_less_real @ W_8 @ Y_36 )
=> ( ord_less_eq_real @ W_8 @ Z_9 ) ) )
=> ( ord_less_eq_real @ Y_36 @ Z_9 ) ) ) ).
thf(fact_1007_less__fun__def,axiom,
! [F_4: int > $o,G_2: int > $o] :
( ( ord_less_int_o @ F_4 @ G_2 )
<=> ( ( ord_less_eq_int_o @ F_4 @ G_2 )
& ~ ( ord_less_eq_int_o @ G_2 @ F_4 ) ) ) ).
thf(fact_1008_less__fun__def,axiom,
! [F_4: nat > $o,G_2: nat > $o] :
( ( ord_less_nat_o @ F_4 @ G_2 )
<=> ( ( ord_less_eq_nat_o @ F_4 @ G_2 )
& ~ ( ord_less_eq_nat_o @ G_2 @ F_4 ) ) ) ).
thf(fact_1009_le__funI,axiom,
! [F_3: int > $o,G_1: int > $o] :
( ! [X_1: int] : ( ord_less_eq_o @ ( F_3 @ X_1 ) @ ( G_1 @ X_1 ) )
=> ( ord_less_eq_int_o @ F_3 @ G_1 ) ) ).
thf(fact_1010_le__funI,axiom,
! [F_3: nat > $o,G_1: nat > $o] :
( ! [X_1: nat] : ( ord_less_eq_o @ ( F_3 @ X_1 ) @ ( G_1 @ X_1 ) )
=> ( ord_less_eq_nat_o @ F_3 @ G_1 ) ) ).
thf(fact_1011_int__induct,axiom,
! [I: int,P: int > $o,K_1: int] :
( ( P @ K_1 )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ K_1 @ I_1 )
=> ( ( P @ I_1 )
=> ( P @ ( plus_plus_int @ I_1 @ one_one_int ) ) ) )
=> ( ! [I_1: int] :
( ( ord_less_eq_int @ I_1 @ K_1 )
=> ( ( P @ I_1 )
=> ( P @ ( minus_minus_int @ I_1 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
thf(fact_1012_Euler_Oaux2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ A @ C )
=> ( ( ord_less_int @ B @ C )
=> ( ( ord_less_eq_int @ A @ B )
| ( ord_less_eq_int @ B @ A ) ) ) ) ).
thf(fact_1013_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_1014_convex__bound__lt,axiom,
! [V_19: rat,U_3: rat,Y_35: rat,X_74: rat,A_240: rat] :
( ( ord_less_rat @ X_74 @ A_240 )
=> ( ( ord_less_rat @ Y_35 @ A_240 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ U_3 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ V_19 )
=> ( ( ( plus_plus_rat @ U_3 @ V_19 )
= one_one_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U_3 @ X_74 ) @ ( times_times_rat @ V_19 @ Y_35 ) ) @ A_240 ) ) ) ) ) ) ).
thf(fact_1015_convex__bound__lt,axiom,
! [V_19: real,U_3: real,Y_35: real,X_74: real,A_240: real] :
( ( ord_less_real @ X_74 @ A_240 )
=> ( ( ord_less_real @ Y_35 @ A_240 )
=> ( ( ord_less_eq_real @ zero_zero_real @ U_3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ V_19 )
=> ( ( ( plus_plus_real @ U_3 @ V_19 )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U_3 @ X_74 ) @ ( times_times_real @ V_19 @ Y_35 ) ) @ A_240 ) ) ) ) ) ) ).
thf(fact_1016_convex__bound__lt,axiom,
! [V_19: int,U_3: int,Y_35: int,X_74: int,A_240: int] :
( ( ord_less_int @ X_74 @ A_240 )
=> ( ( ord_less_int @ Y_35 @ A_240 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U_3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ V_19 )
=> ( ( ( plus_plus_int @ U_3 @ V_19 )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U_3 @ X_74 ) @ ( times_times_int @ V_19 @ Y_35 ) ) @ A_240 ) ) ) ) ) ) ).
thf(fact_1017_number__of__le__ceiling,axiom,
! [V_18: int,X_73: rat] :
( ( ord_less_eq_int @ ( number_number_of_int @ V_18 ) @ ( archim1419379755ng_rat @ X_73 ) )
<=> ( ord_less_rat @ ( minus_minus_rat @ ( number_number_of_rat @ V_18 ) @ one_one_rat ) @ X_73 ) ) ).
thf(fact_1018_number__of__le__ceiling,axiom,
! [V_18: int,X_73: real] :
( ( ord_less_eq_int @ ( number_number_of_int @ V_18 ) @ ( archim856651990g_real @ X_73 ) )
<=> ( ord_less_real @ ( minus_minus_real @ ( number267125858f_real @ V_18 ) @ one_one_real ) @ X_73 ) ) ).
thf(fact_1019_ceiling__less__number__of,axiom,
! [X_72: rat,V_17: int] :
( ( ord_less_int @ ( archim1419379755ng_rat @ X_72 ) @ ( number_number_of_int @ V_17 ) )
<=> ( ord_less_eq_rat @ X_72 @ ( minus_minus_rat @ ( number_number_of_rat @ V_17 ) @ one_one_rat ) ) ) ).
thf(fact_1020_ceiling__less__number__of,axiom,
! [X_72: real,V_17: int] :
( ( ord_less_int @ ( archim856651990g_real @ X_72 ) @ ( number_number_of_int @ V_17 ) )
<=> ( ord_less_eq_real @ X_72 @ ( minus_minus_real @ ( number267125858f_real @ V_17 ) @ one_one_real ) ) ) ).
thf(fact_1021_zero__less__ceiling,axiom,
! [X_71: rat] :
( ( ord_less_int @ zero_zero_int @ ( archim1419379755ng_rat @ X_71 ) )
<=> ( ord_less_rat @ zero_zero_rat @ X_71 ) ) ).
thf(fact_1022_zero__less__ceiling,axiom,
! [X_71: real] :
( ( ord_less_int @ zero_zero_int @ ( archim856651990g_real @ X_71 ) )
<=> ( ord_less_real @ zero_zero_real @ X_71 ) ) ).
thf(fact_1023_ceiling__le__zero,axiom,
! [X_70: rat] :
( ( ord_less_eq_int @ ( archim1419379755ng_rat @ X_70 ) @ zero_zero_int )
<=> ( ord_less_eq_rat @ X_70 @ zero_zero_rat ) ) ).
thf(fact_1024_ceiling__le__zero,axiom,
! [X_70: real] :
( ( ord_less_eq_int @ ( archim856651990g_real @ X_70 ) @ zero_zero_int )
<=> ( ord_less_eq_real @ X_70 @ zero_zero_real ) ) ).
thf(fact_1025_ceiling__zero,axiom,
( ( archim1419379755ng_rat @ zero_zero_rat )
= zero_zero_int ) ).
thf(fact_1026_ceiling__zero,axiom,
( ( archim856651990g_real @ zero_zero_real )
= zero_zero_int ) ).
thf(fact_1027_zero__reorient,axiom,
! [X_69: rat] :
( ( zero_zero_rat = X_69 )
<=> ( X_69 = zero_zero_rat ) ) ).
thf(fact_1028_zero__reorient,axiom,
! [X_69: quickcheck_code_int] :
( ( zero_z891286103de_int = X_69 )
<=> ( X_69 = zero_z891286103de_int ) ) ).
thf(fact_1029_zero__reorient,axiom,
! [X_69: complex] :
( ( zero_zero_complex = X_69 )
<=> ( X_69 = zero_zero_complex ) ) ).
thf(fact_1030_zero__reorient,axiom,
! [X_69: code_code_numeral] :
( ( zero_z126310315umeral = X_69 )
<=> ( X_69 = zero_z126310315umeral ) ) ).
thf(fact_1031_zero__reorient,axiom,
! [X_69: nat] :
( ( zero_zero_nat = X_69 )
<=> ( X_69 = zero_zero_nat ) ) ).
thf(fact_1032_zero__reorient,axiom,
! [X_69: real] :
( ( zero_zero_real = X_69 )
<=> ( X_69 = zero_zero_real ) ) ).
thf(fact_1033_zero__reorient,axiom,
! [X_69: int] :
( ( zero_zero_int = X_69 )
<=> ( X_69 = zero_zero_int ) ) ).
thf(fact_1034_ceiling__less__one,axiom,
! [X_68: rat] :
( ( ord_less_int @ ( archim1419379755ng_rat @ X_68 ) @ one_one_int )
<=> ( ord_less_eq_rat @ X_68 @ zero_zero_rat ) ) ).
thf(fact_1035_ceiling__less__one,axiom,
! [X_68: real] :
( ( ord_less_int @ ( archim856651990g_real @ X_68 ) @ one_one_int )
<=> ( ord_less_eq_real @ X_68 @ zero_zero_real ) ) ).
thf(fact_1036_one__le__ceiling,axiom,
! [X_67: rat] :
( ( ord_less_eq_int @ one_one_int @ ( archim1419379755ng_rat @ X_67 ) )
<=> ( ord_less_rat @ zero_zero_rat @ X_67 ) ) ).
thf(fact_1037_one__le__ceiling,axiom,
! [X_67: real] :
( ( ord_less_eq_int @ one_one_int @ ( archim856651990g_real @ X_67 ) )
<=> ( ord_less_real @ zero_zero_real @ X_67 ) ) ).
thf(fact_1038_divisors__zero,axiom,
! [A_239: rat,B_168: rat] :
( ( ( times_times_rat @ A_239 @ B_168 )
= zero_zero_rat )
=> ( ( A_239 = zero_zero_rat )
| ( B_168 = zero_zero_rat ) ) ) ).
thf(fact_1039_divisors__zero,axiom,
! [A_239: quickcheck_code_int,B_168: quickcheck_code_int] :
( ( ( times_123202395de_int @ A_239 @ B_168 )
= zero_z891286103de_int )
=> ( ( A_239 = zero_z891286103de_int )
| ( B_168 = zero_z891286103de_int ) ) ) ).
thf(fact_1040_divisors__zero,axiom,
! [A_239: complex,B_168: complex] :
( ( ( times_times_complex @ A_239 @ B_168 )
= zero_zero_complex )
=> ( ( A_239 = zero_zero_complex )
| ( B_168 = zero_zero_complex ) ) ) ).
thf(fact_1041_divisors__zero,axiom,
! [A_239: code_code_numeral,B_168: code_code_numeral] :
( ( ( times_1655362735umeral @ A_239 @ B_168 )
= zero_z126310315umeral )
=> ( ( A_239 = zero_z126310315umeral )
| ( B_168 = zero_z126310315umeral ) ) ) ).
thf(fact_1042_divisors__zero,axiom,
! [A_239: nat,B_168: nat] :
( ( ( times_times_nat @ A_239 @ B_168 )
= zero_zero_nat )
=> ( ( A_239 = zero_zero_nat )
| ( B_168 = zero_zero_nat ) ) ) ).
thf(fact_1043_divisors__zero,axiom,
! [A_239: real,B_168: real] :
( ( ( times_times_real @ A_239 @ B_168 )
= zero_zero_real )
=> ( ( A_239 = zero_zero_real )
| ( B_168 = zero_zero_real ) ) ) ).
thf(fact_1044_divisors__zero,axiom,
! [A_239: int,B_168: int] :
( ( ( times_times_int @ A_239 @ B_168 )
= zero_zero_int )
=> ( ( A_239 = zero_zero_int )
| ( B_168 = zero_zero_int ) ) ) ).
thf(fact_1045_no__zero__divisors,axiom,
! [B_167: rat,A_238: rat] :
( ( A_238 != zero_zero_rat )
=> ( ( B_167 != zero_zero_rat )
=> ( ( times_times_rat @ A_238 @ B_167 )
!= zero_zero_rat ) ) ) ).
thf(fact_1046_no__zero__divisors,axiom,
! [B_167: quickcheck_code_int,A_238: quickcheck_code_int] :
( ( A_238 != zero_z891286103de_int )
=> ( ( B_167 != zero_z891286103de_int )
=> ( ( times_123202395de_int @ A_238 @ B_167 )
!= zero_z891286103de_int ) ) ) ).
thf(fact_1047_no__zero__divisors,axiom,
! [B_167: complex,A_238: complex] :
( ( A_238 != zero_zero_complex )
=> ( ( B_167 != zero_zero_complex )
=> ( ( times_times_complex @ A_238 @ B_167 )
!= zero_zero_complex ) ) ) ).
thf(fact_1048_no__zero__divisors,axiom,
! [B_167: code_code_numeral,A_238: code_code_numeral] :
( ( A_238 != zero_z126310315umeral )
=> ( ( B_167 != zero_z126310315umeral )
=> ( ( times_1655362735umeral @ A_238 @ B_167 )
!= zero_z126310315umeral ) ) ) ).
thf(fact_1049_no__zero__divisors,axiom,
! [B_167: nat,A_238: nat] :
( ( A_238 != zero_zero_nat )
=> ( ( B_167 != zero_zero_nat )
=> ( ( times_times_nat @ A_238 @ B_167 )
!= zero_zero_nat ) ) ) ).
thf(fact_1050_no__zero__divisors,axiom,
! [B_167: real,A_238: real] :
( ( A_238 != zero_zero_real )
=> ( ( B_167 != zero_zero_real )
=> ( ( times_times_real @ A_238 @ B_167 )
!= zero_zero_real ) ) ) ).
thf(fact_1051_no__zero__divisors,axiom,
! [B_167: int,A_238: int] :
( ( A_238 != zero_zero_int )
=> ( ( B_167 != zero_zero_int )
=> ( ( times_times_int @ A_238 @ B_167 )
!= zero_zero_int ) ) ) ).
thf(fact_1052_mult__eq__0__iff,axiom,
! [A_237: rat,B_166: rat] :
( ( ( times_times_rat @ A_237 @ B_166 )
= zero_zero_rat )
<=> ( ( A_237 = zero_zero_rat )
| ( B_166 = zero_zero_rat ) ) ) ).
thf(fact_1053_mult__eq__0__iff,axiom,
! [A_237: complex,B_166: complex] :
( ( ( times_times_complex @ A_237 @ B_166 )
= zero_zero_complex )
<=> ( ( A_237 = zero_zero_complex )
| ( B_166 = zero_zero_complex ) ) ) ).
thf(fact_1054_mult__eq__0__iff,axiom,
! [A_237: real,B_166: real] :
( ( ( times_times_real @ A_237 @ B_166 )
= zero_zero_real )
<=> ( ( A_237 = zero_zero_real )
| ( B_166 = zero_zero_real ) ) ) ).
thf(fact_1055_mult__eq__0__iff,axiom,
! [A_237: int,B_166: int] :
( ( ( times_times_int @ A_237 @ B_166 )
= zero_zero_int )
<=> ( ( A_237 = zero_zero_int )
| ( B_166 = zero_zero_int ) ) ) ).
thf(fact_1056_mult__right_Ozero,axiom,
! [X_66: complex] :
( ( times_times_complex @ X_66 @ zero_zero_complex )
= zero_zero_complex ) ).
thf(fact_1057_mult__right_Ozero,axiom,
! [X_66: real] :
( ( times_times_real @ X_66 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_1058_mult_Ozero__right,axiom,
! [A_236: complex] :
( ( times_times_complex @ A_236 @ zero_zero_complex )
= zero_zero_complex ) ).
thf(fact_1059_mult_Ozero__right,axiom,
! [A_236: real] :
( ( times_times_real @ A_236 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_1060_mult__zero__right,axiom,
! [A_235: rat] :
( ( times_times_rat @ A_235 @ zero_zero_rat )
= zero_zero_rat ) ).
thf(fact_1061_mult__zero__right,axiom,
! [A_235: quickcheck_code_int] :
( ( times_123202395de_int @ A_235 @ zero_z891286103de_int )
= zero_z891286103de_int ) ).
thf(fact_1062_mult__zero__right,axiom,
! [A_235: complex] :
( ( times_times_complex @ A_235 @ zero_zero_complex )
= zero_zero_complex ) ).
thf(fact_1063_mult__zero__right,axiom,
! [A_235: code_code_numeral] :
( ( times_1655362735umeral @ A_235 @ zero_z126310315umeral )
= zero_z126310315umeral ) ).
thf(fact_1064_mult__zero__right,axiom,
! [A_235: nat] :
( ( times_times_nat @ A_235 @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_1065_mult__zero__right,axiom,
! [A_235: real] :
( ( times_times_real @ A_235 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_1066_mult__zero__right,axiom,
! [A_235: int] :
( ( times_times_int @ A_235 @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_1067_mult__left_Ozero,axiom,
! [Y_34: complex] :
( ( times_times_complex @ zero_zero_complex @ Y_34 )
= zero_zero_complex ) ).
thf(fact_1068_mult__left_Ozero,axiom,
! [Y_34: real] :
( ( times_times_real @ zero_zero_real @ Y_34 )
= zero_zero_real ) ).
thf(fact_1069_mult_Ozero__left,axiom,
! [B_165: complex] :
( ( times_times_complex @ zero_zero_complex @ B_165 )
= zero_zero_complex ) ).
thf(fact_1070_mult_Ozero__left,axiom,
! [B_165: real] :
( ( times_times_real @ zero_zero_real @ B_165 )
= zero_zero_real ) ).
thf(fact_1071_mult__zero__left,axiom,
! [A_234: rat] :
( ( times_times_rat @ zero_zero_rat @ A_234 )
= zero_zero_rat ) ).
thf(fact_1072_mult__zero__left,axiom,
! [A_234: quickcheck_code_int] :
( ( times_123202395de_int @ zero_z891286103de_int @ A_234 )
= zero_z891286103de_int ) ).
thf(fact_1073_mult__zero__left,axiom,
! [A_234: complex] :
( ( times_times_complex @ zero_zero_complex @ A_234 )
= zero_zero_complex ) ).
thf(fact_1074_mult__zero__left,axiom,
! [A_234: code_code_numeral] :
( ( times_1655362735umeral @ zero_z126310315umeral @ A_234 )
= zero_z126310315umeral ) ).
thf(fact_1075_mult__zero__left,axiom,
! [A_234: nat] :
( ( times_times_nat @ zero_zero_nat @ A_234 )
= zero_zero_nat ) ).
thf(fact_1076_mult__zero__left,axiom,
! [A_234: real] :
( ( times_times_real @ zero_zero_real @ A_234 )
= zero_zero_real ) ).
thf(fact_1077_mult__zero__left,axiom,
! [A_234: int] :
( ( times_times_int @ zero_zero_int @ A_234 )
= zero_zero_int ) ).
thf(fact_1078_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_233: rat] :
( ( times_times_rat @ A_233 @ zero_zero_rat )
= zero_zero_rat ) ).
thf(fact_1079_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_233: quickcheck_code_int] :
( ( times_123202395de_int @ A_233 @ zero_z891286103de_int )
= zero_z891286103de_int ) ).
thf(fact_1080_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_233: complex] :
( ( times_times_complex @ A_233 @ zero_zero_complex )
= zero_zero_complex ) ).
thf(fact_1081_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_233: code_code_numeral] :
( ( times_1655362735umeral @ A_233 @ zero_z126310315umeral )
= zero_z126310315umeral ) ).
thf(fact_1082_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_233: nat] :
( ( times_times_nat @ A_233 @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_1083_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_233: real] :
( ( times_times_real @ A_233 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_1084_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_233: int] :
( ( times_times_int @ A_233 @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_1085_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_232: rat] :
( ( times_times_rat @ zero_zero_rat @ A_232 )
= zero_zero_rat ) ).
thf(fact_1086_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_232: quickcheck_code_int] :
( ( times_123202395de_int @ zero_z891286103de_int @ A_232 )
= zero_z891286103de_int ) ).
thf(fact_1087_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_232: complex] :
( ( times_times_complex @ zero_zero_complex @ A_232 )
= zero_zero_complex ) ).
thf(fact_1088_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_232: code_code_numeral] :
( ( times_1655362735umeral @ zero_z126310315umeral @ A_232 )
= zero_z126310315umeral ) ).
thf(fact_1089_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_232: nat] :
( ( times_times_nat @ zero_zero_nat @ A_232 )
= zero_zero_nat ) ).
thf(fact_1090_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_232: real] :
( ( times_times_real @ zero_zero_real @ A_232 )
= zero_zero_real ) ).
thf(fact_1091_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_232: int] :
( ( times_times_int @ zero_zero_int @ A_232 )
= zero_zero_int ) ).
thf(fact_1092_double__eq__0__iff,axiom,
! [A_231: rat] :
( ( ( plus_plus_rat @ A_231 @ A_231 )
= zero_zero_rat )
<=> ( A_231 = zero_zero_rat ) ) ).
thf(fact_1093_double__eq__0__iff,axiom,
! [A_231: real] :
( ( ( plus_plus_real @ A_231 @ A_231 )
= zero_zero_real )
<=> ( A_231 = zero_zero_real ) ) ).
thf(fact_1094_double__eq__0__iff,axiom,
! [A_231: int] :
( ( ( plus_plus_int @ A_231 @ A_231 )
= zero_zero_int )
<=> ( A_231 = zero_zero_int ) ) ).
thf(fact_1095_add_Ocomm__neutral,axiom,
! [A_230: rat] :
( ( plus_plus_rat @ A_230 @ zero_zero_rat )
= A_230 ) ).
thf(fact_1096_add_Ocomm__neutral,axiom,
! [A_230: quickcheck_code_int] :
( ( plus_p1446045655de_int @ A_230 @ zero_z891286103de_int )
= A_230 ) ).
thf(fact_1097_add_Ocomm__neutral,axiom,
! [A_230: complex] :
( ( plus_plus_complex @ A_230 @ zero_zero_complex )
= A_230 ) ).
thf(fact_1098_add_Ocomm__neutral,axiom,
! [A_230: nat] :
( ( plus_plus_nat @ A_230 @ zero_zero_nat )
= A_230 ) ).
thf(fact_1099_add_Ocomm__neutral,axiom,
! [A_230: real] :
( ( plus_plus_real @ A_230 @ zero_zero_real )
= A_230 ) ).
thf(fact_1100_add_Ocomm__neutral,axiom,
! [A_230: int] :
( ( plus_plus_int @ A_230 @ zero_zero_int )
= A_230 ) ).
thf(fact_1101_add_Ocomm__neutral,axiom,
! [A_230: code_code_numeral] :
( ( plus_p1627245867umeral @ A_230 @ zero_z126310315umeral )
= A_230 ) ).
thf(fact_1102_add__0__right,axiom,
! [A_229: rat] :
( ( plus_plus_rat @ A_229 @ zero_zero_rat )
= A_229 ) ).
thf(fact_1103_add__0__right,axiom,
! [A_229: quickcheck_code_int] :
( ( plus_p1446045655de_int @ A_229 @ zero_z891286103de_int )
= A_229 ) ).
thf(fact_1104_add__0__right,axiom,
! [A_229: complex] :
( ( plus_plus_complex @ A_229 @ zero_zero_complex )
= A_229 ) ).
thf(fact_1105_add__0__right,axiom,
! [A_229: nat] :
( ( plus_plus_nat @ A_229 @ zero_zero_nat )
= A_229 ) ).
thf(fact_1106_add__0__right,axiom,
! [A_229: real] :
( ( plus_plus_real @ A_229 @ zero_zero_real )
= A_229 ) ).
thf(fact_1107_add__0__right,axiom,
! [A_229: int] :
( ( plus_plus_int @ A_229 @ zero_zero_int )
= A_229 ) ).
thf(fact_1108_add__0__right,axiom,
! [A_229: code_code_numeral] :
( ( plus_p1627245867umeral @ A_229 @ zero_z126310315umeral )
= A_229 ) ).
thf(fact_1109_double__zero__sym,axiom,
! [A_228: rat] :
( ( zero_zero_rat
= ( plus_plus_rat @ A_228 @ A_228 ) )
<=> ( A_228 = zero_zero_rat ) ) ).
thf(fact_1110_double__zero__sym,axiom,
! [A_228: real] :
( ( zero_zero_real
= ( plus_plus_real @ A_228 @ A_228 ) )
<=> ( A_228 = zero_zero_real ) ) ).
thf(fact_1111_double__zero__sym,axiom,
! [A_228: int] :
( ( zero_zero_int
= ( plus_plus_int @ A_228 @ A_228 ) )
<=> ( A_228 = zero_zero_int ) ) ).
thf(fact_1112_add__0,axiom,
! [A_227: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A_227 )
= A_227 ) ).
thf(fact_1113_add__0,axiom,
! [A_227: quickcheck_code_int] :
( ( plus_p1446045655de_int @ zero_z891286103de_int @ A_227 )
= A_227 ) ).
thf(fact_1114_add__0,axiom,
! [A_227: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A_227 )
= A_227 ) ).
thf(fact_1115_add__0,axiom,
! [A_227: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_227 )
= A_227 ) ).
thf(fact_1116_add__0,axiom,
! [A_227: real] :
( ( plus_plus_real @ zero_zero_real @ A_227 )
= A_227 ) ).
thf(fact_1117_add__0,axiom,
! [A_227: int] :
( ( plus_plus_int @ zero_zero_int @ A_227 )
= A_227 ) ).
thf(fact_1118_add__0,axiom,
! [A_227: code_code_numeral] :
( ( plus_p1627245867umeral @ zero_z126310315umeral @ A_227 )
= A_227 ) ).
thf(fact_1119_add__0__left,axiom,
! [A_226: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A_226 )
= A_226 ) ).
thf(fact_1120_add__0__left,axiom,
! [A_226: quickcheck_code_int] :
( ( plus_p1446045655de_int @ zero_z891286103de_int @ A_226 )
= A_226 ) ).
thf(fact_1121_add__0__left,axiom,
! [A_226: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A_226 )
= A_226 ) ).
thf(fact_1122_add__0__left,axiom,
! [A_226: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_226 )
= A_226 ) ).
thf(fact_1123_add__0__left,axiom,
! [A_226: real] :
( ( plus_plus_real @ zero_zero_real @ A_226 )
= A_226 ) ).
thf(fact_1124_add__0__left,axiom,
! [A_226: int] :
( ( plus_plus_int @ zero_zero_int @ A_226 )
= A_226 ) ).
thf(fact_1125_add__0__left,axiom,
! [A_226: code_code_numeral] :
( ( plus_p1627245867umeral @ zero_z126310315umeral @ A_226 )
= A_226 ) ).
thf(fact_1126_add__0__iff,axiom,
! [B_164: rat,A_225: rat] :
( ( B_164
= ( plus_plus_rat @ B_164 @ A_225 ) )
<=> ( A_225 = zero_zero_rat ) ) ).
thf(fact_1127_add__0__iff,axiom,
! [B_164: complex,A_225: complex] :
( ( B_164
= ( plus_plus_complex @ B_164 @ A_225 ) )
<=> ( A_225 = zero_zero_complex ) ) ).
thf(fact_1128_add__0__iff,axiom,
! [B_164: nat,A_225: nat] :
( ( B_164
= ( plus_plus_nat @ B_164 @ A_225 ) )
<=> ( A_225 = zero_zero_nat ) ) ).
thf(fact_1129_add__0__iff,axiom,
! [B_164: real,A_225: real] :
( ( B_164
= ( plus_plus_real @ B_164 @ A_225 ) )
<=> ( A_225 = zero_zero_real ) ) ).
thf(fact_1130_add__0__iff,axiom,
! [B_164: int,A_225: int] :
( ( B_164
= ( plus_plus_int @ B_164 @ A_225 ) )
<=> ( A_225 = zero_zero_int ) ) ).
thf(fact_1131_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_224: rat] :
( ( plus_plus_rat @ A_224 @ zero_zero_rat )
= A_224 ) ).
thf(fact_1132_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_224: quickcheck_code_int] :
( ( plus_p1446045655de_int @ A_224 @ zero_z891286103de_int )
= A_224 ) ).
thf(fact_1133_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_224: complex] :
( ( plus_plus_complex @ A_224 @ zero_zero_complex )
= A_224 ) ).
thf(fact_1134_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_224: nat] :
( ( plus_plus_nat @ A_224 @ zero_zero_nat )
= A_224 ) ).
thf(fact_1135_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_224: real] :
( ( plus_plus_real @ A_224 @ zero_zero_real )
= A_224 ) ).
thf(fact_1136_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_224: int] :
( ( plus_plus_int @ A_224 @ zero_zero_int )
= A_224 ) ).
thf(fact_1137_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_224: code_code_numeral] :
( ( plus_p1627245867umeral @ A_224 @ zero_z126310315umeral )
= A_224 ) ).
thf(fact_1138_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_223: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A_223 )
= A_223 ) ).
thf(fact_1139_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_223: quickcheck_code_int] :
( ( plus_p1446045655de_int @ zero_z891286103de_int @ A_223 )
= A_223 ) ).
thf(fact_1140_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_223: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A_223 )
= A_223 ) ).
thf(fact_1141_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_223: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A_223 )
= A_223 ) ).
thf(fact_1142_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_223: real] :
( ( plus_plus_real @ zero_zero_real @ A_223 )
= A_223 ) ).
thf(fact_1143_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_223: int] :
( ( plus_plus_int @ zero_zero_int @ A_223 )
= A_223 ) ).
thf(fact_1144_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_223: code_code_numeral] :
( ( plus_p1627245867umeral @ zero_z126310315umeral @ A_223 )
= A_223 ) ).
thf(fact_1145_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
thf(fact_1146_zero__neq__one,axiom,
zero_z891286103de_int != one_on1684967323de_int ).
thf(fact_1147_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
thf(fact_1148_zero__neq__one,axiom,
zero_z126310315umeral != one_on1645066479umeral ).
thf(fact_1149_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
thf(fact_1150_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
thf(fact_1151_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
thf(fact_1152_one__neq__zero,axiom,
one_one_rat != zero_zero_rat ).
thf(fact_1153_one__neq__zero,axiom,
one_on1684967323de_int != zero_z891286103de_int ).
thf(fact_1154_one__neq__zero,axiom,
one_one_complex != zero_zero_complex ).
thf(fact_1155_one__neq__zero,axiom,
one_on1645066479umeral != zero_z126310315umeral ).
thf(fact_1156_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
thf(fact_1157_one__neq__zero,axiom,
one_one_real != zero_zero_real ).
thf(fact_1158_one__neq__zero,axiom,
one_one_int != zero_zero_int ).
thf(fact_1159_right__minus__eq,axiom,
! [A_222: rat,B_163: rat] :
( ( ( minus_minus_rat @ A_222 @ B_163 )
= zero_zero_rat )
<=> ( A_222 = B_163 ) ) ).
thf(fact_1160_right__minus__eq,axiom,
! [A_222: complex,B_163: complex] :
( ( ( minus_minus_complex @ A_222 @ B_163 )
= zero_zero_complex )
<=> ( A_222 = B_163 ) ) ).
thf(fact_1161_right__minus__eq,axiom,
! [A_222: real,B_163: real] :
( ( ( minus_minus_real @ A_222 @ B_163 )
= zero_zero_real )
<=> ( A_222 = B_163 ) ) ).
thf(fact_1162_right__minus__eq,axiom,
! [A_222: int,B_163: int] :
( ( ( minus_minus_int @ A_222 @ B_163 )
= zero_zero_int )
<=> ( A_222 = B_163 ) ) ).
thf(fact_1163_eq__iff__diff__eq__0,axiom,
! [A_221: rat,B_162: rat] :
( ( A_221 = B_162 )
<=> ( ( minus_minus_rat @ A_221 @ B_162 )
= zero_zero_rat ) ) ).
thf(fact_1164_eq__iff__diff__eq__0,axiom,
! [A_221: complex,B_162: complex] :
( ( A_221 = B_162 )
<=> ( ( minus_minus_complex @ A_221 @ B_162 )
= zero_zero_complex ) ) ).
thf(fact_1165_eq__iff__diff__eq__0,axiom,
! [A_221: real,B_162: real] :
( ( A_221 = B_162 )
<=> ( ( minus_minus_real @ A_221 @ B_162 )
= zero_zero_real ) ) ).
thf(fact_1166_eq__iff__diff__eq__0,axiom,
! [A_221: int,B_162: int] :
( ( A_221 = B_162 )
<=> ( ( minus_minus_int @ A_221 @ B_162 )
= zero_zero_int ) ) ).
thf(fact_1167_diff__self,axiom,
! [A_220: rat] :
( ( minus_minus_rat @ A_220 @ A_220 )
= zero_zero_rat ) ).
thf(fact_1168_diff__self,axiom,
! [A_220: complex] :
( ( minus_minus_complex @ A_220 @ A_220 )
= zero_zero_complex ) ).
thf(fact_1169_diff__self,axiom,
! [A_220: real] :
( ( minus_minus_real @ A_220 @ A_220 )
= zero_zero_real ) ).
thf(fact_1170_diff__self,axiom,
! [A_220: int] :
( ( minus_minus_int @ A_220 @ A_220 )
= zero_zero_int ) ).
thf(fact_1171_diff__0__right,axiom,
! [A_219: rat] :
( ( minus_minus_rat @ A_219 @ zero_zero_rat )
= A_219 ) ).
thf(fact_1172_diff__0__right,axiom,
! [A_219: complex] :
( ( minus_minus_complex @ A_219 @ zero_zero_complex )
= A_219 ) ).
thf(fact_1173_diff__0__right,axiom,
! [A_219: real] :
( ( minus_minus_real @ A_219 @ zero_zero_real )
= A_219 ) ).
thf(fact_1174_diff__0__right,axiom,
! [A_219: int] :
( ( minus_minus_int @ A_219 @ zero_zero_int )
= A_219 ) ).
thf(fact_1175_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
thf(fact_1176_zadd__0__right,axiom,
! [Z_1: int] :
( ( plus_plus_int @ Z_1 @ zero_zero_int )
= Z_1 ) ).
thf(fact_1177_zadd__0,axiom,
! [Z_1: int] :
( ( plus_plus_int @ zero_zero_int @ Z_1 )
= Z_1 ) ).
thf(fact_1178_ceiling__less__cancel,axiom,
! [X_65: rat,Y_33: rat] :
( ( ord_less_int @ ( archim1419379755ng_rat @ X_65 ) @ ( archim1419379755ng_rat @ Y_33 ) )
=> ( ord_less_rat @ X_65 @ Y_33 ) ) ).
thf(fact_1179_ceiling__less__cancel,axiom,
! [X_65: real,Y_33: real] :
( ( ord_less_int @ ( archim856651990g_real @ X_65 ) @ ( archim856651990g_real @ Y_33 ) )
=> ( ord_less_real @ X_65 @ Y_33 ) ) ).
thf(fact_1180_ceiling__mono,axiom,
! [Y_32: real,X_64: real] :
( ( ord_less_eq_real @ Y_32 @ X_64 )
=> ( ord_less_eq_int @ ( archim856651990g_real @ Y_32 ) @ ( archim856651990g_real @ X_64 ) ) ) ).
thf(fact_1181_ceiling__mono,axiom,
! [Y_32: rat,X_64: rat] :
( ( ord_less_eq_rat @ Y_32 @ X_64 )
=> ( ord_less_eq_int @ ( archim1419379755ng_rat @ Y_32 ) @ ( archim1419379755ng_rat @ X_64 ) ) ) ).
thf(fact_1182_ceiling__one,axiom,
( ( archim1419379755ng_rat @ one_one_rat )
= one_one_int ) ).
thf(fact_1183_ceiling__one,axiom,
( ( archim856651990g_real @ one_one_real )
= one_one_int ) ).
thf(fact_1184_ceiling__number__of,axiom,
! [V_16: int] :
( ( archim856651990g_real @ ( number267125858f_real @ V_16 ) )
= ( number_number_of_int @ V_16 ) ) ).
thf(fact_1185_ceiling__number__of,axiom,
! [V_16: int] :
( ( archim1419379755ng_rat @ ( number_number_of_rat @ V_16 ) )
= ( number_number_of_int @ V_16 ) ) ).
thf(fact_1186_zero__le__square,axiom,
! [A_218: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A_218 @ A_218 ) ) ).
thf(fact_1187_zero__le__square,axiom,
! [A_218: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_218 @ A_218 ) ) ).
thf(fact_1188_zero__le__square,axiom,
! [A_218: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_218 @ A_218 ) ) ).
thf(fact_1189_zero__le__mult__iff,axiom,
! [A_217: rat,B_161: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A_217 @ B_161 ) )
<=> ( ( ( ord_less_eq_rat @ zero_zero_rat @ A_217 )
& ( ord_less_eq_rat @ zero_zero_rat @ B_161 ) )
| ( ( ord_less_eq_rat @ A_217 @ zero_zero_rat )
& ( ord_less_eq_rat @ B_161 @ zero_zero_rat ) ) ) ) ).
thf(fact_1190_zero__le__mult__iff,axiom,
! [A_217: real,B_161: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_217 @ B_161 ) )
<=> ( ( ( ord_less_eq_real @ zero_zero_real @ A_217 )
& ( ord_less_eq_real @ zero_zero_real @ B_161 ) )
| ( ( ord_less_eq_real @ A_217 @ zero_zero_real )
& ( ord_less_eq_real @ B_161 @ zero_zero_real ) ) ) ) ).
thf(fact_1191_zero__le__mult__iff,axiom,
! [A_217: int,B_161: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_217 @ B_161 ) )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ A_217 )
& ( ord_less_eq_int @ zero_zero_int @ B_161 ) )
| ( ( ord_less_eq_int @ A_217 @ zero_zero_int )
& ( ord_less_eq_int @ B_161 @ zero_zero_int ) ) ) ) ).
thf(fact_1192_mult__le__0__iff,axiom,
! [A_216: rat,B_160: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A_216 @ B_160 ) @ zero_zero_rat )
<=> ( ( ( ord_less_eq_rat @ zero_zero_rat @ A_216 )
& ( ord_less_eq_rat @ B_160 @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A_216 @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B_160 ) ) ) ) ).
thf(fact_1193_mult__le__0__iff,axiom,
! [A_216: real,B_160: real] :
( ( ord_less_eq_real @ ( times_times_real @ A_216 @ B_160 ) @ zero_zero_real )
<=> ( ( ( ord_less_eq_real @ zero_zero_real @ A_216 )
& ( ord_less_eq_real @ B_160 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A_216 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B_160 ) ) ) ) ).
thf(fact_1194_mult__le__0__iff,axiom,
! [A_216: int,B_160: int] :
( ( ord_less_eq_int @ ( times_times_int @ A_216 @ B_160 ) @ zero_zero_int )
<=> ( ( ( ord_less_eq_int @ zero_zero_int @ A_216 )
& ( ord_less_eq_int @ B_160 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A_216 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B_160 ) ) ) ) ).
thf(fact_1195_mult__nonneg__nonneg,axiom,
! [B_159: rat,A_215: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_215 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_159 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A_215 @ B_159 ) ) ) ) ).
thf(fact_1196_mult__nonneg__nonneg,axiom,
! [B_159: quickcheck_code_int,A_215: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_215 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_159 )
=> ( ord_le258702272de_int @ zero_z891286103de_int @ ( times_123202395de_int @ A_215 @ B_159 ) ) ) ) ).
thf(fact_1197_mult__nonneg__nonneg,axiom,
! [B_159: code_code_numeral,A_215: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_215 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_159 )
=> ( ord_le565307924umeral @ zero_z126310315umeral @ ( times_1655362735umeral @ A_215 @ B_159 ) ) ) ) ).
thf(fact_1198_mult__nonneg__nonneg,axiom,
! [B_159: nat,A_215: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_215 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_159 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A_215 @ B_159 ) ) ) ) ).
thf(fact_1199_mult__nonneg__nonneg,axiom,
! [B_159: real,A_215: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_215 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_159 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_215 @ B_159 ) ) ) ) ).
thf(fact_1200_mult__nonneg__nonneg,axiom,
! [B_159: int,A_215: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_215 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_159 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_215 @ B_159 ) ) ) ) ).
thf(fact_1201_mult__nonneg__nonpos,axiom,
! [B_158: rat,A_214: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_214 )
=> ( ( ord_less_eq_rat @ B_158 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_214 @ B_158 ) @ zero_zero_rat ) ) ) ).
thf(fact_1202_mult__nonneg__nonpos,axiom,
! [B_158: quickcheck_code_int,A_214: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_214 )
=> ( ( ord_le258702272de_int @ B_158 @ zero_z891286103de_int )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ A_214 @ B_158 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1203_mult__nonneg__nonpos,axiom,
! [B_158: code_code_numeral,A_214: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_214 )
=> ( ( ord_le565307924umeral @ B_158 @ zero_z126310315umeral )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ A_214 @ B_158 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1204_mult__nonneg__nonpos,axiom,
! [B_158: nat,A_214: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_214 )
=> ( ( ord_less_eq_nat @ B_158 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_214 @ B_158 ) @ zero_zero_nat ) ) ) ).
thf(fact_1205_mult__nonneg__nonpos,axiom,
! [B_158: real,A_214: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_214 )
=> ( ( ord_less_eq_real @ B_158 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A_214 @ B_158 ) @ zero_zero_real ) ) ) ).
thf(fact_1206_mult__nonneg__nonpos,axiom,
! [B_158: int,A_214: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_214 )
=> ( ( ord_less_eq_int @ B_158 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A_214 @ B_158 ) @ zero_zero_int ) ) ) ).
thf(fact_1207_mult__nonneg__nonpos2,axiom,
! [B_157: rat,A_213: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_213 )
=> ( ( ord_less_eq_rat @ B_157 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ B_157 @ A_213 ) @ zero_zero_rat ) ) ) ).
thf(fact_1208_mult__nonneg__nonpos2,axiom,
! [B_157: quickcheck_code_int,A_213: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_213 )
=> ( ( ord_le258702272de_int @ B_157 @ zero_z891286103de_int )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ B_157 @ A_213 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1209_mult__nonneg__nonpos2,axiom,
! [B_157: code_code_numeral,A_213: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_213 )
=> ( ( ord_le565307924umeral @ B_157 @ zero_z126310315umeral )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ B_157 @ A_213 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1210_mult__nonneg__nonpos2,axiom,
! [B_157: nat,A_213: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_213 )
=> ( ( ord_less_eq_nat @ B_157 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B_157 @ A_213 ) @ zero_zero_nat ) ) ) ).
thf(fact_1211_mult__nonneg__nonpos2,axiom,
! [B_157: real,A_213: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_213 )
=> ( ( ord_less_eq_real @ B_157 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B_157 @ A_213 ) @ zero_zero_real ) ) ) ).
thf(fact_1212_mult__nonneg__nonpos2,axiom,
! [B_157: int,A_213: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_213 )
=> ( ( ord_less_eq_int @ B_157 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B_157 @ A_213 ) @ zero_zero_int ) ) ) ).
thf(fact_1213_mult__nonpos__nonneg,axiom,
! [B_156: rat,A_212: rat] :
( ( ord_less_eq_rat @ A_212 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_156 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_212 @ B_156 ) @ zero_zero_rat ) ) ) ).
thf(fact_1214_mult__nonpos__nonneg,axiom,
! [B_156: quickcheck_code_int,A_212: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_212 @ zero_z891286103de_int )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_156 )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ A_212 @ B_156 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1215_mult__nonpos__nonneg,axiom,
! [B_156: code_code_numeral,A_212: code_code_numeral] :
( ( ord_le565307924umeral @ A_212 @ zero_z126310315umeral )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_156 )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ A_212 @ B_156 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1216_mult__nonpos__nonneg,axiom,
! [B_156: nat,A_212: nat] :
( ( ord_less_eq_nat @ A_212 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_156 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_212 @ B_156 ) @ zero_zero_nat ) ) ) ).
thf(fact_1217_mult__nonpos__nonneg,axiom,
! [B_156: real,A_212: real] :
( ( ord_less_eq_real @ A_212 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_156 )
=> ( ord_less_eq_real @ ( times_times_real @ A_212 @ B_156 ) @ zero_zero_real ) ) ) ).
thf(fact_1218_mult__nonpos__nonneg,axiom,
! [B_156: int,A_212: int] :
( ( ord_less_eq_int @ A_212 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_156 )
=> ( ord_less_eq_int @ ( times_times_int @ A_212 @ B_156 ) @ zero_zero_int ) ) ) ).
thf(fact_1219_mult__nonpos__nonpos,axiom,
! [B_155: rat,A_211: rat] :
( ( ord_less_eq_rat @ A_211 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B_155 @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A_211 @ B_155 ) ) ) ) ).
thf(fact_1220_mult__nonpos__nonpos,axiom,
! [B_155: real,A_211: real] :
( ( ord_less_eq_real @ A_211 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B_155 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_211 @ B_155 ) ) ) ) ).
thf(fact_1221_mult__nonpos__nonpos,axiom,
! [B_155: int,A_211: int] :
( ( ord_less_eq_int @ A_211 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B_155 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_211 @ B_155 ) ) ) ) ).
thf(fact_1222_mult__right__mono,axiom,
! [C_68: rat,A_210: rat,B_154: rat] :
( ( ord_less_eq_rat @ A_210 @ B_154 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_68 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_210 @ C_68 ) @ ( times_times_rat @ B_154 @ C_68 ) ) ) ) ).
thf(fact_1223_mult__right__mono,axiom,
! [C_68: quickcheck_code_int,A_210: quickcheck_code_int,B_154: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_210 @ B_154 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_68 )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ A_210 @ C_68 ) @ ( times_123202395de_int @ B_154 @ C_68 ) ) ) ) ).
thf(fact_1224_mult__right__mono,axiom,
! [C_68: code_code_numeral,A_210: code_code_numeral,B_154: code_code_numeral] :
( ( ord_le565307924umeral @ A_210 @ B_154 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_68 )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ A_210 @ C_68 ) @ ( times_1655362735umeral @ B_154 @ C_68 ) ) ) ) ).
thf(fact_1225_mult__right__mono,axiom,
! [C_68: nat,A_210: nat,B_154: nat] :
( ( ord_less_eq_nat @ A_210 @ B_154 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_68 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_210 @ C_68 ) @ ( times_times_nat @ B_154 @ C_68 ) ) ) ) ).
thf(fact_1226_mult__right__mono,axiom,
! [C_68: real,A_210: real,B_154: real] :
( ( ord_less_eq_real @ A_210 @ B_154 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_68 )
=> ( ord_less_eq_real @ ( times_times_real @ A_210 @ C_68 ) @ ( times_times_real @ B_154 @ C_68 ) ) ) ) ).
thf(fact_1227_mult__right__mono,axiom,
! [C_68: int,A_210: int,B_154: int] :
( ( ord_less_eq_int @ A_210 @ B_154 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_68 )
=> ( ord_less_eq_int @ ( times_times_int @ A_210 @ C_68 ) @ ( times_times_int @ B_154 @ C_68 ) ) ) ) ).
thf(fact_1228_mult__left__mono,axiom,
! [C_67: rat,A_209: rat,B_153: rat] :
( ( ord_less_eq_rat @ A_209 @ B_153 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_67 )
=> ( ord_less_eq_rat @ ( times_times_rat @ C_67 @ A_209 ) @ ( times_times_rat @ C_67 @ B_153 ) ) ) ) ).
thf(fact_1229_mult__left__mono,axiom,
! [C_67: quickcheck_code_int,A_209: quickcheck_code_int,B_153: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_209 @ B_153 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_67 )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ C_67 @ A_209 ) @ ( times_123202395de_int @ C_67 @ B_153 ) ) ) ) ).
thf(fact_1230_mult__left__mono,axiom,
! [C_67: code_code_numeral,A_209: code_code_numeral,B_153: code_code_numeral] :
( ( ord_le565307924umeral @ A_209 @ B_153 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_67 )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ C_67 @ A_209 ) @ ( times_1655362735umeral @ C_67 @ B_153 ) ) ) ) ).
thf(fact_1231_mult__left__mono,axiom,
! [C_67: nat,A_209: nat,B_153: nat] :
( ( ord_less_eq_nat @ A_209 @ B_153 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_67 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C_67 @ A_209 ) @ ( times_times_nat @ C_67 @ B_153 ) ) ) ) ).
thf(fact_1232_mult__left__mono,axiom,
! [C_67: real,A_209: real,B_153: real] :
( ( ord_less_eq_real @ A_209 @ B_153 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_67 )
=> ( ord_less_eq_real @ ( times_times_real @ C_67 @ A_209 ) @ ( times_times_real @ C_67 @ B_153 ) ) ) ) ).
thf(fact_1233_mult__left__mono,axiom,
! [C_67: int,A_209: int,B_153: int] :
( ( ord_less_eq_int @ A_209 @ B_153 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_67 )
=> ( ord_less_eq_int @ ( times_times_int @ C_67 @ A_209 ) @ ( times_times_int @ C_67 @ B_153 ) ) ) ) ).
thf(fact_1234_comm__mult__left__mono,axiom,
! [C_66: rat,A_208: rat,B_152: rat] :
( ( ord_less_eq_rat @ A_208 @ B_152 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_66 )
=> ( ord_less_eq_rat @ ( times_times_rat @ C_66 @ A_208 ) @ ( times_times_rat @ C_66 @ B_152 ) ) ) ) ).
thf(fact_1235_comm__mult__left__mono,axiom,
! [C_66: quickcheck_code_int,A_208: quickcheck_code_int,B_152: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_208 @ B_152 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_66 )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ C_66 @ A_208 ) @ ( times_123202395de_int @ C_66 @ B_152 ) ) ) ) ).
thf(fact_1236_comm__mult__left__mono,axiom,
! [C_66: code_code_numeral,A_208: code_code_numeral,B_152: code_code_numeral] :
( ( ord_le565307924umeral @ A_208 @ B_152 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_66 )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ C_66 @ A_208 ) @ ( times_1655362735umeral @ C_66 @ B_152 ) ) ) ) ).
thf(fact_1237_comm__mult__left__mono,axiom,
! [C_66: nat,A_208: nat,B_152: nat] :
( ( ord_less_eq_nat @ A_208 @ B_152 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_66 )
=> ( ord_less_eq_nat @ ( times_times_nat @ C_66 @ A_208 ) @ ( times_times_nat @ C_66 @ B_152 ) ) ) ) ).
thf(fact_1238_comm__mult__left__mono,axiom,
! [C_66: real,A_208: real,B_152: real] :
( ( ord_less_eq_real @ A_208 @ B_152 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_66 )
=> ( ord_less_eq_real @ ( times_times_real @ C_66 @ A_208 ) @ ( times_times_real @ C_66 @ B_152 ) ) ) ) ).
thf(fact_1239_comm__mult__left__mono,axiom,
! [C_66: int,A_208: int,B_152: int] :
( ( ord_less_eq_int @ A_208 @ B_152 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_66 )
=> ( ord_less_eq_int @ ( times_times_int @ C_66 @ A_208 ) @ ( times_times_int @ C_66 @ B_152 ) ) ) ) ).
thf(fact_1240_mult__right__mono__neg,axiom,
! [C_65: rat,B_151: rat,A_207: rat] :
( ( ord_less_eq_rat @ B_151 @ A_207 )
=> ( ( ord_less_eq_rat @ C_65 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_207 @ C_65 ) @ ( times_times_rat @ B_151 @ C_65 ) ) ) ) ).
thf(fact_1241_mult__right__mono__neg,axiom,
! [C_65: real,B_151: real,A_207: real] :
( ( ord_less_eq_real @ B_151 @ A_207 )
=> ( ( ord_less_eq_real @ C_65 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A_207 @ C_65 ) @ ( times_times_real @ B_151 @ C_65 ) ) ) ) ).
thf(fact_1242_mult__right__mono__neg,axiom,
! [C_65: int,B_151: int,A_207: int] :
( ( ord_less_eq_int @ B_151 @ A_207 )
=> ( ( ord_less_eq_int @ C_65 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A_207 @ C_65 ) @ ( times_times_int @ B_151 @ C_65 ) ) ) ) ).
thf(fact_1243_mult__left__mono__neg,axiom,
! [C_64: rat,B_150: rat,A_206: rat] :
( ( ord_less_eq_rat @ B_150 @ A_206 )
=> ( ( ord_less_eq_rat @ C_64 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ C_64 @ A_206 ) @ ( times_times_rat @ C_64 @ B_150 ) ) ) ) ).
thf(fact_1244_mult__left__mono__neg,axiom,
! [C_64: real,B_150: real,A_206: real] :
( ( ord_less_eq_real @ B_150 @ A_206 )
=> ( ( ord_less_eq_real @ C_64 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C_64 @ A_206 ) @ ( times_times_real @ C_64 @ B_150 ) ) ) ) ).
thf(fact_1245_mult__left__mono__neg,axiom,
! [C_64: int,B_150: int,A_206: int] :
( ( ord_less_eq_int @ B_150 @ A_206 )
=> ( ( ord_less_eq_int @ C_64 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C_64 @ A_206 ) @ ( times_times_int @ C_64 @ B_150 ) ) ) ) ).
thf(fact_1246_mult__mono_H,axiom,
! [C_63: rat,D_12: rat,A_205: rat,B_149: rat] :
( ( ord_less_eq_rat @ A_205 @ B_149 )
=> ( ( ord_less_eq_rat @ C_63 @ D_12 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A_205 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_63 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_205 @ C_63 ) @ ( times_times_rat @ B_149 @ D_12 ) ) ) ) ) ) ).
thf(fact_1247_mult__mono_H,axiom,
! [C_63: quickcheck_code_int,D_12: quickcheck_code_int,A_205: quickcheck_code_int,B_149: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_205 @ B_149 )
=> ( ( ord_le258702272de_int @ C_63 @ D_12 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_205 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_63 )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ A_205 @ C_63 ) @ ( times_123202395de_int @ B_149 @ D_12 ) ) ) ) ) ) ).
thf(fact_1248_mult__mono_H,axiom,
! [C_63: code_code_numeral,D_12: code_code_numeral,A_205: code_code_numeral,B_149: code_code_numeral] :
( ( ord_le565307924umeral @ A_205 @ B_149 )
=> ( ( ord_le565307924umeral @ C_63 @ D_12 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_205 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_63 )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ A_205 @ C_63 ) @ ( times_1655362735umeral @ B_149 @ D_12 ) ) ) ) ) ) ).
thf(fact_1249_mult__mono_H,axiom,
! [C_63: nat,D_12: nat,A_205: nat,B_149: nat] :
( ( ord_less_eq_nat @ A_205 @ B_149 )
=> ( ( ord_less_eq_nat @ C_63 @ D_12 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_205 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_63 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_205 @ C_63 ) @ ( times_times_nat @ B_149 @ D_12 ) ) ) ) ) ) ).
thf(fact_1250_mult__mono_H,axiom,
! [C_63: real,D_12: real,A_205: real,B_149: real] :
( ( ord_less_eq_real @ A_205 @ B_149 )
=> ( ( ord_less_eq_real @ C_63 @ D_12 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_205 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_63 )
=> ( ord_less_eq_real @ ( times_times_real @ A_205 @ C_63 ) @ ( times_times_real @ B_149 @ D_12 ) ) ) ) ) ) ).
thf(fact_1251_mult__mono_H,axiom,
! [C_63: int,D_12: int,A_205: int,B_149: int] :
( ( ord_less_eq_int @ A_205 @ B_149 )
=> ( ( ord_less_eq_int @ C_63 @ D_12 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_205 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_63 )
=> ( ord_less_eq_int @ ( times_times_int @ A_205 @ C_63 ) @ ( times_times_int @ B_149 @ D_12 ) ) ) ) ) ) ).
thf(fact_1252_mult__mono,axiom,
! [C_62: rat,D_11: rat,A_204: rat,B_148: rat] :
( ( ord_less_eq_rat @ A_204 @ B_148 )
=> ( ( ord_less_eq_rat @ C_62 @ D_11 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_148 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_62 )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_204 @ C_62 ) @ ( times_times_rat @ B_148 @ D_11 ) ) ) ) ) ) ).
thf(fact_1253_mult__mono,axiom,
! [C_62: quickcheck_code_int,D_11: quickcheck_code_int,A_204: quickcheck_code_int,B_148: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_204 @ B_148 )
=> ( ( ord_le258702272de_int @ C_62 @ D_11 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_148 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_62 )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ A_204 @ C_62 ) @ ( times_123202395de_int @ B_148 @ D_11 ) ) ) ) ) ) ).
thf(fact_1254_mult__mono,axiom,
! [C_62: code_code_numeral,D_11: code_code_numeral,A_204: code_code_numeral,B_148: code_code_numeral] :
( ( ord_le565307924umeral @ A_204 @ B_148 )
=> ( ( ord_le565307924umeral @ C_62 @ D_11 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_148 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_62 )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ A_204 @ C_62 ) @ ( times_1655362735umeral @ B_148 @ D_11 ) ) ) ) ) ) ).
thf(fact_1255_mult__mono,axiom,
! [C_62: nat,D_11: nat,A_204: nat,B_148: nat] :
( ( ord_less_eq_nat @ A_204 @ B_148 )
=> ( ( ord_less_eq_nat @ C_62 @ D_11 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_148 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_62 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_204 @ C_62 ) @ ( times_times_nat @ B_148 @ D_11 ) ) ) ) ) ) ).
thf(fact_1256_mult__mono,axiom,
! [C_62: real,D_11: real,A_204: real,B_148: real] :
( ( ord_less_eq_real @ A_204 @ B_148 )
=> ( ( ord_less_eq_real @ C_62 @ D_11 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_148 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_62 )
=> ( ord_less_eq_real @ ( times_times_real @ A_204 @ C_62 ) @ ( times_times_real @ B_148 @ D_11 ) ) ) ) ) ) ).
thf(fact_1257_mult__mono,axiom,
! [C_62: int,D_11: int,A_204: int,B_148: int] :
( ( ord_less_eq_int @ A_204 @ B_148 )
=> ( ( ord_less_eq_int @ C_62 @ D_11 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_148 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_62 )
=> ( ord_less_eq_int @ ( times_times_int @ A_204 @ C_62 ) @ ( times_times_int @ B_148 @ D_11 ) ) ) ) ) ) ).
thf(fact_1258_split__mult__pos__le,axiom,
! [B_147: rat,A_203: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A_203 )
& ( ord_less_eq_rat @ zero_zero_rat @ B_147 ) )
| ( ( ord_less_eq_rat @ A_203 @ zero_zero_rat )
& ( ord_less_eq_rat @ B_147 @ zero_zero_rat ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A_203 @ B_147 ) ) ) ).
thf(fact_1259_split__mult__pos__le,axiom,
! [B_147: real,A_203: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A_203 )
& ( ord_less_eq_real @ zero_zero_real @ B_147 ) )
| ( ( ord_less_eq_real @ A_203 @ zero_zero_real )
& ( ord_less_eq_real @ B_147 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A_203 @ B_147 ) ) ) ).
thf(fact_1260_split__mult__pos__le,axiom,
! [B_147: int,A_203: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A_203 )
& ( ord_less_eq_int @ zero_zero_int @ B_147 ) )
| ( ( ord_less_eq_int @ A_203 @ zero_zero_int )
& ( ord_less_eq_int @ B_147 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A_203 @ B_147 ) ) ) ).
thf(fact_1261_split__mult__neg__le,axiom,
! [B_146: rat,A_202: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A_202 )
& ( ord_less_eq_rat @ B_146 @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A_202 @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B_146 ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ A_202 @ B_146 ) @ zero_zero_rat ) ) ).
thf(fact_1262_split__mult__neg__le,axiom,
! [B_146: quickcheck_code_int,A_202: quickcheck_code_int] :
( ( ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_202 )
& ( ord_le258702272de_int @ B_146 @ zero_z891286103de_int ) )
| ( ( ord_le258702272de_int @ A_202 @ zero_z891286103de_int )
& ( ord_le258702272de_int @ zero_z891286103de_int @ B_146 ) ) )
=> ( ord_le258702272de_int @ ( times_123202395de_int @ A_202 @ B_146 ) @ zero_z891286103de_int ) ) ).
thf(fact_1263_split__mult__neg__le,axiom,
! [B_146: code_code_numeral,A_202: code_code_numeral] :
( ( ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_202 )
& ( ord_le565307924umeral @ B_146 @ zero_z126310315umeral ) )
| ( ( ord_le565307924umeral @ A_202 @ zero_z126310315umeral )
& ( ord_le565307924umeral @ zero_z126310315umeral @ B_146 ) ) )
=> ( ord_le565307924umeral @ ( times_1655362735umeral @ A_202 @ B_146 ) @ zero_z126310315umeral ) ) ).
thf(fact_1264_split__mult__neg__le,axiom,
! [B_146: nat,A_202: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A_202 )
& ( ord_less_eq_nat @ B_146 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A_202 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B_146 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A_202 @ B_146 ) @ zero_zero_nat ) ) ).
thf(fact_1265_split__mult__neg__le,axiom,
! [B_146: real,A_202: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A_202 )
& ( ord_less_eq_real @ B_146 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A_202 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B_146 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A_202 @ B_146 ) @ zero_zero_real ) ) ).
thf(fact_1266_split__mult__neg__le,axiom,
! [B_146: int,A_202: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A_202 )
& ( ord_less_eq_int @ B_146 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A_202 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B_146 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A_202 @ B_146 ) @ zero_zero_int ) ) ).
thf(fact_1267_not__square__less__zero,axiom,
! [A_201: rat] :
~ ( ord_less_rat @ ( times_times_rat @ A_201 @ A_201 ) @ zero_zero_rat ) ).
thf(fact_1268_not__square__less__zero,axiom,
! [A_201: real] :
~ ( ord_less_real @ ( times_times_real @ A_201 @ A_201 ) @ zero_zero_real ) ).
thf(fact_1269_not__square__less__zero,axiom,
! [A_201: int] :
~ ( ord_less_int @ ( times_times_int @ A_201 @ A_201 ) @ zero_zero_int ) ).
thf(fact_1270_mult__less__cancel__right__disj,axiom,
! [A_200: rat,C_61: rat,B_145: rat] :
( ( ord_less_rat @ ( times_times_rat @ A_200 @ C_61 ) @ ( times_times_rat @ B_145 @ C_61 ) )
<=> ( ( ( ord_less_rat @ zero_zero_rat @ C_61 )
& ( ord_less_rat @ A_200 @ B_145 ) )
| ( ( ord_less_rat @ C_61 @ zero_zero_rat )
& ( ord_less_rat @ B_145 @ A_200 ) ) ) ) ).
thf(fact_1271_mult__less__cancel__right__disj,axiom,
! [A_200: real,C_61: real,B_145: real] :
( ( ord_less_real @ ( times_times_real @ A_200 @ C_61 ) @ ( times_times_real @ B_145 @ C_61 ) )
<=> ( ( ( ord_less_real @ zero_zero_real @ C_61 )
& ( ord_less_real @ A_200 @ B_145 ) )
| ( ( ord_less_real @ C_61 @ zero_zero_real )
& ( ord_less_real @ B_145 @ A_200 ) ) ) ) ).
thf(fact_1272_mult__less__cancel__right__disj,axiom,
! [A_200: int,C_61: int,B_145: int] :
( ( ord_less_int @ ( times_times_int @ A_200 @ C_61 ) @ ( times_times_int @ B_145 @ C_61 ) )
<=> ( ( ( ord_less_int @ zero_zero_int @ C_61 )
& ( ord_less_int @ A_200 @ B_145 ) )
| ( ( ord_less_int @ C_61 @ zero_zero_int )
& ( ord_less_int @ B_145 @ A_200 ) ) ) ) ).
thf(fact_1273_mult__less__cancel__left__disj,axiom,
! [C_60: rat,A_199: rat,B_144: rat] :
( ( ord_less_rat @ ( times_times_rat @ C_60 @ A_199 ) @ ( times_times_rat @ C_60 @ B_144 ) )
<=> ( ( ( ord_less_rat @ zero_zero_rat @ C_60 )
& ( ord_less_rat @ A_199 @ B_144 ) )
| ( ( ord_less_rat @ C_60 @ zero_zero_rat )
& ( ord_less_rat @ B_144 @ A_199 ) ) ) ) ).
thf(fact_1274_mult__less__cancel__left__disj,axiom,
! [C_60: real,A_199: real,B_144: real] :
( ( ord_less_real @ ( times_times_real @ C_60 @ A_199 ) @ ( times_times_real @ C_60 @ B_144 ) )
<=> ( ( ( ord_less_real @ zero_zero_real @ C_60 )
& ( ord_less_real @ A_199 @ B_144 ) )
| ( ( ord_less_real @ C_60 @ zero_zero_real )
& ( ord_less_real @ B_144 @ A_199 ) ) ) ) ).
thf(fact_1275_mult__less__cancel__left__disj,axiom,
! [C_60: int,A_199: int,B_144: int] :
( ( ord_less_int @ ( times_times_int @ C_60 @ A_199 ) @ ( times_times_int @ C_60 @ B_144 ) )
<=> ( ( ( ord_less_int @ zero_zero_int @ C_60 )
& ( ord_less_int @ A_199 @ B_144 ) )
| ( ( ord_less_int @ C_60 @ zero_zero_int )
& ( ord_less_int @ B_144 @ A_199 ) ) ) ) ).
thf(fact_1276_mult__less__cancel__left__pos,axiom,
! [A_198: rat,B_143: rat,C_59: rat] :
( ( ord_less_rat @ zero_zero_rat @ C_59 )
=> ( ( ord_less_rat @ ( times_times_rat @ C_59 @ A_198 ) @ ( times_times_rat @ C_59 @ B_143 ) )
<=> ( ord_less_rat @ A_198 @ B_143 ) ) ) ).
thf(fact_1277_mult__less__cancel__left__pos,axiom,
! [A_198: real,B_143: real,C_59: real] :
( ( ord_less_real @ zero_zero_real @ C_59 )
=> ( ( ord_less_real @ ( times_times_real @ C_59 @ A_198 ) @ ( times_times_real @ C_59 @ B_143 ) )
<=> ( ord_less_real @ A_198 @ B_143 ) ) ) ).
thf(fact_1278_mult__less__cancel__left__pos,axiom,
! [A_198: int,B_143: int,C_59: int] :
( ( ord_less_int @ zero_zero_int @ C_59 )
=> ( ( ord_less_int @ ( times_times_int @ C_59 @ A_198 ) @ ( times_times_int @ C_59 @ B_143 ) )
<=> ( ord_less_int @ A_198 @ B_143 ) ) ) ).
thf(fact_1279_mult__pos__pos,axiom,
! [B_142: rat,A_197: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_197 )
=> ( ( ord_less_rat @ zero_zero_rat @ B_142 )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A_197 @ B_142 ) ) ) ) ).
thf(fact_1280_mult__pos__pos,axiom,
! [B_142: quickcheck_code_int,A_197: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_197 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ B_142 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ ( times_123202395de_int @ A_197 @ B_142 ) ) ) ) ).
thf(fact_1281_mult__pos__pos,axiom,
! [B_142: code_code_numeral,A_197: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_197 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ B_142 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ ( times_1655362735umeral @ A_197 @ B_142 ) ) ) ) ).
thf(fact_1282_mult__pos__pos,axiom,
! [B_142: nat,A_197: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_197 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_142 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A_197 @ B_142 ) ) ) ) ).
thf(fact_1283_mult__pos__pos,axiom,
! [B_142: real,A_197: real] :
( ( ord_less_real @ zero_zero_real @ A_197 )
=> ( ( ord_less_real @ zero_zero_real @ B_142 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A_197 @ B_142 ) ) ) ) ).
thf(fact_1284_mult__pos__pos,axiom,
! [B_142: int,A_197: int] :
( ( ord_less_int @ zero_zero_int @ A_197 )
=> ( ( ord_less_int @ zero_zero_int @ B_142 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_197 @ B_142 ) ) ) ) ).
thf(fact_1285_mult__pos__neg,axiom,
! [B_141: rat,A_196: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_196 )
=> ( ( ord_less_rat @ B_141 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A_196 @ B_141 ) @ zero_zero_rat ) ) ) ).
thf(fact_1286_mult__pos__neg,axiom,
! [B_141: quickcheck_code_int,A_196: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_196 )
=> ( ( ord_le1860547276de_int @ B_141 @ zero_z891286103de_int )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_196 @ B_141 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1287_mult__pos__neg,axiom,
! [B_141: code_code_numeral,A_196: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_196 )
=> ( ( ord_le1304079648umeral @ B_141 @ zero_z126310315umeral )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_196 @ B_141 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1288_mult__pos__neg,axiom,
! [B_141: nat,A_196: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_196 )
=> ( ( ord_less_nat @ B_141 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A_196 @ B_141 ) @ zero_zero_nat ) ) ) ).
thf(fact_1289_mult__pos__neg,axiom,
! [B_141: real,A_196: real] :
( ( ord_less_real @ zero_zero_real @ A_196 )
=> ( ( ord_less_real @ B_141 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A_196 @ B_141 ) @ zero_zero_real ) ) ) ).
thf(fact_1290_mult__pos__neg,axiom,
! [B_141: int,A_196: int] :
( ( ord_less_int @ zero_zero_int @ A_196 )
=> ( ( ord_less_int @ B_141 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A_196 @ B_141 ) @ zero_zero_int ) ) ) ).
thf(fact_1291_mult__pos__neg2,axiom,
! [B_140: rat,A_195: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_195 )
=> ( ( ord_less_rat @ B_140 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ B_140 @ A_195 ) @ zero_zero_rat ) ) ) ).
thf(fact_1292_mult__pos__neg2,axiom,
! [B_140: quickcheck_code_int,A_195: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_195 )
=> ( ( ord_le1860547276de_int @ B_140 @ zero_z891286103de_int )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ B_140 @ A_195 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1293_mult__pos__neg2,axiom,
! [B_140: code_code_numeral,A_195: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_195 )
=> ( ( ord_le1304079648umeral @ B_140 @ zero_z126310315umeral )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ B_140 @ A_195 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1294_mult__pos__neg2,axiom,
! [B_140: nat,A_195: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_195 )
=> ( ( ord_less_nat @ B_140 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B_140 @ A_195 ) @ zero_zero_nat ) ) ) ).
thf(fact_1295_mult__pos__neg2,axiom,
! [B_140: real,A_195: real] :
( ( ord_less_real @ zero_zero_real @ A_195 )
=> ( ( ord_less_real @ B_140 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B_140 @ A_195 ) @ zero_zero_real ) ) ) ).
thf(fact_1296_mult__pos__neg2,axiom,
! [B_140: int,A_195: int] :
( ( ord_less_int @ zero_zero_int @ A_195 )
=> ( ( ord_less_int @ B_140 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B_140 @ A_195 ) @ zero_zero_int ) ) ) ).
thf(fact_1297_zero__less__mult__pos,axiom,
! [A_194: rat,B_139: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A_194 @ B_139 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A_194 )
=> ( ord_less_rat @ zero_zero_rat @ B_139 ) ) ) ).
thf(fact_1298_zero__less__mult__pos,axiom,
! [A_194: quickcheck_code_int,B_139: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ ( times_123202395de_int @ A_194 @ B_139 ) )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_194 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ B_139 ) ) ) ).
thf(fact_1299_zero__less__mult__pos,axiom,
! [A_194: code_code_numeral,B_139: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ ( times_1655362735umeral @ A_194 @ B_139 ) )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_194 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ B_139 ) ) ) ).
thf(fact_1300_zero__less__mult__pos,axiom,
! [A_194: nat,B_139: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A_194 @ B_139 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A_194 )
=> ( ord_less_nat @ zero_zero_nat @ B_139 ) ) ) ).
thf(fact_1301_zero__less__mult__pos,axiom,
! [A_194: real,B_139: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A_194 @ B_139 ) )
=> ( ( ord_less_real @ zero_zero_real @ A_194 )
=> ( ord_less_real @ zero_zero_real @ B_139 ) ) ) ).
thf(fact_1302_zero__less__mult__pos,axiom,
! [A_194: int,B_139: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_194 @ B_139 ) )
=> ( ( ord_less_int @ zero_zero_int @ A_194 )
=> ( ord_less_int @ zero_zero_int @ B_139 ) ) ) ).
thf(fact_1303_zero__less__mult__pos2,axiom,
! [B_138: rat,A_193: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B_138 @ A_193 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A_193 )
=> ( ord_less_rat @ zero_zero_rat @ B_138 ) ) ) ).
thf(fact_1304_zero__less__mult__pos2,axiom,
! [B_138: quickcheck_code_int,A_193: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ ( times_123202395de_int @ B_138 @ A_193 ) )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_193 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ B_138 ) ) ) ).
thf(fact_1305_zero__less__mult__pos2,axiom,
! [B_138: code_code_numeral,A_193: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ ( times_1655362735umeral @ B_138 @ A_193 ) )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_193 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ B_138 ) ) ) ).
thf(fact_1306_zero__less__mult__pos2,axiom,
! [B_138: nat,A_193: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B_138 @ A_193 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A_193 )
=> ( ord_less_nat @ zero_zero_nat @ B_138 ) ) ) ).
thf(fact_1307_zero__less__mult__pos2,axiom,
! [B_138: real,A_193: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B_138 @ A_193 ) )
=> ( ( ord_less_real @ zero_zero_real @ A_193 )
=> ( ord_less_real @ zero_zero_real @ B_138 ) ) ) ).
thf(fact_1308_zero__less__mult__pos2,axiom,
! [B_138: int,A_193: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B_138 @ A_193 ) )
=> ( ( ord_less_int @ zero_zero_int @ A_193 )
=> ( ord_less_int @ zero_zero_int @ B_138 ) ) ) ).
thf(fact_1309_mult__less__cancel__left__neg,axiom,
! [A_192: rat,B_137: rat,C_58: rat] :
( ( ord_less_rat @ C_58 @ zero_zero_rat )
=> ( ( ord_less_rat @ ( times_times_rat @ C_58 @ A_192 ) @ ( times_times_rat @ C_58 @ B_137 ) )
<=> ( ord_less_rat @ B_137 @ A_192 ) ) ) ).
thf(fact_1310_mult__less__cancel__left__neg,axiom,
! [A_192: real,B_137: real,C_58: real] :
( ( ord_less_real @ C_58 @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C_58 @ A_192 ) @ ( times_times_real @ C_58 @ B_137 ) )
<=> ( ord_less_real @ B_137 @ A_192 ) ) ) ).
thf(fact_1311_mult__less__cancel__left__neg,axiom,
! [A_192: int,B_137: int,C_58: int] :
( ( ord_less_int @ C_58 @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C_58 @ A_192 ) @ ( times_times_int @ C_58 @ B_137 ) )
<=> ( ord_less_int @ B_137 @ A_192 ) ) ) ).
thf(fact_1312_mult__neg__pos,axiom,
! [B_136: rat,A_191: rat] :
( ( ord_less_rat @ A_191 @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ B_136 )
=> ( ord_less_rat @ ( times_times_rat @ A_191 @ B_136 ) @ zero_zero_rat ) ) ) ).
thf(fact_1313_mult__neg__pos,axiom,
! [B_136: quickcheck_code_int,A_191: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_191 @ zero_z891286103de_int )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ B_136 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_191 @ B_136 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1314_mult__neg__pos,axiom,
! [B_136: code_code_numeral,A_191: code_code_numeral] :
( ( ord_le1304079648umeral @ A_191 @ zero_z126310315umeral )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ B_136 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_191 @ B_136 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1315_mult__neg__pos,axiom,
! [B_136: nat,A_191: nat] :
( ( ord_less_nat @ A_191 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B_136 )
=> ( ord_less_nat @ ( times_times_nat @ A_191 @ B_136 ) @ zero_zero_nat ) ) ) ).
thf(fact_1316_mult__neg__pos,axiom,
! [B_136: real,A_191: real] :
( ( ord_less_real @ A_191 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B_136 )
=> ( ord_less_real @ ( times_times_real @ A_191 @ B_136 ) @ zero_zero_real ) ) ) ).
thf(fact_1317_mult__neg__pos,axiom,
! [B_136: int,A_191: int] :
( ( ord_less_int @ A_191 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B_136 )
=> ( ord_less_int @ ( times_times_int @ A_191 @ B_136 ) @ zero_zero_int ) ) ) ).
thf(fact_1318_mult__neg__neg,axiom,
! [B_135: rat,A_190: rat] :
( ( ord_less_rat @ A_190 @ zero_zero_rat )
=> ( ( ord_less_rat @ B_135 @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A_190 @ B_135 ) ) ) ) ).
thf(fact_1319_mult__neg__neg,axiom,
! [B_135: real,A_190: real] :
( ( ord_less_real @ A_190 @ zero_zero_real )
=> ( ( ord_less_real @ B_135 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A_190 @ B_135 ) ) ) ) ).
thf(fact_1320_mult__neg__neg,axiom,
! [B_135: int,A_190: int] :
( ( ord_less_int @ A_190 @ zero_zero_int )
=> ( ( ord_less_int @ B_135 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A_190 @ B_135 ) ) ) ) ).
thf(fact_1321_mult__strict__right__mono,axiom,
! [C_57: rat,A_189: rat,B_134: rat] :
( ( ord_less_rat @ A_189 @ B_134 )
=> ( ( ord_less_rat @ zero_zero_rat @ C_57 )
=> ( ord_less_rat @ ( times_times_rat @ A_189 @ C_57 ) @ ( times_times_rat @ B_134 @ C_57 ) ) ) ) ).
thf(fact_1322_mult__strict__right__mono,axiom,
! [C_57: quickcheck_code_int,A_189: quickcheck_code_int,B_134: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_189 @ B_134 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ C_57 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_189 @ C_57 ) @ ( times_123202395de_int @ B_134 @ C_57 ) ) ) ) ).
thf(fact_1323_mult__strict__right__mono,axiom,
! [C_57: code_code_numeral,A_189: code_code_numeral,B_134: code_code_numeral] :
( ( ord_le1304079648umeral @ A_189 @ B_134 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ C_57 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_189 @ C_57 ) @ ( times_1655362735umeral @ B_134 @ C_57 ) ) ) ) ).
thf(fact_1324_mult__strict__right__mono,axiom,
! [C_57: nat,A_189: nat,B_134: nat] :
( ( ord_less_nat @ A_189 @ B_134 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_57 )
=> ( ord_less_nat @ ( times_times_nat @ A_189 @ C_57 ) @ ( times_times_nat @ B_134 @ C_57 ) ) ) ) ).
thf(fact_1325_mult__strict__right__mono,axiom,
! [C_57: real,A_189: real,B_134: real] :
( ( ord_less_real @ A_189 @ B_134 )
=> ( ( ord_less_real @ zero_zero_real @ C_57 )
=> ( ord_less_real @ ( times_times_real @ A_189 @ C_57 ) @ ( times_times_real @ B_134 @ C_57 ) ) ) ) ).
thf(fact_1326_mult__strict__right__mono,axiom,
! [C_57: int,A_189: int,B_134: int] :
( ( ord_less_int @ A_189 @ B_134 )
=> ( ( ord_less_int @ zero_zero_int @ C_57 )
=> ( ord_less_int @ ( times_times_int @ A_189 @ C_57 ) @ ( times_times_int @ B_134 @ C_57 ) ) ) ) ).
thf(fact_1327_mult__strict__left__mono,axiom,
! [C_56: rat,A_188: rat,B_133: rat] :
( ( ord_less_rat @ A_188 @ B_133 )
=> ( ( ord_less_rat @ zero_zero_rat @ C_56 )
=> ( ord_less_rat @ ( times_times_rat @ C_56 @ A_188 ) @ ( times_times_rat @ C_56 @ B_133 ) ) ) ) ).
thf(fact_1328_mult__strict__left__mono,axiom,
! [C_56: quickcheck_code_int,A_188: quickcheck_code_int,B_133: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_188 @ B_133 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ C_56 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ C_56 @ A_188 ) @ ( times_123202395de_int @ C_56 @ B_133 ) ) ) ) ).
thf(fact_1329_mult__strict__left__mono,axiom,
! [C_56: code_code_numeral,A_188: code_code_numeral,B_133: code_code_numeral] :
( ( ord_le1304079648umeral @ A_188 @ B_133 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ C_56 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ C_56 @ A_188 ) @ ( times_1655362735umeral @ C_56 @ B_133 ) ) ) ) ).
thf(fact_1330_mult__strict__left__mono,axiom,
! [C_56: nat,A_188: nat,B_133: nat] :
( ( ord_less_nat @ A_188 @ B_133 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_56 )
=> ( ord_less_nat @ ( times_times_nat @ C_56 @ A_188 ) @ ( times_times_nat @ C_56 @ B_133 ) ) ) ) ).
thf(fact_1331_mult__strict__left__mono,axiom,
! [C_56: real,A_188: real,B_133: real] :
( ( ord_less_real @ A_188 @ B_133 )
=> ( ( ord_less_real @ zero_zero_real @ C_56 )
=> ( ord_less_real @ ( times_times_real @ C_56 @ A_188 ) @ ( times_times_real @ C_56 @ B_133 ) ) ) ) ).
thf(fact_1332_mult__strict__left__mono,axiom,
! [C_56: int,A_188: int,B_133: int] :
( ( ord_less_int @ A_188 @ B_133 )
=> ( ( ord_less_int @ zero_zero_int @ C_56 )
=> ( ord_less_int @ ( times_times_int @ C_56 @ A_188 ) @ ( times_times_int @ C_56 @ B_133 ) ) ) ) ).
thf(fact_1333_comm__mult__strict__left__mono,axiom,
! [C_55: rat,A_187: rat,B_132: rat] :
( ( ord_less_rat @ A_187 @ B_132 )
=> ( ( ord_less_rat @ zero_zero_rat @ C_55 )
=> ( ord_less_rat @ ( times_times_rat @ C_55 @ A_187 ) @ ( times_times_rat @ C_55 @ B_132 ) ) ) ) ).
thf(fact_1334_comm__mult__strict__left__mono,axiom,
! [C_55: quickcheck_code_int,A_187: quickcheck_code_int,B_132: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_187 @ B_132 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ C_55 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ C_55 @ A_187 ) @ ( times_123202395de_int @ C_55 @ B_132 ) ) ) ) ).
thf(fact_1335_comm__mult__strict__left__mono,axiom,
! [C_55: code_code_numeral,A_187: code_code_numeral,B_132: code_code_numeral] :
( ( ord_le1304079648umeral @ A_187 @ B_132 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ C_55 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ C_55 @ A_187 ) @ ( times_1655362735umeral @ C_55 @ B_132 ) ) ) ) ).
thf(fact_1336_comm__mult__strict__left__mono,axiom,
! [C_55: nat,A_187: nat,B_132: nat] :
( ( ord_less_nat @ A_187 @ B_132 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_55 )
=> ( ord_less_nat @ ( times_times_nat @ C_55 @ A_187 ) @ ( times_times_nat @ C_55 @ B_132 ) ) ) ) ).
thf(fact_1337_comm__mult__strict__left__mono,axiom,
! [C_55: real,A_187: real,B_132: real] :
( ( ord_less_real @ A_187 @ B_132 )
=> ( ( ord_less_real @ zero_zero_real @ C_55 )
=> ( ord_less_real @ ( times_times_real @ C_55 @ A_187 ) @ ( times_times_real @ C_55 @ B_132 ) ) ) ) ).
thf(fact_1338_comm__mult__strict__left__mono,axiom,
! [C_55: int,A_187: int,B_132: int] :
( ( ord_less_int @ A_187 @ B_132 )
=> ( ( ord_less_int @ zero_zero_int @ C_55 )
=> ( ord_less_int @ ( times_times_int @ C_55 @ A_187 ) @ ( times_times_int @ C_55 @ B_132 ) ) ) ) ).
thf(fact_1339_mult__strict__right__mono__neg,axiom,
! [C_54: rat,B_131: rat,A_186: rat] :
( ( ord_less_rat @ B_131 @ A_186 )
=> ( ( ord_less_rat @ C_54 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A_186 @ C_54 ) @ ( times_times_rat @ B_131 @ C_54 ) ) ) ) ).
thf(fact_1340_mult__strict__right__mono__neg,axiom,
! [C_54: real,B_131: real,A_186: real] :
( ( ord_less_real @ B_131 @ A_186 )
=> ( ( ord_less_real @ C_54 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A_186 @ C_54 ) @ ( times_times_real @ B_131 @ C_54 ) ) ) ) ).
thf(fact_1341_mult__strict__right__mono__neg,axiom,
! [C_54: int,B_131: int,A_186: int] :
( ( ord_less_int @ B_131 @ A_186 )
=> ( ( ord_less_int @ C_54 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A_186 @ C_54 ) @ ( times_times_int @ B_131 @ C_54 ) ) ) ) ).
thf(fact_1342_mult__strict__left__mono__neg,axiom,
! [C_53: rat,B_130: rat,A_185: rat] :
( ( ord_less_rat @ B_130 @ A_185 )
=> ( ( ord_less_rat @ C_53 @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ C_53 @ A_185 ) @ ( times_times_rat @ C_53 @ B_130 ) ) ) ) ).
thf(fact_1343_mult__strict__left__mono__neg,axiom,
! [C_53: real,B_130: real,A_185: real] :
( ( ord_less_real @ B_130 @ A_185 )
=> ( ( ord_less_real @ C_53 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C_53 @ A_185 ) @ ( times_times_real @ C_53 @ B_130 ) ) ) ) ).
thf(fact_1344_mult__strict__left__mono__neg,axiom,
! [C_53: int,B_130: int,A_185: int] :
( ( ord_less_int @ B_130 @ A_185 )
=> ( ( ord_less_int @ C_53 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C_53 @ A_185 ) @ ( times_times_int @ C_53 @ B_130 ) ) ) ) ).
thf(fact_1345_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A_184: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A_184 @ A_184 ) )
<=> ( ord_less_eq_rat @ zero_zero_rat @ A_184 ) ) ).
thf(fact_1346_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A_184: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A_184 @ A_184 ) )
<=> ( ord_less_eq_real @ zero_zero_real @ A_184 ) ) ).
thf(fact_1347_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A_184: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A_184 @ A_184 ) )
<=> ( ord_less_eq_int @ zero_zero_int @ A_184 ) ) ).
thf(fact_1348_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A_183: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A_183 @ A_183 ) @ zero_zero_rat )
<=> ( ord_less_eq_rat @ A_183 @ zero_zero_rat ) ) ).
thf(fact_1349_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A_183: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A_183 @ A_183 ) @ zero_zero_real )
<=> ( ord_less_eq_real @ A_183 @ zero_zero_real ) ) ).
thf(fact_1350_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A_183: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_183 @ A_183 ) @ zero_zero_int )
<=> ( ord_less_eq_int @ A_183 @ zero_zero_int ) ) ).
thf(fact_1351_add__nonneg__nonneg,axiom,
! [B_129: rat,A_182: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_182 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_129 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A_182 @ B_129 ) ) ) ) ).
thf(fact_1352_add__nonneg__nonneg,axiom,
! [B_129: quickcheck_code_int,A_182: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_182 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_129 )
=> ( ord_le258702272de_int @ zero_z891286103de_int @ ( plus_p1446045655de_int @ A_182 @ B_129 ) ) ) ) ).
thf(fact_1353_add__nonneg__nonneg,axiom,
! [B_129: nat,A_182: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_182 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_129 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A_182 @ B_129 ) ) ) ) ).
thf(fact_1354_add__nonneg__nonneg,axiom,
! [B_129: real,A_182: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_182 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_129 )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A_182 @ B_129 ) ) ) ) ).
thf(fact_1355_add__nonneg__nonneg,axiom,
! [B_129: int,A_182: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_182 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_129 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A_182 @ B_129 ) ) ) ) ).
thf(fact_1356_add__nonneg__nonneg,axiom,
! [B_129: code_code_numeral,A_182: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_182 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_129 )
=> ( ord_le565307924umeral @ zero_z126310315umeral @ ( plus_p1627245867umeral @ A_182 @ B_129 ) ) ) ) ).
thf(fact_1357_add__nonneg__eq__0__iff,axiom,
! [Y_31: rat,X_63: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X_63 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y_31 )
=> ( ( ( plus_plus_rat @ X_63 @ Y_31 )
= zero_zero_rat )
<=> ( ( X_63 = zero_zero_rat )
& ( Y_31 = zero_zero_rat ) ) ) ) ) ).
thf(fact_1358_add__nonneg__eq__0__iff,axiom,
! [Y_31: quickcheck_code_int,X_63: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ X_63 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ Y_31 )
=> ( ( ( plus_p1446045655de_int @ X_63 @ Y_31 )
= zero_z891286103de_int )
<=> ( ( X_63 = zero_z891286103de_int )
& ( Y_31 = zero_z891286103de_int ) ) ) ) ) ).
thf(fact_1359_add__nonneg__eq__0__iff,axiom,
! [Y_31: nat,X_63: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X_63 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y_31 )
=> ( ( ( plus_plus_nat @ X_63 @ Y_31 )
= zero_zero_nat )
<=> ( ( X_63 = zero_zero_nat )
& ( Y_31 = zero_zero_nat ) ) ) ) ) ).
thf(fact_1360_add__nonneg__eq__0__iff,axiom,
! [Y_31: real,X_63: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_63 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_31 )
=> ( ( ( plus_plus_real @ X_63 @ Y_31 )
= zero_zero_real )
<=> ( ( X_63 = zero_zero_real )
& ( Y_31 = zero_zero_real ) ) ) ) ) ).
thf(fact_1361_add__nonneg__eq__0__iff,axiom,
! [Y_31: int,X_63: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_63 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_31 )
=> ( ( ( plus_plus_int @ X_63 @ Y_31 )
= zero_zero_int )
<=> ( ( X_63 = zero_zero_int )
& ( Y_31 = zero_zero_int ) ) ) ) ) ).
thf(fact_1362_add__nonneg__eq__0__iff,axiom,
! [Y_31: code_code_numeral,X_63: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ X_63 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ Y_31 )
=> ( ( ( plus_p1627245867umeral @ X_63 @ Y_31 )
= zero_z126310315umeral )
<=> ( ( X_63 = zero_z126310315umeral )
& ( Y_31 = zero_z126310315umeral ) ) ) ) ) ).
thf(fact_1363_add__increasing,axiom,
! [B_128: rat,C_52: rat,A_181: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_181 )
=> ( ( ord_less_eq_rat @ B_128 @ C_52 )
=> ( ord_less_eq_rat @ B_128 @ ( plus_plus_rat @ A_181 @ C_52 ) ) ) ) ).
thf(fact_1364_add__increasing,axiom,
! [B_128: quickcheck_code_int,C_52: quickcheck_code_int,A_181: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_181 )
=> ( ( ord_le258702272de_int @ B_128 @ C_52 )
=> ( ord_le258702272de_int @ B_128 @ ( plus_p1446045655de_int @ A_181 @ C_52 ) ) ) ) ).
thf(fact_1365_add__increasing,axiom,
! [B_128: nat,C_52: nat,A_181: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_181 )
=> ( ( ord_less_eq_nat @ B_128 @ C_52 )
=> ( ord_less_eq_nat @ B_128 @ ( plus_plus_nat @ A_181 @ C_52 ) ) ) ) ).
thf(fact_1366_add__increasing,axiom,
! [B_128: real,C_52: real,A_181: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_181 )
=> ( ( ord_less_eq_real @ B_128 @ C_52 )
=> ( ord_less_eq_real @ B_128 @ ( plus_plus_real @ A_181 @ C_52 ) ) ) ) ).
thf(fact_1367_add__increasing,axiom,
! [B_128: int,C_52: int,A_181: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_181 )
=> ( ( ord_less_eq_int @ B_128 @ C_52 )
=> ( ord_less_eq_int @ B_128 @ ( plus_plus_int @ A_181 @ C_52 ) ) ) ) ).
thf(fact_1368_add__increasing,axiom,
! [B_128: code_code_numeral,C_52: code_code_numeral,A_181: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_181 )
=> ( ( ord_le565307924umeral @ B_128 @ C_52 )
=> ( ord_le565307924umeral @ B_128 @ ( plus_p1627245867umeral @ A_181 @ C_52 ) ) ) ) ).
thf(fact_1369_add__increasing2,axiom,
! [B_127: rat,A_180: rat,C_51: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ C_51 )
=> ( ( ord_less_eq_rat @ B_127 @ A_180 )
=> ( ord_less_eq_rat @ B_127 @ ( plus_plus_rat @ A_180 @ C_51 ) ) ) ) ).
thf(fact_1370_add__increasing2,axiom,
! [B_127: quickcheck_code_int,A_180: quickcheck_code_int,C_51: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ C_51 )
=> ( ( ord_le258702272de_int @ B_127 @ A_180 )
=> ( ord_le258702272de_int @ B_127 @ ( plus_p1446045655de_int @ A_180 @ C_51 ) ) ) ) ).
thf(fact_1371_add__increasing2,axiom,
! [B_127: nat,A_180: nat,C_51: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C_51 )
=> ( ( ord_less_eq_nat @ B_127 @ A_180 )
=> ( ord_less_eq_nat @ B_127 @ ( plus_plus_nat @ A_180 @ C_51 ) ) ) ) ).
thf(fact_1372_add__increasing2,axiom,
! [B_127: real,A_180: real,C_51: real] :
( ( ord_less_eq_real @ zero_zero_real @ C_51 )
=> ( ( ord_less_eq_real @ B_127 @ A_180 )
=> ( ord_less_eq_real @ B_127 @ ( plus_plus_real @ A_180 @ C_51 ) ) ) ) ).
thf(fact_1373_add__increasing2,axiom,
! [B_127: int,A_180: int,C_51: int] :
( ( ord_less_eq_int @ zero_zero_int @ C_51 )
=> ( ( ord_less_eq_int @ B_127 @ A_180 )
=> ( ord_less_eq_int @ B_127 @ ( plus_plus_int @ A_180 @ C_51 ) ) ) ) ).
thf(fact_1374_add__increasing2,axiom,
! [B_127: code_code_numeral,A_180: code_code_numeral,C_51: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ C_51 )
=> ( ( ord_le565307924umeral @ B_127 @ A_180 )
=> ( ord_le565307924umeral @ B_127 @ ( plus_p1627245867umeral @ A_180 @ C_51 ) ) ) ) ).
thf(fact_1375_add__nonpos__nonpos,axiom,
! [B_126: rat,A_179: rat] :
( ( ord_less_eq_rat @ A_179 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B_126 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A_179 @ B_126 ) @ zero_zero_rat ) ) ) ).
thf(fact_1376_add__nonpos__nonpos,axiom,
! [B_126: quickcheck_code_int,A_179: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_179 @ zero_z891286103de_int )
=> ( ( ord_le258702272de_int @ B_126 @ zero_z891286103de_int )
=> ( ord_le258702272de_int @ ( plus_p1446045655de_int @ A_179 @ B_126 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1377_add__nonpos__nonpos,axiom,
! [B_126: nat,A_179: nat] :
( ( ord_less_eq_nat @ A_179 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B_126 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A_179 @ B_126 ) @ zero_zero_nat ) ) ) ).
thf(fact_1378_add__nonpos__nonpos,axiom,
! [B_126: real,A_179: real] :
( ( ord_less_eq_real @ A_179 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B_126 @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A_179 @ B_126 ) @ zero_zero_real ) ) ) ).
thf(fact_1379_add__nonpos__nonpos,axiom,
! [B_126: int,A_179: int] :
( ( ord_less_eq_int @ A_179 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B_126 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_179 @ B_126 ) @ zero_zero_int ) ) ) ).
thf(fact_1380_add__nonpos__nonpos,axiom,
! [B_126: code_code_numeral,A_179: code_code_numeral] :
( ( ord_le565307924umeral @ A_179 @ zero_z126310315umeral )
=> ( ( ord_le565307924umeral @ B_126 @ zero_z126310315umeral )
=> ( ord_le565307924umeral @ ( plus_p1627245867umeral @ A_179 @ B_126 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1381_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_178: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A_178 @ A_178 ) )
<=> ( ord_less_rat @ zero_zero_rat @ A_178 ) ) ).
thf(fact_1382_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_178: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A_178 @ A_178 ) )
<=> ( ord_less_real @ zero_zero_real @ A_178 ) ) ).
thf(fact_1383_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_178: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_178 @ A_178 ) )
<=> ( ord_less_int @ zero_zero_int @ A_178 ) ) ).
thf(fact_1384_even__less__0__iff,axiom,
! [A_177: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A_177 @ A_177 ) @ zero_zero_rat )
<=> ( ord_less_rat @ A_177 @ zero_zero_rat ) ) ).
thf(fact_1385_even__less__0__iff,axiom,
! [A_177: real] :
( ( ord_less_real @ ( plus_plus_real @ A_177 @ A_177 ) @ zero_zero_real )
<=> ( ord_less_real @ A_177 @ zero_zero_real ) ) ).
thf(fact_1386_even__less__0__iff,axiom,
! [A_177: int] :
( ( ord_less_int @ ( plus_plus_int @ A_177 @ A_177 ) @ zero_zero_int )
<=> ( ord_less_int @ A_177 @ zero_zero_int ) ) ).
thf(fact_1387_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_176: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A_176 @ A_176 ) @ zero_zero_rat )
<=> ( ord_less_rat @ A_176 @ zero_zero_rat ) ) ).
thf(fact_1388_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_176: real] :
( ( ord_less_real @ ( plus_plus_real @ A_176 @ A_176 ) @ zero_zero_real )
<=> ( ord_less_real @ A_176 @ zero_zero_real ) ) ).
thf(fact_1389_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_176: int] :
( ( ord_less_int @ ( plus_plus_int @ A_176 @ A_176 ) @ zero_zero_int )
<=> ( ord_less_int @ A_176 @ zero_zero_int ) ) ).
thf(fact_1390_add__pos__pos,axiom,
! [B_125: rat,A_175: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_175 )
=> ( ( ord_less_rat @ zero_zero_rat @ B_125 )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A_175 @ B_125 ) ) ) ) ).
thf(fact_1391_add__pos__pos,axiom,
! [B_125: quickcheck_code_int,A_175: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_175 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ B_125 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ ( plus_p1446045655de_int @ A_175 @ B_125 ) ) ) ) ).
thf(fact_1392_add__pos__pos,axiom,
! [B_125: nat,A_175: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_175 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_125 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A_175 @ B_125 ) ) ) ) ).
