TPTP Problem File: SLH0634^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Number_Theoretic_Transform/0006_Preliminary_Lemmas/prob_00387_015019__14079454_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1370 ( 785 unt; 98 typ; 0 def)
% Number of atoms : 3205 (1789 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 10301 ( 252 ~; 91 |; 220 &;8704 @)
% ( 0 <=>;1034 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 474 ( 474 >; 0 *; 0 +; 0 <<)
% Number of symbols : 92 ( 89 usr; 14 con; 0-3 aty)
% Number of variables : 3467 ( 165 ^;3182 !; 120 ?;3467 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:38:06.291
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__Finite____Field__Omod____ring_Itf__a_J,type,
finite_mod_ring_a: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (89)
thf(sy_c_Finite__Field_Oof__int__mod__ring_001tf__a,type,
finite8272632373135393572ring_a: int > finite_mod_ring_a ).
thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
finite_card_complex: set_complex > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
finite_card_int: set_int > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Real__Oreal,type,
finite_card_real: set_real > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
minus_minus_complex: complex > complex > complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Finite____Field__Omod____ring_Itf__a_J,type,
minus_3609261664126569004ring_a: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
plus_plus_complex: complex > complex > complex ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Finite____Field__Omod____ring_Itf__a_J,type,
plus_p6165643967897163644ring_a: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
times_times_complex: complex > complex > complex ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Finite____Field__Omod____ring_Itf__a_J,type,
times_5121417576591743744ring_a: finite_mod_ring_a > finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
uminus1482373934393186551omplex: complex > complex ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Finite____Field__Omod____ring_Itf__a_J,type,
uminus3100561713750211260ring_a: finite_mod_ring_a > finite_mod_ring_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Finite____Field__Omod____ring_Itf__a_J,type,
zero_z7902377541816115708ring_a: finite_mod_ring_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Int__Oint,type,
groups5690904116761175830ex_int: ( complex > int ) > set_complex > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Nat__Onat,type,
groups5693394587270226106ex_nat: ( complex > nat ) > set_complex > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Int__Oint,type,
groups4538972089207619220nt_int: ( int > int ) > set_int > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Nat__Onat,type,
groups4541462559716669496nt_nat: ( int > nat ) > set_int > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Real__Oreal,type,
groups8778361861064173332t_real: ( int > real ) > set_int > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Complex__Ocomplex,type,
groups2073611262835488442omplex: ( nat > complex ) > set_nat > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Finite____Field__Omod____ring_Itf__a_J,type,
groups3558780024651037881ring_a: ( nat > finite_mod_ring_a ) > set_nat > finite_mod_ring_a ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Complex__Ocomplex,type,
groups5754745047067104278omplex: ( real > complex ) > set_real > complex ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Finite____Field__Omod____ring_Itf__a_J,type,
groups465414945397457501ring_a: ( real > finite_mod_ring_a ) > set_real > finite_mod_ring_a ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Int__Oint,type,
groups1932886352136224148al_int: ( real > int ) > set_real > int ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Nat__Onat,type,
groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
groups8097168146408367636l_real: ( real > real ) > set_real > real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
semiri8010041392384452111omplex: nat > complex ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Finite____Field__Omod____ring_Itf__a_J,type,
semiri9180929696517417892ring_a: nat > finite_mod_ring_a ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
power_power_complex: complex > nat > complex ).
thf(sy_c_Power_Opower__class_Opower_001t__Finite____Field__Omod____ring_Itf__a_J,type,
power_6826135765519566523ring_a: finite_mod_ring_a > nat > finite_mod_ring_a ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal,type,
set_or66887138388493659n_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_f,type,
f: nat > finite_mod_ring_a ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_r1,type,
r1: nat ).
thf(sy_v_r2,type,
r2: nat ).
% Relevant facts (1267)
thf(fact_0_sum__swap,axiom,
! [F: nat > nat > finite_mod_ring_a,Y: nat,X: nat] :
( ( groups3558780024651037881ring_a
@ ^ [I: nat] : ( groups3558780024651037881ring_a @ ( F @ I ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ Y ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ X ) )
= ( groups3558780024651037881ring_a
@ ^ [J: nat] :
( groups3558780024651037881ring_a
@ ^ [I: nat] : ( F @ I @ J )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ X ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ Y ) ) ) ).
% sum_swap
thf(fact_1_sum__swap,axiom,
! [F: nat > nat > nat,Y: nat,X: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( groups3542108847815614940at_nat @ ( F @ I ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ Y ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ X ) )
= ( groups3542108847815614940at_nat
@ ^ [J: nat] :
( groups3542108847815614940at_nat
@ ^ [I: nat] : ( F @ I @ J )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ X ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ Y ) ) ) ).
% sum_swap
thf(fact_2_sum__eq,axiom,
! [X: nat,F: nat > finite_mod_ring_a,G: nat > finite_mod_ring_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ X )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( groups3558780024651037881ring_a @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ X ) )
= ( groups3558780024651037881ring_a @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ X ) ) ) ) ).
% sum_eq
thf(fact_3_sum__eq,axiom,
! [X: nat,F: nat > nat,G: nat > nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ X )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ X ) )
= ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ X ) ) ) ) ).
% sum_eq
thf(fact_4_k__bound,axiom,
ord_less_nat @ zero_zero_nat @ k ).
% k_bound
thf(fact_5_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_6_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_7_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_8_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_9_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_10_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_11_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_12_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_13_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_14_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_15_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_16_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_17_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_18_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_19_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_20_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_21_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups3558780024651037881ring_a
@ ^ [Uu: nat] : zero_z7902377541816115708ring_a
@ A2 )
= zero_z7902377541816115708ring_a ) ).
% sum.neutral_const
thf(fact_22_sum_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu: nat] : zero_zero_nat
@ A2 )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_23_sum_Oneutral__const,axiom,
! [A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [Uu: complex] : zero_zero_complex
@ A2 )
= zero_zero_complex ) ).
% sum.neutral_const
thf(fact_24_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_25_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_26_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_27_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_28_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_29_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_30_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_31_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_32_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_33_add__right__cancel,