thf(fact_1393_add__pos__pos,axiom,
! [B_125: real,A_175: real] :
( ( ord_less_real @ zero_zero_real @ A_175 )
=> ( ( ord_less_real @ zero_zero_real @ B_125 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A_175 @ B_125 ) ) ) ) ).
thf(fact_1394_add__pos__pos,axiom,
! [B_125: int,A_175: int] :
( ( ord_less_int @ zero_zero_int @ A_175 )
=> ( ( ord_less_int @ zero_zero_int @ B_125 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_175 @ B_125 ) ) ) ) ).
thf(fact_1395_add__pos__pos,axiom,
! [B_125: code_code_numeral,A_175: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_175 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ B_125 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ ( plus_p1627245867umeral @ A_175 @ B_125 ) ) ) ) ).
thf(fact_1396_pos__add__strict,axiom,
! [B_124: rat,C_50: rat,A_174: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_174 )
=> ( ( ord_less_rat @ B_124 @ C_50 )
=> ( ord_less_rat @ B_124 @ ( plus_plus_rat @ A_174 @ C_50 ) ) ) ) ).
thf(fact_1397_pos__add__strict,axiom,
! [B_124: quickcheck_code_int,C_50: quickcheck_code_int,A_174: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_174 )
=> ( ( ord_le1860547276de_int @ B_124 @ C_50 )
=> ( ord_le1860547276de_int @ B_124 @ ( plus_p1446045655de_int @ A_174 @ C_50 ) ) ) ) ).
thf(fact_1398_pos__add__strict,axiom,
! [B_124: nat,C_50: nat,A_174: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_174 )
=> ( ( ord_less_nat @ B_124 @ C_50 )
=> ( ord_less_nat @ B_124 @ ( plus_plus_nat @ A_174 @ C_50 ) ) ) ) ).
thf(fact_1399_pos__add__strict,axiom,
! [B_124: real,C_50: real,A_174: real] :
( ( ord_less_real @ zero_zero_real @ A_174 )
=> ( ( ord_less_real @ B_124 @ C_50 )
=> ( ord_less_real @ B_124 @ ( plus_plus_real @ A_174 @ C_50 ) ) ) ) ).
thf(fact_1400_pos__add__strict,axiom,
! [B_124: int,C_50: int,A_174: int] :
( ( ord_less_int @ zero_zero_int @ A_174 )
=> ( ( ord_less_int @ B_124 @ C_50 )
=> ( ord_less_int @ B_124 @ ( plus_plus_int @ A_174 @ C_50 ) ) ) ) ).
thf(fact_1401_pos__add__strict,axiom,
! [B_124: code_code_numeral,C_50: code_code_numeral,A_174: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_174 )
=> ( ( ord_le1304079648umeral @ B_124 @ C_50 )
=> ( ord_le1304079648umeral @ B_124 @ ( plus_p1627245867umeral @ A_174 @ C_50 ) ) ) ) ).
thf(fact_1402_add__neg__neg,axiom,
! [B_123: rat,A_173: rat] :
( ( ord_less_rat @ A_173 @ zero_zero_rat )
=> ( ( ord_less_rat @ B_123 @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A_173 @ B_123 ) @ zero_zero_rat ) ) ) ).
thf(fact_1403_add__neg__neg,axiom,
! [B_123: quickcheck_code_int,A_173: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_173 @ zero_z891286103de_int )
=> ( ( ord_le1860547276de_int @ B_123 @ zero_z891286103de_int )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_173 @ B_123 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1404_add__neg__neg,axiom,
! [B_123: nat,A_173: nat] :
( ( ord_less_nat @ A_173 @ zero_zero_nat )
=> ( ( ord_less_nat @ B_123 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A_173 @ B_123 ) @ zero_zero_nat ) ) ) ).
thf(fact_1405_add__neg__neg,axiom,
! [B_123: real,A_173: real] :
( ( ord_less_real @ A_173 @ zero_zero_real )
=> ( ( ord_less_real @ B_123 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A_173 @ B_123 ) @ zero_zero_real ) ) ) ).
thf(fact_1406_add__neg__neg,axiom,
! [B_123: int,A_173: int] :
( ( ord_less_int @ A_173 @ zero_zero_int )
=> ( ( ord_less_int @ B_123 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A_173 @ B_123 ) @ zero_zero_int ) ) ) ).
thf(fact_1407_add__neg__neg,axiom,
! [B_123: code_code_numeral,A_173: code_code_numeral] :
( ( ord_le1304079648umeral @ A_173 @ zero_z126310315umeral )
=> ( ( ord_le1304079648umeral @ B_123 @ zero_z126310315umeral )
=> ( ord_le1304079648umeral @ ( plus_p1627245867umeral @ A_173 @ B_123 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1408_zero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
thf(fact_1409_zero__le__one,axiom,
ord_le258702272de_int @ zero_z891286103de_int @ one_on1684967323de_int ).
thf(fact_1410_zero__le__one,axiom,
ord_le565307924umeral @ zero_z126310315umeral @ one_on1645066479umeral ).
thf(fact_1411_zero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
thf(fact_1412_zero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
thf(fact_1413_zero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
thf(fact_1414_not__one__le__zero,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
thf(fact_1415_not__one__le__zero,axiom,
~ ( ord_le258702272de_int @ one_on1684967323de_int @ zero_z891286103de_int ) ).
thf(fact_1416_not__one__le__zero,axiom,
~ ( ord_le565307924umeral @ one_on1645066479umeral @ zero_z126310315umeral ) ).
thf(fact_1417_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
thf(fact_1418_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
thf(fact_1419_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
thf(fact_1420_zero__less__one,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
thf(fact_1421_zero__less__one,axiom,
ord_le1860547276de_int @ zero_z891286103de_int @ one_on1684967323de_int ).
thf(fact_1422_zero__less__one,axiom,
ord_le1304079648umeral @ zero_z126310315umeral @ one_on1645066479umeral ).
thf(fact_1423_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
thf(fact_1424_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
thf(fact_1425_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
thf(fact_1426_not__one__less__zero,axiom,
~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
thf(fact_1427_not__one__less__zero,axiom,
~ ( ord_le1860547276de_int @ one_on1684967323de_int @ zero_z891286103de_int ) ).
thf(fact_1428_not__one__less__zero,axiom,
~ ( ord_le1304079648umeral @ one_on1645066479umeral @ zero_z126310315umeral ) ).
thf(fact_1429_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
thf(fact_1430_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
thf(fact_1431_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
thf(fact_1432_le__iff__diff__le__0,axiom,
! [A_172: rat,B_122: rat] :
( ( ord_less_eq_rat @ A_172 @ B_122 )
<=> ( ord_less_eq_rat @ ( minus_minus_rat @ A_172 @ B_122 ) @ zero_zero_rat ) ) ).
thf(fact_1433_le__iff__diff__le__0,axiom,
! [A_172: real,B_122: real] :
( ( ord_less_eq_real @ A_172 @ B_122 )
<=> ( ord_less_eq_real @ ( minus_minus_real @ A_172 @ B_122 ) @ zero_zero_real ) ) ).
thf(fact_1434_le__iff__diff__le__0,axiom,
! [A_172: int,B_122: int] :
( ( ord_less_eq_int @ A_172 @ B_122 )
<=> ( ord_less_eq_int @ ( minus_minus_int @ A_172 @ B_122 ) @ zero_zero_int ) ) ).
thf(fact_1435_add__scale__eq__noteq,axiom,
! [C_49: rat,D_10: rat,A_171: rat,B_121: rat,R_6: rat] :
( ( R_6 != zero_zero_rat )
=> ( ( ( A_171 = B_121 )
& ( C_49 != D_10 ) )
=> ( ( plus_plus_rat @ A_171 @ ( times_times_rat @ R_6 @ C_49 ) )
!= ( plus_plus_rat @ B_121 @ ( times_times_rat @ R_6 @ D_10 ) ) ) ) ) ).
thf(fact_1436_add__scale__eq__noteq,axiom,
! [C_49: complex,D_10: complex,A_171: complex,B_121: complex,R_6: complex] :
( ( R_6 != zero_zero_complex )
=> ( ( ( A_171 = B_121 )
& ( C_49 != D_10 ) )
=> ( ( plus_plus_complex @ A_171 @ ( times_times_complex @ R_6 @ C_49 ) )
!= ( plus_plus_complex @ B_121 @ ( times_times_complex @ R_6 @ D_10 ) ) ) ) ) ).
thf(fact_1437_add__scale__eq__noteq,axiom,
! [C_49: nat,D_10: nat,A_171: nat,B_121: nat,R_6: nat] :
( ( R_6 != zero_zero_nat )
=> ( ( ( A_171 = B_121 )
& ( C_49 != D_10 ) )
=> ( ( plus_plus_nat @ A_171 @ ( times_times_nat @ R_6 @ C_49 ) )
!= ( plus_plus_nat @ B_121 @ ( times_times_nat @ R_6 @ D_10 ) ) ) ) ) ).
thf(fact_1438_add__scale__eq__noteq,axiom,
! [C_49: real,D_10: real,A_171: real,B_121: real,R_6: real] :
( ( R_6 != zero_zero_real )
=> ( ( ( A_171 = B_121 )
& ( C_49 != D_10 ) )
=> ( ( plus_plus_real @ A_171 @ ( times_times_real @ R_6 @ C_49 ) )
!= ( plus_plus_real @ B_121 @ ( times_times_real @ R_6 @ D_10 ) ) ) ) ) ).
thf(fact_1439_add__scale__eq__noteq,axiom,
! [C_49: int,D_10: int,A_171: int,B_121: int,R_6: int] :
( ( R_6 != zero_zero_int )
=> ( ( ( A_171 = B_121 )
& ( C_49 != D_10 ) )
=> ( ( plus_plus_int @ A_171 @ ( times_times_int @ R_6 @ C_49 ) )
!= ( plus_plus_int @ B_121 @ ( times_times_int @ R_6 @ D_10 ) ) ) ) ) ).
thf(fact_1440_less__iff__diff__less__0,axiom,
! [A_170: rat,B_120: rat] :
( ( ord_less_rat @ A_170 @ B_120 )
<=> ( ord_less_rat @ ( minus_minus_rat @ A_170 @ B_120 ) @ zero_zero_rat ) ) ).
thf(fact_1441_less__iff__diff__less__0,axiom,
! [A_170: real,B_120: real] :
( ( ord_less_real @ A_170 @ B_120 )
<=> ( ord_less_real @ ( minus_minus_real @ A_170 @ B_120 ) @ zero_zero_real ) ) ).
thf(fact_1442_less__iff__diff__less__0,axiom,
! [A_170: int,B_120: int] :
( ( ord_less_int @ A_170 @ B_120 )
<=> ( ord_less_int @ ( minus_minus_int @ A_170 @ B_120 ) @ zero_zero_int ) ) ).
thf(fact_1443_Euler_Oaux1,axiom,
! [A: int,X: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ A )
=> ( ( X
!= ( minus_minus_int @ A @ one_one_int ) )
=> ( ord_less_int @ X @ ( minus_minus_int @ A @ one_one_int ) ) ) ) ) ).
thf(fact_1444_int__0__less__1,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
thf(fact_1445_zmult__zless__mono2,axiom,
! [K_1: int,I: int,J: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K_1 )
=> ( ord_less_int @ ( times_times_int @ K_1 @ I ) @ ( times_times_int @ K_1 @ J ) ) ) ) ).
thf(fact_1446_odd__nonzero,axiom,
! [Z_1: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z_1 ) @ Z_1 )
!= zero_zero_int ) ).
thf(fact_1447_less__bin__lemma,axiom,
! [K_1: int,L: int] :
( ( ord_less_int @ K_1 @ L )
<=> ( ord_less_int @ ( minus_minus_int @ K_1 @ L ) @ zero_zero_int ) ) ).
thf(fact_1448_one__less__ceiling,axiom,
! [X_62: rat] :
( ( ord_less_int @ one_one_int @ ( archim1419379755ng_rat @ X_62 ) )
<=> ( ord_less_rat @ one_one_rat @ X_62 ) ) ).
thf(fact_1449_one__less__ceiling,axiom,
! [X_62: real] :
( ( ord_less_int @ one_one_int @ ( archim856651990g_real @ X_62 ) )
<=> ( ord_less_real @ one_one_real @ X_62 ) ) ).
thf(fact_1450_ceiling__le__one,axiom,
! [X_61: rat] :
( ( ord_less_eq_int @ ( archim1419379755ng_rat @ X_61 ) @ one_one_int )
<=> ( ord_less_eq_rat @ X_61 @ one_one_rat ) ) ).
thf(fact_1451_ceiling__le__one,axiom,
! [X_61: real] :
( ( ord_less_eq_int @ ( archim856651990g_real @ X_61 ) @ one_one_int )
<=> ( ord_less_eq_real @ X_61 @ one_one_real ) ) ).
thf(fact_1452_number__of__less__ceiling,axiom,
! [V_15: int,X_60: rat] :
( ( ord_less_int @ ( number_number_of_int @ V_15 ) @ ( archim1419379755ng_rat @ X_60 ) )
<=> ( ord_less_rat @ ( number_number_of_rat @ V_15 ) @ X_60 ) ) ).
thf(fact_1453_number__of__less__ceiling,axiom,
! [V_15: int,X_60: real] :
( ( ord_less_int @ ( number_number_of_int @ V_15 ) @ ( archim856651990g_real @ X_60 ) )
<=> ( ord_less_real @ ( number267125858f_real @ V_15 ) @ X_60 ) ) ).
thf(fact_1454_ceiling__le__number__of,axiom,
! [X_59: real,V_14: int] :
( ( ord_less_eq_int @ ( archim856651990g_real @ X_59 ) @ ( number_number_of_int @ V_14 ) )
<=> ( ord_less_eq_real @ X_59 @ ( number267125858f_real @ V_14 ) ) ) ).
thf(fact_1455_ceiling__le__number__of,axiom,
! [X_59: rat,V_14: int] :
( ( ord_less_eq_int @ ( archim1419379755ng_rat @ X_59 ) @ ( number_number_of_int @ V_14 ) )
<=> ( ord_less_eq_rat @ X_59 @ ( number_number_of_rat @ V_14 ) ) ) ).
thf(fact_1456_ceiling__add__one,axiom,
! [X_58: rat] :
( ( archim1419379755ng_rat @ ( plus_plus_rat @ X_58 @ one_one_rat ) )
= ( plus_plus_int @ ( archim1419379755ng_rat @ X_58 ) @ one_one_int ) ) ).
thf(fact_1457_ceiling__add__one,axiom,
! [X_58: real] :
( ( archim856651990g_real @ ( plus_plus_real @ X_58 @ one_one_real ) )
= ( plus_plus_int @ ( archim856651990g_real @ X_58 ) @ one_one_int ) ) ).
thf(fact_1458_ceiling__add__number__of,axiom,
! [X_57: rat,V_13: int] :
( ( archim1419379755ng_rat @ ( plus_plus_rat @ X_57 @ ( number_number_of_rat @ V_13 ) ) )
= ( plus_plus_int @ ( archim1419379755ng_rat @ X_57 ) @ ( number_number_of_int @ V_13 ) ) ) ).
thf(fact_1459_ceiling__add__number__of,axiom,
! [X_57: real,V_13: int] :
( ( archim856651990g_real @ ( plus_plus_real @ X_57 @ ( number267125858f_real @ V_13 ) ) )
= ( plus_plus_int @ ( archim856651990g_real @ X_57 ) @ ( number_number_of_int @ V_13 ) ) ) ).
thf(fact_1460_ceiling__diff__one,axiom,
! [X_56: rat] :
( ( archim1419379755ng_rat @ ( minus_minus_rat @ X_56 @ one_one_rat ) )
= ( minus_minus_int @ ( archim1419379755ng_rat @ X_56 ) @ one_one_int ) ) ).
thf(fact_1461_ceiling__diff__one,axiom,
! [X_56: real] :
( ( archim856651990g_real @ ( minus_minus_real @ X_56 @ one_one_real ) )
= ( minus_minus_int @ ( archim856651990g_real @ X_56 ) @ one_one_int ) ) ).
thf(fact_1462_ceiling__diff__number__of,axiom,
! [X_55: rat,V_12: int] :
( ( archim1419379755ng_rat @ ( minus_minus_rat @ X_55 @ ( number_number_of_rat @ V_12 ) ) )
= ( minus_minus_int @ ( archim1419379755ng_rat @ X_55 ) @ ( number_number_of_int @ V_12 ) ) ) ).
thf(fact_1463_ceiling__diff__number__of,axiom,
! [X_55: real,V_12: int] :
( ( archim856651990g_real @ ( minus_minus_real @ X_55 @ ( number267125858f_real @ V_12 ) ) )
= ( minus_minus_int @ ( archim856651990g_real @ X_55 ) @ ( number_number_of_int @ V_12 ) ) ) ).
thf(fact_1464_mult__left__le__imp__le,axiom,
! [C_48: rat,A_169: rat,B_119: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C_48 @ A_169 ) @ ( times_times_rat @ C_48 @ B_119 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C_48 )
=> ( ord_less_eq_rat @ A_169 @ B_119 ) ) ) ).
thf(fact_1465_mult__left__le__imp__le,axiom,
! [C_48: quickcheck_code_int,A_169: quickcheck_code_int,B_119: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( times_123202395de_int @ C_48 @ A_169 ) @ ( times_123202395de_int @ C_48 @ B_119 ) )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ C_48 )
=> ( ord_le258702272de_int @ A_169 @ B_119 ) ) ) ).
thf(fact_1466_mult__left__le__imp__le,axiom,
! [C_48: code_code_numeral,A_169: code_code_numeral,B_119: code_code_numeral] :
( ( ord_le565307924umeral @ ( times_1655362735umeral @ C_48 @ A_169 ) @ ( times_1655362735umeral @ C_48 @ B_119 ) )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ C_48 )
=> ( ord_le565307924umeral @ A_169 @ B_119 ) ) ) ).
thf(fact_1467_mult__left__le__imp__le,axiom,
! [C_48: nat,A_169: nat,B_119: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C_48 @ A_169 ) @ ( times_times_nat @ C_48 @ B_119 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C_48 )
=> ( ord_less_eq_nat @ A_169 @ B_119 ) ) ) ).
thf(fact_1468_mult__left__le__imp__le,axiom,
! [C_48: real,A_169: real,B_119: real] :
( ( ord_less_eq_real @ ( times_times_real @ C_48 @ A_169 ) @ ( times_times_real @ C_48 @ B_119 ) )
=> ( ( ord_less_real @ zero_zero_real @ C_48 )
=> ( ord_less_eq_real @ A_169 @ B_119 ) ) ) ).
thf(fact_1469_mult__left__le__imp__le,axiom,
! [C_48: int,A_169: int,B_119: int] :
( ( ord_less_eq_int @ ( times_times_int @ C_48 @ A_169 ) @ ( times_times_int @ C_48 @ B_119 ) )
=> ( ( ord_less_int @ zero_zero_int @ C_48 )
=> ( ord_less_eq_int @ A_169 @ B_119 ) ) ) ).
thf(fact_1470_mult__right__le__imp__le,axiom,
! [A_168: rat,C_47: rat,B_118: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A_168 @ C_47 ) @ ( times_times_rat @ B_118 @ C_47 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C_47 )
=> ( ord_less_eq_rat @ A_168 @ B_118 ) ) ) ).
thf(fact_1471_mult__right__le__imp__le,axiom,
! [A_168: quickcheck_code_int,C_47: quickcheck_code_int,B_118: quickcheck_code_int] :
( ( ord_le258702272de_int @ ( times_123202395de_int @ A_168 @ C_47 ) @ ( times_123202395de_int @ B_118 @ C_47 ) )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ C_47 )
=> ( ord_le258702272de_int @ A_168 @ B_118 ) ) ) ).
thf(fact_1472_mult__right__le__imp__le,axiom,
! [A_168: code_code_numeral,C_47: code_code_numeral,B_118: code_code_numeral] :
( ( ord_le565307924umeral @ ( times_1655362735umeral @ A_168 @ C_47 ) @ ( times_1655362735umeral @ B_118 @ C_47 ) )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ C_47 )
=> ( ord_le565307924umeral @ A_168 @ B_118 ) ) ) ).
thf(fact_1473_mult__right__le__imp__le,axiom,
! [A_168: nat,C_47: nat,B_118: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A_168 @ C_47 ) @ ( times_times_nat @ B_118 @ C_47 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C_47 )
=> ( ord_less_eq_nat @ A_168 @ B_118 ) ) ) ).
thf(fact_1474_mult__right__le__imp__le,axiom,
! [A_168: real,C_47: real,B_118: real] :
( ( ord_less_eq_real @ ( times_times_real @ A_168 @ C_47 ) @ ( times_times_real @ B_118 @ C_47 ) )
=> ( ( ord_less_real @ zero_zero_real @ C_47 )
=> ( ord_less_eq_real @ A_168 @ B_118 ) ) ) ).
thf(fact_1475_mult__right__le__imp__le,axiom,
! [A_168: int,C_47: int,B_118: int] :
( ( ord_less_eq_int @ ( times_times_int @ A_168 @ C_47 ) @ ( times_times_int @ B_118 @ C_47 ) )
=> ( ( ord_less_int @ zero_zero_int @ C_47 )
=> ( ord_less_eq_int @ A_168 @ B_118 ) ) ) ).
thf(fact_1476_mult__less__imp__less__left,axiom,
! [C_46: rat,A_167: rat,B_117: rat] :
( ( ord_less_rat @ ( times_times_rat @ C_46 @ A_167 ) @ ( times_times_rat @ C_46 @ B_117 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_46 )
=> ( ord_less_rat @ A_167 @ B_117 ) ) ) ).
thf(fact_1477_mult__less__imp__less__left,axiom,
! [C_46: quickcheck_code_int,A_167: quickcheck_code_int,B_117: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( times_123202395de_int @ C_46 @ A_167 ) @ ( times_123202395de_int @ C_46 @ B_117 ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_46 )
=> ( ord_le1860547276de_int @ A_167 @ B_117 ) ) ) ).
thf(fact_1478_mult__less__imp__less__left,axiom,
! [C_46: code_code_numeral,A_167: code_code_numeral,B_117: code_code_numeral] :
( ( ord_le1304079648umeral @ ( times_1655362735umeral @ C_46 @ A_167 ) @ ( times_1655362735umeral @ C_46 @ B_117 ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_46 )
=> ( ord_le1304079648umeral @ A_167 @ B_117 ) ) ) ).
thf(fact_1479_mult__less__imp__less__left,axiom,
! [C_46: nat,A_167: nat,B_117: nat] :
( ( ord_less_nat @ ( times_times_nat @ C_46 @ A_167 ) @ ( times_times_nat @ C_46 @ B_117 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_46 )
=> ( ord_less_nat @ A_167 @ B_117 ) ) ) ).
thf(fact_1480_mult__less__imp__less__left,axiom,
! [C_46: real,A_167: real,B_117: real] :
( ( ord_less_real @ ( times_times_real @ C_46 @ A_167 ) @ ( times_times_real @ C_46 @ B_117 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_46 )
=> ( ord_less_real @ A_167 @ B_117 ) ) ) ).
thf(fact_1481_mult__less__imp__less__left,axiom,
! [C_46: int,A_167: int,B_117: int] :
( ( ord_less_int @ ( times_times_int @ C_46 @ A_167 ) @ ( times_times_int @ C_46 @ B_117 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_46 )
=> ( ord_less_int @ A_167 @ B_117 ) ) ) ).
thf(fact_1482_mult__left__less__imp__less,axiom,
! [C_45: rat,A_166: rat,B_116: rat] :
( ( ord_less_rat @ ( times_times_rat @ C_45 @ A_166 ) @ ( times_times_rat @ C_45 @ B_116 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_45 )
=> ( ord_less_rat @ A_166 @ B_116 ) ) ) ).
thf(fact_1483_mult__left__less__imp__less,axiom,
! [C_45: quickcheck_code_int,A_166: quickcheck_code_int,B_116: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( times_123202395de_int @ C_45 @ A_166 ) @ ( times_123202395de_int @ C_45 @ B_116 ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_45 )
=> ( ord_le1860547276de_int @ A_166 @ B_116 ) ) ) ).
thf(fact_1484_mult__left__less__imp__less,axiom,
! [C_45: code_code_numeral,A_166: code_code_numeral,B_116: code_code_numeral] :
( ( ord_le1304079648umeral @ ( times_1655362735umeral @ C_45 @ A_166 ) @ ( times_1655362735umeral @ C_45 @ B_116 ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_45 )
=> ( ord_le1304079648umeral @ A_166 @ B_116 ) ) ) ).
thf(fact_1485_mult__left__less__imp__less,axiom,
! [C_45: nat,A_166: nat,B_116: nat] :
( ( ord_less_nat @ ( times_times_nat @ C_45 @ A_166 ) @ ( times_times_nat @ C_45 @ B_116 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_45 )
=> ( ord_less_nat @ A_166 @ B_116 ) ) ) ).
thf(fact_1486_mult__left__less__imp__less,axiom,
! [C_45: real,A_166: real,B_116: real] :
( ( ord_less_real @ ( times_times_real @ C_45 @ A_166 ) @ ( times_times_real @ C_45 @ B_116 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_45 )
=> ( ord_less_real @ A_166 @ B_116 ) ) ) ).
thf(fact_1487_mult__left__less__imp__less,axiom,
! [C_45: int,A_166: int,B_116: int] :
( ( ord_less_int @ ( times_times_int @ C_45 @ A_166 ) @ ( times_times_int @ C_45 @ B_116 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_45 )
=> ( ord_less_int @ A_166 @ B_116 ) ) ) ).
thf(fact_1488_mult__less__imp__less__right,axiom,
! [A_165: rat,C_44: rat,B_115: rat] :
( ( ord_less_rat @ ( times_times_rat @ A_165 @ C_44 ) @ ( times_times_rat @ B_115 @ C_44 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_44 )
=> ( ord_less_rat @ A_165 @ B_115 ) ) ) ).
thf(fact_1489_mult__less__imp__less__right,axiom,
! [A_165: quickcheck_code_int,C_44: quickcheck_code_int,B_115: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( times_123202395de_int @ A_165 @ C_44 ) @ ( times_123202395de_int @ B_115 @ C_44 ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_44 )
=> ( ord_le1860547276de_int @ A_165 @ B_115 ) ) ) ).
thf(fact_1490_mult__less__imp__less__right,axiom,
! [A_165: code_code_numeral,C_44: code_code_numeral,B_115: code_code_numeral] :
( ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_165 @ C_44 ) @ ( times_1655362735umeral @ B_115 @ C_44 ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_44 )
=> ( ord_le1304079648umeral @ A_165 @ B_115 ) ) ) ).
thf(fact_1491_mult__less__imp__less__right,axiom,
! [A_165: nat,C_44: nat,B_115: nat] :
( ( ord_less_nat @ ( times_times_nat @ A_165 @ C_44 ) @ ( times_times_nat @ B_115 @ C_44 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_44 )
=> ( ord_less_nat @ A_165 @ B_115 ) ) ) ).
thf(fact_1492_mult__less__imp__less__right,axiom,
! [A_165: real,C_44: real,B_115: real] :
( ( ord_less_real @ ( times_times_real @ A_165 @ C_44 ) @ ( times_times_real @ B_115 @ C_44 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_44 )
=> ( ord_less_real @ A_165 @ B_115 ) ) ) ).
thf(fact_1493_mult__less__imp__less__right,axiom,
! [A_165: int,C_44: int,B_115: int] :
( ( ord_less_int @ ( times_times_int @ A_165 @ C_44 ) @ ( times_times_int @ B_115 @ C_44 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_44 )
=> ( ord_less_int @ A_165 @ B_115 ) ) ) ).
thf(fact_1494_mult__right__less__imp__less,axiom,
! [A_164: rat,C_43: rat,B_114: rat] :
( ( ord_less_rat @ ( times_times_rat @ A_164 @ C_43 ) @ ( times_times_rat @ B_114 @ C_43 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_43 )
=> ( ord_less_rat @ A_164 @ B_114 ) ) ) ).
thf(fact_1495_mult__right__less__imp__less,axiom,
! [A_164: quickcheck_code_int,C_43: quickcheck_code_int,B_114: quickcheck_code_int] :
( ( ord_le1860547276de_int @ ( times_123202395de_int @ A_164 @ C_43 ) @ ( times_123202395de_int @ B_114 @ C_43 ) )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_43 )
=> ( ord_le1860547276de_int @ A_164 @ B_114 ) ) ) ).
thf(fact_1496_mult__right__less__imp__less,axiom,
! [A_164: code_code_numeral,C_43: code_code_numeral,B_114: code_code_numeral] :
( ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_164 @ C_43 ) @ ( times_1655362735umeral @ B_114 @ C_43 ) )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_43 )
=> ( ord_le1304079648umeral @ A_164 @ B_114 ) ) ) ).
thf(fact_1497_mult__right__less__imp__less,axiom,
! [A_164: nat,C_43: nat,B_114: nat] :
( ( ord_less_nat @ ( times_times_nat @ A_164 @ C_43 ) @ ( times_times_nat @ B_114 @ C_43 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_43 )
=> ( ord_less_nat @ A_164 @ B_114 ) ) ) ).
thf(fact_1498_mult__right__less__imp__less,axiom,
! [A_164: real,C_43: real,B_114: real] :
( ( ord_less_real @ ( times_times_real @ A_164 @ C_43 ) @ ( times_times_real @ B_114 @ C_43 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_43 )
=> ( ord_less_real @ A_164 @ B_114 ) ) ) ).
thf(fact_1499_mult__right__less__imp__less,axiom,
! [A_164: int,C_43: int,B_114: int] :
( ( ord_less_int @ ( times_times_int @ A_164 @ C_43 ) @ ( times_times_int @ B_114 @ C_43 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_43 )
=> ( ord_less_int @ A_164 @ B_114 ) ) ) ).
thf(fact_1500_mult__le__less__imp__less,axiom,
! [C_42: rat,D_9: rat,A_163: rat,B_113: rat] :
( ( ord_less_eq_rat @ A_163 @ B_113 )
=> ( ( ord_less_rat @ C_42 @ D_9 )
=> ( ( ord_less_rat @ zero_zero_rat @ A_163 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_42 )
=> ( ord_less_rat @ ( times_times_rat @ A_163 @ C_42 ) @ ( times_times_rat @ B_113 @ D_9 ) ) ) ) ) ) ).
thf(fact_1501_mult__le__less__imp__less,axiom,
! [C_42: quickcheck_code_int,D_9: quickcheck_code_int,A_163: quickcheck_code_int,B_113: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_163 @ B_113 )
=> ( ( ord_le1860547276de_int @ C_42 @ D_9 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_163 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_42 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_163 @ C_42 ) @ ( times_123202395de_int @ B_113 @ D_9 ) ) ) ) ) ) ).
thf(fact_1502_mult__le__less__imp__less,axiom,
! [C_42: code_code_numeral,D_9: code_code_numeral,A_163: code_code_numeral,B_113: code_code_numeral] :
( ( ord_le565307924umeral @ A_163 @ B_113 )
=> ( ( ord_le1304079648umeral @ C_42 @ D_9 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_163 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_42 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_163 @ C_42 ) @ ( times_1655362735umeral @ B_113 @ D_9 ) ) ) ) ) ) ).
thf(fact_1503_mult__le__less__imp__less,axiom,
! [C_42: nat,D_9: nat,A_163: nat,B_113: nat] :
( ( ord_less_eq_nat @ A_163 @ B_113 )
=> ( ( ord_less_nat @ C_42 @ D_9 )
=> ( ( ord_less_nat @ zero_zero_nat @ A_163 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_42 )
=> ( ord_less_nat @ ( times_times_nat @ A_163 @ C_42 ) @ ( times_times_nat @ B_113 @ D_9 ) ) ) ) ) ) ).
thf(fact_1504_mult__le__less__imp__less,axiom,
! [C_42: real,D_9: real,A_163: real,B_113: real] :
( ( ord_less_eq_real @ A_163 @ B_113 )
=> ( ( ord_less_real @ C_42 @ D_9 )
=> ( ( ord_less_real @ zero_zero_real @ A_163 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_42 )
=> ( ord_less_real @ ( times_times_real @ A_163 @ C_42 ) @ ( times_times_real @ B_113 @ D_9 ) ) ) ) ) ) ).
thf(fact_1505_mult__le__less__imp__less,axiom,
! [C_42: int,D_9: int,A_163: int,B_113: int] :
( ( ord_less_eq_int @ A_163 @ B_113 )
=> ( ( ord_less_int @ C_42 @ D_9 )
=> ( ( ord_less_int @ zero_zero_int @ A_163 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_42 )
=> ( ord_less_int @ ( times_times_int @ A_163 @ C_42 ) @ ( times_times_int @ B_113 @ D_9 ) ) ) ) ) ) ).
thf(fact_1506_mult__less__le__imp__less,axiom,
! [C_41: rat,D_8: rat,A_162: rat,B_112: rat] :
( ( ord_less_rat @ A_162 @ B_112 )
=> ( ( ord_less_eq_rat @ C_41 @ D_8 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A_162 )
=> ( ( ord_less_rat @ zero_zero_rat @ C_41 )
=> ( ord_less_rat @ ( times_times_rat @ A_162 @ C_41 ) @ ( times_times_rat @ B_112 @ D_8 ) ) ) ) ) ) ).
thf(fact_1507_mult__less__le__imp__less,axiom,
! [C_41: quickcheck_code_int,D_8: quickcheck_code_int,A_162: quickcheck_code_int,B_112: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_162 @ B_112 )
=> ( ( ord_le258702272de_int @ C_41 @ D_8 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_162 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ C_41 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_162 @ C_41 ) @ ( times_123202395de_int @ B_112 @ D_8 ) ) ) ) ) ) ).
thf(fact_1508_mult__less__le__imp__less,axiom,
! [C_41: code_code_numeral,D_8: code_code_numeral,A_162: code_code_numeral,B_112: code_code_numeral] :
( ( ord_le1304079648umeral @ A_162 @ B_112 )
=> ( ( ord_le565307924umeral @ C_41 @ D_8 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_162 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ C_41 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_162 @ C_41 ) @ ( times_1655362735umeral @ B_112 @ D_8 ) ) ) ) ) ) ).
thf(fact_1509_mult__less__le__imp__less,axiom,
! [C_41: nat,D_8: nat,A_162: nat,B_112: nat] :
( ( ord_less_nat @ A_162 @ B_112 )
=> ( ( ord_less_eq_nat @ C_41 @ D_8 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_162 )
=> ( ( ord_less_nat @ zero_zero_nat @ C_41 )
=> ( ord_less_nat @ ( times_times_nat @ A_162 @ C_41 ) @ ( times_times_nat @ B_112 @ D_8 ) ) ) ) ) ) ).
thf(fact_1510_mult__less__le__imp__less,axiom,
! [C_41: real,D_8: real,A_162: real,B_112: real] :
( ( ord_less_real @ A_162 @ B_112 )
=> ( ( ord_less_eq_real @ C_41 @ D_8 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_162 )
=> ( ( ord_less_real @ zero_zero_real @ C_41 )
=> ( ord_less_real @ ( times_times_real @ A_162 @ C_41 ) @ ( times_times_real @ B_112 @ D_8 ) ) ) ) ) ) ).
thf(fact_1511_mult__less__le__imp__less,axiom,
! [C_41: int,D_8: int,A_162: int,B_112: int] :
( ( ord_less_int @ A_162 @ B_112 )
=> ( ( ord_less_eq_int @ C_41 @ D_8 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_162 )
=> ( ( ord_less_int @ zero_zero_int @ C_41 )
=> ( ord_less_int @ ( times_times_int @ A_162 @ C_41 ) @ ( times_times_int @ B_112 @ D_8 ) ) ) ) ) ) ).
thf(fact_1512_mult__strict__mono_H,axiom,
! [C_40: rat,D_7: rat,A_161: rat,B_111: rat] :
( ( ord_less_rat @ A_161 @ B_111 )
=> ( ( ord_less_rat @ C_40 @ D_7 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A_161 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_40 )
=> ( ord_less_rat @ ( times_times_rat @ A_161 @ C_40 ) @ ( times_times_rat @ B_111 @ D_7 ) ) ) ) ) ) ).
thf(fact_1513_mult__strict__mono_H,axiom,
! [C_40: quickcheck_code_int,D_7: quickcheck_code_int,A_161: quickcheck_code_int,B_111: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_161 @ B_111 )
=> ( ( ord_le1860547276de_int @ C_40 @ D_7 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ A_161 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_40 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_161 @ C_40 ) @ ( times_123202395de_int @ B_111 @ D_7 ) ) ) ) ) ) ).
thf(fact_1514_mult__strict__mono_H,axiom,
! [C_40: code_code_numeral,D_7: code_code_numeral,A_161: code_code_numeral,B_111: code_code_numeral] :
( ( ord_le1304079648umeral @ A_161 @ B_111 )
=> ( ( ord_le1304079648umeral @ C_40 @ D_7 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ A_161 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_40 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_161 @ C_40 ) @ ( times_1655362735umeral @ B_111 @ D_7 ) ) ) ) ) ) ).
thf(fact_1515_mult__strict__mono_H,axiom,
! [C_40: nat,D_7: nat,A_161: nat,B_111: nat] :
( ( ord_less_nat @ A_161 @ B_111 )
=> ( ( ord_less_nat @ C_40 @ D_7 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A_161 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_40 )
=> ( ord_less_nat @ ( times_times_nat @ A_161 @ C_40 ) @ ( times_times_nat @ B_111 @ D_7 ) ) ) ) ) ) ).
thf(fact_1516_mult__strict__mono_H,axiom,
! [C_40: real,D_7: real,A_161: real,B_111: real] :
( ( ord_less_real @ A_161 @ B_111 )
=> ( ( ord_less_real @ C_40 @ D_7 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A_161 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_40 )
=> ( ord_less_real @ ( times_times_real @ A_161 @ C_40 ) @ ( times_times_real @ B_111 @ D_7 ) ) ) ) ) ) ).
thf(fact_1517_mult__strict__mono_H,axiom,
! [C_40: int,D_7: int,A_161: int,B_111: int] :
( ( ord_less_int @ A_161 @ B_111 )
=> ( ( ord_less_int @ C_40 @ D_7 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A_161 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_40 )
=> ( ord_less_int @ ( times_times_int @ A_161 @ C_40 ) @ ( times_times_int @ B_111 @ D_7 ) ) ) ) ) ) ).
thf(fact_1518_mult__strict__mono,axiom,
! [C_39: rat,D_6: rat,A_160: rat,B_110: rat] :
( ( ord_less_rat @ A_160 @ B_110 )
=> ( ( ord_less_rat @ C_39 @ D_6 )
=> ( ( ord_less_rat @ zero_zero_rat @ B_110 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C_39 )
=> ( ord_less_rat @ ( times_times_rat @ A_160 @ C_39 ) @ ( times_times_rat @ B_110 @ D_6 ) ) ) ) ) ) ).
thf(fact_1519_mult__strict__mono,axiom,
! [C_39: quickcheck_code_int,D_6: quickcheck_code_int,A_160: quickcheck_code_int,B_110: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_160 @ B_110 )
=> ( ( ord_le1860547276de_int @ C_39 @ D_6 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ B_110 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ C_39 )
=> ( ord_le1860547276de_int @ ( times_123202395de_int @ A_160 @ C_39 ) @ ( times_123202395de_int @ B_110 @ D_6 ) ) ) ) ) ) ).
thf(fact_1520_mult__strict__mono,axiom,
! [C_39: code_code_numeral,D_6: code_code_numeral,A_160: code_code_numeral,B_110: code_code_numeral] :
( ( ord_le1304079648umeral @ A_160 @ B_110 )
=> ( ( ord_le1304079648umeral @ C_39 @ D_6 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ B_110 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ C_39 )
=> ( ord_le1304079648umeral @ ( times_1655362735umeral @ A_160 @ C_39 ) @ ( times_1655362735umeral @ B_110 @ D_6 ) ) ) ) ) ) ).
thf(fact_1521_mult__strict__mono,axiom,
! [C_39: nat,D_6: nat,A_160: nat,B_110: nat] :
( ( ord_less_nat @ A_160 @ B_110 )
=> ( ( ord_less_nat @ C_39 @ D_6 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_110 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C_39 )
=> ( ord_less_nat @ ( times_times_nat @ A_160 @ C_39 ) @ ( times_times_nat @ B_110 @ D_6 ) ) ) ) ) ) ).
thf(fact_1522_mult__strict__mono,axiom,
! [C_39: real,D_6: real,A_160: real,B_110: real] :
( ( ord_less_real @ A_160 @ B_110 )
=> ( ( ord_less_real @ C_39 @ D_6 )
=> ( ( ord_less_real @ zero_zero_real @ B_110 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C_39 )
=> ( ord_less_real @ ( times_times_real @ A_160 @ C_39 ) @ ( times_times_real @ B_110 @ D_6 ) ) ) ) ) ) ).
thf(fact_1523_mult__strict__mono,axiom,
! [C_39: int,D_6: int,A_160: int,B_110: int] :
( ( ord_less_int @ A_160 @ B_110 )
=> ( ( ord_less_int @ C_39 @ D_6 )
=> ( ( ord_less_int @ zero_zero_int @ B_110 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C_39 )
=> ( ord_less_int @ ( times_times_int @ A_160 @ C_39 ) @ ( times_times_int @ B_110 @ D_6 ) ) ) ) ) ) ).
thf(fact_1524_mult__le__cancel__left__neg,axiom,
! [A_159: rat,B_109: rat,C_38: rat] :
( ( ord_less_rat @ C_38 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C_38 @ A_159 ) @ ( times_times_rat @ C_38 @ B_109 ) )
<=> ( ord_less_eq_rat @ B_109 @ A_159 ) ) ) ).
thf(fact_1525_mult__le__cancel__left__neg,axiom,
! [A_159: real,B_109: real,C_38: real] :
( ( ord_less_real @ C_38 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C_38 @ A_159 ) @ ( times_times_real @ C_38 @ B_109 ) )
<=> ( ord_less_eq_real @ B_109 @ A_159 ) ) ) ).
thf(fact_1526_mult__le__cancel__left__neg,axiom,
! [A_159: int,B_109: int,C_38: int] :
( ( ord_less_int @ C_38 @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C_38 @ A_159 ) @ ( times_times_int @ C_38 @ B_109 ) )
<=> ( ord_less_eq_int @ B_109 @ A_159 ) ) ) ).
thf(fact_1527_mult__le__cancel__left__pos,axiom,
! [A_158: rat,B_108: rat,C_37: rat] :
( ( ord_less_rat @ zero_zero_rat @ C_37 )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C_37 @ A_158 ) @ ( times_times_rat @ C_37 @ B_108 ) )
<=> ( ord_less_eq_rat @ A_158 @ B_108 ) ) ) ).
thf(fact_1528_mult__le__cancel__left__pos,axiom,
! [A_158: real,B_108: real,C_37: real] :
( ( ord_less_real @ zero_zero_real @ C_37 )
=> ( ( ord_less_eq_real @ ( times_times_real @ C_37 @ A_158 ) @ ( times_times_real @ C_37 @ B_108 ) )
<=> ( ord_less_eq_real @ A_158 @ B_108 ) ) ) ).
thf(fact_1529_mult__le__cancel__left__pos,axiom,
! [A_158: int,B_108: int,C_37: int] :
( ( ord_less_int @ zero_zero_int @ C_37 )
=> ( ( ord_less_eq_int @ ( times_times_int @ C_37 @ A_158 ) @ ( times_times_int @ C_37 @ B_108 ) )
<=> ( ord_less_eq_int @ A_158 @ B_108 ) ) ) ).
thf(fact_1530_add__nonpos__neg,axiom,
! [B_107: rat,A_157: rat] :
( ( ord_less_eq_rat @ A_157 @ zero_zero_rat )
=> ( ( ord_less_rat @ B_107 @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A_157 @ B_107 ) @ zero_zero_rat ) ) ) ).
thf(fact_1531_add__nonpos__neg,axiom,
! [B_107: quickcheck_code_int,A_157: quickcheck_code_int] :
( ( ord_le258702272de_int @ A_157 @ zero_z891286103de_int )
=> ( ( ord_le1860547276de_int @ B_107 @ zero_z891286103de_int )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_157 @ B_107 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1532_add__nonpos__neg,axiom,
! [B_107: nat,A_157: nat] :
( ( ord_less_eq_nat @ A_157 @ zero_zero_nat )
=> ( ( ord_less_nat @ B_107 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A_157 @ B_107 ) @ zero_zero_nat ) ) ) ).
thf(fact_1533_add__nonpos__neg,axiom,
! [B_107: real,A_157: real] :
( ( ord_less_eq_real @ A_157 @ zero_zero_real )
=> ( ( ord_less_real @ B_107 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A_157 @ B_107 ) @ zero_zero_real ) ) ) ).
thf(fact_1534_add__nonpos__neg,axiom,
! [B_107: int,A_157: int] :
( ( ord_less_eq_int @ A_157 @ zero_zero_int )
=> ( ( ord_less_int @ B_107 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A_157 @ B_107 ) @ zero_zero_int ) ) ) ).
thf(fact_1535_add__nonpos__neg,axiom,
! [B_107: code_code_numeral,A_157: code_code_numeral] :
( ( ord_le565307924umeral @ A_157 @ zero_z126310315umeral )
=> ( ( ord_le1304079648umeral @ B_107 @ zero_z126310315umeral )
=> ( ord_le1304079648umeral @ ( plus_p1627245867umeral @ A_157 @ B_107 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1536_add__neg__nonpos,axiom,
! [B_106: rat,A_156: rat] :
( ( ord_less_rat @ A_156 @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B_106 @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A_156 @ B_106 ) @ zero_zero_rat ) ) ) ).
thf(fact_1537_add__neg__nonpos,axiom,
! [B_106: quickcheck_code_int,A_156: quickcheck_code_int] :
( ( ord_le1860547276de_int @ A_156 @ zero_z891286103de_int )
=> ( ( ord_le258702272de_int @ B_106 @ zero_z891286103de_int )
=> ( ord_le1860547276de_int @ ( plus_p1446045655de_int @ A_156 @ B_106 ) @ zero_z891286103de_int ) ) ) ).
thf(fact_1538_add__neg__nonpos,axiom,
! [B_106: nat,A_156: nat] :
( ( ord_less_nat @ A_156 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B_106 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A_156 @ B_106 ) @ zero_zero_nat ) ) ) ).
thf(fact_1539_add__neg__nonpos,axiom,
! [B_106: real,A_156: real] :
( ( ord_less_real @ A_156 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B_106 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A_156 @ B_106 ) @ zero_zero_real ) ) ) ).
thf(fact_1540_add__neg__nonpos,axiom,
! [B_106: int,A_156: int] :
( ( ord_less_int @ A_156 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B_106 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A_156 @ B_106 ) @ zero_zero_int ) ) ) ).
thf(fact_1541_add__neg__nonpos,axiom,
! [B_106: code_code_numeral,A_156: code_code_numeral] :
( ( ord_le1304079648umeral @ A_156 @ zero_z126310315umeral )
=> ( ( ord_le565307924umeral @ B_106 @ zero_z126310315umeral )
=> ( ord_le1304079648umeral @ ( plus_p1627245867umeral @ A_156 @ B_106 ) @ zero_z126310315umeral ) ) ) ).
thf(fact_1542_add__strict__increasing2,axiom,
! [B_105: rat,C_36: rat,A_155: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_155 )
=> ( ( ord_less_rat @ B_105 @ C_36 )
=> ( ord_less_rat @ B_105 @ ( plus_plus_rat @ A_155 @ C_36 ) ) ) ) ).
thf(fact_1543_add__strict__increasing2,axiom,
! [B_105: quickcheck_code_int,C_36: quickcheck_code_int,A_155: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_155 )
=> ( ( ord_le1860547276de_int @ B_105 @ C_36 )
=> ( ord_le1860547276de_int @ B_105 @ ( plus_p1446045655de_int @ A_155 @ C_36 ) ) ) ) ).
thf(fact_1544_add__strict__increasing2,axiom,
! [B_105: nat,C_36: nat,A_155: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_155 )
=> ( ( ord_less_nat @ B_105 @ C_36 )
=> ( ord_less_nat @ B_105 @ ( plus_plus_nat @ A_155 @ C_36 ) ) ) ) ).
thf(fact_1545_add__strict__increasing2,axiom,
! [B_105: real,C_36: real,A_155: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_155 )
=> ( ( ord_less_real @ B_105 @ C_36 )
=> ( ord_less_real @ B_105 @ ( plus_plus_real @ A_155 @ C_36 ) ) ) ) ).
thf(fact_1546_add__strict__increasing2,axiom,
! [B_105: int,C_36: int,A_155: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_155 )
=> ( ( ord_less_int @ B_105 @ C_36 )
=> ( ord_less_int @ B_105 @ ( plus_plus_int @ A_155 @ C_36 ) ) ) ) ).
thf(fact_1547_add__strict__increasing2,axiom,
! [B_105: code_code_numeral,C_36: code_code_numeral,A_155: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_155 )
=> ( ( ord_le1304079648umeral @ B_105 @ C_36 )
=> ( ord_le1304079648umeral @ B_105 @ ( plus_p1627245867umeral @ A_155 @ C_36 ) ) ) ) ).
thf(fact_1548_add__strict__increasing,axiom,
! [B_104: rat,C_35: rat,A_154: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_154 )
=> ( ( ord_less_eq_rat @ B_104 @ C_35 )
=> ( ord_less_rat @ B_104 @ ( plus_plus_rat @ A_154 @ C_35 ) ) ) ) ).
thf(fact_1549_add__strict__increasing,axiom,
! [B_104: quickcheck_code_int,C_35: quickcheck_code_int,A_154: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_154 )
=> ( ( ord_le258702272de_int @ B_104 @ C_35 )
=> ( ord_le1860547276de_int @ B_104 @ ( plus_p1446045655de_int @ A_154 @ C_35 ) ) ) ) ).
thf(fact_1550_add__strict__increasing,axiom,
! [B_104: nat,C_35: nat,A_154: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_154 )
=> ( ( ord_less_eq_nat @ B_104 @ C_35 )
=> ( ord_less_nat @ B_104 @ ( plus_plus_nat @ A_154 @ C_35 ) ) ) ) ).
thf(fact_1551_add__strict__increasing,axiom,
! [B_104: real,C_35: real,A_154: real] :
( ( ord_less_real @ zero_zero_real @ A_154 )
=> ( ( ord_less_eq_real @ B_104 @ C_35 )
=> ( ord_less_real @ B_104 @ ( plus_plus_real @ A_154 @ C_35 ) ) ) ) ).
thf(fact_1552_add__strict__increasing,axiom,
! [B_104: int,C_35: int,A_154: int] :
( ( ord_less_int @ zero_zero_int @ A_154 )
=> ( ( ord_less_eq_int @ B_104 @ C_35 )
=> ( ord_less_int @ B_104 @ ( plus_plus_int @ A_154 @ C_35 ) ) ) ) ).
thf(fact_1553_add__strict__increasing,axiom,
! [B_104: code_code_numeral,C_35: code_code_numeral,A_154: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_154 )
=> ( ( ord_le565307924umeral @ B_104 @ C_35 )
=> ( ord_le1304079648umeral @ B_104 @ ( plus_p1627245867umeral @ A_154 @ C_35 ) ) ) ) ).
thf(fact_1554_add__nonneg__pos,axiom,
! [B_103: rat,A_153: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_153 )
=> ( ( ord_less_rat @ zero_zero_rat @ B_103 )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A_153 @ B_103 ) ) ) ) ).
thf(fact_1555_add__nonneg__pos,axiom,
! [B_103: quickcheck_code_int,A_153: quickcheck_code_int] :
( ( ord_le258702272de_int @ zero_z891286103de_int @ A_153 )
=> ( ( ord_le1860547276de_int @ zero_z891286103de_int @ B_103 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ ( plus_p1446045655de_int @ A_153 @ B_103 ) ) ) ) ).
thf(fact_1556_add__nonneg__pos,axiom,
! [B_103: nat,A_153: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A_153 )
=> ( ( ord_less_nat @ zero_zero_nat @ B_103 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A_153 @ B_103 ) ) ) ) ).
thf(fact_1557_add__nonneg__pos,axiom,
! [B_103: real,A_153: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_153 )
=> ( ( ord_less_real @ zero_zero_real @ B_103 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A_153 @ B_103 ) ) ) ) ).
thf(fact_1558_add__nonneg__pos,axiom,
! [B_103: int,A_153: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_153 )
=> ( ( ord_less_int @ zero_zero_int @ B_103 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_153 @ B_103 ) ) ) ) ).
thf(fact_1559_add__nonneg__pos,axiom,
! [B_103: code_code_numeral,A_153: code_code_numeral] :
( ( ord_le565307924umeral @ zero_z126310315umeral @ A_153 )
=> ( ( ord_le1304079648umeral @ zero_z126310315umeral @ B_103 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ ( plus_p1627245867umeral @ A_153 @ B_103 ) ) ) ) ).
thf(fact_1560_add__pos__nonneg,axiom,
! [B_102: rat,A_152: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_152 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B_102 )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A_152 @ B_102 ) ) ) ) ).
thf(fact_1561_add__pos__nonneg,axiom,
! [B_102: quickcheck_code_int,A_152: quickcheck_code_int] :
( ( ord_le1860547276de_int @ zero_z891286103de_int @ A_152 )
=> ( ( ord_le258702272de_int @ zero_z891286103de_int @ B_102 )
=> ( ord_le1860547276de_int @ zero_z891286103de_int @ ( plus_p1446045655de_int @ A_152 @ B_102 ) ) ) ) ).
thf(fact_1562_add__pos__nonneg,axiom,
! [B_102: nat,A_152: nat] :
( ( ord_less_nat @ zero_zero_nat @ A_152 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B_102 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A_152 @ B_102 ) ) ) ) ).
thf(fact_1563_add__pos__nonneg,axiom,
! [B_102: real,A_152: real] :
( ( ord_less_real @ zero_zero_real @ A_152 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B_102 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A_152 @ B_102 ) ) ) ) ).
thf(fact_1564_add__pos__nonneg,axiom,
! [B_102: int,A_152: int] :
( ( ord_less_int @ zero_zero_int @ A_152 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B_102 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A_152 @ B_102 ) ) ) ) ).
thf(fact_1565_add__pos__nonneg,axiom,
! [B_102: code_code_numeral,A_152: code_code_numeral] :
( ( ord_le1304079648umeral @ zero_z126310315umeral @ A_152 )
=> ( ( ord_le565307924umeral @ zero_z126310315umeral @ B_102 )
=> ( ord_le1304079648umeral @ zero_z126310315umeral @ ( plus_p1627245867umeral @ A_152 @ B_102 ) ) ) ) ).
thf(fact_1566_mult__left__le__one__le,axiom,
! [Y_30: rat,X_54: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X_54 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y_30 )
=> ( ( ord_less_eq_rat @ Y_30 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Y_30 @ X_54 ) @ X_54 ) ) ) ) ).
thf(fact_1567_mult__left__le__one__le,axiom,
! [Y_30: real,X_54: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_54 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_30 )
=> ( ( ord_less_eq_real @ Y_30 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y_30 @ X_54 ) @ X_54 ) ) ) ) ).
thf(fact_1568_mult__left__le__one__le,axiom,
! [Y_30: int,X_54: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_54 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_30 )
=> ( ( ord_less_eq_int @ Y_30 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y_30 @ X_54 ) @ X_54 ) ) ) ) ).
thf(fact_1569_mult__right__le__one__le,axiom,
! [Y_29: rat,X_53: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X_53 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y_29 )
=> ( ( ord_less_eq_rat @ Y_29 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ X_53 @ Y_29 ) @ X_53 ) ) ) ) ).
thf(fact_1570_mult__right__le__one__le,axiom,
! [Y_29: real,X_53: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_53 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y_29 )
=> ( ( ord_less_eq_real @ Y_29 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X_53 @ Y_29 ) @ X_53 ) ) ) ) ).
thf(fact_1571_mult__right__le__one__le,axiom,
! [Y_29: int,X_53: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_53 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y_29 )
=> ( ( ord_less_eq_int @ Y_29 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X_53 @ Y_29 ) @ X_53 ) ) ) ) ).
thf(fact_1572_zero__less__two,axiom,
ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
thf(fact_1573_zero__less__two,axiom,
ord_le1860547276de_int @ zero_z891286103de_int @ ( plus_p1446045655de_int @ one_on1684967323de_int @ one_on1684967323de_int ) ).
thf(fact_1574_zero__less__two,axiom,
ord_le1304079648umeral @ zero_z126310315umeral @ ( plus_p1627245867umeral @ one_on1645066479umeral @ one_on1645066479umeral ) ).
thf(fact_1575_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
thf(fact_1576_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
thf(fact_1577_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
thf(fact_1578_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_1579_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_1580_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_1581_convex__bound__le,axiom,
! [V_11: rat,U_2: rat,Y_28: rat,X_52: rat,A_151: rat] :
( ( ord_less_eq_rat @ X_52 @ A_151 )
=> ( ( ord_less_eq_rat @ Y_28 @ A_151 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ U_2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ V_11 )
=> ( ( ( plus_plus_rat @ U_2 @ V_11 )
= one_one_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U_2 @ X_52 ) @ ( times_times_rat @ V_11 @ Y_28 ) ) @ A_151 ) ) ) ) ) ) ).
thf(fact_1582_convex__bound__le,axiom,
! [V_11: real,U_2: real,Y_28: real,X_52: real,A_151: real] :
( ( ord_less_eq_real @ X_52 @ A_151 )
=> ( ( ord_less_eq_real @ Y_28 @ A_151 )
=> ( ( ord_less_eq_real @ zero_zero_real @ U_2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ V_11 )
=> ( ( ( plus_plus_real @ U_2 @ V_11 )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U_2 @ X_52 ) @ ( times_times_real @ V_11 @ Y_28 ) ) @ A_151 ) ) ) ) ) ) ).
thf(fact_1583_convex__bound__le,axiom,
! [V_11: int,U_2: int,Y_28: int,X_52: int,A_151: int] :
( ( ord_less_eq_int @ X_52 @ A_151 )
=> ( ( ord_less_eq_int @ Y_28 @ A_151 )
=> ( ( ord_less_eq_int @ zero_zero_int @ U_2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ V_11 )
=> ( ( ( plus_plus_int @ U_2 @ V_11 )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U_2 @ X_52 ) @ ( times_times_int @ V_11 @ Y_28 ) ) @ A_151 ) ) ) ) ) ) ).
thf(fact_1584_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_1585_self__quotient__aux1,axiom,
! [R_1: int,Q: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( A
= ( plus_plus_int @ R_1 @ ( times_times_int @ A @ Q ) ) )
=> ( ( ord_less_int @ R_1 @ A )
=> ( ord_less_eq_int @ one_one_int @ Q ) ) ) ) ).
thf(fact_1586_self__quotient__aux2,axiom,
! [R_1: int,Q: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( A
= ( plus_plus_int @ R_1 @ ( times_times_int @ A @ Q ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ord_less_eq_int @ Q @ one_one_int ) ) ) ) ).
thf(fact_1587_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q: int,R_1: int,B_5: int,Q_5: int,R_3: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R_1 )
= ( plus_plus_int @ ( times_times_int @ B_5 @ Q_5 ) @ R_3 ) )
=> ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_5 @ Q_5 ) @ R_3 ) @ zero_zero_int )
=> ( ( ord_less_int @ R_1 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_3 )
=> ( ( ord_less_int @ zero_zero_int @ B_5 )
=> ( ( ord_less_eq_int @ B_5 @ B )
=> ( ord_less_eq_int @ Q_5 @ Q ) ) ) ) ) ) ) ).
thf(fact_1588_unique__quotient__lemma__neg,axiom,
! [B: int,Q_5: int,R_3: int,Q: int,R_1: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q_5 ) @ R_3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R_1 ) )
=> ( ( ord_less_eq_int @ R_1 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R_1 )
=> ( ( ord_less_int @ B @ R_3 )
=> ( ord_less_eq_int @ Q @ Q_5 ) ) ) ) ) ).
thf(fact_1589_zdiv__mono2__lemma,axiom,
! [B: int,Q: int,R_1: int,B_5: int,Q_5: int,R_3: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R_1 )
= ( plus_plus_int @ ( times_times_int @ B_5 @ Q_5 ) @ R_3 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_5 @ Q_5 ) @ R_3 ) )
=> ( ( ord_less_int @ R_3 @ B_5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ( ord_less_int @ zero_zero_int @ B_5 )
=> ( ( ord_less_eq_int @ B_5 @ B )
=> ( ord_less_eq_int @ Q @ Q_5 ) ) ) ) ) ) ) ).
thf(fact_1590_unique__quotient__lemma,axiom,
! [B: int,Q_5: int,R_3: int,Q: int,R_1: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q_5 ) @ R_3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R_1 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_3 )
=> ( ( ord_less_int @ R_3 @ B )
=> ( ( ord_less_int @ R_1 @ B )
=> ( ord_less_eq_int @ Q_5 @ Q ) ) ) ) ) ).
thf(fact_1591_q__neg__lemma,axiom,
! [B_5: int,Q_5: int,R_3: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B_5 @ Q_5 ) @ 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_5 @ zero_zero_int ) ) ) ) ).
thf(fact_1592_q__pos__lemma,axiom,
! [B_5: int,Q_5: int,R_3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B_5 @ Q_5 ) @ 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_5 ) ) ) ) ).
thf(fact_1593_sum__squares__gt__zero__iff,axiom,
! [X_51: rat,Y_27: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X_51 @ X_51 ) @ ( times_times_rat @ Y_27 @ Y_27 ) ) )
<=> ( ( X_51 != zero_zero_rat )
| ( Y_27 != zero_zero_rat ) ) ) ).
thf(fact_1594_sum__squares__gt__zero__iff,axiom,
! [X_51: real,Y_27: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X_51 @ X_51 ) @ ( times_times_real @ Y_27 @ Y_27 ) ) )
<=> ( ( X_51 != zero_zero_real )
| ( Y_27 != zero_zero_real ) ) ) ).
thf(fact_1595_sum__squares__gt__zero__iff,axiom,
! [X_51: int,Y_27: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X_51 @ X_51 ) @ ( times_times_int @ Y_27 @ Y_27 ) ) )
<=> ( ( X_51 != zero_zero_int )
| ( Y_27 != zero_zero_int ) ) ) ).
thf(fact_1596_not__sum__squares__lt__zero,axiom,
! [X_50: rat,Y_26: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X_50 @ X_50 ) @ ( times_times_rat @ Y_26 @ Y_26 ) ) @ zero_zero_rat ) ).
thf(fact_1597_not__sum__squares__lt__zero,axiom,
! [X_50: real,Y_26: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X_50 @ X_50 ) @ ( times_times_real @ Y_26 @ Y_26 ) ) @ zero_zero_real ) ).
thf(fact_1598_not__sum__squares__lt__zero,axiom,
! [X_50: int,Y_26: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X_50 @ X_50 ) @ ( times_times_int @ Y_26 @ Y_26 ) ) @ zero_zero_int ) ).
thf(fact_1599_sum__squares__le__zero__iff,axiom,
! [X_49: rat,Y_25: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X_49 @ X_49 ) @ ( times_times_rat @ Y_25 @ Y_25 ) ) @ zero_zero_rat )
<=> ( ( X_49 = zero_zero_rat )
& ( Y_25 = zero_zero_rat ) ) ) ).
thf(fact_1600_sum__squares__le__zero__iff,axiom,
! [X_49: real,Y_25: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X_49 @ X_49 ) @ ( times_times_real @ Y_25 @ Y_25 ) ) @ zero_zero_real )
<=> ( ( X_49 = zero_zero_real )
& ( Y_25 = zero_zero_real ) ) ) ).
thf(fact_1601_sum__squares__le__zero__iff,axiom,
! [X_49: int,Y_25: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X_49 @ X_49 ) @ ( times_times_int @ Y_25 @ Y_25 ) ) @ zero_zero_int )
<=> ( ( X_49 = zero_zero_int )
& ( Y_25 = zero_zero_int ) ) ) ).
thf(fact_1602_sum__squares__eq__zero__iff,axiom,
! [X_48: rat,Y_24: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ X_48 @ X_48 ) @ ( times_times_rat @ Y_24 @ Y_24 ) )
= zero_zero_rat )
<=> ( ( X_48 = zero_zero_rat )
& ( Y_24 = zero_zero_rat ) ) ) ).
thf(fact_1603_sum__squares__eq__zero__iff,axiom,
! [X_48: real,Y_24: real] :
( ( ( plus_plus_real @ ( times_times_real @ X_48 @ X_48 ) @ ( times_times_real @ Y_24 @ Y_24 ) )
= zero_zero_real )
<=> ( ( X_48 = zero_zero_real )
& ( Y_24 = zero_zero_real ) ) ) ).
thf(fact_1604_sum__squares__eq__zero__iff,axiom,
! [X_48: int,Y_24: int] :
( ( ( plus_plus_int @ ( times_times_int @ X_48 @ X_48 ) @ ( times_times_int @ Y_24 @ Y_24 ) )
= zero_zero_int )
<=> ( ( X_48 = zero_zero_int )
& ( Y_24 = zero_zero_int ) ) ) ).
thf(fact_1605_sum__squares__ge__zero,axiom,
! [X_47: rat,Y_23: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X_47 @ X_47 ) @ ( times_times_rat @ Y_23 @ Y_23 ) ) ) ).
thf(fact_1606_sum__squares__ge__zero,axiom,
! [X_47: real,Y_23: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X_47 @ X_47 ) @ ( times_times_real @ Y_23 @ Y_23 ) ) ) ).
thf(fact_1607_sum__squares__ge__zero,axiom,
! [X_47: int,Y_23: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X_47 @ X_47 ) @ ( times_times_int @ Y_23 @ Y_23 ) ) ) ).
thf(fact_1608_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_1609_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_1610_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_1611_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_1612_pos__zmult__pos,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
thf(fact_1613_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
thf(fact_1614_decr__mult__lemma,axiom,
! [K_1: int,P: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( minus_minus_int @ X_1 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( minus_minus_int @ X_1 @ ( times_times_int @ K_1 @ D ) ) ) ) ) ) ) ).
thf(fact_1615_incr__mult__lemma,axiom,
! [K_1: int,P: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( plus_plus_int @ X_1 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ! [X_1: int] :
( ( P @ X_1 )
=> ( P @ ( plus_plus_int @ X_1 @ ( times_times_int @ K_1 @ D ) ) ) ) ) ) ) ).
thf(fact_1616_field__le__mult__one__interval,axiom,
! [X_46: rat,Y_22: rat] :
( ! [Z: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_rat @ Z @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Z @ X_46 ) @ Y_22 ) ) )
=> ( ord_less_eq_rat @ X_46 @ Y_22 ) ) ).
thf(fact_1617_field__le__mult__one__interval,axiom,
! [X_46: real,Y_22: real] :
( ! [Z: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ Z @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z @ X_46 ) @ Y_22 ) ) )
=> ( ord_less_eq_real @ X_46 @ Y_22 ) ) ).
thf(fact_1618_divmod__int__relI,axiom,
! [A: int,B: int,Q: int,R_1: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q ) @ R_1 ) )
=> ( ( ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
& ( ord_less_int @ R_1 @ B ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ R_1 )
& ( ord_less_eq_int @ R_1 @ zero_zero_int ) ) ) )
=> ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q @ R_1 ) ) ) ) ).
thf(fact_1619_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_1620_divmod__int__rel__0,axiom,
! [B: int] :
( ( B != zero_zero_int )
=> ( divmod_int_rel @ zero_zero_int @ B @ ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) ) ).
thf(fact_1621_self__remainder,axiom,
! [A: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A @ A @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( A != zero_zero_int )
=> ( R_1 = zero_zero_int ) ) ) ).
thf(fact_1622_unique__quotient,axiom,
! [Q_5: int,R_3: int,A: int,B: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q_5 @ R_3 ) )
=> ( ( B != zero_zero_int )
=> ( Q = Q_5 ) ) ) ) ).
thf(fact_1623_unique__remainder,axiom,
! [Q_5: int,R_3: int,A: int,B: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q_5 @ R_3 ) )
=> ( ( B != zero_zero_int )
=> ( R_1 = R_3 ) ) ) ) ).
thf(fact_1624_self__quotient,axiom,
! [A: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A @ A @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( A != zero_zero_int )
=> ( Q = one_one_int ) ) ) ).
thf(fact_1625_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_1626_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_1627_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
thf(fact_1628_field__le__epsilon,axiom,
! [X_45: rat,Y_21: rat] :
( ! [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
=> ( ord_less_eq_rat @ X_45 @ ( plus_plus_rat @ Y_21 @ E ) ) )
=> ( ord_less_eq_rat @ X_45 @ Y_21 ) ) ).
thf(fact_1629_field__le__epsilon,axiom,
! [X_45: real,Y_21: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ X_45 @ ( plus_plus_real @ Y_21 @ E ) ) )
=> ( ord_less_eq_real @ X_45 @ Y_21 ) ) ).
thf(fact_1630_minf_I8_J,axiom,
! [T_8: rat] :
? [Z: rat] :
! [X_1: rat] :
( ( ord_less_rat @ X_1 @ Z )
=> ~ ( ord_less_eq_rat @ T_8 @ X_1 ) ) ).
thf(fact_1631_minf_I8_J,axiom,
! [T_8: quickcheck_code_int] :
? [Z: quickcheck_code_int] :
! [X_1: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_1 @ Z )
=> ~ ( ord_le258702272de_int @ T_8 @ X_1 ) ) ).
thf(fact_1632_minf_I8_J,axiom,
! [T_8: nat] :
? [Z: nat] :
! [X_1: nat] :
( ( ord_less_nat @ X_1 @ Z )
=> ~ ( ord_less_eq_nat @ T_8 @ X_1 ) ) ).
thf(fact_1633_minf_I8_J,axiom,
! [T_8: real] :
? [Z: real] :
! [X_1: real] :
( ( ord_less_real @ X_1 @ Z )
=> ~ ( ord_less_eq_real @ T_8 @ X_1 ) ) ).
thf(fact_1634_minf_I8_J,axiom,
! [T_8: int] :
? [Z: int] :
! [X_1: int] :
( ( ord_less_int @ X_1 @ Z )
=> ~ ( ord_less_eq_int @ T_8 @ X_1 ) ) ).
thf(fact_1635_pinf_I8_J,axiom,
! [T_7: rat] :
? [Z: rat] :
! [X_1: rat] :
( ( ord_less_rat @ Z @ X_1 )
=> ( ord_less_eq_rat @ T_7 @ X_1 ) ) ).
thf(fact_1636_pinf_I8_J,axiom,
! [T_7: quickcheck_code_int] :
? [Z: quickcheck_code_int] :
! [X_1: quickcheck_code_int] :
( ( ord_le1860547276de_int @ Z @ X_1 )
=> ( ord_le258702272de_int @ T_7 @ X_1 ) ) ).
thf(fact_1637_pinf_I8_J,axiom,
! [T_7: nat] :
? [Z: nat] :
! [X_1: nat] :
( ( ord_less_nat @ Z @ X_1 )
=> ( ord_less_eq_nat @ T_7 @ X_1 ) ) ).
thf(fact_1638_pinf_I8_J,axiom,
! [T_7: real] :
? [Z: real] :
! [X_1: real] :
( ( ord_less_real @ Z @ X_1 )
=> ( ord_less_eq_real @ T_7 @ X_1 ) ) ).
thf(fact_1639_pinf_I8_J,axiom,
! [T_7: int] :
? [Z: int] :
! [X_1: int] :
( ( ord_less_int @ Z @ X_1 )
=> ( ord_less_eq_int @ T_7 @ X_1 ) ) ).
thf(fact_1640_minf_I6_J,axiom,
! [T_6: rat] :
? [Z: rat] :
! [X_1: rat] :
( ( ord_less_rat @ X_1 @ Z )
=> ( ord_less_eq_rat @ X_1 @ T_6 ) ) ).
thf(fact_1641_minf_I6_J,axiom,
! [T_6: quickcheck_code_int] :
? [Z: quickcheck_code_int] :
! [X_1: quickcheck_code_int] :
( ( ord_le1860547276de_int @ X_1 @ Z )
=> ( ord_le258702272de_int @ X_1 @ T_6 ) ) ).
thf(fact_1642_minf_I6_J,axiom,
! [T_6: nat] :
? [Z: nat] :
! [X_1: nat] :
( ( ord_less_nat @ X_1 @ Z )
=> ( ord_less_eq_nat @ X_1 @ T_6 ) ) ).
thf(fact_1643_minf_I6_J,axiom,
! [T_6: real] :
? [Z: real] :
! [X_1: real] :
( ( ord_less_real @ X_1 @ Z )
=> ( ord_less_eq_real @ X_1 @ T_6 ) ) ).
thf(fact_1644_minf_I6_J,axiom,
! [T_6: int] :
? [Z: int] :
! [X_1: int] :
( ( ord_less_int @ X_1 @ Z )
=> ( ord_less_eq_int @ X_1 @ T_6 ) ) ).
thf(fact_1645_pinf_I6_J,axiom,
! [T_5: rat] :
? [Z: rat] :
! [X_1: rat] :
( ( ord_less_rat @ Z @ X_1 )
=> ~ ( ord_less_eq_rat @ X_1 @ T_5 ) ) ).
thf(fact_1646_pinf_I6_J,axiom,
! [T_5: quickcheck_code_int] :
? [Z: quickcheck_code_int] :
! [X_1: quickcheck_code_int] :
( ( ord_le1860547276de_int @ Z @ X_1 )
=> ~ ( ord_le258702272de_int @ X_1 @ T_5 ) ) ).
thf(fact_1647_pinf_I6_J,axiom,
! [T_5: nat] :
? [Z: nat] :
! [X_1: nat] :
( ( ord_less_nat @ Z @ X_1 )
=> ~ ( ord_less_eq_nat @ X_1 @ T_5 ) ) ).
thf(fact_1648_pinf_I6_J,axiom,
! [T_5: real] :
? [Z: real] :
! [X_1: real] :
( ( ord_less_real @ Z @ X_1 )
=> ~ ( ord_less_eq_real @ X_1 @ T_5 ) ) ).
thf(fact_1649_pinf_I6_J,axiom,
! [T_5: int] :
? [Z: int] :
! [X_1: int] :
( ( ord_less_int @ Z @ X_1 )
=> ~ ( ord_less_eq_int @ X_1 @ T_5 ) ) ).
thf(fact_1650_minusinfinity,axiom,
! [P: int > $o,P1: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int,K: int] :
( ( P1 @ X_1 )
<=> ( P1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K @ D ) ) ) )
=> ( ? [Z: int] :
! [X_1: int] :
( ( ord_less_int @ X_1 @ Z )
=> ( ( P @ X_1 )
<=> ( P1 @ X_1 ) ) )
=> ( ( ?? @ int @ P1 )
=> ( ?? @ int @ P ) ) ) ) ) ).
thf(fact_1651_plusinfinity,axiom,
! [P: int > $o,P_1: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int,K: int] :
( ( P_1 @ X_1 )
<=> ( P_1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K @ D ) ) ) )
=> ( ? [Z: int] :
! [X_1: int] :
( ( ord_less_int @ Z @ X_1 )
=> ( ( P @ X_1 )
<=> ( P_1 @ X_1 ) ) )
=> ( ( ?? @ int @ P_1 )
=> ( ?? @ int @ P ) ) ) ) ) ).
thf(fact_1652_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_1653_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_1654_floor__le__number__of,axiom,
! [X_44: rat,V_10: int] :
( ( ord_less_eq_int @ ( archim791455193or_rat @ X_44 ) @ ( number_number_of_int @ V_10 ) )
<=> ( ord_less_rat @ X_44 @ ( plus_plus_rat @ ( number_number_of_rat @ V_10 ) @ one_one_rat ) ) ) ).
thf(fact_1655_floor__le__number__of,axiom,
! [X_44: real,V_10: int] :
( ( ord_less_eq_int @ ( archim1246769320r_real @ X_44 ) @ ( number_number_of_int @ V_10 ) )
<=> ( ord_less_real @ X_44 @ ( plus_plus_real @ ( number267125858f_real @ V_10 ) @ one_one_real ) ) ) ).
thf(fact_1656_floor__zero,axiom,
( ( archim791455193or_rat @ zero_zero_rat )
= zero_zero_int ) ).
thf(fact_1657_floor__zero,axiom,
( ( archim1246769320r_real @ zero_zero_real )
= zero_zero_int ) ).
thf(fact_1658_floor__less__cancel,axiom,
! [X_43: rat,Y_20: rat] :
( ( ord_less_int @ ( archim791455193or_rat @ X_43 ) @ ( archim791455193or_rat @ Y_20 ) )
=> ( ord_less_rat @ X_43 @ Y_20 ) ) ).
thf(fact_1659_floor__less__cancel,axiom,
! [X_43: real,Y_20: real] :
( ( ord_less_int @ ( archim1246769320r_real @ X_43 ) @ ( archim1246769320r_real @ Y_20 ) )
=> ( ord_less_real @ X_43 @ Y_20 ) ) ).
thf(fact_1660_floor__mono,axiom,
! [X_42: rat,Y_19: rat] :
( ( ord_less_eq_rat @ X_42 @ Y_19 )
=> ( ord_less_eq_int @ ( archim791455193or_rat @ X_42 ) @ ( archim791455193or_rat @ Y_19 ) ) ) ).
thf(fact_1661_floor__mono,axiom,
! [X_42: real,Y_19: real] :
( ( ord_less_eq_real @ X_42 @ Y_19 )
=> ( ord_less_eq_int @ ( archim1246769320r_real @ X_42 ) @ ( archim1246769320r_real @ Y_19 ) ) ) ).
thf(fact_1662_floor__one,axiom,
( ( archim791455193or_rat @ one_one_rat )
= one_one_int ) ).
thf(fact_1663_floor__one,axiom,
( ( archim1246769320r_real @ one_one_real )
= one_one_int ) ).
thf(fact_1664_floor__number__of,axiom,
! [V_9: int] :
( ( archim791455193or_rat @ ( number_number_of_rat @ V_9 ) )
= ( number_number_of_int @ V_9 ) ) ).
thf(fact_1665_floor__number__of,axiom,
! [V_9: int] :
( ( archim1246769320r_real @ ( number267125858f_real @ V_9 ) )
= ( number_number_of_int @ V_9 ) ) ).
thf(fact_1666_floor__less__zero,axiom,
! [X_41: rat] :
( ( ord_less_int @ ( archim791455193or_rat @ X_41 ) @ zero_zero_int )
<=> ( ord_less_rat @ X_41 @ zero_zero_rat ) ) ).
thf(fact_1667_floor__less__zero,axiom,
! [X_41: real] :
( ( ord_less_int @ ( archim1246769320r_real @ X_41 ) @ zero_zero_int )
<=> ( ord_less_real @ X_41 @ zero_zero_real ) ) ).
thf(fact_1668_zero__le__floor,axiom,
! [X_40: rat] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim791455193or_rat @ X_40 ) )
<=> ( ord_less_eq_rat @ zero_zero_rat @ X_40 ) ) ).
thf(fact_1669_zero__le__floor,axiom,
! [X_40: real] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim1246769320r_real @ X_40 ) )
<=> ( ord_less_eq_real @ zero_zero_real @ X_40 ) ) ).
thf(fact_1670_zero__less__floor,axiom,
! [X_39: rat] :
( ( ord_less_int @ zero_zero_int @ ( archim791455193or_rat @ X_39 ) )
<=> ( ord_less_eq_rat @ one_one_rat @ X_39 ) ) ).
thf(fact_1671_zero__less__floor,axiom,
! [X_39: real] :
( ( ord_less_int @ zero_zero_int @ ( archim1246769320r_real @ X_39 ) )
<=> ( ord_less_eq_real @ one_one_real @ X_39 ) ) ).
thf(fact_1672_floor__le__zero,axiom,
! [X_38: rat] :
( ( ord_less_eq_int @ ( archim791455193or_rat @ X_38 ) @ zero_zero_int )
<=> ( ord_less_rat @ X_38 @ one_one_rat ) ) ).
thf(fact_1673_floor__le__zero,axiom,
! [X_38: real] :
( ( ord_less_eq_int @ ( archim1246769320r_real @ X_38 ) @ zero_zero_int )
<=> ( ord_less_real @ X_38 @ one_one_real ) ) ).
thf(fact_1674_floor__less__one,axiom,
! [X_37: rat] :
( ( ord_less_int @ ( archim791455193or_rat @ X_37 ) @ one_one_int )
<=> ( ord_less_rat @ X_37 @ one_one_rat ) ) ).
thf(fact_1675_floor__less__one,axiom,
! [X_37: real] :
( ( ord_less_int @ ( archim1246769320r_real @ X_37 ) @ one_one_int )
<=> ( ord_less_real @ X_37 @ one_one_real ) ) ).
thf(fact_1676_one__le__floor,axiom,
! [X_36: rat] :
( ( ord_less_eq_int @ one_one_int @ ( archim791455193or_rat @ X_36 ) )
<=> ( ord_less_eq_rat @ one_one_rat @ X_36 ) ) ).
thf(fact_1677_one__le__floor,axiom,
! [X_36: real] :
( ( ord_less_eq_int @ one_one_int @ ( archim1246769320r_real @ X_36 ) )
<=> ( ord_less_eq_real @ one_one_real @ X_36 ) ) ).
thf(fact_1678_floor__less__number__of,axiom,
! [X_35: rat,V_8: int] :
( ( ord_less_int @ ( archim791455193or_rat @ X_35 ) @ ( number_number_of_int @ V_8 ) )
<=> ( ord_less_rat @ X_35 @ ( number_number_of_rat @ V_8 ) ) ) ).
thf(fact_1679_floor__less__number__of,axiom,
! [X_35: real,V_8: int] :
( ( ord_less_int @ ( archim1246769320r_real @ X_35 ) @ ( number_number_of_int @ V_8 ) )
<=> ( ord_less_real @ X_35 @ ( number267125858f_real @ V_8 ) ) ) ).
thf(fact_1680_number__of__le__floor,axiom,
! [V_7: int,X_34: rat] :
( ( ord_less_eq_int @ ( number_number_of_int @ V_7 ) @ ( archim791455193or_rat @ X_34 ) )
<=> ( ord_less_eq_rat @ ( number_number_of_rat @ V_7 ) @ X_34 ) ) ).
thf(fact_1681_number__of__le__floor,axiom,
! [V_7: int,X_34: real] :
( ( ord_less_eq_int @ ( number_number_of_int @ V_7 ) @ ( archim1246769320r_real @ X_34 ) )
<=> ( ord_less_eq_real @ ( number267125858f_real @ V_7 ) @ X_34 ) ) ).
thf(fact_1682_floor__add__one,axiom,
! [X_33: rat] :
( ( archim791455193or_rat @ ( plus_plus_rat @ X_33 @ one_one_rat ) )
= ( plus_plus_int @ ( archim791455193or_rat @ X_33 ) @ one_one_int ) ) ).
thf(fact_1683_floor__add__one,axiom,
! [X_33: real] :
( ( archim1246769320r_real @ ( plus_plus_real @ X_33 @ one_one_real ) )
= ( plus_plus_int @ ( archim1246769320r_real @ X_33 ) @ one_one_int ) ) ).
thf(fact_1684_floor__add__number__of,axiom,
! [X_32: rat,V_6: int] :
( ( archim791455193or_rat @ ( plus_plus_rat @ X_32 @ ( number_number_of_rat @ V_6 ) ) )
= ( plus_plus_int @ ( archim791455193or_rat @ X_32 ) @ ( number_number_of_int @ V_6 ) ) ) ).
thf(fact_1685_floor__add__number__of,axiom,
! [X_32: real,V_6: int] :
( ( archim1246769320r_real @ ( plus_plus_real @ X_32 @ ( number267125858f_real @ V_6 ) ) )
= ( plus_plus_int @ ( archim1246769320r_real @ X_32 ) @ ( number_number_of_int @ V_6 ) ) ) ).
thf(fact_1686_floor__diff__one,axiom,
! [X_31: rat] :
( ( archim791455193or_rat @ ( minus_minus_rat @ X_31 @ one_one_rat ) )
= ( minus_minus_int @ ( archim791455193or_rat @ X_31 ) @ one_one_int ) ) ).
thf(fact_1687_floor__diff__one,axiom,
! [X_31: real] :
( ( archim1246769320r_real @ ( minus_minus_real @ X_31 @ one_one_real ) )
= ( minus_minus_int @ ( archim1246769320r_real @ X_31 ) @ one_one_int ) ) ).
thf(fact_1688_floor__diff__number__of,axiom,
! [X_30: rat,V_5: int] :
( ( archim791455193or_rat @ ( minus_minus_rat @ X_30 @ ( number_number_of_rat @ V_5 ) ) )
= ( minus_minus_int @ ( archim791455193or_rat @ X_30 ) @ ( number_number_of_int @ V_5 ) ) ) ).
thf(fact_1689_floor__diff__number__of,axiom,
! [X_30: real,V_5: int] :
( ( archim1246769320r_real @ ( minus_minus_real @ X_30 @ ( number267125858f_real @ V_5 ) ) )
= ( minus_minus_int @ ( archim1246769320r_real @ X_30 ) @ ( number_number_of_int @ V_5 ) ) ) ).
thf(fact_1690_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_1691_tsub__eq,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( nat_tsub @ X @ Y )
= ( minus_minus_int @ X @ Y ) ) ) ).
thf(fact_1692_number__of__less__floor,axiom,
! [V_4: int,X_29: rat] :
( ( ord_less_int @ ( number_number_of_int @ V_4 ) @ ( archim791455193or_rat @ X_29 ) )
<=> ( ord_less_eq_rat @ ( plus_plus_rat @ ( number_number_of_rat @ V_4 ) @ one_one_rat ) @ X_29 ) ) ).
thf(fact_1693_number__of__less__floor,axiom,
! [V_4: int,X_29: real] :
( ( ord_less_int @ ( number_number_of_int @ V_4 ) @ ( archim1246769320r_real @ X_29 ) )
<=> ( ord_less_eq_real @ ( plus_plus_real @ ( number267125858f_real @ V_4 ) @ one_one_real ) @ X_29 ) ) ).
thf(fact_1694_le__mult__floor,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_int @ ( times_times_int @ ( archim1246769320r_real @ A ) @ ( archim1246769320r_real @ B ) ) @ ( archim1246769320r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
thf(fact_1695_posDivAlg__correct,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( divmod_int_rel @ A @ B @ ( posDivAlg @ A @ B ) ) ) ) ).
thf(fact_1696_incr__lemma,axiom,
! [Z_1: int,X: int,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ Z_1 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z_1 ) ) @ one_one_int ) @ D ) ) ) ) ).
thf(fact_1697_decr__lemma,axiom,
! [X: int,Z_1: int,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z_1 ) ) @ one_one_int ) @ D ) ) @ Z_1 ) ) ).
thf(fact_1698_Ints__odd__less__0,axiom,
! [A_150: real] :
( ( member_real @ A_150 @ ring_1_Ints_real )
=> ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A_150 ) @ A_150 ) @ zero_zero_real )
<=> ( ord_less_real @ A_150 @ zero_zero_real ) ) ) ).
thf(fact_1699_Ints__odd__less__0,axiom,
! [A_150: int] :
( ( member_int @ A_150 @ ring_1_Ints_int )
=> ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A_150 ) @ A_150 ) @ zero_zero_int )
<=> ( ord_less_int @ A_150 @ zero_zero_int ) ) ) ).
thf(fact_1700_Ints__odd__less__0,axiom,
! [A_150: rat] :
( ( member_rat @ A_150 @ ring_1_Ints_rat )
=> ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A_150 ) @ A_150 ) @ zero_zero_rat )
<=> ( ord_less_rat @ A_150 @ zero_zero_rat ) ) ) ).
thf(fact_1701_zminus1__lemma,axiom,
! [A: int,B: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( divmod_int_rel @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R_1 = zero_zero_int ) @ ( uminus_uminus_int @ Q ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q ) @ one_one_int ) ) @ ( if_int @ ( R_1 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R_1 ) ) ) ) ) ).
thf(fact_1702_d22set__mem,axiom,
! [A: int,B: int] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( member_int @ B @ ( d22set @ A ) ) ) ) ).
thf(fact_1703_zero__le__ceiling,axiom,
! [X_28: rat] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim1419379755ng_rat @ X_28 ) )
<=> ( ord_less_rat @ ( number_number_of_rat @ min ) @ X_28 ) ) ).
thf(fact_1704_zero__le__ceiling,axiom,
! [X_28: real] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim856651990g_real @ X_28 ) )
<=> ( ord_less_real @ ( number267125858f_real @ min ) @ X_28 ) ) ).
thf(fact_1705_ceiling__less__zero,axiom,
! [X_27: real] :
( ( ord_less_int @ ( archim856651990g_real @ X_27 ) @ zero_zero_int )
<=> ( ord_less_eq_real @ X_27 @ ( number267125858f_real @ min ) ) ) ).
thf(fact_1706_ceiling__less__zero,axiom,
! [X_27: rat] :
( ( ord_less_int @ ( archim1419379755ng_rat @ X_27 ) @ zero_zero_int )
<=> ( ord_less_eq_rat @ X_27 @ ( number_number_of_rat @ min ) ) ) ).
thf(fact_1707_split__zmod,axiom,
! [P: int > $o,N: int,K_1: int] :
( ( P @ ( div_mod_int @ N @ K_1 ) )
<=> ( ( ( K_1 = zero_zero_int )
=> ( P @ N ) )
& ( ( ord_less_int @ zero_zero_int @ K_1 )
=> ! [I_1: int,J_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J_1 )
& ( ord_less_int @ J_1 @ K_1 )
& ( N
= ( plus_plus_int @ ( times_times_int @ K_1 @ I_1 ) @ J_1 ) ) )
=> ( P @ J_1 ) ) )
& ( ( ord_less_int @ K_1 @ zero_zero_int )
=> ! [I_1: int,J_1: int] :
( ( ( ord_less_int @ K_1 @ J_1 )
& ( ord_less_eq_int @ J_1 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K_1 @ I_1 ) @ J_1 ) ) )
=> ( P @ J_1 ) ) ) ) ) ).
thf(fact_1708_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_1709_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_1710_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_1711_zmod__zminus2__not__zero,axiom,
! [K_1: int,L: int] :
( ( ( div_mod_int @ K_1 @ ( uminus_uminus_int @ L ) )
!= zero_zero_int )
=> ( ( div_mod_int @ K_1 @ L )
!= zero_zero_int ) ) ).
thf(fact_1712_zmod__zminus1__not__zero,axiom,
! [K_1: int,L: int] :
( ( ( div_mod_int @ ( uminus_uminus_int @ K_1 ) @ L )
!= zero_zero_int )
=> ( ( div_mod_int @ K_1 @ L )
!= zero_zero_int ) ) ).
thf(fact_1713_real__mult__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
<=> ( A = B ) ) ) ).
thf(fact_1714_real__mult__right__cancel,axiom,
! [A: real,B: real,C: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
<=> ( A = B ) ) ) ).
thf(fact_1715_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_1716_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_1717_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_1718_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_1719_real__zero__not__eq__one,axiom,
zero_zero_real != one_one_real ).
thf(fact_1720_mod__minus__cong,axiom,
! [A_149: int,B_101: int,A_148: int] :
( ( ( div_mod_int @ A_149 @ B_101 )
= ( div_mod_int @ A_148 @ B_101 ) )
=> ( ( div_mod_int @ ( uminus_uminus_int @ A_149 ) @ B_101 )
= ( div_mod_int @ ( uminus_uminus_int @ A_148 ) @ B_101 ) ) ) ).
thf(fact_1721_mod__mod__trivial,axiom,
! [A_147: quickcheck_code_int,B_100: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( div_mo231679042de_int @ A_147 @ B_100 ) @ B_100 )
= ( div_mo231679042de_int @ A_147 @ B_100 ) ) ).
thf(fact_1722_mod__mod__trivial,axiom,
! [A_147: code_code_numeral,B_100: code_code_numeral] :
( ( div_mo1740067990umeral @ ( div_mo1740067990umeral @ A_147 @ B_100 ) @ B_100 )
= ( div_mo1740067990umeral @ A_147 @ B_100 ) ) ).
thf(fact_1723_mod__mod__trivial,axiom,
! [A_147: nat,B_100: nat] :
( ( div_mod_nat @ ( div_mod_nat @ A_147 @ B_100 ) @ B_100 )
= ( div_mod_nat @ A_147 @ B_100 ) ) ).
thf(fact_1724_mod__mod__trivial,axiom,
! [A_147: int,B_100: int] :
( ( div_mod_int @ ( div_mod_int @ A_147 @ B_100 ) @ B_100 )
= ( div_mod_int @ A_147 @ B_100 ) ) ).
thf(fact_1725_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_1726_zmod__zminus__zminus,axiom,
! [A: int,B: int] :
( ( div_mod_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( div_mod_int @ A @ B ) ) ) ).
thf(fact_1727_mod__minus__eq,axiom,
! [A_146: int,B_99: int] :
( ( div_mod_int @ ( uminus_uminus_int @ A_146 ) @ B_99 )
= ( div_mod_int @ ( uminus_uminus_int @ ( div_mod_int @ A_146 @ B_99 ) ) @ B_99 ) ) ).
thf(fact_1728_zmod__zminus2,axiom,
! [A: int,B: int] :
( ( div_mod_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( div_mod_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
thf(fact_1729_arith__simps_I30_J,axiom,
! [W_7: int] :
( ( uminus_uminus_rat @ ( number_number_of_rat @ W_7 ) )
= ( number_number_of_rat @ ( uminus_uminus_int @ W_7 ) ) ) ).
thf(fact_1730_arith__simps_I30_J,axiom,
! [W_7: int] :
( ( uminus473333897omplex @ ( number528085621omplex @ W_7 ) )
= ( number528085621omplex @ ( uminus_uminus_int @ W_7 ) ) ) ).
thf(fact_1731_arith__simps_I30_J,axiom,
! [W_7: int] :
( ( uminus_uminus_real @ ( number267125858f_real @ W_7 ) )
= ( number267125858f_real @ ( uminus_uminus_int @ W_7 ) ) ) ).
thf(fact_1732_arith__simps_I30_J,axiom,
! [W_7: int] :
( ( uminus_uminus_int @ ( number_number_of_int @ W_7 ) )
= ( number_number_of_int @ ( uminus_uminus_int @ W_7 ) ) ) ).
thf(fact_1733_number__of__minus,axiom,
! [W_6: int] :
( ( number_number_of_rat @ ( uminus_uminus_int @ W_6 ) )
= ( uminus_uminus_rat @ ( number_number_of_rat @ W_6 ) ) ) ).
thf(fact_1734_number__of__minus,axiom,
! [W_6: int] :
( ( number528085621omplex @ ( uminus_uminus_int @ W_6 ) )
= ( uminus473333897omplex @ ( number528085621omplex @ W_6 ) ) ) ).
thf(fact_1735_number__of__minus,axiom,
! [W_6: int] :
( ( number267125858f_real @ ( uminus_uminus_int @ W_6 ) )
= ( uminus_uminus_real @ ( number267125858f_real @ W_6 ) ) ) ).
thf(fact_1736_number__of__minus,axiom,
! [W_6: int] :
( ( number_number_of_int @ ( uminus_uminus_int @ W_6 ) )
= ( uminus_uminus_int @ ( number_number_of_int @ W_6 ) ) ) ).
thf(fact_1737_Int_OMin__def,axiom,
( min
= ( uminus_uminus_int @ one_one_int ) ) ).
thf(fact_1738_mult__Min,axiom,
! [K_1: int] :
( ( times_times_int @ min @ K_1 )
= ( uminus_uminus_int @ K_1 ) ) ).
thf(fact_1739_real__le__refl,axiom,
! [W: real] : ( ord_less_eq_real @ W @ W ) ).
thf(fact_1740_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_1741_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_1742_real__le__trans,axiom,
! [K_1: real,I: real,J: real] :
( ( ord_less_eq_real @ I @ J )
=> ( ( ord_less_eq_real @ J @ K_1 )
=> ( ord_less_eq_real @ I @ K_1 ) ) ) ).
thf(fact_1743_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_1744_abs__ge__minus__self,axiom,
! [A_145: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A_145 ) @ ( abs_abs_rat @ A_145 ) ) ).
thf(fact_1745_abs__ge__minus__self,axiom,
! [A_145: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A_145 ) @ ( abs_abs_real @ A_145 ) ) ).
thf(fact_1746_abs__ge__minus__self,axiom,
! [A_145: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A_145 ) @ ( abs_abs_int @ A_145 ) ) ).
thf(fact_1747_abs__le__iff,axiom,
! [A_144: rat,B_98: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A_144 ) @ B_98 )
<=> ( ( ord_less_eq_rat @ A_144 @ B_98 )
& ( ord_less_eq_rat @ ( uminus_uminus_rat @ A_144 ) @ B_98 ) ) ) ).
thf(fact_1748_abs__le__iff,axiom,
! [A_144: real,B_98: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A_144 ) @ B_98 )
<=> ( ( ord_less_eq_real @ A_144 @ B_98 )
& ( ord_less_eq_real @ ( uminus_uminus_real @ A_144 ) @ B_98 ) ) ) ).
thf(fact_1749_abs__le__iff,axiom,
! [A_144: int,B_98: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A_144 ) @ B_98 )
<=> ( ( ord_less_eq_int @ A_144 @ B_98 )
& ( ord_less_eq_int @ ( uminus_uminus_int @ A_144 ) @ B_98 ) ) ) ).
thf(fact_1750_abs__leI,axiom,
! [A_143: rat,B_97: rat] :
( ( ord_less_eq_rat @ A_143 @ B_97 )
=> ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A_143 ) @ B_97 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ A_143 ) @ B_97 ) ) ) ).
thf(fact_1751_abs__leI,axiom,
! [A_143: real,B_97: real] :
( ( ord_less_eq_real @ A_143 @ B_97 )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ A_143 ) @ B_97 )
=> ( ord_less_eq_real @ ( abs_abs_real @ A_143 ) @ B_97 ) ) ) ).
thf(fact_1752_abs__leI,axiom,
! [A_143: int,B_97: int] :
( ( ord_less_eq_int @ A_143 @ B_97 )
=> ( ( ord_less_eq_int @ ( uminus_uminus_int @ A_143 ) @ B_97 )
=> ( ord_less_eq_int @ ( abs_abs_int @ A_143 ) @ B_97 ) ) ) ).
thf(fact_1753_abs__le__D2,axiom,
! [A_142: rat,B_96: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A_142 ) @ B_96 )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ A_142 ) @ B_96 ) ) ).
thf(fact_1754_abs__le__D2,axiom,
! [A_142: real,B_96: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A_142 ) @ B_96 )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ A_142 ) @ B_96 ) ) ).
thf(fact_1755_abs__le__D2,axiom,
! [A_142: int,B_96: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A_142 ) @ B_96 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ A_142 ) @ B_96 ) ) ).
thf(fact_1756_Ints__minus,axiom,
! [A_141: real] :
( ( member_real @ A_141 @ ring_1_Ints_real )
=> ( member_real @ ( uminus_uminus_real @ A_141 ) @ ring_1_Ints_real ) ) ).
thf(fact_1757_Ints__minus,axiom,
! [A_141: int] :
( ( member_int @ A_141 @ ring_1_Ints_int )
=> ( member_int @ ( uminus_uminus_int @ A_141 ) @ ring_1_Ints_int ) ) ).
thf(fact_1758_Ints__minus,axiom,
! [A_141: rat] :
( ( member_rat @ A_141 @ ring_1_Ints_rat )
=> ( member_rat @ ( uminus_uminus_rat @ A_141 ) @ ring_1_Ints_rat ) ) ).
thf(fact_1759_Ints__minus,axiom,
! [A_141: complex] :
( ( member_complex @ A_141 @ ring_1_Ints_complex )
=> ( member_complex @ ( uminus473333897omplex @ A_141 ) @ ring_1_Ints_complex ) ) ).
thf(fact_1760_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_1761_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_1762_neg__equal__iff__equal,axiom,
! [A_140: rat,B_95: rat] :
( ( ( uminus_uminus_rat @ A_140 )
= ( uminus_uminus_rat @ B_95 ) )
<=> ( A_140 = B_95 ) ) ).
thf(fact_1763_neg__equal__iff__equal,axiom,
! [A_140: complex,B_95: complex] :
( ( ( uminus473333897omplex @ A_140 )
= ( uminus473333897omplex @ B_95 ) )
<=> ( A_140 = B_95 ) ) ).
thf(fact_1764_neg__equal__iff__equal,axiom,
! [A_140: real,B_95: real] :
( ( ( uminus_uminus_real @ A_140 )
= ( uminus_uminus_real @ B_95 ) )
<=> ( A_140 = B_95 ) ) ).
thf(fact_1765_neg__equal__iff__equal,axiom,
! [A_140: int,B_95: int] :
( ( ( uminus_uminus_int @ A_140 )
= ( uminus_uminus_int @ B_95 ) )
<=> ( A_140 = B_95 ) ) ).
thf(fact_1766_minus__equation__iff,axiom,
! [A_139: rat,B_94: rat] :
( ( ( uminus_uminus_rat @ A_139 )
= B_94 )
<=> ( ( uminus_uminus_rat @ B_94 )
= A_139 ) ) ).
thf(fact_1767_minus__equation__iff,axiom,
! [A_139: complex,B_94: complex] :
( ( ( uminus473333897omplex @ A_139 )
= B_94 )
<=> ( ( uminus473333897omplex @ B_94 )
= A_139 ) ) ).
thf(fact_1768_minus__equation__iff,axiom,
! [A_139: real,B_94: real] :
( ( ( uminus_uminus_real @ A_139 )
= B_94 )
<=> ( ( uminus_uminus_real @ B_94 )
= A_139 ) ) ).
thf(fact_1769_minus__equation__iff,axiom,
! [A_139: int,B_94: int] :
( ( ( uminus_uminus_int @ A_139 )
= B_94 )
<=> ( ( uminus_uminus_int @ B_94 )
= A_139 ) ) ).
thf(fact_1770_equation__minus__iff,axiom,
! [A_138: rat,B_93: rat] :
( ( A_138
= ( uminus_uminus_rat @ B_93 ) )
<=> ( B_93
= ( uminus_uminus_rat @ A_138 ) ) ) ).
thf(fact_1771_equation__minus__iff,axiom,
! [A_138: complex,B_93: complex] :
( ( A_138
= ( uminus473333897omplex @ B_93 ) )
<=> ( B_93
= ( uminus473333897omplex @ A_138 ) ) ) ).
thf(fact_1772_equation__minus__iff,axiom,
! [A_138: real,B_93: real] :
( ( A_138
= ( uminus_uminus_real @ B_93 ) )
<=> ( B_93
= ( uminus_uminus_real @ A_138 ) ) ) ).
thf(fact_1773_equation__minus__iff,axiom,
! [A_138: int,B_93: int] :
( ( A_138
= ( uminus_uminus_int @ B_93 ) )
<=> ( B_93
= ( uminus_uminus_int @ A_138 ) ) ) ).
thf(fact_1774_real__mult__commute,axiom,
! [Z_1: real,W: real] :
( ( times_times_real @ Z_1 @ W )
= ( times_times_real @ W @ Z_1 ) ) ).
thf(fact_1775_abs__minus__cancel,axiom,
! [A_137: rat] :
( ( abs_abs_rat @ ( uminus_uminus_rat @ A_137 ) )
= ( abs_abs_rat @ A_137 ) ) ).
thf(fact_1776_abs__minus__cancel,axiom,
! [A_137: real] :
( ( abs_abs_real @ ( uminus_uminus_real @ A_137 ) )
= ( abs_abs_real @ A_137 ) ) ).
thf(fact_1777_abs__minus__cancel,axiom,
! [A_137: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A_137 ) )
= ( abs_abs_int @ A_137 ) ) ).
thf(fact_1778_abs__idempotent,axiom,
! [A_136: rat] :
( ( abs_abs_rat @ ( abs_abs_rat @ A_136 ) )
= ( abs_abs_rat @ A_136 ) ) ).
thf(fact_1779_abs__idempotent,axiom,
! [A_136: real] :
( ( abs_abs_real @ ( abs_abs_real @ A_136 ) )
= ( abs_abs_real @ A_136 ) ) ).
thf(fact_1780_abs__idempotent,axiom,
! [A_136: int] :
( ( abs_abs_int @ ( abs_abs_int @ A_136 ) )
= ( abs_abs_int @ A_136 ) ) ).
thf(fact_1781_zminus__zminus,axiom,
! [Z_1: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ Z_1 ) )
= Z_1 ) ).
thf(fact_1782_minus__minus,axiom,
! [A_135: rat] :
( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A_135 ) )
= A_135 ) ).
thf(fact_1783_minus__minus,axiom,
! [A_135: complex] :
( ( uminus473333897omplex @ ( uminus473333897omplex @ A_135 ) )
= A_135 ) ).
thf(fact_1784_minus__minus,axiom,
! [A_135: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A_135 ) )
= A_135 ) ).
thf(fact_1785_minus__minus,axiom,
! [A_135: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A_135 ) )
= A_135 ) ).
thf(fact_1786_real__mult__1,axiom,
! [Z_1: real] :
( ( times_times_real @ one_one_real @ Z_1 )
= Z_1 ) ).
thf(fact_1787_abs__less__iff,axiom,
! [A_134: rat,B_92: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ A_134 ) @ B_92 )
<=> ( ( ord_less_rat @ A_134 @ B_92 )
& ( ord_less_rat @ ( uminus_uminus_rat @ A_134 ) @ B_92 ) ) ) ).
thf(fact_1788_abs__less__iff,axiom,
! [A_134: real,B_92: real] :
( ( ord_less_real @ ( abs_abs_real @ A_134 ) @ B_92 )
<=> ( ( ord_less_real @ A_134 @ B_92 )
& ( ord_less_real @ ( uminus_uminus_real @ A_134 ) @ B_92 ) ) ) ).
thf(fact_1789_abs__less__iff,axiom,
! [A_134: int,B_92: int] :
( ( ord_less_int @ ( abs_abs_int @ A_134 ) @ B_92 )
<=> ( ( ord_less_int @ A_134 @ B_92 )
& ( ord_less_int @ ( uminus_uminus_int @ A_134 ) @ B_92 ) ) ) ).
thf(fact_1790_less__eq__real__def,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
<=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
thf(fact_1791_real__less__def,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
<=> ( ( ord_less_eq_real @ X @ Y )
& ( X != Y ) ) ) ).
thf(fact_1792_abs__minus__le__zero,axiom,
! [A_133: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A_133 ) ) @ zero_zero_rat ) ).
thf(fact_1793_abs__minus__le__zero,axiom,
! [A_133: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A_133 ) ) @ zero_zero_real ) ).
thf(fact_1794_abs__minus__le__zero,axiom,
! [A_133: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A_133 ) ) @ zero_zero_int ) ).
thf(fact_1795_abs__of__nonpos,axiom,
! [A_132: rat] :
( ( ord_less_eq_rat @ A_132 @ zero_zero_rat )
=> ( ( abs_abs_rat @ A_132 )
= ( uminus_uminus_rat @ A_132 ) ) ) ).
thf(fact_1796_abs__of__nonpos,axiom,
! [A_132: real] :
( ( ord_less_eq_real @ A_132 @ zero_zero_real )
=> ( ( abs_abs_real @ A_132 )
= ( uminus_uminus_real @ A_132 ) ) ) ).
thf(fact_1797_abs__of__nonpos,axiom,
! [A_132: int] :
( ( ord_less_eq_int @ A_132 @ zero_zero_int )
=> ( ( abs_abs_int @ A_132 )
= ( uminus_uminus_int @ A_132 ) ) ) ).
thf(fact_1798_abs__of__neg,axiom,
! [A_131: rat] :
( ( ord_less_rat @ A_131 @ zero_zero_rat )
=> ( ( abs_abs_rat @ A_131 )
= ( uminus_uminus_rat @ A_131 ) ) ) ).
thf(fact_1799_abs__of__neg,axiom,
! [A_131: real] :
( ( ord_less_real @ A_131 @ zero_zero_real )
=> ( ( abs_abs_real @ A_131 )
= ( uminus_uminus_real @ A_131 ) ) ) ).
thf(fact_1800_abs__of__neg,axiom,
! [A_131: int] :
( ( ord_less_int @ A_131 @ zero_zero_int )
=> ( ( abs_abs_int @ A_131 )
= ( uminus_uminus_int @ A_131 ) ) ) ).
thf(fact_1801_abs__if,axiom,
! [A_130: rat] :
( ( ( ord_less_rat @ A_130 @ zero_zero_rat )
=> ( ( abs_abs_rat @ A_130 )
= ( uminus_uminus_rat @ A_130 ) ) )
& ( ~ ( ord_less_rat @ A_130 @ zero_zero_rat )
=> ( ( abs_abs_rat @ A_130 )
= A_130 ) ) ) ).
thf(fact_1802_abs__if,axiom,
! [A_130: real] :
( ( ( ord_less_real @ A_130 @ zero_zero_real )
=> ( ( abs_abs_real @ A_130 )
= ( uminus_uminus_real @ A_130 ) ) )
& ( ~ ( ord_less_real @ A_130 @ zero_zero_real )
=> ( ( abs_abs_real @ A_130 )
= A_130 ) ) ) ).
thf(fact_1803_abs__if,axiom,
! [A_130: int] :
( ( ( ord_less_int @ A_130 @ zero_zero_int )
=> ( ( abs_abs_int @ A_130 )
= ( uminus_uminus_int @ A_130 ) ) )
& ( ~ ( ord_less_int @ A_130 @ zero_zero_int )
=> ( ( abs_abs_int @ A_130 )
= A_130 ) ) ) ).
thf(fact_1804_abs__minus__one,axiom,
( ( abs_abs_rat @ ( number_number_of_rat @ min ) )
= one_one_rat ) ).
thf(fact_1805_abs__minus__one,axiom,
( ( abs_abs_real @ ( number267125858f_real @ min ) )
= one_one_real ) ).
thf(fact_1806_abs__minus__one,axiom,
( ( abs_abs_int @ ( number_number_of_int @ min ) )
= one_one_int ) ).
thf(fact_1807_minus__number__of__mult,axiom,
! [W_5: int,Z_8: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( number_number_of_rat @ W_5 ) ) @ Z_8 )
= ( times_times_rat @ ( number_number_of_rat @ ( uminus_uminus_int @ W_5 ) ) @ Z_8 ) ) ).
thf(fact_1808_minus__number__of__mult,axiom,
! [W_5: int,Z_8: complex] :
( ( times_times_complex @ ( uminus473333897omplex @ ( number528085621omplex @ W_5 ) ) @ Z_8 )
= ( times_times_complex @ ( number528085621omplex @ ( uminus_uminus_int @ W_5 ) ) @ Z_8 ) ) ).
thf(fact_1809_minus__number__of__mult,axiom,
! [W_5: int,Z_8: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( number267125858f_real @ W_5 ) ) @ Z_8 )
= ( times_times_real @ ( number267125858f_real @ ( uminus_uminus_int @ W_5 ) ) @ Z_8 ) ) ).
thf(fact_1810_minus__number__of__mult,axiom,
! [W_5: int,Z_8: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( number_number_of_int @ W_5 ) ) @ Z_8 )
= ( times_times_int @ ( number_number_of_int @ ( uminus_uminus_int @ W_5 ) ) @ Z_8 ) ) ).
thf(fact_1811_mult__minus1,axiom,
! [Z_7: rat] :
( ( times_times_rat @ ( number_number_of_rat @ min ) @ Z_7 )
= ( uminus_uminus_rat @ Z_7 ) ) ).
thf(fact_1812_mult__minus1,axiom,
! [Z_7: complex] :
( ( times_times_complex @ ( number528085621omplex @ min ) @ Z_7 )
= ( uminus473333897omplex @ Z_7 ) ) ).
thf(fact_1813_mult__minus1,axiom,
! [Z_7: real] :
( ( times_times_real @ ( number267125858f_real @ min ) @ Z_7 )
= ( uminus_uminus_real @ Z_7 ) ) ).
thf(fact_1814_mult__minus1,axiom,
! [Z_7: int] :
( ( times_times_int @ ( number_number_of_int @ min ) @ Z_7 )
= ( uminus_uminus_int @ Z_7 ) ) ).
thf(fact_1815_mult__minus1__right,axiom,
! [Z_6: rat] :
( ( times_times_rat @ Z_6 @ ( number_number_of_rat @ min ) )
= ( uminus_uminus_rat @ Z_6 ) ) ).
thf(fact_1816_mult__minus1__right,axiom,
! [Z_6: complex] :
( ( times_times_complex @ Z_6 @ ( number528085621omplex @ min ) )
= ( uminus473333897omplex @ Z_6 ) ) ).
thf(fact_1817_mult__minus1__right,axiom,
! [Z_6: real] :
( ( times_times_real @ Z_6 @ ( number267125858f_real @ min ) )
= ( uminus_uminus_real @ Z_6 ) ) ).
thf(fact_1818_mult__minus1__right,axiom,
! [Z_6: int] :
( ( times_times_int @ Z_6 @ ( number_number_of_int @ min ) )
= ( uminus_uminus_int @ Z_6 ) ) ).
thf(fact_1819_number__of__Min,axiom,
( ( number_number_of_rat @ min )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
thf(fact_1820_number__of__Min,axiom,
( ( number528085621omplex @ min )
= ( uminus473333897omplex @ one_one_complex ) ) ).
thf(fact_1821_number__of__Min,axiom,
( ( number267125858f_real @ min )
= ( uminus_uminus_real @ one_one_real ) ) ).
thf(fact_1822_number__of__Min,axiom,
( ( number_number_of_int @ min )
= ( uminus_uminus_int @ one_one_int ) ) ).
thf(fact_1823_arith__simps_I31_J,axiom,
( ( uminus_uminus_rat @ one_one_rat )
= ( number_number_of_rat @ min ) ) ).
thf(fact_1824_arith__simps_I31_J,axiom,
( ( uminus473333897omplex @ one_one_complex )
= ( number528085621omplex @ min ) ) ).
thf(fact_1825_arith__simps_I31_J,axiom,
( ( uminus_uminus_real @ one_one_real )
= ( number267125858f_real @ min ) ) ).
thf(fact_1826_arith__simps_I31_J,axiom,
( ( uminus_uminus_int @ one_one_int )
= ( number_number_of_int @ min ) ) ).
thf(fact_1827_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_1828_zmod__minus1__right,axiom,
! [A: int] :
( ( div_mod_int @ A @ ( number_number_of_int @ min ) )
= zero_zero_int ) ).
thf(fact_1829_zmod__zminus1__eq__if,axiom,
! [A: int,B: int] :
( ( ( ( div_mod_int @ A @ B )
= zero_zero_int )
=> ( ( div_mod_int @ ( uminus_uminus_int @ A ) @ B )
= zero_zero_int ) )
& ( ( ( div_mod_int @ A @ B )
!= zero_zero_int )
=> ( ( div_mod_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ ( div_mod_int @ A @ B ) ) ) ) ) ).
thf(fact_1830_zmod__zminus2__eq__if,axiom,
! [A: int,B: int] :
( ( ( ( div_mod_int @ A @ B )
= zero_zero_int )
=> ( ( div_mod_int @ A @ ( uminus_uminus_int @ B ) )
= zero_zero_int ) )
& ( ( ( div_mod_int @ A @ B )
!= zero_zero_int )
=> ( ( div_mod_int @ A @ ( uminus_uminus_int @ B ) )
= ( minus_minus_int @ ( div_mod_int @ A @ B ) @ B ) ) ) ) ).
thf(fact_1831_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_1832_mod__0,axiom,
! [A_129: quickcheck_code_int] :
( ( div_mo231679042de_int @ zero_z891286103de_int @ A_129 )
= zero_z891286103de_int ) ).
thf(fact_1833_mod__0,axiom,
! [A_129: code_code_numeral] :
( ( div_mo1740067990umeral @ zero_z126310315umeral @ A_129 )
= zero_z126310315umeral ) ).
thf(fact_1834_mod__0,axiom,
! [A_129: nat] :
( ( div_mod_nat @ zero_zero_nat @ A_129 )
= zero_zero_nat ) ).
thf(fact_1835_mod__0,axiom,
! [A_129: int] :
( ( div_mod_int @ zero_zero_int @ A_129 )
= zero_zero_int ) ).
thf(fact_1836_mod__by__0,axiom,
! [A_128: quickcheck_code_int] :
( ( div_mo231679042de_int @ A_128 @ zero_z891286103de_int )
= A_128 ) ).
thf(fact_1837_mod__by__0,axiom,
! [A_128: code_code_numeral] :
( ( div_mo1740067990umeral @ A_128 @ zero_z126310315umeral )
= A_128 ) ).
thf(fact_1838_mod__by__0,axiom,
! [A_128: nat] :
( ( div_mod_nat @ A_128 @ zero_zero_nat )
= A_128 ) ).
thf(fact_1839_mod__by__0,axiom,
! [A_128: int] :
( ( div_mod_int @ A_128 @ zero_zero_int )
= A_128 ) ).
thf(fact_1840_mod__self,axiom,
! [A_127: quickcheck_code_int] :
( ( div_mo231679042de_int @ A_127 @ A_127 )
= zero_z891286103de_int ) ).
thf(fact_1841_mod__self,axiom,
! [A_127: code_code_numeral] :
( ( div_mo1740067990umeral @ A_127 @ A_127 )
= zero_z126310315umeral ) ).
thf(fact_1842_mod__self,axiom,
! [A_127: nat] :
( ( div_mod_nat @ A_127 @ A_127 )
= zero_zero_nat ) ).
thf(fact_1843_mod__self,axiom,
! [A_127: int] :
( ( div_mod_int @ A_127 @ A_127 )
= zero_zero_int ) ).
thf(fact_1844_abs__eq__0,axiom,
! [A_126: rat] :
( ( ( abs_abs_rat @ A_126 )
= zero_zero_rat )
<=> ( A_126 = zero_zero_rat ) ) ).
thf(fact_1845_abs__eq__0,axiom,
! [A_126: real] :
( ( ( abs_abs_real @ A_126 )
= zero_zero_real )
<=> ( A_126 = zero_zero_real ) ) ).
thf(fact_1846_abs__eq__0,axiom,
! [A_126: int] :
( ( ( abs_abs_int @ A_126 )
= zero_zero_int )
<=> ( A_126 = zero_zero_int ) ) ).
thf(fact_1847_abs__zero,axiom,
( ( abs_abs_rat @ zero_zero_rat )
= zero_zero_rat ) ).
thf(fact_1848_abs__zero,axiom,
( ( abs_abs_real @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_1849_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_1850_abs__le__D1,axiom,
! [A_125: rat,B_91: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A_125 ) @ B_91 )
=> ( ord_less_eq_rat @ A_125 @ B_91 ) ) ).
thf(fact_1851_abs__le__D1,axiom,
! [A_125: real,B_91: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A_125 ) @ B_91 )
=> ( ord_less_eq_real @ A_125 @ B_91 ) ) ).
thf(fact_1852_abs__le__D1,axiom,
! [A_125: int,B_91: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A_125 ) @ B_91 )
=> ( ord_less_eq_int @ A_125 @ B_91 ) ) ).
thf(fact_1853_abs__ge__self,axiom,
! [A_124: rat] : ( ord_less_eq_rat @ A_124 @ ( abs_abs_rat @ A_124 ) ) ).
thf(fact_1854_abs__ge__self,axiom,
! [A_124: real] : ( ord_less_eq_real @ A_124 @ ( abs_abs_real @ A_124 ) ) ).
thf(fact_1855_abs__ge__self,axiom,
! [A_124: int] : ( ord_less_eq_int @ A_124 @ ( abs_abs_int @ A_124 ) ) ).
thf(fact_1856_mod__mult__cong,axiom,
! [B_90: quickcheck_code_int,B_89: quickcheck_code_int,A_123: quickcheck_code_int,C_34: quickcheck_code_int,A_122: quickcheck_code_int] :
( ( ( div_mo231679042de_int @ A_123 @ C_34 )
= ( div_mo231679042de_int @ A_122 @ C_34 ) )
=> ( ( ( div_mo231679042de_int @ B_90 @ C_34 )
= ( div_mo231679042de_int @ B_89 @ C_34 ) )
=> ( ( div_mo231679042de_int @ ( times_123202395de_int @ A_123 @ B_90 ) @ C_34 )
= ( div_mo231679042de_int @ ( times_123202395de_int @ A_122 @ B_89 ) @ C_34 ) ) ) ) ).
thf(fact_1857_mod__mult__cong,axiom,
! [B_90: code_code_numeral,B_89: code_code_numeral,A_123: code_code_numeral,C_34: code_code_numeral,A_122: code_code_numeral] :
( ( ( div_mo1740067990umeral @ A_123 @ C_34 )
= ( div_mo1740067990umeral @ A_122 @ C_34 ) )
=> ( ( ( div_mo1740067990umeral @ B_90 @ C_34 )
= ( div_mo1740067990umeral @ B_89 @ C_34 ) )
=> ( ( div_mo1740067990umeral @ ( times_1655362735umeral @ A_123 @ B_90 ) @ C_34 )
= ( div_mo1740067990umeral @ ( times_1655362735umeral @ A_122 @ B_89 ) @ C_34 ) ) ) ) ).
thf(fact_1858_mod__mult__cong,axiom,
! [B_90: nat,B_89: nat,A_123: nat,C_34: nat,A_122: nat] :
( ( ( div_mod_nat @ A_123 @ C_34 )
= ( div_mod_nat @ A_122 @ C_34 ) )
=> ( ( ( div_mod_nat @ B_90 @ C_34 )
= ( div_mod_nat @ B_89 @ C_34 ) )
=> ( ( div_mod_nat @ ( times_times_nat @ A_123 @ B_90 ) @ C_34 )
= ( div_mod_nat @ ( times_times_nat @ A_122 @ B_89 ) @ C_34 ) ) ) ) ).
thf(fact_1859_mod__mult__cong,axiom,
! [B_90: int,B_89: int,A_123: int,C_34: int,A_122: int] :
( ( ( div_mod_int @ A_123 @ C_34 )
= ( div_mod_int @ A_122 @ C_34 ) )
=> ( ( ( div_mod_int @ B_90 @ C_34 )
= ( div_mod_int @ B_89 @ C_34 ) )
=> ( ( div_mod_int @ ( times_times_int @ A_123 @ B_90 ) @ C_34 )
= ( div_mod_int @ ( times_times_int @ A_122 @ B_89 ) @ C_34 ) ) ) ) ).
thf(fact_1860_zmod__simps_I4_J,axiom,
! [A_121: quickcheck_code_int,C_33: quickcheck_code_int,B_88: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( times_123202395de_int @ ( div_mo231679042de_int @ A_121 @ C_33 ) @ B_88 ) @ C_33 )
= ( div_mo231679042de_int @ ( times_123202395de_int @ A_121 @ B_88 ) @ C_33 ) ) ).
thf(fact_1861_zmod__simps_I4_J,axiom,
! [A_121: code_code_numeral,C_33: code_code_numeral,B_88: code_code_numeral] :
( ( div_mo1740067990umeral @ ( times_1655362735umeral @ ( div_mo1740067990umeral @ A_121 @ C_33 ) @ B_88 ) @ C_33 )
= ( div_mo1740067990umeral @ ( times_1655362735umeral @ A_121 @ B_88 ) @ C_33 ) ) ).
thf(fact_1862_zmod__simps_I4_J,axiom,
! [A_121: nat,C_33: nat,B_88: nat] :
( ( div_mod_nat @ ( times_times_nat @ ( div_mod_nat @ A_121 @ C_33 ) @ B_88 ) @ C_33 )
= ( div_mod_nat @ ( times_times_nat @ A_121 @ B_88 ) @ C_33 ) ) ).
thf(fact_1863_zmod__simps_I4_J,axiom,
! [A_121: int,C_33: int,B_88: int] :
( ( div_mod_int @ ( times_times_int @ ( div_mod_int @ A_121 @ C_33 ) @ B_88 ) @ C_33 )
= ( div_mod_int @ ( times_times_int @ A_121 @ B_88 ) @ C_33 ) ) ).
thf(fact_1864_mod__mult__mult2,axiom,
! [A_120: quickcheck_code_int,C_32: quickcheck_code_int,B_87: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( times_123202395de_int @ A_120 @ C_32 ) @ ( times_123202395de_int @ B_87 @ C_32 ) )
= ( times_123202395de_int @ ( div_mo231679042de_int @ A_120 @ B_87 ) @ C_32 ) ) ).
thf(fact_1865_mod__mult__mult2,axiom,
! [A_120: code_code_numeral,C_32: code_code_numeral,B_87: code_code_numeral] :
( ( div_mo1740067990umeral @ ( times_1655362735umeral @ A_120 @ C_32 ) @ ( times_1655362735umeral @ B_87 @ C_32 ) )
= ( times_1655362735umeral @ ( div_mo1740067990umeral @ A_120 @ B_87 ) @ C_32 ) ) ).
thf(fact_1866_mod__mult__mult2,axiom,
! [A_120: nat,C_32: nat,B_87: nat] :
( ( div_mod_nat @ ( times_times_nat @ A_120 @ C_32 ) @ ( times_times_nat @ B_87 @ C_32 ) )
= ( times_times_nat @ ( div_mod_nat @ A_120 @ B_87 ) @ C_32 ) ) ).
thf(fact_1867_mod__mult__mult2,axiom,
! [A_120: int,C_32: int,B_87: int] :
( ( div_mod_int @ ( times_times_int @ A_120 @ C_32 ) @ ( times_times_int @ B_87 @ C_32 ) )
= ( times_times_int @ ( div_mod_int @ A_120 @ B_87 ) @ C_32 ) ) ).
thf(fact_1868_mod__mult__mult1,axiom,
! [C_31: quickcheck_code_int,A_119: quickcheck_code_int,B_86: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( times_123202395de_int @ C_31 @ A_119 ) @ ( times_123202395de_int @ C_31 @ B_86 ) )
= ( times_123202395de_int @ C_31 @ ( div_mo231679042de_int @ A_119 @ B_86 ) ) ) ).
thf(fact_1869_mod__mult__mult1,axiom,
! [C_31: code_code_numeral,A_119: code_code_numeral,B_86: code_code_numeral] :
( ( div_mo1740067990umeral @ ( times_1655362735umeral @ C_31 @ A_119 ) @ ( times_1655362735umeral @ C_31 @ B_86 ) )
= ( times_1655362735umeral @ C_31 @ ( div_mo1740067990umeral @ A_119 @ B_86 ) ) ) ).
thf(fact_1870_mod__mult__mult1,axiom,
! [C_31: nat,A_119: nat,B_86: nat] :
( ( div_mod_nat @ ( times_times_nat @ C_31 @ A_119 ) @ ( times_times_nat @ C_31 @ B_86 ) )
= ( times_times_nat @ C_31 @ ( div_mod_nat @ A_119 @ B_86 ) ) ) ).
thf(fact_1871_mod__mult__mult1,axiom,
! [C_31: int,A_119: int,B_86: int] :
( ( div_mod_int @ ( times_times_int @ C_31 @ A_119 ) @ ( times_times_int @ C_31 @ B_86 ) )
= ( times_times_int @ C_31 @ ( div_mod_int @ A_119 @ B_86 ) ) ) ).
thf(fact_1872_mod__mult__eq,axiom,
! [A_118: quickcheck_code_int,B_85: quickcheck_code_int,C_30: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( times_123202395de_int @ A_118 @ B_85 ) @ C_30 )
= ( div_mo231679042de_int @ ( times_123202395de_int @ ( div_mo231679042de_int @ A_118 @ C_30 ) @ ( div_mo231679042de_int @ B_85 @ C_30 ) ) @ C_30 ) ) ).
thf(fact_1873_mod__mult__eq,axiom,
! [A_118: code_code_numeral,B_85: code_code_numeral,C_30: code_code_numeral] :
( ( div_mo1740067990umeral @ ( times_1655362735umeral @ A_118 @ B_85 ) @ C_30 )
= ( div_mo1740067990umeral @ ( times_1655362735umeral @ ( div_mo1740067990umeral @ A_118 @ C_30 ) @ ( div_mo1740067990umeral @ B_85 @ C_30 ) ) @ C_30 ) ) ).
thf(fact_1874_mod__mult__eq,axiom,
! [A_118: nat,B_85: nat,C_30: nat] :
( ( div_mod_nat @ ( times_times_nat @ A_118 @ B_85 ) @ C_30 )
= ( div_mod_nat @ ( times_times_nat @ ( div_mod_nat @ A_118 @ C_30 ) @ ( div_mod_nat @ B_85 @ C_30 ) ) @ C_30 ) ) ).
thf(fact_1875_mod__mult__eq,axiom,
! [A_118: int,B_85: int,C_30: int] :
( ( div_mod_int @ ( times_times_int @ A_118 @ B_85 ) @ C_30 )
= ( div_mod_int @ ( times_times_int @ ( div_mod_int @ A_118 @ C_30 ) @ ( div_mod_int @ B_85 @ C_30 ) ) @ C_30 ) ) ).
thf(fact_1876_mod__mult__left__eq,axiom,
! [A_117: quickcheck_code_int,B_84: quickcheck_code_int,C_29: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( times_123202395de_int @ A_117 @ B_84 ) @ C_29 )
= ( div_mo231679042de_int @ ( times_123202395de_int @ ( div_mo231679042de_int @ A_117 @ C_29 ) @ B_84 ) @ C_29 ) ) ).
thf(fact_1877_mod__mult__left__eq,axiom,
! [A_117: code_code_numeral,B_84: code_code_numeral,C_29: code_code_numeral] :
( ( div_mo1740067990umeral @ ( times_1655362735umeral @ A_117 @ B_84 ) @ C_29 )
= ( div_mo1740067990umeral @ ( times_1655362735umeral @ ( div_mo1740067990umeral @ A_117 @ C_29 ) @ B_84 ) @ C_29 ) ) ).
thf(fact_1878_mod__mult__left__eq,axiom,
! [A_117: nat,B_84: nat,C_29: nat] :
( ( div_mod_nat @ ( times_times_nat @ A_117 @ B_84 ) @ C_29 )
= ( div_mod_nat @ ( times_times_nat @ ( div_mod_nat @ A_117 @ C_29 ) @ B_84 ) @ C_29 ) ) ).
thf(fact_1879_mod__mult__left__eq,axiom,
! [A_117: int,B_84: int,C_29: int] :
( ( div_mod_int @ ( times_times_int @ A_117 @ B_84 ) @ C_29 )
= ( div_mod_int @ ( times_times_int @ ( div_mod_int @ A_117 @ C_29 ) @ B_84 ) @ C_29 ) ) ).
thf(fact_1880_mod__mult__right__eq,axiom,
! [A_116: quickcheck_code_int,B_83: quickcheck_code_int,C_28: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( times_123202395de_int @ A_116 @ B_83 ) @ C_28 )
= ( div_mo231679042de_int @ ( times_123202395de_int @ A_116 @ ( div_mo231679042de_int @ B_83 @ C_28 ) ) @ C_28 ) ) ).
thf(fact_1881_mod__mult__right__eq,axiom,
! [A_116: code_code_numeral,B_83: code_code_numeral,C_28: code_code_numeral] :
( ( div_mo1740067990umeral @ ( times_1655362735umeral @ A_116 @ B_83 ) @ C_28 )
= ( div_mo1740067990umeral @ ( times_1655362735umeral @ A_116 @ ( div_mo1740067990umeral @ B_83 @ C_28 ) ) @ C_28 ) ) ).
thf(fact_1882_mod__mult__right__eq,axiom,
! [A_116: nat,B_83: nat,C_28: nat] :
( ( div_mod_nat @ ( times_times_nat @ A_116 @ B_83 ) @ C_28 )
= ( div_mod_nat @ ( times_times_nat @ A_116 @ ( div_mod_nat @ B_83 @ C_28 ) ) @ C_28 ) ) ).
thf(fact_1883_mod__mult__right__eq,axiom,
! [A_116: int,B_83: int,C_28: int] :
( ( div_mod_int @ ( times_times_int @ A_116 @ B_83 ) @ C_28 )
= ( div_mod_int @ ( times_times_int @ A_116 @ ( div_mod_int @ B_83 @ C_28 ) ) @ C_28 ) ) ).
thf(fact_1884_mod__add__cong,axiom,
! [B_82: quickcheck_code_int,B_81: quickcheck_code_int,A_115: quickcheck_code_int,C_27: quickcheck_code_int,A_114: quickcheck_code_int] :
( ( ( div_mo231679042de_int @ A_115 @ C_27 )
= ( div_mo231679042de_int @ A_114 @ C_27 ) )
=> ( ( ( div_mo231679042de_int @ B_82 @ C_27 )
= ( div_mo231679042de_int @ B_81 @ C_27 ) )
=> ( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_115 @ B_82 ) @ C_27 )
= ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_114 @ B_81 ) @ C_27 ) ) ) ) ).
thf(fact_1885_mod__add__cong,axiom,
! [B_82: code_code_numeral,B_81: code_code_numeral,A_115: code_code_numeral,C_27: code_code_numeral,A_114: code_code_numeral] :
( ( ( div_mo1740067990umeral @ A_115 @ C_27 )
= ( div_mo1740067990umeral @ A_114 @ C_27 ) )
=> ( ( ( div_mo1740067990umeral @ B_82 @ C_27 )
= ( div_mo1740067990umeral @ B_81 @ C_27 ) )
=> ( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_115 @ B_82 ) @ C_27 )
= ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_114 @ B_81 ) @ C_27 ) ) ) ) ).
thf(fact_1886_mod__add__cong,axiom,
! [B_82: nat,B_81: nat,A_115: nat,C_27: nat,A_114: nat] :
( ( ( div_mod_nat @ A_115 @ C_27 )
= ( div_mod_nat @ A_114 @ C_27 ) )
=> ( ( ( div_mod_nat @ B_82 @ C_27 )
= ( div_mod_nat @ B_81 @ C_27 ) )
=> ( ( div_mod_nat @ ( plus_plus_nat @ A_115 @ B_82 ) @ C_27 )
= ( div_mod_nat @ ( plus_plus_nat @ A_114 @ B_81 ) @ C_27 ) ) ) ) ).
thf(fact_1887_mod__add__cong,axiom,
! [B_82: int,B_81: int,A_115: int,C_27: int,A_114: int] :
( ( ( div_mod_int @ A_115 @ C_27 )
= ( div_mod_int @ A_114 @ C_27 ) )
=> ( ( ( div_mod_int @ B_82 @ C_27 )
= ( div_mod_int @ B_81 @ C_27 ) )
=> ( ( div_mod_int @ ( plus_plus_int @ A_115 @ B_82 ) @ C_27 )
= ( div_mod_int @ ( plus_plus_int @ A_114 @ B_81 ) @ C_27 ) ) ) ) ).
thf(fact_1888_zmod__simps_I1_J,axiom,
! [A_113: quickcheck_code_int,C_26: quickcheck_code_int,B_80: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ ( div_mo231679042de_int @ A_113 @ C_26 ) @ B_80 ) @ C_26 )
= ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_113 @ B_80 ) @ C_26 ) ) ).
thf(fact_1889_zmod__simps_I1_J,axiom,
! [A_113: code_code_numeral,C_26: code_code_numeral,B_80: code_code_numeral] :
( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ ( div_mo1740067990umeral @ A_113 @ C_26 ) @ B_80 ) @ C_26 )
= ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_113 @ B_80 ) @ C_26 ) ) ).
thf(fact_1890_zmod__simps_I1_J,axiom,
! [A_113: nat,C_26: nat,B_80: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ ( div_mod_nat @ A_113 @ C_26 ) @ B_80 ) @ C_26 )
= ( div_mod_nat @ ( plus_plus_nat @ A_113 @ B_80 ) @ C_26 ) ) ).
thf(fact_1891_zmod__simps_I1_J,axiom,
! [A_113: int,C_26: int,B_80: int] :
( ( div_mod_int @ ( plus_plus_int @ ( div_mod_int @ A_113 @ C_26 ) @ B_80 ) @ C_26 )
= ( div_mod_int @ ( plus_plus_int @ A_113 @ B_80 ) @ C_26 ) ) ).
thf(fact_1892_zmod__simps_I2_J,axiom,
! [A_112: quickcheck_code_int,B_79: quickcheck_code_int,C_25: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_112 @ ( div_mo231679042de_int @ B_79 @ C_25 ) ) @ C_25 )
= ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_112 @ B_79 ) @ C_25 ) ) ).
thf(fact_1893_zmod__simps_I2_J,axiom,
! [A_112: code_code_numeral,B_79: code_code_numeral,C_25: code_code_numeral] :
( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_112 @ ( div_mo1740067990umeral @ B_79 @ C_25 ) ) @ C_25 )
= ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_112 @ B_79 ) @ C_25 ) ) ).
thf(fact_1894_zmod__simps_I2_J,axiom,
! [A_112: nat,B_79: nat,C_25: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ A_112 @ ( div_mod_nat @ B_79 @ C_25 ) ) @ C_25 )
= ( div_mod_nat @ ( plus_plus_nat @ A_112 @ B_79 ) @ C_25 ) ) ).
thf(fact_1895_zmod__simps_I2_J,axiom,
! [A_112: int,B_79: int,C_25: int] :
( ( div_mod_int @ ( plus_plus_int @ A_112 @ ( div_mod_int @ B_79 @ C_25 ) ) @ C_25 )
= ( div_mod_int @ ( plus_plus_int @ A_112 @ B_79 ) @ C_25 ) ) ).
thf(fact_1896_mod__add__eq,axiom,
! [A_111: quickcheck_code_int,B_78: quickcheck_code_int,C_24: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_111 @ B_78 ) @ C_24 )
= ( div_mo231679042de_int @ ( plus_p1446045655de_int @ ( div_mo231679042de_int @ A_111 @ C_24 ) @ ( div_mo231679042de_int @ B_78 @ C_24 ) ) @ C_24 ) ) ).
thf(fact_1897_mod__add__eq,axiom,
! [A_111: code_code_numeral,B_78: code_code_numeral,C_24: code_code_numeral] :
( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_111 @ B_78 ) @ C_24 )
= ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ ( div_mo1740067990umeral @ A_111 @ C_24 ) @ ( div_mo1740067990umeral @ B_78 @ C_24 ) ) @ C_24 ) ) ).
thf(fact_1898_mod__add__eq,axiom,
! [A_111: nat,B_78: nat,C_24: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ A_111 @ B_78 ) @ C_24 )
= ( div_mod_nat @ ( plus_plus_nat @ ( div_mod_nat @ A_111 @ C_24 ) @ ( div_mod_nat @ B_78 @ C_24 ) ) @ C_24 ) ) ).
thf(fact_1899_mod__add__eq,axiom,
! [A_111: int,B_78: int,C_24: int] :
( ( div_mod_int @ ( plus_plus_int @ A_111 @ B_78 ) @ C_24 )
= ( div_mod_int @ ( plus_plus_int @ ( div_mod_int @ A_111 @ C_24 ) @ ( div_mod_int @ B_78 @ C_24 ) ) @ C_24 ) ) ).
thf(fact_1900_mod__add__left__eq,axiom,
! [A_110: quickcheck_code_int,B_77: quickcheck_code_int,C_23: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_110 @ B_77 ) @ C_23 )
= ( div_mo231679042de_int @ ( plus_p1446045655de_int @ ( div_mo231679042de_int @ A_110 @ C_23 ) @ B_77 ) @ C_23 ) ) ).
thf(fact_1901_mod__add__left__eq,axiom,
! [A_110: code_code_numeral,B_77: code_code_numeral,C_23: code_code_numeral] :
( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_110 @ B_77 ) @ C_23 )
= ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ ( div_mo1740067990umeral @ A_110 @ C_23 ) @ B_77 ) @ C_23 ) ) ).
thf(fact_1902_mod__add__left__eq,axiom,
! [A_110: nat,B_77: nat,C_23: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ A_110 @ B_77 ) @ C_23 )
= ( div_mod_nat @ ( plus_plus_nat @ ( div_mod_nat @ A_110 @ C_23 ) @ B_77 ) @ C_23 ) ) ).
thf(fact_1903_mod__add__left__eq,axiom,
! [A_110: int,B_77: int,C_23: int] :
( ( div_mod_int @ ( plus_plus_int @ A_110 @ B_77 ) @ C_23 )
= ( div_mod_int @ ( plus_plus_int @ ( div_mod_int @ A_110 @ C_23 ) @ B_77 ) @ C_23 ) ) ).
thf(fact_1904_mod__add__right__eq,axiom,
! [A_109: quickcheck_code_int,B_76: quickcheck_code_int,C_22: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_109 @ B_76 ) @ C_22 )
= ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_109 @ ( div_mo231679042de_int @ B_76 @ C_22 ) ) @ C_22 ) ) ).
thf(fact_1905_mod__add__right__eq,axiom,
! [A_109: code_code_numeral,B_76: code_code_numeral,C_22: code_code_numeral] :
( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_109 @ B_76 ) @ C_22 )
= ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_109 @ ( div_mo1740067990umeral @ B_76 @ C_22 ) ) @ C_22 ) ) ).
thf(fact_1906_mod__add__right__eq,axiom,
! [A_109: nat,B_76: nat,C_22: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ A_109 @ B_76 ) @ C_22 )
= ( div_mod_nat @ ( plus_plus_nat @ A_109 @ ( div_mod_nat @ B_76 @ C_22 ) ) @ C_22 ) ) ).
thf(fact_1907_mod__add__right__eq,axiom,
! [A_109: int,B_76: int,C_22: int] :
( ( div_mod_int @ ( plus_plus_int @ A_109 @ B_76 ) @ C_22 )
= ( div_mod_int @ ( plus_plus_int @ A_109 @ ( div_mod_int @ B_76 @ C_22 ) ) @ C_22 ) ) ).
thf(fact_1908_mod__add__self1,axiom,
! [B_75: quickcheck_code_int,A_108: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ B_75 @ A_108 ) @ B_75 )
= ( div_mo231679042de_int @ A_108 @ B_75 ) ) ).
thf(fact_1909_mod__add__self1,axiom,
! [B_75: code_code_numeral,A_108: code_code_numeral] :
( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ B_75 @ A_108 ) @ B_75 )
= ( div_mo1740067990umeral @ A_108 @ B_75 ) ) ).
thf(fact_1910_mod__add__self1,axiom,
! [B_75: nat,A_108: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ B_75 @ A_108 ) @ B_75 )
= ( div_mod_nat @ A_108 @ B_75 ) ) ).
thf(fact_1911_mod__add__self1,axiom,
! [B_75: int,A_108: int] :
( ( div_mod_int @ ( plus_plus_int @ B_75 @ A_108 ) @ B_75 )
= ( div_mod_int @ A_108 @ B_75 ) ) ).
thf(fact_1912_mod__add__self2,axiom,
! [A_107: quickcheck_code_int,B_74: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_107 @ B_74 ) @ B_74 )
= ( div_mo231679042de_int @ A_107 @ B_74 ) ) ).
thf(fact_1913_mod__add__self2,axiom,
! [A_107: code_code_numeral,B_74: code_code_numeral] :
( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_107 @ B_74 ) @ B_74 )
= ( div_mo1740067990umeral @ A_107 @ B_74 ) ) ).
thf(fact_1914_mod__add__self2,axiom,
! [A_107: nat,B_74: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ A_107 @ B_74 ) @ B_74 )
= ( div_mod_nat @ A_107 @ B_74 ) ) ).
thf(fact_1915_mod__add__self2,axiom,
! [A_107: int,B_74: int] :
( ( div_mod_int @ ( plus_plus_int @ A_107 @ B_74 ) @ B_74 )
= ( div_mod_int @ A_107 @ B_74 ) ) ).
thf(fact_1916_abs__mult__self,axiom,
! [A_106: rat] :
( ( times_times_rat @ ( abs_abs_rat @ A_106 ) @ ( abs_abs_rat @ A_106 ) )
= ( times_times_rat @ A_106 @ A_106 ) ) ).
thf(fact_1917_abs__mult__self,axiom,
! [A_106: real] :
( ( times_times_real @ ( abs_abs_real @ A_106 ) @ ( abs_abs_real @ A_106 ) )
= ( times_times_real @ A_106 @ A_106 ) ) ).
thf(fact_1918_abs__mult__self,axiom,
! [A_106: int] :
( ( times_times_int @ ( abs_abs_int @ A_106 ) @ ( abs_abs_int @ A_106 ) )
= ( times_times_int @ A_106 @ A_106 ) ) ).
thf(fact_1919_abs__mult,axiom,
! [A_105: rat,B_73: rat] :
( ( abs_abs_rat @ ( times_times_rat @ A_105 @ B_73 ) )
= ( times_times_rat @ ( abs_abs_rat @ A_105 ) @ ( abs_abs_rat @ B_73 ) ) ) ).
thf(fact_1920_abs__mult,axiom,
! [A_105: real,B_73: real] :
( ( abs_abs_real @ ( times_times_real @ A_105 @ B_73 ) )
= ( times_times_real @ ( abs_abs_real @ A_105 ) @ ( abs_abs_real @ B_73 ) ) ) ).
thf(fact_1921_abs__mult,axiom,
! [A_105: int,B_73: int] :
( ( abs_abs_int @ ( times_times_int @ A_105 @ B_73 ) )
= ( times_times_int @ ( abs_abs_int @ A_105 ) @ ( abs_abs_int @ B_73 ) ) ) ).
thf(fact_1922_abs__add__abs,axiom,
! [A_104: rat,B_72: rat] :
( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A_104 ) @ ( abs_abs_rat @ B_72 ) ) )
= ( plus_plus_rat @ ( abs_abs_rat @ A_104 ) @ ( abs_abs_rat @ B_72 ) ) ) ).
thf(fact_1923_abs__add__abs,axiom,
! [A_104: real,B_72: real] :
( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A_104 ) @ ( abs_abs_real @ B_72 ) ) )
= ( plus_plus_real @ ( abs_abs_real @ A_104 ) @ ( abs_abs_real @ B_72 ) ) ) ).
thf(fact_1924_abs__add__abs,axiom,
! [A_104: int,B_72: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A_104 ) @ ( abs_abs_int @ B_72 ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A_104 ) @ ( abs_abs_int @ B_72 ) ) ) ).
thf(fact_1925_mod__diff__cong,axiom,
! [B_71: int,B_70: int,A_103: int,C_21: int,A_102: int] :
( ( ( div_mod_int @ A_103 @ C_21 )
= ( div_mod_int @ A_102 @ C_21 ) )
=> ( ( ( div_mod_int @ B_71 @ C_21 )
= ( div_mod_int @ B_70 @ C_21 ) )
=> ( ( div_mod_int @ ( minus_minus_int @ A_103 @ B_71 ) @ C_21 )
= ( div_mod_int @ ( minus_minus_int @ A_102 @ B_70 ) @ C_21 ) ) ) ) ).
thf(fact_1926_mod__diff__eq,axiom,
! [A_101: int,B_69: int,C_20: int] :
( ( div_mod_int @ ( minus_minus_int @ A_101 @ B_69 ) @ C_20 )
= ( div_mod_int @ ( minus_minus_int @ ( div_mod_int @ A_101 @ C_20 ) @ ( div_mod_int @ B_69 @ C_20 ) ) @ C_20 ) ) ).
thf(fact_1927_mod__diff__left__eq,axiom,
! [A_100: int,B_68: int,C_19: int] :
( ( div_mod_int @ ( minus_minus_int @ A_100 @ B_68 ) @ C_19 )
= ( div_mod_int @ ( minus_minus_int @ ( div_mod_int @ A_100 @ C_19 ) @ B_68 ) @ C_19 ) ) ).
thf(fact_1928_mod__diff__right__eq,axiom,
! [A_99: int,B_67: int,C_18: int] :
( ( div_mod_int @ ( minus_minus_int @ A_99 @ B_67 ) @ C_18 )
= ( div_mod_int @ ( minus_minus_int @ A_99 @ ( div_mod_int @ B_67 @ C_18 ) ) @ C_18 ) ) ).
thf(fact_1929_abs__one,axiom,
( ( abs_abs_rat @ one_one_rat )
= one_one_rat ) ).
thf(fact_1930_abs__one,axiom,
( ( abs_abs_real @ one_one_real )
= one_one_real ) ).
thf(fact_1931_abs__one,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
thf(fact_1932_abs__minus__commute,axiom,
! [A_98: rat,B_66: rat] :
( ( abs_abs_rat @ ( minus_minus_rat @ A_98 @ B_66 ) )
= ( abs_abs_rat @ ( minus_minus_rat @ B_66 @ A_98 ) ) ) ).
thf(fact_1933_abs__minus__commute,axiom,
! [A_98: real,B_66: real] :
( ( abs_abs_real @ ( minus_minus_real @ A_98 @ B_66 ) )
= ( abs_abs_real @ ( minus_minus_real @ B_66 @ A_98 ) ) ) ).
thf(fact_1934_abs__minus__commute,axiom,
! [A_98: int,B_66: int] :
( ( abs_abs_int @ ( minus_minus_int @ A_98 @ B_66 ) )
= ( abs_abs_int @ ( minus_minus_int @ B_66 @ A_98 ) ) ) ).
thf(fact_1935_floor__minus,axiom,
! [X_26: real] :
( ( archim1246769320r_real @ ( uminus_uminus_real @ X_26 ) )
= ( uminus_uminus_int @ ( archim856651990g_real @ X_26 ) ) ) ).
thf(fact_1936_floor__minus,axiom,
! [X_26: rat] :
( ( archim791455193or_rat @ ( uminus_uminus_rat @ X_26 ) )
= ( uminus_uminus_int @ ( archim1419379755ng_rat @ X_26 ) ) ) ).
thf(fact_1937_ceiling__minus,axiom,
! [X_25: real] :
( ( archim856651990g_real @ ( uminus_uminus_real @ X_25 ) )
= ( uminus_uminus_int @ ( archim1246769320r_real @ X_25 ) ) ) ).
thf(fact_1938_ceiling__minus,axiom,
! [X_25: rat] :
( ( archim1419379755ng_rat @ ( uminus_uminus_rat @ X_25 ) )
= ( uminus_uminus_int @ ( archim791455193or_rat @ X_25 ) ) ) ).
thf(fact_1939_ceiling__def,axiom,
! [X_24: real] :
( ( archim856651990g_real @ X_24 )
= ( uminus_uminus_int @ ( archim1246769320r_real @ ( uminus_uminus_real @ X_24 ) ) ) ) ).
thf(fact_1940_ceiling__def,axiom,
! [X_24: rat] :
( ( archim1419379755ng_rat @ X_24 )
= ( uminus_uminus_int @ ( archim791455193or_rat @ ( uminus_uminus_rat @ X_24 ) ) ) ) ).
thf(fact_1941_neg__equal__zero,axiom,
! [A_97: rat] :
( ( ( uminus_uminus_rat @ A_97 )
= A_97 )
<=> ( A_97 = zero_zero_rat ) ) ).
thf(fact_1942_neg__equal__zero,axiom,
! [A_97: real] :
( ( ( uminus_uminus_real @ A_97 )
= A_97 )
<=> ( A_97 = zero_zero_real ) ) ).
thf(fact_1943_neg__equal__zero,axiom,
! [A_97: int] :
( ( ( uminus_uminus_int @ A_97 )
= A_97 )
<=> ( A_97 = zero_zero_int ) ) ).
thf(fact_1944_neg__equal__0__iff__equal,axiom,
! [A_96: rat] :
( ( ( uminus_uminus_rat @ A_96 )
= zero_zero_rat )
<=> ( A_96 = zero_zero_rat ) ) ).
thf(fact_1945_neg__equal__0__iff__equal,axiom,
! [A_96: complex] :
( ( ( uminus473333897omplex @ A_96 )
= zero_zero_complex )
<=> ( A_96 = zero_zero_complex ) ) ).
thf(fact_1946_neg__equal__0__iff__equal,axiom,
! [A_96: real] :
( ( ( uminus_uminus_real @ A_96 )
= zero_zero_real )
<=> ( A_96 = zero_zero_real ) ) ).
thf(fact_1947_neg__equal__0__iff__equal,axiom,
! [A_96: int] :
( ( ( uminus_uminus_int @ A_96 )
= zero_zero_int )
<=> ( A_96 = zero_zero_int ) ) ).
thf(fact_1948_equal__neg__zero,axiom,
! [A_95: rat] :
( ( A_95
= ( uminus_uminus_rat @ A_95 ) )
<=> ( A_95 = zero_zero_rat ) ) ).
thf(fact_1949_equal__neg__zero,axiom,
! [A_95: real] :
( ( A_95
= ( uminus_uminus_real @ A_95 ) )
<=> ( A_95 = zero_zero_real ) ) ).
thf(fact_1950_equal__neg__zero,axiom,
! [A_95: int] :
( ( A_95
= ( uminus_uminus_int @ A_95 ) )
<=> ( A_95 = zero_zero_int ) ) ).
thf(fact_1951_neg__0__equal__iff__equal,axiom,
! [A_94: rat] :
( ( zero_zero_rat
= ( uminus_uminus_rat @ A_94 ) )
<=> ( zero_zero_rat = A_94 ) ) ).
thf(fact_1952_neg__0__equal__iff__equal,axiom,
! [A_94: complex] :
( ( zero_zero_complex
= ( uminus473333897omplex @ A_94 ) )
<=> ( zero_zero_complex = A_94 ) ) ).
thf(fact_1953_neg__0__equal__iff__equal,axiom,
! [A_94: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A_94 ) )
<=> ( zero_zero_real = A_94 ) ) ).
thf(fact_1954_neg__0__equal__iff__equal,axiom,
! [A_94: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A_94 ) )
<=> ( zero_zero_int = A_94 ) ) ).
thf(fact_1955_minus__zero,axiom,
( ( uminus_uminus_rat @ zero_zero_rat )
= zero_zero_rat ) ).
thf(fact_1956_minus__zero,axiom,
( ( uminus473333897omplex @ zero_zero_complex )
= zero_zero_complex ) ).
thf(fact_1957_minus__zero,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_1958_minus__zero,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_1959_compl__le__swap2,axiom,
! [Y_18: int > $o,X_23: int > $o] :
( ( ord_less_eq_int_o @ ( uminus_uminus_int_o @ Y_18 ) @ X_23 )
=> ( ord_less_eq_int_o @ ( uminus_uminus_int_o @ X_23 ) @ Y_18 ) ) ).
thf(fact_1960_compl__le__swap2,axiom,
! [Y_18: nat > $o,X_23: nat > $o] :
( ( ord_less_eq_nat_o @ ( uminus_uminus_nat_o @ Y_18 ) @ X_23 )
=> ( ord_less_eq_nat_o @ ( uminus_uminus_nat_o @ X_23 ) @ Y_18 ) ) ).
thf(fact_1961_compl__le__swap1,axiom,
! [Y_17: int > $o,X_22: int > $o] :
( ( ord_less_eq_int_o @ Y_17 @ ( uminus_uminus_int_o @ X_22 ) )
=> ( ord_less_eq_int_o @ X_22 @ ( uminus_uminus_int_o @ Y_17 ) ) ) ).
thf(fact_1962_compl__le__swap1,axiom,
! [Y_17: nat > $o,X_22: nat > $o] :
( ( ord_less_eq_nat_o @ Y_17 @ ( uminus_uminus_nat_o @ X_22 ) )
=> ( ord_less_eq_nat_o @ X_22 @ ( uminus_uminus_nat_o @ Y_17 ) ) ) ).
thf(fact_1963_compl__mono,axiom,
! [X_21: int > $o,Y_16: int > $o] :
( ( ord_less_eq_int_o @ X_21 @ Y_16 )
=> ( ord_less_eq_int_o @ ( uminus_uminus_int_o @ Y_16 ) @ ( uminus_uminus_int_o @ X_21 ) ) ) ).
thf(fact_1964_compl__mono,axiom,
! [X_21: nat > $o,Y_16: nat > $o] :
( ( ord_less_eq_nat_o @ X_21 @ Y_16 )
=> ( ord_less_eq_nat_o @ ( uminus_uminus_nat_o @ Y_16 ) @ ( uminus_uminus_nat_o @ X_21 ) ) ) ).
thf(fact_1965_compl__le__compl__iff,axiom,
! [X_20: int > $o,Y_15: int > $o] :
( ( ord_less_eq_int_o @ ( uminus_uminus_int_o @ X_20 ) @ ( uminus_uminus_int_o @ Y_15 ) )
<=> ( ord_less_eq_int_o @ Y_15 @ X_20 ) ) ).
thf(fact_1966_compl__le__compl__iff,axiom,
! [X_20: nat > $o,Y_15: nat > $o] :
( ( ord_less_eq_nat_o @ ( uminus_uminus_nat_o @ X_20 ) @ ( uminus_uminus_nat_o @ Y_15 ) )
<=> ( ord_less_eq_nat_o @ Y_15 @ X_20 ) ) ).
thf(fact_1967_le__imp__neg__le,axiom,
! [A_93: rat,B_65: rat] :
( ( ord_less_eq_rat @ A_93 @ B_65 )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B_65 ) @ ( uminus_uminus_rat @ A_93 ) ) ) ).
thf(fact_1968_le__imp__neg__le,axiom,
! [A_93: real,B_65: real] :
( ( ord_less_eq_real @ A_93 @ B_65 )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B_65 ) @ ( uminus_uminus_real @ A_93 ) ) ) ).
thf(fact_1969_le__imp__neg__le,axiom,
! [A_93: int,B_65: int] :
( ( ord_less_eq_int @ A_93 @ B_65 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B_65 ) @ ( uminus_uminus_int @ A_93 ) ) ) ).
thf(fact_1970_neg__le__iff__le,axiom,
! [B_64: rat,A_92: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B_64 ) @ ( uminus_uminus_rat @ A_92 ) )
<=> ( ord_less_eq_rat @ A_92 @ B_64 ) ) ).
thf(fact_1971_neg__le__iff__le,axiom,
! [B_64: real,A_92: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B_64 ) @ ( uminus_uminus_real @ A_92 ) )
<=> ( ord_less_eq_real @ A_92 @ B_64 ) ) ).
thf(fact_1972_neg__le__iff__le,axiom,
! [B_64: int,A_92: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B_64 ) @ ( uminus_uminus_int @ A_92 ) )
<=> ( ord_less_eq_int @ A_92 @ B_64 ) ) ).
thf(fact_1973_minus__le__iff,axiom,
! [A_91: rat,B_63: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A_91 ) @ B_63 )
<=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B_63 ) @ A_91 ) ) ).
thf(fact_1974_minus__le__iff,axiom,
! [A_91: real,B_63: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A_91 ) @ B_63 )
<=> ( ord_less_eq_real @ ( uminus_uminus_real @ B_63 ) @ A_91 ) ) ).
thf(fact_1975_minus__le__iff,axiom,
! [A_91: int,B_63: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A_91 ) @ B_63 )
<=> ( ord_less_eq_int @ ( uminus_uminus_int @ B_63 ) @ A_91 ) ) ).
thf(fact_1976_le__minus__iff,axiom,
! [A_90: rat,B_62: rat] :
( ( ord_less_eq_rat @ A_90 @ ( uminus_uminus_rat @ B_62 ) )
<=> ( ord_less_eq_rat @ B_62 @ ( uminus_uminus_rat @ A_90 ) ) ) ).
thf(fact_1977_le__minus__iff,axiom,
! [A_90: real,B_62: real] :
( ( ord_less_eq_real @ A_90 @ ( uminus_uminus_real @ B_62 ) )
<=> ( ord_less_eq_real @ B_62 @ ( uminus_uminus_real @ A_90 ) ) ) ).
thf(fact_1978_le__minus__iff,axiom,
! [A_90: int,B_62: int] :
( ( ord_less_eq_int @ A_90 @ ( uminus_uminus_int @ B_62 ) )
<=> ( ord_less_eq_int @ B_62 @ ( uminus_uminus_int @ A_90 ) ) ) ).
thf(fact_1979_compl__less__swap2,axiom,
! [Y_14: int > $o,X_19: int > $o] :
( ( ord_less_int_o @ ( uminus_uminus_int_o @ Y_14 ) @ X_19 )
=> ( ord_less_int_o @ ( uminus_uminus_int_o @ X_19 ) @ Y_14 ) ) ).
thf(fact_1980_compl__less__swap2,axiom,
! [Y_14: nat > $o,X_19: nat > $o] :
( ( ord_less_nat_o @ ( uminus_uminus_nat_o @ Y_14 ) @ X_19 )
=> ( ord_less_nat_o @ ( uminus_uminus_nat_o @ X_19 ) @ Y_14 ) ) ).
thf(fact_1981_compl__less__swap1,axiom,
! [Y_13: int > $o,X_18: int > $o] :
( ( ord_less_int_o @ Y_13 @ ( uminus_uminus_int_o @ X_18 ) )
=> ( ord_less_int_o @ X_18 @ ( uminus_uminus_int_o @ Y_13 ) ) ) ).
thf(fact_1982_compl__less__swap1,axiom,
! [Y_13: nat > $o,X_18: nat > $o] :
( ( ord_less_nat_o @ Y_13 @ ( uminus_uminus_nat_o @ X_18 ) )
=> ( ord_less_nat_o @ X_18 @ ( uminus_uminus_nat_o @ Y_13 ) ) ) ).
thf(fact_1983_compl__less__compl__iff,axiom,
! [X_17: int > $o,Y_12: int > $o] :
( ( ord_less_int_o @ ( uminus_uminus_int_o @ X_17 ) @ ( uminus_uminus_int_o @ Y_12 ) )
<=> ( ord_less_int_o @ Y_12 @ X_17 ) ) ).
thf(fact_1984_compl__less__compl__iff,axiom,
! [X_17: nat > $o,Y_12: nat > $o] :
( ( ord_less_nat_o @ ( uminus_uminus_nat_o @ X_17 ) @ ( uminus_uminus_nat_o @ Y_12 ) )
<=> ( ord_less_nat_o @ Y_12 @ X_17 ) ) ).
thf(fact_1985_neg__less__iff__less,axiom,
! [B_61: rat,A_89: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ B_61 ) @ ( uminus_uminus_rat @ A_89 ) )
<=> ( ord_less_rat @ A_89 @ B_61 ) ) ).
thf(fact_1986_neg__less__iff__less,axiom,
! [B_61: real,A_89: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B_61 ) @ ( uminus_uminus_real @ A_89 ) )
<=> ( ord_less_real @ A_89 @ B_61 ) ) ).
thf(fact_1987_neg__less__iff__less,axiom,
! [B_61: int,A_89: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B_61 ) @ ( uminus_uminus_int @ A_89 ) )
<=> ( ord_less_int @ A_89 @ B_61 ) ) ).
thf(fact_1988_minus__less__iff,axiom,
! [A_88: rat,B_60: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A_88 ) @ B_60 )
<=> ( ord_less_rat @ ( uminus_uminus_rat @ B_60 ) @ A_88 ) ) ).
thf(fact_1989_minus__less__iff,axiom,
! [A_88: real,B_60: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A_88 ) @ B_60 )
<=> ( ord_less_real @ ( uminus_uminus_real @ B_60 ) @ A_88 ) ) ).
thf(fact_1990_minus__less__iff,axiom,
! [A_88: int,B_60: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A_88 ) @ B_60 )
<=> ( ord_less_int @ ( uminus_uminus_int @ B_60 ) @ A_88 ) ) ).
thf(fact_1991_less__minus__iff,axiom,
! [A_87: rat,B_59: rat] :
( ( ord_less_rat @ A_87 @ ( uminus_uminus_rat @ B_59 ) )
<=> ( ord_less_rat @ B_59 @ ( uminus_uminus_rat @ A_87 ) ) ) ).
thf(fact_1992_less__minus__iff,axiom,
! [A_87: real,B_59: real] :
( ( ord_less_real @ A_87 @ ( uminus_uminus_real @ B_59 ) )
<=> ( ord_less_real @ B_59 @ ( uminus_uminus_real @ A_87 ) ) ) ).
thf(fact_1993_less__minus__iff,axiom,
! [A_87: int,B_59: int] :
( ( ord_less_int @ A_87 @ ( uminus_uminus_int @ B_59 ) )
<=> ( ord_less_int @ B_59 @ ( uminus_uminus_int @ A_87 ) ) ) ).
thf(fact_1994_square__eq__iff,axiom,
! [A_86: rat,B_58: rat] :
( ( ( times_times_rat @ A_86 @ A_86 )
= ( times_times_rat @ B_58 @ B_58 ) )
<=> ( ( A_86 = B_58 )
| ( A_86
= ( uminus_uminus_rat @ B_58 ) ) ) ) ).
thf(fact_1995_square__eq__iff,axiom,
! [A_86: complex,B_58: complex] :
( ( ( times_times_complex @ A_86 @ A_86 )
= ( times_times_complex @ B_58 @ B_58 ) )
<=> ( ( A_86 = B_58 )
| ( A_86
= ( uminus473333897omplex @ B_58 ) ) ) ) ).
thf(fact_1996_square__eq__iff,axiom,
! [A_86: real,B_58: real] :
( ( ( times_times_real @ A_86 @ A_86 )
= ( times_times_real @ B_58 @ B_58 ) )
<=> ( ( A_86 = B_58 )
| ( A_86
= ( uminus_uminus_real @ B_58 ) ) ) ) ).
thf(fact_1997_square__eq__iff,axiom,
! [A_86: int,B_58: int] :
( ( ( times_times_int @ A_86 @ A_86 )
= ( times_times_int @ B_58 @ B_58 ) )
<=> ( ( A_86 = B_58 )
| ( A_86
= ( uminus_uminus_int @ B_58 ) ) ) ) ).
thf(fact_1998_minus__mult__minus,axiom,
! [A_85: rat,B_57: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A_85 ) @ ( uminus_uminus_rat @ B_57 ) )
= ( times_times_rat @ A_85 @ B_57 ) ) ).
thf(fact_1999_minus__mult__minus,axiom,
! [A_85: complex,B_57: complex] :
( ( times_times_complex @ ( uminus473333897omplex @ A_85 ) @ ( uminus473333897omplex @ B_57 ) )
= ( times_times_complex @ A_85 @ B_57 ) ) ).
thf(fact_2000_minus__mult__minus,axiom,
! [A_85: real,B_57: real] :
( ( times_times_real @ ( uminus_uminus_real @ A_85 ) @ ( uminus_uminus_real @ B_57 ) )
= ( times_times_real @ A_85 @ B_57 ) ) ).
thf(fact_2001_minus__mult__minus,axiom,
! [A_85: int,B_57: int] :
( ( times_times_int @ ( uminus_uminus_int @ A_85 ) @ ( uminus_uminus_int @ B_57 ) )
= ( times_times_int @ A_85 @ B_57 ) ) ).
thf(fact_2002_minus__mult__commute,axiom,
! [A_84: rat,B_56: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A_84 ) @ B_56 )
= ( times_times_rat @ A_84 @ ( uminus_uminus_rat @ B_56 ) ) ) ).
thf(fact_2003_minus__mult__commute,axiom,
! [A_84: complex,B_56: complex] :
( ( times_times_complex @ ( uminus473333897omplex @ A_84 ) @ B_56 )
= ( times_times_complex @ A_84 @ ( uminus473333897omplex @ B_56 ) ) ) ).
thf(fact_2004_minus__mult__commute,axiom,
! [A_84: real,B_56: real] :
( ( times_times_real @ ( uminus_uminus_real @ A_84 ) @ B_56 )
= ( times_times_real @ A_84 @ ( uminus_uminus_real @ B_56 ) ) ) ).
thf(fact_2005_minus__mult__commute,axiom,
! [A_84: int,B_56: int] :
( ( times_times_int @ ( uminus_uminus_int @ A_84 ) @ B_56 )
= ( times_times_int @ A_84 @ ( uminus_uminus_int @ B_56 ) ) ) ).
thf(fact_2006_mult__left_Ominus,axiom,
! [X_16: complex,Y_11: complex] :
( ( times_times_complex @ ( uminus473333897omplex @ X_16 ) @ Y_11 )
= ( uminus473333897omplex @ ( times_times_complex @ X_16 @ Y_11 ) ) ) ).
thf(fact_2007_mult__left_Ominus,axiom,
! [X_16: real,Y_11: real] :
( ( times_times_real @ ( uminus_uminus_real @ X_16 ) @ Y_11 )
= ( uminus_uminus_real @ ( times_times_real @ X_16 @ Y_11 ) ) ) ).
thf(fact_2008_mult_Ominus__left,axiom,
! [A_83: complex,B_55: complex] :
( ( times_times_complex @ ( uminus473333897omplex @ A_83 ) @ B_55 )
= ( uminus473333897omplex @ ( times_times_complex @ A_83 @ B_55 ) ) ) ).
thf(fact_2009_mult_Ominus__left,axiom,
! [A_83: real,B_55: real] :
( ( times_times_real @ ( uminus_uminus_real @ A_83 ) @ B_55 )
= ( uminus_uminus_real @ ( times_times_real @ A_83 @ B_55 ) ) ) ).
thf(fact_2010_mult__right_Ominus,axiom,
! [Xa_2: complex,X_15: complex] :
( ( times_times_complex @ Xa_2 @ ( uminus473333897omplex @ X_15 ) )
= ( uminus473333897omplex @ ( times_times_complex @ Xa_2 @ X_15 ) ) ) ).
thf(fact_2011_mult__right_Ominus,axiom,
! [Xa_2: real,X_15: real] :
( ( times_times_real @ Xa_2 @ ( uminus_uminus_real @ X_15 ) )
= ( uminus_uminus_real @ ( times_times_real @ Xa_2 @ X_15 ) ) ) ).
thf(fact_2012_mult_Ominus__right,axiom,
! [A_82: complex,B_54: complex] :
( ( times_times_complex @ A_82 @ ( uminus473333897omplex @ B_54 ) )
= ( uminus473333897omplex @ ( times_times_complex @ A_82 @ B_54 ) ) ) ).
thf(fact_2013_mult_Ominus__right,axiom,
! [A_82: real,B_54: real] :
( ( times_times_real @ A_82 @ ( uminus_uminus_real @ B_54 ) )
= ( uminus_uminus_real @ ( times_times_real @ A_82 @ B_54 ) ) ) ).
thf(fact_2014_minus__mult__left,axiom,
! [A_81: rat,B_53: rat] :
( ( uminus_uminus_rat @ ( times_times_rat @ A_81 @ B_53 ) )
= ( times_times_rat @ ( uminus_uminus_rat @ A_81 ) @ B_53 ) ) ).
thf(fact_2015_minus__mult__left,axiom,
! [A_81: complex,B_53: complex] :
( ( uminus473333897omplex @ ( times_times_complex @ A_81 @ B_53 ) )
= ( times_times_complex @ ( uminus473333897omplex @ A_81 ) @ B_53 ) ) ).
thf(fact_2016_minus__mult__left,axiom,
! [A_81: real,B_53: real] :
( ( uminus_uminus_real @ ( times_times_real @ A_81 @ B_53 ) )
= ( times_times_real @ ( uminus_uminus_real @ A_81 ) @ B_53 ) ) ).
thf(fact_2017_minus__mult__left,axiom,
! [A_81: int,B_53: int] :
( ( uminus_uminus_int @ ( times_times_int @ A_81 @ B_53 ) )
= ( times_times_int @ ( uminus_uminus_int @ A_81 ) @ B_53 ) ) ).
thf(fact_2018_minus__mult__right,axiom,
! [A_80: rat,B_52: rat] :
( ( uminus_uminus_rat @ ( times_times_rat @ A_80 @ B_52 ) )
= ( times_times_rat @ A_80 @ ( uminus_uminus_rat @ B_52 ) ) ) ).
thf(fact_2019_minus__mult__right,axiom,
! [A_80: complex,B_52: complex] :
( ( uminus473333897omplex @ ( times_times_complex @ A_80 @ B_52 ) )
= ( times_times_complex @ A_80 @ ( uminus473333897omplex @ B_52 ) ) ) ).
thf(fact_2020_minus__mult__right,axiom,
! [A_80: real,B_52: real] :
( ( uminus_uminus_real @ ( times_times_real @ A_80 @ B_52 ) )
= ( times_times_real @ A_80 @ ( uminus_uminus_real @ B_52 ) ) ) ).
thf(fact_2021_minus__mult__right,axiom,
! [A_80: int,B_52: int] :
( ( uminus_uminus_int @ ( times_times_int @ A_80 @ B_52 ) )
= ( times_times_int @ A_80 @ ( uminus_uminus_int @ B_52 ) ) ) ).
thf(fact_2022_minus__add__cancel,axiom,
! [A_79: rat,B_51: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A_79 ) @ ( plus_plus_rat @ A_79 @ B_51 ) )
= B_51 ) ).
thf(fact_2023_minus__add__cancel,axiom,
! [A_79: complex,B_51: complex] :
( ( plus_plus_complex @ ( uminus473333897omplex @ A_79 ) @ ( plus_plus_complex @ A_79 @ B_51 ) )
= B_51 ) ).
thf(fact_2024_minus__add__cancel,axiom,
! [A_79: real,B_51: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A_79 ) @ ( plus_plus_real @ A_79 @ B_51 ) )
= B_51 ) ).
thf(fact_2025_minus__add__cancel,axiom,
! [A_79: int,B_51: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A_79 ) @ ( plus_plus_int @ A_79 @ B_51 ) )
= B_51 ) ).
thf(fact_2026_add__minus__cancel,axiom,
! [A_78: rat,B_50: rat] :
( ( plus_plus_rat @ A_78 @ ( plus_plus_rat @ ( uminus_uminus_rat @ A_78 ) @ B_50 ) )
= B_50 ) ).
thf(fact_2027_add__minus__cancel,axiom,
! [A_78: complex,B_50: complex] :
( ( plus_plus_complex @ A_78 @ ( plus_plus_complex @ ( uminus473333897omplex @ A_78 ) @ B_50 ) )
= B_50 ) ).
thf(fact_2028_add__minus__cancel,axiom,
! [A_78: real,B_50: real] :
( ( plus_plus_real @ A_78 @ ( plus_plus_real @ ( uminus_uminus_real @ A_78 ) @ B_50 ) )
= B_50 ) ).
thf(fact_2029_add__minus__cancel,axiom,
! [A_78: int,B_50: int] :
( ( plus_plus_int @ A_78 @ ( plus_plus_int @ ( uminus_uminus_int @ A_78 ) @ B_50 ) )
= B_50 ) ).
thf(fact_2030_minus__add,axiom,
! [A_77: rat,B_49: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A_77 @ B_49 ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ B_49 ) @ ( uminus_uminus_rat @ A_77 ) ) ) ).
thf(fact_2031_minus__add,axiom,
! [A_77: complex,B_49: complex] :
( ( uminus473333897omplex @ ( plus_plus_complex @ A_77 @ B_49 ) )
= ( plus_plus_complex @ ( uminus473333897omplex @ B_49 ) @ ( uminus473333897omplex @ A_77 ) ) ) ).
thf(fact_2032_minus__add,axiom,
! [A_77: real,B_49: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A_77 @ B_49 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B_49 ) @ ( uminus_uminus_real @ A_77 ) ) ) ).
thf(fact_2033_minus__add,axiom,
! [A_77: int,B_49: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A_77 @ B_49 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B_49 ) @ ( uminus_uminus_int @ A_77 ) ) ) ).
thf(fact_2034_minus__add__distrib,axiom,
! [A_76: rat,B_48: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A_76 @ B_48 ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ A_76 ) @ ( uminus_uminus_rat @ B_48 ) ) ) ).
thf(fact_2035_minus__add__distrib,axiom,
! [A_76: complex,B_48: complex] :
( ( uminus473333897omplex @ ( plus_plus_complex @ A_76 @ B_48 ) )
= ( plus_plus_complex @ ( uminus473333897omplex @ A_76 ) @ ( uminus473333897omplex @ B_48 ) ) ) ).
thf(fact_2036_minus__add__distrib,axiom,
! [A_76: real,B_48: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A_76 @ B_48 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A_76 ) @ ( uminus_uminus_real @ B_48 ) ) ) ).
thf(fact_2037_minus__add__distrib,axiom,
! [A_76: int,B_48: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A_76 @ B_48 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A_76 ) @ ( uminus_uminus_int @ B_48 ) ) ) ).
thf(fact_2038_zmod__self,axiom,
! [A: int] :
( ( div_mod_int @ A @ A )
= zero_zero_int ) ).
thf(fact_2039_zmod__zero,axiom,
! [B: int] :
( ( div_mod_int @ zero_zero_int @ B )
= zero_zero_int ) ).
thf(fact_2040_minus__diff__minus,axiom,
! [A_75: rat,B_47: rat] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ A_75 ) @ ( uminus_uminus_rat @ B_47 ) )
= ( uminus_uminus_rat @ ( minus_minus_rat @ A_75 @ B_47 ) ) ) ).
thf(fact_2041_minus__diff__minus,axiom,
! [A_75: complex,B_47: complex] :
( ( minus_minus_complex @ ( uminus473333897omplex @ A_75 ) @ ( uminus473333897omplex @ B_47 ) )
= ( uminus473333897omplex @ ( minus_minus_complex @ A_75 @ B_47 ) ) ) ).
thf(fact_2042_minus__diff__minus,axiom,
! [A_75: real,B_47: real] :
( ( minus_minus_real @ ( uminus_uminus_real @ A_75 ) @ ( uminus_uminus_real @ B_47 ) )
= ( uminus_uminus_real @ ( minus_minus_real @ A_75 @ B_47 ) ) ) ).
thf(fact_2043_minus__diff__minus,axiom,
! [A_75: int,B_47: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ A_75 ) @ ( uminus_uminus_int @ B_47 ) )
= ( uminus_uminus_int @ ( minus_minus_int @ A_75 @ B_47 ) ) ) ).
thf(fact_2044_minus__diff__eq,axiom,
! [A_74: rat,B_46: rat] :
( ( uminus_uminus_rat @ ( minus_minus_rat @ A_74 @ B_46 ) )
= ( minus_minus_rat @ B_46 @ A_74 ) ) ).
thf(fact_2045_minus__diff__eq,axiom,
! [A_74: complex,B_46: complex] :
( ( uminus473333897omplex @ ( minus_minus_complex @ A_74 @ B_46 ) )
= ( minus_minus_complex @ B_46 @ A_74 ) ) ).
thf(fact_2046_minus__diff__eq,axiom,
! [A_74: real,B_46: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A_74 @ B_46 ) )
= ( minus_minus_real @ B_46 @ A_74 ) ) ).
thf(fact_2047_minus__diff__eq,axiom,
! [A_74: int,B_46: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A_74 @ B_46 ) )
= ( minus_minus_int @ B_46 @ A_74 ) ) ).
thf(fact_2048_zmod__simps_I3_J,axiom,
! [A: int,B: int,C: int] :
( ( div_mod_int @ ( times_times_int @ A @ ( div_mod_int @ B @ C ) ) @ C )
= ( div_mod_int @ ( times_times_int @ A @ B ) @ C ) ) ).
thf(fact_2049_zmod__zmult1__eq,axiom,
! [A: int,B: int,C: int] :
( ( div_mod_int @ ( times_times_int @ A @ B ) @ C )
= ( div_mod_int @ ( times_times_int @ A @ ( div_mod_int @ B @ C ) ) @ C ) ) ).
thf(fact_2050_zminus__0,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_2051_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_2052_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_2053_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_2054_rel__simps_I7_J,axiom,
~ ( ord_less_int @ min @ min ) ).
thf(fact_2055_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_2056_rel__simps_I24_J,axiom,
ord_less_eq_int @ min @ min ).
thf(fact_2057_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_2058_abs__number__of,axiom,
! [X_14: int] :
( ( ( ord_less_rat @ ( number_number_of_rat @ X_14 ) @ zero_zero_rat )
=> ( ( abs_abs_rat @ ( number_number_of_rat @ X_14 ) )
= ( uminus_uminus_rat @ ( number_number_of_rat @ X_14 ) ) ) )
& ( ~ ( ord_less_rat @ ( number_number_of_rat @ X_14 ) @ zero_zero_rat )
=> ( ( abs_abs_rat @ ( number_number_of_rat @ X_14 ) )
= ( number_number_of_rat @ X_14 ) ) ) ) ).
thf(fact_2059_abs__number__of,axiom,
! [X_14: int] :
( ( ( ord_less_real @ ( number267125858f_real @ X_14 ) @ zero_zero_real )
=> ( ( abs_abs_real @ ( number267125858f_real @ X_14 ) )
= ( uminus_uminus_real @ ( number267125858f_real @ X_14 ) ) ) )
& ( ~ ( ord_less_real @ ( number267125858f_real @ X_14 ) @ zero_zero_real )
=> ( ( abs_abs_real @ ( number267125858f_real @ X_14 ) )
= ( number267125858f_real @ X_14 ) ) ) ) ).
thf(fact_2060_abs__number__of,axiom,
! [X_14: int] :
( ( ( ord_less_int @ ( number_number_of_int @ X_14 ) @ zero_zero_int )
=> ( ( abs_abs_int @ ( number_number_of_int @ X_14 ) )
= ( uminus_uminus_int @ ( number_number_of_int @ X_14 ) ) ) )
& ( ~ ( ord_less_int @ ( number_number_of_int @ X_14 ) @ zero_zero_int )
=> ( ( abs_abs_int @ ( number_number_of_int @ X_14 ) )
= ( number_number_of_int @ X_14 ) ) ) ) ).
thf(fact_2061_Ints__0,axiom,
member_real @ zero_zero_real @ ring_1_Ints_real ).
thf(fact_2062_Ints__0,axiom,
member_int @ zero_zero_int @ ring_1_Ints_int ).
thf(fact_2063_Ints__0,axiom,
member_rat @ zero_zero_rat @ ring_1_Ints_rat ).
thf(fact_2064_Ints__0,axiom,
member_complex @ zero_zero_complex @ ring_1_Ints_complex ).
thf(fact_2065_Ints__mult,axiom,
! [B_45: real,A_73: real] :
( ( member_real @ A_73 @ ring_1_Ints_real )
=> ( ( member_real @ B_45 @ ring_1_Ints_real )
=> ( member_real @ ( times_times_real @ A_73 @ B_45 ) @ ring_1_Ints_real ) ) ) ).
thf(fact_2066_Ints__mult,axiom,
! [B_45: int,A_73: int] :
( ( member_int @ A_73 @ ring_1_Ints_int )
=> ( ( member_int @ B_45 @ ring_1_Ints_int )
=> ( member_int @ ( times_times_int @ A_73 @ B_45 ) @ ring_1_Ints_int ) ) ) ).
thf(fact_2067_Ints__mult,axiom,
! [B_45: rat,A_73: rat] :
( ( member_rat @ A_73 @ ring_1_Ints_rat )
=> ( ( member_rat @ B_45 @ ring_1_Ints_rat )
=> ( member_rat @ ( times_times_rat @ A_73 @ B_45 ) @ ring_1_Ints_rat ) ) ) ).
thf(fact_2068_Ints__mult,axiom,
! [B_45: complex,A_73: complex] :
( ( member_complex @ A_73 @ ring_1_Ints_complex )
=> ( ( member_complex @ B_45 @ ring_1_Ints_complex )
=> ( member_complex @ ( times_times_complex @ A_73 @ B_45 ) @ ring_1_Ints_complex ) ) ) ).
thf(fact_2069_Ints__add,axiom,
! [B_44: real,A_72: real] :
( ( member_real @ A_72 @ ring_1_Ints_real )
=> ( ( member_real @ B_44 @ ring_1_Ints_real )
=> ( member_real @ ( plus_plus_real @ A_72 @ B_44 ) @ ring_1_Ints_real ) ) ) ).
thf(fact_2070_Ints__add,axiom,
! [B_44: int,A_72: int] :
( ( member_int @ A_72 @ ring_1_Ints_int )
=> ( ( member_int @ B_44 @ ring_1_Ints_int )
=> ( member_int @ ( plus_plus_int @ A_72 @ B_44 ) @ ring_1_Ints_int ) ) ) ).
thf(fact_2071_Ints__add,axiom,
! [B_44: rat,A_72: rat] :
( ( member_rat @ A_72 @ ring_1_Ints_rat )
=> ( ( member_rat @ B_44 @ ring_1_Ints_rat )
=> ( member_rat @ ( plus_plus_rat @ A_72 @ B_44 ) @ ring_1_Ints_rat ) ) ) ).
thf(fact_2072_Ints__add,axiom,
! [B_44: complex,A_72: complex] :
( ( member_complex @ A_72 @ ring_1_Ints_complex )
=> ( ( member_complex @ B_44 @ ring_1_Ints_complex )
=> ( member_complex @ ( plus_plus_complex @ A_72 @ B_44 ) @ ring_1_Ints_complex ) ) ) ).
thf(fact_2073_Ints__number__of,axiom,
! [W_4: int] : ( member_real @ ( number267125858f_real @ W_4 ) @ ring_1_Ints_real ) ).
thf(fact_2074_Ints__number__of,axiom,
! [W_4: int] : ( member_int @ ( number_number_of_int @ W_4 ) @ ring_1_Ints_int ) ).
thf(fact_2075_Ints__number__of,axiom,
! [W_4: int] : ( member_rat @ ( number_number_of_rat @ W_4 ) @ ring_1_Ints_rat ) ).
thf(fact_2076_Ints__number__of,axiom,
! [W_4: int] : ( member_complex @ ( number528085621omplex @ W_4 ) @ ring_1_Ints_complex ) ).
thf(fact_2077_Ints__1,axiom,
member_real @ one_one_real @ ring_1_Ints_real ).
thf(fact_2078_Ints__1,axiom,
member_int @ one_one_int @ ring_1_Ints_int ).
thf(fact_2079_Ints__1,axiom,
member_rat @ one_one_rat @ ring_1_Ints_rat ).
thf(fact_2080_Ints__1,axiom,
member_complex @ one_one_complex @ ring_1_Ints_complex ).
thf(fact_2081_Ints__diff,axiom,
! [B_43: real,A_71: real] :
( ( member_real @ A_71 @ ring_1_Ints_real )
=> ( ( member_real @ B_43 @ ring_1_Ints_real )
=> ( member_real @ ( minus_minus_real @ A_71 @ B_43 ) @ ring_1_Ints_real ) ) ) ).
thf(fact_2082_Ints__diff,axiom,
! [B_43: int,A_71: int] :
( ( member_int @ A_71 @ ring_1_Ints_int )
=> ( ( member_int @ B_43 @ ring_1_Ints_int )
=> ( member_int @ ( minus_minus_int @ A_71 @ B_43 ) @ ring_1_Ints_int ) ) ) ).
thf(fact_2083_Ints__diff,axiom,
! [B_43: rat,A_71: rat] :
( ( member_rat @ A_71 @ ring_1_Ints_rat )
=> ( ( member_rat @ B_43 @ ring_1_Ints_rat )
=> ( member_rat @ ( minus_minus_rat @ A_71 @ B_43 ) @ ring_1_Ints_rat ) ) ) ).
thf(fact_2084_Ints__diff,axiom,
! [B_43: complex,A_71: complex] :
( ( member_complex @ A_71 @ ring_1_Ints_complex )
=> ( ( member_complex @ B_43 @ ring_1_Ints_complex )
=> ( member_complex @ ( minus_minus_complex @ A_71 @ B_43 ) @ ring_1_Ints_complex ) ) ) ).
thf(fact_2085_mod__mult__self1__is__0,axiom,
! [B_42: quickcheck_code_int,A_70: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( times_123202395de_int @ B_42 @ A_70 ) @ B_42 )
= zero_z891286103de_int ) ).
thf(fact_2086_mod__mult__self1__is__0,axiom,
! [B_42: code_code_numeral,A_70: code_code_numeral] :
( ( div_mo1740067990umeral @ ( times_1655362735umeral @ B_42 @ A_70 ) @ B_42 )
= zero_z126310315umeral ) ).
thf(fact_2087_mod__mult__self1__is__0,axiom,
! [B_42: nat,A_70: nat] :
( ( div_mod_nat @ ( times_times_nat @ B_42 @ A_70 ) @ B_42 )
= zero_zero_nat ) ).
thf(fact_2088_mod__mult__self1__is__0,axiom,
! [B_42: int,A_70: int] :
( ( div_mod_int @ ( times_times_int @ B_42 @ A_70 ) @ B_42 )
= zero_zero_int ) ).
thf(fact_2089_mod__mult__self2__is__0,axiom,
! [A_69: quickcheck_code_int,B_41: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( times_123202395de_int @ A_69 @ B_41 ) @ B_41 )
= zero_z891286103de_int ) ).
thf(fact_2090_mod__mult__self2__is__0,axiom,
! [A_69: code_code_numeral,B_41: code_code_numeral] :
( ( div_mo1740067990umeral @ ( times_1655362735umeral @ A_69 @ B_41 ) @ B_41 )
= zero_z126310315umeral ) ).
thf(fact_2091_mod__mult__self2__is__0,axiom,
! [A_69: nat,B_41: nat] :
( ( div_mod_nat @ ( times_times_nat @ A_69 @ B_41 ) @ B_41 )
= zero_zero_nat ) ).
thf(fact_2092_mod__mult__self2__is__0,axiom,
! [A_69: int,B_41: int] :
( ( div_mod_int @ ( times_times_int @ A_69 @ B_41 ) @ B_41 )
= zero_zero_int ) ).
thf(fact_2093_abs__of__nonneg,axiom,
! [A_68: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A_68 )
=> ( ( abs_abs_rat @ A_68 )
= A_68 ) ) ).
thf(fact_2094_abs__of__nonneg,axiom,
! [A_68: real] :
( ( ord_less_eq_real @ zero_zero_real @ A_68 )
=> ( ( abs_abs_real @ A_68 )
= A_68 ) ) ).
thf(fact_2095_abs__of__nonneg,axiom,
! [A_68: int] :
( ( ord_less_eq_int @ zero_zero_int @ A_68 )
=> ( ( abs_abs_int @ A_68 )
= A_68 ) ) ).
thf(fact_2096_abs__le__zero__iff,axiom,
! [A_67: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A_67 ) @ zero_zero_rat )
<=> ( A_67 = zero_zero_rat ) ) ).
thf(fact_2097_abs__le__zero__iff,axiom,
! [A_67: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A_67 ) @ zero_zero_real )
<=> ( A_67 = zero_zero_real ) ) ).
thf(fact_2098_abs__le__zero__iff,axiom,
! [A_67: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A_67 ) @ zero_zero_int )
<=> ( A_67 = zero_zero_int ) ) ).
thf(fact_2099_abs__ge__zero,axiom,
! [A_66: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A_66 ) ) ).
thf(fact_2100_abs__ge__zero,axiom,
! [A_66: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A_66 ) ) ).
thf(fact_2101_abs__ge__zero,axiom,
! [A_66: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A_66 ) ) ).
thf(fact_2102_abs__of__pos,axiom,
! [A_65: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_65 )
=> ( ( abs_abs_rat @ A_65 )
= A_65 ) ) ).
thf(fact_2103_abs__of__pos,axiom,
! [A_65: real] :
( ( ord_less_real @ zero_zero_real @ A_65 )
=> ( ( abs_abs_real @ A_65 )
= A_65 ) ) ).
thf(fact_2104_abs__of__pos,axiom,
! [A_65: int] :
( ( ord_less_int @ zero_zero_int @ A_65 )
=> ( ( abs_abs_int @ A_65 )
= A_65 ) ) ).
thf(fact_2105_zero__less__abs__iff,axiom,
! [A_64: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A_64 ) )
<=> ( A_64 != zero_zero_rat ) ) ).
thf(fact_2106_zero__less__abs__iff,axiom,
! [A_64: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A_64 ) )
<=> ( A_64 != zero_zero_real ) ) ).
thf(fact_2107_zero__less__abs__iff,axiom,
! [A_64: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A_64 ) )
<=> ( A_64 != zero_zero_int ) ) ).
thf(fact_2108_abs__not__less__zero,axiom,
! [A_63: rat] :
~ ( ord_less_rat @ ( abs_abs_rat @ A_63 ) @ zero_zero_rat ) ).
thf(fact_2109_abs__not__less__zero,axiom,
! [A_63: real] :
~ ( ord_less_real @ ( abs_abs_real @ A_63 ) @ zero_zero_real ) ).
thf(fact_2110_abs__not__less__zero,axiom,
! [A_63: int] :
~ ( ord_less_int @ ( abs_abs_int @ A_63 ) @ zero_zero_int ) ).
thf(fact_2111_mod__by__1,axiom,
! [A_62: quickcheck_code_int] :
( ( div_mo231679042de_int @ A_62 @ one_on1684967323de_int )
= zero_z891286103de_int ) ).
thf(fact_2112_mod__by__1,axiom,
! [A_62: code_code_numeral] :
( ( div_mo1740067990umeral @ A_62 @ one_on1645066479umeral )
= zero_z126310315umeral ) ).
thf(fact_2113_mod__by__1,axiom,
! [A_62: nat] :
( ( div_mod_nat @ A_62 @ one_one_nat )
= zero_zero_nat ) ).
thf(fact_2114_mod__by__1,axiom,
! [A_62: int] :
( ( div_mod_int @ A_62 @ one_one_int )
= zero_zero_int ) ).
thf(fact_2115_abs__mult__less,axiom,
! [B_40: rat,D_5: rat,A_61: rat,C_17: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ A_61 ) @ C_17 )
=> ( ( ord_less_rat @ ( abs_abs_rat @ B_40 ) @ D_5 )
=> ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A_61 ) @ ( abs_abs_rat @ B_40 ) ) @ ( times_times_rat @ C_17 @ D_5 ) ) ) ) ).
thf(fact_2116_abs__mult__less,axiom,
! [B_40: real,D_5: real,A_61: real,C_17: real] :
( ( ord_less_real @ ( abs_abs_real @ A_61 ) @ C_17 )
=> ( ( ord_less_real @ ( abs_abs_real @ B_40 ) @ D_5 )
=> ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A_61 ) @ ( abs_abs_real @ B_40 ) ) @ ( times_times_real @ C_17 @ D_5 ) ) ) ) ).
thf(fact_2117_abs__mult__less,axiom,
! [B_40: int,D_5: int,A_61: int,C_17: int] :
( ( ord_less_int @ ( abs_abs_int @ A_61 ) @ C_17 )
=> ( ( ord_less_int @ ( abs_abs_int @ B_40 ) @ D_5 )
=> ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A_61 ) @ ( abs_abs_int @ B_40 ) ) @ ( times_times_int @ C_17 @ D_5 ) ) ) ) ).
thf(fact_2118_mod__mult__self1,axiom,
! [A_60: quickcheck_code_int,C_16: quickcheck_code_int,B_39: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_60 @ ( times_123202395de_int @ C_16 @ B_39 ) ) @ B_39 )
= ( div_mo231679042de_int @ A_60 @ B_39 ) ) ).
thf(fact_2119_mod__mult__self1,axiom,
! [A_60: code_code_numeral,C_16: code_code_numeral,B_39: code_code_numeral] :
( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_60 @ ( times_1655362735umeral @ C_16 @ B_39 ) ) @ B_39 )
= ( div_mo1740067990umeral @ A_60 @ B_39 ) ) ).
thf(fact_2120_mod__mult__self1,axiom,
! [A_60: nat,C_16: nat,B_39: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ A_60 @ ( times_times_nat @ C_16 @ B_39 ) ) @ B_39 )
= ( div_mod_nat @ A_60 @ B_39 ) ) ).
thf(fact_2121_mod__mult__self1,axiom,
! [A_60: int,C_16: int,B_39: int] :
( ( div_mod_int @ ( plus_plus_int @ A_60 @ ( times_times_int @ C_16 @ B_39 ) ) @ B_39 )
= ( div_mod_int @ A_60 @ B_39 ) ) ).
thf(fact_2122_mod__mult__self2,axiom,
! [A_59: quickcheck_code_int,B_38: quickcheck_code_int,C_15: quickcheck_code_int] :
( ( div_mo231679042de_int @ ( plus_p1446045655de_int @ A_59 @ ( times_123202395de_int @ B_38 @ C_15 ) ) @ B_38 )
= ( div_mo231679042de_int @ A_59 @ B_38 ) ) ).
thf(fact_2123_mod__mult__self2,axiom,
! [A_59: code_code_numeral,B_38: code_code_numeral,C_15: code_code_numeral] :
( ( div_mo1740067990umeral @ ( plus_p1627245867umeral @ A_59 @ ( times_1655362735umeral @ B_38 @ C_15 ) ) @ B_38 )
= ( div_mo1740067990umeral @ A_59 @ B_38 ) ) ).
thf(fact_2124_mod__mult__self2,axiom,
! [A_59: nat,B_38: nat,C_15: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ A_59 @ ( times_times_nat @ B_38 @ C_15 ) ) @ B_38 )
= ( div_mod_nat @ A_59 @ B_38 ) ) ).
thf(fact_2125_mod__mult__self2,axiom,
! [A_59: int,B_38: int,C_15: int] :
( ( div_mod_int @ ( plus_plus_int @ A_59 @ ( times_times_int @ B_38 @ C_15 ) ) @ B_38 )
= ( div_mod_int @ A_59 @ B_38 ) ) ).
thf(fact_2126_abs__triangle__ineq,axiom,
! [A_58: rat,B_37: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A_58 @ B_37 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A_58 ) @ ( abs_abs_rat @ B_37 ) ) ) ).
thf(fact_2127_abs__triangle__ineq,axiom,
! [A_58: real,B_37: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A_58 @ B_37 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A_58 ) @ ( abs_abs_real @ B_37 ) ) ) ).
thf(fact_2128_abs__triangle__ineq,axiom,
! [A_58: int,B_37: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A_58 @ B_37 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A_58 ) @ ( abs_abs_int @ B_37 ) ) ) ).
thf(fact_2129_abs__triangle__ineq3,axiom,
! [A_57: rat,B_36: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A_57 ) @ ( abs_abs_rat @ B_36 ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A_57 @ B_36 ) ) ) ).
thf(fact_2130_abs__triangle__ineq3,axiom,
! [A_57: real,B_36: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A_57 ) @ ( abs_abs_real @ B_36 ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A_57 @ B_36 ) ) ) ).
thf(fact_2131_abs__triangle__ineq3,axiom,
! [A_57: int,B_36: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A_57 ) @ ( abs_abs_int @ B_36 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A_57 @ B_36 ) ) ) ).
thf(fact_2132_abs__triangle__ineq2,axiom,
! [A_56: rat,B_35: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A_56 ) @ ( abs_abs_rat @ B_35 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A_56 @ B_35 ) ) ) ).
thf(fact_2133_abs__triangle__ineq2,axiom,
! [A_56: real,B_35: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A_56 ) @ ( abs_abs_real @ B_35 ) ) @ ( abs_abs_real @ ( minus_minus_real @ A_56 @ B_35 ) ) ) ).
thf(fact_2134_abs__triangle__ineq2,axiom,
! [A_56: int,B_35: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A_56 ) @ ( abs_abs_int @ B_35 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A_56 @ B_35 ) ) ) ).
thf(fact_2135_abs__triangle__ineq2__sym,axiom,
! [A_55: rat,B_34: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A_55 ) @ ( abs_abs_rat @ B_34 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B_34 @ A_55 ) ) ) ).
thf(fact_2136_abs__triangle__ineq2__sym,axiom,
! [A_55: real,B_34: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A_55 ) @ ( abs_abs_real @ B_34 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B_34 @ A_55 ) ) ) ).
thf(fact_2137_abs__triangle__ineq2__sym,axiom,
! [A_55: int,B_34: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A_55 ) @ ( abs_abs_int @ B_34 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B_34 @ A_55 ) ) ) ).
thf(fact_2138_abs__diff__less__iff,axiom,
! [X_13: rat,A_54: rat,R_5: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X_13 @ A_54 ) ) @ R_5 )
<=> ( ( ord_less_rat @ ( minus_minus_rat @ A_54 @ R_5 ) @ X_13 )
& ( ord_less_rat @ X_13 @ ( plus_plus_rat @ A_54 @ R_5 ) ) ) ) ).
thf(fact_2139_abs__diff__less__iff,axiom,
! [X_13: real,A_54: real,R_5: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X_13 @ A_54 ) ) @ R_5 )
<=> ( ( ord_less_real @ ( minus_minus_real @ A_54 @ R_5 ) @ X_13 )
& ( ord_less_real @ X_13 @ ( plus_plus_real @ A_54 @ R_5 ) ) ) ) ).
thf(fact_2140_abs__diff__less__iff,axiom,
! [X_13: int,A_54: int,R_5: int] :
( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X_13 @ A_54 ) ) @ R_5 )
<=> ( ( ord_less_int @ ( minus_minus_int @ A_54 @ R_5 ) @ X_13 )
& ( ord_less_int @ X_13 @ ( plus_plus_int @ A_54 @ R_5 ) ) ) ) ).
thf(fact_2141_d22set__le__swap,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( member_int @ B @ ( d22set @ A ) ) ) ).
thf(fact_2142_minus__le__self__iff,axiom,
! [A_53: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A_53 ) @ A_53 )
<=> ( ord_less_eq_rat @ zero_zero_rat @ A_53 ) ) ).
thf(fact_2143_minus__le__self__iff,axiom,
! [A_53: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A_53 ) @ A_53 )
<=> ( ord_less_eq_real @ zero_zero_real @ A_53 ) ) ).
thf(fact_2144_minus__le__self__iff,axiom,
! [A_53: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A_53 ) @ A_53 )
<=> ( ord_less_eq_int @ zero_zero_int @ A_53 ) ) ).
thf(fact_2145_neg__le__0__iff__le,axiom,
! [A_52: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A_52 ) @ zero_zero_rat )
<=> ( ord_less_eq_rat @ zero_zero_rat @ A_52 ) ) ).
thf(fact_2146_neg__le__0__iff__le,axiom,
! [A_52: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A_52 ) @ zero_zero_real )
<=> ( ord_less_eq_real @ zero_zero_real @ A_52 ) ) ).
thf(fact_2147_neg__le__0__iff__le,axiom,
! [A_52: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A_52 ) @ zero_zero_int )
<=> ( ord_less_eq_int @ zero_zero_int @ A_52 ) ) ).
thf(fact_2148_le__minus__self__iff,axiom,
! [A_51: rat] :
( ( ord_less_eq_rat @ A_51 @ ( uminus_uminus_rat @ A_51 ) )
<=> ( ord_less_eq_rat @ A_51 @ zero_zero_rat ) ) ).
thf(fact_2149_le__minus__self__iff,axiom,
! [A_51: real] :
( ( ord_less_eq_real @ A_51 @ ( uminus_uminus_real @ A_51 ) )
<=> ( ord_less_eq_real @ A_51 @ zero_zero_real ) ) ).
thf(fact_2150_le__minus__self__iff,axiom,
! [A_51: int] :
( ( ord_less_eq_int @ A_51 @ ( uminus_uminus_int @ A_51 ) )
<=> ( ord_less_eq_int @ A_51 @ zero_zero_int ) ) ).
thf(fact_2151_neg__0__le__iff__le,axiom,
! [A_50: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A_50 ) )
<=> ( ord_less_eq_rat @ A_50 @ zero_zero_rat ) ) ).
thf(fact_2152_neg__0__le__iff__le,axiom,
! [A_50: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A_50 ) )
<=> ( ord_less_eq_real @ A_50 @ zero_zero_real ) ) ).
thf(fact_2153_neg__0__le__iff__le,axiom,
! [A_50: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A_50 ) )
<=> ( ord_less_eq_int @ A_50 @ zero_zero_int ) ) ).
thf(fact_2154_d22set__le,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( d22set @ A ) )
=> ( ord_less_eq_int @ B @ A ) ) ).
thf(fact_2155_neg__less__nonneg,axiom,
! [A_49: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A_49 ) @ A_49 )
<=> ( ord_less_rat @ zero_zero_rat @ A_49 ) ) ).
thf(fact_2156_neg__less__nonneg,axiom,
! [A_49: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A_49 ) @ A_49 )
<=> ( ord_less_real @ zero_zero_real @ A_49 ) ) ).
thf(fact_2157_neg__less__nonneg,axiom,
! [A_49: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A_49 ) @ A_49 )
<=> ( ord_less_int @ zero_zero_int @ A_49 ) ) ).
thf(fact_2158_neg__less__0__iff__less,axiom,
! [A_48: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A_48 ) @ zero_zero_rat )
<=> ( ord_less_rat @ zero_zero_rat @ A_48 ) ) ).
thf(fact_2159_neg__less__0__iff__less,axiom,
! [A_48: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A_48 ) @ zero_zero_real )
<=> ( ord_less_real @ zero_zero_real @ A_48 ) ) ).
thf(fact_2160_neg__less__0__iff__less,axiom,
! [A_48: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A_48 ) @ zero_zero_int )
<=> ( ord_less_int @ zero_zero_int @ A_48 ) ) ).
thf(fact_2161_less__minus__self__iff,axiom,
! [A_47: rat] :
( ( ord_less_rat @ A_47 @ ( uminus_uminus_rat @ A_47 ) )
<=> ( ord_less_rat @ A_47 @ zero_zero_rat ) ) ).
thf(fact_2162_less__minus__self__iff,axiom,
! [A_47: real] :
( ( ord_less_real @ A_47 @ ( uminus_uminus_real @ A_47 ) )
<=> ( ord_less_real @ A_47 @ zero_zero_real ) ) ).
thf(fact_2163_less__minus__self__iff,axiom,
! [A_47: int] :
( ( ord_less_int @ A_47 @ ( uminus_uminus_int @ A_47 ) )
<=> ( ord_less_int @ A_47 @ zero_zero_int ) ) ).
thf(fact_2164_neg__0__less__iff__less,axiom,
! [A_46: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A_46 ) )
<=> ( ord_less_rat @ A_46 @ zero_zero_rat ) ) ).
thf(fact_2165_neg__0__less__iff__less,axiom,
! [A_46: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A_46 ) )
<=> ( ord_less_real @ A_46 @ zero_zero_real ) ) ).
thf(fact_2166_neg__0__less__iff__less,axiom,
! [A_46: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A_46 ) )
<=> ( ord_less_int @ A_46 @ zero_zero_int ) ) ).
thf(fact_2167_right__minus,axiom,
! [A_45: rat] :
( ( plus_plus_rat @ A_45 @ ( uminus_uminus_rat @ A_45 ) )
= zero_zero_rat ) ).
thf(fact_2168_right__minus,axiom,
! [A_45: complex] :
( ( plus_plus_complex @ A_45 @ ( uminus473333897omplex @ A_45 ) )
= zero_zero_complex ) ).
thf(fact_2169_right__minus,axiom,
! [A_45: real] :
( ( plus_plus_real @ A_45 @ ( uminus_uminus_real @ A_45 ) )
= zero_zero_real ) ).
thf(fact_2170_right__minus,axiom,
! [A_45: int] :
( ( plus_plus_int @ A_45 @ ( uminus_uminus_int @ A_45 ) )
= zero_zero_int ) ).
thf(fact_2171_eq__neg__iff__add__eq__0,axiom,
! [A_44: rat,B_33: rat] :
( ( A_44
= ( uminus_uminus_rat @ B_33 ) )
<=> ( ( plus_plus_rat @ A_44 @ B_33 )
= zero_zero_rat ) ) ).
thf(fact_2172_eq__neg__iff__add__eq__0,axiom,
! [A_44: complex,B_33: complex] :
( ( A_44
= ( uminus473333897omplex @ B_33 ) )
<=> ( ( plus_plus_complex @ A_44 @ B_33 )
= zero_zero_complex ) ) ).
thf(fact_2173_eq__neg__iff__add__eq__0,axiom,
! [A_44: real,B_33: real] :
( ( A_44
= ( uminus_uminus_real @ B_33 ) )
<=> ( ( plus_plus_real @ A_44 @ B_33 )
= zero_zero_real ) ) ).
thf(fact_2174_eq__neg__iff__add__eq__0,axiom,
! [A_44: int,B_33: int] :
( ( A_44
= ( uminus_uminus_int @ B_33 ) )
<=> ( ( plus_plus_int @ A_44 @ B_33 )
= zero_zero_int ) ) ).
thf(fact_2175_left__minus,axiom,
! [A_43: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A_43 ) @ A_43 )
= zero_zero_rat ) ).
thf(fact_2176_left__minus,axiom,
! [A_43: complex] :
( ( plus_plus_complex @ ( uminus473333897omplex @ A_43 ) @ A_43 )
= zero_zero_complex ) ).
thf(fact_2177_left__minus,axiom,
! [A_43: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A_43 ) @ A_43 )
= zero_zero_real ) ).
thf(fact_2178_left__minus,axiom,
! [A_43: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A_43 ) @ A_43 )
= zero_zero_int ) ).
thf(fact_2179_ab__left__minus,axiom,
! [A_42: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A_42 ) @ A_42 )
= zero_zero_rat ) ).
thf(fact_2180_ab__left__minus,axiom,
! [A_42: complex] :
( ( plus_plus_complex @ ( uminus473333897omplex @ A_42 ) @ A_42 )
= zero_zero_complex ) ).
thf(fact_2181_ab__left__minus,axiom,
! [A_42: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A_42 ) @ A_42 )
= zero_zero_real ) ).
thf(fact_2182_ab__left__minus,axiom,
! [A_42: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A_42 ) @ A_42 )
= zero_zero_int ) ).
thf(fact_2183_add__eq__0__iff,axiom,
! [X_12: rat,Y_10: rat] :
( ( ( plus_plus_rat @ X_12 @ Y_10 )
= zero_zero_rat )
<=> ( Y_10
= ( uminus_uminus_rat @ X_12 ) ) ) ).
thf(fact_2184_add__eq__0__iff,axiom,
! [X_12: complex,Y_10: complex] :
( ( ( plus_plus_complex @ X_12 @ Y_10 )
= zero_zero_complex )
<=> ( Y_10
= ( uminus473333897omplex @ X_12 ) ) ) ).
thf(fact_2185_add__eq__0__iff,axiom,
! [X_12: real,Y_10: real] :
( ( ( plus_plus_real @ X_12 @ Y_10 )
= zero_zero_real )
<=> ( Y_10
= ( uminus_uminus_real @ X_12 ) ) ) ).
thf(fact_2186_add__eq__0__iff,axiom,
! [X_12: int,Y_10: int] :
( ( ( plus_plus_int @ X_12 @ Y_10 )
= zero_zero_int )
<=> ( Y_10
= ( uminus_uminus_int @ X_12 ) ) ) ).
thf(fact_2187_minus__unique,axiom,
! [A_41: rat,B_32: rat] :
( ( ( plus_plus_rat @ A_41 @ B_32 )
= zero_zero_rat )
=> ( ( uminus_uminus_rat @ A_41 )
= B_32 ) ) ).
thf(fact_2188_minus__unique,axiom,
! [A_41: complex,B_32: complex] :
( ( ( plus_plus_complex @ A_41 @ B_32 )
= zero_zero_complex )
=> ( ( uminus473333897omplex @ A_41 )
= B_32 ) ) ).
thf(fact_2189_minus__unique,axiom,
! [A_41: real,B_32: real] :
( ( ( plus_plus_real @ A_41 @ B_32 )
= zero_zero_real )
=> ( ( uminus_uminus_real @ A_41 )
= B_32 ) ) ).
thf(fact_2190_minus__unique,axiom,
! [A_41: int,B_32: int] :
( ( ( plus_plus_int @ A_41 @ B_32 )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A_41 )
= B_32 ) ) ).
thf(fact_2191_diff__0,axiom,
! [A_40: rat] :
( ( minus_minus_rat @ zero_zero_rat @ A_40 )
= ( uminus_uminus_rat @ A_40 ) ) ).
thf(fact_2192_diff__0,axiom,
! [A_40: complex] :
( ( minus_minus_complex @ zero_zero_complex @ A_40 )
= ( uminus473333897omplex @ A_40 ) ) ).
thf(fact_2193_diff__0,axiom,
! [A_40: real] :
( ( minus_minus_real @ zero_zero_real @ A_40 )
= ( uminus_uminus_real @ A_40 ) ) ).
thf(fact_2194_diff__0,axiom,
! [A_40: int] :
( ( minus_minus_int @ zero_zero_int @ A_40 )
= ( uminus_uminus_int @ A_40 ) ) ).
thf(fact_2195_square__eq__1__iff,axiom,
! [X_11: rat] :
( ( ( times_times_rat @ X_11 @ X_11 )
= one_one_rat )
<=> ( ( X_11 = one_one_rat )
| ( X_11
= ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
thf(fact_2196_square__eq__1__iff,axiom,
! [X_11: complex] :
( ( ( times_times_complex @ X_11 @ X_11 )
= one_one_complex )
<=> ( ( X_11 = one_one_complex )
| ( X_11
= ( uminus473333897omplex @ one_one_complex ) ) ) ) ).
thf(fact_2197_square__eq__1__iff,axiom,
! [X_11: real] :
( ( ( times_times_real @ X_11 @ X_11 )
= one_one_real )
<=> ( ( X_11 = one_one_real )
| ( X_11
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
thf(fact_2198_square__eq__1__iff,axiom,
! [X_11: int] :
( ( ( times_times_int @ X_11 @ X_11 )
= one_one_int )
<=> ( ( X_11 = one_one_int )
| ( X_11
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
thf(fact_2199_comm__ring__1__class_Onormalizing__ring__rules_I1_J,axiom,
! [X_10: rat] :
( ( uminus_uminus_rat @ X_10 )
= ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X_10 ) ) ).
thf(fact_2200_comm__ring__1__class_Onormalizing__ring__rules_I1_J,axiom,
! [X_10: complex] :
( ( uminus473333897omplex @ X_10 )
= ( times_times_complex @ ( uminus473333897omplex @ one_one_complex ) @ X_10 ) ) ).
thf(fact_2201_comm__ring__1__class_Onormalizing__ring__rules_I1_J,axiom,
! [X_10: real] :
( ( uminus_uminus_real @ X_10 )
= ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ X_10 ) ) ).
thf(fact_2202_comm__ring__1__class_Onormalizing__ring__rules_I1_J,axiom,
! [X_10: int] :
( ( uminus_uminus_int @ X_10 )
= ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ X_10 ) ) ).
thf(fact_2203_diff__minus__eq__add,axiom,
! [A_39: rat,B_31: rat] :
( ( minus_minus_rat @ A_39 @ ( uminus_uminus_rat @ B_31 ) )
= ( plus_plus_rat @ A_39 @ B_31 ) ) ).
thf(fact_2204_diff__minus__eq__add,axiom,
! [A_39: complex,B_31: complex] :
( ( minus_minus_complex @ A_39 @ ( uminus473333897omplex @ B_31 ) )
= ( plus_plus_complex @ A_39 @ B_31 ) ) ).
thf(fact_2205_diff__minus__eq__add,axiom,
! [A_39: real,B_31: real] :
( ( minus_minus_real @ A_39 @ ( uminus_uminus_real @ B_31 ) )
= ( plus_plus_real @ A_39 @ B_31 ) ) ).
thf(fact_2206_diff__minus__eq__add,axiom,
! [A_39: int,B_31: int] :
( ( minus_minus_int @ A_39 @ ( uminus_uminus_int @ B_31 ) )
= ( plus_plus_int @ A_39 @ B_31 ) ) ).
thf(fact_2207_ab__diff__minus,axiom,
! [A_38: rat,B_30: rat] :
( ( minus_minus_rat @ A_38 @ B_30 )
= ( plus_plus_rat @ A_38 @ ( uminus_uminus_rat @ B_30 ) ) ) ).
thf(fact_2208_ab__diff__minus,axiom,
! [A_38: complex,B_30: complex] :
( ( minus_minus_complex @ A_38 @ B_30 )
= ( plus_plus_complex @ A_38 @ ( uminus473333897omplex @ B_30 ) ) ) ).
thf(fact_2209_ab__diff__minus,axiom,
! [A_38: real,B_30: real] :
( ( minus_minus_real @ A_38 @ B_30 )
= ( plus_plus_real @ A_38 @ ( uminus_uminus_real @ B_30 ) ) ) ).
thf(fact_2210_ab__diff__minus,axiom,
! [A_38: int,B_30: int] :
( ( minus_minus_int @ A_38 @ B_30 )
= ( plus_plus_int @ A_38 @ ( uminus_uminus_int @ B_30 ) ) ) ).
thf(fact_2211_Groups_Odiff__def,axiom,
! [A_37: rat,B_29: rat] :
( ( minus_minus_rat @ A_37 @ B_29 )
= ( plus_plus_rat @ A_37 @ ( uminus_uminus_rat @ B_29 ) ) ) ).
thf(fact_2212_Groups_Odiff__def,axiom,
! [A_37: complex,B_29: complex] :
( ( minus_minus_complex @ A_37 @ B_29 )
= ( plus_plus_complex @ A_37 @ ( uminus473333897omplex @ B_29 ) ) ) ).
thf(fact_2213_Groups_Odiff__def,axiom,
! [A_37: real,B_29: real] :
( ( minus_minus_real @ A_37 @ B_29 )
= ( plus_plus_real @ A_37 @ ( uminus_uminus_real @ B_29 ) ) ) ).
thf(fact_2214_Groups_Odiff__def,axiom,
! [A_37: int,B_29: int] :
( ( minus_minus_int @ A_37 @ B_29 )
= ( plus_plus_int @ A_37 @ ( uminus_uminus_int @ B_29 ) ) ) ).
thf(fact_2215_comm__ring__1__class_Onormalizing__ring__rules_I2_J,axiom,
! [X_9: rat,Y_9: rat] :
( ( minus_minus_rat @ X_9 @ Y_9 )
= ( plus_plus_rat @ X_9 @ ( uminus_uminus_rat @ Y_9 ) ) ) ).
thf(fact_2216_comm__ring__1__class_Onormalizing__ring__rules_I2_J,axiom,
! [X_9: complex,Y_9: complex] :
( ( minus_minus_complex @ X_9 @ Y_9 )
= ( plus_plus_complex @ X_9 @ ( uminus473333897omplex @ Y_9 ) ) ) ).
thf(fact_2217_comm__ring__1__class_Onormalizing__ring__rules_I2_J,axiom,
! [X_9: real,Y_9: real] :
( ( minus_minus_real @ X_9 @ Y_9 )
= ( plus_plus_real @ X_9 @ ( uminus_uminus_real @ Y_9 ) ) ) ).
thf(fact_2218_comm__ring__1__class_Onormalizing__ring__rules_I2_J,axiom,
! [X_9: int,Y_9: int] :
( ( minus_minus_int @ X_9 @ Y_9 )
= ( plus_plus_int @ X_9 @ ( uminus_uminus_int @ Y_9 ) ) ) ).
thf(fact_2219_pos__mod__bound,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( div_mod_int @ A @ B ) @ B ) ) ).
thf(fact_2220_neg__mod__bound,axiom,
! [A: int,B: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ B @ ( div_mod_int @ A @ B ) ) ) ).
thf(fact_2221_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_2222_zmod__le__nonneg__dividend,axiom,
! [K_1: int,M: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ord_less_eq_int @ ( div_mod_int @ M @ K_1 ) @ M ) ) ).
thf(fact_2223_zmod__eq__0__iff,axiom,
! [M: int,D: int] :
( ( ( div_mod_int @ M @ D )
= zero_zero_int )
<=> ? [Q_2: int] :
( M
= ( times_times_int @ D @ Q_2 ) ) ) ).
thf(fact_2224_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_2225_zadd__zminus__inverse2,axiom,
! [Z_1: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ Z_1 ) @ Z_1 )
= zero_zero_int ) ).
thf(fact_2226_bin__less__0__simps_I2_J,axiom,
ord_less_int @ min @ zero_zero_int ).
thf(fact_2227_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_2228_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_2229_zmod__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( div_mod_int @ ( number_number_of_int @ min ) @ B )
= ( minus_minus_int @ B @ one_one_int ) ) ) ).
thf(fact_2230_abs__eq__mult,axiom,
! [B_28: rat,A_36: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A_36 )
| ( ord_less_eq_rat @ A_36 @ zero_zero_rat ) )
& ( ( ord_less_eq_rat @ zero_zero_rat @ B_28 )
| ( ord_less_eq_rat @ B_28 @ zero_zero_rat ) ) )
=> ( ( abs_abs_rat @ ( times_times_rat @ A_36 @ B_28 ) )
= ( times_times_rat @ ( abs_abs_rat @ A_36 ) @ ( abs_abs_rat @ B_28 ) ) ) ) ).
thf(fact_2231_abs__eq__mult,axiom,
! [B_28: real,A_36: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A_36 )
| ( ord_less_eq_real @ A_36 @ zero_zero_real ) )
& ( ( ord_less_eq_real @ zero_zero_real @ B_28 )
| ( ord_less_eq_real @ B_28 @ zero_zero_real ) ) )
=> ( ( abs_abs_real @ ( times_times_real @ A_36 @ B_28 ) )
= ( times_times_real @ ( abs_abs_real @ A_36 ) @ ( abs_abs_real @ B_28 ) ) ) ) ).
thf(fact_2232_abs__eq__mult,axiom,
! [B_28: int,A_36: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A_36 )
| ( ord_less_eq_int @ A_36 @ zero_zero_int ) )
& ( ( ord_less_eq_int @ zero_zero_int @ B_28 )
| ( ord_less_eq_int @ B_28 @ zero_zero_int ) ) )
=> ( ( abs_abs_int @ ( times_times_int @ A_36 @ B_28 ) )
= ( times_times_int @ ( abs_abs_int @ A_36 ) @ ( abs_abs_int @ B_28 ) ) ) ) ).
thf(fact_2233_abs__mult__pos,axiom,
! [Y_8: rat,X_8: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X_8 )
=> ( ( times_times_rat @ ( abs_abs_rat @ Y_8 ) @ X_8 )
= ( abs_abs_rat @ ( times_times_rat @ Y_8 @ X_8 ) ) ) ) ).
thf(fact_2234_abs__mult__pos,axiom,
! [Y_8: real,X_8: real] :
( ( ord_less_eq_real @ zero_zero_real @ X_8 )
=> ( ( times_times_real @ ( abs_abs_real @ Y_8 ) @ X_8 )
= ( abs_abs_real @ ( times_times_real @ Y_8 @ X_8 ) ) ) ) ).
thf(fact_2235_abs__mult__pos,axiom,
! [Y_8: int,X_8: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_8 )
=> ( ( times_times_int @ ( abs_abs_int @ Y_8 ) @ X_8 )
= ( abs_abs_int @ ( times_times_int @ Y_8 @ X_8 ) ) ) ) ).
thf(fact_2236_abs__diff__triangle__ineq,axiom,
! [A_35: rat,B_27: rat,C_14: rat,D_4: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A_35 @ B_27 ) @ ( plus_plus_rat @ C_14 @ D_4 ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A_35 @ C_14 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B_27 @ D_4 ) ) ) ) ).
thf(fact_2237_abs__diff__triangle__ineq,axiom,
! [A_35: real,B_27: real,C_14: real,D_4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A_35 @ B_27 ) @ ( plus_plus_real @ C_14 @ D_4 ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A_35 @ C_14 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B_27 @ D_4 ) ) ) ) ).
thf(fact_2238_abs__diff__triangle__ineq,axiom,
! [A_35: int,B_27: int,C_14: int,D_4: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A_35 @ B_27 ) @ ( plus_plus_int @ C_14 @ D_4 ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A_35 @ C_14 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B_27 @ D_4 ) ) ) ) ).
thf(fact_2239_abs__triangle__ineq4,axiom,
! [A_34: rat,B_26: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A_34 @ B_26 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A_34 ) @ ( abs_abs_rat @ B_26 ) ) ) ).
thf(fact_2240_abs__triangle__ineq4,axiom,
! [A_34: real,B_26: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A_34 @ B_26 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A_34 ) @ ( abs_abs_real @ B_26 ) ) ) ).
thf(fact_2241_abs__triangle__ineq4,axiom,
! [A_34: int,B_26: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A_34 @ B_26 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A_34 ) @ ( abs_abs_int @ B_26 ) ) ) ).
thf(fact_2242_Ints__double__eq__0__iff,axiom,
! [A_33: real] :
( ( member_real @ A_33 @ ring_1_Ints_real )
=> ( ( ( plus_plus_real @ A_33 @ A_33 )
= zero_zero_real )
<=> ( A_33 = zero_zero_real ) ) ) ).
thf(fact_2243_Ints__double__eq__0__iff,axiom,
! [A_33: int] :
( ( member_int @ A_33 @ ring_1_Ints_int )
=> ( ( ( plus_plus_int @ A_33 @ A_33 )
= zero_zero_int )
<=> ( A_33 = zero_zero_int ) ) ) ).
thf(fact_2244_Ints__double__eq__0__iff,axiom,
! [A_33: rat] :
( ( member_rat @ A_33 @ ring_1_Ints_rat )
=> ( ( ( plus_plus_rat @ A_33 @ A_33 )
= zero_zero_rat )
<=> ( A_33 = zero_zero_rat ) ) ) ).
thf(fact_2245_Ints__double__eq__0__iff,axiom,
! [A_33: complex] :
( ( member_complex @ A_33 @ ring_1_Ints_complex )
=> ( ( ( plus_plus_complex @ A_33 @ A_33 )
= zero_zero_complex )
<=> ( A_33 = zero_zero_complex ) ) ) ).
thf(fact_2246_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_2247_mod__neg__neg__trivial,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ A )
=> ( ( div_mod_int @ A @ B )
= A ) ) ) ).
thf(fact_2248_neg__mod__conj,axiom,
! [A: int,B: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( div_mod_int @ A @ B ) @ zero_zero_int )
& ( ord_less_int @ B @ ( div_mod_int @ A @ B ) ) ) ) ).
thf(fact_2249_neg__mod__sign,axiom,
! [A: int,B: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( div_mod_int @ A @ B ) @ zero_zero_int ) ) ).
thf(fact_2250_mod__pos__pos__trivial,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ B )
=> ( ( div_mod_int @ A @ B )
= A ) ) ) ).
thf(fact_2251_pos__mod__conj,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( div_mod_int @ A @ B ) )
& ( ord_less_int @ ( div_mod_int @ A @ B ) @ B ) ) ) ).
thf(fact_2252_pos__mod__sign,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( div_mod_int @ A @ B ) ) ) ).
thf(fact_2253_posDivAlg__0,axiom,
! [B: int] :
( ( posDivAlg @ zero_zero_int @ B )
= ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) ).
thf(fact_2254_diff__number__of__eq,axiom,
! [V_3: int,W_3: int] :
( ( minus_minus_rat @ ( number_number_of_rat @ V_3 ) @ ( number_number_of_rat @ W_3 ) )
= ( number_number_of_rat @ ( plus_plus_int @ V_3 @ ( uminus_uminus_int @ W_3 ) ) ) ) ).
thf(fact_2255_diff__number__of__eq,axiom,
! [V_3: int,W_3: int] :
( ( minus_minus_complex @ ( number528085621omplex @ V_3 ) @ ( number528085621omplex @ W_3 ) )
= ( number528085621omplex @ ( plus_plus_int @ V_3 @ ( uminus_uminus_int @ W_3 ) ) ) ) ).
thf(fact_2256_diff__number__of__eq,axiom,
! [V_3: int,W_3: int] :
( ( minus_minus_real @ ( number267125858f_real @ V_3 ) @ ( number267125858f_real @ W_3 ) )
= ( number267125858f_real @ ( plus_plus_int @ V_3 @ ( uminus_uminus_int @ W_3 ) ) ) ) ).
thf(fact_2257_diff__number__of__eq,axiom,
! [V_3: int,W_3: int] :
( ( minus_minus_int @ ( number_number_of_int @ V_3 ) @ ( number_number_of_int @ W_3 ) )
= ( number_number_of_int @ ( plus_plus_int @ V_3 @ ( uminus_uminus_int @ W_3 ) ) ) ) ).
thf(fact_2258_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_2259_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_2260_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_2261_divmod__int__rel__mod,axiom,
! [A: int,B: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( B != zero_zero_int )
=> ( ( div_mod_int @ A @ B )
= R_1 ) ) ) ).
thf(fact_2262_d22set__g__1,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( d22set @ A ) )
=> ( ord_less_int @ one_one_int @ B ) ) ).
thf(fact_2263_mod__pos__neg__trivial,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int )
=> ( ( div_mod_int @ A @ B )
= ( plus_plus_int @ A @ B ) ) ) ) ).
thf(fact_2264_add__number__of__diff2,axiom,
! [V_2: int,C_13: rat,W_2: int] :
( ( plus_plus_rat @ ( number_number_of_rat @ V_2 ) @ ( minus_minus_rat @ C_13 @ ( number_number_of_rat @ W_2 ) ) )
= ( plus_plus_rat @ ( number_number_of_rat @ ( plus_plus_int @ V_2 @ ( uminus_uminus_int @ W_2 ) ) ) @ C_13 ) ) ).
thf(fact_2265_add__number__of__diff2,axiom,
! [V_2: int,C_13: complex,W_2: int] :
( ( plus_plus_complex @ ( number528085621omplex @ V_2 ) @ ( minus_minus_complex @ C_13 @ ( number528085621omplex @ W_2 ) ) )
= ( plus_plus_complex @ ( number528085621omplex @ ( plus_plus_int @ V_2 @ ( uminus_uminus_int @ W_2 ) ) ) @ C_13 ) ) ).
thf(fact_2266_add__number__of__diff2,axiom,
! [V_2: int,C_13: real,W_2: int] :
( ( plus_plus_real @ ( number267125858f_real @ V_2 ) @ ( minus_minus_real @ C_13 @ ( number267125858f_real @ W_2 ) ) )
= ( plus_plus_real @ ( number267125858f_real @ ( plus_plus_int @ V_2 @ ( uminus_uminus_int @ W_2 ) ) ) @ C_13 ) ) ).
thf(fact_2267_add__number__of__diff2,axiom,
! [V_2: int,C_13: int,W_2: int] :
( ( plus_plus_int @ ( number_number_of_int @ V_2 ) @ ( minus_minus_int @ C_13 @ ( number_number_of_int @ W_2 ) ) )
= ( plus_plus_int @ ( number_number_of_int @ ( plus_plus_int @ V_2 @ ( uminus_uminus_int @ W_2 ) ) ) @ C_13 ) ) ).
thf(fact_2268_Ints__odd__nonzero,axiom,
! [A_32: real] :
( ( member_real @ A_32 @ ring_1_Ints_real )
=> ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A_32 ) @ A_32 )
!= zero_zero_real ) ) ).
thf(fact_2269_Ints__odd__nonzero,axiom,
! [A_32: int] :
( ( member_int @ A_32 @ ring_1_Ints_int )
=> ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A_32 ) @ A_32 )
!= zero_zero_int ) ) ).
thf(fact_2270_Ints__odd__nonzero,axiom,
! [A_32: rat] :
( ( member_rat @ A_32 @ ring_1_Ints_rat )
=> ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A_32 ) @ A_32 )
!= zero_zero_rat ) ) ).
thf(fact_2271_Ints__odd__nonzero,axiom,
! [A_32: complex] :
( ( member_complex @ A_32 @ ring_1_Ints_complex )
=> ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A_32 ) @ A_32 )
!= zero_zero_complex ) ) ).
thf(fact_2272_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_2273_zmult2__lemma__aux2,axiom,
! [Q: int,B: int,R_1: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ B @ R_1 )
=> ( ( ord_less_eq_int @ R_1 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ ( div_mod_int @ Q @ C ) ) @ R_1 ) @ zero_zero_int ) ) ) ) ).
thf(fact_2274_zmult2__lemma__aux1,axiom,
! [Q: int,B: int,R_1: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ B @ R_1 )
=> ( ( ord_less_eq_int @ R_1 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ ( div_mod_int @ Q @ C ) ) @ R_1 ) ) ) ) ) ).
thf(fact_2275_zmult2__lemma__aux4,axiom,
! [Q: int,B: int,R_1: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ( ord_less_int @ R_1 @ B )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B @ ( div_mod_int @ Q @ C ) ) @ R_1 ) @ ( times_times_int @ B @ C ) ) ) ) ) ).
thf(fact_2276_zmult2__lemma__aux3,axiom,
! [Q: int,B: int,R_1: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ zero_zero_int @ R_1 )
=> ( ( ord_less_int @ R_1 @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B @ ( div_mod_int @ Q @ C ) ) @ R_1 ) ) ) ) ) ).
thf(fact_2277_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_2278_pdivmod__posDivAlg,axiom,
! [K_1: int,L: int] :
( ( ( L = zero_zero_int )
=> ( ( pdivmod @ K_1 @ L )
= ( product_Pair_int_int @ zero_zero_int @ ( abs_abs_int @ K_1 ) ) ) )
& ( ( L != zero_zero_int )
=> ( ( pdivmod @ K_1 @ L )
= ( posDivAlg @ ( abs_abs_int @ K_1 ) @ ( abs_abs_int @ L ) ) ) ) ) ).
thf(fact_2279_z3mod__def,axiom,
! [K_1: int,L: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( z3mod @ K_1 @ L )
= ( div_mod_int @ K_1 @ L ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( z3mod @ K_1 @ L )
= ( div_mod_int @ K_1 @ ( uminus_uminus_int @ L ) ) ) ) ) ).
thf(fact_2280_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_2281_nat0__intermed__int__val,axiom,
! [K_1: int,F: nat > int,N: nat] :
( ! [I_1: nat] :
( ( ord_less_nat @ I_1 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I_1 @ one_one_nat ) ) @ ( F @ I_1 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K_1 )
=> ( ( ord_less_eq_int @ K_1 @ ( F @ N ) )
=> ? [I_1: nat] :
( ( ord_less_eq_nat @ I_1 @ N )
& ( ( F @ I_1 )
= K_1 ) ) ) ) ) ).
thf(fact_2282_int__val__lemma,axiom,
! [K_1: int,F: nat > int,N: nat] :
( ! [I_1: nat] :
( ( ord_less_nat @ I_1 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I_1 @ one_one_nat ) ) @ ( F @ I_1 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K_1 )
=> ( ( ord_less_eq_int @ K_1 @ ( F @ N ) )
=> ? [I_1: nat] :
( ( ord_less_eq_nat @ I_1 @ N )
& ( ( F @ I_1 )
= K_1 ) ) ) ) ) ).
thf(fact_2283_nat__intermed__int__val,axiom,
! [K_1: int,F: nat > int,N: nat,M: nat] :
( ! [I_1: nat] :
( ( ( ord_less_eq_nat @ M @ I_1 )
& ( ord_less_nat @ I_1 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I_1 @ one_one_nat ) ) @ ( F @ I_1 ) ) ) @ one_one_int ) )
=> ( ( ord_less_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K_1 )
=> ( ( ord_less_eq_int @ K_1 @ ( F @ N ) )
=> ? [I_1: nat] :
( ( ord_less_eq_nat @ M @ I_1 )
& ( ord_less_eq_nat @ I_1 @ N )
& ( ( F @ I_1 )
= K_1 ) ) ) ) ) ) ).
thf(fact_2284_negDivAlg__minus1,axiom,
! [B: int] :
( ( negDivAlg @ ( number_number_of_int @ min ) @ B )
= ( product_Pair_int_int @ ( number_number_of_int @ min ) @ ( minus_minus_int @ B @ one_one_int ) ) ) ).
thf(fact_2285_zmult2__lemma,axiom,
! [C: int,A: int,B: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( B != zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( divmod_int_rel @ A @ ( times_times_int @ B @ C ) @ ( product_Pair_int_int @ ( div_div_int @ Q @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ ( div_mod_int @ Q @ C ) ) @ R_1 ) ) ) ) ) ) ).
thf(fact_2286_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_2287_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_2288_mod__eq__0__iff,axiom,
! [M: nat,D: nat] :
( ( ( div_mod_nat @ M @ D )
= zero_zero_nat )
<=> ? [Q_2: nat] :
( M
= ( times_times_nat @ D @ Q_2 ) ) ) ).
thf(fact_2289_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_2290_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_2291_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_2292_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_2293_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( div_mod_nat @ M @ N )
= M ) ) ).
thf(fact_2294_mod__lemma,axiom,
! [Q: nat,R_1: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ( ord_less_nat @ R_1 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( div_mod_nat @ Q @ C ) ) @ R_1 ) @ ( times_times_nat @ B @ C ) ) ) ) ).
thf(fact_2295_split__mod,axiom,
! [P: nat > $o,N: nat,K_1: nat] :
( ( P @ ( div_mod_nat @ N @ K_1 ) )
<=> ( ( ( K_1 = zero_zero_nat )
=> ( P @ N ) )
& ( ( K_1 != zero_zero_nat )
=> ! [I_1: nat,J_1: nat] :
( ( ord_less_nat @ J_1 @ K_1 )
=> ( ( N
= ( plus_plus_nat @ ( times_times_nat @ K_1 @ I_1 ) @ J_1 ) )
=> ( P @ J_1 ) ) ) ) ) ) ).
thf(fact_2296_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_2297_abs__real__def,axiom,
! [A: real] :
( ( ( ord_less_real @ A @ zero_zero_real )
=> ( ( abs_abs_real @ A )
= ( uminus_uminus_real @ A ) ) )
& ( ~ ( ord_less_real @ A @ zero_zero_real )
=> ( ( abs_abs_real @ A )
= A ) ) ) ).
thf(fact_2298_mod__mult__self3,axiom,
! [K_1: nat,N: nat,M: nat] :
( ( div_mod_nat @ ( plus_plus_nat @ ( times_times_nat @ K_1 @ N ) @ M ) @ N )
= ( div_mod_nat @ M @ N ) ) ).
thf(fact_2299_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_2300_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( div_mod_nat @ M @ N ) @ M ) ).
thf(fact_2301_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_2302_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_2303_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_2304_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_2305_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_2306_minus__real__def,axiom,
! [X: real,Y: real] :
( ( minus_minus_real @ X @ Y )
= ( plus_plus_real @ X @ ( uminus_uminus_real @ Y ) ) ) ).
thf(fact_2307_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_2308_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_2309_termination__basic__simps_I5_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
thf(fact_2310_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_2311_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_2312_zdiv__zero,axiom,
! [B: int] :
( ( div_div_int @ zero_zero_int @ B )
= zero_zero_int ) ).
thf(fact_2313_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_2314_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_2315_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_2316_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_2317_real__add__minus__iff,axiom,
! [X: real,A: real] :
( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
<=> ( X = A ) ) ).
thf(fact_2318_zdiv__zminus2,axiom,
! [A: int,B: int] :
( ( div_div_int @ A @ ( uminus_uminus_int @ B ) )
= ( div_div_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
thf(fact_2319_zdiv__zminus__zminus,axiom,
! [A: int,B: int] :
( ( div_div_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( div_div_int @ A @ B ) ) ).
thf(fact_2320_div__by__0,axiom,
! [A_31: quickcheck_code_int] :
( ( div_di1430059507de_int @ A_31 @ zero_z891286103de_int )
= zero_z891286103de_int ) ).
thf(fact_2321_div__by__0,axiom,
! [A_31: code_code_numeral] :
( ( div_di1218280263umeral @ A_31 @ zero_z126310315umeral )
= zero_z126310315umeral ) ).
thf(fact_2322_div__by__0,axiom,
! [A_31: nat] :
( ( div_div_nat @ A_31 @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_2323_div__by__0,axiom,
! [A_31: int] :
( ( div_div_int @ A_31 @ zero_zero_int )
= zero_zero_int ) ).
thf(fact_2324_div__0,axiom,
! [A_30: quickcheck_code_int] :
( ( div_di1430059507de_int @ zero_z891286103de_int @ A_30 )
= zero_z891286103de_int ) ).
thf(fact_2325_div__0,axiom,
! [A_30: code_code_numeral] :
( ( div_di1218280263umeral @ zero_z126310315umeral @ A_30 )
= zero_z126310315umeral ) ).
thf(fact_2326_div__0,axiom,
! [A_30: nat] :
( ( div_div_nat @ zero_zero_nat @ A_30 )
= zero_zero_nat ) ).
thf(fact_2327_div__0,axiom,
! [A_30: int] :
( ( div_div_int @ zero_zero_int @ A_30 )
= zero_zero_int ) ).
thf(fact_2328_div__by__1,axiom,
! [A_29: quickcheck_code_int] :
( ( div_di1430059507de_int @ A_29 @ one_on1684967323de_int )
= A_29 ) ).
thf(fact_2329_div__by__1,axiom,
! [A_29: code_code_numeral] :
( ( div_di1218280263umeral @ A_29 @ one_on1645066479umeral )
= A_29 ) ).
thf(fact_2330_div__by__1,axiom,
! [A_29: nat] :
( ( div_div_nat @ A_29 @ one_one_nat )
= A_29 ) ).
thf(fact_2331_div__by__1,axiom,
! [A_29: int] :
( ( div_div_int @ A_29 @ one_one_int )
= A_29 ) ).
thf(fact_2332_div__neg__pos__less0,axiom,
! [B: int,A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( div_div_int @ A @ B ) @ zero_zero_int ) ) ) ).
thf(fact_2333_neg__imp__zdiv__neg__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( div_div_int @ A @ B ) @ zero_zero_int )
<=> ( ord_less_int @ zero_zero_int @ A ) ) ) ).
thf(fact_2334_pos__imp__zdiv__neg__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( div_div_int @ A @ B ) @ zero_zero_int )
<=> ( ord_less_int @ A @ zero_zero_int ) ) ) ).
thf(fact_2335_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_2336_zdiv__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( div_div_int @ A @ A )
= one_one_int ) ) ).
thf(fact_2337_zmod__zdiv__trivial,axiom,
! [A: int,B: int] :
( ( div_div_int @ ( div_mod_int @ A @ B ) @ B )
= zero_zero_int ) ).
thf(fact_2338_DIVISION__BY__ZERO,axiom,
! [A: int] :
( ( ( div_div_int @ A @ zero_zero_int )
= zero_zero_int )
& ( ( div_mod_int @ A @ zero_zero_int )
= A ) ) ).
thf(fact_2339_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_2340_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_2341_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_2342_zdiv__zadd1__eq,axiom,
! [A: int,B: int,C: int] :
( ( div_div_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( plus_plus_int @ ( div_div_int @ A @ C ) @ ( div_div_int @ B @ C ) ) @ ( div_div_int @ ( plus_plus_int @ ( div_mod_int @ A @ C ) @ ( div_mod_int @ B @ C ) ) @ C ) ) ) ).
thf(fact_2343_pdivmod__def,axiom,
! [K_1: int,L: int] :
( ( pdivmod @ K_1 @ L )
= ( product_Pair_int_int @ ( div_div_int @ ( abs_abs_int @ K_1 ) @ ( abs_abs_int @ L ) ) @ ( div_mod_int @ ( abs_abs_int @ K_1 ) @ ( abs_abs_int @ L ) ) ) ) ).
thf(fact_2344_div__mult__mult1__if,axiom,
! [A_28: quickcheck_code_int,B_25: quickcheck_code_int,C_12: quickcheck_code_int] :
( ( ( C_12 = zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( times_123202395de_int @ C_12 @ A_28 ) @ ( times_123202395de_int @ C_12 @ B_25 ) )
= zero_z891286103de_int ) )
& ( ( C_12 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( times_123202395de_int @ C_12 @ A_28 ) @ ( times_123202395de_int @ C_12 @ B_25 ) )
= ( div_di1430059507de_int @ A_28 @ B_25 ) ) ) ) ).
thf(fact_2345_div__mult__mult1__if,axiom,
! [A_28: code_code_numeral,B_25: code_code_numeral,C_12: code_code_numeral] :
( ( ( C_12 = zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( times_1655362735umeral @ C_12 @ A_28 ) @ ( times_1655362735umeral @ C_12 @ B_25 ) )
= zero_z126310315umeral ) )
& ( ( C_12 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( times_1655362735umeral @ C_12 @ A_28 ) @ ( times_1655362735umeral @ C_12 @ B_25 ) )
= ( div_di1218280263umeral @ A_28 @ B_25 ) ) ) ) ).
thf(fact_2346_div__mult__mult1__if,axiom,
! [A_28: nat,B_25: nat,C_12: nat] :
( ( ( C_12 = zero_zero_nat )
=> ( ( div_div_nat @ ( times_times_nat @ C_12 @ A_28 ) @ ( times_times_nat @ C_12 @ B_25 ) )
= zero_zero_nat ) )
& ( ( C_12 != zero_zero_nat )
=> ( ( div_div_nat @ ( times_times_nat @ C_12 @ A_28 ) @ ( times_times_nat @ C_12 @ B_25 ) )
= ( div_div_nat @ A_28 @ B_25 ) ) ) ) ).
thf(fact_2347_div__mult__mult1__if,axiom,
! [A_28: int,B_25: int,C_12: int] :
( ( ( C_12 = zero_zero_int )
=> ( ( div_div_int @ ( times_times_int @ C_12 @ A_28 ) @ ( times_times_int @ C_12 @ B_25 ) )
= zero_zero_int ) )
& ( ( C_12 != zero_zero_int )
=> ( ( div_div_int @ ( times_times_int @ C_12 @ A_28 ) @ ( times_times_int @ C_12 @ B_25 ) )
= ( div_div_int @ A_28 @ B_25 ) ) ) ) ).
thf(fact_2348_div__mult__self2__is__id,axiom,
! [A_27: quickcheck_code_int,B_24: quickcheck_code_int] :
( ( B_24 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( times_123202395de_int @ A_27 @ B_24 ) @ B_24 )
= A_27 ) ) ).
thf(fact_2349_div__mult__self2__is__id,axiom,
! [A_27: code_code_numeral,B_24: code_code_numeral] :
( ( B_24 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( times_1655362735umeral @ A_27 @ B_24 ) @ B_24 )
= A_27 ) ) ).
thf(fact_2350_div__mult__self2__is__id,axiom,
! [A_27: nat,B_24: nat] :
( ( B_24 != zero_zero_nat )
=> ( ( div_div_nat @ ( times_times_nat @ A_27 @ B_24 ) @ B_24 )
= A_27 ) ) ).
thf(fact_2351_div__mult__self2__is__id,axiom,
! [A_27: int,B_24: int] :
( ( B_24 != zero_zero_int )
=> ( ( div_div_int @ ( times_times_int @ A_27 @ B_24 ) @ B_24 )
= A_27 ) ) ).
thf(fact_2352_div__mult__self1__is__id,axiom,
! [A_26: quickcheck_code_int,B_23: quickcheck_code_int] :
( ( B_23 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( times_123202395de_int @ B_23 @ A_26 ) @ B_23 )
= A_26 ) ) ).
thf(fact_2353_div__mult__self1__is__id,axiom,
! [A_26: code_code_numeral,B_23: code_code_numeral] :
( ( B_23 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( times_1655362735umeral @ B_23 @ A_26 ) @ B_23 )
= A_26 ) ) ).
thf(fact_2354_div__mult__self1__is__id,axiom,
! [A_26: nat,B_23: nat] :
( ( B_23 != zero_zero_nat )
=> ( ( div_div_nat @ ( times_times_nat @ B_23 @ A_26 ) @ B_23 )
= A_26 ) ) ).
thf(fact_2355_div__mult__self1__is__id,axiom,
! [A_26: int,B_23: int] :
( ( B_23 != zero_zero_int )
=> ( ( div_div_int @ ( times_times_int @ B_23 @ A_26 ) @ B_23 )
= A_26 ) ) ).
thf(fact_2356_div__mult__mult2,axiom,
! [A_25: quickcheck_code_int,B_22: quickcheck_code_int,C_11: quickcheck_code_int] :
( ( C_11 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( times_123202395de_int @ A_25 @ C_11 ) @ ( times_123202395de_int @ B_22 @ C_11 ) )
= ( div_di1430059507de_int @ A_25 @ B_22 ) ) ) ).
thf(fact_2357_div__mult__mult2,axiom,
! [A_25: code_code_numeral,B_22: code_code_numeral,C_11: code_code_numeral] :
( ( C_11 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( times_1655362735umeral @ A_25 @ C_11 ) @ ( times_1655362735umeral @ B_22 @ C_11 ) )
= ( div_di1218280263umeral @ A_25 @ B_22 ) ) ) ).
thf(fact_2358_div__mult__mult2,axiom,
! [A_25: nat,B_22: nat,C_11: nat] :
( ( C_11 != zero_zero_nat )
=> ( ( div_div_nat @ ( times_times_nat @ A_25 @ C_11 ) @ ( times_times_nat @ B_22 @ C_11 ) )
= ( div_div_nat @ A_25 @ B_22 ) ) ) ).
thf(fact_2359_div__mult__mult2,axiom,
! [A_25: int,B_22: int,C_11: int] :
( ( C_11 != zero_zero_int )
=> ( ( div_div_int @ ( times_times_int @ A_25 @ C_11 ) @ ( times_times_int @ B_22 @ C_11 ) )
= ( div_div_int @ A_25 @ B_22 ) ) ) ).
thf(fact_2360_div__mult__mult1,axiom,
! [A_24: quickcheck_code_int,B_21: quickcheck_code_int,C_10: quickcheck_code_int] :
( ( C_10 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( times_123202395de_int @ C_10 @ A_24 ) @ ( times_123202395de_int @ C_10 @ B_21 ) )
= ( div_di1430059507de_int @ A_24 @ B_21 ) ) ) ).
thf(fact_2361_div__mult__mult1,axiom,
! [A_24: code_code_numeral,B_21: code_code_numeral,C_10: code_code_numeral] :
( ( C_10 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( times_1655362735umeral @ C_10 @ A_24 ) @ ( times_1655362735umeral @ C_10 @ B_21 ) )
= ( div_di1218280263umeral @ A_24 @ B_21 ) ) ) ).
thf(fact_2362_div__mult__mult1,axiom,
! [A_24: nat,B_21: nat,C_10: nat] :
( ( C_10 != zero_zero_nat )
=> ( ( div_div_nat @ ( times_times_nat @ C_10 @ A_24 ) @ ( times_times_nat @ C_10 @ B_21 ) )
= ( div_div_nat @ A_24 @ B_21 ) ) ) ).
thf(fact_2363_div__mult__mult1,axiom,
! [A_24: int,B_21: int,C_10: int] :
( ( C_10 != zero_zero_int )
=> ( ( div_div_int @ ( times_times_int @ C_10 @ A_24 ) @ ( times_times_int @ C_10 @ B_21 ) )
= ( div_div_int @ A_24 @ B_21 ) ) ) ).
thf(fact_2364_div__self,axiom,
! [A_23: quickcheck_code_int] :
( ( A_23 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ A_23 @ A_23 )
= one_on1684967323de_int ) ) ).
thf(fact_2365_div__self,axiom,
! [A_23: code_code_numeral] :
( ( A_23 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ A_23 @ A_23 )
= one_on1645066479umeral ) ) ).
thf(fact_2366_div__self,axiom,
! [A_23: nat] :
( ( A_23 != zero_zero_nat )
=> ( ( div_div_nat @ A_23 @ A_23 )
= one_one_nat ) ) ).
thf(fact_2367_div__self,axiom,
! [A_23: int] :
( ( A_23 != zero_zero_int )
=> ( ( div_div_int @ A_23 @ A_23 )
= one_one_int ) ) ).
thf(fact_2368_mod__div__trivial,axiom,
! [A_22: quickcheck_code_int,B_20: quickcheck_code_int] :
( ( div_di1430059507de_int @ ( div_mo231679042de_int @ A_22 @ B_20 ) @ B_20 )
= zero_z891286103de_int ) ).
thf(fact_2369_mod__div__trivial,axiom,
! [A_22: code_code_numeral,B_20: code_code_numeral] :
( ( div_di1218280263umeral @ ( div_mo1740067990umeral @ A_22 @ B_20 ) @ B_20 )
= zero_z126310315umeral ) ).
thf(fact_2370_mod__div__trivial,axiom,
! [A_22: nat,B_20: nat] :
( ( div_div_nat @ ( div_mod_nat @ A_22 @ B_20 ) @ B_20 )
= zero_zero_nat ) ).
thf(fact_2371_mod__div__trivial,axiom,
! [A_22: int,B_20: int] :
( ( div_div_int @ ( div_mod_int @ A_22 @ B_20 ) @ B_20 )
= zero_zero_int ) ).
thf(fact_2372_negDivAlg__div__mod,axiom,
! [L: int,K_1: int] :
( ( ord_less_int @ K_1 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ L )
=> ( ( negDivAlg @ K_1 @ L )
= ( product_Pair_int_int @ ( div_div_int @ K_1 @ L ) @ ( div_mod_int @ K_1 @ L ) ) ) ) ) ).
thf(fact_2373_zdiv__eq__0__iff,axiom,
! [I: int,K_1: int] :
( ( ( div_div_int @ I @ K_1 )
= zero_zero_int )
<=> ( ( K_1 = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K_1 ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K_1 @ I ) ) ) ) ).
thf(fact_2374_pos__imp__zdiv__nonneg__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( div_div_int @ A @ B ) )
<=> ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
thf(fact_2375_pos__imp__zdiv__pos__iff,axiom,
! [I: int,K_1: int] :
( ( ord_less_int @ zero_zero_int @ K_1 )
=> ( ( ord_less_int @ zero_zero_int @ ( div_div_int @ I @ K_1 ) )
<=> ( ord_less_eq_int @ K_1 @ I ) ) ) ).
thf(fact_2376_nonneg1__imp__zdiv__pos__iff,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( div_div_int @ A @ B ) )
<=> ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
thf(fact_2377_zdiv__mono2,axiom,
! [B: int,B_5: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B_5 )
=> ( ( ord_less_eq_int @ B_5 @ B )
=> ( ord_less_eq_int @ ( div_div_int @ A @ B ) @ ( div_div_int @ A @ B_5 ) ) ) ) ) ).
thf(fact_2378_div__nonneg__neg__le0,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( div_div_int @ A @ B ) @ zero_zero_int ) ) ) ).
thf(fact_2379_div__pos__pos__trivial,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ B )
=> ( ( div_div_int @ A @ B )
= zero_zero_int ) ) ) ).
thf(fact_2380_neg__imp__zdiv__nonneg__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( div_div_int @ A @ B ) )
<=> ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
thf(fact_2381_div__nonpos__pos__le0,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( div_div_int @ A @ B ) @ zero_zero_int ) ) ) ).
thf(fact_2382_zdiv__mono2__neg,axiom,
! [B: int,B_5: int,A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B_5 )
=> ( ( ord_less_eq_int @ B_5 @ B )
=> ( ord_less_eq_int @ ( div_div_int @ A @ B_5 ) @ ( div_div_int @ A @ B ) ) ) ) ) ).
thf(fact_2383_div__neg__neg__trivial,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ A )
=> ( ( div_div_int @ A @ B )
= zero_zero_int ) ) ) ).
thf(fact_2384_zdiv__mono1,axiom,
! [B: int,A: int,A_5: int] :
( ( ord_less_eq_int @ A @ A_5 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( div_div_int @ A @ B ) @ ( div_div_int @ A_5 @ B ) ) ) ) ).
thf(fact_2385_zdiv__mono1__neg,axiom,
! [B: int,A: int,A_5: int] :
( ( ord_less_eq_int @ A @ A_5 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( div_div_int @ A_5 @ B ) @ ( div_div_int @ A @ B ) ) ) ) ).
thf(fact_2386_int__div__less__self,axiom,
! [K_1: int,X: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K_1 )
=> ( ord_less_int @ ( div_div_int @ X @ K_1 ) @ X ) ) ) ).
thf(fact_2387_zdiv__zmult2__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( div_div_int @ A @ ( times_times_int @ B @ C ) )
= ( div_div_int @ ( div_div_int @ A @ B ) @ C ) ) ) ).
thf(fact_2388_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_2389_zmod__zdiv__equality,axiom,
! [A: int,B: int] :
( A
= ( plus_plus_int @ ( times_times_int @ B @ ( div_div_int @ A @ B ) ) @ ( div_mod_int @ A @ B ) ) ) ).
thf(fact_2390_zdiv__zmult1__eq,axiom,
! [A: int,B: int,C: int] :
( ( div_div_int @ ( times_times_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ ( div_div_int @ B @ C ) ) @ ( div_div_int @ ( times_times_int @ A @ ( div_mod_int @ B @ C ) ) @ C ) ) ) ).
thf(fact_2391_zdiv__zmod__equality,axiom,
! [B: int,A: int,K_1: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( div_div_int @ A @ B ) ) @ ( div_mod_int @ A @ B ) ) @ K_1 )
= ( plus_plus_int @ A @ K_1 ) ) ).
thf(fact_2392_zdiv__zmod__equality2,axiom,
! [A: int,B: int,K_1: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( div_div_int @ A @ B ) @ B ) @ ( div_mod_int @ A @ B ) ) @ K_1 )
= ( plus_plus_int @ A @ K_1 ) ) ).
thf(fact_2393_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_2394_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_2395_zdiv__minus1__right,axiom,
! [A: int] :
( ( div_div_int @ A @ ( number_number_of_int @ min ) )
= ( uminus_uminus_int @ A ) ) ).
thf(fact_2396_div__mult__self2,axiom,
! [A_21: quickcheck_code_int,C_9: quickcheck_code_int,B_19: quickcheck_code_int] :
( ( B_19 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( plus_p1446045655de_int @ A_21 @ ( times_123202395de_int @ B_19 @ C_9 ) ) @ B_19 )
= ( plus_p1446045655de_int @ C_9 @ ( div_di1430059507de_int @ A_21 @ B_19 ) ) ) ) ).
thf(fact_2397_div__mult__self2,axiom,
! [A_21: code_code_numeral,C_9: code_code_numeral,B_19: code_code_numeral] :
( ( B_19 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( plus_p1627245867umeral @ A_21 @ ( times_1655362735umeral @ B_19 @ C_9 ) ) @ B_19 )
= ( plus_p1627245867umeral @ C_9 @ ( div_di1218280263umeral @ A_21 @ B_19 ) ) ) ) ).
thf(fact_2398_div__mult__self2,axiom,
! [A_21: nat,C_9: nat,B_19: nat] :
( ( B_19 != zero_zero_nat )
=> ( ( div_div_nat @ ( plus_plus_nat @ A_21 @ ( times_times_nat @ B_19 @ C_9 ) ) @ B_19 )
= ( plus_plus_nat @ C_9 @ ( div_div_nat @ A_21 @ B_19 ) ) ) ) ).
thf(fact_2399_div__mult__self2,axiom,
! [A_21: int,C_9: int,B_19: int] :
( ( B_19 != zero_zero_int )
=> ( ( div_div_int @ ( plus_plus_int @ A_21 @ ( times_times_int @ B_19 @ C_9 ) ) @ B_19 )
= ( plus_plus_int @ C_9 @ ( div_div_int @ A_21 @ B_19 ) ) ) ) ).
thf(fact_2400_div__mult__self1,axiom,
! [A_20: quickcheck_code_int,C_8: quickcheck_code_int,B_18: quickcheck_code_int] :
( ( B_18 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( plus_p1446045655de_int @ A_20 @ ( times_123202395de_int @ C_8 @ B_18 ) ) @ B_18 )
= ( plus_p1446045655de_int @ C_8 @ ( div_di1430059507de_int @ A_20 @ B_18 ) ) ) ) ).
thf(fact_2401_div__mult__self1,axiom,
! [A_20: code_code_numeral,C_8: code_code_numeral,B_18: code_code_numeral] :
( ( B_18 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( plus_p1627245867umeral @ A_20 @ ( times_1655362735umeral @ C_8 @ B_18 ) ) @ B_18 )
= ( plus_p1627245867umeral @ C_8 @ ( div_di1218280263umeral @ A_20 @ B_18 ) ) ) ) ).
thf(fact_2402_div__mult__self1,axiom,
! [A_20: nat,C_8: nat,B_18: nat] :
( ( B_18 != zero_zero_nat )
=> ( ( div_div_nat @ ( plus_plus_nat @ A_20 @ ( times_times_nat @ C_8 @ B_18 ) ) @ B_18 )
= ( plus_plus_nat @ C_8 @ ( div_div_nat @ A_20 @ B_18 ) ) ) ) ).
thf(fact_2403_div__mult__self1,axiom,
! [A_20: int,C_8: int,B_18: int] :
( ( B_18 != zero_zero_int )
=> ( ( div_div_int @ ( plus_plus_int @ A_20 @ ( times_times_int @ C_8 @ B_18 ) ) @ B_18 )
= ( plus_plus_int @ C_8 @ ( div_div_int @ A_20 @ B_18 ) ) ) ) ).
thf(fact_2404_divmod__int__rel__div,axiom,
! [A: int,B: int,Q: int,R_1: int] :
( ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( B != zero_zero_int )
=> ( ( div_div_int @ A @ B )
= Q ) ) ) ).
thf(fact_2405_div__add__self2,axiom,
! [A_19: quickcheck_code_int,B_17: quickcheck_code_int] :
( ( B_17 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( plus_p1446045655de_int @ A_19 @ B_17 ) @ B_17 )
= ( plus_p1446045655de_int @ ( div_di1430059507de_int @ A_19 @ B_17 ) @ one_on1684967323de_int ) ) ) ).
thf(fact_2406_div__add__self2,axiom,
! [A_19: code_code_numeral,B_17: code_code_numeral] :
( ( B_17 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( plus_p1627245867umeral @ A_19 @ B_17 ) @ B_17 )
= ( plus_p1627245867umeral @ ( div_di1218280263umeral @ A_19 @ B_17 ) @ one_on1645066479umeral ) ) ) ).
thf(fact_2407_div__add__self2,axiom,
! [A_19: nat,B_17: nat] :
( ( B_17 != zero_zero_nat )
=> ( ( div_div_nat @ ( plus_plus_nat @ A_19 @ B_17 ) @ B_17 )
= ( plus_plus_nat @ ( div_div_nat @ A_19 @ B_17 ) @ one_one_nat ) ) ) ).
thf(fact_2408_div__add__self2,axiom,
! [A_19: int,B_17: int] :
( ( B_17 != zero_zero_int )
=> ( ( div_div_int @ ( plus_plus_int @ A_19 @ B_17 ) @ B_17 )
= ( plus_plus_int @ ( div_div_int @ A_19 @ B_17 ) @ one_one_int ) ) ) ).
thf(fact_2409_div__add__self1,axiom,
! [A_18: quickcheck_code_int,B_16: quickcheck_code_int] :
( ( B_16 != zero_z891286103de_int )
=> ( ( div_di1430059507de_int @ ( plus_p1446045655de_int @ B_16 @ A_18 ) @ B_16 )
= ( plus_p1446045655de_int @ ( div_di1430059507de_int @ A_18 @ B_16 ) @ one_on1684967323de_int ) ) ) ).
thf(fact_2410_div__add__self1,axiom,
! [A_18: code_code_numeral,B_16: code_code_numeral] :
( ( B_16 != zero_z126310315umeral )
=> ( ( div_di1218280263umeral @ ( plus_p1627245867umeral @ B_16 @ A_18 ) @ B_16 )
= ( plus_p1627245867umeral @ ( div_di1218280263umeral @ A_18 @ B_16 ) @ one_on1645066479umeral ) ) ) ).
thf(fact_2411_div__add__self1,axiom,
! [A_18: nat,B_16: nat] :
( ( B_16 != zero_zero_nat )
=> ( ( div_div_nat @ ( plus_plus_nat @ B_16 @ A_18 ) @ B_16 )
= ( plus_plus_nat @ ( div_div_nat @ A_18 @ B_16 ) @ one_one_nat ) ) ) ).
thf(fact_2412_div__add__self1,axiom,
! [A_18: int,B_16: int] :
( ( B_16 != zero_zero_int )
=> ( ( div_div_int @ ( plus_plus_int @ B_16 @ A_18 ) @ B_16 )
= ( plus_plus_int @ ( div_div_int @ A_18 @ B_16 ) @ one_one_int ) ) ) ).
thf(fact_2413_semiring__div__class_Omod__div__equality_H,axiom,
! [A_17: quickcheck_code_int,B_15: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( div_mo231679042de_int @ A_17 @ B_15 ) @ ( times_123202395de_int @ ( div_di1430059507de_int @ A_17 @ B_15 ) @ B_15 ) )
= A_17 ) ).
thf(fact_2414_semiring__div__class_Omod__div__equality_H,axiom,
! [A_17: code_code_numeral,B_15: code_code_numeral] :
( ( plus_p1627245867umeral @ ( div_mo1740067990umeral @ A_17 @ B_15 ) @ ( times_1655362735umeral @ ( div_di1218280263umeral @ A_17 @ B_15 ) @ B_15 ) )
= A_17 ) ).
thf(fact_2415_semiring__div__class_Omod__div__equality_H,axiom,
! [A_17: nat,B_15: nat] :
( ( plus_plus_nat @ ( div_mod_nat @ A_17 @ B_15 ) @ ( times_times_nat @ ( div_div_nat @ A_17 @ B_15 ) @ B_15 ) )
= A_17 ) ).
thf(fact_2416_semiring__div__class_Omod__div__equality_H,axiom,
! [A_17: int,B_15: int] :
( ( plus_plus_int @ ( div_mod_int @ A_17 @ B_15 ) @ ( times_times_int @ ( div_div_int @ A_17 @ B_15 ) @ B_15 ) )
= A_17 ) ).
thf(fact_2417_mod__div__equality2,axiom,
! [B_14: quickcheck_code_int,A_16: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( times_123202395de_int @ B_14 @ ( div_di1430059507de_int @ A_16 @ B_14 ) ) @ ( div_mo231679042de_int @ A_16 @ B_14 ) )
= A_16 ) ).
thf(fact_2418_mod__div__equality2,axiom,
! [B_14: code_code_numeral,A_16: code_code_numeral] :
( ( plus_p1627245867umeral @ ( times_1655362735umeral @ B_14 @ ( div_di1218280263umeral @ A_16 @ B_14 ) ) @ ( div_mo1740067990umeral @ A_16 @ B_14 ) )
= A_16 ) ).
thf(fact_2419_mod__div__equality2,axiom,
! [B_14: nat,A_16: nat] :
( ( plus_plus_nat @ ( times_times_nat @ B_14 @ ( div_div_nat @ A_16 @ B_14 ) ) @ ( div_mod_nat @ A_16 @ B_14 ) )
= A_16 ) ).
thf(fact_2420_mod__div__equality2,axiom,
! [B_14: int,A_16: int] :
( ( plus_plus_int @ ( times_times_int @ B_14 @ ( div_div_int @ A_16 @ B_14 ) ) @ ( div_mod_int @ A_16 @ B_14 ) )
= A_16 ) ).
thf(fact_2421_mod__div__equality,axiom,
! [A_15: quickcheck_code_int,B_13: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( times_123202395de_int @ ( div_di1430059507de_int @ A_15 @ B_13 ) @ B_13 ) @ ( div_mo231679042de_int @ A_15 @ B_13 ) )
= A_15 ) ).
thf(fact_2422_mod__div__equality,axiom,
! [A_15: code_code_numeral,B_13: code_code_numeral] :
( ( plus_p1627245867umeral @ ( times_1655362735umeral @ ( div_di1218280263umeral @ A_15 @ B_13 ) @ B_13 ) @ ( div_mo1740067990umeral @ A_15 @ B_13 ) )
= A_15 ) ).
thf(fact_2423_mod__div__equality,axiom,
! [A_15: nat,B_13: nat] :
( ( plus_plus_nat @ ( times_times_nat @ ( div_div_nat @ A_15 @ B_13 ) @ B_13 ) @ ( div_mod_nat @ A_15 @ B_13 ) )
= A_15 ) ).
thf(fact_2424_mod__div__equality,axiom,
! [A_15: int,B_13: int] :
( ( plus_plus_int @ ( times_times_int @ ( div_div_int @ A_15 @ B_13 ) @ B_13 ) @ ( div_mod_int @ A_15 @ B_13 ) )
= A_15 ) ).
thf(fact_2425_div__mod__equality2,axiom,
! [B_12: quickcheck_code_int,A_14: quickcheck_code_int,C_7: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ ( times_123202395de_int @ B_12 @ ( div_di1430059507de_int @ A_14 @ B_12 ) ) @ ( div_mo231679042de_int @ A_14 @ B_12 ) ) @ C_7 )
= ( plus_p1446045655de_int @ A_14 @ C_7 ) ) ).
thf(fact_2426_div__mod__equality2,axiom,
! [B_12: code_code_numeral,A_14: code_code_numeral,C_7: code_code_numeral] :
( ( plus_p1627245867umeral @ ( plus_p1627245867umeral @ ( times_1655362735umeral @ B_12 @ ( div_di1218280263umeral @ A_14 @ B_12 ) ) @ ( div_mo1740067990umeral @ A_14 @ B_12 ) ) @ C_7 )
= ( plus_p1627245867umeral @ A_14 @ C_7 ) ) ).
thf(fact_2427_div__mod__equality2,axiom,
! [B_12: nat,A_14: nat,C_7: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B_12 @ ( div_div_nat @ A_14 @ B_12 ) ) @ ( div_mod_nat @ A_14 @ B_12 ) ) @ C_7 )
= ( plus_plus_nat @ A_14 @ C_7 ) ) ).
thf(fact_2428_div__mod__equality2,axiom,
! [B_12: int,A_14: int,C_7: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B_12 @ ( div_div_int @ A_14 @ B_12 ) ) @ ( div_mod_int @ A_14 @ B_12 ) ) @ C_7 )
= ( plus_plus_int @ A_14 @ C_7 ) ) ).
thf(fact_2429_div__mod__equality,axiom,
! [A_13: quickcheck_code_int,B_11: quickcheck_code_int,C_6: quickcheck_code_int] :
( ( plus_p1446045655de_int @ ( plus_p1446045655de_int @ ( times_123202395de_int @ ( div_di1430059507de_int @ A_13 @ B_11 ) @ B_11 ) @ ( div_mo231679042de_int @ A_13 @ B_11 ) ) @ C_6 )
= ( plus_p1446045655de_int @ A_13 @ C_6 ) ) ).
thf(fact_2430_div__mod__equality,axiom,
! [A_13: code_code_numeral,B_11: code_code_numeral,C_6: code_code_numeral] :
( ( plus_p1627245867umeral @ ( plus_p1627245867umeral @ ( times_1655362735umeral @ ( div_di1218280263umeral @ A_13 @ B_11 ) @ B_11 ) @ ( div_mo1740067990umeral @ A_13 @ B_11 ) ) @ C_6 )
= ( plus_p1627245867umeral @ A_13 @ C_6 ) ) ).
thf(fact_2431_div__mod__equality,axiom,
! [A_13: nat,B_11: nat,C_6: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( div_div_nat @ A_13 @ B_11 ) @ B_11 ) @ ( div_mod_nat @ A_13 @ B_11 ) ) @ C_6 )
= ( plus_plus_nat @ A_13 @ C_6 ) ) ).
thf(fact_2432_div__mod__equality,axiom,
! [A_13: int,B_11: int,C_6: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( div_div_int @ A_13 @ B_11 ) @ B_11 ) @ ( div_mod_int @ A_13 @ B_11 ) ) @ C_6 )
= ( plus_plus_int @ A_13 @ C_6 ) ) ).
thf(fact_2433_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_2434_div__eq__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( div_div_int @ ( number_number_of_int @ min ) @ B )
= ( number_number_of_int @ min ) ) ) ).
thf(fact_2435_zcong__zmod__aux,axiom,
! [A: int,B: int,M: int] :
( ( minus_minus_int @ A @ B )
= ( plus_plus_int @ ( times_times_int @ M @ ( minus_minus_int @ ( div_div_int @ A @ M ) @ ( div_div_int @ B @ M ) ) ) @ ( minus_minus_int @ ( div_mod_int @ A @ M ) @ ( div_mod_int @ B @ M ) ) ) ) ).
thf(fact_2436_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_2437_divmod__int__rel__div__mod,axiom,
! [A: int,B: int] :
( ( B != zero_zero_int )
=> ( divmod_int_rel @ A @ B @ ( product_Pair_int_int @ ( div_div_int @ A @ B ) @ ( div_mod_int @ A @ B ) ) ) ) ).
thf(fact_2438_negDivAlg__correct,axiom,
! [B: int,A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( divmod_int_rel @ A @ B @ ( negDivAlg @ A @ B ) ) ) ) ).
thf(fact_2439_split__zdiv,axiom,
! [P: int > $o,N: int,K_1: int] :
( ( P @ ( div_div_int @ N @ K_1 ) )
<=> ( ( ( K_1 = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K_1 )
=> ! [I_1: int,J_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J_1 )
& ( ord_less_int @ J_1 @ K_1 )
& ( N
= ( plus_plus_int @ ( times_times_int @ K_1 @ I_1 ) @ J_1 ) ) )
=> ( P @ I_1 ) ) )
& ( ( ord_less_int @ K_1 @ zero_zero_int )
=> ! [I_1: int,J_1: int] :
( ( ( ord_less_int @ K_1 @ J_1 )
& ( ord_less_eq_int @ J_1 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K_1 @ I_1 ) @ J_1 ) ) )
=> ( P @ I_1 ) ) ) ) ) ).
thf(fact_2440_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_2441_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_2442_zmod__zmult2__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( div_mod_int @ A @ ( times_times_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ B @ ( div_mod_int @ ( div_div_int @ A @ B ) @ C ) ) @ ( div_mod_int @ A @ B ) ) ) ) ).
thf(fact_2443_zdiv__zminus2__eq__if,axiom,
! [A: int,B: int] :
( ( B != zero_zero_int )
=> ( ( ( ( div_mod_int @ A @ B )
= zero_zero_int )
=> ( ( div_div_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( div_div_int @ A @ B ) ) ) )
& ( ( ( div_mod_int @ A @ B )
!= zero_zero_int )
=> ( ( div_div_int @ A @ ( uminus_uminus_int @ B ) )
= ( minus_minus_int @ ( uminus_uminus_int @ ( div_div_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
thf(fact_2444_zdiv__zminus1__eq__if,axiom,
! [A: int,B: int] :
( ( B != zero_zero_int )
=> ( ( ( ( div_mod_int @ A @ B )
= zero_zero_int )
=> ( ( div_div_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( div_div_int @ A @ B ) ) ) )
& ( ( ( div_mod_int @ A @ B )
!= zero_zero_int )
=> ( ( div_div_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ ( uminus_uminus_int @ ( div_div_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
thf(fact_2445_zadd1__lemma,axiom,
! [B: int,Bq: int,Br: int,A: int,C: int,Aq: int,Ar: int] :
( ( divmod_int_rel @ A @ C @ ( product_Pair_int_int @ Aq @ Ar ) )
=> ( ( divmod_int_rel @ B @ C @ ( product_Pair_int_int @ Bq @ Br ) )
=> ( ( C != zero_zero_int )
=> ( divmod_int_rel @ ( plus_plus_int @ A @ B ) @ C @ ( product_Pair_int_int @ ( plus_plus_int @ ( plus_plus_int @ Aq @ Bq ) @ ( div_div_int @ ( plus_plus_int @ Ar @ Br ) @ C ) ) @ ( div_mod_int @ ( plus_plus_int @ Ar @ Br ) @ C ) ) ) ) ) ) ).
thf(fact_2446_posDivAlg__div__mod,axiom,
! [L: int,K_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ( ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( posDivAlg @ K_1 @ L )
= ( product_Pair_int_int @ ( div_div_int @ K_1 @ L ) @ ( div_mod_int @ K_1 @ L ) ) ) ) ) ).
thf(fact_2447_div__pos__neg__trivial,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int )
=> ( ( div_div_int @ A @ B )
= ( number_number_of_int @ min ) ) ) ) ).
thf(fact_2448_split__pos__lemma,axiom,
! [P: int > int > $o,N: int,K_1: int] :
( ( ord_less_int @ zero_zero_int @ K_1 )
=> ( ( P @ ( div_div_int @ N @ K_1 ) @ ( div_mod_int @ N @ K_1 ) )
<=> ! [I_1: int,J_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J_1 )
& ( ord_less_int @ J_1 @ K_1 )
& ( N
= ( plus_plus_int @ ( times_times_int @ K_1 @ I_1 ) @ J_1 ) ) )
=> ( P @ I_1 @ J_1 ) ) ) ) ).
thf(fact_2449_split__neg__lemma,axiom,
! [P: int > int > $o,N: int,K_1: int] :
( ( ord_less_int @ K_1 @ zero_zero_int )
=> ( ( P @ ( div_div_int @ N @ K_1 ) @ ( div_mod_int @ N @ K_1 ) )
<=> ! [I_1: int,J_1: int] :
( ( ( ord_less_int @ K_1 @ J_1 )
& ( ord_less_eq_int @ J_1 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K_1 @ I_1 ) @ J_1 ) ) )
=> ( P @ I_1 @ J_1 ) ) ) ) ).
thf(fact_2450_zmult1__lemma,axiom,
! [A: int,B: int,C: int,Q: int,R_1: int] :
( ( divmod_int_rel @ B @ C @ ( product_Pair_int_int @ Q @ R_1 ) )
=> ( ( C != zero_zero_int )
=> ( divmod_int_rel @ ( times_times_int @ A @ B ) @ C @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A @ Q ) @ ( div_div_int @ ( times_times_int @ A @ R_1 ) @ C ) ) @ ( div_mod_int @ ( times_times_int @ A @ R_1 ) @ C ) ) ) ) ) ).
thf(fact_2451_z3div__def,axiom,
! [K_1: int,L: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( z3div @ K_1 @ L )
= ( div_div_int @ K_1 @ L ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ L )
=> ( ( z3div @ K_1 @ L )
= ( uminus_uminus_int @ ( div_div_int @ K_1 @ ( uminus_uminus_int @ L ) ) ) ) ) ) ).
thf(fact_2452_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_2453_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_2454_mod__div__decomp,axiom,
! [A_12: quickcheck_code_int,B_10: quickcheck_code_int] :
( A_12
= ( plus_p1446045655de_int @ ( times_123202395de_int @ ( div_di1430059507de_int @ A_12 @ B_10 ) @ B_10 ) @ ( div_mo231679042de_int @ A_12 @ B_10 ) ) ) ).
thf(fact_2455_mod__div__decomp,axiom,
! [A_12: code_code_numeral,B_10: code_code_numeral] :
( A_12
= ( plus_p1627245867umeral @ ( times_1655362735umeral @ ( div_di1218280263umeral @ A_12 @ B_10 ) @ B_10 ) @ ( div_mo1740067990umeral @ A_12 @ B_10 ) ) ) ).
thf(fact_2456_mod__div__decomp,axiom,
! [A_12: nat,B_10: nat] :
( A_12
= ( plus_plus_nat @ ( times_times_nat @ ( div_div_nat @ A_12 @ B_10 ) @ B_10 ) @ ( div_mod_nat @ A_12 @ B_10 ) ) ) ).
thf(fact_2457_mod__div__decomp,axiom,
! [A_12: int,B_10: int] :
( A_12
= ( plus_plus_int @ ( times_times_int @ ( div_div_int @ A_12 @ B_10 ) @ B_10 ) @ ( div_mod_int @ A_12 @ B_10 ) ) ) ).
thf(fact_2458_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_2459_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_2460_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_2461_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_2462_mult__le__cancel1,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_2463_nat__mult__div__cancel__disj,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( ( K_1 = zero_zero_nat )
=> ( ( div_div_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
= zero_zero_nat ) )
& ( ( K_1 != zero_zero_nat )
=> ( ( div_div_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
= ( div_div_nat @ M @ N ) ) ) ) ).
thf(fact_2464_div__mult2__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( div_div_nat @ A @ ( times_times_nat @ B @ C ) )
= ( div_div_nat @ ( div_div_nat @ A @ B ) @ C ) ) ).
thf(fact_2465_mod__mult__distrib2,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( times_times_nat @ K_1 @ ( div_mod_nat @ M @ N ) )
= ( div_mod_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) ) ) ).
thf(fact_2466_mod__mult__distrib,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( times_times_nat @ ( div_mod_nat @ M @ N ) @ K_1 )
= ( div_mod_nat @ ( times_times_nat @ M @ K_1 ) @ ( times_times_nat @ N @ K_1 ) ) ) ).
thf(fact_2467_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_2468_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_2469_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_2470_diff__mult__distrib2,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( times_times_nat @ K_1 @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) ) ) ).
thf(fact_2471_diff__mult__distrib,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K_1 )
= ( minus_minus_nat @ ( times_times_nat @ M @ K_1 ) @ ( times_times_nat @ N @ K_1 ) ) ) ).
thf(fact_2472_diff__commute,axiom,
! [I: nat,J: nat,K_1: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K_1 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K_1 ) @ J ) ) ).
thf(fact_2473_nat__mult__commute,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ N )
= ( times_times_nat @ N @ M ) ) ).
thf(fact_2474_nat__mult__assoc,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( times_times_nat @ ( times_times_nat @ M @ N ) @ K_1 )
= ( times_times_nat @ M @ ( times_times_nat @ N @ K_1 ) ) ) ).
thf(fact_2475_div__add1__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( div_div_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( plus_plus_nat @ ( div_div_nat @ A @ C ) @ ( div_div_nat @ B @ C ) ) @ ( div_div_nat @ ( plus_plus_nat @ ( div_mod_nat @ A @ C ) @ ( div_mod_nat @ B @ C ) ) @ C ) ) ) ).
thf(fact_2476_div__mult1__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( div_div_nat @ ( times_times_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( div_div_nat @ B @ C ) ) @ ( div_div_nat @ ( times_times_nat @ A @ ( div_mod_nat @ B @ C ) ) @ C ) ) ) ).
thf(fact_2477_mod__mult2__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( div_mod_nat @ A @ ( times_times_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ B @ ( div_mod_nat @ ( div_div_nat @ A @ B ) @ C ) ) @ ( div_mod_nat @ A @ B ) ) ) ).
thf(fact_2478_div__le__mono,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( div_div_nat @ M @ K_1 ) @ ( div_div_nat @ N @ K_1 ) ) ) ).
thf(fact_2479_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( div_div_nat @ M @ N ) @ M ) ).
thf(fact_2480_nat__mult__div__cancel1,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ( div_div_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
= ( div_div_nat @ M @ N ) ) ) ).
thf(fact_2481_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_2482_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_2483_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( div_div_nat @ M @ N )
= zero_zero_nat ) ) ).
thf(fact_2484_split__div,axiom,
! [P: nat > $o,N: nat,K_1: nat] :
( ( P @ ( div_div_nat @ N @ K_1 ) )
<=> ( ( ( K_1 = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( K_1 != zero_zero_nat )
=> ! [I_1: nat,J_1: nat] :
( ( ord_less_nat @ J_1 @ K_1 )
=> ( ( N
= ( plus_plus_nat @ ( times_times_nat @ K_1 @ I_1 ) @ J_1 ) )
=> ( P @ I_1 ) ) ) ) ) ) ).
thf(fact_2485_div__le__mono2,axiom,
! [K_1: nat,N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( div_div_nat @ K_1 @ N ) @ ( div_div_nat @ K_1 @ M ) ) ) ) ).
thf(fact_2486_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_2487_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_2488_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_2489_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_2490_nat__mult__eq__cancel__disj,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K_1 @ M )
= ( times_times_nat @ K_1 @ N ) )
<=> ( ( K_1 = zero_zero_nat )
| ( M = N ) ) ) ).
thf(fact_2491_mult__cancel1,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K_1 @ M )
= ( times_times_nat @ K_1 @ N ) )
<=> ( ( M = N )
| ( K_1 = zero_zero_nat ) ) ) ).
thf(fact_2492_mult__cancel2,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ( times_times_nat @ M @ K_1 )
= ( times_times_nat @ N @ K_1 ) )
<=> ( ( M = N )
| ( K_1 = zero_zero_nat ) ) ) ).
thf(fact_2493_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_2494_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
thf(fact_2495_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_2496_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_2497_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_2498_less__not__refl3,axiom,
! [S_1: nat,T: nat] :
( ( ord_less_nat @ S_1 @ T )
=> ( S_1 != T ) ) ).
thf(fact_2499_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
thf(fact_2500_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_2501_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
thf(fact_2502_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
<=> ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
thf(fact_2503_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
thf(fact_2504_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_2505_less__imp__diff__less,axiom,
! [N: nat,J: nat,K_1: nat] :
( ( ord_less_nat @ J @ K_1 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K_1 ) ) ).
thf(fact_2506_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
thf(fact_2507_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
thf(fact_2508_diff__diff__left,axiom,
! [I: nat,J: nat,K_1: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K_1 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K_1 ) ) ) ).
thf(fact_2509_Nat_Odiff__cancel,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K_1 @ M ) @ ( plus_plus_nat @ K_1 @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_2510_diff__cancel2,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K_1 ) @ ( plus_plus_nat @ N @ K_1 ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_2511_nat__add__right__cancel,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ( plus_plus_nat @ M @ K_1 )
= ( plus_plus_nat @ N @ K_1 ) )
<=> ( M = N ) ) ).
thf(fact_2512_nat__add__left__cancel,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ( plus_plus_nat @ K_1 @ M )
= ( plus_plus_nat @ K_1 @ N ) )
<=> ( M = N ) ) ).
thf(fact_2513_nat__add__assoc,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ M @ N ) @ K_1 )
= ( plus_plus_nat @ M @ ( plus_plus_nat @ N @ K_1 ) ) ) ).
thf(fact_2514_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_2515_nat__add__commute,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ N )
= ( plus_plus_nat @ N @ M ) ) ).
thf(fact_2516_add__mult__distrib2,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( times_times_nat @ K_1 @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) ) ) ).
thf(fact_2517_add__mult__distrib,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K_1 )
= ( plus_plus_nat @ ( times_times_nat @ M @ K_1 ) @ ( times_times_nat @ N @ K_1 ) ) ) ).
thf(fact_2518_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K_1: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K_1 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K_1 ) ) ).
thf(fact_2519_Nat_Odiff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
thf(fact_2520_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_2521_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_2522_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_2523_eq__diff__iff,axiom,
! [N: nat,K_1: nat,M: nat] :
( ( ord_less_eq_nat @ K_1 @ M )
=> ( ( ord_less_eq_nat @ K_1 @ N )
=> ( ( ( minus_minus_nat @ M @ K_1 )
= ( minus_minus_nat @ N @ K_1 ) )
<=> ( M = N ) ) ) ) ).
thf(fact_2524_Nat_Odiff__diff__eq,axiom,
! [N: nat,K_1: nat,M: nat] :
( ( ord_less_eq_nat @ K_1 @ M )
=> ( ( ord_less_eq_nat @ K_1 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K_1 ) @ ( minus_minus_nat @ N @ K_1 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_2525_le__diff__iff,axiom,
! [N: nat,K_1: nat,M: nat] :
( ( ord_less_eq_nat @ K_1 @ M )
=> ( ( ord_less_eq_nat @ K_1 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K_1 ) @ ( minus_minus_nat @ N @ K_1 ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ) ) ).
thf(fact_2526_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
thf(fact_2527_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_2528_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_2529_le__trans,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K_1 )
=> ( ord_less_eq_nat @ I @ K_1 ) ) ) ).
thf(fact_2530_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
thf(fact_2531_mult__le__mono,axiom,
! [K_1: nat,L: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K_1 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K_1 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
thf(fact_2532_mult__le__mono2,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K_1 @ I ) @ ( times_times_nat @ K_1 @ J ) ) ) ).
thf(fact_2533_mult__le__mono1,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K_1 ) @ ( times_times_nat @ J @ K_1 ) ) ) ).
thf(fact_2534_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
thf(fact_2535_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
thf(fact_2536_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
thf(fact_2537_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_2538_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
thf(fact_2539_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_2540_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_2541_mult__less__cancel1,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
& ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_2542_mult__less__cancel2,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K_1 ) @ ( times_times_nat @ N @ K_1 ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
& ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_2543_nat__mult__eq__cancel1,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ( ( times_times_nat @ K_1 @ M )
= ( times_times_nat @ K_1 @ N ) )
<=> ( M = N ) ) ) ).
thf(fact_2544_nat__mult__less__cancel1,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ( ord_less_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_2545_mult__less__mono1,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K_1 ) @ ( times_times_nat @ J @ K_1 ) ) ) ) ).
thf(fact_2546_mult__less__mono2,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ord_less_nat @ ( times_times_nat @ K_1 @ I ) @ ( times_times_nat @ K_1 @ J ) ) ) ) ).
thf(fact_2547_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_2548_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
thf(fact_2549_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_2550_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
<=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_2551_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
thf(fact_2552_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_2553_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_2554_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
thf(fact_2555_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
thf(fact_2556_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_2557_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
thf(fact_2558_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
thf(fact_2559_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_2560_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_2561_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
thf(fact_2562_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_2563_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
thf(fact_2564_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
thf(fact_2565_nat__add__left__cancel__less,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K_1 @ M ) @ ( plus_plus_nat @ K_1 @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_2566_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_2567_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_2568_add__less__mono1,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K_1 ) @ ( plus_plus_nat @ J @ K_1 ) ) ) ).
thf(fact_2569_add__less__mono,axiom,
! [K_1: nat,L: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K_1 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K_1 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
thf(fact_2570_less__add__eq__less,axiom,
! [M: nat,N: nat,K_1: nat,L: nat] :
( ( ord_less_nat @ K_1 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K_1 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_2571_add__lessD1,axiom,
! [I: nat,J: nat,K_1: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K_1 )
=> ( ord_less_nat @ I @ K_1 ) ) ).
thf(fact_2572_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_2573_less__diff__conv,axiom,
! [I: nat,J: nat,K_1: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K_1 ) )
<=> ( ord_less_nat @ ( plus_plus_nat @ I @ K_1 ) @ J ) ) ).
thf(fact_2574_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_2575_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_2576_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
thf(fact_2577_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_2578_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_2579_nat__less__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
<=> ( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ).
thf(fact_2580_less__diff__iff,axiom,
! [N: nat,K_1: nat,M: nat] :
( ( ord_less_eq_nat @ K_1 @ M )
=> ( ( ord_less_eq_nat @ K_1 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K_1 ) @ ( minus_minus_nat @ N @ K_1 ) )
<=> ( ord_less_nat @ M @ N ) ) ) ) ).
thf(fact_2581_diff__less__mono,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
thf(fact_2582_diff__diff__right,axiom,
! [I: nat,K_1: nat,J: nat] :
( ( ord_less_eq_nat @ K_1 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K_1 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K_1 ) @ J ) ) ) ).
thf(fact_2583_le__diff__conv,axiom,
! [J: nat,K_1: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K_1 ) @ I )
<=> ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K_1 ) ) ) ).
thf(fact_2584_le__add__diff,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ord_less_eq_nat @ K_1 @ N )
=> ( ord_less_eq_nat @ M @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K_1 ) ) ) ).
thf(fact_2585_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_2586_add__diff__assoc,axiom,
! [I: nat,K_1: nat,J: nat] :
( ( ord_less_eq_nat @ K_1 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K_1 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K_1 ) ) ) ).
thf(fact_2587_le__diff__conv2,axiom,
! [I: nat,K_1: nat,J: nat] :
( ( ord_less_eq_nat @ K_1 @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K_1 ) )
<=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K_1 ) @ J ) ) ) ).
thf(fact_2588_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_2589_le__imp__diff__is__add,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K_1 )
<=> ( J
= ( plus_plus_nat @ K_1 @ I ) ) ) ) ).
thf(fact_2590_diff__add__assoc,axiom,
! [I: nat,K_1: nat,J: nat] :
( ( ord_less_eq_nat @ K_1 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K_1 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K_1 ) ) ) ) ).
thf(fact_2591_add__diff__assoc2,axiom,
! [I: nat,K_1: nat,J: nat] :
( ( ord_less_eq_nat @ K_1 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K_1 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K_1 ) ) ) ).
thf(fact_2592_diff__add__assoc2,axiom,
! [I: nat,K_1: nat,J: nat] :
( ( ord_less_eq_nat @ K_1 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K_1 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K_1 ) @ I ) ) ) ).
thf(fact_2593_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_2594_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_2595_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_2596_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_2597_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_2598_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_2599_add__leE,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K_1 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K_1 @ N ) ) ) ).
thf(fact_2600_add__leD1,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K_1 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_2601_add__leD2,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K_1 ) @ N )
=> ( ord_less_eq_nat @ K_1 @ N ) ) ).
thf(fact_2602_add__le__mono,axiom,
! [K_1: nat,L: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K_1 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K_1 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
thf(fact_2603_add__le__mono1,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K_1 ) @ ( plus_plus_nat @ J @ K_1 ) ) ) ).
thf(fact_2604_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_2605_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_2606_nat__add__left__cancel__le,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K_1 @ M ) @ ( plus_plus_nat @ K_1 @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_2607_le__iff__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
<=> ? [K: nat] :
( N
= ( plus_plus_nat @ M @ K ) ) ) ).
thf(fact_2608_le__Suc__ex__iff,axiom,
! [K_1: nat,L: nat] :
( ( ord_less_eq_nat @ K_1 @ L )
<=> ? [N_1: nat] :
( L
= ( plus_plus_nat @ K_1 @ N_1 ) ) ) ).
thf(fact_2609_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
thf(fact_2610_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
thf(fact_2611_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_2612_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_2613_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
<=> ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D_2: nat] :
( ( A
= ( plus_plus_nat @ B @ D_2 ) )
=> ( P @ D_2 ) ) ) ) ).
thf(fact_2614_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
<=> ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D_2: nat] :
( ( A
= ( plus_plus_nat @ B @ D_2 ) )
& ~ ( P @ D_2 ) ) ) ) ).
thf(fact_2615_nat__mult__le__cancel1,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_2616_mult__le__cancel2,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K_1 ) @ ( times_times_nat @ N @ K_1 ) )
<=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
thf(fact_2617_ex__least__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ( P @ N )
=> ? [K: nat] :
( ( ord_less_nat @ K @ N )
& ! [I_1: nat] :
( ( ord_less_eq_nat @ I_1 @ K )
=> ~ ( P @ I_1 ) )
& ( P @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ).
thf(fact_2618_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_2619_mod__eqD,axiom,
! [M: nat,D: nat,R_1: nat] :
( ( ( div_mod_nat @ M @ D )
= R_1 )
=> ? [Q_2: nat] :
( M
= ( plus_plus_nat @ R_1 @ ( times_times_nat @ Q_2 @ D ) ) ) ) ).
thf(fact_2620_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_2621_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_2622_mono__nat__linear__lb,axiom,
! [M: nat,K_1: nat,F: nat > nat] :
( ! [M_2: nat,N_1: nat] :
( ( ord_less_nat @ M_2 @ N_1 )
=> ( ord_less_nat @ ( F @ M_2 ) @ ( F @ N_1 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K_1 ) @ ( F @ ( plus_plus_nat @ M @ K_1 ) ) ) ) ).
thf(fact_2623_ex__least__nat__le,axiom,
! [N: nat,P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ( P @ N )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N )
& ! [I_1: nat] :
( ( ord_less_nat @ I_1 @ K )
=> ~ ( P @ I_1 ) )
& ( P @ K ) ) ) ) ).
thf(fact_2624_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
& ( ( plus_plus_nat @ I @ K )
= J ) ) ) ).
thf(fact_2625_mult__left__cancel,axiom,
! [N: nat,M: nat,K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ( ( times_times_nat @ K_1 @ N )
= ( times_times_nat @ K_1 @ M ) )
=> ( N = M ) ) ) ).
thf(fact_2626_divmod__nat__rel__mult1__eq,axiom,
! [A: nat,B: nat,C: nat,Q: nat,R_1: nat] :
( ( divmod_nat_rel @ B @ C @ ( product_Pair_nat_nat @ Q @ R_1 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( divmod_nat_rel @ ( times_times_nat @ A @ B ) @ C @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ Q ) @ ( div_div_nat @ ( times_times_nat @ A @ R_1 ) @ C ) ) @ ( div_mod_nat @ ( times_times_nat @ A @ R_1 ) @ C ) ) ) ) ) ).
thf(fact_2627_divmod__nat__rel__mult2__eq,axiom,
! [C: nat,A: nat,B: nat,Q: nat,R_1: nat] :
( ( divmod_nat_rel @ A @ B @ ( product_Pair_nat_nat @ Q @ R_1 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( divmod_nat_rel @ A @ ( times_times_nat @ B @ C ) @ ( product_Pair_nat_nat @ ( div_div_nat @ Q @ C ) @ ( plus_plus_nat @ ( times_times_nat @ B @ ( div_mod_nat @ Q @ C ) ) @ R_1 ) ) ) ) ) ) ).
thf(fact_2628_lemma__interval,axiom,
! [B: real,A: real,X: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D_2: real] :
( ( ord_less_real @ zero_zero_real @ D_2 )
& ! [Y_1: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y_1 ) ) @ D_2 )
=> ( ( ord_less_eq_real @ A @ Y_1 )
& ( ord_less_eq_real @ Y_1 @ B ) ) ) ) ) ) ).
thf(fact_2629_le__Suc__ex,axiom,
! [K_1: nat,L: nat] :
( ( ord_less_eq_nat @ K_1 @ L )
=> ? [N_1: nat] :
( L
= ( plus_plus_nat @ K_1 @ N_1 ) ) ) ).
thf(fact_2630_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_2631_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_2632_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_2633_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_2634_natceiling__number__of__eq,axiom,
! [N: int] :
( ( natceiling @ ( number267125858f_real @ N ) )
= ( number_number_of_nat @ N ) ) ).
thf(fact_2635_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_2636_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_2637_natceiling__zero,axiom,
( ( natceiling @ zero_zero_real )
= zero_zero_nat ) ).
thf(fact_2638_zero__le__natceiling,axiom,
! [X: real] : ( ord_less_eq_nat @ zero_zero_nat @ ( natceiling @ X ) ) ).
thf(fact_2639_natceiling__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( natceiling @ X ) @ ( natceiling @ Y ) ) ) ).
thf(fact_2640_natceiling__one,axiom,
( ( natceiling @ one_one_real )
= one_one_nat ) ).
thf(fact_2641_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_2642_natceiling__neg,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( natceiling @ X )
= zero_zero_nat ) ) ).
thf(fact_2643_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_2644_divmod__nat__rel__add1__eq,axiom,
! [B: nat,Bq: nat,Br: nat,A: nat,C: nat,Aq: nat,Ar: nat] :
( ( divmod_nat_rel @ A @ C @ ( product_Pair_nat_nat @ Aq @ Ar ) )
=> ( ( divmod_nat_rel @ B @ C @ ( product_Pair_nat_nat @ Bq @ Br ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( divmod_nat_rel @ ( plus_plus_nat @ A @ B ) @ C @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( plus_plus_nat @ Aq @ Bq ) @ ( div_div_nat @ ( plus_plus_nat @ Ar @ Br ) @ C ) ) @ ( div_mod_nat @ ( plus_plus_nat @ Ar @ Br ) @ C ) ) ) ) ) ) ).
thf(fact_2645_lemma__interval__lt,axiom,
! [B: real,A: real,X: real] :
( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ? [D_2: real] :
( ( ord_less_real @ zero_zero_real @ D_2 )
& ! [Y_1: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y_1 ) ) @ D_2 )
=> ( ( ord_less_real @ A @ Y_1 )
& ( ord_less_real @ Y_1 @ B ) ) ) ) ) ) ).
thf(fact_2646_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_2647_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_2648_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_2649_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_2650_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_2651_divmod__int__def,axiom,
! [B: int,A: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( divmod_int @ A @ B )
= ( posDivAlg @ A @ B ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( A = zero_zero_int )
=> ( ( divmod_int @ A @ B )
= ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) )
& ( ( A != zero_zero_int )
=> ( ( divmod_int @ A @ B )
= ( negateSnd @ ( negDivAlg @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( divmod_int @ A @ B )
= ( negDivAlg @ A @ B ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ B )
=> ( ( divmod_int @ A @ B )
= ( negateSnd @ ( posDivAlg @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ) ) ) ) ) ).
thf(fact_2652_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_2653_real__of__nat__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( real_nat @ N ) ) ).
thf(fact_2654_natfloor__real__of__nat,axiom,
! [N: nat] :
( ( natfloor @ ( real_nat @ N ) )
= N ) ).
thf(fact_2655_Ints__real__of__nat,axiom,
! [N: nat] : ( member_real @ ( real_nat @ N ) @ ring_1_Ints_real ) ).
thf(fact_2656_natfloor__number__of__eq,axiom,
! [N: int] :
( ( natfloor @ ( number267125858f_real @ N ) )
= ( number_number_of_nat @ N ) ) ).
thf(fact_2657_real__of__nat__inject,axiom,
! [N: nat,M: nat] :
( ( ( real_nat @ N )
= ( real_nat @ M ) )
<=> ( N = M ) ) ).
thf(fact_2658_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_2659_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_2660_le__natfloor,axiom,
! [X: nat,A: real] :
( ( ord_less_eq_real @ ( real_nat @ X ) @ A )
=> ( ord_less_eq_nat @ X @ ( natfloor @ A ) ) ) ).
thf(fact_2661_abs__real__of__nat__cancel,axiom,
! [X: nat] :
( ( abs_abs_real @ ( real_nat @ X ) )
= ( real_nat @ X ) ) ).
thf(fact_2662_not__neg__0,axiom,
~ ( nat_neg @ zero_zero_int ) ).
thf(fact_2663_not__neg__1,axiom,
~ ( nat_neg @ one_one_int ) ).
thf(fact_2664_natceiling__real__of__nat,axiom,
! [N: nat] :
( ( natceiling @ ( real_nat @ N ) )
= N ) ).
thf(fact_2665_le__natfloor__eq,axiom,
! [A: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_nat @ A @ ( natfloor @ X ) )
<=> ( ord_less_eq_real @ ( real_nat @ A ) @ X ) ) ) ).
thf(fact_2666_real__natfloor__add__one__gt,axiom,
! [X: real] : ( ord_less_real @ X @ ( plus_plus_real @ ( real_nat @ ( natfloor @ X ) ) @ one_one_real ) ) ).
thf(fact_2667_natfloor__subtract,axiom,
! [A: nat,X: real] :
( ( ord_less_eq_real @ ( real_nat @ A ) @ X )
=> ( ( natfloor @ ( minus_minus_real @ X @ ( real_nat @ A ) ) )
= ( minus_minus_nat @ ( natfloor @ X ) @ A ) ) ) ).
thf(fact_2668_real__natfloor__gt__diff__one,axiom,
! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ one_one_real ) @ ( real_nat @ ( natfloor @ X ) ) ) ).
thf(fact_2669_real__of__nat__zero,axiom,
( ( real_nat @ zero_zero_nat )
= zero_zero_real ) ).
thf(fact_2670_real__of__nat__zero__iff,axiom,
! [N: nat] :
( ( ( real_nat @ N )
= zero_zero_real )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_2671_not__real__of__nat__less__zero,axiom,
! [N: nat] :
~ ( ord_less_real @ ( real_nat @ N ) @ zero_zero_real ) ).
thf(fact_2672_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_2673_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_2674_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_2675_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_2676_natfloor__add,axiom,
! [A: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( natfloor @ ( plus_plus_real @ X @ ( real_nat @ A ) ) )
= ( plus_plus_nat @ ( natfloor @ X ) @ A ) ) ) ).
thf(fact_2677_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_2678_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_2679_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_2680_real__of__nat__1,axiom,
( ( real_nat @ one_one_nat )
= one_one_real ) ).
thf(fact_2681_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_2682_real__natceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( real_nat @ ( natceiling @ X ) ) ) ).
thf(fact_2683_neg__def,axiom,
! [Z_5: int] :
( ( nat_neg @ Z_5 )
<=> ( ord_less_int @ Z_5 @ zero_zero_int ) ) ).
thf(fact_2684_not__neg__eq__ge__0,axiom,
! [X: int] :
( ~ ( nat_neg @ X )
<=> ( ord_less_eq_int @ zero_zero_int @ X ) ) ).
thf(fact_2685_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_2686_neg__number__of__Min,axiom,
nat_neg @ ( number_number_of_int @ min ) ).
thf(fact_2687_natfloor__zero,axiom,
( ( natfloor @ zero_zero_real )
= zero_zero_nat ) ).
thf(fact_2688_zero__le__natfloor,axiom,
! [X: real] : ( ord_less_eq_nat @ zero_zero_nat @ ( natfloor @ X ) ) ).
thf(fact_2689_natfloor__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_nat @ ( natfloor @ X ) @ ( natfloor @ Y ) ) ) ).
thf(fact_2690_natfloor__one,axiom,
( ( natfloor @ one_one_real )
= one_one_nat ) ).
thf(fact_2691_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_2692_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_2693_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_2694_natceiling__le,axiom,
! [X: real,A: nat] :
( ( ord_less_eq_real @ X @ ( real_nat @ A ) )
=> ( ord_less_eq_nat @ ( natceiling @ X ) @ A ) ) ).
thf(fact_2695_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_2696_nat__number__of__add__left,axiom,
! [V_1: int,K_1: nat,V: int] :
( ( ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K_1 ) )
= ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K_1 ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V ) )
=> ( ( ( nat_neg @ ( number_number_of_int @ V_1 ) )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K_1 ) )
= ( plus_plus_nat @ ( number_number_of_nat @ V ) @ K_1 ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ V_1 ) )
=> ( ( plus_plus_nat @ ( number_number_of_nat @ V ) @ ( plus_plus_nat @ ( number_number_of_nat @ V_1 ) @ K_1 ) )
= ( plus_plus_nat @ ( number_number_of_nat @ ( plus_plus_int @ V @ V_1 ) ) @ K_1 ) ) ) ) ) ) ).
thf(fact_2697_natfloor__neg,axiom,
! [X: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( natfloor @ X )
= zero_zero_nat ) ) ).
thf(fact_2698_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_2699_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_2700_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_2701_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_2702_natceiling__le__eq,axiom,
! [A: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_nat @ ( natceiling @ X ) @ A )
<=> ( ord_less_eq_real @ X @ ( real_nat @ A ) ) ) ) ).
thf(fact_2703_divmod__int__rel__neg,axiom,
! [A: int,B: int,Qr: product_prod_int_int] :
( ( divmod_int_rel @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) @ Qr )
=> ( divmod_int_rel @ A @ B @ ( negateSnd @ Qr ) ) ) ).
thf(fact_2704_natceiling__subtract,axiom,
! [A: nat,X: real] :
( ( ord_less_eq_real @ ( real_nat @ A ) @ X )
=> ( ( natceiling @ ( minus_minus_real @ X @ ( real_nat @ A ) ) )
= ( minus_minus_nat @ ( natceiling @ X ) @ A ) ) ) ).
thf(fact_2705_divmod__int__correct,axiom,
! [A: int,B: int] :
( ( B != zero_zero_int )
=> ( divmod_int_rel @ A @ B @ ( divmod_int @ A @ B ) ) ) ).
thf(fact_2706_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_2707_natceiling__add,axiom,
! [A: nat,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( natceiling @ ( plus_plus_real @ X @ ( real_nat @ A ) ) )
= ( plus_plus_nat @ ( natceiling @ X ) @ A ) ) ) ).
thf(fact_2708_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_2709_le__mult__natfloor,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( natfloor @ A ) @ ( natfloor @ B ) ) @ ( natfloor @ ( times_times_real @ A @ B ) ) ) ) ) ).
thf(fact_2710_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_2711_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_2712_reals__Archimedean2,axiom,
! [X: real] :
? [N_1: nat] : ( ord_less_real @ X @ ( real_nat @ N_1 ) ) ).
thf(fact_2713_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_2714_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_2715_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_2716_div__neg__neg,axiom,
! [B: int,A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ( div_div_int @ A @ B )
= ( product_fst_int_int @ ( negateSnd @ ( posDivAlg @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ) ) ) ) ).
thf(fact_2717_mod__neg__neg,axiom,
! [B: int,A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ( div_mod_int @ A @ B )
= ( product_snd_int_int @ ( negateSnd @ ( posDivAlg @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ) ) ) ) ).
thf(fact_2718_prod__eqI,axiom,
! [P_6: produc914805421l_real,Q_3: produc914805421l_real] :
( ( ( produc1935615926l_real @ P_6 )
= ( produc1935615926l_real @ Q_3 ) )
=> ( ( ( produc556554744l_real @ P_6 )
= ( produc556554744l_real @ Q_3 ) )
=> ( P_6 = Q_3 ) ) ) ).
thf(fact_2719_prod__eqI,axiom,
! [P_6: product_prod_nat_nat,Q_3: product_prod_nat_nat] :
( ( ( product_fst_nat_nat @ P_6 )
= ( product_fst_nat_nat @ Q_3 ) )
=> ( ( ( product_snd_nat_nat @ P_6 )
= ( product_snd_nat_nat @ Q_3 ) )
=> ( P_6 = Q_3 ) ) ) ).
thf(fact_2720_prod__eqI,axiom,
! [P_6: product_prod_int_int,Q_3: product_prod_int_int] :
( ( ( product_fst_int_int @ P_6 )
= ( product_fst_int_int @ Q_3 ) )
=> ( ( ( product_snd_int_int @ P_6 )
= ( product_snd_int_int @ Q_3 ) )
=> ( P_6 = Q_3 ) ) ) ).
thf(fact_2721_prod__eq__iff,axiom,
! [S_4: produc914805421l_real,T_4: produc914805421l_real] :
( ( S_4 = T_4 )
<=> ( ( ( produc1935615926l_real @ S_4 )
= ( produc1935615926l_real @ T_4 ) )
& ( ( produc556554744l_real @ S_4 )
= ( produc556554744l_real @ T_4 ) ) ) ) ).
thf(fact_2722_prod__eq__iff,axiom,
! [S_4: product_prod_nat_nat,T_4: product_prod_nat_nat] :
( ( S_4 = T_4 )
<=> ( ( ( product_fst_nat_nat @ S_4 )
= ( product_fst_nat_nat @ T_4 ) )
& ( ( product_snd_nat_nat @ S_4 )
= ( product_snd_nat_nat @ T_4 ) ) ) ) ).
thf(fact_2723_prod__eq__iff,axiom,
! [S_4: product_prod_int_int,T_4: product_prod_int_int] :
( ( S_4 = T_4 )
<=> ( ( ( product_fst_int_int @ S_4 )
= ( product_fst_int_int @ T_4 ) )
& ( ( product_snd_int_int @ S_4 )
= ( product_snd_int_int @ T_4 ) ) ) ) ).
thf(fact_2724_surjective__pairing,axiom,
! [T_3: produc914805421l_real] :
( T_3
= ( produc865579683l_real @ ( produc1935615926l_real @ T_3 ) @ ( produc556554744l_real @ T_3 ) ) ) ).
thf(fact_2725_surjective__pairing,axiom,
! [T_3: product_prod_nat_nat] :
( T_3
= ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T_3 ) @ ( product_snd_nat_nat @ T_3 ) ) ) ).
thf(fact_2726_surjective__pairing,axiom,
! [T_3: product_prod_int_int] :
( T_3
= ( product_Pair_int_int @ ( product_fst_int_int @ T_3 ) @ ( product_snd_int_int @ T_3 ) ) ) ).
thf(fact_2727_surjective__pairing,axiom,
! [T_3: produc975137661_int_o] :
( T_3
= ( produc398918003_int_o @ ( produc2019076742_int_o @ T_3 ) @ ( produc883676360_int_o @ T_3 ) ) ) ).
thf(fact_2728_surjective__pairing,axiom,
! [T_3: produc1137372701nt_int] :
( T_3
= ( produc883642259nt_int @ ( produc1096438694nt_int @ T_3 ) @ ( produc1011150824nt_int @ T_3 ) ) ) ).
thf(fact_2729_surjective__pairing,axiom,
! [T_3: produc167071911de_int] :
( T_3
= ( produc1318306967de_int @ ( produc1968959274de_int @ T_3 ) @ ( produc221809260de_int @ T_3 ) ) ) ).
thf(fact_2730_surjective__pairing,axiom,
! [T_3: produc1359518119umeral] :
( T_3
= ( produc2136830103umeral @ ( produc310214954umeral @ T_3 ) @ ( produc1857364076umeral @ T_3 ) ) ) ).
thf(fact_2731_surjective__pairing,axiom,
! [T_3: produc1322466333at_nat] :
( T_3
= ( produc494345619at_nat @ ( produc707142054at_nat @ T_3 ) @ ( produc621854184at_nat @ T_3 ) ) ) ).
thf(fact_2732_surjective__pairing,axiom,
! [T_3: produc393999548nt_int] :
( T_3
= ( produc282740534nt_int @ ( produc2110120675nt_int @ T_3 ) @ ( produc8341793nt_int @ T_3 ) ) ) ).
thf(fact_2733_pair__collapse,axiom,
! [P_5: produc914805421l_real] :
( ( produc865579683l_real @ ( produc1935615926l_real @ P_5 ) @ ( produc556554744l_real @ P_5 ) )
= P_5 ) ).
thf(fact_2734_pair__collapse,axiom,
! [P_5: product_prod_nat_nat] :
( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ P_5 ) @ ( product_snd_nat_nat @ P_5 ) )
= P_5 ) ).
thf(fact_2735_pair__collapse,axiom,
! [P_5: product_prod_int_int] :
( ( product_Pair_int_int @ ( product_fst_int_int @ P_5 ) @ ( product_snd_int_int @ P_5 ) )
= P_5 ) ).
thf(fact_2736_pair__collapse,axiom,
! [P_5: produc975137661_int_o] :
( ( produc398918003_int_o @ ( produc2019076742_int_o @ P_5 ) @ ( produc883676360_int_o @ P_5 ) )
= P_5 ) ).
thf(fact_2737_pair__collapse,axiom,
! [P_5: produc1137372701nt_int] :
( ( produc883642259nt_int @ ( produc1096438694nt_int @ P_5 ) @ ( produc1011150824nt_int @ P_5 ) )
= P_5 ) ).
thf(fact_2738_pair__collapse,axiom,
! [P_5: produc167071911de_int] :
( ( produc1318306967de_int @ ( produc1968959274de_int @ P_5 ) @ ( produc221809260de_int @ P_5 ) )
= P_5 ) ).
thf(fact_2739_pair__collapse,axiom,
! [P_5: produc1359518119umeral] :
( ( produc2136830103umeral @ ( produc310214954umeral @ P_5 ) @ ( produc1857364076umeral @ P_5 ) )
= P_5 ) ).
thf(fact_2740_pair__collapse,axiom,
! [P_5: produc1322466333at_nat] :
( ( produc494345619at_nat @ ( produc707142054at_nat @ P_5 ) @ ( produc621854184at_nat @ P_5 ) )
= P_5 ) ).
thf(fact_2741_pair__collapse,axiom,
! [P_5: produc393999548nt_int] :
( ( produc282740534nt_int @ ( produc2110120675nt_int @ P_5 ) @ ( produc8341793nt_int @ P_5 ) )
= P_5 ) ).
thf(fact_2742_divide__zero__left,axiom,
! [A_11: rat] :
( ( inverse_divide_rat @ zero_zero_rat @ A_11 )
= zero_zero_rat ) ).
thf(fact_2743_divide__zero__left,axiom,
! [A_11: complex] :
( ( invers1025623611omplex @ zero_zero_complex @ A_11 )
= zero_zero_complex ) ).
thf(fact_2744_divide__zero__left,axiom,
! [A_11: real] :
( ( inverse_divide_real @ zero_zero_real @ A_11 )
= zero_zero_real ) ).
thf(fact_2745_divide__zero,axiom,
! [A_10: rat] :
( ( inverse_divide_rat @ A_10 @ zero_zero_rat )
= zero_zero_rat ) ).
thf(fact_2746_divide__zero,axiom,
! [A_10: complex] :
( ( invers1025623611omplex @ A_10 @ zero_zero_complex )
= zero_zero_complex ) ).
thf(fact_2747_divide__zero,axiom,
! [A_10: real] :
( ( inverse_divide_real @ A_10 @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_2748_divide_Ozero,axiom,
! [Y_7: complex] :
( ( invers1025623611omplex @ zero_zero_complex @ Y_7 )
= zero_zero_complex ) ).
thf(fact_2749_divide_Ozero,axiom,
! [Y_7: real] :
( ( inverse_divide_real @ zero_zero_real @ Y_7 )
= zero_zero_real ) ).
thf(fact_2750_times__divide__times__eq,axiom,
! [X_7: rat,Y_6: rat,Z_4: rat,W_1: rat] :
( ( times_times_rat @ ( inverse_divide_rat @ X_7 @ Y_6 ) @ ( inverse_divide_rat @ Z_4 @ W_1 ) )
= ( inverse_divide_rat @ ( times_times_rat @ X_7 @ Z_4 ) @ ( times_times_rat @ Y_6 @ W_1 ) ) ) ).
thf(fact_2751_times__divide__times__eq,axiom,
! [X_7: complex,Y_6: complex,Z_4: complex,W_1: complex] :
( ( times_times_complex @ ( invers1025623611omplex @ X_7 @ Y_6 ) @ ( invers1025623611omplex @ Z_4 @ W_1 ) )
= ( invers1025623611omplex @ ( times_times_complex @ X_7 @ Z_4 ) @ ( times_times_complex @ Y_6 @ W_1 ) ) ) ).
thf(fact_2752_times__divide__times__eq,axiom,
! [X_7: real,Y_6: real,Z_4: real,W_1: real] :
( ( times_times_real @ ( inverse_divide_real @ X_7 @ Y_6 ) @ ( inverse_divide_real @ Z_4 @ W_1 ) )
= ( inverse_divide_real @ ( times_times_real @ X_7 @ Z_4 ) @ ( times_times_real @ Y_6 @ W_1 ) ) ) ).
thf(fact_2753_times__divide__eq__right,axiom,
! [A_9: rat,B_9: rat,C_5: rat] :
( ( times_times_rat @ A_9 @ ( inverse_divide_rat @ B_9 @ C_5 ) )
= ( inverse_divide_rat @ ( times_times_rat @ A_9 @ B_9 ) @ C_5 ) ) ).
thf(fact_2754_times__divide__eq__right,axiom,
! [A_9: complex,B_9: complex,C_5: complex] :
( ( times_times_complex @ A_9 @ ( invers1025623611omplex @ B_9 @ C_5 ) )
= ( invers1025623611omplex @ ( times_times_complex @ A_9 @ B_9 ) @ C_5 ) ) ).
thf(fact_2755_times__divide__eq__right,axiom,
! [A_9: real,B_9: real,C_5: real] :
( ( times_times_real @ A_9 @ ( inverse_divide_real @ B_9 @ C_5 ) )
= ( inverse_divide_real @ ( times_times_real @ A_9 @ B_9 ) @ C_5 ) ) ).
thf(fact_2756_add__divide__distrib,axiom,
! [A_8: rat,B_8: rat,C_4: rat] :
( ( inverse_divide_rat @ ( plus_plus_rat @ A_8 @ B_8 ) @ C_4 )
= ( plus_plus_rat @ ( inverse_divide_rat @ A_8 @ C_4 ) @ ( inverse_divide_rat @ B_8 @ C_4 ) ) ) ).
thf(fact_2757_add__divide__distrib,axiom,
! [A_8: complex,B_8: complex,C_4: complex] :
( ( invers1025623611omplex @ ( plus_plus_complex @ A_8 @ B_8 ) @ C_4 )
= ( plus_plus_complex @ ( invers1025623611omplex @ A_8 @ C_4 ) @ ( invers1025623611omplex @ B_8 @ C_4 ) ) ) ).
thf(fact_2758_add__divide__distrib,axiom,
! [A_8: real,B_8: real,C_4: real] :
( ( inverse_divide_real @ ( plus_plus_real @ A_8 @ B_8 ) @ C_4 )
= ( plus_plus_real @ ( inverse_divide_real @ A_8 @ C_4 ) @ ( inverse_divide_real @ B_8 @ C_4 ) ) ) ).
thf(fact_2759_divide_Oadd,axiom,
! [X_6: complex,Y_5: complex,Ya_2: complex] :
( ( invers1025623611omplex @ ( plus_plus_complex @ X_6 @ Y_5 ) @ Ya_2 )
= ( plus_plus_complex @ ( invers1025623611omplex @ X_6 @ Ya_2 ) @ ( invers1025623611omplex @ Y_5 @ Ya_2 ) ) ) ).
thf(fact_2760_divide_Oadd,axiom,
! [X_6: real,Y_5: real,Ya_2: real] :
( ( inverse_divide_real @ ( plus_plus_real @ X_6 @ Y_5 ) @ Ya_2 )
= ( plus_plus_real @ ( inverse_divide_real @ X_6 @ Ya_2 ) @ ( inverse_divide_real @ Y_5 @ Ya_2 ) ) ) ).
thf(fact_2761_divide__1,axiom,
! [A_7: rat] :
( ( inverse_divide_rat @ A_7 @ one_one_rat )
= A_7 ) ).
thf(fact_2762_divide__1,axiom,
! [A_7: complex] :
( ( invers1025623611omplex @ A_7 @ one_one_complex )
= A_7 ) ).
thf(fact_2763_divide__1,axiom,
! [A_7: real] :
( ( inverse_divide_real @ A_7 @ one_one_real )
= A_7 ) ).
thf(fact_2764_diff__divide__distrib,axiom,
! [A_6: rat,B_7: rat,C_3: rat] :
( ( inverse_divide_rat @ ( minus_minus_rat @ A_6 @ B_7 ) @ C_3 )
= ( minus_minus_rat @ ( inverse_divide_rat @ A_6 @ C_3 ) @ ( inverse_divide_rat @ B_7 @ C_3 ) ) ) ).
thf(fact_2765_diff__divide__distrib,axiom,
! [A_6: complex,B_7: complex,C_3: complex] :
( ( invers1025623611omplex @ ( minus_minus_complex @ A_6 @ B_7 ) @ C_3 )
= ( minus_minus_complex @ ( invers1025623611omplex @ A_6 @ C_3 ) @ ( invers1025623611omplex @ B_7 @ C_3 ) ) ) ).
thf(fact_2766_diff__divide__distrib,axiom,
! [A_6: real,B_7: real,C_3: real] :
( ( inverse_divide_real @ ( minus_minus_real @ A_6 @ B_7 ) @ C_3 )
= ( minus_minus_real @ ( inverse_divide_real @ A_6 @ C_3 ) @ ( inverse_divide_real @ B_7 @ C_3 ) ) ) ).
thf(fact_2767_divide_Odiff,axiom,
! [X_5: complex,Y_4: complex,Ya_1: complex] :
( ( invers1025623611omplex @ ( minus_minus_complex @ X_5 @ Y_4 ) @ Ya_1 )
= ( minus_minus_complex @ ( invers1025623611omplex @ X_5 @ Ya_1 ) @ ( invers1025623611omplex @ Y_4 @ Ya_1 ) ) ) ).
thf(fact_2768_divide_Odiff,axiom,
! [X_5: real,Y_4: real,Ya_1: real] :
( ( inverse_divide_real @ ( minus_minus_real @ X_5 @ Y_4 ) @ Ya_1 )
= ( minus_minus_real @ ( inverse_divide_real @ X_5 @ Ya_1 ) @ ( inverse_divide_real @ Y_4 @ Ya_1 ) ) ) ).
thf(fact_2769_real__divide__square__eq,axiom,
! [R_1: real,A: real] :
( ( inverse_divide_real @ ( times_times_real @ R_1 @ A ) @ ( times_times_real @ R_1 @ R_1 ) )
= ( inverse_divide_real @ A @ R_1 ) ) ).
thf(fact_2770_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_2771_lemma__MVT,axiom,
! [F: real > real,A: real,B: real] :
( ( minus_minus_real @ ( F @ A ) @ ( times_times_real @ ( inverse_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) ) @ A ) )
= ( minus_minus_real @ ( F @ B ) @ ( times_times_real @ ( inverse_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) ) @ B ) ) ) ).
thf(fact_2772_mod__int__def,axiom,
! [A: int,B: int] :
( ( div_mod_int @ A @ B )
= ( product_snd_int_int @ ( divmod_int @ A @ B ) ) ) ).
thf(fact_2773_div__int__def,axiom,
! [A: int,B: int] :
( ( div_div_int @ A @ B )
= ( product_fst_int_int @ ( divmod_int @ A @ B ) ) ) ).
thf(fact_2774_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_2775_mod__neg__pos,axiom,
! [B: int,A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( div_mod_int @ A @ B )
= ( product_snd_int_int @ ( negDivAlg @ A @ B ) ) ) ) ) ).
thf(fact_2776_div__neg__pos,axiom,
! [B: int,A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( div_div_int @ A @ B )
= ( product_fst_int_int @ ( negDivAlg @ A @ B ) ) ) ) ) ).
thf(fact_2777_mod__pos__pos,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( div_mod_int @ A @ B )
= ( product_snd_int_int @ ( posDivAlg @ A @ B ) ) ) ) ) ).
thf(fact_2778_div__pos__pos,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( div_div_int @ A @ B )
= ( product_fst_int_int @ ( posDivAlg @ A @ B ) ) ) ) ) ).
thf(fact_2779_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_2780_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_2781_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_2782_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_2783_mod__pos__neg,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ( div_mod_int @ A @ B )
= ( product_snd_int_int @ ( negateSnd @ ( negDivAlg @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ) ) ) ) ).
thf(fact_2784_real__of__nat__div__aux,axiom,
! [X: nat,D: nat] :
( ( ord_less_nat @ zero_zero_nat @ D )
=> ( ( inverse_divide_real @ ( real_nat @ X ) @ ( real_nat @ D ) )
= ( plus_plus_real @ ( real_nat @ ( div_div_nat @ X @ D ) ) @ ( inverse_divide_real @ ( real_nat @ ( div_mod_nat @ X @ D ) ) @ ( real_nat @ D ) ) ) ) ) ).
thf(fact_2785_div__pos__neg,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ( div_div_int @ A @ B )
= ( product_fst_int_int @ ( negateSnd @ ( negDivAlg @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ) ) ) ) ).
thf(fact_2786_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_2787_pair__lessI2,axiom,
! [S_1: nat,T: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ S_1 @ T )
=> ( member180897546at_nat @ ( produc494345619at_nat @ ( product_Pair_nat_nat @ A @ S_1 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ pair_less ) ) ) ).
thf(fact_2788_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_2789_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_2790_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_2791_mod__nat__def,axiom,
! [M: nat,N: nat] :
( ( div_mod_nat @ M @ N )
= ( product_snd_nat_nat @ ( divmod_nat @ M @ N ) ) ) ).
thf(fact_2792_div__nat__def,axiom,
! [M: nat,N: nat] :
( ( div_div_nat @ M @ N )
= ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) ) ) ).
thf(fact_2793_arctan__minus,axiom,
! [X: real] :
( ( arctan @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( arctan @ X ) ) ) ).
thf(fact_2794_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_2795_arctan__monotone,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
thf(fact_2796_arctan__zero__zero,axiom,
( ( arctan @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_2797_arctan__monotone_H,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
thf(fact_2798_natceiling__def,axiom,
! [X: real] :
( ( natceiling @ X )
= ( nat_1 @ ( archim856651990g_real @ X ) ) ) ).
thf(fact_2799_nat__0,axiom,
( ( nat_1 @ zero_zero_int )
= zero_zero_nat ) ).
thf(fact_2800_transfer__nat__int__numerals_I1_J,axiom,
( zero_zero_nat
= ( nat_1 @ zero_zero_int ) ) ).
thf(fact_2801_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_2802_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_2803_all__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_2804_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_2805_nat__number__of__def,axiom,
! [V: int] :
( ( number_number_of_nat @ V )
= ( nat_1 @ ( number_number_of_int @ V ) ) ) ).
thf(fact_2806_nat__number__of,axiom,
! [W: int] :
( ( nat_1 @ ( number_number_of_int @ W ) )
= ( number_number_of_nat @ W ) ) ).
thf(fact_2807_transfer__nat__int__numerals_I2_J,axiom,
( one_one_nat
= ( nat_1 @ one_one_int ) ) ).
thf(fact_2808_neg__nat,axiom,
! [Z_1: int] :
( ( nat_neg @ Z_1 )
=> ( ( nat_1 @ Z_1 )
= zero_zero_nat ) ) ).
thf(fact_2809_natfloor__def,axiom,
! [X: real] :
( ( natfloor @ X )
= ( nat_1 @ ( archim1246769320r_real @ X ) ) ) ).
thf(fact_2810_divmod__nat__rel__divmod__nat,axiom,
! [M: nat,N: nat] : ( divmod_nat_rel @ M @ N @ ( divmod_nat @ M @ N ) ) ).
thf(fact_2811_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_2812_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_2813_nat__0__iff,axiom,
! [I: int] :
( ( ( nat_1 @ I )
= zero_zero_nat )
<=> ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
thf(fact_2814_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_2815_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_2816_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_2817_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_2818_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_2819_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_2820_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_2821_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_2822_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_2823_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_2824_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_2825_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_2826_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_2827_divmod__nat__zero,axiom,
! [M: nat] :
( ( divmod_nat @ M @ zero_zero_nat )
= ( product_Pair_nat_nat @ zero_zero_nat @ M ) ) ).
thf(fact_2828_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_2829_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_2830_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_2831_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_2832_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_2833_pair__lessI1,axiom,
! [S_1: nat,T: nat,A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( member180897546at_nat @ ( produc494345619at_nat @ ( product_Pair_nat_nat @ A @ S_1 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ pair_less ) ) ).
thf(fact_2834_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_2835_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_2836_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_2837_nat__aux__def,axiom,
! [I: int,N: nat] :
( ( nat_aux @ I @ N )
= ( plus_plus_nat @ ( nat_1 @ I ) @ N ) ) ).
thf(fact_2838_transfer__morphism__nat__int,axiom,
nat_tr876908586nt_nat @ nat_1 @ ( ord_less_eq_int @ zero_zero_int ) ).
thf(fact_2839_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_2840_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_2841_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_2842_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_2843_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
thf(fact_2844_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
thf(fact_2845_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
thf(fact_2846_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
thf(fact_2847_nat_Oinject,axiom,
! [Nat: nat,Nat_3: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat_3 ) )
<=> ( Nat = Nat_3 ) ) ).
thf(fact_2848_Suc__n__not__n,axiom,
! [N: nat] :
( ( suc @ N )
!= N ) ).
thf(fact_2849_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
thf(fact_2850_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_2851_nat_Osimps_I2_J,axiom,
! [Nat_3: nat] :
( zero_zero_nat
!= ( suc @ Nat_3 ) ) ).
thf(fact_2852_Suc__not__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_2853_nat_Osimps_I3_J,axiom,
! [Nat_2: nat] :
( ( suc @ Nat_2 )
!= zero_zero_nat ) ).
thf(fact_2854_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
thf(fact_2855_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
thf(fact_2856_not__less__eq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
<=> ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
thf(fact_2857_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
<=> ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
thf(fact_2858_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_2859_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_2860_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
thf(fact_2861_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_2862_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
thf(fact_2863_less__trans__Suc,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K_1 )
=> ( ord_less_nat @ ( suc @ I ) @ K_1 ) ) ) ).
thf(fact_2864_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
thf(fact_2865_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_2866_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_2867_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
thf(fact_2868_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
thf(fact_2869_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_2870_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_2871_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_2872_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_2873_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
<=> ( ord_less_eq_nat @ N @ M ) ) ).
thf(fact_2874_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_2875_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_eq_nat @ M @ N )
<=> ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
thf(fact_2876_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
thf(fact_2877_Suc__mult__cancel1,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K_1 ) @ M )
= ( times_times_nat @ ( suc @ K_1 ) @ N ) )
<=> ( M = N ) ) ).
thf(fact_2878_Suc__diff__diff,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K_1 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K_1 ) ) ).
thf(fact_2879_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
thf(fact_2880_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_2881_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_2882_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_2883_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
<=> ? [M_2: nat] :
( N
= ( suc @ M_2 ) ) ) ).
thf(fact_2884_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_2885_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_2886_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_2887_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
thf(fact_2888_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
thf(fact_2889_less__iff__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
<=> ? [K: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K ) ) ) ) ).
thf(fact_2890_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
thf(fact_2891_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_2892_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_2893_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
thf(fact_2894_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
thf(fact_2895_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_2896_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_2897_less__eq__Suc__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
<=> ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
thf(fact_2898_Suc__mult__less__cancel1,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K_1 ) @ M ) @ ( times_times_nat @ ( suc @ K_1 ) @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_2899_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
thf(fact_2900_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
thf(fact_2901_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_2902_Suc__mult__le__cancel1,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K_1 ) @ M ) @ ( times_times_nat @ ( suc @ K_1 ) @ N ) )
<=> ( ord_less_eq_nat @ M @ N ) ) ).
thf(fact_2903_Suc__eq__plus1__left,axiom,
! [N: nat] :
( ( suc @ N )
= ( plus_plus_nat @ one_one_nat @ N ) ) ).
thf(fact_2904_Suc__eq__plus1,axiom,
! [N: nat] :
( ( suc @ N )
= ( plus_plus_nat @ N @ one_one_nat ) ) ).
thf(fact_2905_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_2906_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
thf(fact_2907_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_2908_div__1,axiom,
! [M: nat] :
( ( div_div_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
thf(fact_2909_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_2910_mod__1,axiom,
! [M: nat] :
( ( div_mod_nat @ M @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
thf(fact_2911_one__less__m,axiom,
! [M: nat,K_1: nat] :
( ( M
!= ( times_times_nat @ M @ K_1 ) )
=> ( ( M
!= ( suc @ zero_zero_nat ) )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M ) ) ) ).
thf(fact_2912_one__less__k,axiom,
! [M: nat,K_1: nat] :
( ( M
!= ( times_times_nat @ M @ K_1 ) )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ K_1 ) )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ K_1 ) ) ) ).
thf(fact_2913_prod__mn__less__k,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ( ( times_times_nat @ M @ N )
= K_1 )
=> ( ord_less_nat @ N @ K_1 ) ) ) ) ) ).
thf(fact_2914_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_2915_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_2916_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_2917_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_2918_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_2919_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
thf(fact_2920_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_2921_real__of__nat__Suc__gt__zero,axiom,
! [N: nat] : ( ord_less_real @ zero_zero_real @ ( real_nat @ ( suc @ N ) ) ) ).
thf(fact_2922_nat__1,axiom,
( ( nat_1 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
thf(fact_2923_real__of__nat__one,axiom,
( ( real_nat @ ( suc @ zero_zero_nat ) )
= one_one_real ) ).
thf(fact_2924_diff__Suc__diff__eq1,axiom,
! [M: nat,K_1: nat,J: nat] :
( ( ord_less_eq_nat @ K_1 @ J )
=> ( ( minus_minus_nat @ M @ ( suc @ ( minus_minus_nat @ J @ K_1 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ M @ K_1 ) @ ( suc @ J ) ) ) ) ).
thf(fact_2925_diff__Suc__diff__eq2,axiom,
! [M: nat,K_1: nat,J: nat] :
( ( ord_less_eq_nat @ K_1 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K_1 ) ) @ M )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K_1 @ M ) ) ) ) ).
thf(fact_2926_real__of__nat__Suc,axiom,
! [N: nat] :
( ( real_nat @ ( suc @ N ) )
= ( plus_plus_real @ ( real_nat @ N ) @ one_one_real ) ) ).
thf(fact_2927_mod__mult__self4,axiom,
! [K_1: nat,N: nat,M: nat] :
( ( div_mod_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K_1 @ N ) @ M ) ) @ N )
= ( div_mod_nat @ ( suc @ M ) @ N ) ) ).
thf(fact_2928_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
thf(fact_2929_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
thf(fact_2930_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_2931_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_2932_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_2933_Suc__times__mod__eq,axiom,
! [M: nat,K_1: nat] :
( ( ord_less_nat @ one_one_nat @ K_1 )
=> ( ( div_mod_nat @ ( suc @ ( times_times_nat @ K_1 @ M ) ) @ K_1 )
= one_one_nat ) ) ).
thf(fact_2934_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_2935_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_2936_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_2937_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_2938_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_2939_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_2940_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_2941_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_2942_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_2943_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_2944_div__Suc,axiom,
! [A: nat,C: nat] :
( ( div_div_nat @ ( suc @ A ) @ C )
= ( plus_plus_nat @ ( plus_plus_nat @ ( div_div_nat @ A @ C ) @ ( div_div_nat @ ( suc @ zero_zero_nat ) @ C ) ) @ ( div_div_nat @ ( plus_plus_nat @ ( div_mod_nat @ A @ C ) @ ( div_mod_nat @ ( suc @ zero_zero_nat ) @ C ) ) @ C ) ) ) ).
thf(fact_2945_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_2946_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_2947_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_2948_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_2949_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K ) ) ) ) ).
thf(fact_2950_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_2951_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M_2: nat] :
( N
= ( suc @ M_2 ) ) ) ).
thf(fact_2952_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_2953_zero__induct__lemma,axiom,
! [I: nat,P: nat > $o,K_1: nat] :
( ( P @ K_1 )
=> ( ! [N_1: nat] :
( ( P @ ( suc @ N_1 ) )
=> ( P @ N_1 ) )
=> ( P @ ( minus_minus_nat @ K_1 @ I ) ) ) ) ).
thf(fact_2954_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_2955_lessE,axiom,
! [I: nat,K_1: nat] :
( ( ord_less_nat @ I @ K_1 )
=> ( ( K_1
!= ( suc @ I ) )
=> ~ ! [J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( K_1
!= ( suc @ J_1 ) ) ) ) ) ).
thf(fact_2956_Suc__lessE,axiom,
! [I: nat,K_1: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K_1 )
=> ~ ! [J_1: nat] :
( ( ord_less_nat @ I @ J_1 )
=> ( K_1
!= ( suc @ J_1 ) ) ) ) ).
thf(fact_2957_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M_2: nat] :
( N
= ( suc @ M_2 ) ) ) ).
thf(fact_2958_succ__def,axiom,
! [K_1: int] :
( ( succ @ K_1 )
= ( plus_plus_int @ K_1 @ one_one_int ) ) ).
thf(fact_2959_diff__bin__simps_I2_J,axiom,
! [K_1: int] :
( ( minus_minus_int @ K_1 @ min )
= ( succ @ K_1 ) ) ).
thf(fact_2960_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_2961_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_2962_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat_1: nat] :
( Y
!= ( suc @ Nat_1 ) ) ) ).
thf(fact_2963_zero__induct,axiom,
! [P: nat > $o,K_1: nat] :
( ( P @ K_1 )
=> ( ! [N_1: nat] :
( ( P @ ( suc @ N_1 ) )
=> ( P @ N_1 ) )
=> ( P @ zero_zero_nat ) ) ) ).
thf(fact_2964_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_2965_int__power__div__base,axiom,
! [K_1: int,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_int @ zero_zero_int @ K_1 )
=> ( ( div_div_int @ ( power_power_int @ K_1 @ M ) @ K_1 )
= ( power_power_int @ K_1 @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
thf(fact_2966_pair__leqI2,axiom,
! [S_1: nat,T: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ S_1 @ T )
=> ( member180897546at_nat @ ( produc494345619at_nat @ ( product_Pair_nat_nat @ A @ S_1 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ pair_leq ) ) ) ).
thf(fact_2967_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_2968_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_2969_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_2970_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_2971_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_2972_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_2973_Euler_Oaux__1,axiom,
! [A: int,P_3: int] :
( ( ord_less_int @ zero_zero_int @ P_3 )
=> ( ( power_power_int @ A @ ( nat_1 @ P_3 ) )
= ( times_times_int @ A @ ( power_power_int @ A @ ( minus_minus_nat @ ( nat_1 @ P_3 ) @ one_one_nat ) ) ) ) ) ).
thf(fact_2974_pair__leqI1,axiom,
! [S_1: nat,T: nat,A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( member180897546at_nat @ ( produc494345619at_nat @ ( product_Pair_nat_nat @ A @ S_1 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ pair_leq ) ) ).
thf(fact_2975_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_2976_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_2977_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_2978_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_2979_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_2980_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_2981_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
thf(fact_2982_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_2983_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_2984_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_2985_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_2986_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_2987_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_2988_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_2989_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_2990_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_2991_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_2992_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_2993_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_2994_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_2995_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_2996_realpow__pos__nth,axiom,
! [A: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ N )
= A ) ) ) ) ).
thf(fact_2997_realpow__pos__nth__unique,axiom,
! [A: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X_1: real] :
( ( ord_less_real @ zero_zero_real @ X_1 )
& ( ( power_power_real @ X_1 @ N )
= A )
& ! [Y_1: real] :
( ( ( ord_less_real @ zero_zero_real @ Y_1 )
& ( ( power_power_real @ Y_1 @ N )
= A ) )
=> ( Y_1 = X_1 ) ) ) ) ) ).
thf(fact_2998_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_2999_realpow__pos__nth2,axiom,
! [N: nat,A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ ( suc @ N ) )
= A ) ) ) ).
thf(fact_3000_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_3001_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_3002_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_3003_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_3004_neg__odd__power,axiom,
! [A: int,X: int] :
( ( member_int @ X @ zOdd )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( nat_1 @ X ) )
= ( uminus_uminus_int @ ( power_power_int @ A @ ( nat_1 @ X ) ) ) ) ) ) ).
thf(fact_3005_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_3006_even__odd__conj,axiom,
! [X: int] :
~ ( ( member_int @ X @ zOdd )
& ( member_int @ X @ zEven ) ) ).
thf(fact_3007_odd__iff__not__even,axiom,
! [X: int] :
( ( member_int @ X @ zOdd )
<=> ~ ( member_int @ X @ zEven ) ) ).
thf(fact_3008_even__odd__disj,axiom,
! [X: int] :
( ( member_int @ X @ zOdd )
| ( member_int @ X @ zEven ) ) ).
thf(fact_3009_not__odd__impl__even,axiom,
! [X: int] :
( ~ ( member_int @ X @ zOdd )
=> ( member_int @ X @ zEven ) ) ).
thf(fact_3010_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_3011_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_3012_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_3013_even__plus__odd__prop2,axiom,
! [A: int,B: int] :
( ( member_int @ ( plus_plus_int @ A @ B ) @ zOdd )
=> ( ( member_int @ A @ zEven )
=> ( member_int @ B @ zOdd ) ) ) ).
thf(fact_3014_even__plus__odd__prop1,axiom,
! [A: int,B: int] :
( ( member_int @ ( plus_plus_int @ A @ B ) @ zOdd )
=> ( ( member_int @ A @ zOdd )
=> ( member_int @ B @ zEven ) ) ) ).
thf(fact_3015_IntNatAux_Oeven__plus__odd,axiom,
! [B: int,A: int] :
( ( member_int @ A @ zEven )
=> ( ( member_int @ B @ zOdd )
=> ( member_int @ ( plus_plus_int @ A @ B ) @ zOdd ) ) ) ).
thf(fact_3016_IntNatAux_Oodd__plus__odd,axiom,
! [B: int,A: int] :
( ( member_int @ A @ zOdd )
=> ( ( member_int @ B @ zOdd )
=> ( member_int @ ( plus_plus_int @ A @ B ) @ zEven ) ) ) ).
thf(fact_3017_real__of__int__inject,axiom,
! [X: int,Y: int] :
( ( ( real_int @ X )
= ( real_int @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_3018_pi__neq__zero,axiom,
pi != zero_zero_real ).
thf(fact_3019_floor__real__of__int,axiom,
! [N: int] :
( ( archim1246769320r_real @ ( real_int @ N ) )
= N ) ).
thf(fact_3020_real__of__int__floor__cancel,axiom,
! [X: real] :
( ( ( real_int @ ( archim1246769320r_real @ X ) )
= X )
<=> ? [N_1: int] :
( X
= ( real_int @ N_1 ) ) ) ).
thf(fact_3021_odd__minus__one__even,axiom,
! [X: int] :
( ( member_int @ X @ zOdd )
=> ( member_int @ ( minus_minus_int @ X @ one_one_int ) @ zEven ) ) ).
thf(fact_3022_ceiling__real__of__int,axiom,
! [N: int] :
( ( archim856651990g_real @ ( real_int @ N ) )
= N ) ).
thf(fact_3023_real__of__int__ceiling__cancel,axiom,
! [X: real] :
( ( ( real_int @ ( archim856651990g_real @ X ) )
= X )
<=> ? [N_1: int] :
( X
= ( real_int @ N_1 ) ) ) ).
thf(fact_3024_one__not__even,axiom,
~ ( member_int @ one_one_int @ zEven ) ).
thf(fact_3025_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_3026_even__times__either,axiom,
! [Y: int,X: int] :
( ( member_int @ X @ zEven )
=> ( member_int @ ( times_times_int @ X @ Y ) @ zEven ) ) ).
thf(fact_3027_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_3028_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_3029_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_3030_odd__mult__odd__prop,axiom,
! [X: int,Y: int] :
( ( member_int @ ( times_times_int @ X @ Y ) @ zOdd )
=> ( member_int @ X @ zOdd ) ) ).
thf(fact_3031_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_3032_real__of__int__zero__cancel,axiom,
! [X: int] :
( ( ( real_int @ X )
= zero_zero_real )
<=> ( X = zero_zero_int ) ) ).
thf(fact_3033_real__of__int__zero,axiom,
( ( real_int @ zero_zero_int )
= zero_zero_real ) ).
thf(fact_3034_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_3035_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_3036_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_3037_real__of__one,axiom,
( ( real_int @ one_one_int )
= one_one_real ) ).
thf(fact_3038_real__number__of,axiom,
! [V: int] :
( ( real_int @ ( number_number_of_int @ V ) )
= ( number267125858f_real @ V ) ) ).
thf(fact_3039_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_3040_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_3041_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_3042_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_3043_pi__ge__zero,axiom,
ord_less_eq_real @ zero_zero_real @ pi ).
thf(fact_3044_pi__gt__zero,axiom,
ord_less_real @ zero_zero_real @ pi ).
thf(fact_3045_pi__not__less__zero,axiom,
~ ( ord_less_real @ pi @ zero_zero_real ) ).
thf(fact_3046_real__of__int__minus,axiom,
! [X: int] :
( ( real_int @ ( uminus_uminus_int @ X ) )
= ( uminus_uminus_real @ ( real_int @ X ) ) ) ).
thf(fact_3047_Ints__real__of__int,axiom,
! [X: int] : ( member_real @ ( real_int @ X ) @ ring_1_Ints_real ) ).
thf(fact_3048_real__of__int__abs,axiom,
! [X: int] :
( ( real_int @ ( abs_abs_int @ X ) )
= ( abs_abs_real @ ( real_int @ X ) ) ) ).
thf(fact_3049_real__of__int__floor__le,axiom,
! [R_1: real] : ( ord_less_eq_real @ ( real_int @ ( archim1246769320r_real @ R_1 ) ) @ R_1 ) ).
thf(fact_3050_real__of__int__ceiling__ge,axiom,
! [R_1: real] : ( ord_less_eq_real @ R_1 @ ( real_int @ ( archim856651990g_real @ R_1 ) ) ) ).
thf(fact_3051_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_3052_floor__less__eq,axiom,
! [X: real,A: int] :
( ( ord_less_int @ ( archim1246769320r_real @ X ) @ A )
<=> ( ord_less_real @ X @ ( real_int @ A ) ) ) ).
thf(fact_3053_le__floor,axiom,
! [A: int,X: real] :
( ( ord_less_eq_real @ ( real_int @ A ) @ X )
=> ( ord_less_eq_int @ A @ ( archim1246769320r_real @ X ) ) ) ).
thf(fact_3054_le__floor__eq,axiom,
! [A: int,X: real] :
( ( ord_less_eq_int @ A @ ( archim1246769320r_real @ X ) )
<=> ( ord_less_eq_real @ ( real_int @ A ) @ X ) ) ).
thf(fact_3055_real__le__floor,axiom,
! [A: int,X: real] :
( ( ord_less_eq_int @ A @ ( archim1246769320r_real @ X ) )
=> ( ord_less_eq_real @ ( real_int @ A ) @ X ) ) ).
thf(fact_3056_less__ceiling__eq,axiom,
! [A: int,X: real] :
( ( ord_less_int @ A @ ( archim856651990g_real @ X ) )
<=> ( ord_less_real @ ( real_int @ A ) @ X ) ) ).
thf(fact_3057_floor__add,axiom,
! [X: real,A: int] :
( ( archim1246769320r_real @ ( plus_plus_real @ X @ ( real_int @ A ) ) )
= ( plus_plus_int @ ( archim1246769320r_real @ X ) @ A ) ) ).
thf(fact_3058_ceiling__le__eq,axiom,
! [X: real,A: int] :
( ( ord_less_eq_int @ ( archim856651990g_real @ X ) @ A )
<=> ( ord_less_eq_real @ X @ ( real_int @ A ) ) ) ).
thf(fact_3059_ceiling__le,axiom,
! [X: real,A: int] :
( ( ord_less_eq_real @ X @ ( real_int @ A ) )
=> ( ord_less_eq_int @ ( archim856651990g_real @ X ) @ A ) ) ).
thf(fact_3060_ceiling__le__real,axiom,
! [X: real,A: int] :
( ( ord_less_eq_int @ ( archim856651990g_real @ X ) @ A )
=> ( ord_less_eq_real @ X @ ( real_int @ A ) ) ) ).
thf(fact_3061_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_3062_floor__subtract,axiom,
! [X: real,A: int] :
( ( archim1246769320r_real @ ( minus_minus_real @ X @ ( real_int @ A ) ) )
= ( minus_minus_int @ ( archim1246769320r_real @ X ) @ A ) ) ).
thf(fact_3063_ceiling__add,axiom,
! [X: real,A: int] :
( ( archim856651990g_real @ ( plus_plus_real @ X @ ( real_int @ A ) ) )
= ( plus_plus_int @ ( archim856651990g_real @ X ) @ A ) ) ).
thf(fact_3064_floor__minus__real__of__int,axiom,
! [N: int] :
( ( archim1246769320r_real @ ( uminus_uminus_real @ ( real_int @ N ) ) )
= ( uminus_uminus_int @ N ) ) ).
thf(fact_3065_ceiling__subtract,axiom,
! [X: real,A: int] :
( ( archim856651990g_real @ ( minus_minus_real @ X @ ( real_int @ A ) ) )
= ( minus_minus_int @ ( archim856651990g_real @ X ) @ A ) ) ).
thf(fact_3066_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_3067_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_3068_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_3069_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_3070_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_3071_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_3072_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_3073_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_3074_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_3075_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_3076_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_3077_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_3078_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_3079_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_3080_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_3081_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_3082_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_3083_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_3084_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_3085_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_3086_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_3087_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_3088_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_3089_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_3090_real__of__int__div__aux,axiom,
! [X: int,D: int] :
( ( D != zero_zero_int )
=> ( ( inverse_divide_real @ ( real_int @ X ) @ ( real_int @ D ) )
= ( plus_plus_real @ ( real_int @ ( div_div_int @ X @ D ) ) @ ( inverse_divide_real @ ( real_int @ ( div_mod_int @ X @ D ) ) @ ( real_int @ D ) ) ) ) ) ).
thf(fact_3091_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_3092_less__floor__eq,axiom,
! [A: int,X: real] :
( ( ord_less_int @ A @ ( archim1246769320r_real @ X ) )
<=> ( ord_less_eq_real @ ( plus_plus_real @ ( real_int @ A ) @ one_one_real ) @ X ) ) ).
thf(fact_3093_floor__le__eq,axiom,
! [X: real,A: int] :
( ( ord_less_eq_int @ ( archim1246769320r_real @ X ) @ A )
<=> ( ord_less_real @ X @ ( plus_plus_real @ ( real_int @ A ) @ one_one_real ) ) ) ).
thf(fact_3094_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_3095_ceiling__less__eq,axiom,
! [X: real,A: int] :
( ( ord_less_int @ ( archim856651990g_real @ X ) @ A )
<=> ( ord_less_eq_real @ X @ ( minus_minus_real @ ( real_int @ A ) @ one_one_real ) ) ) ).
thf(fact_3096_le__ceiling__eq,axiom,
! [A: int,X: real] :
( ( ord_less_eq_int @ A @ ( archim856651990g_real @ X ) )
<=> ( ord_less_real @ ( minus_minus_real @ ( real_int @ A ) @ one_one_real ) @ X ) ) ).
thf(fact_3097_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_3098_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_3099_neg__even__power,axiom,
! [A: int,X: int] :
( ( member_int @ X @ zEven )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( nat_1 @ X ) )
= ( power_power_int @ A @ ( nat_1 @ X ) ) ) ) ) ).
thf(fact_3100_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_3101_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_3102_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_3103_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_3104_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_3105_cos__npi2,axiom,
! [N: nat] :
( ( cos @ ( times_times_real @ pi @ ( real_nat @ N ) ) )
= ( power_power_real @ ( number267125858f_real @ min ) @ N ) ) ).
thf(fact_3106_cos__npi,axiom,
! [N: nat] :
( ( cos @ ( times_times_real @ ( real_nat @ N ) @ pi ) )
= ( power_power_real @ ( number267125858f_real @ min ) @ N ) ) ).
thf(fact_3107_cos__minus,axiom,
! [X: real] :
( ( cos @ ( uminus_uminus_real @ X ) )
= ( cos @ X ) ) ).
thf(fact_3108_cos__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( cos @ X ) @ one_one_real ) ).
thf(fact_3109_cos__zero,axiom,
( ( cos @ zero_zero_real )
= one_one_real ) ).
thf(fact_3110_cos__arctan__not__zero,axiom,
! [X: real] :
( ( cos @ ( arctan @ X ) )
!= zero_zero_real ) ).
thf(fact_3111_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_3112_cos__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( number267125858f_real @ min ) @ ( cos @ X ) ) ).
thf(fact_3113_abs__cos__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos @ X ) ) @ one_one_real ) ).
thf(fact_3114_cos__pi,axiom,
( ( cos @ pi )
= ( number267125858f_real @ min ) ) ).
thf(fact_3115_cos__periodic__pi,axiom,
! [X: real] :
( ( cos @ ( plus_plus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( cos @ X ) ) ) ).
thf(fact_3116_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_3117_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_3118_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_3119_cos__arccos__abs,axiom,
! [Y: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
=> ( ( cos @ ( arccos @ Y ) )
= Y ) ) ).
thf(fact_3120_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_3121_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_3122_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_3123_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_3124_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_3125_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_3126_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_3127_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_3128_Pls__def,axiom,
pls = zero_zero_int ).
thf(fact_3129_tan__zero,axiom,
( ( tan @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_3130_rel__simps_I2_J,axiom,
~ ( ord_less_int @ pls @ pls ) ).
thf(fact_3131_rel__simps_I19_J,axiom,
ord_less_eq_int @ pls @ pls ).
thf(fact_3132_mult__Pls,axiom,
! [W: int] :
( ( times_times_int @ pls @ W )
= pls ) ).
thf(fact_3133_add__Pls,axiom,
! [K_1: int] :
( ( plus_plus_int @ pls @ K_1 )
= K_1 ) ).
thf(fact_3134_add__Pls__right,axiom,
! [K_1: int] :
( ( plus_plus_int @ K_1 @ pls )
= K_1 ) ).
thf(fact_3135_diff__bin__simps_I1_J,axiom,
! [K_1: int] :
( ( minus_minus_int @ K_1 @ pls )
= K_1 ) ).
thf(fact_3136_rel__simps_I40_J,axiom,
min != pls ).
thf(fact_3137_rel__simps_I37_J,axiom,
pls != min ).
thf(fact_3138_minus__Pls,axiom,
( ( uminus_uminus_int @ pls )
= pls ) ).
thf(fact_3139_tan__minus,axiom,
! [X: real] :
( ( tan @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( tan @ X ) ) ) ).
thf(fact_3140_tan__arctan,axiom,
! [Y: real] :
( ( tan @ ( arctan @ Y ) )
= Y ) ).
thf(fact_3141_semiring__norm_I113_J,axiom,
( zero_zero_nat
= ( number_number_of_nat @ pls ) ) ).
thf(fact_3142_nat__number__of__Pls,axiom,
( ( number_number_of_nat @ pls )
= zero_zero_nat ) ).
thf(fact_3143_bin__less__0__simps_I1_J,axiom,
~ ( ord_less_int @ pls @ zero_zero_int ) ).
thf(fact_3144_zero__is__num__zero,axiom,
( zero_zero_int
= ( number_number_of_int @ pls ) ) ).
thf(fact_3145_rel__simps_I6_J,axiom,
ord_less_int @ min @ pls ).
thf(fact_3146_rel__simps_I3_J,axiom,
~ ( ord_less_int @ pls @ min ) ).
thf(fact_3147_tan__pi,axiom,
( ( tan @ pi )
= zero_zero_real ) ).
thf(fact_3148_rel__simps_I23_J,axiom,
ord_less_eq_int @ min @ pls ).
thf(fact_3149_rel__simps_I20_J,axiom,
~ ( ord_less_eq_int @ pls @ min ) ).
thf(fact_3150_eq__number__of__Pls__Min,axiom,
( ( number_number_of_int @ pls )
!= ( number_number_of_int @ min ) ) ).
thf(fact_3151_tan__periodic__pi,axiom,
! [X: real] :
( ( tan @ ( plus_plus_real @ X @ pi ) )
= ( tan @ X ) ) ).
thf(fact_3152_not__neg__number__of__Pls,axiom,
~ ( nat_neg @ ( number_number_of_int @ pls ) ) ).
thf(fact_3153_succ__Min,axiom,
( ( succ @ min )
= pls ) ).
thf(fact_3154_eq__0__number__of,axiom,
! [V: int] :
( ( zero_zero_nat
= ( number_number_of_nat @ V ) )
<=> ( ord_less_eq_int @ V @ pls ) ) ).
thf(fact_3155_eq__number__of__0,axiom,
! [V: int] :
( ( ( number_number_of_nat @ V )
= zero_zero_nat )
<=> ( ord_less_eq_int @ V @ pls ) ) ).
thf(fact_3156_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_3157_number__of2,axiom,
ord_less_eq_int @ zero_zero_int @ ( number_number_of_int @ pls ) ).
thf(fact_3158_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_3159_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_3160_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_3161_tan__npi,axiom,
! [N: nat] :
( ( tan @ ( times_times_real @ ( real_nat @ N ) @ pi ) )
= zero_zero_real ) ).
thf(fact_3162_tan__periodic__n,axiom,
! [X: real,N: int] :
( ( tan @ ( plus_plus_real @ X @ ( times_times_real @ ( number267125858f_real @ N ) @ pi ) ) )
= ( tan @ X ) ) ).
thf(fact_3163_tan__periodic__nat,axiom,
! [X: real,N: nat] :
( ( tan @ ( plus_plus_real @ X @ ( times_times_real @ ( real_nat @ N ) @ pi ) ) )
= ( tan @ X ) ) ).
thf(fact_3164_tan__periodic__int,axiom,
! [X: real,I: int] :
( ( tan @ ( plus_plus_real @ X @ ( times_times_real @ ( real_int @ I ) @ pi ) ) )
= ( tan @ X ) ) ).
thf(fact_3165_nat__number__of__mult__left,axiom,
! [V_1: int,K_1: nat,V: int] :
( ( ( ord_less_int @ V @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V ) @ ( times_times_nat @ ( number_number_of_nat @ V_1 ) @ K_1 ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_int @ V @ pls )
=> ( ( times_times_nat @ ( number_number_of_nat @ V ) @ ( times_times_nat @ ( number_number_of_nat @ V_1 ) @ K_1 ) )
= ( times_times_nat @ ( number_number_of_nat @ ( times_times_int @ V @ V_1 ) ) @ K_1 ) ) ) ) ).
thf(fact_3166_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_3167_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_3168_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_3169_rel__simps_I51_J,axiom,
! [K_1: int,L: int] :
( ( ( bit1 @ K_1 )
= ( bit1 @ L ) )
<=> ( K_1 = L ) ) ).
thf(fact_3170_minus__Bit1,axiom,
! [K_1: int] :
( ( uminus_uminus_int @ ( bit1 @ K_1 ) )
= ( bit1 @ ( pred @ ( uminus_uminus_int @ K_1 ) ) ) ) ).
thf(fact_3171_rel__simps_I46_J,axiom,
! [K_1: int] :
( ( bit1 @ K_1 )
!= pls ) ).
thf(fact_3172_rel__simps_I39_J,axiom,
! [L: int] :
( pls
!= ( bit1 @ L ) ) ).
thf(fact_3173_rel__simps_I17_J,axiom,
! [K_1: int,L: int] :
( ( ord_less_int @ ( bit1 @ K_1 ) @ ( bit1 @ L ) )
<=> ( ord_less_int @ K_1 @ L ) ) ).
thf(fact_3174_less__int__code_I16_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit1 @ K1 ) @ ( bit1 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_3175_rel__simps_I34_J,axiom,
! [K_1: int,L: int] :
( ( ord_less_eq_int @ ( bit1 @ K_1 ) @ ( bit1 @ L ) )
<=> ( ord_less_eq_int @ K_1 @ L ) ) ).
thf(fact_3176_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_3177_rel__simps_I47_J,axiom,
! [K_1: int] :
( ( ( bit1 @ K_1 )
= min )
<=> ( K_1 = min ) ) ).
thf(fact_3178_rel__simps_I43_J,axiom,
! [L: int] :
( ( min
= ( bit1 @ L ) )
<=> ( min = L ) ) ).
thf(fact_3179_Bit1__Min,axiom,
( ( bit1 @ min )
= min ) ).
thf(fact_3180_succ__pred,axiom,
! [X: int] :
( ( succ @ ( pred @ X ) )
= X ) ).
thf(fact_3181_rel__simps_I12_J,axiom,
! [K_1: int] :
( ( ord_less_int @ ( bit1 @ K_1 ) @ pls )
<=> ( ord_less_int @ K_1 @ pls ) ) ).
thf(fact_3182_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_3183_rel__simps_I22_J,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ pls @ ( bit1 @ K_1 ) )
<=> ( ord_less_eq_int @ pls @ K_1 ) ) ).
thf(fact_3184_Bit1__def,axiom,
! [K_1: int] :
( ( bit1 @ K_1 )
= ( plus_plus_int @ ( plus_plus_int @ one_one_int @ K_1 ) @ K_1 ) ) ).
thf(fact_3185_rel__simps_I13_J,axiom,
! [K_1: int] :
( ( ord_less_int @ ( bit1 @ K_1 ) @ min )
<=> ( ord_less_int @ K_1 @ min ) ) ).
thf(fact_3186_rel__simps_I9_J,axiom,
! [K_1: int] :
( ( ord_less_int @ min @ ( bit1 @ K_1 ) )
<=> ( ord_less_int @ min @ K_1 ) ) ).
thf(fact_3187_rel__simps_I30_J,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ ( bit1 @ K_1 ) @ min )
<=> ( ord_less_eq_int @ K_1 @ min ) ) ).
thf(fact_3188_rel__simps_I26_J,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ min @ ( bit1 @ K_1 ) )
<=> ( ord_less_eq_int @ min @ K_1 ) ) ).
thf(fact_3189_neg__number__of__Bit1,axiom,
! [W: int] :
( ( nat_neg @ ( number_number_of_int @ ( bit1 @ W ) ) )
<=> ( nat_neg @ ( number_number_of_int @ W ) ) ) ).
thf(fact_3190_succ__Pls,axiom,
( ( succ @ pls )
= ( bit1 @ pls ) ) ).
thf(fact_3191_le__iff__pred__less,axiom,
! [K_1: int,L: int] :
( ( ord_less_eq_int @ K_1 @ L )
<=> ( ord_less_int @ ( pred @ K_1 ) @ L ) ) ).
thf(fact_3192_pred__Pls,axiom,
( ( pred @ pls )
= min ) ).
thf(fact_3193_pred__def,axiom,
! [K_1: int] :
( ( pred @ K_1 )
= ( minus_minus_int @ K_1 @ one_one_int ) ) ).
thf(fact_3194_add__Min,axiom,
! [K_1: int] :
( ( plus_plus_int @ min @ K_1 )
= ( pred @ K_1 ) ) ).
thf(fact_3195_add__Min__right,axiom,
! [K_1: int] :
( ( plus_plus_int @ K_1 @ min )
= ( pred @ K_1 ) ) ).
thf(fact_3196_rel__simps_I5_J,axiom,
! [K_1: int] :
( ( ord_less_int @ pls @ ( bit1 @ K_1 ) )
<=> ( ord_less_eq_int @ pls @ K_1 ) ) ).
thf(fact_3197_rel__simps_I29_J,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ ( bit1 @ K_1 ) @ pls )
<=> ( ord_less_int @ K_1 @ pls ) ) ).
thf(fact_3198_one__is__num__one,axiom,
( one_one_int
= ( number_number_of_int @ ( bit1 @ pls ) ) ) ).
thf(fact_3199_Numeral1__eq1__nat,axiom,
( one_one_nat
= ( number_number_of_nat @ ( bit1 @ pls ) ) ) ).
thf(fact_3200_nat__numeral__1__eq__1,axiom,
( ( number_number_of_nat @ ( bit1 @ pls ) )
= one_one_nat ) ).
thf(fact_3201_diff__bin__simps_I4_J,axiom,
! [L: int] :
( ( minus_minus_int @ pls @ ( bit1 @ L ) )
= ( bit1 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_3202_minus__Min,axiom,
( ( uminus_uminus_int @ min )
= ( bit1 @ pls ) ) ).
thf(fact_3203_numeral__3__eq__3,axiom,
( ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) )
= ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
thf(fact_3204_numeral__1__eq__Suc__0,axiom,
( ( number_number_of_nat @ ( bit1 @ pls ) )
= ( suc @ zero_zero_nat ) ) ).
thf(fact_3205_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) @ N ) ) ).
thf(fact_3206_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_3207_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_3208_power3__minus,axiom,
! [A: int] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( uminus_uminus_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3209_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_3210_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_3211_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_3212_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_3213_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_3214_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_3215_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_3216_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_3217_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_3218_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_3219_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_3220_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_3221_neg__zmod__mult__2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( div_mod_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) )
= ( minus_minus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( div_mod_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).
thf(fact_3222_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_3223_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_3224_rel__simps_I48_J,axiom,
! [K_1: int,L: int] :
( ( ( bit0 @ K_1 )
= ( bit0 @ L ) )
<=> ( K_1 = L ) ) ).
thf(fact_3225_rel__simps_I49_J,axiom,
! [K_1: int,L: int] :
( ( bit0 @ K_1 )
!= ( bit1 @ L ) ) ).
thf(fact_3226_rel__simps_I50_J,axiom,
! [K_1: int,L: int] :
( ( bit1 @ K_1 )
!= ( bit0 @ L ) ) ).
thf(fact_3227_Bit0__Pls,axiom,
( ( bit0 @ pls )
= pls ) ).
thf(fact_3228_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls
= ( bit0 @ L ) )
<=> ( pls = L ) ) ).
thf(fact_3229_rel__simps_I44_J,axiom,
! [K_1: int] :
( ( ( bit0 @ K_1 )
= pls )
<=> ( K_1 = pls ) ) ).
thf(fact_3230_less__int__code_I13_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit0 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_3231_rel__simps_I14_J,axiom,
! [K_1: int,L: int] :
( ( ord_less_int @ ( bit0 @ K_1 ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K_1 @ L ) ) ).
thf(fact_3232_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_3233_rel__simps_I31_J,axiom,
! [K_1: int,L: int] :
( ( ord_less_eq_int @ ( bit0 @ K_1 ) @ ( bit0 @ L ) )
<=> ( ord_less_eq_int @ K_1 @ L ) ) ).
thf(fact_3234_mult__Bit0,axiom,
! [K_1: int,L: int] :
( ( times_times_int @ ( bit0 @ K_1 ) @ L )
= ( bit0 @ ( times_times_int @ K_1 @ L ) ) ) ).
thf(fact_3235_Bit0__def,axiom,
! [K_1: int] :
( ( bit0 @ K_1 )
= ( plus_plus_int @ K_1 @ K_1 ) ) ).
thf(fact_3236_add__Bit0__Bit0,axiom,
! [K_1: int,L: int] :
( ( plus_plus_int @ ( bit0 @ K_1 ) @ ( bit0 @ L ) )
= ( bit0 @ ( plus_plus_int @ K_1 @ L ) ) ) ).
thf(fact_3237_diff__bin__simps_I7_J,axiom,
! [K_1: int,L: int] :
( ( minus_minus_int @ ( bit0 @ K_1 ) @ ( bit0 @ L ) )
= ( bit0 @ ( minus_minus_int @ K_1 @ L ) ) ) ).
thf(fact_3238_rel__simps_I42_J,axiom,
! [L: int] :
( min
!= ( bit0 @ L ) ) ).
thf(fact_3239_rel__simps_I45_J,axiom,
! [K_1: int] :
( ( bit0 @ K_1 )
!= min ) ).
thf(fact_3240_minus__Bit0,axiom,
! [K_1: int] :
( ( uminus_uminus_int @ ( bit0 @ K_1 ) )
= ( bit0 @ ( uminus_uminus_int @ K_1 ) ) ) ).
thf(fact_3241_less__int__code_I15_J,axiom,
! [K1: int,K2: int] :
( ( ord_less_int @ ( bit1 @ K1 ) @ ( bit0 @ K2 ) )
<=> ( ord_less_int @ K1 @ K2 ) ) ).
thf(fact_3242_rel__simps_I16_J,axiom,
! [K_1: int,L: int] :
( ( ord_less_int @ ( bit1 @ K_1 ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K_1 @ L ) ) ).
thf(fact_3243_rel__simps_I10_J,axiom,
! [K_1: int] :
( ( ord_less_int @ ( bit0 @ K_1 ) @ pls )
<=> ( ord_less_int @ K_1 @ pls ) ) ).
thf(fact_3244_rel__simps_I4_J,axiom,
! [K_1: int] :
( ( ord_less_int @ pls @ ( bit0 @ K_1 ) )
<=> ( ord_less_int @ pls @ K_1 ) ) ).
thf(fact_3245_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_3246_rel__simps_I32_J,axiom,
! [K_1: int,L: int] :
( ( ord_less_eq_int @ ( bit0 @ K_1 ) @ ( bit1 @ L ) )
<=> ( ord_less_eq_int @ K_1 @ L ) ) ).
thf(fact_3247_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_3248_rel__simps_I27_J,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ ( bit0 @ K_1 ) @ pls )
<=> ( ord_less_eq_int @ K_1 @ pls ) ) ).
thf(fact_3249_rel__simps_I21_J,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ pls @ ( bit0 @ K_1 ) )
<=> ( ord_less_eq_int @ pls @ K_1 ) ) ).
thf(fact_3250_add__Bit1__Bit0,axiom,
! [K_1: int,L: int] :
( ( plus_plus_int @ ( bit1 @ K_1 ) @ ( bit0 @ L ) )
= ( bit1 @ ( plus_plus_int @ K_1 @ L ) ) ) ).
thf(fact_3251_add__Bit0__Bit1,axiom,
! [K_1: int,L: int] :
( ( plus_plus_int @ ( bit0 @ K_1 ) @ ( bit1 @ L ) )
= ( bit1 @ ( plus_plus_int @ K_1 @ L ) ) ) ).
thf(fact_3252_diff__bin__simps_I9_J,axiom,
! [K_1: int,L: int] :
( ( minus_minus_int @ ( bit1 @ K_1 ) @ ( bit0 @ L ) )
= ( bit1 @ ( minus_minus_int @ K_1 @ L ) ) ) ).
thf(fact_3253_diff__bin__simps_I10_J,axiom,
! [K_1: int,L: int] :
( ( minus_minus_int @ ( bit1 @ K_1 ) @ ( bit1 @ L ) )
= ( bit0 @ ( minus_minus_int @ K_1 @ L ) ) ) ).
thf(fact_3254_diff__bin__simps_I3_J,axiom,
! [L: int] :
( ( minus_minus_int @ pls @ ( bit0 @ L ) )
= ( bit0 @ ( minus_minus_int @ pls @ L ) ) ) ).
thf(fact_3255_rel__simps_I8_J,axiom,
! [K_1: int] :
( ( ord_less_int @ min @ ( bit0 @ K_1 ) )
<=> ( ord_less_int @ min @ K_1 ) ) ).
thf(fact_3256_rel__simps_I28_J,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ ( bit0 @ K_1 ) @ min )
<=> ( ord_less_eq_int @ K_1 @ min ) ) ).
thf(fact_3257_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_3258_neg__number__of__Bit0,axiom,
! [W: int] :
( ( nat_neg @ ( number_number_of_int @ ( bit0 @ W ) ) )
<=> ( nat_neg @ ( number_number_of_int @ W ) ) ) ).
thf(fact_3259_pred__Bit1,axiom,
! [K_1: int] :
( ( pred @ ( bit1 @ K_1 ) )
= ( bit0 @ K_1 ) ) ).
thf(fact_3260_pred__Bit0,axiom,
! [K_1: int] :
( ( pred @ ( bit0 @ K_1 ) )
= ( bit1 @ ( pred @ K_1 ) ) ) ).
thf(fact_3261_pred__Min,axiom,
( ( pred @ min )
= ( bit0 @ min ) ) ).
thf(fact_3262_succ__Bit1,axiom,
! [K_1: int] :
( ( succ @ ( bit1 @ K_1 ) )
= ( bit0 @ ( succ @ K_1 ) ) ) ).
thf(fact_3263_succ__Bit0,axiom,
! [K_1: int] :
( ( succ @ ( bit0 @ K_1 ) )
= ( bit1 @ K_1 ) ) ).
thf(fact_3264_rel__simps_I15_J,axiom,
! [K_1: int,L: int] :
( ( ord_less_int @ ( bit0 @ K_1 ) @ ( bit1 @ L ) )
<=> ( ord_less_eq_int @ K_1 @ L ) ) ).
thf(fact_3265_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_3266_rel__simps_I33_J,axiom,
! [K_1: int,L: int] :
( ( ord_less_eq_int @ ( bit1 @ K_1 ) @ ( bit0 @ L ) )
<=> ( ord_less_int @ K_1 @ L ) ) ).
thf(fact_3267_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_3268_mult__Bit1,axiom,
! [K_1: int,L: int] :
( ( times_times_int @ ( bit1 @ K_1 ) @ L )
= ( plus_plus_int @ ( bit0 @ ( times_times_int @ K_1 @ L ) ) @ L ) ) ).
thf(fact_3269_rel__simps_I11_J,axiom,
! [K_1: int] :
( ( ord_less_int @ ( bit0 @ K_1 ) @ min )
<=> ( ord_less_eq_int @ K_1 @ min ) ) ).
thf(fact_3270_rel__simps_I25_J,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ min @ ( bit0 @ K_1 ) )
<=> ( ord_less_int @ min @ K_1 ) ) ).
thf(fact_3271_diff__bin__simps_I6_J,axiom,
! [L: int] :
( ( minus_minus_int @ min @ ( bit1 @ L ) )
= ( bit0 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_3272_diff__bin__simps_I5_J,axiom,
! [L: int] :
( ( minus_minus_int @ min @ ( bit0 @ L ) )
= ( bit1 @ ( minus_minus_int @ min @ L ) ) ) ).
thf(fact_3273_diff__bin__simps_I8_J,axiom,
! [K_1: int,L: int] :
( ( minus_minus_int @ ( bit0 @ K_1 ) @ ( bit1 @ L ) )
= ( bit1 @ ( minus_minus_int @ ( pred @ K_1 ) @ L ) ) ) ).
thf(fact_3274_add__Bit1__Bit1,axiom,
! [K_1: int,L: int] :
( ( plus_plus_int @ ( bit1 @ K_1 ) @ ( bit1 @ L ) )
= ( bit0 @ ( plus_plus_int @ K_1 @ ( succ @ L ) ) ) ) ).
thf(fact_3275_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_3276_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_3277_numeral__2__eq__2,axiom,
( ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_3278_semiring__norm_I115_J,axiom,
( ( suc @ ( suc @ zero_zero_nat ) )
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3279_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_3280_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( suc @ ( suc @ N ) ) ) ).
thf(fact_3281_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
thf(fact_3282_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_3283_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_3284_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_3285_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3286_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_3287_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_3288_cos__two__neq__zero,axiom,
( ( cos @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= zero_zero_real ) ).
thf(fact_3289_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_3290_power2__ge__self,axiom,
! [X: int] : ( ord_less_eq_int @ X @ ( power_power_int @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3291_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_3292_pi__ge__two,axiom,
ord_less_eq_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ).
thf(fact_3293_pi__less__4,axiom,
ord_less_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3294_cube__square,axiom,
! [A: int] :
( ( times_times_int @ A @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( power_power_int @ A @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3295_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_3296_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_3297_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_3298_pi__half__neq__two,axiom,
( ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3299_power2__eq__iff__abs__eq,axiom,
! [A: int,B: int] :
( ( ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( power_power_int @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
<=> ( ( abs_abs_int @ A )
= ( abs_abs_int @ B ) ) ) ).
thf(fact_3300_abs__power2__distrib,axiom,
! [A: int] :
( ( abs_abs_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( power_power_int @ ( abs_abs_int @ A ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3301_zEvenI,axiom,
! [X: int,K_1: int] :
( ( X
= ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_1 ) )
=> ( member_int @ X @ zEven ) ) ).
thf(fact_3302_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_3303_nat__2,axiom,
( ( nat_1 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_3304_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_3305_cos__two__le__zero,axiom,
ord_less_eq_real @ ( cos @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_real ).
thf(fact_3306_cos__two__less__zero,axiom,
ord_less_real @ ( cos @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ zero_zero_real ).
thf(fact_3307_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_3308_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_3309_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_3310_two__realpow__gt,axiom,
! [N: nat] : ( ord_less_real @ ( real_nat @ N ) @ ( power_power_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) ).
thf(fact_3311_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_3312_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_3313_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_3314_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_3315_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_3316_pi__half__neq__zero,axiom,
( ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= zero_zero_real ) ).
thf(fact_3317_pi__half__less__two,axiom,
ord_less_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_3318_real__average__minus__second,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( inverse_divide_real @ ( plus_plus_real @ B @ A ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A )
= ( inverse_divide_real @ ( minus_minus_real @ B @ A ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3319_real__average__minus__first,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( inverse_divide_real @ ( plus_plus_real @ A @ B ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ A )
= ( inverse_divide_real @ ( minus_minus_real @ B @ A ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3320_power2__eq1__iff,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= one_one_int )
=> ( ( abs_abs_int @ A )
= one_one_int ) ) ).
thf(fact_3321_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_3322_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_3323_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_3324_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_3325_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_3326_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_3327_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_3328_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_3329_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_3330_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_3331_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_3332_pi__half__ge__zero,axiom,
ord_less_eq_real @ zero_zero_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3333_pi__half__gt__zero,axiom,
ord_less_real @ zero_zero_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3334_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_3335_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_3336_zadd__power2,axiom,
! [A: int,B: int] :
( ( power_power_int @ ( plus_plus_int @ A @ B ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) @ B ) ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3337_zadd__power3,axiom,
! [A: int,B: int] :
( ( power_power_int @ ( plus_plus_int @ A @ B ) @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ B ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) @ A ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3338_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_3339_zspecial__product,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3340_m2pi__less__pi,axiom,
ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ) ) @ pi ).
thf(fact_3341_cos__pi__half,axiom,
( ( cos @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= zero_zero_real ) ).
thf(fact_3342_cos__two__pi,axiom,
( ( cos @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ) )
= one_one_real ) ).
thf(fact_3343_cos__periodic,axiom,
! [X: real] :
( ( cos @ ( plus_plus_real @ X @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ) ) )
= ( cos @ X ) ) ).
thf(fact_3344_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_3345_zOddI,axiom,
! [X: int,K_1: int] :
( ( X
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ K_1 ) @ one_one_int ) )
=> ( member_int @ X @ zOdd ) ) ).
thf(fact_3346_tan__periodic,axiom,
! [X: real] :
( ( tan @ ( plus_plus_real @ X @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ) ) )
= ( tan @ X ) ) ).
thf(fact_3347_tan__45,axiom,
( ( tan @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
= one_one_real ) ).
thf(fact_3348_arctan__ubound,axiom,
! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3349_arctan1__eq__pi4,axiom,
( ( arctan @ one_one_real )
= ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3350_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_3351_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_3352_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_3353_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_3354_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_3355_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_3356_zdiff__power3,axiom,
! [A: int,B: int] :
( ( power_power_int @ ( minus_minus_int @ A @ B ) @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) )
= ( minus_minus_int @ ( plus_plus_int @ ( minus_minus_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ B ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) @ A ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3357_zdiff__power2,axiom,
! [A: int,B: int] :
( ( power_power_int @ ( minus_minus_int @ A @ B ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( plus_plus_int @ ( minus_minus_int @ ( power_power_int @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) @ B ) ) @ ( power_power_int @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_3358_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_3359_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_3360_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_3361_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_3362_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_3363_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_3364_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_3365_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_3366_arctan__lbound,axiom,
! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( arctan @ Y ) ) ).
thf(fact_3367_neg__zdiv__mult__2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( div_div_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) )
= ( div_div_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
thf(fact_3368_pos__zdiv__mult__2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( div_div_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) )
= ( div_div_int @ B @ A ) ) ) ).
thf(fact_3369_pos__zmod__mult__2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( div_mod_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ A ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( div_mod_int @ B @ A ) ) ) ) ) ).
thf(fact_3370_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_3371_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_3372_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_3373_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_3374_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_3375_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_3376_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_3377_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_3378_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_3379_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_3380_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_3381_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_3382_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_3383_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_3384_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_3385_zOddE,axiom,
! [X: int] :
( ( member_int @ X @ zOdd )
=> ~ ! [K: int] :
( X
!= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ K ) @ one_one_int ) ) ) ).
thf(fact_3386_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_3387_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_3388_zEvenE,axiom,
! [X: int] :
( ( member_int @ X @ zEven )
=> ~ ! [K: int] :
( X
!= ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ K ) ) ) ).
thf(fact_3389_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_3390_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_3391_negDivAlg__eqn,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) )
=> ( ( negDivAlg @ A @ B )
= ( product_Pair_int_int @ ( number_number_of_int @ min ) @ ( plus_plus_int @ A @ B ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) )
=> ( ( negDivAlg @ A @ B )
= ( adjust @ B @ ( negDivAlg @ A @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) ) ) ) ) ) ).
thf(fact_3392_posDivAlg__eqn,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ( ord_less_int @ A @ B )
=> ( ( posDivAlg @ A @ B )
= ( product_Pair_int_int @ zero_zero_int @ A ) ) )
& ( ~ ( ord_less_int @ A @ B )
=> ( ( posDivAlg @ A @ B )
= ( adjust @ B @ ( posDivAlg @ A @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) ) ) ) ) ) ).
thf(fact_3393_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_3394_posDivAlg_Osimps,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_int @ A @ B )
| ( ord_less_eq_int @ B @ zero_zero_int ) )
=> ( ( posDivAlg @ A @ B )
= ( product_Pair_int_int @ zero_zero_int @ A ) ) )
& ( ~ ( ( ord_less_int @ A @ B )
| ( ord_less_eq_int @ B @ zero_zero_int ) )
=> ( ( posDivAlg @ A @ B )
= ( adjust @ B @ ( posDivAlg @ A @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) ) ) ) ) ).
thf(fact_3395_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_3396_negDivAlg_Osimps,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) )
| ( ord_less_eq_int @ B @ zero_zero_int ) )
=> ( ( negDivAlg @ A @ B )
= ( product_Pair_int_int @ ( number_number_of_int @ min ) @ ( plus_plus_int @ A @ B ) ) ) )
& ( ~ ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) )
| ( ord_less_eq_int @ B @ zero_zero_int ) )
=> ( ( negDivAlg @ A @ B )
= ( adjust @ B @ ( negDivAlg @ A @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) ) ) ) ) ).
thf(fact_3397_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_3398_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_3399_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_3400_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_3401_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_3402_real__of__nat__fact__not__zero,axiom,
! [N: nat] :
( ( real_nat @ ( fact_fact_nat @ N ) )
!= zero_zero_real ) ).
thf(fact_3403_real__of__nat__fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( real_nat @ ( fact_fact_nat @ N ) ) ) ).
thf(fact_3404_real__of__nat__fact__gt__zero,axiom,
! [N: nat] : ( ord_less_real @ zero_zero_real @ ( real_nat @ ( fact_fact_nat @ N ) ) ) ).
thf(fact_3405_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_3406_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_3407_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_3408_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_3409_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_3410_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_3411_odd__1__nat,axiom,
~ ( even_odd_even_nat @ one_one_nat ) ).
thf(fact_3412_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_3413_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_3414_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_3415_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_3416_even__Suc,axiom,
! [X: nat] :
( ( even_odd_even_nat @ ( suc @ X ) )
<=> ~ ( even_odd_even_nat @ X ) ) ).
thf(fact_3417_even__zero__nat,axiom,
even_odd_even_nat @ zero_zero_nat ).
thf(fact_3418_odd__pos,axiom,
! [N: nat] :
( ~ ( even_odd_even_nat @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
thf(fact_3419_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_3420_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_3421_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_3422_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_3423_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_3424_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_3425_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_3426_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_3427_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_3428_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_3429_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_3430_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_3431_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_3432_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_3433_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_3434_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_3435_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_3436_fact__nonzero__nat,axiom,
! [N: nat] :
( ( fact_fact_nat @ N )
!= zero_zero_nat ) ).
thf(fact_3437_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_3438_fact__1__nat,axiom,
( ( fact_fact_nat @ one_one_nat )
= one_one_nat ) ).
thf(fact_3439_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_3440_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_3441_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_3442_Fact_Ofact__0__nat,axiom,
( ( fact_fact_nat @ zero_zero_nat )
= ( suc @ zero_zero_nat ) ) ).
thf(fact_3443_fact__Suc__0__nat,axiom,
( ( fact_fact_nat @ ( suc @ zero_zero_nat ) )
= ( suc @ zero_zero_nat ) ) ).
thf(fact_3444_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_3445_fact__gt__zero__nat,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( fact_fact_nat @ N ) ) ).
thf(fact_3446_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_3447_Fact_Ofact__Suc,axiom,
! [X: nat] :
( ( fact_fact_nat @ ( suc @ X ) )
= ( times_times_nat @ ( suc @ X ) @ ( fact_fact_nat @ X ) ) ) ).
thf(fact_3448_fact__ge__one__nat,axiom,
! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( fact_fact_nat @ N ) ) ).
thf(fact_3449_fact__ge__Suc__0__nat,axiom,
! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( fact_fact_nat @ N ) ) ).
thf(fact_3450_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_3451_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_3452_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_3453_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_3454_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_3455_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_3456_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_3457_WilsonRuss_Oinv__def,axiom,
! [P_3: int,A: int] :
( ( inv @ P_3 @ A )
= ( div_mod_int @ ( power_power_int @ A @ ( nat_1 @ ( minus_minus_int @ P_3 @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ P_3 ) ) ).
thf(fact_3458_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_3459_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_3460_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_3461_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_3462_sin__zero,axiom,
( ( sin @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_3463_sin__minus,axiom,
! [X: real] :
( ( sin @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( sin @ X ) ) ) ).
thf(fact_3464_sin__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( sin @ X ) @ one_one_real ) ).
thf(fact_3465_sin__pi,axiom,
( ( sin @ pi )
= zero_zero_real ) ).
thf(fact_3466_ln__less__self,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ ( ln @ X ) @ X ) ) ).
thf(fact_3467_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_3468_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_3469_ln__one,axiom,
( ( ln @ one_one_real )
= zero_zero_real ) ).
thf(fact_3470_real__sgn__eq,axiom,
! [X: real] :
( ( sgn_sgn_real @ X )
= ( inverse_divide_real @ X @ ( abs_abs_real @ X ) ) ) ).
thf(fact_3471_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_3472_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_3473_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_3474_sin__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( number267125858f_real @ min ) @ ( sin @ X ) ) ).
thf(fact_3475_cos__one__sin__zero,axiom,
! [X: real] :
( ( ( cos @ X )
= one_one_real )
=> ( ( sin @ X )
= zero_zero_real ) ) ).
thf(fact_3476_abs__sin__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin @ X ) ) @ one_one_real ) ).
thf(fact_3477_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_3478_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_3479_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_3480_sin__periodic__pi2,axiom,
! [X: real] :
( ( sin @ ( plus_plus_real @ pi @ X ) )
= ( uminus_uminus_real @ ( sin @ X ) ) ) ).
thf(fact_3481_sin__periodic__pi,axiom,
! [X: real] :
( ( sin @ ( plus_plus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( sin @ X ) ) ) ).
thf(fact_3482_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_3483_tan__def,axiom,
! [X: real] :
( ( tan @ X )
= ( inverse_divide_real @ ( sin @ X ) @ ( cos @ X ) ) ) ).
thf(fact_3484_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_3485_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( ln @ X ) ) ) ).
thf(fact_3486_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_3487_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_3488_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_3489_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_3490_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_3491_real__sgn__pos,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( sgn_sgn_real @ X )
= one_one_real ) ) ).
thf(fact_3492_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_3493_sin__npi2,axiom,
! [N: nat] :
( ( sin @ ( times_times_real @ pi @ ( real_nat @ N ) ) )
= zero_zero_real ) ).
thf(fact_3494_sin__npi,axiom,
! [N: nat] :
( ( sin @ ( times_times_real @ ( real_nat @ N ) @ pi ) )
= zero_zero_real ) ).
thf(fact_3495_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_3496_sin__zero__abs__cos__one,axiom,
! [X: real] :
( ( ( sin @ X )
= zero_zero_real )
=> ( ( abs_abs_real @ ( cos @ X ) )
= one_one_real ) ) ).
thf(fact_3497_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_3498_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_3499_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_3500_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_3501_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_3502_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_3503_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_3504_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_3505_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_3506_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_3507_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_3508_real__sgn__neg,axiom,
! [X: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( sgn_sgn_real @ X )
= ( number267125858f_real @ min ) ) ) ).
thf(fact_3509_sgn__real__def,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( sgn_sgn_real @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( sgn_sgn_real @ A )
= one_one_real ) )
& ( ~ ( ord_less_real @ zero_zero_real @ A )
=> ( ( sgn_sgn_real @ A )
= ( uminus_uminus_real @ one_one_real ) ) ) ) ) ) ).
thf(fact_3510_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_3511_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_3512_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_3513_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_3514_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_3515_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_3516_sin__two__pi,axiom,
( ( sin @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ) )
= zero_zero_real ) ).
thf(fact_3517_sin__periodic,axiom,
! [X: real] :
( ( sin @ ( plus_plus_real @ X @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ pi ) ) )
= ( sin @ X ) ) ).
thf(fact_3518_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_3519_sin__pi__half,axiom,
( ( sin @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= one_one_real ) ).
thf(fact_3520_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_3521_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_3522_sin__cos__squared__add2__mult,axiom,
! [R_1: real,A: real] :
( ( plus_plus_real @ ( power_power_real @ ( times_times_real @ R_1 @ ( cos @ A ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( times_times_real @ R_1 @ ( sin @ A ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( power_power_real @ R_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3523_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_3524_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_3525_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_3526_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_3527_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_3528_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_3529_cos__sin__eq,axiom,
! [X: real] :
( ( cos @ X )
= ( sin @ ( minus_minus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ X ) ) ) ).
thf(fact_3530_sin__cos__eq,axiom,
! [X: real] :
( ( sin @ X )
= ( cos @ ( minus_minus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ X ) ) ) ).
thf(fact_3531_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_3532_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_3533_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_3534_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_3535_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_3536_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_3537_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_3538_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_3539_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_3540_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_3541_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_3542_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_3543_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_3544_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_3545_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_3546_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_3547_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_3548_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_3549_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_3550_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_3551_INVERSE__ZERO,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_3552_real__sqrt__minus,axiom,
! [X: real] :
( ( sqrt @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).
thf(fact_3553_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_3554_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_3555_real__sqrt__eq__0__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= zero_zero_real )
<=> ( X = zero_zero_real ) ) ).
thf(fact_3556_real__sqrt__zero,axiom,
( ( sqrt @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_3557_real__sqrt__inverse,axiom,
! [X: real] :
( ( sqrt @ ( inverse_inverse_real @ X ) )
= ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).
thf(fact_3558_real__sqrt__eq__iff,axiom,
! [X: real,Y: real] :
( ( ( sqrt @ X )
= ( sqrt @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_3559_real__sqrt__divide,axiom,
! [X: real,Y: real] :
( ( sqrt @ ( inverse_divide_real @ X @ Y ) )
= ( inverse_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
thf(fact_3560_real__sqrt__one,axiom,
( ( sqrt @ one_one_real )
= one_one_real ) ).
thf(fact_3561_real__sqrt__eq__1__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= one_one_real )
<=> ( X = one_one_real ) ) ).
thf(fact_3562_real__sqrt__power,axiom,
! [X: real,K_1: nat] :
( ( sqrt @ ( power_power_real @ X @ K_1 ) )
= ( power_power_real @ ( sqrt @ X ) @ K_1 ) ) ).
thf(fact_3563_real__sqrt__mult,axiom,
! [X: real,Y: real] :
( ( sqrt @ ( times_times_real @ X @ Y ) )
= ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
thf(fact_3564_real__sqrt__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
thf(fact_3565_real__sqrt__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
<=> ( ord_less_real @ X @ Y ) ) ).
thf(fact_3566_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_3567_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_3568_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_3569_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_3570_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_3571_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_3572_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_3573_real__sqrt__not__eq__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( sqrt @ X )
!= zero_zero_real ) ) ).
thf(fact_3574_real__sqrt__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
thf(fact_3575_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_3576_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_3577_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_3578_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_3579_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_3580_real__sqrt__abs2,axiom,
! [X: real] :
( ( sqrt @ ( times_times_real @ X @ X ) )
= ( abs_abs_real @ X ) ) ).
thf(fact_3581_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_3582_divide__real__def,axiom,
! [X: real,Y: real] :
( ( inverse_divide_real @ X @ Y )
= ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ).
thf(fact_3583_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_3584_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_3585_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_3586_real__mult__inverse__left,axiom,
! [X: real] :
( ( X != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ X ) @ X )
= one_one_real ) ) ).
thf(fact_3587_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_3588_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_3589_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_3590_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_3591_ln__inverse,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln @ ( inverse_inverse_real @ X ) )
= ( uminus_uminus_real @ ( ln @ X ) ) ) ) ).
thf(fact_3592_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_3593_real__sqrt__two__ge__zero,axiom,
ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3594_real__sqrt__two__gt__zero,axiom,
ord_less_real @ zero_zero_real @ ( sqrt @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3595_real__sqrt__two__gt__one,axiom,
ord_less_real @ one_one_real @ ( sqrt @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ).
thf(fact_3596_real__sqrt__abs,axiom,
! [X: real] :
( ( sqrt @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( abs_abs_real @ X ) ) ).
thf(fact_3597_tan__60,axiom,
( ( tan @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit1 @ ( bit1 @ pls ) ) ) ) )
= ( sqrt @ ( number267125858f_real @ ( bit1 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3598_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_3599_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_3600_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_3601_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_3602_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_3603_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( plus_plus_real @ B @ D ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ D @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ).
thf(fact_3604_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_3605_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_3606_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_3607_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_3608_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_3609_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_3610_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_3611_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_3612_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_3613_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_3614_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_3615_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_3616_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_3617_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_3618_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_3619_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_3620_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_3621_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_3622_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_3623_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_3624_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_3625_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_3626_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_3627_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_3628_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_3629_Bolzano__bisect__diff,axiom,
! [P: produc914805421l_real > $o,N: nat,A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( minus_minus_real @ ( produc556554744l_real @ ( bolzano_bisect @ P @ A @ B @ N ) ) @ ( produc1935615926l_real @ ( bolzano_bisect @ P @ A @ B @ N ) ) )
= ( inverse_divide_real @ ( minus_minus_real @ B @ A ) @ ( power_power_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) ) ) ).
thf(fact_3630_Bolzano__bisect_Osimps_I1_J,axiom,
! [P: produc914805421l_real > $o,A: real,B: real] :
( ( bolzano_bisect @ P @ A @ B @ zero_zero_nat )
= ( produc865579683l_real @ A @ B ) ) ).
thf(fact_3631_Bolzano__bisect__le,axiom,
! [P: produc914805421l_real > $o,A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ! [N_1: nat] : ( ord_less_eq_real @ ( produc1935615926l_real @ ( bolzano_bisect @ P @ A @ B @ N_1 ) ) @ ( produc556554744l_real @ ( bolzano_bisect @ P @ A @ B @ N_1 ) ) ) ) ).
thf(fact_3632_DERIV__sin,axiom,
! [X: real] : ( deriv_real @ sin @ X @ ( cos @ X ) ) ).
thf(fact_3633_DERIV__cos,axiom,
! [X: real] : ( deriv_real @ cos @ X @ ( uminus_uminus_real @ ( sin @ X ) ) ) ).
thf(fact_3634_Bolzano__bisect__fst__le__Suc,axiom,
! [P: produc914805421l_real > $o,A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ! [N_1: nat] : ( ord_less_eq_real @ ( produc1935615926l_real @ ( bolzano_bisect @ P @ A @ B @ N_1 ) ) @ ( produc1935615926l_real @ ( bolzano_bisect @ P @ A @ B @ ( suc @ N_1 ) ) ) ) ) ).
thf(fact_3635_Bolzano__bisect__Suc__le__snd,axiom,
! [P: produc914805421l_real > $o,A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ! [N_1: nat] : ( ord_less_eq_real @ ( produc556554744l_real @ ( bolzano_bisect @ P @ A @ B @ ( suc @ N_1 ) ) ) @ ( produc556554744l_real @ ( bolzano_bisect @ P @ A @ B @ N_1 ) ) ) ) ).
thf(fact_3636_DERIV__ln,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( deriv_real @ ln @ X @ ( inverse_inverse_real @ X ) ) ) ).
thf(fact_3637_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_3638_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_3639_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_3640_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_3641_DERIV__real__sqrt__generic,axiom,
! [D_1: real,X: real] :
( ( X != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ X )
=> ( D_1
= ( inverse_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
=> ( ( ( ord_less_real @ X @ zero_zero_real )
=> ( D_1
= ( inverse_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
=> ( deriv_real @ sqrt @ X @ D_1 ) ) ) ) ).
thf(fact_3642_DERIV__const__average,axiom,
! [V: real > real,K_1: real,A: real,B: real] :
( ( A != B )
=> ( ! [X_1: real] : ( deriv_real @ V @ X_1 @ K_1 )
=> ( ( V @ ( inverse_divide_real @ ( plus_plus_real @ A @ B ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( inverse_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_3643_DERIV__local__max,axiom,
! [D: real,F: real > real,X: real,L: real] :
( ( deriv_real @ F @ X @ L )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y_1: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y_1 ) ) @ D )
=> ( ord_less_eq_real @ ( F @ Y_1 ) @ ( F @ X ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
thf(fact_3644_DERIV__local__min,axiom,
! [D: real,F: real > real,X: real,L: real] :
( ( deriv_real @ F @ X @ L )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y_1: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y_1 ) ) @ D )
=> ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y_1 ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
thf(fact_3645_DERIV__local__const,axiom,
! [D: real,F: real > real,X: real,L: real] :
( ( deriv_real @ F @ X @ L )
=> ( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Y_1: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y_1 ) ) @ D )
=> ( ( F @ X )
= ( F @ Y_1 ) ) )
=> ( L = zero_zero_real ) ) ) ) ).
thf(fact_3646_not__P__Bolzano__bisect,axiom,
! [N: nat,A: real,B: real,P: produc914805421l_real > $o] :
( ! [A_2: real,B_4: real,C_2: real] :
( ( P @ ( produc865579683l_real @ A_2 @ B_4 ) )
=> ( ( P @ ( produc865579683l_real @ B_4 @ C_2 ) )
=> ( ( ord_less_eq_real @ A_2 @ B_4 )
=> ( ( ord_less_eq_real @ B_4 @ C_2 )
=> ( P @ ( produc865579683l_real @ A_2 @ C_2 ) ) ) ) ) )
=> ( ~ ( P @ ( produc865579683l_real @ A @ B ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ~ ( P @ ( produc865579683l_real @ ( produc1935615926l_real @ ( bolzano_bisect @ P @ A @ B @ N ) ) @ ( produc556554744l_real @ ( bolzano_bisect @ P @ A @ B @ N ) ) ) ) ) ) ) ).
thf(fact_3647_not__P__Bolzano__bisect_H,axiom,
! [A: real,B: real,P: produc914805421l_real > $o] :
( ! [A_2: real,B_4: real,C_2: real] :
( ( ( P @ ( produc865579683l_real @ A_2 @ B_4 ) )
& ( P @ ( produc865579683l_real @ B_4 @ C_2 ) )
& ( ord_less_eq_real @ A_2 @ B_4 )
& ( ord_less_eq_real @ B_4 @ C_2 ) )
=> ( P @ ( produc865579683l_real @ A_2 @ C_2 ) ) )
=> ( ~ ( P @ ( produc865579683l_real @ A @ B ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ! [N_1: nat] :
~ ( P @ ( produc865579683l_real @ ( produc1935615926l_real @ ( bolzano_bisect @ P @ A @ B @ N_1 ) ) @ ( produc556554744l_real @ ( bolzano_bisect @ P @ A @ B @ N_1 ) ) ) ) ) ) ) ).
thf(fact_3648_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_3649_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_3650_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_3651_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_3652_DERIV__const__ratio__const2,axiom,
! [F: real > real,K_1: real,A: real,B: real] :
( ( A != B )
=> ( ! [X_1: real] : ( deriv_real @ F @ X_1 @ K_1 )
=> ( ( inverse_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
= K_1 ) ) ) ).
thf(fact_3653_DERIV__const__ratio__const,axiom,
! [F: real > real,K_1: real,A: real,B: real] :
( ( A != B )
=> ( ! [X_1: real] : ( deriv_real @ F @ X_1 @ K_1 )
=> ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
= ( times_times_real @ ( minus_minus_real @ B @ A ) @ K_1 ) ) ) ) ).
thf(fact_3654_DERIV__real__root__generic,axiom,
! [D_1: real,X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( X != zero_zero_real )
=> ( ( ( even_odd_even_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( D_1
= ( inverse_inverse_real @ ( times_times_real @ ( real_nat @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
=> ( ( ( even_odd_even_nat @ N )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( D_1
= ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( real_nat @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
=> ( ( ~ ( even_odd_even_nat @ N )
=> ( D_1
= ( inverse_inverse_real @ ( times_times_real @ ( real_nat @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
=> ( deriv_real @ ( root @ N ) @ X @ D_1 ) ) ) ) ) ) ).
thf(fact_3655_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_3656_ln__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( ln @ X ) @ X ) ) ).
thf(fact_3657_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_3658_real__root__zero,axiom,
! [N: nat] :
( ( root @ N @ zero_zero_real )
= zero_zero_real ) ).
thf(fact_3659_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ zero_zero_nat ) @ X )
= X ) ).
thf(fact_3660_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_3661_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_3662_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_3663_odd__real__root__pow,axiom,
! [X: real,N: nat] :
( ~ ( even_odd_even_nat @ N )
=> ( ( power_power_real @ ( root @ N @ X ) @ N )
= X ) ) ).
thf(fact_3664_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_3665_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_3666_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_3667_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_3668_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_3669_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_3670_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_3671_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_3672_real__root__power,axiom,
! [X: real,K_1: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( power_power_real @ X @ K_1 ) )
= ( power_power_real @ ( root @ N @ X ) @ K_1 ) ) ) ).
thf(fact_3673_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_3674_real__root__one,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ one_one_real )
= one_one_real ) ) ).
thf(fact_3675_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_3676_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_3677_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_3678_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_3679_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_3680_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_3681_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_3682_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_3683_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_3684_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_3685_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_3686_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_3687_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_3688_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_3689_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_3690_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_3691_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_3692_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_3693_real__root__pos__pos,axiom,
! [X_4: real,N_5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_5 )
=> ( ( ord_less_real @ zero_zero_real @ X_4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N_5 @ X_4 ) ) ) ) ).
thf(fact_3694_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_3695_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_3696_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_3697_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_3698_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_3699_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_3700_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_3701_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_3702_sqrt__def,axiom,
( sqrt
= ( root @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_3703_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_3704_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_3705_negDivAlg_Opsimps,axiom,
! [A: int,B: int] :
( ( accp_P2006205492nt_int @ negDivAlg_rel @ ( product_Pair_int_int @ A @ B ) )
=> ( ( ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) )
| ( ord_less_eq_int @ B @ zero_zero_int ) )
=> ( ( negDivAlg @ A @ B )
= ( product_Pair_int_int @ ( number_number_of_int @ min ) @ ( plus_plus_int @ A @ B ) ) ) )
& ( ~ ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) )
| ( ord_less_eq_int @ B @ zero_zero_int ) )
=> ( ( negDivAlg @ A @ B )
= ( adjust @ B @ ( negDivAlg @ A @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) ) ) ) ) ) ).
thf(fact_3706_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_3707_MVT2,axiom,
! [F: real > real,F_1: real > real,A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A @ X_1 )
& ( ord_less_eq_real @ X_1 @ B ) )
=> ( deriv_real @ F @ X_1 @ ( F_1 @ X_1 ) ) )
=> ? [Z: real] :
( ( ord_less_real @ A @ Z )
& ( ord_less_real @ Z @ B )
& ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
= ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F_1 @ Z ) ) ) ) ) ) ).
thf(fact_3708_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_3709_exp__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
thf(fact_3710_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_3711_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_3712_exp__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
=> ( ord_less_real @ X @ Y ) ) ).
thf(fact_3713_exp__inj__iff,axiom,
! [X: real,Y: real] :
( ( ( exp_real @ X )
= ( exp_real @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_3714_abs__exp__cancel,axiom,
! [X: real] :
( ( abs_abs_real @ ( exp_real @ X ) )
= ( exp_real @ X ) ) ).
thf(fact_3715_ln__unique,axiom,
! [Y: real,X: real] :
( ( ( exp_real @ Y )
= X )
=> ( ( ln @ X )
= Y ) ) ).
thf(fact_3716_exp__ln__eq,axiom,
! [U: real,X: real] :
( ( ( exp_real @ U )
= X )
=> ( ( ln @ X )
= U ) ) ).
thf(fact_3717_ln__exp,axiom,
! [X: real] :
( ( ln @ ( exp_real @ X ) )
= X ) ).
thf(fact_3718_not__exp__le__zero,axiom,
! [X: real] :
~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).
thf(fact_3719_exp__ge__zero,axiom,
! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).
thf(fact_3720_exp__gt__zero,axiom,
! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).
thf(fact_3721_not__exp__less__zero,axiom,
! [X: real] :
~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).
thf(fact_3722_exp__eq__one__iff,axiom,
! [X: real] :
( ( ( exp_real @ X )
= one_one_real )
<=> ( X = zero_zero_real ) ) ).
thf(fact_3723_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_3724_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_3725_exp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
thf(fact_3726_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_3727_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_3728_exp__ge__add__one__self,axiom,
! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).
thf(fact_3729_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_3730_exp__ln__iff,axiom,
! [X: real] :
( ( ( exp_real @ ( ln @ X ) )
= X )
<=> ( ord_less_real @ zero_zero_real @ X ) ) ).
thf(fact_3731_exp__ln,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( exp_real @ ( ln @ X ) )
= X ) ) ).
thf(fact_3732_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_3733_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_3734_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_3735_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_3736_posDivAlg_Opsimps,axiom,
! [A: int,B: int] :
( ( accp_P2006205492nt_int @ posDivAlg_rel @ ( product_Pair_int_int @ A @ B ) )
=> ( ( ( ( ord_less_int @ A @ B )
| ( ord_less_eq_int @ B @ zero_zero_int ) )
=> ( ( posDivAlg @ A @ B )
= ( product_Pair_int_int @ zero_zero_int @ A ) ) )
& ( ~ ( ( ord_less_int @ A @ B )
| ( ord_less_eq_int @ B @ zero_zero_int ) )
=> ( ( posDivAlg @ A @ B )
= ( adjust @ B @ ( posDivAlg @ A @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ B ) ) ) ) ) ) ) ).
thf(fact_3737_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_3738_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_3739_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_3740_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_3741_exp__total,axiom,
! [Y: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ? [X_1: real] :
( ( exp_real @ X_1 )
= Y ) ) ).
thf(fact_3742_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_3743_log__one,axiom,
! [A: real] :
( ( log @ A @ one_one_real )
= zero_zero_real ) ).
thf(fact_3744_Log_Olog__def,axiom,
! [A: real,X: real] :
( ( log @ A @ X )
= ( inverse_divide_real @ ( ln @ X ) @ ( ln @ A ) ) ) ).
thf(fact_3745_log__less__cancel__iff,axiom,
! [Y: real,X: real,A: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
<=> ( ord_less_real @ X @ Y ) ) ) ) ) ).
thf(fact_3746_log__eq__one,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ A )
= one_one_real ) ) ) ).
thf(fact_3747_log__ln,axiom,
! [X: real] :
( ( ln @ X )
= ( log @ ( exp_real @ one_one_real ) @ X ) ) ).
thf(fact_3748_log__le__cancel__iff,axiom,
! [Y: real,X: real,A: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
<=> ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).
thf(fact_3749_log__nat__power,axiom,
! [B: real,N: nat,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ B @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( real_nat @ N ) @ ( log @ B @ X ) ) ) ) ).
thf(fact_3750_log__mult,axiom,
! [Y: real,X: real,A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( log @ A @ ( times_times_real @ X @ Y ) )
= ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
thf(fact_3751_log__divide,axiom,
! [Y: real,X: real,A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( log @ A @ ( inverse_divide_real @ X @ Y ) )
= ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
thf(fact_3752_log__inverse,axiom,
! [X: real,A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ A @ ( inverse_inverse_real @ X ) )
= ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).
thf(fact_3753_log__eq__div__ln__mult__log,axiom,
! [X: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ A @ X )
= ( times_times_real @ ( inverse_divide_real @ ( ln @ B ) @ ( ln @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).
thf(fact_3754_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_3755_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_3756_Code__Numeral_Oint__of__code,axiom,
! [K_1: code_code_numeral] :
( ( ( K_1 = zero_z126310315umeral )
=> ( ( code_int_of @ K_1 )
= zero_zero_int ) )
& ( ( K_1 != zero_z126310315umeral )
=> ( ( ( ( div_mo1740067990umeral @ K_1 @ ( number1443263063umeral @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_z126310315umeral )
=> ( ( code_int_of @ K_1 )
= ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( code_int_of @ ( div_di1218280263umeral @ K_1 @ ( number1443263063umeral @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) )
& ( ( ( div_mo1740067990umeral @ K_1 @ ( number1443263063umeral @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= zero_z126310315umeral )
=> ( ( code_int_of @ K_1 )
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( code_int_of @ ( div_di1218280263umeral @ K_1 @ ( number1443263063umeral @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
thf(fact_3757_inv__inv,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_eq_int @ ( number_number_of_int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P_3 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ P_3 )
=> ( ( inv @ P_3 @ ( inv @ P_3 @ A ) )
= A ) ) ) ) ) ).
thf(fact_3758_even__neg,axiom,
! [X: int] :
( ( even_odd_even_int @ ( uminus_uminus_int @ X ) )
<=> ( even_odd_even_int @ X ) ) ).
thf(fact_3759_code__numeral__zero__minus__one,axiom,
( ( minus_1690775515umeral @ zero_z126310315umeral @ one_on1645066479umeral )
= zero_z126310315umeral ) ).
thf(fact_3760_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_3761_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_3762_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_3763_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_3764_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_3765_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_3766_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_3767_anything__times__even,axiom,
! [X: int,Y: int] :
( ( even_odd_even_int @ Y )
=> ( even_odd_even_int @ ( times_times_int @ X @ Y ) ) ) ).
thf(fact_3768_even__times__anything,axiom,
! [Y: int,X: int] :
( ( even_odd_even_int @ X )
=> ( even_odd_even_int @ ( times_times_int @ X @ Y ) ) ) ).
thf(fact_3769_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_3770_odd__one__int,axiom,
~ ( even_odd_even_int @ one_one_int ) ).
thf(fact_3771_even__zero__int,axiom,
even_odd_even_int @ zero_zero_int ).
thf(fact_3772_odd__pow,axiom,
! [N: nat,X: int] :
( ~ ( even_odd_even_int @ X )
=> ~ ( even_odd_even_int @ ( power_power_int @ X @ N ) ) ) ).
thf(fact_3773_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_3774_zero__code__numeral__code,axiom,
( zero_z126310315umeral
= ( number1443263063umeral @ pls ) ) ).
thf(fact_3775_zprime__2,axiom,
zprime @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_3776_inv__less__p__minus__1,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ord_less_int @ ( inv @ P_3 @ A ) @ ( minus_minus_int @ P_3 @ one_one_int ) ) ) ) ) ).
thf(fact_3777_inv__g__1,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ord_less_int @ one_one_int @ ( inv @ P_3 @ A ) ) ) ) ) ).
thf(fact_3778_inv__not__p__minus__1,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( inv @ P_3 @ A )
!= ( minus_minus_int @ P_3 @ one_one_int ) ) ) ) ) ).
thf(fact_3779_inv__not__1,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( inv @ P_3 @ A )
!= one_one_int ) ) ) ) ).
thf(fact_3780_inv__distinct,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( A
!= ( inv @ P_3 @ A ) ) ) ) ) ).
thf(fact_3781_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_3782_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_3783_inv__not__0,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( inv @ P_3 @ A )
!= zero_zero_int ) ) ) ) ).
thf(fact_3784_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_3785_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_3786_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_3787_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_3788_one__code__numeral__code,axiom,
( one_on1645066479umeral
= ( number1443263063umeral @ ( bit1 @ pls ) ) ) ).
thf(fact_3789_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_3790_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_3791_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_3792_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_3793_wset__mem__inv__mem,axiom,
! [B: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_eq_int @ ( number_number_of_int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P_3 )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( member_int @ B @ ( wset @ A @ P_3 ) )
=> ( member_int @ ( inv @ P_3 @ B ) @ ( wset @ A @ P_3 ) ) ) ) ) ) ).
thf(fact_3794_wset__inv__mem__mem,axiom,
! [B: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_eq_int @ ( number_number_of_int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P_3 )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ B @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( member_int @ ( inv @ P_3 @ B ) @ ( wset @ A @ P_3 ) )
=> ( member_int @ B @ ( wset @ A @ P_3 ) ) ) ) ) ) ) ) ).
thf(fact_3795_Euler__part2,axiom,
! [A: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ( zcong @ A @ zero_zero_int @ P_3 )
=> ( zcong @ zero_zero_int @ ( power_power_int @ A @ ( nat_1 @ ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ P_3 ) ) ) ) ).
thf(fact_3796_zcong__id,axiom,
! [M: int] : ( zcong @ M @ zero_zero_int @ M ) ).
thf(fact_3797_IntPrimes_Ozcong__zero,axiom,
! [A: int,B: int] :
( ( zcong @ A @ B @ zero_zero_int )
<=> ( A = B ) ) ).
thf(fact_3798_zcong__zmod,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
<=> ( zcong @ ( div_mod_int @ A @ M ) @ ( div_mod_int @ B @ M ) @ M ) ) ).
thf(fact_3799_zcong__1,axiom,
! [A: int,B: int] : ( zcong @ A @ B @ one_one_int ) ).
thf(fact_3800_zcong__zmult,axiom,
! [C: int,D: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ C @ D @ M )
=> ( zcong @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) @ M ) ) ) ).
thf(fact_3801_zcong__scalar2,axiom,
! [K_1: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( zcong @ ( times_times_int @ K_1 @ A ) @ ( times_times_int @ K_1 @ B ) @ M ) ) ).
thf(fact_3802_zcong__scalar,axiom,
! [K_1: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( zcong @ ( times_times_int @ A @ K_1 ) @ ( times_times_int @ B @ K_1 ) @ M ) ) ).
thf(fact_3803_zcong__zmult__prop2,axiom,
! [C: int,D: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ C @ ( times_times_int @ D @ A ) @ M )
<=> ( zcong @ C @ ( times_times_int @ D @ B ) @ M ) ) ) ).
thf(fact_3804_zcong__zmult__prop1,axiom,
! [C: int,D: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ C @ ( times_times_int @ A @ D ) @ M )
<=> ( zcong @ C @ ( times_times_int @ B @ D ) @ M ) ) ) ).
thf(fact_3805_zcong__zmult__self,axiom,
! [A: int,M: int,B: int] : ( zcong @ ( times_times_int @ A @ M ) @ ( times_times_int @ B @ M ) @ M ) ).
thf(fact_3806_zcong__zadd,axiom,
! [C: int,D: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ C @ D @ M )
=> ( zcong @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) @ M ) ) ) ).
thf(fact_3807_zcong__shift,axiom,
! [C: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( zcong @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) @ M ) ) ).
thf(fact_3808_zcong__zdiff,axiom,
! [C: int,D: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ C @ D @ M )
=> ( zcong @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) @ M ) ) ) ).
thf(fact_3809_zcong__zminus,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ ( uminus_uminus_int @ M ) )
<=> ( zcong @ A @ B @ M ) ) ).
thf(fact_3810_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_3811_zcong__sym,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
<=> ( zcong @ B @ A @ M ) ) ).
thf(fact_3812_zcong__refl,axiom,
! [K_1: int,M: int] : ( zcong @ K_1 @ K_1 @ M ) ).
thf(fact_3813_zcong__eq__trans,axiom,
! [D: int,C: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( B = C )
=> ( ( zcong @ C @ D @ M )
=> ( zcong @ A @ D @ M ) ) ) ) ).
thf(fact_3814_zcong__trans,axiom,
! [C: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ B @ C @ M )
=> ( zcong @ A @ C @ M ) ) ) ).
thf(fact_3815_zcong__not,axiom,
! [B: int,M: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ M )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ A )
=> ~ ( zcong @ A @ B @ M ) ) ) ) ) ).
thf(fact_3816_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_3817_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_3818_zcong__iff__lin,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
<=> ? [K: int] :
( B
= ( plus_plus_int @ A @ ( times_times_int @ M @ K ) ) ) ) ).
thf(fact_3819_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_3820_zcong__zless__0,axiom,
! [M: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ M )
=> ( ( zcong @ A @ zero_zero_int @ M )
=> ( A = zero_zero_int ) ) ) ) ).
thf(fact_3821_zcong__zless__imp__eq,axiom,
! [B: int,M: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ M )
=> ( ( zcong @ A @ B @ M )
=> ( A = B ) ) ) ) ) ) ).
thf(fact_3822_zcong__zmod__eq,axiom,
! [A: int,B: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( zcong @ A @ B @ M )
<=> ( ( div_mod_int @ A @ M )
= ( div_mod_int @ B @ M ) ) ) ) ).
thf(fact_3823_inv__not__p__minus__1__aux,axiom,
! [A: int,P_3: int] :
( ( zcong @ ( times_times_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) ) @ one_one_int @ P_3 )
<=> ( zcong @ A @ ( minus_minus_int @ P_3 @ one_one_int ) @ P_3 ) ) ).
thf(fact_3824_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_3825_wset__mem__mem,axiom,
! [P_3: int,A: int] :
( ( ord_less_int @ one_one_int @ A )
=> ( member_int @ A @ ( wset @ A @ P_3 ) ) ) ).
thf(fact_3826_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_3827_zcong__zprime__prod__zero__contra,axiom,
! [B: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
& ~ ( zcong @ B @ zero_zero_int @ P_3 ) )
=> ~ ( zcong @ ( times_times_int @ A @ B ) @ zero_zero_int @ P_3 ) ) ) ) ).
thf(fact_3828_zcong__zprime__prod__zero,axiom,
! [B: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( zcong @ ( times_times_int @ A @ B ) @ zero_zero_int @ P_3 )
=> ( ( zcong @ A @ zero_zero_int @ P_3 )
| ( zcong @ B @ zero_zero_int @ P_3 ) ) ) ) ) ).
thf(fact_3829_wset__subset,axiom,
! [B: int,P_3: int,A: int] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( member_int @ B @ ( wset @ ( minus_minus_int @ A @ one_one_int ) @ P_3 ) )
=> ( member_int @ B @ ( wset @ A @ P_3 ) ) ) ) ).
thf(fact_3830_zcong__square,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( zcong @ ( times_times_int @ A @ A ) @ one_one_int @ P_3 )
=> ( ( zcong @ A @ one_one_int @ P_3 )
| ( zcong @ A @ ( minus_minus_int @ P_3 @ one_one_int ) @ P_3 ) ) ) ) ) ).
thf(fact_3831_zcong__square__zless,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ P_3 )
=> ( ( zcong @ ( times_times_int @ A @ A ) @ one_one_int @ P_3 )
=> ( ( A = one_one_int )
| ( A
= ( minus_minus_int @ P_3 @ one_one_int ) ) ) ) ) ) ) ).
thf(fact_3832_wset__g__1,axiom,
! [B: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( member_int @ B @ ( wset @ A @ P_3 ) )
=> ( ord_less_int @ one_one_int @ B ) ) ) ) ).
thf(fact_3833_wset__less,axiom,
! [B: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( member_int @ B @ ( wset @ A @ P_3 ) )
=> ( ord_less_int @ B @ ( minus_minus_int @ P_3 @ one_one_int ) ) ) ) ) ).
thf(fact_3834_inv__is__inv,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ P_3 )
=> ( zcong @ ( times_times_int @ A @ ( inv @ P_3 @ A ) ) @ one_one_int @ P_3 ) ) ) ) ).
thf(fact_3835_wset__mem__imp__or,axiom,
! [B: int,P_3: int,A: int] :
( ( ord_less_int @ one_one_int @ A )
=> ( ~ ( member_int @ B @ ( wset @ ( minus_minus_int @ A @ one_one_int ) @ P_3 ) )
=> ( ( member_int @ B @ ( wset @ A @ P_3 ) )
=> ( ( B = A )
| ( B
= ( inv @ P_3 @ A ) ) ) ) ) ) ).
thf(fact_3836_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_3837_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_3838_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_3839_neg__one__power__eq__mod__m,axiom,
! [J: nat,K_1: nat,M: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ M )
=> ( ( zcong @ ( power_power_int @ ( number_number_of_int @ min ) @ J ) @ ( power_power_int @ ( number_number_of_int @ min ) @ K_1 ) @ M )
=> ( ( power_power_int @ ( number_number_of_int @ min ) @ J )
= ( power_power_int @ ( number_number_of_int @ min ) @ K_1 ) ) ) ) ).
thf(fact_3840_wset__mem,axiom,
! [B: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( member_int @ B @ ( wset @ A @ P_3 ) ) ) ) ) ) ).
thf(fact_3841_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_3842_Euler__Criterion,axiom,
! [A: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( zcong @ ( legendre @ A @ P_3 ) @ ( power_power_int @ A @ ( nat_1 @ ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ P_3 ) ) ) ).
thf(fact_3843_zfact__prop,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A )
=> ( zcong @ ( zfact @ ( minus_minus_int @ P_3 @ one_one_int ) ) @ ( power_power_int @ A @ ( nat_1 @ ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ P_3 ) ) ) ) ) ).
thf(fact_3844_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_3845_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_3846_Legendre__def,axiom,
! [A: int,P_3: int] :
( ( ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ( legendre @ A @ P_3 )
= zero_zero_int ) )
& ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ( ( quadRes @ P_3 @ A )
=> ( ( legendre @ A @ P_3 )
= one_one_int ) )
& ( ~ ( quadRes @ P_3 @ A )
=> ( ( legendre @ A @ P_3 )
= ( number_number_of_int @ min ) ) ) ) ) ) ).
thf(fact_3847_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_3848_zcong__zless__unique,axiom,
! [A: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ? [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
& ( ord_less_int @ X_1 @ M )
& ( zcong @ A @ X_1 @ M )
& ! [Y_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Y_1 )
& ( ord_less_int @ Y_1 @ M )
& ( zcong @ A @ Y_1 @ M ) )
=> ( Y_1 = X_1 ) ) ) ) ).
thf(fact_3849_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_3850_mod__mod__is__mod,axiom,
! [X: int,M: int] : ( zcong @ X @ ( div_mod_int @ X @ M ) @ M ) ).
thf(fact_3851_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_3852_MultInvPair__distinct,axiom,
! [J: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ~ ( zcong @ J @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A )
=> ~ ( zcong @ J @ ( times_times_int @ A @ ( multInv @ P_3 @ J ) ) @ P_3 ) ) ) ) ) ) ).
thf(fact_3853_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_3854_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_3855_phi__prime,axiom,
! [P_3: int] :
( ( zprime @ P_3 )
=> ( ( phi @ P_3 )
= ( nat_1 @ ( minus_minus_int @ P_3 @ one_one_int ) ) ) ) ).
thf(fact_3856_aux______1,axiom,
! [J: int,A: int,P_3: int,K_1: int] :
( ( zcong @ J @ ( times_times_int @ A @ ( multInv @ P_3 @ K_1 ) ) @ P_3 )
=> ( zcong @ ( times_times_int @ J @ K_1 ) @ ( times_times_int @ ( times_times_int @ A @ ( multInv @ P_3 @ K_1 ) ) @ K_1 ) @ P_3 ) ) ).
thf(fact_3857_aux______3,axiom,
! [J: int,K_1: int,A: int,P_3: int] :
( ( zcong @ ( times_times_int @ J @ K_1 ) @ A @ P_3 )
=> ( zcong @ ( times_times_int @ ( times_times_int @ ( multInv @ P_3 @ J ) @ J ) @ K_1 ) @ ( times_times_int @ ( multInv @ P_3 @ J ) @ A ) @ P_3 ) ) ).
thf(fact_3858_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_3859_MultInv__zcong__prop1,axiom,
! [A: int,J: int,K_1: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zcong @ J @ K_1 @ P_3 )
=> ( zcong @ ( times_times_int @ A @ ( multInv @ P_3 @ J ) ) @ ( times_times_int @ A @ ( multInv @ P_3 @ K_1 ) ) @ P_3 ) ) ) ).
thf(fact_3860_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_3861_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_3862_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_3863_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_3864_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_3865_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_3866_MultInv__zcong__prop3,axiom,
! [J: int,K_1: int,A: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ~ ( zcong @ K_1 @ zero_zero_int @ P_3 )
=> ( ~ ( zcong @ J @ zero_zero_int @ P_3 )
=> ( ( zcong @ ( times_times_int @ A @ ( multInv @ P_3 @ J ) ) @ ( times_times_int @ A @ ( multInv @ P_3 @ K_1 ) ) @ P_3 )
=> ( zcong @ J @ K_1 @ P_3 ) ) ) ) ) ) ) ).
thf(fact_3867_MultInv__zcong__prop2,axiom,
! [A: int,J: int,K_1: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ K_1 @ zero_zero_int @ P_3 )
=> ( ~ ( zcong @ J @ zero_zero_int @ P_3 )
=> ( ( zcong @ J @ ( times_times_int @ A @ ( multInv @ P_3 @ K_1 ) ) @ P_3 )
=> ( zcong @ K_1 @ ( times_times_int @ A @ ( multInv @ P_3 @ J ) ) @ P_3 ) ) ) ) ) ) ).
thf(fact_3868_aux______2,axiom,
! [J: int,A: int,K_1: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ K_1 @ zero_zero_int @ P_3 )
=> ( ( zcong @ ( times_times_int @ J @ K_1 ) @ ( times_times_int @ ( times_times_int @ A @ ( multInv @ P_3 @ K_1 ) ) @ K_1 ) @ P_3 )
=> ( zcong @ ( times_times_int @ J @ K_1 ) @ A @ P_3 ) ) ) ) ) ).
thf(fact_3869_aux______4,axiom,
! [K_1: int,A: int,J: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ( zprime @ P_3 )
=> ( ~ ( zcong @ J @ zero_zero_int @ P_3 )
=> ( ( zcong @ ( times_times_int @ ( times_times_int @ ( multInv @ P_3 @ J ) @ J ) @ K_1 ) @ ( times_times_int @ ( multInv @ P_3 @ J ) @ A ) @ P_3 )
=> ( zcong @ K_1 @ ( times_times_int @ A @ ( multInv @ P_3 @ J ) ) @ P_3 ) ) ) ) ) ).
thf(fact_3870_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_3871_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_3872_complex__inverse,axiom,
! [A: real,B: real] :
( ( invers1449016382omplex @ ( complex_1 @ A @ B ) )
= ( complex_1 @ ( inverse_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( inverse_divide_real @ ( uminus_uminus_real @ B ) @ ( plus_plus_real @ ( power_power_real @ A @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ B @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ).
thf(fact_3873_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1621563631at_int @ N ) ) @ ( semiri1621563631at_int @ M ) ) ).
thf(fact_3874_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1621563631at_int @ ( suc @ N ) ) ) @ ( semiri1621563631at_int @ M ) ) ).
thf(fact_3875_complex__zero__def,axiom,
( zero_zero_complex
= ( complex_1 @ zero_zero_real @ zero_zero_real ) ) ).
thf(fact_3876_Complex__eq__0,axiom,
! [A: real,B: real] :
( ( ( complex_1 @ A @ B )
= zero_zero_complex )
<=> ( ( A = zero_zero_real )
& ( B = zero_zero_real ) ) ) ).
thf(fact_3877_complex__divide__def,axiom,
! [X: complex,Y: complex] :
( ( invers1025623611omplex @ X @ Y )
= ( times_times_complex @ X @ ( invers1449016382omplex @ Y ) ) ) ).
thf(fact_3878_abs__int__eq,axiom,
! [M: nat] :
( ( abs_abs_int @ ( semiri1621563631at_int @ M ) )
= ( semiri1621563631at_int @ M ) ) ).
thf(fact_3879_real__of__nat__def,axiom,
real_nat = semiri132038758t_real ).
thf(fact_3880_real__eq__of__nat,axiom,
real_nat = semiri132038758t_real ).
thf(fact_3881_nat__int,axiom,
! [N: nat] :
( ( nat_1 @ ( semiri1621563631at_int @ N ) )
= N ) ).
thf(fact_3882_complex__add,axiom,
! [A: real,B: real,C: real,D: real] :
( ( plus_plus_complex @ ( complex_1 @ A @ B ) @ ( complex_1 @ C @ D ) )
= ( complex_1 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
thf(fact_3883_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_3884_complex__minus,axiom,
! [A: real,B: real] :
( ( uminus473333897omplex @ ( complex_1 @ A @ B ) )
= ( complex_1 @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
thf(fact_3885_not__neg__int,axiom,
! [N: nat] :
~ ( nat_neg @ ( semiri1621563631at_int @ N ) ) ).
thf(fact_3886_complex__diff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( minus_minus_complex @ ( complex_1 @ A @ B ) @ ( complex_1 @ C @ D ) )
= ( complex_1 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).
thf(fact_3887_StandardRes__SR__prop,axiom,
! [X: int,P_3: int] :
( ( member_int @ X @ ( sr @ P_3 ) )
=> ( ( standardRes @ P_3 @ X )
= X ) ) ).
thf(fact_3888_StandardRes__prop1,axiom,
! [X: int,M: int] : ( zcong @ X @ ( standardRes @ M @ X ) @ M ) ).
thf(fact_3889_StandardRes__def,axiom,
! [M: int,X: int] :
( ( standardRes @ M @ X )
= ( div_mod_int @ X @ M ) ) ).
thf(fact_3890_int__eq__0__conv,axiom,
! [N: nat] :
( ( ( semiri1621563631at_int @ N )
= zero_zero_int )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_3891_int__0,axiom,
( ( semiri1621563631at_int @ zero_zero_nat )
= zero_zero_int ) ).
thf(fact_3892_transfer__int__nat__numerals_I1_J,axiom,
( zero_zero_int
= ( semiri1621563631at_int @ zero_zero_nat ) ) ).
thf(fact_3893_int__less__0__conv,axiom,
! [K_1: nat] :
~ ( ord_less_int @ ( semiri1621563631at_int @ K_1 ) @ zero_zero_int ) ).
thf(fact_3894_zero__zle__int,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1621563631at_int @ N ) ) ).
thf(fact_3895_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_3896_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_3897_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_3898_zless__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1621563631at_int @ M ) @ ( semiri1621563631at_int @ N ) )
<=> ( ord_less_nat @ M @ N ) ) ).
thf(fact_3899_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_3900_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_3901_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_3902_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_3903_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_3904_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_3905_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_3906_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_3907_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_3908_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_3909_int__1,axiom,
( ( semiri1621563631at_int @ one_one_nat )
= one_one_int ) ).
thf(fact_3910_transfer__int__nat__numerals_I2_J,axiom,
( one_one_int
= ( semiri1621563631at_int @ one_one_nat ) ) ).
thf(fact_3911_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_3912_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1621563631at_int @ N ) @ ( uminus_uminus_int @ ( semiri1621563631at_int @ M ) ) ) ).
thf(fact_3913_Complex__eq__number__of,axiom,
! [A: real,B: real,W: int] :
( ( ( complex_1 @ A @ B )
= ( number528085621omplex @ W ) )
<=> ( ( A
= ( number267125858f_real @ W ) )
& ( B = zero_zero_real ) ) ) ).
thf(fact_3914_Complex__eq__1,axiom,
! [A: real,B: real] :
( ( ( complex_1 @ A @ B )
= one_one_complex )
<=> ( ( A = one_one_real )
& ( B = zero_zero_real ) ) ) ).
thf(fact_3915_complex__one__def,axiom,
( one_one_complex
= ( complex_1 @ one_one_real @ zero_zero_real ) ) ).
thf(fact_3916_real__of__int__of__nat__eq,axiom,
! [N: nat] :
( ( real_int @ ( semiri1621563631at_int @ N ) )
= ( real_nat @ N ) ) ).
thf(fact_3917_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_3918_zdiv__int,axiom,
! [A: nat,B: nat] :
( ( semiri1621563631at_int @ ( div_div_nat @ A @ B ) )
= ( div_div_int @ ( semiri1621563631at_int @ A ) @ ( semiri1621563631at_int @ B ) ) ) ).
thf(fact_3919_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_3920_zmod__int,axiom,
! [A: nat,B: nat] :
( ( semiri1621563631at_int @ ( div_mod_nat @ A @ B ) )
= ( div_mod_int @ ( semiri1621563631at_int @ A ) @ ( semiri1621563631at_int @ B ) ) ) ).
thf(fact_3921_zpower__int,axiom,
! [M: nat,N: nat] :
( ( power_power_int @ ( semiri1621563631at_int @ M ) @ N )
= ( semiri1621563631at_int @ ( power_power_nat @ M @ N ) ) ) ).
thf(fact_3922_int__power,axiom,
! [M: nat,N: nat] :
( ( semiri1621563631at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri1621563631at_int @ M ) @ N ) ) ).
thf(fact_3923_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_3924_not__neg__nat,axiom,
! [Z_1: int] :
( ~ ( nat_neg @ Z_1 )
=> ( ( semiri1621563631at_int @ ( nat_1 @ Z_1 ) )
= Z_1 ) ) ).
thf(fact_3925_floor__real__of__nat,axiom,
! [N: nat] :
( ( archim1246769320r_real @ ( real_nat @ N ) )
= ( semiri1621563631at_int @ N ) ) ).
thf(fact_3926_Parity_Otransfer__int__nat__relations,axiom,
! [X: nat] :
( ( even_odd_even_int @ ( semiri1621563631at_int @ X ) )
<=> ( even_odd_even_nat @ X ) ) ).
thf(fact_3927_even__nat__def,axiom,
! [X: nat] :
( ( even_odd_even_nat @ X )
<=> ( even_odd_even_int @ ( semiri1621563631at_int @ X ) ) ) ).
thf(fact_3928_ceiling__real__of__nat,axiom,
! [N: nat] :
( ( archim856651990g_real @ ( real_nat @ N ) )
= ( semiri1621563631at_int @ N ) ) ).
thf(fact_3929_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_3930_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_3931_StandardRes__prop3,axiom,
! [X: int,P_3: int] :
( ~ ( zcong @ X @ zero_zero_int @ P_3 )
<=> ( ( standardRes @ P_3 @ X )
!= zero_zero_int ) ) ).
thf(fact_3932_StandardRes__eq__zcong,axiom,
! [M: int,X: int] :
( ( ( standardRes @ M @ X )
= zero_zero_int )
<=> ( zcong @ X @ zero_zero_int @ M ) ) ).
thf(fact_3933_int__le__0__conv,axiom,
! [N: nat] :
( ( ord_less_eq_int @ ( semiri1621563631at_int @ N ) @ zero_zero_int )
<=> ( N = zero_zero_nat ) ) ).
thf(fact_3934_int__Suc0__eq__1,axiom,
( ( semiri1621563631at_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
thf(fact_3935_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_3936_int__Suc,axiom,
! [M: nat] :
( ( semiri1621563631at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1621563631at_int @ M ) ) ) ).
thf(fact_3937_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_3938_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_3939_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_3940_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_3941_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_3942_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1621563631at_int @ N ) ) @ zero_zero_int ) ).
thf(fact_3943_nat__zminus__int,axiom,
! [N: nat] :
( ( nat_1 @ ( uminus_uminus_int @ ( semiri1621563631at_int @ N ) ) )
= zero_zero_nat ) ).
thf(fact_3944_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_3945_complex__mult,axiom,
! [A: real,B: real,C: real,D: real] :
( ( times_times_complex @ ( complex_1 @ A @ B ) @ ( complex_1 @ C @ D ) )
= ( complex_1 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
thf(fact_3946_neg__zminus__int,axiom,
! [N: nat] : ( nat_neg @ ( uminus_uminus_int @ ( semiri1621563631at_int @ ( suc @ N ) ) ) ) ).
thf(fact_3947_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_3948_zmult__zless__mono2__lemma,axiom,
! [K_1: nat,I: int,J: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ord_less_int @ ( times_times_int @ ( semiri1621563631at_int @ K_1 ) @ I ) @ ( times_times_int @ ( semiri1621563631at_int @ K_1 ) @ J ) ) ) ) ).
thf(fact_3949_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_3950_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_3951_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_3952_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_3953_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1621563631at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
thf(fact_3954_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1621563631at_int @ ( suc @ N ) ) ) ) ).
thf(fact_3955_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_3956_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_3957_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_3958_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_3959_floor__minus__real__of__nat,axiom,
! [N: nat] :
( ( archim1246769320r_real @ ( uminus_uminus_real @ ( real_nat @ N ) ) )
= ( uminus_uminus_int @ ( semiri1621563631at_int @ N ) ) ) ).
thf(fact_3960_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_3961_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_3962_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_3963_SRStar__mult__prop2,axiom,
! [X: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ( member_int @ X @ ( sRStar @ P_3 ) )
=> ( member_int @ ( standardRes @ P_3 @ ( times_times_int @ A @ ( multInv @ P_3 @ X ) ) ) @ ( sRStar @ P_3 ) ) ) ) ) ) ).
thf(fact_3964_Nat__Transfer_Otransfer__int__nat__relations_I1_J,axiom,
! [X: nat,Y: nat] :
( ( ( semiri1621563631at_int @ X )
= ( semiri1621563631at_int @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_3965_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_3966_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1621563631at_int @ M )
= ( semiri1621563631at_int @ N ) )
<=> ( M = N ) ) ).
thf(fact_3967_complex__diff__def,axiom,
! [X: complex,Y: complex] :
( ( minus_minus_complex @ X @ Y )
= ( plus_plus_complex @ X @ ( uminus473333897omplex @ Y ) ) ) ).
thf(fact_3968_real__norm__def,axiom,
! [R_1: real] :
( ( norm_norm_real @ R_1 )
= ( abs_abs_real @ R_1 ) ) ).
thf(fact_3969_StandardRes__SRStar__prop3,axiom,
! [X: int,P_3: int] :
( ( member_int @ X @ ( sRStar @ P_3 ) )
=> ( ( standardRes @ P_3 @ X )
= X ) ) ).
thf(fact_3970_complex__mod__triangle__ineq2,axiom,
! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( norm_norm_complex @ ( plus_plus_complex @ B @ A ) ) @ ( norm_norm_complex @ B ) ) @ ( norm_norm_complex @ A ) ) ).
thf(fact_3971_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_3972_SRStar__SR__prop,axiom,
! [X: int,P_3: int] :
( ( member_int @ X @ ( sRStar @ P_3 ) )
=> ( member_int @ X @ ( sr @ P_3 ) ) ) ).
thf(fact_3973_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_3974_StandardRes__SRStar__prop1a,axiom,
! [X: int,P_3: int] :
( ( member_int @ X @ ( sRStar @ P_3 ) )
=> ~ ( zcong @ X @ zero_zero_int @ P_3 ) ) ).
thf(fact_3975_cmod__unit__one,axiom,
! [A: real] :
( ( norm_norm_complex @ ( complex_1 @ ( cos @ A ) @ ( sin @ A ) ) )
= one_one_real ) ).
thf(fact_3976_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_3977_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_3978_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_3979_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_3980_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_3981_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_3982_zero__less__imp__eq__int,axiom,
! [K_1: int] :
( ( ord_less_int @ zero_zero_int @ K_1 )
=> ? [N_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ N_1 )
& ( K_1
= ( semiri1621563631at_int @ N_1 ) ) ) ) ).
thf(fact_3983_complex_Osize_I1_J,axiom,
! [Real1: real,Real2: real] :
( ( complex_size @ ( complex_1 @ Real1 @ Real2 ) )
= zero_zero_nat ) ).
thf(fact_3984_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_3985_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
thf(fact_3986_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_3987_zero__le__imp__eq__int,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ? [N_1: nat] :
( K_1
= ( semiri1621563631at_int @ N_1 ) ) ) ).
thf(fact_3988_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_3989_nonneg__int__cases,axiom,
! [K_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ~ ! [N_1: nat] :
( K_1
!= ( semiri1621563631at_int @ N_1 ) ) ) ).
thf(fact_3990_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_3991_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_3992_int__nat__abs,axiom,
! [X: int] :
( ( semiri1621563631at_int @ ( nat_1 @ ( abs_abs_int @ X ) ) )
= ( abs_abs_int @ X ) ) ).
thf(fact_3993_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_3994_complex_Osize_I2_J,axiom,
! [Real1: real,Real2: real] :
( ( size_size_complex @ ( complex_1 @ Real1 @ Real2 ) )
= zero_zero_nat ) ).
thf(fact_3995_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_3996_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_3997_MultInvPair__card__two,axiom,
! [J: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A )
=> ( ~ ( zcong @ J @ zero_zero_int @ P_3 )
=> ( ( finite_card_int @ ( multInvPair @ A @ P_3 @ J ) )
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_3998_SetS__elems__card,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A )
=> ! [X_1: int > $o] :
( ( member_int_o @ X_1 @ ( setS @ A @ P_3 ) )
=> ( ( finite_card_int @ X_1 )
= ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_3999_power2__i,axiom,
( ( power_power_complex @ ii @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( number528085621omplex @ min ) ) ).
thf(fact_4000_complex__i__not__zero,axiom,
ii != zero_zero_complex ).
thf(fact_4001_complex__i__not__one,axiom,
ii != one_one_complex ).
thf(fact_4002_complex__i__not__number__of,axiom,
! [W: int] :
( ii
!= ( number528085621omplex @ W ) ) ).
thf(fact_4003_complex__i__mult__minus,axiom,
! [X: complex] :
( ( times_times_complex @ ii @ ( times_times_complex @ ii @ X ) )
= ( uminus473333897omplex @ X ) ) ).
thf(fact_4004_inverse__i,axiom,
( ( invers1449016382omplex @ ii )
= ( uminus473333897omplex @ ii ) ) ).
thf(fact_4005_i__def,axiom,
( ii
= ( complex_1 @ zero_zero_real @ one_one_real ) ) ).
thf(fact_4006_Complex__eq__i,axiom,
! [X: real,Y: real] :
( ( ( complex_1 @ X @ Y )
= ii )
<=> ( ( X = zero_zero_real )
& ( Y = one_one_real ) ) ) ).
thf(fact_4007_Complex__mult__i,axiom,
! [A: real,B: real] :
( ( times_times_complex @ ( complex_1 @ A @ B ) @ ii )
= ( complex_1 @ ( uminus_uminus_real @ B ) @ A ) ) ).
thf(fact_4008_i__mult__Complex,axiom,
! [A: real,B: real] :
( ( times_times_complex @ ii @ ( complex_1 @ A @ B ) )
= ( complex_1 @ ( uminus_uminus_real @ B ) @ A ) ) ).
thf(fact_4009_i__squared,axiom,
( ( times_times_complex @ ii @ ii )
= ( number528085621omplex @ min ) ) ).
thf(fact_4010_i__mult__eq2,axiom,
( ( times_times_complex @ ii @ ii )
= ( uminus473333897omplex @ one_one_complex ) ) ).
thf(fact_4011_SetS__card,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A )
=> ( ( semiri1621563631at_int @ ( finite_card_int_o @ ( setS @ A @ P_3 ) ) )
= ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_4012_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_4013_MultInvPair__prop2,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ( comple1092985777_int_o @ ( setS @ A @ P_3 ) )
= ( sRStar @ P_3 ) ) ) ) ) ).
thf(fact_4014_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_4015_complex__of__real__def,axiom,
! [R_1: real] :
( ( of_real_complex @ R_1 )
= ( complex_1 @ R_1 @ zero_zero_real ) ) ).
thf(fact_4016_sgn__eq,axiom,
! [Z_1: complex] :
( ( sgn_sgn_complex @ Z_1 )
= ( invers1025623611omplex @ Z_1 @ ( of_real_complex @ ( norm_norm_complex @ Z_1 ) ) ) ) ).
thf(fact_4017_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_4018_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_4019_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_4020_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_4021_i__mult__eq,axiom,
( ( times_times_complex @ ii @ ii )
= ( of_real_complex @ ( number267125858f_real @ min ) ) ) ).
thf(fact_4022_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_4023_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_4024_complex__of__real__minus__one,axiom,
( ( of_real_complex @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus473333897omplex @ one_one_complex ) ) ).
thf(fact_4025_cmod__complex__polar,axiom,
! [R_1: real,A: real] :
( ( norm_norm_complex @ ( times_times_complex @ ( of_real_complex @ R_1 ) @ ( complex_1 @ ( cos @ A ) @ ( sin @ A ) ) ) )
= ( abs_abs_real @ R_1 ) ) ).
thf(fact_4026_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_4027_expi__add,axiom,
! [A: complex,B: complex] :
( ( expi @ ( plus_plus_complex @ A @ B ) )
= ( times_times_complex @ ( expi @ A ) @ ( expi @ B ) ) ) ).
thf(fact_4028_expi__zero,axiom,
( ( expi @ zero_zero_complex )
= one_one_complex ) ).
thf(fact_4029_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_4030_Union__SetS__finite,axiom,
! [A: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( finite_finite_int @ ( comple1092985777_int_o @ ( setS @ A @ P_3 ) ) ) ) ).
thf(fact_4031_wset__fin,axiom,
! [A: int,P_3: int] : ( finite_finite_int @ ( wset @ A @ P_3 ) ) ).
thf(fact_4032_d22set__fin,axiom,
! [A: int] : ( finite_finite_int @ ( d22set @ A ) ) ).
thf(fact_4033_SetS__elems__finite,axiom,
! [A: int,P_3: int,X_1: int > $o] :
( ( member_int_o @ X_1 @ ( setS @ A @ P_3 ) )
=> ( finite_finite_int @ X_1 ) ) ).
thf(fact_4034_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_4035_SetS__finite,axiom,
! [A: int,P_3: int] :
( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( finite_finite_int_o @ ( setS @ A @ P_3 ) ) ) ).
thf(fact_4036_DERIV__neg__imp__decreasing,axiom,
! [F: real > real,A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A @ X_1 )
& ( ord_less_eq_real @ X_1 @ B ) )
=> ? [Y_1: real] :
( ( deriv_real @ F @ X_1 @ Y_1 )
& ( ord_less_real @ Y_1 @ zero_zero_real ) ) )
=> ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
thf(fact_4037_DERIV__pos__imp__increasing,axiom,
! [F: real > real,A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A @ X_1 )
& ( ord_less_eq_real @ X_1 @ B ) )
=> ? [Y_1: real] :
( ( deriv_real @ F @ X_1 @ Y_1 )
& ( ord_less_real @ zero_zero_real @ Y_1 ) ) )
=> ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
thf(fact_4038_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_4039_coprime__commute,axiom,
! [A: nat,B: nat] :
( ( coprime @ A @ B )
<=> ( coprime @ B @ A ) ) ).
thf(fact_4040_coprime__lmul2,axiom,
! [D: nat,A: nat,B: nat] :
( ( coprime @ D @ ( times_times_nat @ A @ B ) )
=> ( coprime @ D @ B ) ) ).
thf(fact_4041_coprime__rmul2,axiom,
! [D: nat,A: nat,B: nat] :
( ( coprime @ D @ ( times_times_nat @ A @ B ) )
=> ( coprime @ D @ A ) ) ).
thf(fact_4042_coprime__mul,axiom,
! [B: nat,D: nat,A: nat] :
( ( coprime @ D @ A )
=> ( ( coprime @ D @ B )
=> ( coprime @ D @ ( times_times_nat @ A @ B ) ) ) ) ).
thf(fact_4043_coprime__mul__eq,axiom,
! [D: nat,A: nat,B: nat] :
( ( coprime @ D @ ( times_times_nat @ A @ B ) )
<=> ( ( coprime @ D @ A )
& ( coprime @ D @ B ) ) ) ).
thf(fact_4044_coprime__1_H,axiom,
! [A: nat] : ( coprime @ one_one_nat @ A ) ).
thf(fact_4045_coprime__1,axiom,
! [A: nat] : ( coprime @ A @ one_one_nat ) ).
thf(fact_4046_coprime__refl,axiom,
! [N: nat] :
( ( coprime @ N @ N )
<=> ( N = one_one_nat ) ) ).
thf(fact_4047_coprime__exp__imp,axiom,
! [N: nat,A: nat,B: nat] :
( ( coprime @ A @ B )
=> ( coprime @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
thf(fact_4048_coprime__exp,axiom,
! [N: nat,D: nat,A: nat] :
( ( coprime @ D @ A )
=> ( coprime @ D @ ( power_power_nat @ A @ N ) ) ) ).
thf(fact_4049_coprime__Suc0_H,axiom,
! [A: nat] : ( coprime @ ( suc @ zero_zero_nat ) @ A ) ).
thf(fact_4050_coprime__Suc0,axiom,
! [A: nat] : ( coprime @ A @ ( suc @ zero_zero_nat ) ) ).
thf(fact_4051_coprime__0,axiom,
! [D: nat] :
( ( coprime @ D @ zero_zero_nat )
<=> ( D = one_one_nat ) ) ).
thf(fact_4052_coprime__0_H,axiom,
! [D: nat] :
( ( coprime @ zero_zero_nat @ D )
<=> ( D = one_one_nat ) ) ).
thf(fact_4053_coprime__plus1,axiom,
! [N: nat] : ( coprime @ ( plus_plus_nat @ N @ one_one_nat ) @ N ) ).
thf(fact_4054_coprime__exp2,axiom,
! [A: nat,N: nat,B: nat] :
( ( coprime @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
<=> ( coprime @ A @ B ) ) ).
thf(fact_4055_coprime__minus1,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( coprime @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
thf(fact_4056_coprime__bezout,axiom,
! [A: nat,B: nat] :
( ( coprime @ A @ B )
<=> ? [X_1: nat,Y_1: nat] :
( ( ( minus_minus_nat @ ( times_times_nat @ A @ X_1 ) @ ( times_times_nat @ B @ Y_1 ) )
= one_one_nat )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X_1 ) @ ( times_times_nat @ A @ Y_1 ) )
= one_one_nat ) ) ) ).
thf(fact_4057_nat_Osize_I2_J,axiom,
! [Nat: nat] :
( ( nat_size @ ( suc @ Nat ) )
= ( plus_plus_nat @ ( nat_size @ Nat ) @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_4058_coprime__bezout__strong,axiom,
! [A: nat,B: nat] :
( ( coprime @ A @ B )
=> ( ( B != one_one_nat )
=> ? [X_1: nat,Y_1: nat] :
( ( times_times_nat @ A @ X_1 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y_1 ) @ one_one_nat ) ) ) ) ).
thf(fact_4059_nat_Osize_I1_J,axiom,
( ( nat_size @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_4060_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_4061_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_4062_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_4063_nat__size,axiom,
! [N: nat] :
( ( size_size_nat @ N )
= N ) ).
thf(fact_4064_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_4065_nat_Osize_I3_J,axiom,
( ( size_size_nat @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_4066_ResSet__SRStar__prop,axiom,
! [P_3: int] : ( resSet @ P_3 @ ( sRStar @ P_3 ) ) ).
thf(fact_4067_code__numeral_Osize_I1_J,axiom,
( ( code_c271388182l_size @ zero_z126310315umeral )
= zero_zero_nat ) ).
thf(fact_4068_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_4069_complex__cnj__complex__of__real,axiom,
! [X: real] :
( ( cnj @ ( of_real_complex @ X ) )
= ( of_real_complex @ X ) ) ).
thf(fact_4070_complex__cnj__divide,axiom,
! [X: complex,Y: complex] :
( ( cnj @ ( invers1025623611omplex @ X @ Y ) )
= ( invers1025623611omplex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
thf(fact_4071_complex__cnj__power,axiom,
! [X: complex,N: nat] :
( ( cnj @ ( power_power_complex @ X @ N ) )
= ( power_power_complex @ ( cnj @ X ) @ N ) ) ).
thf(fact_4072_complex__cnj__of__nat,axiom,
! [N: nat] :
( ( cnj @ ( semiri2020571505omplex @ N ) )
= ( semiri2020571505omplex @ N ) ) ).
thf(fact_4073_complex__cnj__one,axiom,
( ( cnj @ one_one_complex )
= one_one_complex ) ).
thf(fact_4074_complex__cnj__mult,axiom,
! [X: complex,Y: complex] :
( ( cnj @ ( times_times_complex @ X @ Y ) )
= ( times_times_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
thf(fact_4075_cnj_Oadd,axiom,
! [X: complex,Y: complex] :
( ( cnj @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
thf(fact_4076_complex__cnj__number__of,axiom,
! [W: int] :
( ( cnj @ ( number528085621omplex @ W ) )
= ( number528085621omplex @ W ) ) ).
thf(fact_4077_cnj_Ozero,axiom,
( ( cnj @ zero_zero_complex )
= zero_zero_complex ) ).
thf(fact_4078_complex__cnj__zero__iff,axiom,
! [Z_1: complex] :
( ( ( cnj @ Z_1 )
= zero_zero_complex )
<=> ( Z_1 = zero_zero_complex ) ) ).
thf(fact_4079_complex__cnj__cnj,axiom,
! [Z_1: complex] :
( ( cnj @ ( cnj @ Z_1 ) )
= Z_1 ) ).
thf(fact_4080_complex__cnj__cancel__iff,axiom,
! [X: complex,Y: complex] :
( ( ( cnj @ X )
= ( cnj @ Y ) )
<=> ( X = Y ) ) ).
thf(fact_4081_complex__cnj__inverse,axiom,
! [X: complex] :
( ( cnj @ ( invers1449016382omplex @ X ) )
= ( invers1449016382omplex @ ( cnj @ X ) ) ) ).
thf(fact_4082_complex__mod__cnj,axiom,
! [Z_1: complex] :
( ( norm_norm_complex @ ( cnj @ Z_1 ) )
= ( norm_norm_complex @ Z_1 ) ) ).
thf(fact_4083_cnj_Odiff,axiom,
! [X: complex,Y: complex] :
( ( cnj @ ( minus_minus_complex @ X @ Y ) )
= ( minus_minus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
thf(fact_4084_cnj_Ominus,axiom,
! [X: complex] :
( ( cnj @ ( uminus473333897omplex @ X ) )
= ( uminus473333897omplex @ ( cnj @ X ) ) ) ).
thf(fact_4085_complex__cnj,axiom,
! [A: real,B: real] :
( ( cnj @ ( complex_1 @ A @ B ) )
= ( complex_1 @ A @ ( uminus_uminus_real @ B ) ) ) ).
thf(fact_4086_complex__cnj__i,axiom,
( ( cnj @ ii )
= ( uminus473333897omplex @ ii ) ) ).
thf(fact_4087_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_4088_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_4089_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_4090_complex__diff__cnj,axiom,
! [Z_1: complex] :
( ( minus_minus_complex @ Z_1 @ ( cnj @ Z_1 ) )
= ( times_times_complex @ ( of_real_complex @ ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( im @ Z_1 ) ) ) @ ii ) ) ).
thf(fact_4091_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_4092_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_4093_Re__complex__of__real,axiom,
! [Z_1: real] :
( ( re @ ( of_real_complex @ Z_1 ) )
= Z_1 ) ).
thf(fact_4094_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_4095_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_4096_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_4097_complex__eq__iff,axiom,
! [X: complex,Y: complex] :
( ( X = Y )
<=> ( ( ( re @ X )
= ( re @ Y ) )
& ( ( im @ X )
= ( im @ Y ) ) ) ) ).
thf(fact_4098_complex__eqI,axiom,
! [X: complex,Y: complex] :
( ( ( re @ X )
= ( re @ Y ) )
=> ( ( ( im @ X )
= ( im @ Y ) )
=> ( X = Y ) ) ) ).
thf(fact_4099_complex__surj,axiom,
! [Z_1: complex] :
( ( complex_1 @ ( re @ Z_1 ) @ ( im @ Z_1 ) )
= Z_1 ) ).
thf(fact_4100_Re,axiom,
! [X: real,Y: real] :
( ( re @ ( complex_1 @ X @ Y ) )
= X ) ).
thf(fact_4101_Im,axiom,
! [X: real,Y: real] :
( ( im @ ( complex_1 @ X @ Y ) )
= Y ) ).
thf(fact_4102_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_4103_complex__minus__def,axiom,
! [X: complex] :
( ( uminus473333897omplex @ X )
= ( complex_1 @ ( uminus_uminus_real @ ( re @ X ) ) @ ( uminus_uminus_real @ ( im @ X ) ) ) ) ).
thf(fact_4104_cnj__def,axiom,
! [Z_1: complex] :
( ( cnj @ Z_1 )
= ( complex_1 @ ( re @ Z_1 ) @ ( uminus_uminus_real @ ( im @ Z_1 ) ) ) ) ).
thf(fact_4105_complex__Re__cnj,axiom,
! [X: complex] :
( ( re @ ( cnj @ X ) )
= ( re @ X ) ) ).
thf(fact_4106_complex__Re__le__cmod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( re @ X ) @ ( norm_norm_complex @ X ) ) ).
thf(fact_4107_complex__Im__cnj,axiom,
! [X: complex] :
( ( im @ ( cnj @ X ) )
= ( uminus_uminus_real @ ( im @ X ) ) ) ).
thf(fact_4108_complex__Re__i,axiom,
( ( re @ ii )
= zero_zero_real ) ).
thf(fact_4109_Im__complex__of__real,axiom,
! [Z_1: real] :
( ( im @ ( of_real_complex @ Z_1 ) )
= zero_zero_real ) ).
thf(fact_4110_complex__Im__i,axiom,
( ( im @ ii )
= one_one_real ) ).
thf(fact_4111_code__numeral_Osimps_I3_J,axiom,
! [Code_numeral_3: code_code_numeral] :
( ( code_S1047413653umeral @ Code_numeral_3 )
!= zero_z126310315umeral ) ).
thf(fact_4112_code__numeral_Osimps_I2_J,axiom,
! [Code_numeral_2: code_code_numeral] :
( zero_z126310315umeral
!= ( code_S1047413653umeral @ Code_numeral_2 ) ) ).
thf(fact_4113_Re_Ozero,axiom,
( ( re @ zero_zero_complex )
= zero_zero_real ) ).
thf(fact_4114_Im_Ozero,axiom,
( ( im @ zero_zero_complex )
= zero_zero_real ) ).
thf(fact_4115_complex__Im__one,axiom,
( ( im @ one_one_complex )
= zero_zero_real ) ).
thf(fact_4116_Re_Oadd,axiom,
! [X: complex,Y: complex] :
( ( re @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).
thf(fact_4117_Im_Oadd,axiom,
! [X: complex,Y: complex] :
( ( im @ ( plus_plus_complex @ X @ Y ) )
= ( plus_plus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).
thf(fact_4118_complex__Re__one,axiom,
( ( re @ one_one_complex )
= one_one_real ) ).
thf(fact_4119_complex__Im__number__of,axiom,
! [V: int] :
( ( im @ ( number528085621omplex @ V ) )
= zero_zero_real ) ).
thf(fact_4120_Re_Ominus,axiom,
! [X: complex] :
( ( re @ ( uminus473333897omplex @ X ) )
= ( uminus_uminus_real @ ( re @ X ) ) ) ).
thf(fact_4121_complex__Re__number__of,axiom,
! [V: int] :
( ( re @ ( number528085621omplex @ V ) )
= ( number267125858f_real @ V ) ) ).
thf(fact_4122_Im_Ominus,axiom,
! [X: complex] :
( ( im @ ( uminus473333897omplex @ X ) )
= ( uminus_uminus_real @ ( im @ X ) ) ) ).
thf(fact_4123_Re_Odiff,axiom,
! [X: complex,Y: complex] :
( ( re @ ( minus_minus_complex @ X @ Y ) )
= ( minus_minus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).
thf(fact_4124_Im_Odiff,axiom,
! [X: complex,Y: complex] :
( ( im @ ( minus_minus_complex @ X @ Y ) )
= ( minus_minus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).
thf(fact_4125_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_4126_complex__Im__of__nat,axiom,
! [N: nat] :
( ( im @ ( semiri2020571505omplex @ N ) )
= zero_zero_real ) ).
thf(fact_4127_complex__Re__of__nat,axiom,
! [N: nat] :
( ( re @ ( semiri2020571505omplex @ N ) )
= ( semiri132038758t_real @ N ) ) ).
thf(fact_4128_abs__Re__le__cmod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( norm_norm_complex @ X ) ) ).
thf(fact_4129_abs__Im__le__cmod,axiom,
! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( norm_norm_complex @ X ) ) ).
thf(fact_4130_complex__In__mult__cnj__zero,axiom,
! [Z_1: complex] :
( ( im @ ( times_times_complex @ Z_1 @ ( cnj @ Z_1 ) ) )
= zero_zero_real ) ).
thf(fact_4131_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_4132_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_4133_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_4134_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_4135_Suc__code__numeral__minus__one,axiom,
! [N: code_code_numeral] :
( ( minus_1690775515umeral @ ( code_S1047413653umeral @ N ) @ one_on1645066479umeral )
= N ) ).
thf(fact_4136_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_4137_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_4138_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_4139_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_4140_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_4141_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_4142_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_4143_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_4144_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_4145_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_4146_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_4147_code__numeral_Oexhaust,axiom,
! [Y: code_code_numeral] :
( ( Y != zero_z126310315umeral )
=> ~ ! [Code_numeral: code_code_numeral] :
( Y
!= ( code_S1047413653umeral @ Code_numeral ) ) ) ).
thf(fact_4148_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_4149_Re__cis,axiom,
! [A: real] :
( ( re @ ( cis @ A ) )
= ( cos @ A ) ) ).
thf(fact_4150_Im__cis,axiom,
! [A: real] :
( ( im @ ( cis @ A ) )
= ( sin @ A ) ) ).
thf(fact_4151_cis__mult,axiom,
! [A: real,B: real] :
( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
= ( cis @ ( plus_plus_real @ A @ B ) ) ) ).
thf(fact_4152_cis__zero,axiom,
( ( cis @ zero_zero_real )
= one_one_complex ) ).
thf(fact_4153_cis__inverse,axiom,
! [A: real] :
( ( invers1449016382omplex @ ( cis @ A ) )
= ( cis @ ( uminus_uminus_real @ A ) ) ) ).
thf(fact_4154_cis__divide,axiom,
! [A: real,B: real] :
( ( invers1025623611omplex @ ( cis @ A ) @ ( cis @ B ) )
= ( cis @ ( minus_minus_real @ A @ B ) ) ) ).
thf(fact_4155_cis__def,axiom,
! [A: real] :
( ( cis @ A )
= ( complex_1 @ ( cos @ A ) @ ( sin @ A ) ) ) ).
thf(fact_4156_DeMoivre,axiom,
! [A: real,N: nat] :
( ( power_power_complex @ ( cis @ A ) @ N )
= ( cis @ ( times_times_real @ ( real_nat @ N ) @ A ) ) ) ).
thf(fact_4157_cis__real__of__nat__Suc__mult,axiom,
! [N: nat,A: real] :
( ( cis @ ( times_times_real @ ( real_nat @ ( suc @ N ) ) @ A ) )
= ( times_times_complex @ ( cis @ A ) @ ( cis @ ( times_times_real @ ( real_nat @ N ) @ A ) ) ) ) ).
thf(fact_4158_code__numeral_Osize_I3_J,axiom,
( ( size_s945831648umeral @ zero_z126310315umeral )
= zero_zero_nat ) ).
thf(fact_4159_cos__n__Re__cis__pow__n,axiom,
! [N: nat,A: real] :
( ( cos @ ( times_times_real @ ( real_nat @ N ) @ A ) )
= ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
thf(fact_4160_sin__n__Im__cis__pow__n,axiom,
! [N: nat,A: real] :
( ( sin @ ( times_times_real @ ( real_nat @ N ) @ A ) )
= ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
thf(fact_4161_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_4162_dvd_Oorder__refl,axiom,
! [X: nat] : ( dvd_dvd_nat @ X @ X ) ).
thf(fact_4163_dvd__1__left,axiom,
! [K_1: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K_1 ) ).
thf(fact_4164_coprime__divisors,axiom,
! [E_1: nat,B: nat,D: nat,A: nat] :
( ( dvd_dvd_nat @ D @ A )
=> ( ( dvd_dvd_nat @ E_1 @ B )
=> ( ( coprime @ A @ B )
=> ( coprime @ D @ E_1 ) ) ) ) ).
thf(fact_4165_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_4166_gcd__lcm__complete__lattice__nat_Onot__top__less,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ~ ( dvd_dvd_nat @ A @ zero_zero_nat ) ) ).
thf(fact_4167_gcd__lcm__complete__lattice__nat_Otop__greatest,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
thf(fact_4168_gcd__lcm__complete__lattice__nat_Oless__top,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
<=> ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ~ ( dvd_dvd_nat @ zero_zero_nat @ A ) ) ) ).
thf(fact_4169_gcd__lcm__complete__lattice__nat_Otop__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
<=> ( A = zero_zero_nat ) ) ).
thf(fact_4170_gcd__lcm__complete__lattice__nat_Otop__le,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
thf(fact_4171_gcd__lcm__complete__lattice__nat_Ole__bot,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( A = one_one_nat ) ) ).
thf(fact_4172_gcd__lcm__complete__lattice__nat_Obot__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
<=> ( A = one_one_nat ) ) ).
thf(fact_4173_gcd__lcm__complete__lattice__nat_Obot__less,axiom,
! [A: nat] :
( ( A != one_one_nat )
<=> ( ( dvd_dvd_nat @ one_one_nat @ A )
& ~ ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
thf(fact_4174_gcd__lcm__complete__lattice__nat_Obot__least,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
thf(fact_4175_gcd__lcm__complete__lattice__nat_Onot__less__bot,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ A @ one_one_nat )
& ~ ( dvd_dvd_nat @ one_one_nat @ A ) ) ).
thf(fact_4176_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
<=> ( M = one_one_nat ) ) ).
thf(fact_4177_divides__mul__r,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
thf(fact_4178_divides__mul__l,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( dvd_dvd_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
thf(fact_4179_divides__add__revr,axiom,
! [B: nat,D: nat,A: nat] :
( ( dvd_dvd_nat @ D @ A )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
=> ( dvd_dvd_nat @ D @ B ) ) ) ).
thf(fact_4180_dvd__reduce,axiom,
! [K_1: nat,N: nat] :
( ( dvd_dvd_nat @ K_1 @ ( plus_plus_nat @ N @ K_1 ) )
<=> ( dvd_dvd_nat @ K_1 @ N ) ) ).
thf(fact_4181_dvd_Oless__irrefl,axiom,
! [X: nat] :
~ ( ( dvd_dvd_nat @ X @ X )
& ~ ( dvd_dvd_nat @ X @ X ) ) ).
thf(fact_4182_dvd_Oeq__iff,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
<=> ( ( dvd_dvd_nat @ X @ Y )
& ( dvd_dvd_nat @ Y @ X ) ) ) ).
thf(fact_4183_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_4184_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_4185_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_4186_dvd_Oneq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) ) ) ) ).
thf(fact_4187_dvd_Oeq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( dvd_dvd_nat @ X @ Y ) ) ).
thf(fact_4188_dvd_Oantisym__conv,axiom,
! [Y: nat,X: nat] :
( ( dvd_dvd_nat @ Y @ X )
=> ( ( dvd_dvd_nat @ X @ Y )
<=> ( X = Y ) ) ) ).
thf(fact_4189_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_4190_dvd_Ole__neq__trans,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( A != B )
=> ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) ) ) ) ).
thf(fact_4191_dvd_Oord__eq__le__trans,axiom,
! [C: nat,A: nat,B: nat] :
( ( A = B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
thf(fact_4192_dvd_Oord__le__eq__trans,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( B = C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
thf(fact_4193_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
thf(fact_4194_dvd_Oantisym,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( dvd_dvd_nat @ Y @ X )
=> ( X = Y ) ) ) ).
thf(fact_4195_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_4196_dvd_Oord__eq__less__trans,axiom,
! [C: nat,A: nat,B: nat] :
( ( A = B )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ~ ( dvd_dvd_nat @ C @ B ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ~ ( dvd_dvd_nat @ C @ A ) ) ) ) ).
thf(fact_4197_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_4198_dvd_Oless__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( X != Y ) ) ).
thf(fact_4199_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_4200_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_4201_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_4202_dvd_Oless__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( X != Y ) ) ).
thf(fact_4203_dvd_Oless__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ~ ( dvd_dvd_nat @ Y @ X ) )
=> ( Y != X ) ) ).
thf(fact_4204_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_4205_dvd_Oord__less__eq__trans,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) )
=> ( ( B = C )
=> ( ( dvd_dvd_nat @ A @ C )
& ~ ( dvd_dvd_nat @ C @ A ) ) ) ) ).
thf(fact_4206_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_4207_dvd_Oless__asym_H,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) )
=> ~ ( ( dvd_dvd_nat @ B @ A )
& ~ ( dvd_dvd_nat @ A @ B ) ) ) ).
thf(fact_4208_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_4209_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_4210_divides__antisym,axiom,
! [X: nat,Y: nat] :
( ( ( dvd_dvd_nat @ X @ Y )
& ( dvd_dvd_nat @ Y @ X ) )
<=> ( X = Y ) ) ).
thf(fact_4211_dvd__diff__nat,axiom,
! [N: nat,K_1: nat,M: nat] :
( ( dvd_dvd_nat @ K_1 @ M )
=> ( ( dvd_dvd_nat @ K_1 @ N )
=> ( dvd_dvd_nat @ K_1 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
thf(fact_4212_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
<=> ( M
= ( suc @ zero_zero_nat ) ) ) ).
thf(fact_4213_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_4214_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_4215_divides__ge,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( B = zero_zero_nat )
| ( ord_less_eq_nat @ A @ B ) ) ) ).
thf(fact_4216_divides__le,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( ord_less_eq_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
thf(fact_4217_nat__mult__dvd__cancel__disj_H,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ M @ K_1 ) @ ( times_times_nat @ N @ K_1 ) )
<=> ( ( K_1 = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
thf(fact_4218_nat__mult__dvd__cancel__disj,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
<=> ( ( K_1 = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
thf(fact_4219_dvd__diffD1,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K_1 @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K_1 @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K_1 @ N ) ) ) ) ).
thf(fact_4220_dvd__diffD,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K_1 @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K_1 @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K_1 @ M ) ) ) ) ).
thf(fact_4221_divides__rev,axiom,
! [A: nat,N: nat,B: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( N != zero_zero_nat )
=> ( dvd_dvd_nat @ A @ B ) ) ) ).
thf(fact_4222_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_4223_pow__divides__eq__nat,axiom,
! [A: nat,B: nat,N: nat] :
( ( N != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
<=> ( dvd_dvd_nat @ A @ B ) ) ) ).
thf(fact_4224_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_4225_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_4226_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_4227_coprime__divprod,axiom,
! [D: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ D @ ( times_times_nat @ A @ B ) )
=> ( ( coprime @ D @ A )
=> ( dvd_dvd_nat @ D @ B ) ) ) ).
thf(fact_4228_coprime,axiom,
! [A: nat,B: nat] :
( ( coprime @ A @ B )
<=> ! [D_2: nat] :
( ( ( dvd_dvd_nat @ D_2 @ A )
& ( dvd_dvd_nat @ D_2 @ B ) )
<=> ( D_2 = one_one_nat ) ) ) ).
thf(fact_4229_dvd__imp__le,axiom,
! [K_1: nat,N: nat] :
( ( dvd_dvd_nat @ K_1 @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K_1 @ N ) ) ) ).
thf(fact_4230_dvd__mult__cancel,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
thf(fact_4231_nat__mult__dvd__cancel1,axiom,
! [M: nat,N: nat,K_1: nat] :
( ( ord_less_nat @ zero_zero_nat @ K_1 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) )
<=> ( dvd_dvd_nat @ M @ N ) ) ) ).
thf(fact_4232_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_4233_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_4234_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_4235_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_4236_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_4237_even__dvd,axiom,
! [N: nat] :
( ( even_odd_even_nat @ N )
<=> ( dvd_dvd_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) ).
thf(fact_4238_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_4239_real__of__nat__div,axiom,
! [N: nat,D: nat] :
( ( ord_less_nat @ zero_zero_nat @ D )
=> ( ( dvd_dvd_nat @ D @ N )
=> ( ( real_nat @ ( div_div_nat @ N @ D ) )
= ( inverse_divide_real @ ( real_nat @ N ) @ ( real_nat @ D ) ) ) ) ) ).
thf(fact_4240_dvd_Olift__Suc__mono__le,axiom,
! [N: nat,N_4: nat,F: nat > nat] :
( ! [N_1: nat] : ( dvd_dvd_nat @ ( F @ N_1 ) @ ( F @ ( suc @ N_1 ) ) )
=> ( ( ord_less_eq_nat @ N @ N_4 )
=> ( dvd_dvd_nat @ ( F @ N ) @ ( F @ N_4 ) ) ) ) ).
thf(fact_4241_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_4242_dvd_Olift__Suc__mono__less,axiom,
! [N: nat,N_4: 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_4 )
=> ( ( dvd_dvd_nat @ ( F @ N ) @ ( F @ N_4 ) )
& ~ ( dvd_dvd_nat @ ( F @ N_4 ) @ ( F @ N ) ) ) ) ) ).
thf(fact_4243_zdvd__self__abs2,axiom,
! [D: int] : ( dvd_dvd_int @ ( abs_abs_int @ D ) @ D ) ).
thf(fact_4244_zdvd__self__abs1,axiom,
! [D: int] : ( dvd_dvd_int @ D @ ( abs_abs_int @ D ) ) ).
thf(fact_4245_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_4246_zdvd__zmod__imp__zdvd,axiom,
! [K_1: int,M: int,N: int] :
( ( dvd_dvd_int @ K_1 @ ( div_mod_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K_1 @ N )
=> ( dvd_dvd_int @ K_1 @ M ) ) ) ).
thf(fact_4247_zdvd__antisym__abs,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( abs_abs_int @ A )
= ( abs_abs_int @ B ) ) ) ) ).
thf(fact_4248_zdvd__zdiffD,axiom,
! [K_1: int,M: int,N: int] :
( ( dvd_dvd_int @ K_1 @ ( minus_minus_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K_1 @ N )
=> ( dvd_dvd_int @ K_1 @ M ) ) ) ).
thf(fact_4249_uminus__dvd__conv_I1_J,axiom,
! [D: int,T: int] :
( ( dvd_dvd_int @ D @ T )
<=> ( dvd_dvd_int @ ( uminus_uminus_int @ D ) @ T ) ) ).
thf(fact_4250_uminus__dvd__conv_I2_J,axiom,
! [D: int,T: int] :
( ( dvd_dvd_int @ D @ T )
<=> ( dvd_dvd_int @ D @ ( uminus_uminus_int @ T ) ) ) ).
thf(fact_4251_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_4252_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_4253_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_4254_zdvd__mono,axiom,
! [M: int,T: int,K_1: int] :
( ( K_1 != zero_zero_int )
=> ( ( dvd_dvd_int @ M @ T )
<=> ( dvd_dvd_int @ ( times_times_int @ K_1 @ M ) @ ( times_times_int @ K_1 @ T ) ) ) ) ).
thf(fact_4255_zdvd__mult__cancel,axiom,
! [K_1: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K_1 @ M ) @ ( times_times_int @ K_1 @ N ) )
=> ( ( K_1 != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
thf(fact_4256_zdvd__period,axiom,
! [C: int,X: int,T: int,A: int,D: int] :
( ( dvd_dvd_int @ A @ D )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
<=> ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
thf(fact_4257_zdvd__reduce,axiom,
! [K_1: int,N: int,M: int] :
( ( dvd_dvd_int @ K_1 @ ( plus_plus_int @ N @ ( times_times_int @ K_1 @ M ) ) )
<=> ( dvd_dvd_int @ K_1 @ N ) ) ).
thf(fact_4258_zcong__zero__equiv__div,axiom,
! [A: int,M: int] :
( ( zcong @ A @ zero_zero_int @ M )
<=> ( dvd_dvd_int @ M @ A ) ) ).
thf(fact_4259_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_4260_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_4261_zdvd__int,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
<=> ( dvd_dvd_int @ ( semiri1621563631at_int @ X ) @ ( semiri1621563631at_int @ Y ) ) ) ).
thf(fact_4262_pow__divides__pow__int,axiom,
! [A: int,N: nat,B: int] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( N != zero_zero_nat )
=> ( dvd_dvd_int @ A @ B ) ) ) ).
thf(fact_4263_pow__divides__eq__int,axiom,
! [A: int,B: int,N: nat] :
( ( N != zero_zero_nat )
=> ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
<=> ( dvd_dvd_int @ A @ B ) ) ) ).
thf(fact_4264_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_4265_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
<=> ( ( abs_abs_int @ X )
= one_one_int ) ) ).
thf(fact_4266_zcong__def,axiom,
! [A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
<=> ( dvd_dvd_int @ M @ ( minus_minus_int @ A @ B ) ) ) ).
thf(fact_4267_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_4268_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_4269_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_4270_zprime__zdvd__power,axiom,
! [A: int,N: nat,P_3: int] :
( ( zprime @ P_3 )
=> ( ( dvd_dvd_int @ P_3 @ ( power_power_int @ A @ N ) )
=> ( dvd_dvd_int @ P_3 @ A ) ) ) ).
thf(fact_4271_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_4272_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_4273_zmod__zdvd__zmod,axiom,
! [A: int,B: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( dvd_dvd_int @ M @ B )
=> ( ( div_mod_int @ ( div_mod_int @ A @ B ) @ M )
= ( div_mod_int @ A @ M ) ) ) ) ).
thf(fact_4274_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_4275_dvd__imp__le__int,axiom,
! [D: int,I: int] :
( ( I != zero_zero_int )
=> ( ( dvd_dvd_int @ D @ I )
=> ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
thf(fact_4276_zprime__power__zdvd__cancel__right,axiom,
! [N: nat,A: int,B: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ~ ( dvd_dvd_int @ P_3 @ B )
=> ( ( dvd_dvd_int @ ( power_power_int @ P_3 @ N ) @ ( times_times_int @ A @ B ) )
=> ( dvd_dvd_int @ ( power_power_int @ P_3 @ N ) @ A ) ) ) ) ).
thf(fact_4277_zprime__power__zdvd__cancel__left,axiom,
! [N: nat,B: int,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ~ ( dvd_dvd_int @ P_3 @ A )
=> ( ( dvd_dvd_int @ ( power_power_int @ P_3 @ N ) @ ( times_times_int @ A @ B ) )
=> ( dvd_dvd_int @ ( power_power_int @ P_3 @ N ) @ B ) ) ) ) ).
thf(fact_4278_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_4279_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_4280_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_4281_real__of__int__div,axiom,
! [N: int,D: int] :
( ( D != zero_zero_int )
=> ( ( dvd_dvd_int @ D @ N )
=> ( ( real_int @ ( div_div_int @ N @ D ) )
= ( inverse_divide_real @ ( real_int @ N ) @ ( real_int @ D ) ) ) ) ) ).
thf(fact_4282_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_4283_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_4284_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_4285_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_4286_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_4287_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_4288_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_4289_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_4290_zprime__factor__exists,axiom,
! [A: int] :
( ( ord_less_int @ one_one_int @ A )
=> ? [P_4: int] :
( ( zprime @ P_4 )
& ( dvd_dvd_int @ P_4 @ A ) ) ) ).
thf(fact_4291_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_4292_one__code__int__code,axiom,
( one_on1684967323de_int
= ( number1226105091de_int @ ( bit1 @ pls ) ) ) ).
thf(fact_4293_Quickcheck__Narrowing_Oint__of__code,axiom,
! [K_1: quickcheck_code_int] :
( ( ( K_1 = zero_z891286103de_int )
=> ( ( quickcheck_int_of @ K_1 )
= zero_zero_int ) )
& ( ( K_1 != zero_z891286103de_int )
=> ( ( ( ( div_mo231679042de_int @ K_1 @ ( number1226105091de_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
= zero_z891286103de_int )
=> ( ( quickcheck_int_of @ K_1 )
= ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( quickcheck_int_of @ ( div_di1430059507de_int @ K_1 @ ( number1226105091de_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) )
& ( ( ( div_mo231679042de_int @ K_1 @ ( number1226105091de_int @ ( bit0 @ ( bit1 @ pls ) ) ) )
!= zero_z891286103de_int )
=> ( ( quickcheck_int_of @ K_1 )
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ ( quickcheck_int_of @ ( div_di1430059507de_int @ K_1 @ ( number1226105091de_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
thf(fact_4294_distinct__prime__coprime,axiom,
! [Q: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( prime @ Q )
=> ( ( P_3 != Q )
=> ( coprime @ P_3 @ Q ) ) ) ) ).
thf(fact_4295_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_4296_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_4297_prime__1,axiom,
~ ( prime @ one_one_nat ) ).
thf(fact_4298_prime__0,axiom,
~ ( prime @ zero_zero_nat ) ).
thf(fact_4299_int__of__number,axiom,
! [K_1: int] :
( ( quickcheck_int_of @ ( number1226105091de_int @ K_1 ) )
= ( number_number_of_int @ K_1 ) ) ).
thf(fact_4300_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_4301_Quickcheck__Narrowing_Oint__of__inject,axiom,
! [K_1: quickcheck_code_int,L: quickcheck_code_int] :
( ( ( quickcheck_int_of @ K_1 )
= ( quickcheck_int_of @ L ) )
<=> ( K_1 = L ) ) ).
thf(fact_4302_primes__eq,axiom,
! [Q: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( prime @ Q )
=> ( ( dvd_dvd_nat @ P_3 @ Q )
=> ( P_3 = Q ) ) ) ) ).
thf(fact_4303_prime__Suc0,axiom,
~ ( prime @ ( suc @ zero_zero_nat ) ) ).
thf(fact_4304_prime__g__zero,axiom,
! [P_3: nat] :
( ( prime @ P_3 )
=> ( ord_less_nat @ zero_zero_nat @ P_3 ) ) ).
thf(fact_4305_prime__divprod__eq,axiom,
! [A: nat,B: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( dvd_dvd_nat @ P_3 @ ( times_times_nat @ A @ B ) )
<=> ( ( dvd_dvd_nat @ P_3 @ A )
| ( dvd_dvd_nat @ P_3 @ B ) ) ) ) ).
thf(fact_4306_prime__divprod,axiom,
! [A: nat,B: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( dvd_dvd_nat @ P_3 @ ( times_times_nat @ A @ B ) )
=> ( ( dvd_dvd_nat @ P_3 @ A )
| ( dvd_dvd_nat @ P_3 @ B ) ) ) ) ).
thf(fact_4307_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_4308_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_4309_prime__dvd__power,axiom,
! [A: nat,N: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( dvd_dvd_nat @ P_3 @ ( power_power_nat @ A @ N ) )
=> ( dvd_dvd_nat @ P_3 @ A ) ) ) ).
thf(fact_4310_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_4311_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_4312_prime__exp,axiom,
! [P_3: nat,N: nat] :
( ( prime @ ( power_power_nat @ P_3 @ N ) )
<=> ( ( prime @ P_3 )
& ( N = one_one_nat ) ) ) ).
thf(fact_4313_coprime__prime,axiom,
! [P_3: nat,A: nat,B: nat] :
( ( coprime @ A @ B )
=> ~ ( ( prime @ P_3 )
& ( dvd_dvd_nat @ P_3 @ A )
& ( dvd_dvd_nat @ P_3 @ B ) ) ) ).
thf(fact_4314_prime__coprime__strong,axiom,
! [N: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( dvd_dvd_nat @ P_3 @ N )
| ( coprime @ P_3 @ N ) ) ) ).
thf(fact_4315_coprime__prime__eq,axiom,
! [A: nat,B: nat] :
( ( coprime @ A @ B )
<=> ! [P_4: nat] :
~ ( ( prime @ P_4 )
& ( dvd_dvd_nat @ P_4 @ A )
& ( dvd_dvd_nat @ P_4 @ B ) ) ) ).
thf(fact_4316_prime__impl__zprime__int,axiom,
! [A: nat] :
( ( prime @ A )
=> ( zprime @ ( semiri1621563631at_int @ A ) ) ) ).
thf(fact_4317_zero__code__int__code,axiom,
( zero_z891286103de_int
= ( number1226105091de_int @ pls ) ) ).
thf(fact_4318_prime__g__one,axiom,
! [P_3: nat] :
( ( prime @ P_3 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ P_3 ) ) ).
thf(fact_4319_prime__nd__one,axiom,
! [P_3: nat] :
( ( prime @ P_3 )
=> ~ ( dvd_dvd_nat @ P_3 @ ( suc @ zero_zero_nat ) ) ) ).
thf(fact_4320_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_4321_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_4322_divides__primepow,axiom,
! [D: nat,K_1: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( dvd_dvd_nat @ D @ ( power_power_nat @ P_3 @ K_1 ) )
<=> ? [I_1: nat] :
( ( ord_less_eq_nat @ I_1 @ K_1 )
& ( D
= ( power_power_nat @ P_3 @ I_1 ) ) ) ) ) ).
thf(fact_4323_prime__power__dvd__cancel__right,axiom,
! [N: nat,A: nat,B: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ~ ( dvd_dvd_nat @ P_3 @ B )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ P_3 @ N ) @ ( times_times_nat @ A @ B ) )
=> ( dvd_dvd_nat @ ( power_power_nat @ P_3 @ N ) @ A ) ) ) ) ).
thf(fact_4324_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_4325_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_4326_two__is__prime,axiom,
prime @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_4327_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_4328_prime__divprod__pow,axiom,
! [N: nat,A: nat,B: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( coprime @ A @ B )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ P_3 @ N ) @ ( times_times_nat @ A @ B ) )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ P_3 @ N ) @ A )
| ( dvd_dvd_nat @ ( power_power_nat @ P_3 @ N ) @ B ) ) ) ) ) ).
thf(fact_4329_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_4330_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_4331_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_4332_nat__of__def,axiom,
! [I: quickcheck_code_int] :
( ( quickcheck_nat_of @ I )
= ( nat_1 @ ( quickcheck_int_of @ I ) ) ) ).
thf(fact_4333_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_4334_prime__power__exp,axiom,
! [X: nat,K_1: nat,N: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( N != zero_zero_nat )
=> ( ( ( power_power_nat @ X @ N )
= ( power_power_nat @ P_3 @ K_1 ) )
=> ? [I_1: nat] :
( X
= ( power_power_nat @ P_3 @ I_1 ) ) ) ) ) ).
thf(fact_4335_prime__factor,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ? [P_4: nat] :
( ( prime @ P_4 )
& ( dvd_dvd_nat @ P_4 @ N ) ) ) ).
thf(fact_4336_bezout__prime,axiom,
! [A: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ~ ( dvd_dvd_nat @ P_3 @ A )
=> ? [X_1: nat,Y_1: nat] :
( ( times_times_nat @ A @ X_1 )
= ( plus_plus_nat @ ( times_times_nat @ P_3 @ Y_1 ) @ one_one_nat ) ) ) ) ).
thf(fact_4337_not__prime__ex__mk,axiom,
! [N: nat] :
( ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
& ~ ( prime @ N ) )
=> ? [M_2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M_2 )
& ( ord_less_nat @ ( suc @ zero_zero_nat ) @ K )
& ( ord_less_nat @ M_2 @ N )
& ( ord_less_nat @ K @ N )
& ( N
= ( times_times_nat @ M_2 @ K ) ) ) ) ).
thf(fact_4338_euclid,axiom,
! [N: nat] :
? [P_4: nat] :
( ( prime @ P_4 )
& ( ord_less_nat @ N @ P_4 ) ) ).
thf(fact_4339_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_4340_DeMoivre2,axiom,
! [R_1: real,A: real,N: nat] :
( ( power_power_complex @ ( rcis @ R_1 @ A ) @ N )
= ( rcis @ ( power_power_real @ R_1 @ N ) @ ( times_times_real @ ( real_nat @ N ) @ A ) ) ) ).
thf(fact_4341_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_4342_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_4343_fact__1__int,axiom,
( ( fact_fact_int @ one_one_int )
= one_one_int ) ).
thf(fact_4344_fact__neg__int,axiom,
! [M: int] :
( ( ord_less_int @ M @ zero_zero_int )
=> ( ( fact_fact_int @ M )
= zero_zero_int ) ) ).
thf(fact_4345_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_4346_fact__ge__zero__int,axiom,
! [M: int] : ( ord_less_eq_int @ zero_zero_int @ ( fact_fact_int @ M ) ) ).
thf(fact_4347_fact__nonzero__int,axiom,
! [N: int] :
( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( fact_fact_int @ N )
!= zero_zero_int ) ) ).
thf(fact_4348_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_4349_fact__0__int,axiom,
( ( fact_fact_int @ zero_zero_int )
= one_one_int ) ).
thf(fact_4350_transfer__int__nat__factorial,axiom,
! [X: nat] :
( ( fact_fact_int @ ( semiri1621563631at_int @ X ) )
= ( semiri1621563631at_int @ ( fact_fact_nat @ X ) ) ) ).
thf(fact_4351_rcis__zero__arg,axiom,
! [R_1: real] :
( ( rcis @ R_1 @ zero_zero_real )
= ( of_real_complex @ R_1 ) ) ).
thf(fact_4352_complex__mod__rcis,axiom,
! [R_1: real,A: real] :
( ( norm_norm_complex @ ( rcis @ R_1 @ A ) )
= ( abs_abs_real @ R_1 ) ) ).
thf(fact_4353_cis__rcis__eq,axiom,
! [A: real] :
( ( cis @ A )
= ( rcis @ one_one_real @ A ) ) ).
thf(fact_4354_rcis__zero__mod,axiom,
! [A: real] :
( ( rcis @ zero_zero_real @ A )
= zero_zero_complex ) ).
thf(fact_4355_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_4356_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_4357_fact__mono__int__aux,axiom,
! [M: int,K_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ( ord_less_eq_int @ ( fact_fact_int @ M ) @ ( fact_fact_int @ ( plus_plus_int @ M @ K_1 ) ) ) ) ).
thf(fact_4358_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_4359_Re__rcis,axiom,
! [R_1: real,A: real] :
( ( re @ ( rcis @ R_1 @ A ) )
= ( times_times_real @ R_1 @ ( cos @ A ) ) ) ).
thf(fact_4360_Im__rcis,axiom,
! [R_1: real,A: real] :
( ( im @ ( rcis @ R_1 @ A ) )
= ( times_times_real @ R_1 @ ( sin @ A ) ) ) ).
thf(fact_4361_rcis__mult,axiom,
! [R1: real,A: real,R2: real,B: real] :
( ( times_times_complex @ ( rcis @ R1 @ A ) @ ( rcis @ R2 @ B ) )
= ( rcis @ ( times_times_real @ R1 @ R2 ) @ ( plus_plus_real @ A @ B ) ) ) ).
thf(fact_4362_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_4363_rcis__divide,axiom,
! [R1: real,A: real,R2: real,B: real] :
( ( invers1025623611omplex @ ( rcis @ R1 @ A ) @ ( rcis @ R2 @ B ) )
= ( rcis @ ( inverse_divide_real @ R1 @ R2 ) @ ( minus_minus_real @ A @ B ) ) ) ).
thf(fact_4364_rcis__def,axiom,
! [R_1: real,A: real] :
( ( rcis @ R_1 @ A )
= ( times_times_complex @ ( of_real_complex @ R_1 ) @ ( cis @ A ) ) ) ).
thf(fact_4365_fact__less__mono__int__aux,axiom,
! [M: int,K_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ( ( ord_less_int @ zero_zero_int @ M )
=> ( ord_less_int @ ( fact_fact_int @ M ) @ ( fact_fact_int @ ( plus_plus_int @ ( plus_plus_int @ M @ one_one_int ) @ K_1 ) ) ) ) ) ).
thf(fact_4366_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_4367_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_4368_rcis__inverse,axiom,
! [R_1: real,A: real] :
( ( invers1449016382omplex @ ( rcis @ R_1 @ A ) )
= ( rcis @ ( inverse_divide_real @ one_one_real @ R_1 ) @ ( uminus_uminus_real @ A ) ) ) ).
thf(fact_4369_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_4370_Bnor__prime,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ A @ P_3 )
=> ( ( finite_card_int @ ( bnorRset @ A @ P_3 ) )
= ( nat_1 @ A ) ) ) ) ).
thf(fact_4371_Bnor__mem__zle,axiom,
! [B: int,A: int,M: int] :
( ( member_int @ B @ ( bnorRset @ A @ M ) )
=> ( ord_less_eq_int @ B @ A ) ) ).
thf(fact_4372_Bnor__mem__zle__swap,axiom,
! [M: int,A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( member_int @ B @ ( bnorRset @ A @ M ) ) ) ).
thf(fact_4373_Bnor__fin,axiom,
! [A: int,M: int] : ( finite_finite_int @ ( bnorRset @ A @ M ) ) ).
thf(fact_4374_Bnor__mem__zg,axiom,
! [B: int,A: int,M: int] :
( ( member_int @ B @ ( bnorRset @ A @ M ) )
=> ( ord_less_int @ zero_zero_int @ B ) ) ).
thf(fact_4375_lemma__BOLZANO,axiom,
! [A: real,B: real,P: produc914805421l_real > $o] :
( ! [A_2: real,B_4: real,C_2: real] :
( ( ( P @ ( produc865579683l_real @ A_2 @ B_4 ) )
& ( P @ ( produc865579683l_real @ B_4 @ C_2 ) )
& ( ord_less_eq_real @ A_2 @ B_4 )
& ( ord_less_eq_real @ B_4 @ C_2 ) )
=> ( P @ ( produc865579683l_real @ A_2 @ C_2 ) ) )
=> ( ! [X_1: real] :
? [D_2: real] :
( ( ord_less_real @ zero_zero_real @ D_2 )
& ! [A_2: real,B_4: real] :
( ( ( ord_less_eq_real @ A_2 @ X_1 )
& ( ord_less_eq_real @ X_1 @ B_4 )
& ( ord_less_real @ ( minus_minus_real @ B_4 @ A_2 ) @ D_2 ) )
=> ( P @ ( produc865579683l_real @ A_2 @ B_4 ) ) ) )
=> ( ( ord_less_eq_real @ A @ B )
=> ( P @ ( produc865579683l_real @ A @ B ) ) ) ) ) ).
thf(fact_4376_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_4377_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_4378_gcd__dvd2__nat,axiom,
! [M: nat,N: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ M @ N ) @ N ) ).
thf(fact_4379_gcd__dvd1__nat,axiom,
! [M: nat,N: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ M @ N ) @ M ) ).
thf(fact_4380_gcd__dvd__prod__nat,axiom,
! [M: nat,N: nat,K_1: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ M @ N ) @ ( times_times_nat @ K_1 @ N ) ) ).
thf(fact_4381_coprime__exp2__nat,axiom,
! [N: nat,M: nat,A: nat,B: nat] :
( ( ( gcd_gcd_nat @ A @ B )
= one_one_nat )
=> ( ( gcd_gcd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ M ) )
= one_one_nat ) ) ).
thf(fact_4382_gcd__le2__nat,axiom,
! [A: nat,B: nat] :
( ( B != zero_zero_nat )
=> ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
thf(fact_4383_gcd__le1__nat,axiom,
! [B: nat,A: nat] :
( ( A != zero_zero_nat )
=> ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
thf(fact_4384_gcd__Suc__0,axiom,
! [M: nat] :
( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( suc @ zero_zero_nat ) ) ).
thf(fact_4385_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_4386_gcd__exp__nat,axiom,
! [A: nat,N: nat,B: nat] :
( ( gcd_gcd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( power_power_nat @ ( gcd_gcd_nat @ A @ B ) @ N ) ) ).
thf(fact_4387_gcd__semilattice__nat_Oinf_Oidem,axiom,
! [A: nat] :
( ( gcd_gcd_nat @ A @ A )
= A ) ).
thf(fact_4388_gcd__idem__nat,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ X @ X )
= X ) ).
thf(fact_4389_gcd__commute__nat,axiom,
! [A: nat,B: nat] :
( ( gcd_gcd_nat @ A @ B )
= ( gcd_gcd_nat @ B @ A ) ) ).
thf(fact_4390_gcd__semilattice__nat_Oinf__commute,axiom,
! [X: nat,Y: nat] :
( ( gcd_gcd_nat @ X @ Y )
= ( gcd_gcd_nat @ Y @ X ) ) ).
thf(fact_4391_gcd__semilattice__nat_Oinf_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( gcd_gcd_nat @ A @ ( gcd_gcd_nat @ A @ B ) )
= ( gcd_gcd_nat @ A @ B ) ) ).
thf(fact_4392_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_4393_gcd__left__commute__nat,axiom,
! [B: nat,A: nat,C: nat] :
( ( gcd_gcd_nat @ B @ ( gcd_gcd_nat @ A @ C ) )
= ( gcd_gcd_nat @ A @ ( gcd_gcd_nat @ B @ C ) ) ) ).
thf(fact_4394_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_4395_gcd__assoc__nat,axiom,
! [A: nat,B: nat,C: nat] :
( ( gcd_gcd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
= ( gcd_gcd_nat @ A @ ( gcd_gcd_nat @ B @ C ) ) ) ).
thf(fact_4396_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_4397_gcd__mult__distrib__nat,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( times_times_nat @ K_1 @ ( gcd_gcd_nat @ M @ N ) )
= ( gcd_gcd_nat @ ( times_times_nat @ K_1 @ M ) @ ( times_times_nat @ K_1 @ N ) ) ) ).
thf(fact_4398_gcd__add1__nat,axiom,
! [M: nat,N: nat] :
( ( gcd_gcd_nat @ ( plus_plus_nat @ M @ N ) @ N )
= ( gcd_gcd_nat @ M @ N ) ) ).
thf(fact_4399_gcd__add2__nat,axiom,
! [M: nat,N: nat] :
( ( gcd_gcd_nat @ M @ ( plus_plus_nat @ M @ N ) )
= ( gcd_gcd_nat @ M @ N ) ) ).
thf(fact_4400_gcd__semilattice__nat_Oless__infI2,axiom,
! [A: nat,B: nat,X: nat] :
( ( ( dvd_dvd_nat @ B @ X )
& ~ ( dvd_dvd_nat @ X @ B ) )
=> ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ X )
& ~ ( dvd_dvd_nat @ X @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
thf(fact_4401_gcd__semilattice__nat_Oless__infI1,axiom,
! [B: nat,A: nat,X: nat] :
( ( ( dvd_dvd_nat @ A @ X )
& ~ ( dvd_dvd_nat @ X @ A ) )
=> ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ X )
& ~ ( dvd_dvd_nat @ X @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
thf(fact_4402_gcd__semilattice__nat_Ole__infE,axiom,
! [X: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ X @ ( gcd_gcd_nat @ A @ B ) )
=> ~ ( ( dvd_dvd_nat @ X @ A )
=> ~ ( dvd_dvd_nat @ X @ B ) ) ) ).
thf(fact_4403_dvd__gcd__D2__nat,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K_1 @ ( gcd_gcd_nat @ M @ N ) )
=> ( dvd_dvd_nat @ K_1 @ N ) ) ).
thf(fact_4404_dvd__gcd__D1__nat,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K_1 @ ( gcd_gcd_nat @ M @ N ) )
=> ( dvd_dvd_nat @ K_1 @ M ) ) ).
thf(fact_4405_gcd__semilattice__nat_Oinf__mono,axiom,
! [B: nat,D: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ B @ D )
=> ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ ( gcd_gcd_nat @ C @ D ) ) ) ) ).
thf(fact_4406_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_4407_gcd__semilattice__nat_Ole__infI,axiom,
! [B: nat,X: nat,A: nat] :
( ( dvd_dvd_nat @ X @ A )
=> ( ( dvd_dvd_nat @ X @ B )
=> ( dvd_dvd_nat @ X @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
thf(fact_4408_gcd__greatest__nat,axiom,
! [N: nat,K_1: nat,M: nat] :
( ( dvd_dvd_nat @ K_1 @ M )
=> ( ( dvd_dvd_nat @ K_1 @ N )
=> ( dvd_dvd_nat @ K_1 @ ( gcd_gcd_nat @ M @ N ) ) ) ) ).
thf(fact_4409_gcd__proj2__if__dvd__nat,axiom,
! [Y: nat,X: nat] :
( ( dvd_dvd_nat @ Y @ X )
=> ( ( gcd_gcd_nat @ X @ Y )
= Y ) ) ).
thf(fact_4410_gcd__proj1__if__dvd__nat,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( gcd_gcd_nat @ X @ Y )
= X ) ) ).
thf(fact_4411_gcd__semilattice__nat_Ole__infI2,axiom,
! [A: nat,B: nat,X: nat] :
( ( dvd_dvd_nat @ B @ X )
=> ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ X ) ) ).
thf(fact_4412_gcd__semilattice__nat_Ole__infI1,axiom,
! [B: nat,A: nat,X: nat] :
( ( dvd_dvd_nat @ A @ X )
=> ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ X ) ) ).
thf(fact_4413_gcd__unique__nat,axiom,
! [B: nat,D: nat,A: nat] :
( ( ( dvd_dvd_nat @ D @ A )
& ( dvd_dvd_nat @ D @ B )
& ! [E: nat] :
( ( ( dvd_dvd_nat @ E @ A )
& ( dvd_dvd_nat @ E @ B ) )
=> ( dvd_dvd_nat @ E @ D ) ) )
<=> ( D
= ( gcd_gcd_nat @ A @ B ) ) ) ).
thf(fact_4414_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_4415_gcd__greatest__iff__nat,axiom,
! [K_1: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K_1 @ ( gcd_gcd_nat @ M @ N ) )
<=> ( ( dvd_dvd_nat @ K_1 @ M )
& ( dvd_dvd_nat @ K_1 @ N ) ) ) ).
thf(fact_4416_gcd__semilattice__nat_Ole__iff__inf,axiom,
! [X: nat,Y: nat] :
( ( dvd_dvd_nat @ X @ Y )
<=> ( ( gcd_gcd_nat @ X @ Y )
= X ) ) ).
thf(fact_4417_gcd__semilattice__nat_Oinf__le2,axiom,
! [X: nat,Y: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ X @ Y ) @ Y ) ).
thf(fact_4418_gcd__semilattice__nat_Oinf__le1,axiom,
! [X: nat,Y: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ X @ Y ) @ X ) ).
thf(fact_4419_gcd__lcm__complete__lattice__nat_Oinf__bot__left,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ one_one_nat @ X )
= one_one_nat ) ).
thf(fact_4420_gcd__1__nat,axiom,
! [M: nat] :
( ( gcd_gcd_nat @ M @ one_one_nat )
= one_one_nat ) ).
thf(fact_4421_gcd__lcm__complete__lattice__nat_Oinf__bot__right,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ X @ one_one_nat )
= one_one_nat ) ).
thf(fact_4422_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_4423_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_4424_gcd__0__nat,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ X @ zero_zero_nat )
= X ) ).
thf(fact_4425_gcd__0__left__nat,axiom,
! [X: nat] :
( ( gcd_gcd_nat @ zero_zero_nat @ X )
= X ) ).
thf(fact_4426_gcd__red__nat,axiom,
! [X: nat,Y: nat] :
( ( gcd_gcd_nat @ X @ Y )
= ( gcd_gcd_nat @ Y @ ( div_mod_nat @ X @ Y ) ) ) ).
thf(fact_4427_coprime__Suc__nat,axiom,
! [N: nat] :
( ( gcd_gcd_nat @ ( suc @ N ) @ N )
= one_one_nat ) ).
thf(fact_4428_gcd__add__mult__nat,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( gcd_gcd_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ K_1 @ M ) @ N ) )
= ( gcd_gcd_nat @ M @ N ) ) ).
thf(fact_4429_coprime__common__divisor__nat,axiom,
! [X: nat,A: nat,B: nat] :
( ( ( gcd_gcd_nat @ A @ B )
= one_one_nat )
=> ( ( dvd_dvd_nat @ X @ A )
=> ( ( dvd_dvd_nat @ X @ B )
=> ( X = one_one_nat ) ) ) ) ).
thf(fact_4430_coprime__nat,axiom,
! [A: nat,B: nat] :
( ( ( gcd_gcd_nat @ A @ B )
= one_one_nat )
<=> ! [D_2: nat] :
( ( ( dvd_dvd_nat @ D_2 @ A )
& ( dvd_dvd_nat @ D_2 @ B ) )
<=> ( D_2 = one_one_nat ) ) ) ).
thf(fact_4431_coprime__plus__one__nat,axiom,
! [N: nat] :
( ( gcd_gcd_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ N )
= one_one_nat ) ).
thf(fact_4432_coprime__rmult__nat,axiom,
! [D: nat,A: nat,B: nat] :
( ( ( gcd_gcd_nat @ D @ ( times_times_nat @ A @ B ) )
= one_one_nat )
=> ( ( gcd_gcd_nat @ D @ B )
= one_one_nat ) ) ).
thf(fact_4433_coprime__lmult__nat,axiom,
! [D: nat,A: nat,B: nat] :
( ( ( gcd_gcd_nat @ D @ ( times_times_nat @ A @ B ) )
= one_one_nat )
=> ( ( gcd_gcd_nat @ D @ A )
= one_one_nat ) ) ).
thf(fact_4434_coprime__crossproduct__nat,axiom,
! [B: nat,C: nat,A: nat,D: nat] :
( ( ( gcd_gcd_nat @ A @ D )
= one_one_nat )
=> ( ( ( gcd_gcd_nat @ B @ C )
= one_one_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ D ) )
<=> ( ( A = B )
& ( C = D ) ) ) ) ) ).
thf(fact_4435_coprime__mult__nat,axiom,
! [B: nat,D: nat,A: nat] :
( ( ( gcd_gcd_nat @ D @ A )
= one_one_nat )
=> ( ( ( gcd_gcd_nat @ D @ B )
= one_one_nat )
=> ( ( gcd_gcd_nat @ D @ ( times_times_nat @ A @ B ) )
= one_one_nat ) ) ) ).
thf(fact_4436_gcd__mult__cancel__nat,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ( gcd_gcd_nat @ K_1 @ N )
= one_one_nat )
=> ( ( gcd_gcd_nat @ ( times_times_nat @ K_1 @ M ) @ N )
= ( gcd_gcd_nat @ M @ N ) ) ) ).
thf(fact_4437_coprime__mul__eq__nat,axiom,
! [D: nat,A: nat,B: nat] :
( ( ( gcd_gcd_nat @ D @ ( times_times_nat @ A @ B ) )
= one_one_nat )
<=> ( ( ( gcd_gcd_nat @ D @ A )
= one_one_nat )
& ( ( gcd_gcd_nat @ D @ B )
= one_one_nat ) ) ) ).
thf(fact_4438_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_4439_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_4440_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_4441_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_4442_coprime__exp__nat,axiom,
! [N: nat,D: nat,A: nat] :
( ( ( gcd_gcd_nat @ D @ A )
= one_one_nat )
=> ( ( gcd_gcd_nat @ D @ ( power_power_nat @ A @ N ) )
= one_one_nat ) ) ).
thf(fact_4443_gcd__coprime__nat,axiom,
! [B_5: nat,A_5: nat,A: nat,B: nat] :
( ( ( gcd_gcd_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A
= ( times_times_nat @ A_5 @ ( gcd_gcd_nat @ A @ B ) ) )
=> ( ( B
= ( times_times_nat @ B_5 @ ( gcd_gcd_nat @ A @ B ) ) )
=> ( ( gcd_gcd_nat @ A_5 @ B_5 )
= one_one_nat ) ) ) ) ).
thf(fact_4444_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_4445_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_4446_coprime__dvd__mult__iff__nat,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ( gcd_gcd_nat @ K_1 @ N )
= one_one_nat )
=> ( ( dvd_dvd_nat @ K_1 @ ( times_times_nat @ M @ N ) )
<=> ( dvd_dvd_nat @ K_1 @ M ) ) ) ).
thf(fact_4447_coprime__dvd__mult__nat,axiom,
! [M: nat,K_1: nat,N: nat] :
( ( ( gcd_gcd_nat @ K_1 @ N )
= one_one_nat )
=> ( ( dvd_dvd_nat @ K_1 @ ( times_times_nat @ M @ N ) )
=> ( dvd_dvd_nat @ K_1 @ M ) ) ) ).
thf(fact_4448_div__gcd__coprime__nat,axiom,
! [B: nat,A: nat] :
( ( ( A != zero_zero_nat )
| ( B != zero_zero_nat ) )
=> ( ( gcd_gcd_nat @ ( div_div_nat @ A @ ( gcd_gcd_nat @ A @ B ) ) @ ( div_div_nat @ B @ ( gcd_gcd_nat @ A @ B ) ) )
= one_one_nat ) ) ).
thf(fact_4449_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_4450_coprime__Suc__0__nat,axiom,
! [A: nat,B: nat] :
( ( ( gcd_gcd_nat @ A @ B )
= one_one_nat )
<=> ! [D_2: nat] :
( ( ( dvd_dvd_nat @ D_2 @ A )
& ( dvd_dvd_nat @ D_2 @ B ) )
<=> ( D_2
= ( suc @ zero_zero_nat ) ) ) ) ).
thf(fact_4451_gcd__coprime__exists__nat,axiom,
! [A: nat,B: nat] :
( ( ( gcd_gcd_nat @ A @ B )
!= zero_zero_nat )
=> ? [A_3: nat,B_2: nat] :
( ( A
= ( times_times_nat @ A_3 @ ( gcd_gcd_nat @ A @ B ) ) )
& ( B
= ( times_times_nat @ B_2 @ ( gcd_gcd_nat @ A @ B ) ) )
& ( ( gcd_gcd_nat @ A_3 @ B_2 )
= one_one_nat ) ) ) ).
thf(fact_4452_bezout__nat,axiom,
! [B: nat,A: nat] :
( ( A != zero_zero_nat )
=> ? [X_1: nat,Y_1: nat] :
( ( times_times_nat @ A @ X_1 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y_1 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
thf(fact_4453_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_4454_ln__powr__bound,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( ln @ X ) @ ( inverse_divide_real @ ( powr @ X @ A ) @ A ) ) ) ) ).
thf(fact_4455_ln__powr__bound2,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( powr @ ( ln @ X ) @ A ) @ ( times_times_real @ ( powr @ A @ A ) @ X ) ) ) ) ).
thf(fact_4456_powr__one__eq__one,axiom,
! [A: real] :
( ( powr @ one_one_real @ A )
= one_one_real ) ).
thf(fact_4457_powr__not__zero,axiom,
! [X: real,A: real] :
( ( powr @ X @ A )
!= zero_zero_real ) ).
thf(fact_4458_powr__powr__swap,axiom,
! [X: real,A: real,B: real] :
( ( powr @ ( powr @ X @ A ) @ B )
= ( powr @ ( powr @ X @ B ) @ A ) ) ).
thf(fact_4459_powr__powr,axiom,
! [X: real,A: real,B: real] :
( ( powr @ ( powr @ X @ A ) @ B )
= ( powr @ X @ ( times_times_real @ A @ B ) ) ) ).
thf(fact_4460_powr__ge__pzero,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr @ X @ Y ) ) ).
thf(fact_4461_powr__less__mono2__neg,axiom,
! [Y: real,X: real,A: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( powr @ Y @ A ) @ ( powr @ X @ A ) ) ) ) ) ).
thf(fact_4462_powr__less__mono2,axiom,
! [Y: real,X: real,A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( powr @ X @ A ) @ ( powr @ Y @ A ) ) ) ) ) ).
thf(fact_4463_powr__gt__zero,axiom,
! [X: real,A: real] : ( ord_less_real @ zero_zero_real @ ( powr @ X @ A ) ) ).
thf(fact_4464_powr__mono,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( powr @ X @ A ) @ ( powr @ X @ B ) ) ) ) ).
thf(fact_4465_powr__zero__eq__one,axiom,
! [X: real] :
( ( powr @ X @ zero_zero_real )
= one_one_real ) ).
thf(fact_4466_powr__less__cancel,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_real @ ( powr @ X @ A ) @ ( powr @ X @ B ) )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ A @ B ) ) ) ).
thf(fact_4467_powr__less__mono,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( powr @ X @ A ) @ ( powr @ X @ B ) ) ) ) ).
thf(fact_4468_powr__less__cancel__iff,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ ( powr @ X @ A ) @ ( powr @ X @ B ) )
<=> ( ord_less_real @ A @ B ) ) ) ).
thf(fact_4469_powr__add,axiom,
! [X: real,A: real,B: real] :
( ( powr @ X @ ( plus_plus_real @ A @ B ) )
= ( times_times_real @ ( powr @ X @ A ) @ ( powr @ X @ B ) ) ) ).
thf(fact_4470_powr__divide2,axiom,
! [X: real,A: real,B: real] :
( ( inverse_divide_real @ ( powr @ X @ A ) @ ( powr @ X @ B ) )
= ( powr @ X @ ( minus_minus_real @ A @ B ) ) ) ).
thf(fact_4471_powr__minus,axiom,
! [X: real,A: real] :
( ( powr @ X @ ( uminus_uminus_real @ A ) )
= ( inverse_inverse_real @ ( powr @ X @ A ) ) ) ).
thf(fact_4472_powr__mono2,axiom,
! [Y: real,X: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( powr @ X @ A ) @ ( powr @ Y @ A ) ) ) ) ) ).
thf(fact_4473_powr__mult,axiom,
! [A: real,Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( powr @ ( times_times_real @ X @ Y ) @ A )
= ( times_times_real @ ( powr @ X @ A ) @ ( powr @ Y @ A ) ) ) ) ) ).
thf(fact_4474_ge__one__powr__ge__zero,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( powr @ X @ A ) ) ) ) ).
thf(fact_4475_powr__le__cancel__iff,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ ( powr @ X @ A ) @ ( powr @ X @ B ) )
<=> ( ord_less_eq_real @ A @ B ) ) ) ).
thf(fact_4476_powr__one__gt__zero__iff,axiom,
! [X: real] :
( ( ( powr @ X @ one_one_real )
= X )
<=> ( ord_less_real @ zero_zero_real @ X ) ) ).
thf(fact_4477_powr__divide,axiom,
! [A: real,Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( powr @ ( inverse_divide_real @ X @ Y ) @ A )
= ( inverse_divide_real @ ( powr @ X @ A ) @ ( powr @ Y @ A ) ) ) ) ) ).
thf(fact_4478_powr__minus__divide,axiom,
! [X: real,A: real] :
( ( powr @ X @ ( uminus_uminus_real @ A ) )
= ( inverse_divide_real @ one_one_real @ ( powr @ X @ A ) ) ) ).
thf(fact_4479_powr__def,axiom,
! [X: real,A: real] :
( ( powr @ X @ A )
= ( exp_real @ ( times_times_real @ A @ ( ln @ X ) ) ) ) ).
thf(fact_4480_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_4481_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_4482_log__powr__cancel,axiom,
! [Y: real,A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ ( powr @ A @ Y ) )
= Y ) ) ) ).
thf(fact_4483_powr__log__cancel,axiom,
! [X: real,A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr @ A @ ( log @ A @ X ) )
= X ) ) ) ) ).
thf(fact_4484_log__powr,axiom,
! [B: real,Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( log @ B @ ( powr @ X @ Y ) )
= ( times_times_real @ Y @ ( log @ B @ X ) ) ) ) ) ).
thf(fact_4485_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_4486_coprime__pow,axiom,
! [C: nat,N: nat,A: nat,B: nat] :
( ( coprime @ A @ B )
=> ( ( ( times_times_nat @ A @ B )
= ( power_power_nat @ C @ N ) )
=> ? [R: nat,S_2: nat] :
( ( A
= ( power_power_nat @ R @ N ) )
& ( B
= ( power_power_nat @ S_2 @ N ) ) ) ) ) ).
thf(fact_4487_prime__power__mult,axiom,
! [X: nat,Y: nat,K_1: nat,P_3: nat] :
( ( prime @ P_3 )
=> ( ( ( times_times_nat @ X @ Y )
= ( power_power_nat @ P_3 @ K_1 ) )
=> ? [I_1: nat,J_1: nat] :
( ( X
= ( power_power_nat @ P_3 @ I_1 ) )
& ( Y
= ( power_power_nat @ P_3 @ J_1 ) ) ) ) ) ).
thf(fact_4488_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_4489_zgcd__zdvd2,axiom,
! [I: int,J: int] : ( dvd_dvd_int @ ( legacy_zgcd @ I @ J ) @ J ) ).
thf(fact_4490_zgcd__zdvd1,axiom,
! [I: int,J: int] : ( dvd_dvd_int @ ( legacy_zgcd @ I @ J ) @ I ) ).
thf(fact_4491_zgcd__zmult__distrib2__abs,axiom,
! [K_1: int,M: int,N: int] :
( ( legacy_zgcd @ ( times_times_int @ K_1 @ M ) @ ( times_times_int @ K_1 @ N ) )
= ( times_times_int @ ( abs_abs_int @ K_1 ) @ ( legacy_zgcd @ M @ N ) ) ) ).
thf(fact_4492_zgcd__pos,axiom,
! [I: int,J: int] : ( ord_less_eq_int @ zero_zero_int @ ( legacy_zgcd @ I @ J ) ) ).
thf(fact_4493_zgcd__self,axiom,
! [M: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( legacy_zgcd @ M @ M )
= M ) ) ).
thf(fact_4494_zgcd__geq__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( legacy_zgcd @ X @ Y ) ) ).
thf(fact_4495_zgcd__0,axiom,
! [M: int] :
( ( legacy_zgcd @ M @ zero_zero_int )
= ( abs_abs_int @ M ) ) ).
thf(fact_4496_zgcd__0__left,axiom,
! [M: int] :
( ( legacy_zgcd @ zero_zero_int @ M )
= ( abs_abs_int @ M ) ) ).
thf(fact_4497_zgcd__zdvd__zgcd__zmult,axiom,
! [M: int,N: int,K_1: int] : ( dvd_dvd_int @ ( legacy_zgcd @ M @ N ) @ ( legacy_zgcd @ ( times_times_int @ K_1 @ M ) @ N ) ) ).
thf(fact_4498_zgcd__1__power__left__distrib,axiom,
! [N: nat,A: int,B: int] :
( ( ( legacy_zgcd @ A @ B )
= one_one_int )
=> ( ( legacy_zgcd @ ( power_power_int @ A @ N ) @ B )
= one_one_int ) ) ).
thf(fact_4499_zgcd__1__power__distrib,axiom,
! [N: nat,A: int,B: int] :
( ( ( legacy_zgcd @ A @ B )
= one_one_int )
=> ( ( legacy_zgcd @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= one_one_int ) ) ).
thf(fact_4500_zgcd__power__distrib,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( legacy_zgcd @ A @ B ) @ N )
= ( legacy_zgcd @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
thf(fact_4501_zgcd__commute,axiom,
! [I: int,J: int] :
( ( legacy_zgcd @ I @ J )
= ( legacy_zgcd @ J @ I ) ) ).
thf(fact_4502_zgcd__left__commute,axiom,
! [K_1: int,M: int,N: int] :
( ( legacy_zgcd @ K_1 @ ( legacy_zgcd @ M @ N ) )
= ( legacy_zgcd @ M @ ( legacy_zgcd @ K_1 @ N ) ) ) ).
thf(fact_4503_zgcd__assoc,axiom,
! [K_1: int,M: int,N: int] :
( ( legacy_zgcd @ ( legacy_zgcd @ K_1 @ M ) @ N )
= ( legacy_zgcd @ K_1 @ ( legacy_zgcd @ M @ N ) ) ) ).
thf(fact_4504_zgcd__zminus2,axiom,
! [I: int,J: int] :
( ( legacy_zgcd @ I @ ( uminus_uminus_int @ J ) )
= ( legacy_zgcd @ I @ J ) ) ).
thf(fact_4505_zgcd__zminus,axiom,
! [I: int,J: int] :
( ( legacy_zgcd @ ( uminus_uminus_int @ I ) @ J )
= ( legacy_zgcd @ I @ J ) ) ).
thf(fact_4506_zgcd__zadd__zmult,axiom,
! [M: int,N: int,K_1: int] :
( ( legacy_zgcd @ ( plus_plus_int @ M @ ( times_times_int @ N @ K_1 ) ) @ N )
= ( legacy_zgcd @ M @ N ) ) ).
thf(fact_4507_zgcd__greatest,axiom,
! [N: int,K_1: int,M: int] :
( ( dvd_dvd_int @ K_1 @ M )
=> ( ( dvd_dvd_int @ K_1 @ N )
=> ( dvd_dvd_int @ K_1 @ ( legacy_zgcd @ M @ N ) ) ) ) ).
thf(fact_4508_zgcd__greatest__iff,axiom,
! [K_1: int,M: int,N: int] :
( ( dvd_dvd_int @ K_1 @ ( legacy_zgcd @ M @ N ) )
<=> ( ( dvd_dvd_int @ K_1 @ M )
& ( dvd_dvd_int @ K_1 @ N ) ) ) ).
thf(fact_4509_zgcd__zgcd__zmult,axiom,
! [N: int,K_1: int,M: int] :
( ( ( legacy_zgcd @ K_1 @ M )
= one_one_int )
=> ( ( ( legacy_zgcd @ N @ M )
= one_one_int )
=> ( ( legacy_zgcd @ ( times_times_int @ K_1 @ N ) @ M )
= one_one_int ) ) ) ).
thf(fact_4510_zgcd__zmult__cancel,axiom,
! [M: int,K_1: int,N: int] :
( ( ( legacy_zgcd @ K_1 @ N )
= one_one_int )
=> ( ( legacy_zgcd @ ( times_times_int @ K_1 @ M ) @ N )
= ( legacy_zgcd @ M @ N ) ) ) ).
thf(fact_4511_zgcd__1,axiom,
! [M: int] :
( ( legacy_zgcd @ M @ one_one_int )
= one_one_int ) ).
thf(fact_4512_zgcd__1__left,axiom,
! [M: int] :
( ( legacy_zgcd @ one_one_int @ M )
= one_one_int ) ).
thf(fact_4513_zgcd0,axiom,
! [I: int,J: int] :
( ( ( legacy_zgcd @ I @ J )
= zero_zero_int )
<=> ( ( I = zero_zero_int )
& ( J = zero_zero_int ) ) ) ).
thf(fact_4514_zgcd__eq,axiom,
! [M: int,N: int] :
( ( legacy_zgcd @ M @ N )
= ( legacy_zgcd @ N @ ( div_mod_int @ M @ N ) ) ) ).
thf(fact_4515_zgcd__zmult__eq__self2,axiom,
! [N: int,K_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ( ( legacy_zgcd @ ( times_times_int @ K_1 @ N ) @ K_1 )
= K_1 ) ) ).
thf(fact_4516_zgcd__zmult__distrib2,axiom,
! [M: int,N: int,K_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ( ( times_times_int @ K_1 @ ( legacy_zgcd @ M @ N ) )
= ( legacy_zgcd @ ( times_times_int @ K_1 @ M ) @ ( times_times_int @ K_1 @ N ) ) ) ) ).
thf(fact_4517_zgcd__zmult__eq__self,axiom,
! [N: int,K_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ K_1 )
=> ( ( legacy_zgcd @ K_1 @ ( times_times_int @ K_1 @ N ) )
= K_1 ) ) ).
thf(fact_4518_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_4519_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_4520_div__zgcd__relprime,axiom,
! [B: int,A: int] :
( ( ( A != zero_zero_int )
| ( B != zero_zero_int ) )
=> ( ( legacy_zgcd @ ( div_div_int @ A @ ( legacy_zgcd @ A @ B ) ) @ ( div_div_int @ B @ ( legacy_zgcd @ A @ B ) ) )
= one_one_int ) ) ).
thf(fact_4521_zcong__zgcd__zmult__zmod,axiom,
! [N: int,A: int,B: int,M: int] :
( ( zcong @ A @ B @ M )
=> ( ( zcong @ A @ B @ N )
=> ( ( ( legacy_zgcd @ M @ N )
= one_one_int )
=> ( zcong @ A @ B @ ( times_times_int @ M @ N ) ) ) ) ) ).
thf(fact_4522_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_4523_zgcd__zmult__zdvd__zgcd,axiom,
! [M: int,K_1: int,N: int] :
( ( ( legacy_zgcd @ K_1 @ N )
= one_one_int )
=> ( dvd_dvd_int @ ( legacy_zgcd @ ( times_times_int @ K_1 @ M ) @ N ) @ ( legacy_zgcd @ M @ N ) ) ) ).
thf(fact_4524_zrelprime__zdvd__zmult,axiom,
! [M: int,N: int,K_1: int] :
( ( ( legacy_zgcd @ N @ K_1 )
= one_one_int )
=> ( ( dvd_dvd_int @ K_1 @ ( times_times_int @ M @ N ) )
=> ( dvd_dvd_int @ K_1 @ M ) ) ) ).
thf(fact_4525_zrelprime__dvd__mult,axiom,
! [K_1: int,I: int,J: int] :
( ( ( legacy_zgcd @ I @ J )
= one_one_int )
=> ( ( dvd_dvd_int @ I @ ( times_times_int @ K_1 @ J ) )
=> ( dvd_dvd_int @ I @ K_1 ) ) ) ).
thf(fact_4526_zgcd__code,axiom,
! [K_1: int,L: int] :
( ( legacy_zgcd @ K_1 @ L )
= ( abs_abs_int @ ( if_int @ ( L = zero_zero_int ) @ K_1 @ ( legacy_zgcd @ L @ ( div_mod_int @ ( abs_abs_int @ K_1 ) @ ( abs_abs_int @ L ) ) ) ) ) ) ).
thf(fact_4527_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_4528_zgcd1__iff__no__common__primedivisor,axiom,
! [A: int,B: int] :
( ( ( legacy_zgcd @ A @ B )
= one_one_int )
<=> ~ ? [P_4: int] :
( ( zprime @ P_4 )
& ( dvd_dvd_int @ P_4 @ A )
& ( dvd_dvd_int @ P_4 @ B ) ) ) ).
thf(fact_4529_zgcd__zcong__zgcd,axiom,
! [B: int,A: int,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( legacy_zgcd @ A @ M )
= one_one_int )
=> ( ( zcong @ A @ B @ M )
=> ( ( legacy_zgcd @ B @ M )
= one_one_int ) ) ) ) ).
thf(fact_4530_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_4531_zcong__cancel2,axiom,
! [A: int,B: int,K_1: int,M: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ( legacy_zgcd @ K_1 @ M )
= one_one_int )
=> ( ( zcong @ ( times_times_int @ K_1 @ A ) @ ( times_times_int @ K_1 @ B ) @ M )
<=> ( zcong @ A @ B @ M ) ) ) ) ).
thf(fact_4532_zcong__cancel,axiom,
! [A: int,B: int,K_1: int,M: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ( legacy_zgcd @ K_1 @ M )
= one_one_int )
=> ( ( zcong @ ( times_times_int @ A @ K_1 ) @ ( times_times_int @ B @ K_1 ) @ M )
<=> ( zcong @ A @ B @ M ) ) ) ) ).
thf(fact_4533_zrelprime__zdvd__zmult__aux,axiom,
! [M: int,N: int,K_1: int] :
( ( ( legacy_zgcd @ N @ K_1 )
= one_one_int )
=> ( ( dvd_dvd_int @ K_1 @ ( times_times_int @ M @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( dvd_dvd_int @ K_1 @ M ) ) ) ) ).
thf(fact_4534_Bnor__mem__if,axiom,
! [A: int,B: int,M: int] :
( ( ( legacy_zgcd @ B @ M )
= one_one_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( member_int @ B @ ( bnorRset @ A @ M ) ) ) ) ) ).
thf(fact_4535_int__relprime__odd__power__divisors,axiom,
! [A: int,B: int,C: int,X: int] :
( ( ( times_times_int @ A @ B )
= ( power_power_int @ C @ ( nat_1 @ X ) ) )
=> ( ( ord_less_nat @ one_one_nat @ ( nat_1 @ X ) )
=> ( ( member_int @ X @ zOdd )
=> ( ( ( legacy_zgcd @ A @ B )
= one_one_int )
=> ? [K: int] :
( A
= ( power_power_int @ K @ ( nat_1 @ X ) ) ) ) ) ) ) ).
thf(fact_4536_xzgcd__correct,axiom,
! [M: int,K_1: int,N: int] :
( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( legacy_zgcd @ M @ N )
= K_1 )
<=> ? [S_2: int,T_1: int] :
( ( xzgcd @ M @ N )
= ( produc282740534nt_int @ K_1 @ ( product_Pair_int_int @ S_2 @ T_1 ) ) ) ) ) ).
thf(fact_4537_int__relprime__power__divisors,axiom,
! [A: int,B: int,C: int,N: nat] :
( ( ( times_times_int @ A @ B )
= ( power_power_int @ C @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ( legacy_zgcd @ A @ B )
= one_one_int )
=> ? [K: int] :
( ( abs_abs_int @ A )
= ( power_power_int @ K @ N ) ) ) ) ) ).
thf(fact_4538_xzgcd__correct__aux2,axiom,
! [M: int,N: int,R_3: int,R_1: int,S_3: int,S_1: int,T_2: int,T: int,K_1: int] :
( ? [Sn: int,Tn: int] :
( ( xzgcda @ M @ N @ R_3 @ R_1 @ S_3 @ S_1 @ T_2 @ T )
= ( produc282740534nt_int @ K_1 @ ( product_Pair_int_int @ Sn @ Tn ) ) )
=> ( ( ord_less_int @ zero_zero_int @ R_1 )
=> ( ( legacy_zgcd @ R_3 @ R_1 )
= K_1 ) ) ) ).
thf(fact_4539_zcong__lineq__unique,axiom,
! [B: int,A: int,N: int] :
( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( legacy_zgcd @ A @ N )
= one_one_int )
=> ? [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
& ( ord_less_int @ X_1 @ N )
& ( zcong @ ( times_times_int @ A @ X_1 ) @ B @ N )
& ! [Y_1: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Y_1 )
& ( ord_less_int @ Y_1 @ N )
& ( zcong @ ( times_times_int @ A @ Y_1 ) @ B @ N ) )
=> ( Y_1 = X_1 ) ) ) ) ) ).
thf(fact_4540_zcong__lineq__ex,axiom,
! [A: int,N: int] :
( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( legacy_zgcd @ A @ N )
= one_one_int )
=> ? [X_1: int] : ( zcong @ ( times_times_int @ A @ X_1 ) @ one_one_int @ N ) ) ) ).
thf(fact_4541_xzgcd__correct__aux1,axiom,
! [M: int,N: int,S_3: int,S_1: int,T_2: int,T: int,R_3: int,R_1: int,K_1: int] :
( ( ( legacy_zgcd @ R_3 @ R_1 )
= K_1 )
=> ( ( ord_less_int @ zero_zero_int @ R_1 )
=> ? [Sn: int,Tn: int] :
( ( xzgcda @ M @ N @ R_3 @ R_1 @ S_3 @ S_1 @ T_2 @ T )
= ( produc282740534nt_int @ K_1 @ ( product_Pair_int_int @ Sn @ Tn ) ) ) ) ) ).
thf(fact_4542_int__triple__relprime__odd__power__divisors,axiom,
! [A: int,B: int,C: int,D: int,X: int] :
( ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( power_power_int @ D @ ( nat_1 @ X ) ) )
=> ( ( ord_less_nat @ one_one_nat @ ( nat_1 @ X ) )
=> ( ( member_int @ X @ zOdd )
=> ( ( ( legacy_zgcd @ A @ B )
= one_one_int )
=> ( ( ( legacy_zgcd @ B @ C )
= one_one_int )
=> ( ( ( legacy_zgcd @ C @ A )
= one_one_int )
=> ? [K: int,L_1: int,M_2: int] :
( ( A
= ( power_power_int @ K @ ( nat_1 @ X ) ) )
& ( B
= ( power_power_int @ L_1 @ ( nat_1 @ X ) ) )
& ( C
= ( power_power_int @ M_2 @ ( nat_1 @ X ) ) ) ) ) ) ) ) ) ) ).
thf(fact_4543_zgcd__ex__linear,axiom,
! [M: int,K_1: int,N: int] :
( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( legacy_zgcd @ M @ N )
= K_1 )
=> ? [S_2: int,T_1: int] :
( K_1
= ( plus_plus_int @ ( times_times_int @ S_2 @ M ) @ ( times_times_int @ T_1 @ N ) ) ) ) ) ).
thf(fact_4544_int__triple__relprime__power__divisors,axiom,
! [A: int,B: int,C: int,D: int,N: nat] :
( ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( power_power_int @ D @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ( legacy_zgcd @ A @ B )
= one_one_int )
=> ( ( ( legacy_zgcd @ B @ C )
= one_one_int )
=> ( ( ( legacy_zgcd @ C @ A )
= one_one_int )
=> ? [K: int,L_1: int,M_2: int] :
( ( ( abs_abs_int @ A )
= ( power_power_int @ K @ N ) )
& ( ( abs_abs_int @ B )
= ( power_power_int @ L_1 @ N ) )
& ( ( abs_abs_int @ C )
= ( power_power_int @ M_2 @ N ) ) ) ) ) ) ) ) ).
thf(fact_4545_make__zrelprime,axiom,
! [B: int,A: int] :
( ( ( A != zero_zero_int )
| ( B != zero_zero_int ) )
=> ? [C_2: int,D_2: int] :
( ( A
= ( times_times_int @ ( legacy_zgcd @ A @ B ) @ C_2 ) )
& ( B
= ( times_times_int @ ( legacy_zgcd @ A @ B ) @ D_2 ) )
& ( ( legacy_zgcd @ C_2 @ D_2 )
= one_one_int ) ) ) ).
thf(fact_4546_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_4547_dist__real__def,axiom,
! [X: real,Y: real] :
( ( dist_dist_real @ X @ Y )
= ( abs_abs_real @ ( minus_minus_real @ X @ Y ) ) ) ).
thf(fact_4548_dist__complex__def,axiom,
! [X: complex,Y: complex] :
( ( dist_dist_complex @ X @ Y )
= ( norm_norm_complex @ ( minus_minus_complex @ X @ Y ) ) ) ).
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_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_4551_DERIV__nonneg__imp__nonincreasing,axiom,
! [F: real > real,A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A @ X_1 )
& ( ord_less_eq_real @ X_1 @ B ) )
=> ? [Y_1: real] :
( ( deriv_real @ F @ X_1 @ Y_1 )
& ( ord_less_eq_real @ zero_zero_real @ Y_1 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
thf(fact_4552_DERIV__nonpos__imp__nonincreasing,axiom,
! [F: real > real,A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A @ X_1 )
& ( ord_less_eq_real @ X_1 @ B ) )
=> ? [Y_1: real] :
( ( deriv_real @ F @ X_1 @ Y_1 )
& ( ord_less_eq_real @ Y_1 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
thf(fact_4553_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_4554_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_4555_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_4556_Quickcheck__Narrowing_Oof__int__inject,axiom,
! [N: int,M: int] :
( ( ( quickcheck_of_int @ N )
= ( quickcheck_of_int @ M ) )
<=> ( N = M ) ) ).
thf(fact_4557_int__of__inverse,axiom,
! [X: quickcheck_code_int] :
( ( quickcheck_of_int @ ( quickcheck_int_of @ X ) )
= X ) ).
thf(fact_4558_of__int__int__of,axiom,
! [K_1: quickcheck_code_int] :
( ( quickcheck_of_int @ ( quickcheck_int_of @ K_1 ) )
= K_1 ) ).
thf(fact_4559_int__of__of__int,axiom,
! [N: int] :
( ( quickcheck_int_of @ ( quickcheck_of_int @ N ) )
= N ) ).
thf(fact_4560_number__of__code__int__def,axiom,
number1226105091de_int = quickcheck_of_int ).
thf(fact_4561_fact__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( fact @ M ) @ ( fact @ N ) ) ) ).
thf(fact_4562_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_4563_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_4564_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_4565_zero__code__int__def,axiom,
( zero_z891286103de_int
= ( quickcheck_of_int @ zero_zero_int ) ) ).
thf(fact_4566_fact__lt,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( fact @ N ) ) ).
thf(fact_4567_fact_Osimps_I1_J,axiom,
( ( fact @ zero_zero_nat )
= one_one_nat ) ).
thf(fact_4568_fact_Osimps_I2_J,axiom,
! [N: nat] :
( ( fact @ ( suc @ N ) )
= ( times_times_nat @ ( suc @ N ) @ ( fact @ N ) ) ) ).
thf(fact_4569_fact__le,axiom,
! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( fact @ N ) ) ).
thf(fact_4570_one__code__int__def,axiom,
( one_on1684967323de_int
= ( quickcheck_of_int @ one_one_int ) ) ).
thf(fact_4571_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_4572_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_4573_weak__decr__stable,axiom,
! [F: nat > nat] :
( ! [I_1: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ I_1 ) ) @ ( F @ I_1 ) )
=> ? [I_1: nat] :
! [K: nat] :
( ( F @ ( plus_plus_nat @ I_1 @ K ) )
= ( F @ I_1 ) ) ) ).
thf(fact_4574_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_4575_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_4576_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_4577_norRRset__def,axiom,
! [M: int] :
( ( norRRset @ M )
= ( bnorRset @ ( minus_minus_int @ M @ one_one_int ) @ M ) ) ).
thf(fact_4578_of__int__int__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_int @ ( semiri1621563631at_int @ N ) )
= ( semiri1621563631at_int @ N ) ) ).
thf(fact_4579_real__eq__of__int,axiom,
real_int = ring_1_of_int_real ).
thf(fact_4580_real__of__int__def,axiom,
real_int = ring_1_of_int_real ).
thf(fact_4581_complex__cnj__of__int,axiom,
! [Z_1: int] :
( ( cnj @ ( ring_11397209091omplex @ Z_1 ) )
= ( ring_11397209091omplex @ Z_1 ) ) ).
thf(fact_4582_complex__Re__of__int,axiom,
! [Z_1: int] :
( ( re @ ( ring_11397209091omplex @ Z_1 ) )
= ( ring_1_of_int_real @ Z_1 ) ) ).
thf(fact_4583_complex__number__of__def,axiom,
! [W: int] :
( ( number528085621omplex @ W )
= ( ring_11397209091omplex @ W ) ) ).
thf(fact_4584_int__number__of__def,axiom,
! [W: int] :
( ( number_number_of_int @ W )
= ( ring_1_of_int_int @ W ) ) ).
thf(fact_4585_number__of__real__def,axiom,
! [X: int] :
( ( number267125858f_real @ X )
= ( ring_1_of_int_real @ X ) ) ).
thf(fact_4586_complex__Im__of__int,axiom,
! [Z_1: int] :
( ( im @ ( ring_11397209091omplex @ Z_1 ) )
= zero_zero_real ) ).
thf(fact_4587_phi__def,axiom,
! [M: int] :
( ( phi @ M )
= ( finite_card_int @ ( norRRset @ M ) ) ) ).
thf(fact_4588_norR__mem__unique,axiom,
! [A: int,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ( legacy_zgcd @ A @ M )
= one_one_int )
=> ? [X_1: int] :
( ( zcong @ A @ X_1 @ M )
& ( member_int @ X_1 @ ( norRRset @ M ) )
& ! [Y_1: int] :
( ( ( zcong @ A @ Y_1 @ M )
& ( member_int @ Y_1 @ ( norRRset @ M ) ) )
=> ( Y_1 = X_1 ) ) ) ) ) ).
thf(fact_4589_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_4590_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_4591_RRset__gcd,axiom,
! [A: int,A_1: int > $o,M: int] :
( ( is_RRset @ A_1 @ M )
=> ( ( member_int @ A @ A_1 )
=> ( ( legacy_zgcd @ A @ M )
= one_one_int ) ) ) ).
thf(fact_4592_RRset__zcong__eq,axiom,
! [A: int,B: int,A_1: int > $o,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( is_RRset @ A_1 @ M )
=> ( ( zcong @ A @ B @ M )
=> ( ( member_int @ A @ A_1 )
=> ( ( member_int @ B @ A_1 )
=> ( A = B ) ) ) ) ) ) ).
thf(fact_4593_RRset2norRR__correct,axiom,
! [A: int,A_1: int > $o,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( is_RRset @ A_1 @ M )
=> ( ( member_int @ A @ A_1 )
=> ( ( zcong @ A @ ( rRset2norRR @ A_1 @ M @ A ) @ M )
& ( member_int @ ( rRset2norRR @ A_1 @ M @ A ) @ ( norRRset @ M ) ) ) ) ) ) ).
thf(fact_4594_RRset2norRR__correct2,axiom,
! [A: int,A_1: int > $o,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( is_RRset @ A_1 @ M )
=> ( ( member_int @ A @ A_1 )
=> ( member_int @ ( rRset2norRR @ A_1 @ M @ A ) @ ( norRRset @ M ) ) ) ) ) ).
thf(fact_4595_RRset2norRR__correct1,axiom,
! [A: int,A_1: int > $o,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( is_RRset @ A_1 @ M )
=> ( ( member_int @ A @ A_1 )
=> ( zcong @ A @ ( rRset2norRR @ A_1 @ M @ A ) @ M ) ) ) ) ).
thf(fact_4596_is__RRset__def,axiom,
! [A_1: int > $o,M: int] :
( ( is_RRset @ A_1 @ M )
<=> ( ( member_int_o @ A_1 @ ( rsetR @ M ) )
& ( ( finite_card_int @ A_1 )
= ( phi @ M ) ) ) ) ).
thf(fact_4597_RsetR__fin,axiom,
! [A_1: int > $o,M: int] :
( ( member_int_o @ A_1 @ ( rsetR @ M ) )
=> ( finite_finite_int @ A_1 ) ) ).
thf(fact_4598_Bnor__in__RsetR,axiom,
! [A: int,M: int] :
( ( ord_less_int @ A @ M )
=> ( member_int_o @ ( bnorRset @ A @ M ) @ ( rsetR @ M ) ) ) ).
thf(fact_4599_RRset2norRR__eq__norR,axiom,
! [A_1: int > $o,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( is_RRset @ A_1 @ M )
=> ( ( image_int_int @ ( rRset2norRR @ A_1 @ M ) @ A_1 )
= ( norRRset @ M ) ) ) ) ).
thf(fact_4600_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_4601_ResSet__image,axiom,
! [F: int > int,A_1: int > $o,M: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( resSet @ M @ A_1 )
=> ( ! [X_1: int] :
( ( member_int @ X_1 @ A_1 )
=> ! [Xa: int] :
( ( member_int @ Xa @ A_1 )
=> ( ( zcong @ ( F @ X_1 ) @ ( F @ Xa ) @ M )
=> ( X_1 = Xa ) ) ) )
=> ( resSet @ M @ ( image_int_int @ F @ A_1 ) ) ) ) ) ).
thf(fact_4602_transfer__nat__int__set__relations_I1_J,axiom,
! [A_1: nat > $o] :
( ( finite_finite_nat @ A_1 )
<=> ( finite_finite_int @ ( image_nat_int @ semiri1621563631at_int @ A_1 ) ) ) ).
thf(fact_4603_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J,axiom,
! [A_1: nat > $o] :
( ( finite_card_nat @ A_1 )
= ( finite_card_int @ ( image_nat_int @ semiri1621563631at_int @ A_1 ) ) ) ).
thf(fact_4604_transfer__nat__int__set__relations_I2_J,axiom,
! [X: nat,A_1: nat > $o] :
( ( member_nat @ X @ A_1 )
<=> ( member_int @ ( semiri1621563631at_int @ X ) @ ( image_nat_int @ semiri1621563631at_int @ A_1 ) ) ) ).
thf(fact_4605_transfer__nat__int__set__relations_I3_J,axiom,
! [A_1: nat > $o,B_1: nat > $o] :
( ( A_1 = B_1 )
<=> ( ( image_nat_int @ semiri1621563631at_int @ A_1 )
= ( image_nat_int @ semiri1621563631at_int @ B_1 ) ) ) ).
thf(fact_4606_transfer__int__nat__set__return__embed,axiom,
! [A_1: nat > $o] :
( ( image_int_nat @ nat_1 @ ( image_nat_int @ semiri1621563631at_int @ A_1 ) )
= A_1 ) ).
thf(fact_4607_transfer__nat__int__set__relations_I4_J,axiom,
! [A_1: nat > $o,B_1: nat > $o] :
( ( ord_less_nat_o @ A_1 @ B_1 )
<=> ( ord_less_int_o @ ( image_nat_int @ semiri1621563631at_int @ A_1 ) @ ( image_nat_int @ semiri1621563631at_int @ B_1 ) ) ) ).
thf(fact_4608_transfer__nat__int__set__relations_I5_J,axiom,
! [A_1: nat > $o,B_1: nat > $o] :
( ( ord_less_eq_nat_o @ A_1 @ B_1 )
<=> ( ord_less_eq_int_o @ ( image_nat_int @ semiri1621563631at_int @ A_1 ) @ ( image_nat_int @ semiri1621563631at_int @ B_1 ) ) ) ).
thf(fact_4609_SetS__def,axiom,
! [A: int,P_3: int] :
( ( setS @ A @ P_3 )
= ( image_int_int_o @ ( multInvPair @ A @ P_3 ) @ ( sRStar @ P_3 ) ) ) ).
thf(fact_4610_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_4611_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_4612_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_4613_transfer__int__nat__set__relations_I4_J,axiom,
! [B_1: int > $o,A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ( nat_nat_set @ B_1 )
=> ( ( ord_less_int_o @ A_1 @ B_1 )
<=> ( ord_less_nat_o @ ( image_int_nat @ nat_1 @ A_1 ) @ ( image_int_nat @ nat_1 @ B_1 ) ) ) ) ) ).
thf(fact_4614_gcd__eq__nitpick__gcd,axiom,
! [X: nat,Y: nat] :
( ( gcd_gcd_nat @ X @ Y )
= ( nat_gcd @ X @ Y ) ) ).
thf(fact_4615_Nat__Transfer_Otransfer__nat__int__set__function__closures_I6_J,axiom,
! [X: int,A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ( member_int @ X @ A_1 )
=> ( ord_less_eq_int @ zero_zero_int @ X ) ) ) ).
thf(fact_4616_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_4617_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_4618_transfer__int__nat__set__relations_I3_J,axiom,
! [B_1: int > $o,A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ( nat_nat_set @ B_1 )
=> ( ( A_1 = B_1 )
<=> ( ( image_int_nat @ nat_1 @ A_1 )
= ( image_int_nat @ nat_1 @ B_1 ) ) ) ) ) ).
thf(fact_4619_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_4620_transfer__nat__int__set__return__embed,axiom,
! [A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ( image_nat_int @ semiri1621563631at_int @ ( image_int_nat @ nat_1 @ A_1 ) )
= A_1 ) ) ).
thf(fact_4621_transfer__int__nat__set__relations_I1_J,axiom,
! [A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ( finite_finite_int @ A_1 )
<=> ( finite_finite_nat @ ( image_int_nat @ nat_1 @ A_1 ) ) ) ) ).
thf(fact_4622_Nat__Transfer_Otransfer__int__nat__set__functions_I1_J,axiom,
! [A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ( finite_card_int @ A_1 )
= ( finite_card_nat @ ( image_int_nat @ nat_1 @ A_1 ) ) ) ) ).
thf(fact_4623_transfer__int__nat__set__relations_I5_J,axiom,
! [B_1: int > $o,A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ( nat_nat_set @ B_1 )
=> ( ( ord_less_eq_int_o @ A_1 @ B_1 )
<=> ( ord_less_eq_nat_o @ ( image_int_nat @ nat_1 @ A_1 ) @ ( image_int_nat @ nat_1 @ B_1 ) ) ) ) ) ).
thf(fact_4624_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_4625_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_4626_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_4627_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_4628_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_4629_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_4630_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_4631_finite__greaterThanLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( ord_gr1297742076an_int @ L @ U ) ) ).
thf(fact_4632_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_4633_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( ord_gr660468384an_nat @ L @ U ) ) ).
thf(fact_4634_Rats__real__nat,axiom,
! [N: nat] : ( member_real @ ( real_nat @ N ) @ field_1210416355s_real ) ).
thf(fact_4635_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( ord_gr660468384an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
thf(fact_4636_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_4637_DERIV__isconst3,axiom,
! [F: real > real,Y: real,X: real,A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( member_real @ X @ ( ord_gr788844697n_real @ A @ B ) )
=> ( ( member_real @ Y @ ( ord_gr788844697n_real @ A @ B ) )
=> ( ! [X_1: real] :
( ( member_real @ X_1 @ ( ord_gr788844697n_real @ A @ B ) )
=> ( deriv_real @ F @ X_1 @ zero_zero_real ) )
=> ( ( F @ X )
= ( F @ Y ) ) ) ) ) ) ).
thf(fact_4638_iszero__rat,axiom,
! [K_1: int] :
( ( iszero_rat @ ( number_number_of_rat @ K_1 ) )
<=> ( iszero_int @ ( number_number_of_int @ K_1 ) ) ) ).
thf(fact_4639_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_4640_Sup__greaterThanLessThan,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ Y @ X )
=> ( ( comple124823625p_real @ ( ord_gr788844697n_real @ Y @ X ) )
= X ) ) ).
thf(fact_4641_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_4642_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_4643_Frct__code__post_I4_J,axiom,
! [K_1: int] :
( ( frct @ ( product_Pair_int_int @ ( number_number_of_int @ K_1 ) @ one_one_int ) )
= ( number_number_of_rat @ K_1 ) ) ).
thf(fact_4644_Frct__code__post_I2_J,axiom,
! [K_1: int] :
( ( frct @ ( product_Pair_int_int @ K_1 @ zero_zero_int ) )
= zero_zero_rat ) ).
thf(fact_4645_Frct__code__post_I1_J,axiom,
! [K_1: int] :
( ( frct @ ( product_Pair_int_int @ zero_zero_int @ K_1 ) )
= zero_zero_rat ) ).
thf(fact_4646_Frct__code__post_I3_J,axiom,
( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
= one_one_rat ) ).
thf(fact_4647_Frct__code__post_I6_J,axiom,
! [K_1: int,L: int] :
( ( frct @ ( product_Pair_int_int @ ( number_number_of_int @ K_1 ) @ ( number_number_of_int @ L ) ) )
= ( inverse_divide_rat @ ( number_number_of_rat @ K_1 ) @ ( number_number_of_rat @ L ) ) ) ).
thf(fact_4648_Frct__code__post_I5_J,axiom,
! [K_1: int] :
( ( frct @ ( product_Pair_int_int @ one_one_int @ ( number_number_of_int @ K_1 ) ) )
= ( inverse_divide_rat @ one_one_rat @ ( number_number_of_rat @ K_1 ) ) ) ).
thf(fact_4649_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_4650_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_4651_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_4652_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_4653_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_4654_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_4655_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_4656_diff__rat__def,axiom,
! [Q: rat,R_1: rat] :
( ( minus_minus_rat @ Q @ R_1 )
= ( plus_plus_rat @ Q @ ( uminus_uminus_rat @ R_1 ) ) ) ).
thf(fact_4657_quotient__of__inject__eq,axiom,
! [A: rat,B: rat] :
( ( ( quotient_of @ A )
= ( quotient_of @ B ) )
<=> ( A = B ) ) ).
thf(fact_4658_quotient__of__inject,axiom,
! [A: rat,B: rat] :
( ( ( quotient_of @ A )
= ( quotient_of @ B ) )
=> ( A = B ) ) ).
thf(fact_4659_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_4660_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_4661_rat__one__code,axiom,
( ( quotient_of @ one_one_rat )
= ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
thf(fact_4662_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_4663_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_4664_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_4665_rat__zero__code,axiom,
( ( quotient_of @ zero_zero_rat )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
thf(fact_4666_quotient__of__number_I3_J,axiom,
! [K_1: int] :
( ( quotient_of @ ( number_number_of_rat @ K_1 ) )
= ( product_Pair_int_int @ ( number_number_of_int @ K_1 ) @ one_one_int ) ) ).
thf(fact_4667_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_4668_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_4669_Fract__1__number__of,axiom,
! [K_1: int] :
( ( fract @ one_one_int @ ( number_number_of_int @ K_1 ) )
= ( inverse_divide_rat @ one_one_rat @ ( number_number_of_rat @ K_1 ) ) ) ).
thf(fact_4670_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_4671_quotient__of__eq,axiom,
! [A: int,B: int,P_3: int,Q: int] :
( ( ( quotient_of @ ( fract @ A @ B ) )
= ( product_Pair_int_int @ P_3 @ Q ) )
=> ( ( fract @ P_3 @ Q )
= ( fract @ A @ B ) ) ) ).
thf(fact_4672_minus__rat,axiom,
! [A: int,B: int] :
( ( uminus_uminus_rat @ ( fract @ A @ B ) )
= ( fract @ ( uminus_uminus_int @ A ) @ B ) ) ).
thf(fact_4673_divide__rat,axiom,
! [A: int,B: int,C: int,D: int] :
( ( inverse_divide_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ).
thf(fact_4674_Fract__of__int__quotient,axiom,
! [K_1: int,L: int] :
( ( fract @ K_1 @ L )
= ( inverse_divide_rat @ ( ring_1_of_int_rat @ K_1 ) @ ( ring_1_of_int_rat @ L ) ) ) ).
thf(fact_4675_Fract__of__int__eq,axiom,
! [K_1: int] :
( ( fract @ K_1 @ one_one_int )
= ( ring_1_of_int_rat @ K_1 ) ) ).
thf(fact_4676_of__int__rat,axiom,
! [K_1: int] :
( ( ring_1_of_int_rat @ K_1 )
= ( fract @ K_1 @ one_one_int ) ) ).
thf(fact_4677_rat__number__collapse_I1_J,axiom,
! [K_1: int] :
( ( fract @ zero_zero_int @ K_1 )
= zero_zero_rat ) ).
thf(fact_4678_eq__rat_I3_J,axiom,
! [A: int,C: int] :
( ( fract @ zero_zero_int @ A )
= ( fract @ zero_zero_int @ C ) ) ).
thf(fact_4679_rat__number__collapse_I4_J,axiom,
! [K_1: int] :
( ( fract @ K_1 @ zero_zero_int )
= zero_zero_rat ) ).
thf(fact_4680_one__rat,axiom,
( one_one_rat
= ( fract @ one_one_int @ one_one_int ) ) ).
thf(fact_4681_rat__number__collapse_I2_J,axiom,
( ( fract @ one_one_int @ one_one_int )
= one_one_rat ) ).
thf(fact_4682_eq__rat_I2_J,axiom,
! [A: int] :
( ( fract @ A @ zero_zero_int )
= ( fract @ zero_zero_int @ one_one_int ) ) ).
thf(fact_4683_Fract__of__nat__eq,axiom,
! [K_1: nat] :
( ( fract @ ( semiri1621563631at_int @ K_1 ) @ one_one_int )
= ( semiri151668891at_rat @ K_1 ) ) ).
thf(fact_4684_of__nat__rat,axiom,
! [K_1: nat] :
( ( semiri151668891at_rat @ K_1 )
= ( fract @ ( semiri1621563631at_int @ K_1 ) @ one_one_int ) ) ).
thf(fact_4685_rat__number__of__def,axiom,
! [W: int] :
( ( number_number_of_rat @ W )
= ( fract @ W @ one_one_int ) ) ).
thf(fact_4686_real__floor__code,axiom,
! [X: rat] :
( ( archim1246769320r_real @ ( ratreal @ X ) )
= ( archim791455193or_rat @ X ) ) ).
thf(fact_4687_floor__Fract,axiom,
! [A: int,B: int] :
( ( archim791455193or_rat @ ( fract @ A @ B ) )
= ( div_div_int @ A @ B ) ) ).
thf(fact_4688_inverse__rat,axiom,
! [A: int,B: int] :
( ( inverse_inverse_rat @ ( fract @ A @ B ) )
= ( fract @ B @ A ) ) ).
thf(fact_4689_minus__rat__cancel,axiom,
! [A: int,B: int] :
( ( fract @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( fract @ A @ B ) ) ).
thf(fact_4690_mult__rat__cancel,axiom,
! [A: int,B: int,C: int] :
( ( C != zero_zero_int )
=> ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( fract @ A @ B ) ) ) ).
thf(fact_4691_eq__rat_I1_J,axiom,
! [A: int,C: int,D: int,B: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( ( fract @ A @ B )
= ( fract @ C @ D ) )
<=> ( ( times_times_int @ A @ D )
= ( times_times_int @ C @ B ) ) ) ) ) ).
thf(fact_4692_mult__rat,axiom,
! [A: int,B: int,C: int,D: int] :
( ( times_times_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ).
thf(fact_4693_abs__rat,axiom,
! [A: int,B: int] :
( ( abs_abs_rat @ ( fract @ A @ B ) )
= ( fract @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
thf(fact_4694_normalize__eq,axiom,
! [A: int,B: int,P_3: int,Q: int] :
( ( ( normalize @ ( product_Pair_int_int @ A @ B ) )
= ( product_Pair_int_int @ P_3 @ Q ) )
=> ( ( fract @ P_3 @ Q )
= ( fract @ A @ B ) ) ) ).
thf(fact_4695_real__less__code,axiom,
! [X: rat,Y: rat] :
( ( ord_less_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
<=> ( ord_less_rat @ X @ Y ) ) ).
thf(fact_4696_zero__real__code,axiom,
( zero_zero_real
= ( ratreal @ zero_zero_rat ) ) ).
thf(fact_4697_Ratreal__number__collapse_I1_J,axiom,
( ( ratreal @ zero_zero_rat )
= zero_zero_real ) ).
thf(fact_4698_number__of__real__code,axiom,
! [K_1: int] :
( ( number267125858f_real @ K_1 )
= ( ratreal @ ( number_number_of_rat @ K_1 ) ) ) ).
thf(fact_4699_Ratreal__number__collapse_I3_J,axiom,
! [K_1: int] :
( ( ratreal @ ( number_number_of_rat @ K_1 ) )
= ( number267125858f_real @ K_1 ) ) ).
thf(fact_4700_one__real__code,axiom,
( one_one_real
= ( ratreal @ one_one_rat ) ) ).
thf(fact_4701_Ratreal__number__collapse_I2_J,axiom,
( ( ratreal @ one_one_rat )
= one_one_real ) ).
thf(fact_4702_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_4703_real__plus__code,axiom,
! [X: rat,Y: rat] :
( ( plus_plus_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
= ( ratreal @ ( plus_plus_rat @ X @ Y ) ) ) ).
thf(fact_4704_real__minus__code,axiom,
! [X: rat,Y: rat] :
( ( minus_minus_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
= ( ratreal @ ( minus_minus_rat @ X @ Y ) ) ) ).
thf(fact_4705_real__uminus__code,axiom,
! [X: rat] :
( ( uminus_uminus_real @ ( ratreal @ X ) )
= ( ratreal @ ( uminus_uminus_rat @ X ) ) ) ).
thf(fact_4706_real__divide__code,axiom,
! [X: rat,Y: rat] :
( ( inverse_divide_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
= ( ratreal @ ( inverse_divide_rat @ X @ Y ) ) ) ).
thf(fact_4707_real__times__code,axiom,
! [X: rat,Y: rat] :
( ( times_times_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
= ( ratreal @ ( times_times_rat @ X @ Y ) ) ) ).
thf(fact_4708_real__inverse__code,axiom,
! [X: rat] :
( ( inverse_inverse_real @ ( ratreal @ X ) )
= ( ratreal @ ( inverse_inverse_rat @ X ) ) ) ).
thf(fact_4709_zero__rat,axiom,
( zero_zero_rat
= ( fract @ zero_zero_int @ one_one_int ) ) ).
thf(fact_4710_rat__number__expand_I3_J,axiom,
! [K_1: int] :
( ( number_number_of_rat @ K_1 )
= ( fract @ ( number_number_of_int @ K_1 ) @ one_one_int ) ) ).
thf(fact_4711_rat__number__collapse_I3_J,axiom,
! [K_1: int] :
( ( fract @ ( number_number_of_int @ K_1 ) @ one_one_int )
= ( number_number_of_rat @ K_1 ) ) ).
thf(fact_4712_Fract__number__of__quotient,axiom,
! [K_1: int,L: int] :
( ( fract @ ( number_number_of_int @ K_1 ) @ ( number_number_of_int @ L ) )
= ( inverse_divide_rat @ ( number_number_of_rat @ K_1 ) @ ( number_number_of_rat @ L ) ) ) ).
thf(fact_4713_quotient__of__Fract,axiom,
! [A: int,B: int] :
( ( quotient_of @ ( fract @ A @ B ) )
= ( normalize @ ( product_Pair_int_int @ A @ B ) ) ) ).
thf(fact_4714_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_4715_less__rat,axiom,
! [A: int,C: int,D: int,B: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
<=> ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
thf(fact_4716_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_4717_sgn__rat,axiom,
! [A: int,B: int] :
( ( sgn_sgn_rat @ ( fract @ A @ B ) )
= ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).
thf(fact_4718_Fract__less__zero__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
<=> ( ord_less_int @ A @ zero_zero_int ) ) ) ).
thf(fact_4719_zero__less__Fract__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
<=> ( ord_less_int @ zero_zero_int @ A ) ) ) ).
thf(fact_4720_le__rat,axiom,
! [A: int,C: int,D: int,B: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
<=> ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
thf(fact_4721_add__rat,axiom,
! [A: int,C: int,D: int,B: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
thf(fact_4722_diff__rat,axiom,
! [A: int,C: int,D: int,B: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
thf(fact_4723_one__less__Fract__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
<=> ( ord_less_int @ B @ A ) ) ) ).
thf(fact_4724_Fract__less__one__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
<=> ( ord_less_int @ A @ B ) ) ) ).
thf(fact_4725_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_4726_Fract__le__zero__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
<=> ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
thf(fact_4727_zero__le__Fract__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
<=> ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
thf(fact_4728_rat__floor__lemma,axiom,
! [A: int,B: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( div_div_int @ A @ B ) ) @ ( fract @ A @ B ) )
& ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( div_div_int @ A @ B ) @ one_one_int ) ) ) ) ).
thf(fact_4729_one__le__Fract__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
<=> ( ord_less_eq_int @ B @ A ) ) ) ).
thf(fact_4730_Fract__le__one__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
<=> ( ord_less_eq_int @ A @ B ) ) ) ).
thf(fact_4731_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_4732_adjust__eq,axiom,
! [B: int,Q: int,R_1: int] :
( ( adjust @ B @ ( product_Pair_int_int @ Q @ R_1 ) )
= ( if_Pro1731782967nt_int @ ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ R_1 @ B ) ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Q ) @ one_one_int ) @ ( minus_minus_int @ R_1 @ B ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Q ) @ R_1 ) ) ) ).
thf(fact_4733_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_4734_finite__Collect__less__nat,axiom,
! [K_1: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N_1: nat] : ( ord_less_nat @ N_1 @ K_1 ) ) ) ).
thf(fact_4735_finite__Collect__le__nat,axiom,
! [K_1: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N_1: nat] : ( ord_less_eq_nat @ N_1 @ K_1 ) ) ) ).
thf(fact_4736_transfer__morphism__int__nat,axiom,
( nat_tr160667106at_int @ semiri1621563631at_int
@ ^ [N_1: nat] : $true ) ).
thf(fact_4737_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_4738_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_4739_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_4740_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_4741_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_4742_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_4743_DERIV__cos__add,axiom,
! [K_1: real,Xa_1: real] :
( deriv_real
@ ^ [X_1: real] : ( cos @ ( plus_plus_real @ X_1 @ K_1 ) )
@ Xa_1
@ ( uminus_uminus_real @ ( sin @ ( plus_plus_real @ Xa_1 @ K_1 ) ) ) ) ).
thf(fact_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_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_4746_DERIV__sin__add,axiom,
! [K_1: real,Xa_1: real] :
( deriv_real
@ ^ [X_1: real] : ( sin @ ( plus_plus_real @ X_1 @ K_1 ) )
@ Xa_1
@ ( cos @ ( plus_plus_real @ Xa_1 @ K_1 ) ) ) ).
thf(fact_4747_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_4748_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_4749_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_4750_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_4751_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_4752_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_4753_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_4754_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_4755_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_4756_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_4757_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_4758_primes__infinite,axiom,
~ ( finite_finite_nat @ ( collect_nat @ prime ) ) ).
thf(fact_4759_cnj_OCauchy,axiom,
! [X_2: nat > complex] :
( ( cauchy_complex @ X_2 )
=> ( cauchy_complex
@ ^ [N_1: nat] : ( cnj @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_4760_Re_OCauchy,axiom,
! [X_2: nat > complex] :
( ( cauchy_complex @ X_2 )
=> ( cauchy_real
@ ^ [N_1: nat] : ( re @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_4761_Im_OCauchy,axiom,
! [X_2: nat > complex] :
( ( cauchy_complex @ X_2 )
=> ( cauchy_real
@ ^ [N_1: nat] : ( im @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_4762_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_4763_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_4764_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I_1: nat] : ( ord_less_nat @ I_1 @ N ) ) )
= N ) ).
thf(fact_4765_card__bdd__nat__set__l,axiom,
! [X: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [Y_1: nat] : ( ord_less_nat @ Y_1 @ X ) ) )
= X ) ).
thf(fact_4766_bdd__nat__set__l__finite,axiom,
! [X: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [Y_1: nat] : ( ord_less_nat @ Y_1 @ X ) ) ) ).
thf(fact_4767_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K: nat] : ( (&) @ ( P @ K ) @ ( ord_less_nat @ K @ I ) ) ) ) ).
thf(fact_4768_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_4769_card__less__Suc2,axiom,
! [I: nat,M_3: nat > $o] :
( ~ ( member_nat @ zero_zero_nat @ M_3 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] : ( (&) @ ( member_nat @ ( suc @ K ) @ M_3 ) @ ( ord_less_nat @ K @ I ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] : ( (&) @ ( member_nat @ K @ M_3 ) @ ( ord_less_nat @ K @ ( suc @ I ) ) ) ) ) ) ) ).
thf(fact_4770_card__less,axiom,
! [I: nat,M_3: nat > $o] :
( ( member_nat @ zero_zero_nat @ M_3 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] : ( (&) @ ( member_nat @ K @ M_3 ) @ ( ord_less_nat @ K @ ( suc @ I ) ) ) ) )
!= zero_zero_nat ) ) ).
thf(fact_4771_card__less__Suc,axiom,
! [I: nat,M_3: nat > $o] :
( ( member_nat @ zero_zero_nat @ M_3 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] : ( (&) @ ( member_nat @ ( suc @ K ) @ M_3 ) @ ( ord_less_nat @ K @ I ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] : ( (&) @ ( member_nat @ K @ M_3 ) @ ( ord_less_nat @ K @ ( suc @ I ) ) ) ) ) ) ) ).
thf(fact_4772_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_4773_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_4774_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_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_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_4777_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_4778_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_4779_eq__number__of__Suc,axiom,
! [V: int,N: nat] :
( ( ( number_number_of_nat @ V )
= ( suc @ N ) )
<=> ( ~ ( nat_neg @ ( number_number_of_int @ ( pred @ V ) ) )
& ( ~ ( nat_neg @ ( number_number_of_int @ ( pred @ V ) ) )
=> ( ( nat_1 @ ( number_number_of_int @ ( pred @ V ) ) )
= N ) ) ) ) ).
thf(fact_4780_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_4781_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_4782_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_4783_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_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_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_4786_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_4787_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_4788_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_4789_RsetR__zmult__mono,axiom,
! [X: int,A_1: int > $o,M: int] :
( ( member_int_o @ A_1 @ ( rsetR @ M ) )
=> ( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( legacy_zgcd @ X @ M )
= one_one_int )
=> ( member_int_o
@ ( image_int_int
@ ^ [A_2: int] : ( times_times_int @ A_2 @ X )
@ A_1 )
@ ( rsetR @ M ) ) ) ) ) ).
thf(fact_4790_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_4791_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_4792_DERIV__log,axiom,
! [B: real,X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( deriv_real @ ( log @ B ) @ X @ ( inverse_divide_real @ one_one_real @ ( times_times_real @ ( ln @ B ) @ X ) ) ) ) ).
thf(fact_4793_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_4794_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_4795_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_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_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_4798_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_4799_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_4800_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_4801_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_4802_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_4803_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_4804_norm__frac_Opsimps,axiom,
! [A: int,B: int] :
( ( accp_P2006205492nt_int @ norm_frac_rel @ ( product_Pair_int_int @ A @ B ) )
=> ( ( ( ord_less_int @ B @ zero_zero_int )
=> ( ( norm_frac @ A @ B )
= ( norm_frac @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) )
& ( ~ ( ord_less_int @ B @ zero_zero_int )
=> ( ( ( ( A = zero_zero_int )
| ( B = zero_zero_int ) )
=> ( ( norm_frac @ A @ B )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) )
& ( ~ ( ( A = zero_zero_int )
| ( B = zero_zero_int ) )
=> ( ( norm_frac @ A @ B )
= ( product_Pair_int_int @ ( div_div_int @ A @ ( int_gcd @ A @ B ) ) @ ( div_div_int @ B @ ( int_gcd @ A @ B ) ) ) ) ) ) ) ) ) ).
thf(fact_4805_monoseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( monoseq_real
@ ^ [N_1: nat] : ( times_times_real @ ( inverse_divide_real @ one_one_real @ ( real_nat @ ( plus_plus_nat @ ( times_times_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) ) ) ).
thf(fact_4806_monoseq__realpow,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( monoseq_real @ ( power_power_real @ X ) ) ) ) ).
thf(fact_4807_norm__frac_Osimps,axiom,
! [A: int,B: int] :
( ( ( ord_less_int @ B @ zero_zero_int )
=> ( ( norm_frac @ A @ B )
= ( norm_frac @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) )
& ( ~ ( ord_less_int @ B @ zero_zero_int )
=> ( ( ( ( A = zero_zero_int )
| ( B = zero_zero_int ) )
=> ( ( norm_frac @ A @ B )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) )
& ( ~ ( ( A = zero_zero_int )
| ( B = zero_zero_int ) )
=> ( ( norm_frac @ A @ B )
= ( product_Pair_int_int @ ( div_div_int @ A @ ( int_gcd @ A @ B ) ) @ ( div_div_int @ B @ ( int_gcd @ A @ B ) ) ) ) ) ) ) ) ).
thf(fact_4808_arg__def,axiom,
! [Z_1: complex] :
( ( arg @ Z_1 )
= ( hilbert_Eps_real
@ ^ [A_2: real] :
( (&)
@ ( ( re @ ( sgn_sgn_complex @ Z_1 ) )
= ( cos @ A_2 ) )
@ ( (&)
@ ( ( im @ ( sgn_sgn_complex @ Z_1 ) )
= ( sin @ A_2 ) )
@ ( (&) @ ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A_2 ) @ ( ord_less_eq_real @ A_2 @ pi ) ) ) ) ) ) ).
thf(fact_4809_Union__SetS__setprod__prop1,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A )
=> ( zcong
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( comple1092985777_int_o @ ( setS @ A @ P_3 ) ) )
@ ( power_power_int @ A @ ( nat_1 @ ( div_div_int @ ( minus_minus_int @ P_3 @ one_one_int ) @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
@ P_3 ) ) ) ) ) ).
thf(fact_4810_d22set__prod__zfact,axiom,
! [A: int] :
( ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( d22set @ A ) )
= ( zfact @ A ) ) ).
thf(fact_4811_Bnor__prod__zgcd,axiom,
! [A: int,M: int] :
( ( ord_less_int @ A @ M )
=> ( ( legacy_zgcd
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( bnorRset @ A @ M ) )
@ M )
= one_one_int ) ) ).
thf(fact_4812_Bnor__prod__power,axiom,
! [A: int,M: int,X: int] :
( ( X != zero_zero_int )
=> ( ( ord_less_int @ A @ M )
=> ( ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( image_int_int
@ ^ [A_2: int] : ( times_times_int @ A_2 @ X )
@ ( bnorRset @ A @ M ) ) )
= ( times_times_int
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( bnorRset @ A @ M ) )
@ ( power_power_int @ X @ ( finite_card_int @ ( bnorRset @ A @ M ) ) ) ) ) ) ) ).
thf(fact_4813_SetS__setprod__prop,axiom,
! [X: int > $o,A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ~ ( quadRes @ P_3 @ A )
=> ( ( member_int_o @ X @ ( setS @ A @ P_3 ) )
=> ( zcong
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ X )
@ A
@ P_3 ) ) ) ) ) ) ).
thf(fact_4814_wset__zcong__prod__1,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_eq_int @ ( number_number_of_int @ ( bit1 @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ P_3 )
=> ( ( ord_less_int @ A @ ( minus_minus_int @ P_3 @ one_one_int ) )
=> ( zcong
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( wset @ A @ P_3 ) )
@ one_one_int
@ P_3 ) ) ) ) ).
thf(fact_4815_Union__SetS__setprod__prop2,axiom,
! [A: int,P_3: int] :
( ( zprime @ P_3 )
=> ( ( ord_less_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ P_3 )
=> ( ~ ( zcong @ A @ zero_zero_int @ P_3 )
=> ( ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ ( comple1092985777_int_o @ ( setS @ A @ P_3 ) ) )
= ( zfact @ ( minus_minus_int @ P_3 @ one_one_int ) ) ) ) ) ) ).
thf(fact_4816_transfer__nat__int__sum__prod__closure_I2_J,axiom,
! [F: int > int,A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ! [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X_1 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( big_co1548731110nt_int @ F @ A_1 ) ) ) ) ).
thf(fact_4817_arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( arctan @ X )
= ( suminf_real
@ ^ [K: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ K ) @ ( times_times_real @ ( inverse_divide_real @ one_one_real @ ( real_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
thf(fact_4818_DERIV__arctan__series,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( deriv_real
@ ^ [X_3: real] :
( suminf_real
@ ^ [K: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ K ) @ ( times_times_real @ ( inverse_divide_real @ one_one_real @ ( real_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X_3 @ ( plus_plus_nat @ ( times_times_nat @ K @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) ) )
@ X
@ ( suminf_real
@ ^ [K: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ K ) @ ( power_power_real @ X @ ( times_times_nat @ K @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_4819_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_4820_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_4821_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_4822_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_4823_lemma__sin__minus,axiom,
! [X: real] :
( ( uminus_uminus_real @ ( sin @ X ) )
= ( suminf_real
@ ^ [N_1: nat] : ( uminus_uminus_real @ ( times_times_real @ ( sin_coeff @ N_1 ) @ ( power_power_real @ X @ N_1 ) ) ) ) ) ).
thf(fact_4824_exp__first__two__terms,axiom,
! [X: real] :
( ( exp_real @ X )
= ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X )
@ ( suminf_real
@ ^ [N_1: nat] : ( times_times_real @ ( inverse_inverse_real @ ( real_nat @ ( fact_fact_nat @ ( plus_plus_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ) ) ).
thf(fact_4825_pi__series,axiom,
( ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
= ( suminf_real
@ ^ [K: nat] : ( inverse_divide_real @ ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ K ) @ one_one_real ) @ ( real_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) ) ) ).
thf(fact_4826_ln__series,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
=> ( ( ln @ X )
= ( suminf_real
@ ^ [N_1: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ N_1 ) @ ( inverse_divide_real @ one_one_real @ ( real_nat @ ( plus_plus_nat @ N_1 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N_1 ) ) ) ) ) ) ) ).
thf(fact_4827_summable__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( summable_real
@ ^ [K: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ K ) @ ( times_times_real @ ( inverse_divide_real @ one_one_real @ ( real_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) ) ) ) ).
thf(fact_4828_summable__rabs__cancel,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N_1: nat] : ( abs_abs_real @ ( F @ N_1 ) ) )
=> ( summable_real @ F ) ) ).
thf(fact_4829_summable__rabs__comparison__test,axiom,
! [F: nat > real,G: nat > real] :
( ? [N_2: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ N_2 @ N_1 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N_1 ) ) @ ( G @ N_1 ) ) )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N_1: nat] : ( abs_abs_real @ ( F @ N_1 ) ) ) ) ) ).
thf(fact_4830_summable__rabs,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N_1: nat] : ( abs_abs_real @ ( F @ N_1 ) ) )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
@ ( suminf_real
@ ^ [N_1: nat] : ( abs_abs_real @ ( F @ N_1 ) ) ) ) ) ).
thf(fact_4831_summable__sin,axiom,
! [X: real] :
( summable_real
@ ^ [N_1: nat] : ( times_times_real @ ( sin_coeff @ N_1 ) @ ( power_power_real @ X @ N_1 ) ) ) ).
thf(fact_4832_summable__cos,axiom,
! [X: real] :
( summable_real
@ ^ [N_1: nat] : ( times_times_real @ ( cos_coeff @ N_1 ) @ ( power_power_real @ X @ N_1 ) ) ) ).
thf(fact_4833_rat__floor__code,axiom,
! [P_3: rat] :
( ( archim791455193or_rat @ P_3 )
= ( produc1298267108nt_int @ div_div_int @ ( quotient_of @ P_3 ) ) ) ).
thf(fact_4834_rat__uminus__code,axiom,
! [P_3: rat] :
( ( quotient_of @ ( uminus_uminus_rat @ P_3 ) )
= ( produc1518849193nt_int
@ ^ [A_2: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A_2 ) )
@ ( quotient_of @ P_3 ) ) ) ).
thf(fact_4835_summable__exp,axiom,
! [X: real] :
( summable_real
@ ^ [N_1: nat] : ( times_times_real @ ( inverse_inverse_real @ ( real_nat @ ( fact_fact_nat @ N_1 ) ) ) @ ( power_power_real @ X @ N_1 ) ) ) ).
thf(fact_4836_rat__abs__code,axiom,
! [P_3: rat] :
( ( quotient_of @ ( abs_abs_rat @ P_3 ) )
= ( produc1518849193nt_int
@ ^ [A_2: int] : ( product_Pair_int_int @ ( abs_abs_int @ A_2 ) )
@ ( quotient_of @ P_3 ) ) ) ).
thf(fact_4837_divmod__int__rel__def,axiom,
! [A: int,B: int] :
( ( divmod_int_rel @ A @ B )
= ( produc450523309_int_o
@ ^ [Q_2: int,R: int] :
( (&)
@ ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q_2 ) @ R ) )
@ ( (&) @ ( (=>) @ ( ord_less_int @ zero_zero_int @ B ) @ ( (&) @ ( ord_less_eq_int @ zero_zero_int @ R ) @ ( ord_less_int @ R @ B ) ) ) @ ( (=>) @ ( (~) @ ( ord_less_int @ zero_zero_int @ B ) ) @ ( (&) @ ( ord_less_int @ B @ R ) @ ( ord_less_eq_int @ R @ zero_zero_int ) ) ) ) ) ) ) ).
thf(fact_4838_sum2sq__def,axiom,
( twoSqu2072599593sum2sq
= ( produc1298267108nt_int
@ ^ [A_2: int,B_4: int] : ( plus_plus_int @ ( power_power_int @ A_2 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_int @ B_4 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_4839_rat__inverse__code,axiom,
! [P_3: rat] :
( ( quotient_of @ ( inverse_inverse_rat @ P_3 ) )
= ( produc1518849193nt_int
@ ^ [A_2: int,B_4: int] : ( if_Pro1731782967nt_int @ ( A_2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A_2 ) @ B_4 ) @ ( abs_abs_int @ A_2 ) ) )
@ ( quotient_of @ P_3 ) ) ) ).
thf(fact_4840_exp__tail__after__first__two__terms__summable,axiom,
! [X: real] :
( summable_real
@ ^ [N_1: nat] : ( times_times_real @ ( inverse_inverse_real @ ( real_nat @ ( fact_fact_nat @ ( plus_plus_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ) ).
thf(fact_4841_adjust__def,axiom,
! [B: int] :
( ( adjust @ B )
= ( produc1518849193nt_int
@ ^ [Q_2: int,R: int] : ( if_Pro1731782967nt_int @ ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ R @ B ) ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Q_2 ) @ one_one_int ) @ ( minus_minus_int @ R @ B ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ Q_2 ) @ R ) ) ) ) ).
thf(fact_4842_DERIV__power__series_H,axiom,
! [X0: real,F: nat > real,R_2: real] :
( ! [X_1: real] :
( ( member_real @ X_1 @ ( ord_gr788844697n_real @ ( uminus_uminus_real @ R_2 ) @ R_2 ) )
=> ( summable_real
@ ^ [N_1: nat] : ( times_times_real @ ( times_times_real @ ( F @ N_1 ) @ ( real_nat @ ( suc @ N_1 ) ) ) @ ( power_power_real @ X_1 @ N_1 ) ) ) )
=> ( ( member_real @ X0 @ ( ord_gr788844697n_real @ ( uminus_uminus_real @ R_2 ) @ R_2 ) )
=> ( ( ord_less_real @ zero_zero_real @ R_2 )
=> ( deriv_real
@ ^ [X_1: real] :
( suminf_real
@ ^ [N_1: nat] : ( times_times_real @ ( F @ N_1 ) @ ( power_power_real @ X_1 @ ( suc @ N_1 ) ) ) )
@ X0
@ ( suminf_real
@ ^ [N_1: nat] : ( times_times_real @ ( times_times_real @ ( F @ N_1 ) @ ( real_nat @ ( suc @ N_1 ) ) ) @ ( power_power_real @ X0 @ N_1 ) ) ) ) ) ) ) ).
thf(fact_4843_Nitpick_OFrac__def,axiom,
( frac
= ( produc450523309_int_o
@ ^ [A_2: int,B_4: int] :
( (&) @ ( ord_less_int @ zero_zero_int @ B_4 )
@ ( ( int_gcd @ A_2 @ B_4 )
= one_one_int ) ) ) ) ).
thf(fact_4844_cnj_Osuminf,axiom,
! [X_2: nat > complex] :
( ( summable_complex @ X_2 )
=> ( ( cnj @ ( suminf_complex @ X_2 ) )
= ( suminf_complex
@ ^ [N_1: nat] : ( cnj @ ( X_2 @ N_1 ) ) ) ) ) ).
thf(fact_4845_cnj_Osummable,axiom,
! [X_2: nat > complex] :
( ( summable_complex @ X_2 )
=> ( summable_complex
@ ^ [N_1: nat] : ( cnj @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_4846_Re_Osummable,axiom,
! [X_2: nat > complex] :
( ( summable_complex @ X_2 )
=> ( summable_real
@ ^ [N_1: nat] : ( re @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_4847_Im_Osummable,axiom,
! [X_2: nat > complex] :
( ( summable_complex @ X_2 )
=> ( summable_real
@ ^ [N_1: nat] : ( im @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_4848_Re_Osuminf,axiom,
! [X_2: nat > complex] :
( ( summable_complex @ X_2 )
=> ( ( re @ ( suminf_complex @ X_2 ) )
= ( suminf_real
@ ^ [N_1: nat] : ( re @ ( X_2 @ N_1 ) ) ) ) ) ).
thf(fact_4849_Im_Osuminf,axiom,
! [X_2: nat > complex] :
( ( summable_complex @ X_2 )
=> ( ( im @ ( suminf_complex @ X_2 ) )
= ( suminf_real
@ ^ [N_1: nat] : ( im @ ( X_2 @ N_1 ) ) ) ) ) ).
thf(fact_4850_ratrel__def,axiom,
( ratrel
= ( collec50511176nt_int
@ ( produc141074865_int_o
@ ^ [X_1: product_prod_int_int,Y_1: product_prod_int_int] :
( (&)
@ ( (~)
@ ( ( product_snd_int_int @ X_1 )
= zero_zero_int ) )
@ ( (&)
@ ( (~)
@ ( ( product_snd_int_int @ Y_1 )
= zero_zero_int ) )
@ ( ( times_times_int @ ( product_fst_int_int @ X_1 ) @ ( product_snd_int_int @ Y_1 ) )
= ( times_times_int @ ( product_fst_int_int @ Y_1 ) @ ( product_snd_int_int @ X_1 ) ) ) ) ) ) ) ) ).
thf(fact_4851_divmod__nat__if,axiom,
! [M: nat,N: nat] :
( ( ( ( N = zero_zero_nat )
| ( ord_less_nat @ M @ N ) )
=> ( ( divmod_nat @ M @ N )
= ( product_Pair_nat_nat @ zero_zero_nat @ M ) ) )
& ( ~ ( ( N = zero_zero_nat )
| ( ord_less_nat @ M @ N ) )
=> ( ( divmod_nat @ M @ N )
= ( produc1391996073at_nat
@ ^ [Q_2: nat] : ( product_Pair_nat_nat @ ( suc @ Q_2 ) )
@ ( divmod_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
thf(fact_4852_Bolzano__bisect_Osimps_I2_J,axiom,
! [P: produc914805421l_real > $o,A: real,B: real,N: nat] :
( ( bolzano_bisect @ P @ A @ B @ ( suc @ N ) )
= ( produc595218619l_real
@ ^ [X_1: real,Y_1: real] : ( if_Pro313124157l_real @ ( P @ ( produc865579683l_real @ X_1 @ ( inverse_divide_real @ ( plus_plus_real @ X_1 @ Y_1 ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) @ ( produc865579683l_real @ ( inverse_divide_real @ ( plus_plus_real @ X_1 @ Y_1 ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ Y_1 ) @ ( produc865579683l_real @ X_1 @ ( inverse_divide_real @ ( plus_plus_real @ X_1 @ Y_1 ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
@ ( bolzano_bisect @ P @ A @ B @ N ) ) ) ).
thf(fact_4853_suminf__exist__split,axiom,
! [A: nat > real,R_1: real] :
( ( ord_less_real @ zero_zero_real @ R_1 )
=> ( ( summable_real @ A )
=> ? [N_2: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ N_2 @ N_1 )
=> ( ord_less_real
@ ( abs_abs_real
@ ( suminf_real
@ ^ [I_1: nat] : ( A @ ( plus_plus_nat @ I_1 @ N_1 ) ) ) )
@ R_1 ) ) ) ) ).
thf(fact_4854_divmod__int__pdivmod,axiom,
! [L: int,K_1: int] :
( ( ( K_1 = zero_zero_int )
=> ( ( divmod_int @ K_1 @ L )
= ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) )
& ( ( K_1 != zero_zero_int )
=> ( ( ( L = zero_zero_int )
=> ( ( divmod_int @ K_1 @ L )
= ( product_Pair_int_int @ zero_zero_int @ K_1 ) ) )
& ( ( L != zero_zero_int )
=> ( ( divmod_int @ K_1 @ L )
= ( produc713050258nt_int @ ( times_times_int @ ( sgn_sgn_int @ L ) )
@ ( if_Pro1731782967nt_int @ ( (|) @ ( (&) @ ( ord_less_int @ zero_zero_int @ L ) @ ( ord_less_eq_int @ zero_zero_int @ K_1 ) ) @ ( (&) @ ( ord_less_int @ L @ zero_zero_int ) @ ( ord_less_int @ K_1 @ zero_zero_int ) ) ) @ ( pdivmod @ K_1 @ L )
@ ( produc1518849193nt_int
@ ^ [R: int,S_2: int] : ( if_Pro1731782967nt_int @ ( S_2 = zero_zero_int ) @ ( product_Pair_int_int @ ( uminus_uminus_int @ R ) @ zero_zero_int ) @ ( product_Pair_int_int @ ( minus_minus_int @ ( uminus_uminus_int @ R ) @ one_one_int ) @ ( minus_minus_int @ ( abs_abs_int @ L ) @ S_2 ) ) )
@ ( pdivmod @ K_1 @ L ) ) ) ) ) ) ) ) ) ).
thf(fact_4855_negateSnd__def,axiom,
( negateSnd
= ( produc713050258nt_int @ uminus_uminus_int ) ) ).
thf(fact_4856_divmod__int__code,axiom,
! [L: int,K_1: int] :
( ( ( K_1 = zero_zero_int )
=> ( ( divmod_int @ K_1 @ L )
= ( product_Pair_int_int @ zero_zero_int @ zero_zero_int ) ) )
& ( ( K_1 != zero_zero_int )
=> ( ( ( L = zero_zero_int )
=> ( ( divmod_int @ K_1 @ L )
= ( product_Pair_int_int @ zero_zero_int @ K_1 ) ) )
& ( ( L != zero_zero_int )
=> ( ( divmod_int @ K_1 @ L )
= ( produc713050258nt_int @ ( times_times_int @ ( sgn_sgn_int @ L ) )
@ ( if_Pro1731782967nt_int
@ ( ( sgn_sgn_int @ K_1 )
= ( sgn_sgn_int @ L ) )
@ ( pdivmod @ K_1 @ L )
@ ( produc1518849193nt_int
@ ^ [R: int,S_2: int] : ( if_Pro1731782967nt_int @ ( S_2 = zero_zero_int ) @ ( product_Pair_int_int @ ( uminus_uminus_int @ R ) @ zero_zero_int ) @ ( product_Pair_int_int @ ( minus_minus_int @ ( uminus_uminus_int @ R ) @ one_one_int ) @ ( minus_minus_int @ ( abs_abs_int @ L ) @ S_2 ) ) )
@ ( pdivmod @ K_1 @ L ) ) ) ) ) ) ) ) ) ).
thf(fact_4857_summable__le2,axiom,
! [F: nat > real,G: nat > real] :
( ! [N_1: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N_1 ) ) @ ( G @ N_1 ) )
=> ( ( summable_real @ G )
=> ( ( summable_real @ F )
& ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
thf(fact_4858_suminf__gt__zero,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N_1: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N_1 ) )
=> ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
thf(fact_4859_suminf__ge__zero,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N_1 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
thf(fact_4860_suminf__0__le,axiom,
! [F: nat > real] :
( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N_1 ) )
=> ( ( summable_real @ F )
=> ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
thf(fact_4861_int__ge__less__than__def,axiom,
! [D: int] :
( ( int_ge_less_than @ D )
= ( collec1347809874nt_int
@ ( produc450523309_int_o
@ ^ [Z_2: int,Z: int] : ( (&) @ ( ord_less_eq_int @ D @ Z_2 ) @ ( ord_less_int @ Z_2 @ Z ) ) ) ) ) ).
thf(fact_4862_int__ge__less__than2__def,axiom,
! [D: int] :
( ( int_ge_less_than2 @ D )
= ( collec1347809874nt_int
@ ( produc450523309_int_o
@ ^ [Z_2: int,Z: int] : ( (&) @ ( ord_less_eq_int @ D @ Z ) @ ( ord_less_int @ Z_2 @ Z ) ) ) ) ) ).
thf(fact_4863_sin__paired,axiom,
! [X: real] :
( sums_real
@ ^ [N_1: nat] : ( times_times_real @ ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ N_1 ) @ ( real_nat @ ( fact_fact_nat @ ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) @ one_one_nat ) ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) @ one_one_nat ) ) )
@ ( sin @ X ) ) ).
thf(fact_4864_sin__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N_1: nat] : ( times_times_real @ ( sin_coeff @ N_1 ) @ ( power_power_real @ X @ N_1 ) )
@ ( sin @ X ) ) ).
thf(fact_4865_cos__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N_1: nat] : ( times_times_real @ ( cos_coeff @ N_1 ) @ ( power_power_real @ X @ N_1 ) )
@ ( cos @ X ) ) ).
thf(fact_4866_power__half__series,axiom,
( sums_real
@ ^ [N_1: nat] : ( power_power_real @ ( inverse_divide_real @ one_one_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( suc @ N_1 ) )
@ one_one_real ) ).
thf(fact_4867_Ln_Oaux2,axiom,
! [X: real] :
( sums_real
@ ^ [N_1: nat] : ( times_times_real @ ( inverse_divide_real @ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( power_power_real @ ( inverse_divide_real @ one_one_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ N_1 ) )
@ ( power_power_real @ X @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ).
thf(fact_4868_sums__if_H,axiom,
! [G: nat > real,X: real] :
( ( sums_real @ G @ X )
=> ( sums_real
@ ^ [N_1: nat] : ( if_real @ ( even_odd_even_nat @ N_1 ) @ zero_zero_real @ ( G @ ( div_div_nat @ ( minus_minus_nat @ N_1 @ one_one_nat ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
@ X ) ) ).
thf(fact_4869_sums__if,axiom,
! [F: nat > real,Y: real,G: nat > real,X: real] :
( ( sums_real @ G @ X )
=> ( ( sums_real @ F @ Y )
=> ( sums_real
@ ^ [N_1: nat] : ( if_real @ ( even_odd_even_nat @ N_1 ) @ ( F @ ( div_div_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( G @ ( div_div_nat @ ( minus_minus_nat @ N_1 @ one_one_nat ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) )
@ ( plus_plus_real @ X @ Y ) ) ) ) ).
thf(fact_4870_cos__paired,axiom,
! [X: real] :
( sums_real
@ ^ [N_1: nat] : ( times_times_real @ ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ N_1 ) @ ( real_nat @ ( fact_fact_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) ) )
@ ( cos @ X ) ) ).
thf(fact_4871_cnj_Osums,axiom,
! [X_2: nat > complex,A: complex] :
( ( sums_complex @ X_2 @ A )
=> ( sums_complex
@ ^ [N_1: nat] : ( cnj @ ( X_2 @ N_1 ) )
@ ( cnj @ A ) ) ) ).
thf(fact_4872_Re_Osums,axiom,
! [X_2: nat > complex,A: complex] :
( ( sums_complex @ X_2 @ A )
=> ( sums_real
@ ^ [N_1: nat] : ( re @ ( X_2 @ N_1 ) )
@ ( re @ A ) ) ) ).
thf(fact_4873_Im_Osums,axiom,
! [X_2: nat > complex,A: complex] :
( ( sums_complex @ X_2 @ A )
=> ( sums_real
@ ^ [N_1: nat] : ( im @ ( X_2 @ N_1 ) )
@ ( im @ A ) ) ) ).
thf(fact_4874_sin__fdiffs2,axiom,
! [N: nat] :
( ( diffs_real @ sin_coeff @ N )
= ( cos_coeff @ N ) ) ).
thf(fact_4875_sin__fdiffs,axiom,
( ( diffs_real @ sin_coeff )
= cos_coeff ) ).
thf(fact_4876_exp__fdiffs,axiom,
! [X_1: nat] :
( ( diffs_real
@ ^ [N_1: nat] : ( inverse_inverse_real @ ( real_nat @ ( fact_fact_nat @ N_1 ) ) )
@ X_1 )
= ( inverse_inverse_real @ ( real_nat @ ( fact_fact_nat @ X_1 ) ) ) ) ).
thf(fact_4877_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
<=> ( nat_case_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
thf(fact_4878_cos__fdiffs,axiom,
! [X_1: nat] :
( ( diffs_real @ cos_coeff @ X_1 )
= ( uminus_uminus_real @ ( sin_coeff @ X_1 ) ) ) ).
thf(fact_4879_cos__fdiffs2,axiom,
! [N: nat] :
( ( diffs_real @ cos_coeff @ N )
= ( uminus_uminus_real @ ( sin_coeff @ N ) ) ) ).
thf(fact_4880_diff__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( nat_case_nat @ zero_zero_nat
@ ^ [K: nat] : K
@ ( minus_minus_nat @ M @ N ) ) ) ).
thf(fact_4881_summable__le,axiom,
! [F: nat > real,G: nat > real] :
( ! [N_1: nat] : ( ord_less_eq_real @ ( F @ N_1 ) @ ( G @ N_1 ) )
=> ( ( summable_real @ F )
=> ( ( summable_real @ G )
=> ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
thf(fact_4882_arctan__def,axiom,
! [Y: real] :
( ( arctan @ Y )
= ( the_real
@ ^ [X_1: real] :
( (&) @ ( ord_less_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ X_1 )
@ ( (&) @ ( ord_less_real @ X_1 @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
@ ( ( tan @ X_1 )
= Y ) ) ) ) ) ).
thf(fact_4883_arcsin__def,axiom,
! [Y: real] :
( ( arcsin @ Y )
= ( the_real
@ ^ [X_1: real] :
( (&) @ ( ord_less_eq_real @ ( uminus_uminus_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ X_1 )
@ ( (&) @ ( ord_less_eq_real @ X_1 @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) )
@ ( ( sin @ X_1 )
= Y ) ) ) ) ) ).
thf(fact_4884_ln__def,axiom,
! [X: real] :
( ( ln @ X )
= ( the_real
@ ^ [U_1: real] :
( ( exp_real @ U_1 )
= X ) ) ) ).
thf(fact_4885_root__def,axiom,
! [N: nat,X: real] :
( ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( root @ N @ X )
= ( the_real
@ ^ [U_1: real] :
( (&) @ ( ord_less_real @ zero_zero_real @ U_1 )
@ ( ( power_power_real @ U_1 @ N )
= X ) ) ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ord_less_real @ X @ zero_zero_real )
=> ( ( root @ N @ X )
= ( uminus_uminus_real
@ ( the_real
@ ^ [U_1: real] :
( (&) @ ( ord_less_real @ zero_zero_real @ U_1 )
@ ( ( power_power_real @ U_1 @ N )
= ( uminus_uminus_real @ X ) ) ) ) ) ) )
& ( ~ ( ord_less_real @ X @ zero_zero_real )
=> ( ( root @ N @ X )
= zero_zero_real ) ) ) ) ) ).
thf(fact_4886_arccos__def,axiom,
! [Y: real] :
( ( arccos @ Y )
= ( the_real
@ ^ [X_1: real] :
( (&) @ ( ord_less_eq_real @ zero_zero_real @ X_1 )
@ ( (&) @ ( ord_less_eq_real @ X_1 @ pi )
@ ( ( cos @ X_1 )
= Y ) ) ) ) ) ).
thf(fact_4887_pi__def,axiom,
( pi
= ( times_times_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) )
@ ( the_real
@ ^ [X_1: real] :
( (&) @ ( ord_less_eq_real @ zero_zero_real @ X_1 )
@ ( (&) @ ( ord_less_eq_real @ X_1 @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
@ ( ( cos @ X_1 )
= zero_zero_real ) ) ) ) ) ) ).
thf(fact_4888_pi__half,axiom,
( ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
= ( the_real
@ ^ [X_1: real] :
( (&) @ ( ord_less_eq_real @ zero_zero_real @ X_1 )
@ ( (&) @ ( ord_less_eq_real @ X_1 @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) )
@ ( ( cos @ X_1 )
= zero_zero_real ) ) ) ) ) ).
thf(fact_4889_divmod__nat__def,axiom,
! [M: nat,N: nat] :
( ( divmod_nat @ M @ N )
= ( the_Pr588456374at_nat @ ( divmod_nat_rel @ M @ N ) ) ) ).
thf(fact_4890_floor__real__def,axiom,
! [X: real] :
( ( archim1246769320r_real @ X )
= ( the_int
@ ^ [Z: int] : ( (&) @ ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X ) @ ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z @ one_one_int ) ) ) ) ) ) ).
thf(fact_4891_floor__rat__def,axiom,
! [X: rat] :
( ( archim791455193or_rat @ X )
= ( the_int
@ ^ [Z: int] : ( (&) @ ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X ) @ ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z @ one_one_int ) ) ) ) ) ) ).
thf(fact_4892_RRset2norRR__def,axiom,
! [A: int,A_1: int > $o,M: int] :
( ( ( ( ord_less_int @ one_one_int @ M )
& ( is_RRset @ A_1 @ M )
& ( member_int @ A @ A_1 ) )
=> ( ( rRset2norRR @ A_1 @ M @ A )
= ( hilbert_Eps_int
@ ^ [B_4: int] : ( (&) @ ( zcong @ A @ B_4 @ M ) @ ( member_int @ B_4 @ ( norRRset @ M ) ) ) ) ) )
& ( ~ ( ( ord_less_int @ one_one_int @ M )
& ( is_RRset @ A_1 @ M )
& ( member_int @ A @ A_1 ) )
=> ( ( rRset2norRR @ A_1 @ M @ A )
= zero_zero_int ) ) ) ).
thf(fact_4893_aux__some,axiom,
! [A: int,A_1: int > $o,M: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( is_RRset @ A_1 @ M )
=> ( ( member_int @ A @ A_1 )
=> ( ( zcong @ A
@ ( hilbert_Eps_int
@ ^ [B_4: int] : ( (&) @ ( zcong @ A @ B_4 @ M ) @ ( member_int @ B_4 @ ( norRRset @ M ) ) ) )
@ M )
& ( member_int
@ ( hilbert_Eps_int
@ ^ [B_4: int] : ( (&) @ ( zcong @ A @ B_4 @ M ) @ ( member_int @ B_4 @ ( norRRset @ M ) ) ) )
@ ( norRRset @ M ) ) ) ) ) ) ).
thf(fact_4894_Maclaurin__sin__bound,axiom,
! [X: real,N: nat] :
( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real @ ( sin @ X )
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( if_real @ ( even_odd_even_nat @ M_2 ) @ zero_zero_real @ ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ ( div_div_nat @ ( minus_minus_nat @ M_2 @ ( suc @ zero_zero_nat ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) ) @ ( power_power_real @ X @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) ) ) )
@ ( times_times_real @ ( inverse_inverse_real @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).
thf(fact_4895_finite__less__ub,axiom,
! [U: nat,F: nat > nat] :
( ! [N_1: nat] : ( ord_less_eq_nat @ N_1 @ ( F @ N_1 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N_1: nat] : ( ord_less_eq_nat @ ( F @ N_1 ) @ U ) ) ) ) ).
thf(fact_4896_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( ord_at4362885an_nat @ L @ U ) ) ).
thf(fact_4897_sumr__diff__mult__const,axiom,
! [F: nat > real,N: nat,R_1: real] :
( ( minus_minus_real @ ( big_co604158596t_real @ F @ ( ord_at4362885an_nat @ zero_zero_nat @ N ) ) @ ( times_times_real @ ( real_nat @ N ) @ R_1 ) )
= ( big_co604158596t_real
@ ^ [I_1: nat] : ( minus_minus_real @ ( F @ I_1 ) @ R_1 )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) ) ) ).
thf(fact_4898_sumr__offset4,axiom,
! [K_1: nat,N_1: nat,F_2: nat > real] :
( ( big_co604158596t_real @ F_2 @ ( ord_at4362885an_nat @ zero_zero_nat @ ( plus_plus_nat @ N_1 @ K_1 ) ) )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( F_2 @ ( plus_plus_nat @ M_2 @ K_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N_1 ) )
@ ( big_co604158596t_real @ F_2 @ ( ord_at4362885an_nat @ zero_zero_nat @ K_1 ) ) ) ) ).
thf(fact_4899_sumr__offset2,axiom,
! [K_1: nat,N: nat,F_2: nat > real] :
( ( big_co604158596t_real
@ ^ [M_2: nat] : ( F_2 @ ( plus_plus_nat @ M_2 @ K_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
= ( minus_minus_real @ ( big_co604158596t_real @ F_2 @ ( ord_at4362885an_nat @ zero_zero_nat @ ( plus_plus_nat @ N @ K_1 ) ) ) @ ( big_co604158596t_real @ F_2 @ ( ord_at4362885an_nat @ zero_zero_nat @ K_1 ) ) ) ) ).
thf(fact_4900_sumr__one__lb__realpow__zero,axiom,
! [F: nat > real,N: nat] :
( ( big_co604158596t_real
@ ^ [N_1: nat] : ( times_times_real @ ( F @ N_1 ) @ ( power_power_real @ zero_zero_real @ N_1 ) )
@ ( ord_at4362885an_nat @ ( suc @ zero_zero_nat ) @ N ) )
= zero_zero_real ) ).
thf(fact_4901_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_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_4903_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( ord_at4362885an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
thf(fact_4904_sumr__geometric,axiom,
! [N: nat,X: real] :
( ( X != one_one_real )
=> ( ( big_co604158596t_real @ ( power_power_real @ X ) @ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
= ( inverse_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ N ) @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ) ).
thf(fact_4905_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( ord_at4362885an_nat @ ( suc @ L ) @ U )
= ( ord_gr660468384an_nat @ L @ U ) ) ).
thf(fact_4906_image__add__atLeastLessThan,axiom,
! [K_1: nat,I: nat,J: nat] :
( ( image_nat_nat
@ ^ [N_1: nat] : ( plus_plus_nat @ N_1 @ K_1 )
@ ( ord_at4362885an_nat @ I @ J ) )
= ( ord_at4362885an_nat @ ( plus_plus_nat @ I @ K_1 ) @ ( plus_plus_nat @ J @ K_1 ) ) ) ).
thf(fact_4907_image__Suc__atLeastLessThan,axiom,
! [I: nat,J: nat] :
( ( image_nat_nat @ suc @ ( ord_at4362885an_nat @ I @ J ) )
= ( ord_at4362885an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
thf(fact_4908_Maclaurin__zero,axiom,
! [Diff: nat > real > real,N: nat,X: real] :
( ( X = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ M_2 @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ ( power_power_real @ X @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
= ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).
thf(fact_4909_subset__card__intvl__is__intvl,axiom,
! [A_1: nat > $o,K_1: nat] :
( ( ord_less_eq_nat_o @ A_1 @ ( ord_at4362885an_nat @ K_1 @ ( plus_plus_nat @ K_1 @ ( finite_card_nat @ A_1 ) ) ) )
=> ( A_1
= ( ord_at4362885an_nat @ K_1 @ ( plus_plus_nat @ K_1 @ ( finite_card_nat @ A_1 ) ) ) ) ) ).
thf(fact_4910_sum__split__even__odd,axiom,
! [F: nat > real,G: nat > real,N: nat] :
( ( big_co604158596t_real
@ ^ [I_1: nat] : ( if_real @ ( even_odd_even_nat @ I_1 ) @ ( F @ I_1 ) @ ( G @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( F @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ I_1 ) @ one_one_nat ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) ) ) ) ).
thf(fact_4911_sumr__minus__one__realpow__zero,axiom,
! [N: nat] :
( ( big_co604158596t_real
@ ^ [I_1: nat] : ( power_power_real @ ( number267125858f_real @ min ) @ ( suc @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) )
= zero_zero_real ) ).
thf(fact_4912_lemma__STAR__cos2,axiom,
! [N: nat] :
( ( big_co604158596t_real
@ ^ [N_1: nat] : ( if_real @ ( even_odd_even_nat @ N_1 ) @ ( times_times_real @ ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ ( div_div_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( real_nat @ ( fact_fact_nat @ N_1 ) ) ) @ ( power_power_real @ zero_zero_real @ N_1 ) ) @ zero_zero_real )
@ ( ord_at4362885an_nat @ one_one_nat @ N ) )
= zero_zero_real ) ).
thf(fact_4913_sumr__cos__zero__one,axiom,
! [N: nat] :
( ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( if_real @ ( even_odd_even_nat @ M_2 ) @ ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ ( div_div_nat @ M_2 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ zero_zero_real ) @ ( power_power_real @ zero_zero_real @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( suc @ N ) ) )
= one_one_real ) ).
thf(fact_4914_Maclaurin__sin__expansion2,axiom,
! [N: nat,X: real] :
? [T_1: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T_1 ) @ ( abs_abs_real @ X ) )
& ( ( sin @ X )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( if_real @ ( even_odd_even_nat @ M_2 ) @ zero_zero_real @ ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ ( div_div_nat @ ( minus_minus_nat @ M_2 @ ( suc @ zero_zero_nat ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) ) @ ( power_power_real @ X @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( sin @ ( plus_plus_real @ T_1 @ ( times_times_real @ ( times_times_real @ ( inverse_divide_real @ one_one_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( real_nat @ N ) ) @ pi ) ) ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
thf(fact_4915_Maclaurin__cos__expansion,axiom,
! [N: nat,X: real] :
? [T_1: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T_1 ) @ ( abs_abs_real @ X ) )
& ( ( cos @ X )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( if_real @ ( even_odd_even_nat @ M_2 ) @ ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ ( div_div_nat @ M_2 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ zero_zero_real ) @ ( power_power_real @ X @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( cos @ ( plus_plus_real @ T_1 @ ( times_times_real @ ( times_times_real @ ( inverse_divide_real @ one_one_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( real_nat @ N ) ) @ pi ) ) ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
thf(fact_4916_finite__atLeastLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( ord_at641636577an_int @ L @ U ) ) ).
thf(fact_4917_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite_int @ ( ord_at641636577an_int @ zero_zero_int @ U ) ) ).
thf(fact_4918_Sup__atLeastLessThan,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ Y @ X )
=> ( ( comple124823625p_real @ ( ord_at1496968948n_real @ Y @ X ) )
= X ) ) ).
thf(fact_4919_card__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( finite_card_int @ ( ord_at641636577an_int @ zero_zero_int @ U ) )
= ( nat_1 @ U ) ) ).
thf(fact_4920_card__atLeastLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( ord_at641636577an_int @ L @ U ) )
= ( nat_1 @ ( minus_minus_int @ U @ L ) ) ) ).
thf(fact_4921_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( ord_at641636577an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
= ( ord_gr1297742076an_int @ L @ U ) ) ).
thf(fact_4922_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image_int_int
@ ^ [X_1: int] : ( plus_plus_int @ X_1 @ L )
@ ( ord_at641636577an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
= ( ord_at641636577an_int @ L @ U ) ) ).
thf(fact_4923_Maclaurin__sin__expansion,axiom,
! [X: real,N: nat] :
? [T_1: real] :
( ( sin @ X )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( if_real @ ( even_odd_even_nat @ M_2 ) @ zero_zero_real @ ( inverse_divide_real @ ( power_power_real @ ( number267125858f_real @ min ) @ ( div_div_nat @ ( minus_minus_nat @ M_2 @ ( suc @ zero_zero_nat ) ) @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) ) @ ( power_power_real @ X @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( sin @ ( plus_plus_real @ T_1 @ ( times_times_real @ ( times_times_real @ ( inverse_divide_real @ one_one_real @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ ( real_nat @ N ) ) @ pi ) ) ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ X @ N ) ) ) ) ).
thf(fact_4924_Maclaurin__lemma2,axiom,
! [B_1: real,K_1: nat,Diff: nat > real > real,H: real,N: nat] :
( ! [M_2: nat,T_1: real] :
( ( ( ord_less_nat @ M_2 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T_1 )
& ( ord_less_eq_real @ T_1 @ H ) )
=> ( deriv_real @ ( Diff @ M_2 ) @ T_1 @ ( Diff @ ( suc @ M_2 ) @ T_1 ) ) )
=> ( ( N
= ( suc @ K_1 ) )
=> ! [M_2: nat,T_1: real] :
( ( ( ord_less_nat @ M_2 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T_1 )
& ( ord_less_eq_real @ T_1 @ H ) )
=> ( deriv_real
@ ^ [U_1: real] :
( minus_minus_real @ ( Diff @ M_2 @ U_1 )
@ ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [P_4: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ ( plus_plus_nat @ M_2 @ P_4 ) @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ P_4 ) ) ) @ ( power_power_real @ U_1 @ P_4 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M_2 ) ) )
@ ( times_times_real @ B_1 @ ( inverse_divide_real @ ( power_power_real @ U_1 @ ( minus_minus_nat @ N @ M_2 ) ) @ ( real_nat @ ( fact_fact_nat @ ( minus_minus_nat @ N @ M_2 ) ) ) ) ) ) )
@ T_1
@ ( minus_minus_real @ ( Diff @ ( suc @ M_2 ) @ T_1 )
@ ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [P_4: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M_2 ) @ P_4 ) @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ P_4 ) ) ) @ ( power_power_real @ T_1 @ P_4 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ ( suc @ M_2 ) ) ) )
@ ( times_times_real @ B_1 @ ( inverse_divide_real @ ( power_power_real @ T_1 @ ( minus_minus_nat @ N @ ( suc @ M_2 ) ) ) @ ( real_nat @ ( fact_fact_nat @ ( minus_minus_nat @ N @ ( suc @ M_2 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(fact_4925_Maclaurin__exp__lt,axiom,
! [N: nat,X: real] :
( ( X != zero_zero_real )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [T_1: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T_1 ) )
& ( ord_less_real @ ( abs_abs_real @ T_1 ) @ ( abs_abs_real @ X ) )
& ( ( exp_real @ X )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( inverse_divide_real @ ( power_power_real @ X @ M_2 ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( exp_real @ T_1 ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
thf(fact_4926_Maclaurin__exp__le,axiom,
! [N: nat,X: real] :
? [T_1: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T_1 ) @ ( abs_abs_real @ X ) )
& ( ( exp_real @ X )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( inverse_divide_real @ ( power_power_real @ X @ M_2 ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( exp_real @ T_1 ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
thf(fact_4927_sumr__pos__lt__pair,axiom,
! [K_1: nat,F: nat > real] :
( ( summable_real @ F )
=> ( ! [D_2: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K_1 @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D_2 ) ) ) @ ( F @ ( plus_plus_nat @ K_1 @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D_2 ) @ one_one_nat ) ) ) ) )
=> ( ord_less_real @ ( big_co604158596t_real @ F @ ( ord_at4362885an_nat @ zero_zero_nat @ K_1 ) ) @ ( suminf_real @ F ) ) ) ) ).
thf(fact_4928_Maclaurin__lemma,axiom,
! [F: real > real,J: nat > real,N: nat,H: real] :
( ( ord_less_real @ zero_zero_real @ H )
=> ? [B_6: real] :
( ( F @ H )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( inverse_divide_real @ ( J @ M_2 ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ ( power_power_real @ H @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ B_6 @ ( inverse_divide_real @ ( power_power_real @ H @ N ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) ) ) ) ) ).
thf(fact_4929_series__pos__less,axiom,
! [N: nat,F: nat > real] :
( ( summable_real @ F )
=> ( ! [M_2: nat] :
( ( ord_less_eq_nat @ N @ M_2 )
=> ( ord_less_real @ zero_zero_real @ ( F @ M_2 ) ) )
=> ( ord_less_real @ ( big_co604158596t_real @ F @ ( ord_at4362885an_nat @ zero_zero_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).
thf(fact_4930_series__pos__le,axiom,
! [N: nat,F: nat > real] :
( ( summable_real @ F )
=> ( ! [M_2: nat] :
( ( ord_less_eq_nat @ N @ M_2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ M_2 ) ) )
=> ( ord_less_eq_real @ ( big_co604158596t_real @ F @ ( ord_at4362885an_nat @ zero_zero_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).
thf(fact_4931_Maclaurin__all__lt__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_4932_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_4933_Maclaurin__bi__le,axiom,
! [X: real,N: nat,Diff: nat > real > real,F: real > real] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M_2: nat,T_1: real] :
( ( ( ord_less_nat @ M_2 @ N )
& ( ord_less_eq_real @ ( abs_abs_real @ T_1 ) @ ( abs_abs_real @ X ) ) )
=> ( deriv_real @ ( Diff @ M_2 ) @ T_1 @ ( Diff @ ( suc @ M_2 ) @ T_1 ) ) )
=> ? [T_1: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T_1 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ M_2 @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ ( power_power_real @ X @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( Diff @ N @ T_1 ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
thf(fact_4934_Maclaurin__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_4935_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_4936_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_4937_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_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__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_4940_Maclaurin__all__le__objl,axiom,
! [N: nat,X: real,Diff: nat > real > real,F: real > real] :
( ( ( ( Diff @ zero_zero_nat )
= F )
& ! [M_2: nat,X_1: real] : ( deriv_real @ ( Diff @ M_2 ) @ X_1 @ ( Diff @ ( suc @ M_2 ) @ X_1 ) ) )
=> ? [T_1: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T_1 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ M_2 @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ ( power_power_real @ X @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( Diff @ N @ T_1 ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
thf(fact_4941_Maclaurin__all__le,axiom,
! [N: nat,X: real,Diff: nat > real > real,F: real > real] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M_2: nat,X_1: real] : ( deriv_real @ ( Diff @ M_2 ) @ X_1 @ ( Diff @ ( suc @ M_2 ) @ X_1 ) )
=> ? [T_1: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T_1 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( big_co604158596t_real
@ ^ [M_2: nat] : ( times_times_real @ ( inverse_divide_real @ ( Diff @ M_2 @ zero_zero_real ) @ ( real_nat @ ( fact_fact_nat @ M_2 ) ) ) @ ( power_power_real @ X @ M_2 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ N ) )
@ ( times_times_real @ ( inverse_divide_real @ ( Diff @ N @ T_1 ) @ ( real_nat @ ( fact_fact_nat @ N ) ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
thf(fact_4942_real__setsum__nat__ivl__bounded,axiom,
! [F: nat > real,K_2: real,N: nat] :
( ! [P_4: nat] :
( ( ord_less_nat @ P_4 @ N )
=> ( ord_less_eq_real @ ( F @ P_4 ) @ K_2 ) )
=> ( ord_less_eq_real @ ( big_co604158596t_real @ F @ ( ord_at4362885an_nat @ zero_zero_nat @ N ) ) @ ( times_times_real @ ( real_nat @ N ) @ K_2 ) ) ) ).
thf(fact_4943_DERIV__sumr,axiom,
! [F: nat > real > real,X: real,F_1: nat > real > real,N: nat,M: nat] :
( ! [R: nat] :
( ( ( ord_less_eq_nat @ M @ R )
& ( ord_less_nat @ R @ ( plus_plus_nat @ M @ N ) ) )
=> ( deriv_real @ ( F @ R ) @ X @ ( F_1 @ R @ X ) ) )
=> ( deriv_real
@ ^ [X_1: real] :
( big_co604158596t_real
@ ^ [N_1: nat] : ( F @ N_1 @ X_1 )
@ ( ord_at4362885an_nat @ M @ N ) )
@ X
@ ( big_co604158596t_real
@ ^ [R: nat] : ( F_1 @ R @ X )
@ ( ord_at4362885an_nat @ M @ N ) ) ) ) ).
thf(fact_4944_suminf__le,axiom,
! [X: real,F: nat > real] :
( ( summable_real @ F )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( big_co604158596t_real @ F @ ( ord_at4362885an_nat @ zero_zero_nat @ N_1 ) ) @ X )
=> ( ord_less_eq_real @ ( suminf_real @ F ) @ X ) ) ) ).
thf(fact_4945_transfer__nat__int__sum__prod__closure_I1_J,axiom,
! [F: int > int,A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ! [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X_1 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( big_co230513141nt_int @ F @ A_1 ) ) ) ) ).
thf(fact_4946_pos__summable,axiom,
! [X: real,F: nat > real] :
( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N_1 ) )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( big_co604158596t_real @ F @ ( ord_at4362885an_nat @ zero_zero_nat @ N_1 ) ) @ X )
=> ( summable_real @ F ) ) ) ).
thf(fact_4947_arith__series__int,axiom,
! [A: int,D: int,N: nat] :
( ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) )
@ ( big_co1024481617at_int
@ ^ [I_1: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1621563631at_int @ I_1 ) @ D ) )
@ ( ord_lessThan_nat @ N ) ) )
= ( times_times_int @ ( semiri1621563631at_int @ N ) @ ( plus_plus_int @ A @ ( plus_plus_int @ A @ ( times_times_int @ ( semiri1621563631at_int @ ( minus_minus_nat @ N @ one_one_nat ) ) @ D ) ) ) ) ) ).
thf(fact_4948_BseqI2,axiom,
! [K_2: real,K_1: real,F: nat > real] :
( ! [N_1: nat] :
( ( ord_less_eq_real @ K_1 @ ( F @ N_1 ) )
& ( ord_less_eq_real @ ( F @ N_1 ) @ K_2 ) )
=> ( bseq_real @ F ) ) ).
thf(fact_4949_finite__lessThan,axiom,
! [K_1: nat] : ( finite_finite_nat @ ( ord_lessThan_nat @ K_1 ) ) ).
thf(fact_4950_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( ord_lessThan_nat @ U ) )
= U ) ).
thf(fact_4951_atLeast0LessThan,axiom,
! [N: nat] :
( ( ord_at4362885an_nat @ zero_zero_nat @ N )
= ( ord_lessThan_nat @ N ) ) ).
thf(fact_4952_finite__nat__iff__bounded,axiom,
! [S: nat > $o] :
( ( finite_finite_nat @ S )
<=> ? [K: nat] : ( ord_less_eq_nat_o @ S @ ( ord_lessThan_nat @ K ) ) ) ).
thf(fact_4953_arith__series__nat,axiom,
! [A: nat,D: nat,N: nat] :
( ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) )
@ ( big_co387207925at_nat
@ ^ [I_1: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I_1 @ D ) )
@ ( ord_lessThan_nat @ N ) ) )
= ( times_times_nat @ N @ ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ ( times_times_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ D ) ) ) ) ) ).
thf(fact_4954_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_at641636577an_int @ zero_zero_int @ U )
= ( image_nat_int @ semiri1621563631at_int @ ( ord_lessThan_nat @ ( nat_1 @ U ) ) ) ) ) ).
thf(fact_4955_finite__nat__bounded,axiom,
! [S: nat > $o] :
( ( finite_finite_nat @ S )
=> ? [K: nat] : ( ord_less_eq_nat_o @ S @ ( ord_lessThan_nat @ K ) ) ) ).
thf(fact_4956_sum__diff__distrib,axiom,
! [N: nat,Q_1: nat > nat,P: nat > nat] :
( ! [X_1: nat] : ( ord_less_eq_nat @ ( Q_1 @ X_1 ) @ ( P @ X_1 ) )
=> ( ( minus_minus_nat @ ( big_co387207925at_nat @ P @ ( ord_lessThan_nat @ N ) ) @ ( big_co387207925at_nat @ Q_1 @ ( ord_lessThan_nat @ N ) ) )
= ( big_co387207925at_nat
@ ^ [X_1: nat] : ( minus_minus_nat @ ( P @ X_1 ) @ ( Q_1 @ X_1 ) )
@ ( ord_lessThan_nat @ N ) ) ) ) ).
thf(fact_4957_Sup__lessThan,axiom,
! [X: real] :
( ( comple124823625p_real @ ( ord_lessThan_real @ X ) )
= X ) ).
thf(fact_4958_transfer__nat__int__set__cong,axiom,
! [P_1: int > $o,P: int > $o] :
( ! [X_1: int] :
( ( ord_less_eq_int @ zero_zero_int @ X_1 )
=> ( ( P @ X_1 )
<=> ( P_1 @ X_1 ) ) )
=> ( ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( P @ X_1 ) ) )
= ( collect_int
@ ^ [X_1: int] : ( (&) @ ( ord_less_eq_int @ zero_zero_int @ X_1 ) @ ( P_1 @ X_1 ) ) ) ) ) ).
thf(fact_4959_normalize__def,axiom,
! [P_3: product_prod_int_int] :
( ( ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P_3 ) )
=> ( ( normalize @ P_3 )
= ( product_Pair_int_int @ ( div_div_int @ ( product_fst_int_int @ P_3 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P_3 ) @ ( product_snd_int_int @ P_3 ) ) ) @ ( div_div_int @ ( product_snd_int_int @ P_3 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P_3 ) @ ( product_snd_int_int @ P_3 ) ) ) ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P_3 ) )
=> ( ( ( ( product_snd_int_int @ P_3 )
= zero_zero_int )
=> ( ( normalize @ P_3 )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) )
& ( ( ( product_snd_int_int @ P_3 )
!= zero_zero_int )
=> ( ( normalize @ P_3 )
= ( product_Pair_int_int @ ( div_div_int @ ( product_fst_int_int @ P_3 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P_3 ) @ ( product_snd_int_int @ P_3 ) ) ) ) @ ( div_div_int @ ( product_snd_int_int @ P_3 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P_3 ) @ ( product_snd_int_int @ P_3 ) ) ) ) ) ) ) ) ) ) ).
thf(fact_4960_gcd__dvd1__int,axiom,
! [X: int,Y: int] : ( dvd_dvd_int @ ( gcd_gcd_int @ X @ Y ) @ X ) ).
thf(fact_4961_gcd__dvd2__int,axiom,
! [X: int,Y: int] : ( dvd_dvd_int @ ( gcd_gcd_int @ X @ Y ) @ Y ) ).
thf(fact_4962_coprime__exp2__int,axiom,
! [N: nat,M: nat,A: int,B: int] :
( ( ( gcd_gcd_int @ A @ B )
= one_one_int )
=> ( ( gcd_gcd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ M ) )
= one_one_int ) ) ).
thf(fact_4963_invertible__coprime__int,axiom,
! [X: int,Y: int,M: int] :
( ( ( div_mod_int @ ( times_times_int @ X @ Y ) @ M )
= one_one_int )
=> ( ( gcd_gcd_int @ X @ M )
= one_one_int ) ) ).
thf(fact_4964_coprime__dvd__mult__int,axiom,
! [M: int,K_1: int,N: int] :
( ( ( gcd_gcd_int @ K_1 @ N )
= one_one_int )
=> ( ( dvd_dvd_int @ K_1 @ ( times_times_int @ M @ N ) )
=> ( dvd_dvd_int @ K_1 @ M ) ) ) ).
thf(fact_4965_coprime__dvd__mult__iff__int,axiom,
! [M: int,K_1: int,N: int] :
( ( ( gcd_gcd_int @ K_1 @ N )
= one_one_int )
=> ( ( dvd_dvd_int @ K_1 @ ( times_times_int @ M @ N ) )
<=> ( dvd_dvd_int @ K_1 @ M ) ) ) ).
thf(fact_4966_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_4967_gcd__coprime__int,axiom,
! [B_5: int,A_5: int,A: int,B: int] :
( ( ( gcd_gcd_int @ A @ B )
!= zero_zero_int )
=> ( ( A
= ( times_times_int @ A_5 @ ( gcd_gcd_int @ A @ B ) ) )
=> ( ( B
= ( times_times_int @ B_5 @ ( gcd_gcd_int @ A @ B ) ) )
=> ( ( gcd_gcd_int @ A_5 @ B_5 )
= one_one_int ) ) ) ) ).
thf(fact_4968_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_4969_gcd__add__mult__int,axiom,
! [M: int,K_1: int,N: int] :
( ( gcd_gcd_int @ M @ ( plus_plus_int @ ( times_times_int @ K_1 @ M ) @ N ) )
= ( gcd_gcd_int @ M @ N ) ) ).
thf(fact_4970_gcd__add2__int,axiom,
! [M: int,N: int] :
( ( gcd_gcd_int @ M @ ( plus_plus_int @ M @ N ) )
= ( gcd_gcd_int @ M @ 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_coprime__minus__one__int,axiom,
! [N: int] :
( ( gcd_gcd_int @ ( minus_minus_int @ N @ one_one_int ) @ N )
= one_one_int ) ).
thf(fact_4973_coprime__plus__one__int,axiom,
! [N: int] :
( ( gcd_gcd_int @ ( plus_plus_int @ N @ one_one_int ) @ N )
= one_one_int ) ).
thf(fact_4974_coprime__exp__int,axiom,
! [N: nat,D: int,A: int] :
( ( ( gcd_gcd_int @ D @ A )
= one_one_int )
=> ( ( gcd_gcd_int @ D @ ( power_power_int @ A @ N ) )
= one_one_int ) ) ).
thf(fact_4975_coprime__divisors__nat,axiom,
! [E_1: int,B: int,D: int,A: int] :
( ( dvd_dvd_int @ D @ A )
=> ( ( dvd_dvd_int @ E_1 @ B )
=> ( ( ( gcd_gcd_int @ A @ B )
= one_one_int )
=> ( ( gcd_gcd_int @ D @ E_1 )
= one_one_int ) ) ) ) ).
thf(fact_4976_gcd__1__int,axiom,
! [M: int] :
( ( gcd_gcd_int @ M @ one_one_int )
= one_one_int ) ).
thf(fact_4977_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_4978_gcd__red__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ X @ Y )
= ( gcd_gcd_int @ Y @ ( div_mod_int @ X @ Y ) ) ) ).
thf(fact_4979_gcd__greatest__iff__int,axiom,
! [K_1: int,M: int,N: int] :
( ( dvd_dvd_int @ K_1 @ ( gcd_gcd_int @ M @ N ) )
<=> ( ( dvd_dvd_int @ K_1 @ M )
& ( dvd_dvd_int @ K_1 @ N ) ) ) ).
thf(fact_4980_gcd__greatest__int,axiom,
! [N: int,K_1: int,M: int] :
( ( dvd_dvd_int @ K_1 @ M )
=> ( ( dvd_dvd_int @ K_1 @ N )
=> ( dvd_dvd_int @ K_1 @ ( gcd_gcd_int @ M @ N ) ) ) ) ).
thf(fact_4981_dvd__gcd__D1__int,axiom,
! [I: int,M: int,N: int] :
( ( dvd_dvd_int @ I @ ( gcd_gcd_int @ M @ N ) )
=> ( dvd_dvd_int @ I @ M ) ) ).
thf(fact_4982_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_4983_gcd__assoc__int,axiom,
! [A: int,B: int,C: int] :
( ( gcd_gcd_int @ ( gcd_gcd_int @ A @ B ) @ C )
= ( gcd_gcd_int @ A @ ( gcd_gcd_int @ B @ C ) ) ) ).
thf(fact_4984_gcd__int_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( gcd_gcd_int @ B @ ( gcd_gcd_int @ A @ C ) )
= ( gcd_gcd_int @ A @ ( gcd_gcd_int @ B @ C ) ) ) ).
thf(fact_4985_gcd__commute__int,axiom,
! [A: int,B: int] :
( ( gcd_gcd_int @ A @ B )
= ( gcd_gcd_int @ B @ A ) ) ).
thf(fact_4986_gcd__exp__int,axiom,
! [A: int,N: nat,B: int] :
( ( gcd_gcd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( power_power_int @ ( gcd_gcd_int @ A @ B ) @ N ) ) ).
thf(fact_4987_gcd__mult__distrib__int,axiom,
! [K_1: int,M: int,N: int] :
( ( times_times_int @ ( abs_abs_int @ K_1 ) @ ( gcd_gcd_int @ M @ N ) )
= ( gcd_gcd_int @ ( times_times_int @ K_1 @ M ) @ ( times_times_int @ K_1 @ N ) ) ) ).
thf(fact_4988_abs__gcd__int,axiom,
! [X: int,Y: int] :
( ( abs_abs_int @ ( gcd_gcd_int @ X @ Y ) )
= ( gcd_gcd_int @ X @ Y ) ) ).
thf(fact_4989_gcd__idem__int,axiom,
! [X: int] :
( ( gcd_gcd_int @ X @ X )
= ( abs_abs_int @ X ) ) ).
thf(fact_4990_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_4991_gcd__abs2__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ X @ ( abs_abs_int @ Y ) )
= ( gcd_gcd_int @ X @ Y ) ) ).
thf(fact_4992_gcd__abs1__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ ( abs_abs_int @ X ) @ Y )
= ( gcd_gcd_int @ X @ Y ) ) ).
thf(fact_4993_gcd__neg2__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) )
= ( gcd_gcd_int @ X @ Y ) ) ).
thf(fact_4994_gcd__neg1__int,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y )
= ( gcd_gcd_int @ X @ Y ) ) ).
thf(fact_4995_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_4996_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_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_gcd__0__left__int,axiom,
! [X: int] :
( ( gcd_gcd_int @ zero_zero_int @ X )
= ( abs_abs_int @ X ) ) ).
thf(fact_4999_gcd__0__int,axiom,
! [X: int] :
( ( gcd_gcd_int @ X @ zero_zero_int )
= ( abs_abs_int @ X ) ) ).
thf(fact_5000_coprime__rmult__int,axiom,
! [D: int,A: int,B: int] :
( ( ( gcd_gcd_int @ D @ ( times_times_int @ A @ B ) )
= one_one_int )
=> ( ( gcd_gcd_int @ D @ B )
= one_one_int ) ) ).
thf(fact_5001_coprime__lmult__int,axiom,
! [D: int,A: int,B: int] :
( ( ( gcd_gcd_int @ D @ ( times_times_int @ A @ B ) )
= one_one_int )
=> ( ( gcd_gcd_int @ D @ A )
= one_one_int ) ) ).
thf(fact_5002_coprime__mult__int,axiom,
! [B: int,D: int,A: int] :
( ( ( gcd_gcd_int @ D @ A )
= one_one_int )
=> ( ( ( gcd_gcd_int @ D @ B )
= one_one_int )
=> ( ( gcd_gcd_int @ D @ ( times_times_int @ A @ B ) )
= one_one_int ) ) ) ).
thf(fact_5003_gcd__mult__cancel__int,axiom,
! [M: int,K_1: int,N: int] :
( ( ( gcd_gcd_int @ K_1 @ N )
= one_one_int )
=> ( ( gcd_gcd_int @ ( times_times_int @ K_1 @ M ) @ N )
= ( gcd_gcd_int @ M @ N ) ) ) ).
thf(fact_5004_coprime__mul__eq__int,axiom,
! [D: int,A: int,B: int] :
( ( ( gcd_gcd_int @ D @ ( times_times_int @ A @ B ) )
= one_one_int )
<=> ( ( ( gcd_gcd_int @ D @ A )
= one_one_int )
& ( ( gcd_gcd_int @ D @ B )
= one_one_int ) ) ) ).
thf(fact_5005_Fract__coprime,axiom,
! [A: int,B: int] :
( ( fract @ ( div_div_int @ A @ ( gcd_gcd_int @ A @ B ) ) @ ( div_div_int @ B @ ( gcd_gcd_int @ A @ B ) ) )
= ( fract @ A @ B ) ) ).
thf(fact_5006_div__gcd__coprime__int,axiom,
! [B: int,A: int] :
( ( ( A != zero_zero_int )
| ( B != zero_zero_int ) )
=> ( ( gcd_gcd_int @ ( div_div_int @ A @ ( gcd_gcd_int @ A @ B ) ) @ ( div_div_int @ B @ ( gcd_gcd_int @ A @ B ) ) )
= one_one_int ) ) ).
thf(fact_5007_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_5008_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_5009_gcd__ge__0__int,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y ) ) ).
thf(fact_5010_gcd__le1__int,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
thf(fact_5011_gcd__le2__int,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
thf(fact_5012_gcd__unique__int,axiom,
! [B: int,A: int,D: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ D )
& ( dvd_dvd_int @ D @ A )
& ( dvd_dvd_int @ D @ B )
& ! [E: int] :
( ( ( dvd_dvd_int @ E @ A )
& ( dvd_dvd_int @ E @ B ) )
=> ( dvd_dvd_int @ E @ D ) ) )
<=> ( D
= ( gcd_gcd_int @ A @ B ) ) ) ).
thf(fact_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_coprime__crossproduct__int,axiom,
! [B: int,C: int,A: int,D: int] :
( ( ( gcd_gcd_int @ A @ D )
= one_one_int )
=> ( ( ( gcd_gcd_int @ B @ C )
= one_one_int )
=> ( ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ C ) )
= ( times_times_int @ ( abs_abs_int @ B ) @ ( abs_abs_int @ D ) ) )
<=> ( ( ( abs_abs_int @ A )
= ( abs_abs_int @ B ) )
& ( ( abs_abs_int @ C )
= ( abs_abs_int @ D ) ) ) ) ) ) ).
thf(fact_5015_gcd__code__int,axiom,
! [K_1: int,L: int] :
( ( gcd_gcd_int @ K_1 @ L )
= ( abs_abs_int @ ( if_int @ ( L = zero_zero_int ) @ K_1 @ ( gcd_gcd_int @ L @ ( div_mod_int @ ( abs_abs_int @ K_1 ) @ ( abs_abs_int @ L ) ) ) ) ) ) ).
thf(fact_5016_coprime__common__divisor__int,axiom,
! [X: int,A: int,B: int] :
( ( ( gcd_gcd_int @ A @ B )
= one_one_int )
=> ( ( dvd_dvd_int @ X @ A )
=> ( ( dvd_dvd_int @ X @ B )
=> ( ( abs_abs_int @ X )
= one_one_int ) ) ) ) ).
thf(fact_5017_quotient__of__coprime,axiom,
! [R_1: rat,P_3: int,Q: int] :
( ( ( quotient_of @ R_1 )
= ( product_Pair_int_int @ P_3 @ Q ) )
=> ( ( gcd_gcd_int @ P_3 @ Q )
= one_one_int ) ) ).
thf(fact_5018_normalize__coprime,axiom,
! [R_1: product_prod_int_int,P_3: int,Q: int] :
( ( ( normalize @ R_1 )
= ( product_Pair_int_int @ P_3 @ Q ) )
=> ( ( gcd_gcd_int @ P_3 @ Q )
= one_one_int ) ) ).
thf(fact_5019_coprime__int,axiom,
! [A: int,B: int] :
( ( ( gcd_gcd_int @ A @ B )
= one_one_int )
<=> ! [D_2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ D_2 )
& ( dvd_dvd_int @ D_2 @ A )
& ( dvd_dvd_int @ D_2 @ B ) )
<=> ( D_2 = one_one_int ) ) ) ).
thf(fact_5020_transfer__nat__int__gcd_I1_J,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( gcd_gcd_nat @ ( nat_1 @ X ) @ ( nat_1 @ Y ) )
= ( nat_1 @ ( gcd_gcd_int @ X @ Y ) ) ) ) ) ).
thf(fact_5021_gcd__int__def,axiom,
! [X: int,Y: int] :
( ( gcd_gcd_int @ X @ Y )
= ( semiri1621563631at_int @ ( gcd_gcd_nat @ ( nat_1 @ ( abs_abs_int @ X ) ) @ ( nat_1 @ ( abs_abs_int @ Y ) ) ) ) ) ).
thf(fact_5022_normalize__stable,axiom,
! [P_3: int,Q: int] :
( ( ord_less_int @ zero_zero_int @ Q )
=> ( ( ( gcd_gcd_int @ P_3 @ Q )
= one_one_int )
=> ( ( normalize @ ( product_Pair_int_int @ P_3 @ Q ) )
= ( product_Pair_int_int @ P_3 @ Q ) ) ) ) ).
thf(fact_5023_quotient__of__def,axiom,
! [X: rat] :
( ( quotient_of @ X )
= ( the_Pr2103884470nt_int
@ ^ [Pair: product_prod_int_int] :
( (&)
@ ( X
= ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
@ ( (&) @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
@ ( ( gcd_gcd_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) )
= one_one_int ) ) ) ) ) ).
thf(fact_5024_quotient__of__unique,axiom,
! [R_1: rat] :
? [X_1: product_prod_int_int] :
( ( R_1
= ( fract @ ( product_fst_int_int @ X_1 ) @ ( product_snd_int_int @ X_1 ) ) )
& ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X_1 ) )
& ( ( gcd_gcd_int @ ( product_fst_int_int @ X_1 ) @ ( product_snd_int_int @ X_1 ) )
= one_one_int )
& ! [Y_1: product_prod_int_int] :
( ( ( R_1
= ( fract @ ( product_fst_int_int @ Y_1 ) @ ( product_snd_int_int @ Y_1 ) ) )
& ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y_1 ) )
& ( ( gcd_gcd_int @ ( product_fst_int_int @ Y_1 ) @ ( product_snd_int_int @ Y_1 ) )
= one_one_int ) )
=> ( Y_1 = X_1 ) ) ) ).
thf(fact_5025_Rat__cases__nonzero,axiom,
! [Q: rat] :
( ! [A_2: int,B_4: int] :
( ( Q
= ( fract @ A_2 @ B_4 ) )
=> ( ( ord_less_int @ zero_zero_int @ B_4 )
=> ( ( A_2 != zero_zero_int )
=> ( ( gcd_gcd_int @ A_2 @ B_4 )
!= one_one_int ) ) ) )
=> ( Q = zero_zero_rat ) ) ).
thf(fact_5026_Rat__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_5027_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_5028_card__setsum__aux,axiom,
! [N: nat,S: ( int > $o ) > $o] :
( ( finite_finite_int_o @ S )
=> ( ! [X_1: int > $o] :
( ( member_int_o @ X_1 @ S )
=> ( finite_finite_int @ X_1 ) )
=> ( ! [X_1: int > $o] :
( ( member_int_o @ X_1 @ S )
=> ( ( finite_card_int @ X_1 )
= N ) )
=> ( ( big_co1971440592_o_nat @ finite_card_int @ S )
= ( big_co1971440592_o_nat
@ ^ [X_1: int > $o] : N
@ S ) ) ) ) ) ).
thf(fact_5029_zOdd__def,axiom,
( zOdd
= ( collect_int
@ ^ [X_1: int] :
? [K: int] :
( X_1
= ( plus_plus_int @ ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ K ) @ one_one_int ) ) ) ) ).
thf(fact_5030_Rats__eq__int__div__int,axiom,
( field_1210416355s_real
= ( collect_real
@ ^ [Uu: real] :
? [I_1: int,J_1: int] :
( (&)
@ ( Uu
= ( inverse_divide_real @ ( real_int @ I_1 ) @ ( real_int @ J_1 ) ) )
@ ( (~) @ ( J_1 = zero_zero_int ) ) ) ) ) ).
thf(fact_5031_zEven__def,axiom,
( zEven
= ( collect_int
@ ^ [X_1: int] :
? [K: int] :
( X_1
= ( times_times_int @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) @ K ) ) ) ) ).
thf(fact_5032_summable__Leibniz_I5_J,axiom,
! [A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ( monoseq_real @ A )
=> ( tendsto_nat_real
@ ^ [N_1: nat] :
( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) @ one_one_nat ) ) )
@ ( suminf_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) ) )
@ sequentially ) ) ) ).
thf(fact_5033_LIMSEQ__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_5034_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_5035_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_5036_LIMSEQ__divide__realpow__zero,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( tendsto_nat_real
@ ^ [N_1: nat] : ( inverse_divide_real @ A @ ( power_power_real @ X @ N_1 ) )
@ zero_zero_real
@ sequentially ) ) ).
thf(fact_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__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_5039_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_5040_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_5041_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_5042_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_5043_monoseq__le,axiom,
! [X: real,A: nat > real] :
( ( monoseq_real @ A )
=> ( ( tendsto_nat_real @ A @ X @ sequentially )
=> ( ( ! [N_1: nat] : ( ord_less_eq_real @ ( A @ N_1 ) @ X )
& ! [M_2: nat,N_1: nat] :
( ( ord_less_eq_nat @ M_2 @ N_1 )
=> ( ord_less_eq_real @ ( A @ M_2 ) @ ( A @ N_1 ) ) ) )
| ( ! [N_1: nat] : ( ord_less_eq_real @ X @ ( A @ N_1 ) )
& ! [M_2: nat,N_1: nat] :
( ( ord_less_eq_nat @ M_2 @ N_1 )
=> ( ord_less_eq_real @ ( A @ N_1 ) @ ( A @ M_2 ) ) ) ) ) ) ) ).
thf(fact_5044_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_5045_LIMSEQ__le__const2,axiom,
! [A: real,X_2: nat > real,X: real] :
( ( tendsto_nat_real @ X_2 @ X @ sequentially )
=> ( ? [N_2: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ N_2 @ N_1 )
=> ( ord_less_eq_real @ ( X_2 @ N_1 ) @ A ) )
=> ( ord_less_eq_real @ X @ A ) ) ) ).
thf(fact_5046_LIMSEQ__le__const,axiom,
! [A: real,X_2: nat > real,X: real] :
( ( tendsto_nat_real @ X_2 @ X @ sequentially )
=> ( ? [N_2: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ N_2 @ N_1 )
=> ( ord_less_eq_real @ A @ ( X_2 @ N_1 ) ) )
=> ( ord_less_eq_real @ A @ X ) ) ) ).
thf(fact_5047_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_5048_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R_1: real] :
( tendsto_nat_real
@ ^ [N_1: nat] : ( times_times_real @ R_1 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( real_nat @ ( suc @ N_1 ) ) ) ) ) )
@ R_1
@ sequentially ) ).
thf(fact_5049_summable__Leibniz_I1_J,axiom,
! [A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ( monoseq_real @ A )
=> ( summable_real
@ ^ [N_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ N_1 ) @ ( A @ N_1 ) ) ) ) ) ).
thf(fact_5050_zeroseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( tendsto_nat_real
@ ^ [N_1: nat] : ( times_times_real @ ( inverse_divide_real @ one_one_real @ ( real_nat @ ( plus_plus_nat @ ( times_times_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N_1 @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) ) @ one_one_nat ) ) )
@ zero_zero_real
@ sequentially ) ) ).
thf(fact_5051_Rats__eq__int__div__nat,axiom,
( field_1210416355s_real
= ( collect_real
@ ^ [Uu: real] :
? [I_1: int,N_1: nat] :
( (&)
@ ( Uu
= ( inverse_divide_real @ ( real_int @ I_1 ) @ ( real_nat @ N_1 ) ) )
@ ( (~) @ ( N_1 = zero_zero_nat ) ) ) ) ) ).
thf(fact_5052_summable__Leibniz_I4_J,axiom,
! [A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ( monoseq_real @ A )
=> ( tendsto_nat_real
@ ^ [N_1: nat] :
( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) ) )
@ ( suminf_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) ) )
@ sequentially ) ) ) ).
thf(fact_5053_summable__Leibniz_H_I4_J,axiom,
! [N: nat,A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N_1 ) )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N_1 ) ) @ ( A @ N_1 ) )
=> ( ord_less_eq_real
@ ( suminf_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) ) )
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
thf(fact_5054_summable__Leibniz_H_I5_J,axiom,
! [A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N_1 ) )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N_1 ) ) @ ( A @ N_1 ) )
=> ( tendsto_nat_real
@ ^ [N_1: nat] :
( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) @ one_one_nat ) ) )
@ ( suminf_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) ) )
@ sequentially ) ) ) ) ).
thf(fact_5055_cnj_OLIMSEQ,axiom,
! [X_2: nat > complex,A: complex] :
( ( tendsto_nat_complex @ X_2 @ A @ sequentially )
=> ( tendsto_nat_complex
@ ^ [N_1: nat] : ( cnj @ ( X_2 @ N_1 ) )
@ ( cnj @ A )
@ sequentially ) ) ).
thf(fact_5056_Re_OLIMSEQ,axiom,
! [X_2: nat > complex,A: complex] :
( ( tendsto_nat_complex @ X_2 @ A @ sequentially )
=> ( tendsto_nat_real
@ ^ [N_1: nat] : ( re @ ( X_2 @ N_1 ) )
@ ( re @ A )
@ sequentially ) ) ).
thf(fact_5057_Im_OLIMSEQ,axiom,
! [X_2: nat > complex,A: complex] :
( ( tendsto_nat_complex @ X_2 @ A @ sequentially )
=> ( tendsto_nat_real
@ ^ [N_1: nat] : ( im @ ( X_2 @ N_1 ) )
@ ( im @ A )
@ sequentially ) ) ).
thf(fact_5058_LIMSEQ__Complex,axiom,
! [Y_2: nat > real,B: real,X_2: nat > real,A: real] :
( ( tendsto_nat_real @ X_2 @ A @ sequentially )
=> ( ( tendsto_nat_real @ Y_2 @ B @ sequentially )
=> ( tendsto_nat_complex
@ ^ [N_1: nat] : ( complex_1 @ ( X_2 @ N_1 ) @ ( Y_2 @ N_1 ) )
@ ( complex_1 @ A @ B )
@ sequentially ) ) ) ).
thf(fact_5059_Bolzano__nest__unique,axiom,
! [P_2: produc914805421l_real > $o,A_4: real,B_3: real] :
( ( ord_less_eq_real @ A_4 @ B_3 )
=> ( ( ord_less_eq_real @ A_4 @ B_3 )
=> ( ( ord_less_eq_real @ A_4 @ B_3 )
=> ( ( tendsto_nat_real
@ ^ [N_1: nat] : ( minus_minus_real @ ( produc1935615926l_real @ ( bolzano_bisect @ P_2 @ A_4 @ B_3 @ N_1 ) ) @ ( produc556554744l_real @ ( bolzano_bisect @ P_2 @ A_4 @ B_3 @ N_1 ) ) )
@ zero_zero_real
@ sequentially )
=> ? [L_1: real] :
( ! [N_1: nat] : ( ord_less_eq_real @ ( produc1935615926l_real @ ( bolzano_bisect @ P_2 @ A_4 @ B_3 @ N_1 ) ) @ L_1 )
& ( tendsto_nat_real
@ ^ [N_1: nat] : ( produc1935615926l_real @ ( bolzano_bisect @ P_2 @ A_4 @ B_3 @ N_1 ) )
@ L_1
@ sequentially )
& ! [N_1: nat] : ( ord_less_eq_real @ L_1 @ ( produc556554744l_real @ ( bolzano_bisect @ P_2 @ A_4 @ B_3 @ N_1 ) ) )
& ( tendsto_nat_real
@ ^ [N_1: nat] : ( produc556554744l_real @ ( bolzano_bisect @ P_2 @ A_4 @ B_3 @ N_1 ) )
@ L_1
@ sequentially ) ) ) ) ) ) ).
thf(fact_5060_summable__Leibniz_H_I3_J,axiom,
! [A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N_1 ) )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N_1 ) ) @ ( A @ N_1 ) )
=> ( tendsto_nat_real
@ ^ [N_1: nat] :
( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) ) )
@ ( suminf_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) ) )
@ sequentially ) ) ) ) ).
thf(fact_5061_summable__Leibniz_H_I2_J,axiom,
! [N: nat,A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N_1 ) )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N_1 ) ) @ ( A @ N_1 ) )
=> ( ord_less_eq_real
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N ) ) )
@ ( suminf_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) ) ) ) ) ) ) ).
thf(fact_5062_summable__Leibniz_I3_J,axiom,
! [A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ( monoseq_real @ A )
=> ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
=> ! [N_1: nat] :
( member_real
@ ( suminf_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) ) )
@ ( ord_at1589558736t_real
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) @ one_one_nat ) ) )
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) ) ) ) ) ) ) ) ).
thf(fact_5063_summable__Leibniz_I2_J,axiom,
! [A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ( monoseq_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
=> ! [N_1: nat] :
( member_real
@ ( suminf_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) ) )
@ ( ord_at1589558736t_real
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) ) )
@ ( big_co604158596t_real
@ ^ [I_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ I_1 ) @ ( A @ I_1 ) )
@ ( ord_at4362885an_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( number_number_of_nat @ ( bit0 @ ( bit1 @ pls ) ) ) @ N_1 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
thf(fact_5064_Sup__atLeastAtMost,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( comple124823625p_real @ ( ord_at1589558736t_real @ Y @ X ) )
= X ) ) ).
thf(fact_5065_summable,axiom,
! [A: nat > real] :
( ( tendsto_nat_real @ A @ zero_zero_real @ sequentially )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N_1 ) )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N_1 ) ) @ ( A @ N_1 ) )
=> ( summable_real
@ ^ [N_1: nat] : ( times_times_real @ ( power_power_real @ ( number267125858f_real @ min ) @ N_1 ) @ ( A @ N_1 ) ) ) ) ) ) ).
thf(fact_5066_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( ord_at238088361st_nat @ L @ U ) ) ).
thf(fact_5067_finite__atLeastAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( ord_at875362053st_int @ L @ U ) ) ).
thf(fact_5068_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_5069_all__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ! [M_2: nat] :
( ( ord_less_eq_nat @ M_2 @ N )
=> ( P @ M_2 ) )
<=> ! [X_1: nat] :
( ( member_nat @ X_1 @ ( ord_at238088361st_nat @ zero_zero_nat @ N ) )
=> ( P @ X_1 ) ) ) ).
thf(fact_5070_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_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_1: nat,I: nat,J: nat] :
( ( image_nat_nat
@ ^ [N_1: nat] : ( plus_plus_nat @ N_1 @ K_1 )
@ ( ord_at238088361st_nat @ I @ J ) )
= ( ord_at238088361st_nat @ ( plus_plus_nat @ I @ K_1 ) @ ( plus_plus_nat @ J @ K_1 ) ) ) ).
thf(fact_5074_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_5075_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_5076_cnj_Ocont,axiom,
! [A: complex] : ( tendst1507391555omplex @ cnj @ ( cnj @ A ) @ ( at_complex @ A ) ) ).
thf(fact_5077_Re_Ocont,axiom,
! [A: complex] : ( tendsto_complex_real @ re @ ( re @ A ) @ ( at_complex @ A ) ) ).
thf(fact_5078_Im_Ocont,axiom,
! [A: complex] : ( tendsto_complex_real @ im @ ( im @ A ) @ ( at_complex @ A ) ) ).
thf(fact_5079_card__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( ord_at238088361st_nat @ L @ U ) )
= ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
thf(fact_5080_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( ord_at641636577an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
= ( ord_at875362053st_int @ L @ U ) ) ).
thf(fact_5081_setsum__shift__lb__Suc0__0,axiom,
! [K_1: nat,F: nat > nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( big_co387207925at_nat @ F @ ( ord_at238088361st_nat @ ( suc @ zero_zero_nat ) @ K_1 ) )
= ( big_co387207925at_nat @ F @ ( ord_at238088361st_nat @ zero_zero_nat @ K_1 ) ) ) ) ).
thf(fact_5082_card__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( ord_at875362053st_int @ L @ U ) )
= ( nat_1 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).
thf(fact_5083_fact__div__fact,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( div_div_nat @ ( fact_fact_nat @ M ) @ ( fact_fact_nat @ N ) )
= ( big_co1705425894at_nat
@ ^ [X_1: nat] : X_1
@ ( ord_at238088361st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
thf(fact_5084_fact__altdef__int,axiom,
! [N: int] :
( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( fact_fact_int @ N )
= ( big_co1548731110nt_int
@ ^ [I_1: int] : I_1
@ ( ord_at875362053st_int @ one_one_int @ N ) ) ) ) ).
thf(fact_5085_LIM__cos__div__sin,axiom,
( tendsto_real_real
@ ^ [X_1: real] : ( inverse_divide_real @ ( cos @ X_1 ) @ ( sin @ X_1 ) )
@ zero_zero_real
@ ( at_real @ ( inverse_divide_real @ pi @ ( number267125858f_real @ ( bit0 @ ( bit1 @ pls ) ) ) ) ) ) ).
thf(fact_5086_LIM__fun__not__zero,axiom,
! [F: real > real,L: real,C: real] :
( ( tendsto_real_real @ F @ L @ ( at_real @ C ) )
=> ( ( L != zero_zero_real )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ! [X_1: real] :
( ( ( X_1 != C )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X_1 ) ) @ R ) )
=> ( ( F @ X_1 )
!= zero_zero_real ) ) ) ) ) ).
thf(fact_5087_LIM__fun__gt__zero,axiom,
! [F: real > real,L: real,C: real] :
( ( tendsto_real_real @ F @ L @ ( at_real @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ L )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ! [X_1: real] :
( ( ( X_1 != C )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X_1 ) ) @ R ) )
=> ( ord_less_real @ zero_zero_real @ ( F @ X_1 ) ) ) ) ) ) ).
thf(fact_5088_LIM__fun__less__zero,axiom,
! [F: real > real,L: real,C: real] :
( ( tendsto_real_real @ F @ L @ ( at_real @ C ) )
=> ( ( ord_less_real @ L @ zero_zero_real )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ! [X_1: real] :
( ( ( X_1 != C )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X_1 ) ) @ R ) )
=> ( ord_less_real @ ( F @ X_1 ) @ zero_zero_real ) ) ) ) ) ).
thf(fact_5089_aset_I8_J,axiom,
! [T: int,A_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A_1 )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ T @ X_1 )
=> ( ord_less_eq_int @ T @ ( plus_plus_int @ X_1 @ D_1 ) ) ) ) ) ).
thf(fact_5090_aset_I6_J,axiom,
! [T: int,A_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A_1 )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ X_1 @ T )
=> ( ord_less_eq_int @ ( plus_plus_int @ X_1 @ D_1 ) @ T ) ) ) ) ) ).
thf(fact_5091_bset_I8_J,axiom,
! [T: int,B_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B_1 )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ T @ X_1 )
=> ( ord_less_eq_int @ T @ ( minus_minus_int @ X_1 @ D_1 ) ) ) ) ) ) ).
thf(fact_5092_bset_I6_J,axiom,
! [T: int,B_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B_1 )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_eq_int @ X_1 @ T )
=> ( ord_less_eq_int @ ( minus_minus_int @ X_1 @ D_1 ) @ T ) ) ) ) ).
thf(fact_5093_bset_I3_J,axiom,
! [T: int,B_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B_1 )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( X_1 = T )
=> ( ( minus_minus_int @ X_1 @ D_1 )
= T ) ) ) ) ) ).
thf(fact_5094_aset_I3_J,axiom,
! [T: int,A_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A_1 )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( X_1 = T )
=> ( ( plus_plus_int @ X_1 @ D_1 )
= T ) ) ) ) ) ).
thf(fact_5095_aset_I7_J,axiom,
! [T: int,A_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A_1 )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ T @ X_1 )
=> ( ord_less_int @ T @ ( plus_plus_int @ X_1 @ D_1 ) ) ) ) ) ).
thf(fact_5096_bset_I5_J,axiom,
! [T: int,B_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B_1 )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ X_1 @ T )
=> ( ord_less_int @ ( minus_minus_int @ X_1 @ D_1 ) @ T ) ) ) ) ).
thf(fact_5097_aset_I5_J,axiom,
! [T: int,A_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ( member_int @ T @ A_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A_1 )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ X_1 @ T )
=> ( ord_less_int @ ( plus_plus_int @ X_1 @ D_1 ) @ T ) ) ) ) ) ).
thf(fact_5098_bset_I7_J,axiom,
! [T: int,B_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ( member_int @ T @ B_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B_1 )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( ord_less_int @ T @ X_1 )
=> ( ord_less_int @ T @ ( minus_minus_int @ X_1 @ D_1 ) ) ) ) ) ) ).
thf(fact_5099_aset_I4_J,axiom,
! [T: int,A_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ( member_int @ T @ A_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A_1 )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( X_1 != T )
=> ( ( plus_plus_int @ X_1 @ D_1 )
!= T ) ) ) ) ) ).
thf(fact_5100_bset_I4_J,axiom,
! [T: int,B_1: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ( member_int @ T @ B_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B_1 )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( X_1 != T )
=> ( ( minus_minus_int @ X_1 @ D_1 )
!= T ) ) ) ) ) ).
thf(fact_5101_periodic__finite__ex,axiom,
! [P: int > $o,D: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X_1: int,K: int] :
( ( P @ X_1 )
<=> ( P @ ( minus_minus_int @ X_1 @ ( times_times_int @ K @ D ) ) ) )
=> ( ( ?? @ int @ P )
<=> ? [X_1: int] :
( ( member_int @ X_1 @ ( ord_at875362053st_int @ one_one_int @ D ) )
& ( P @ X_1 ) ) ) ) ) ).
thf(fact_5102_aset_I10_J,axiom,
! [T: int,A_1: int > $o,D: int,D_1: int] :
( ( dvd_dvd_int @ D @ D_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A_1 )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X_1 @ T ) )
=> ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X_1 @ D_1 ) @ T ) ) ) ) ) ).
thf(fact_5103_bset_I10_J,axiom,
! [T: int,B_1: int > $o,D: int,D_1: int] :
( ( dvd_dvd_int @ D @ D_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B_1 )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X_1 @ T ) )
=> ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X_1 @ D_1 ) @ T ) ) ) ) ) ).
thf(fact_5104_bset_I9_J,axiom,
! [T: int,B_1: int > $o,D: int,D_1: int] :
( ( dvd_dvd_int @ D @ D_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B_1 )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X_1 @ T ) )
=> ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X_1 @ D_1 ) @ T ) ) ) ) ) ).
thf(fact_5105_aset_I9_J,axiom,
! [T: int,A_1: int > $o,D: int,D_1: int] :
( ( dvd_dvd_int @ D @ D_1 )
=> ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A_1 )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X_1 @ T ) )
=> ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X_1 @ D_1 ) @ T ) ) ) ) ) ).
thf(fact_5106_LIMSEQ__inverse__zero,axiom,
! [X_2: nat > real] :
( ! [R: real] :
? [N_2: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ N_2 @ N_1 )
=> ( ord_less_real @ R @ ( X_2 @ N_1 ) ) )
=> ( tendsto_nat_real
@ ^ [N_1: nat] : ( inverse_inverse_real @ ( X_2 @ N_1 ) )
@ zero_zero_real
@ sequentially ) ) ).
thf(fact_5107_cnj_OscaleR,axiom,
! [R_1: real,X: complex] :
( ( cnj @ ( scaleR1652505878omplex @ R_1 @ X ) )
= ( scaleR1652505878omplex @ R_1 @ ( cnj @ X ) ) ) ).
thf(fact_5108_Re_OscaleR,axiom,
! [R_1: real,X: complex] :
( ( re @ ( scaleR1652505878omplex @ R_1 @ X ) )
= ( scaleR_scaleR_real @ R_1 @ ( re @ X ) ) ) ).
thf(fact_5109_Im_OscaleR,axiom,
! [R_1: real,X: complex] :
( ( im @ ( scaleR1652505878omplex @ R_1 @ X ) )
= ( scaleR_scaleR_real @ R_1 @ ( im @ X ) ) ) ).
thf(fact_5110_real__scaleR__def,axiom,
! [A: real,X: real] :
( ( scaleR_scaleR_real @ A @ X )
= ( times_times_real @ A @ X ) ) ).
thf(fact_5111_complex__scaleR,axiom,
! [R_1: real,A: real,B: real] :
( ( scaleR1652505878omplex @ R_1 @ ( complex_1 @ A @ B ) )
= ( complex_1 @ ( times_times_real @ R_1 @ A ) @ ( times_times_real @ R_1 @ B ) ) ) ).
thf(fact_5112_complex__Im__scaleR,axiom,
! [R_1: real,X: complex] :
( ( im @ ( scaleR1652505878omplex @ R_1 @ X ) )
= ( times_times_real @ R_1 @ ( im @ X ) ) ) ).
thf(fact_5113_complex__Re__scaleR,axiom,
! [R_1: real,X: complex] :
( ( re @ ( scaleR1652505878omplex @ R_1 @ X ) )
= ( times_times_real @ R_1 @ ( re @ X ) ) ) ).
thf(fact_5114_complex__sgn__def,axiom,
! [X: complex] :
( ( sgn_sgn_complex @ X )
= ( scaleR1652505878omplex @ ( inverse_inverse_real @ ( norm_norm_complex @ X ) ) @ X ) ) ).
thf(fact_5115_complex__scaleR__def,axiom,
! [R_1: real,X: complex] :
( ( scaleR1652505878omplex @ R_1 @ X )
= ( complex_1 @ ( times_times_real @ R_1 @ ( re @ X ) ) @ ( times_times_real @ R_1 @ ( im @ X ) ) ) ) ).
thf(fact_5116_trivial__limit__sequentially,axiom,
~ ( trivial_limit_nat @ sequentially ) ).
thf(fact_5117_pred__nat__def,axiom,
( pred_nat
= ( collec1979865426at_nat
@ ( produc1038563245_nat_o
@ ^ [M_2: nat,N_1: nat] :
( N_1
= ( suc @ M_2 ) ) ) ) ) ).
thf(fact_5118_gcd__coprime__exists__int,axiom,
! [A: int,B: int] :
( ( ( gcd_gcd_int @ A @ B )
!= zero_zero_int )
=> ? [A_3: int,B_2: int] :
( ( A
= ( times_times_int @ A_3 @ ( gcd_gcd_int @ A @ B ) ) )
& ( B
= ( times_times_int @ B_2 @ ( gcd_gcd_int @ A @ B ) ) )
& ( ( gcd_gcd_int @ A_3 @ B_2 )
= one_one_int ) ) ) ).
thf(fact_5119_min__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__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_5121_min__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
thf(fact_5122_min__0R,axiom,
! [N: nat] :
( ( ord_min_nat @ N @ zero_zero_nat )
= zero_zero_nat ) ).
thf(fact_5123_min__0L,axiom,
! [N: nat] :
( ( ord_min_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
thf(fact_5124_min__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_5125_min__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ M @ ( suc @ N ) )
= ( nat_case_nat @ zero_zero_nat
@ ^ [M_1: nat] : ( suc @ ( ord_min_nat @ M_1 @ N ) )
@ M ) ) ).
thf(fact_5126_min__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_min_nat @ ( suc @ N ) @ M )
= ( nat_case_nat @ zero_zero_nat
@ ^ [M_1: nat] : ( suc @ ( ord_min_nat @ N @ M_1 ) )
@ M ) ) ).
thf(fact_5127_vanishes__mult__bounded,axiom,
! [Y_2: nat > rat,X_2: nat > rat] :
( ? [A_2: rat] :
( ( ord_less_rat @ zero_zero_rat @ A_2 )
& ! [N_1: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X_2 @ N_1 ) ) @ A_2 ) )
=> ( ( vanishes @ Y_2 )
=> ( vanishes
@ ^ [N_1: nat] : ( times_times_rat @ ( X_2 @ N_1 ) @ ( Y_2 @ N_1 ) ) ) ) ) ).
thf(fact_5128_SetInterval_Otransfer__int__nat__set__functions,axiom,
! [N: int,M: int] :
( ( nat_is_nat @ M )
=> ( ( nat_is_nat @ N )
=> ( ( ord_at875362053st_int @ M @ N )
= ( image_nat_int @ semiri1621563631at_int @ ( ord_at238088361st_nat @ ( nat_1 @ M ) @ ( nat_1 @ N ) ) ) ) ) ) ).
thf(fact_5129_is__nat__def,axiom,
! [X: int] :
( ( nat_is_nat @ X )
<=> ( ord_less_eq_int @ zero_zero_int @ X ) ) ).
thf(fact_5130_Nat__Transfer_Otransfer__int__nat__function__closures_I9_J,axiom,
! [Z_1: nat] : ( nat_is_nat @ ( semiri1621563631at_int @ Z_1 ) ) ).
thf(fact_5131_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_5132_Nat__Transfer_Otransfer__int__nat__function__closures_I6_J,axiom,
nat_is_nat @ one_one_int ).
thf(fact_5133_Nat__Transfer_Otransfer__int__nat__function__closures_I5_J,axiom,
nat_is_nat @ zero_zero_int ).
thf(fact_5134_Nat__Transfer_Otransfer__int__nat__set__function__closures_I6_J,axiom,
! [X: int,A_1: int > $o] :
( ( nat_nat_set @ A_1 )
=> ( ( member_int @ X @ A_1 )
=> ( nat_is_nat @ X ) ) ) ).
thf(fact_5135_Nat__Transfer_Otransfer__int__nat__function__closures_I4_J,axiom,
! [N: nat,X: int] :
( ( nat_is_nat @ X )
=> ( nat_is_nat @ ( power_power_int @ X @ N ) ) ) ).
thf(fact_5136_Divides_Otransfer__int__nat__function__closures_I2_J,axiom,
! [Y: int,X: int] :
( ( nat_is_nat @ X )
=> ( ( nat_is_nat @ Y )
=> ( nat_is_nat @ ( div_mod_int @ X @ Y ) ) ) ) ).
thf(fact_5137_Divides_Otransfer__int__nat__function__closures_I1_J,axiom,
! [Y: int,X: int] :
( ( nat_is_nat @ X )
=> ( ( nat_is_nat @ Y )
=> ( nat_is_nat @ ( div_div_int @ X @ Y ) ) ) ) ).
thf(fact_5138_Nat__Transfer_Otransfer__int__nat__function__closures_I3_J,axiom,
! [Y: int,X: int] :
( ( nat_is_nat @ X )
=> ( ( nat_is_nat @ Y )
=> ( nat_is_nat @ ( nat_tsub @ X @ Y ) ) ) ) ).
thf(fact_5139_Nat__Transfer_Otransfer__int__nat__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_5140_vanishes__minus,axiom,
! [X_2: nat > rat] :
( ( vanishes @ X_2 )
=> ( vanishes
@ ^ [N_1: nat] : ( uminus_uminus_rat @ ( X_2 @ N_1 ) ) ) ) ).
thf(fact_5141_vanishes__const,axiom,
! [C: rat] :
( ( vanishes
@ ^ [N_1: nat] : C )
<=> ( C = zero_zero_rat ) ) ).
thf(fact_5142_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_5143_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_5144_SetInterval_Otransfer__int__nat__set__function__closures,axiom,
! [Y: int,X: int] :
( ( nat_is_nat @ X )
=> ( nat_nat_set @ ( ord_at875362053st_int @ X @ Y ) ) ) ).
thf(fact_5145_Nat__Transfer_Otransfer__int__nat__function__closures_I8_J,axiom,
nat_is_nat @ ( number_number_of_int @ ( bit1 @ ( bit1 @ pls ) ) ) ).
thf(fact_5146_transfer__int__nat__gcd__closures_I1_J,axiom,
! [Y: int,X: int] :
( ( nat_is_nat @ X )
=> ( ( nat_is_nat @ Y )
=> ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y ) ) ) ) ).
thf(fact_5147_transfer__int__nat__factorial__closure,axiom,
! [X: int] :
( ( nat_is_nat @ X )
=> ( ord_less_eq_int @ zero_zero_int @ ( fact_fact_int @ X ) ) ) ).
thf(fact_5148_transfer__int__nat__set__relations_I2_J,axiom,
! [A_1: int > $o,X: int] :
( ( nat_is_nat @ X )
=> ( ( nat_nat_set @ A_1 )
=> ( ( member_int @ X @ A_1 )
<=> ( member_nat @ ( nat_1 @ X ) @ ( image_int_nat @ nat_1 @ A_1 ) ) ) ) ) ).
thf(fact_5149_Nat__Transfer_Otransfer__int__nat__function__closures_I7_J,axiom,
nat_is_nat @ ( number_number_of_int @ ( bit0 @ ( bit1 @ pls ) ) ) ).
thf(fact_5150_vanishesD,axiom,
! [R_1: rat,X_2: nat > rat] :
( ( vanishes @ X_2 )
=> ( ( ord_less_rat @ zero_zero_rat @ R_1 )
=> ? [K: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ K @ N_1 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X_2 @ N_1 ) ) @ R_1 ) ) ) ) ).
thf(fact_5151_vanishes__def,axiom,
! [X_2: nat > rat] :
( ( vanishes @ X_2 )
<=> ! [R: rat] :
( ( ord_less_rat @ zero_zero_rat @ R )
=> ? [K: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ K @ N_1 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X_2 @ N_1 ) ) @ R ) ) ) ) ).
thf(fact_5152_vanishesI,axiom,
! [X_2: nat > rat] :
( ! [R: rat] :
( ( ord_less_rat @ zero_zero_rat @ R )
=> ? [K: nat] :
! [N_1: nat] :
( ( ord_less_eq_nat @ K @ N_1 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X_2 @ N_1 ) ) @ R ) ) )
=> ( vanishes @ X_2 ) ) ).
thf(fact_5153_zcongm__def,axiom,
! [M: int,X_1: int,Xa: int] :
( ( zcongm @ M @ X_1 @ Xa )
<=> ( zcong @ X_1 @ Xa @ M ) ) ).
thf(fact_5154_cppi,axiom,
! [A_1: int > $o,P_1: int > $o,P: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ? [Z: int] :
! [X_1: int] :
( ( ord_less_int @ Z @ X_1 )
=> ( ( P @ X_1 )
<=> ( P_1 @ X_1 ) ) )
=> ( ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A_1 )
=> ( X_1
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X_1 )
=> ( P @ ( plus_plus_int @ X_1 @ D_1 ) ) ) )
=> ( ! [X_1: int,K: int] :
( ( P_1 @ X_1 )
<=> ( P_1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K @ D_1 ) ) ) )
=> ( ( ?? @ int @ P )
<=> ( ? [X_1: int] :
( ( member_int @ X_1 @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
& ( P_1 @ X_1 ) )
| ? [X_1: int] :
( ( member_int @ X_1 @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
& ? [Xa: int] :
( ( member_int @ Xa @ A_1 )
& ( P @ ( minus_minus_int @ Xa @ X_1 ) ) ) ) ) ) ) ) ) ) ).
thf(fact_5155_bijzcong__zcong__prod,axiom,
! [A_1: int > $o,B_1: int > $o,M: int] :
( ( member1329254762_int_o @ ( produc398918003_int_o @ A_1 @ B_1 ) @ ( bijR_int_int @ ( zcongm @ M ) ) )
=> ( zcong
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ A_1 )
@ ( big_co1548731110nt_int
@ ^ [X_1: int] : X_1
@ B_1 )
@ M ) ) ).
thf(fact_5156_cpmi,axiom,
! [B_1: int > $o,P_1: int > $o,P: int > $o,D_1: int] :
( ( ord_less_int @ zero_zero_int @ D_1 )
=> ( ? [Z: int] :
! [X_1: int] :
( ( ord_less_int @ X_1 @ Z )
=> ( ( P @ X_1 )
<=> ( P_1 @ X_1 ) ) )
=> ( ! [X_1: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B_1 )
=> ( X_1
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X_1 )
=> ( P @ ( minus_minus_int @ X_1 @ D_1 ) ) ) )
=> ( ! [X_1: int,K: int] :
( ( P_1 @ X_1 )
<=> ( P_1 @ ( minus_minus_int @ X_1 @ ( times_times_int @ K @ D_1 ) ) ) )
=> ( ( ?? @ int @ P )
<=> ( ? [X_1: int] :
( ( member_int @ X_1 @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
& ( P_1 @ X_1 ) )
| ? [X_1: int] :
( ( member_int @ X_1 @ ( ord_at875362053st_int @ one_one_int @ D_1 ) )
& ? [Xa: int] :
( ( member_int @ Xa @ B_1 )
& ( P @ ( plus_plus_int @ Xa @ X_1 ) ) ) ) ) ) ) ) ) ) ).
thf(fact_5157_increasing__LIMSEQ,axiom,
! [L: real,F: nat > real] :
( ! [N_1: nat] : ( ord_less_eq_real @ ( F @ N_1 ) @ ( F @ ( suc @ N_1 ) ) )
=> ( ! [N_1: nat] : ( ord_less_eq_real @ ( F @ N_1 ) @ L )
=> ( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [N_1: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N_1 ) @ E ) ) )
=> ( tendsto_nat_real @ F @ L @ sequentially ) ) ) ) ).
thf(fact_5158_isCont__real__root,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( isCont_real_real @ ( root @ N ) @ X ) ) ).
thf(fact_5159_isCont__ln,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( isCont_real_real @ ln @ X ) ) ).
thf(fact_5160_isCont__abs,axiom,
! [A: real] : ( isCont_real_real @ abs_abs_real @ A ) ).
thf(fact_5161_Im_OisCont,axiom,
! [A: complex] : ( isCont_complex_real @ im @ A ) ).
thf(fact_5162_Re_OisCont,axiom,
! [A: complex] : ( isCont_complex_real @ re @ A ) ).
thf(fact_5163_cnj_OisCont,axiom,
! [A: complex] : ( isCont156215680omplex @ cnj @ A ) ).
thf(fact_5164_isCont__real__sqrt,axiom,
! [X: real] : ( isCont_real_real @ sqrt @ X ) ).
thf(fact_5165_isCont__cos,axiom,
! [X: real] : ( isCont_real_real @ cos @ X ) ).
thf(fact_5166_isCont__arctan,axiom,
! [X: real] : ( isCont_real_real @ arctan @ X ) ).
thf(fact_5167_isCont__sin,axiom,
! [X: real] : ( isCont_real_real @ sin @ X ) ).
thf(fact_5168_isCont__tan,axiom,
! [X: real] :
( ( ( cos @ X )
!= zero_zero_real )
=> ( isCont_real_real @ tan @ X ) ) ).
thf(fact_5169_isCont__root__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( isCont_real_real @ ( root @ N ) @ zero_zero_real ) ) ).
thf(fact_5170_isCont__root__neg,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( isCont_real_real @ ( root @ N ) @ X ) ) ) ).
thf(fact_5171_isCont__root__pos,axiom,
! [X: real,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( isCont_real_real @ ( root @ N ) @ X ) ) ) ).
thf(fact_5172_isCont__arccos,axiom,
! [X: real] :
( ( ord_less_real @ ( number267125858f_real @ min ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( isCont_real_real @ arccos @ X ) ) ) ).
thf(fact_5173_isCont__arcsin,axiom,
! [X: real] :
( ( ord_less_real @ ( number267125858f_real @ min ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( isCont_real_real @ arcsin @ X ) ) ) ).
thf(fact_5174_LIM__less__bound,axiom,
! [F: real > real,B: real,X: real] :
( ( ord_less_real @ B @ X )
=> ( ! [X_1: real] :
( ( member_real @ X_1 @ ( ord_gr788844697n_real @ B @ X ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X_1 ) ) )
=> ( ( isCont_real_real @ F @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).
thf(fact_5175_DERIV__inverse__function,axiom,
! [B: real,A: real,F: real > real,G: real > real,X: real,D_1: real] :
( ( deriv_real @ F @ ( G @ X ) @ D_1 )
=> ( ( D_1 != zero_zero_real )
=> ( ( ord_less_real @ A @ X )
=> ( ( ord_less_real @ X @ B )
=> ( ! [Y_1: real] :
( ( ( ord_less_real @ A @ Y_1 )
& ( ord_less_real @ Y_1 @ B ) )
=> ( ( F @ ( G @ Y_1 ) )
= Y_1 ) )
=> ( ( isCont_real_real @ G @ X )
=> ( deriv_real @ G @ X @ ( inverse_inverse_real @ D_1 ) ) ) ) ) ) ) ) ).
thf(fact_5176_isCont__inv__fun,axiom,
! [G: real > real,F: real > real,X: real,D: real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
=> ( ( G @ ( F @ Z ) )
= Z ) )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
=> ( isCont_real_real @ F @ Z ) )
=> ( isCont_real_real @ G @ ( F @ X ) ) ) ) ) ).
thf(fact_5177_DERIV__isconst2,axiom,
! [X: real,F: real > real,A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A @ X_1 )
& ( ord_less_eq_real @ X_1 @ B ) )
=> ( isCont_real_real @ F @ X_1 ) )
=> ( ! [X_1: real] :
( ( ( ord_less_real @ A @ X_1 )
& ( ord_less_real @ X_1 @ B ) )
=> ( deriv_real @ F @ X_1 @ zero_zero_real ) )
=> ( ( ord_less_eq_real @ A @ X )
=> ( ( ord_less_eq_real @ X @ B )
=> ( ( F @ X )
= ( F @ A ) ) ) ) ) ) ) ).
thf(fact_5178_DERIV__isconst__end,axiom,
! [F: real > real,A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A @ X_1 )
& ( ord_less_eq_real @ X_1 @ B ) )
=> ( isCont_real_real @ F @ X_1 ) )
=> ( ! [X_1: real] :
( ( ( ord_less_real @ A @ X_1 )
& ( ord_less_real @ X_1 @ B ) )
=> ( deriv_real @ F @ X_1 @ zero_zero_real ) )
=> ( ( F @ B )
= ( F @ A ) ) ) ) ) ).
thf(fact_5179_DERIV__isconst1,axiom,
! [F: real > real,A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ! [X_1: real] :
( ( ( ord_less_eq_real @ A @ X_1 )
& ( ord_less_eq_real @ X_1 @ B ) )
=> ( isCont_real_real @ F @ X_1 ) )
=> ( ! [X_1: real] :
( ( ( ord_less_real @ A @ X_1 )
& ( ord_less_real @ X_1 @ B ) )
=> ( deriv_real @ F @ X_1 @ zero_zero_real ) )
=> ! [X_1: real] :
( ( ( ord_less_eq_real @ A @ X_1 )
& ( ord_less_eq_real @ X_1 @ B ) )
=> ( ( F @ X_1 )
= ( F @ A ) ) ) ) ) ) ).
thf(fact_5180_lemma__isCont__inj2,axiom,
! [G: real > real,F: real > real,X: real,D: real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
=> ( ( G @ ( F @ Z ) )
= Z ) )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
=> ( isCont_real_real @ F @ Z ) )
=> ? [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
& ( ord_less_real @ ( F @ Z ) @ ( F @ X ) ) ) ) ) ) ).
thf(fact_5181_isCont__inv__fun__inv,axiom,
! [G: real > real,F: real > real,X: real,D: real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
=> ( ( G @ ( F @ Z ) )
= Z ) )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
=> ( isCont_real_real @ F @ Z ) )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ! [Y_1: real] :
( ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ ( minus_minus_real @ Y_1 @ ( F @ X ) ) ) )
& ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ Y_1 @ ( F @ X ) ) ) @ E ) )
=> ( ( F @ ( G @ Y_1 ) )
= Y_1 ) ) ) ) ) ) ).
thf(fact_5182_lemma__isCont__inj,axiom,
! [G: real > real,F: real > real,X: real,D: real] :
( ( ord_less_real @ zero_zero_real @ D )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
=> ( ( G @ ( F @ Z ) )
= Z ) )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
=> ( isCont_real_real @ F @ Z ) )
=> ? [Z: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ X ) ) @ D )
& ( ord_less_real @ ( F @ X ) @ ( F @ Z ) ) ) ) ) ) ).
thf(fact_5183_SetInterval_Otransfer__nat__int__set__functions_I1_J,axiom,
! [N: nat] :
( ( ord_atMost_nat @ N )
= ( image_int_nat @ nat_1 @ ( ord_at875362053st_int @ zero_zero_int @ ( semiri1621563631at_int @ N ) ) ) ) ).
thf(fact_5184_max__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_5185_finite__atMost,axiom,
! [K_1: nat] : ( finite_finite_nat @ ( ord_atMost_nat @ K_1 ) ) ).
thf(fact_5186_atLeast0AtMost,axiom,
! [N: nat] :
( ( ord_at238088361st_nat @ zero_zero_nat @ N )
= ( ord_atMost_nat @ N ) ) ).
thf(fact_5187_card__atMost,axiom,
! [U: nat] :
( ( finite_card_nat @ ( ord_atMost_nat @ U ) )
= ( suc @ U ) ) ).
thf(fact_5188_max__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
thf(fact_5189_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
thf(fact_5190_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
thf(fact_5191_nat__minus__add__max,axiom,
! [N: nat,M: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
= ( ord_max_nat @ N @ M ) ) ).
thf(fact_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__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_5194_lessThan__Suc__atMost,axiom,
! [K_1: nat] :
( ( ord_lessThan_nat @ ( suc @ K_1 ) )
= ( ord_atMost_nat @ K_1 ) ) ).
thf(fact_5195_finite__nat__iff__bounded__le,axiom,
! [S: nat > $o] :
( ( finite_finite_nat @ S )
<=> ? [K: nat] : ( ord_less_eq_nat_o @ S @ ( ord_atMost_nat @ K ) ) ) ).
%----Helper facts (15)
thf(help_If_1_1_If_000tc__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_000tc__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__Int__Oint_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_1_1_If_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_1_1_If_000tc__RealDef__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_000tc__RealDef__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__RealDef__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_1_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] :
( ( if_Pro1731782967nt_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y: product_prod_int_int] :
( ( if_Pro1731782967nt_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_1_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T,axiom,
! [X: produc914805421l_real,Y: produc914805421l_real] :
( ( if_Pro313124157l_real @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T,axiom,
! [X: produc914805421l_real,Y: produc914805421l_real] :
( ( if_Pro313124157l_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_3_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( times_times_int @ ( twoSqu2072599593sum2sq @ ( product_Pair_int_int @ a @ b ) ) @ ( twoSqu2072599593sum2sq @ ( product_Pair_int_int @ p @ q ) ) )
= ( twoSqu2072599593sum2sq @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ a @ p ) @ ( times_times_int @ b @ q ) ) @ ( minus_minus_int @ ( times_times_int @ a @ q ) @ ( times_times_int @ b @ p ) ) ) ) ) ).
%------------------------------------------------------------------------------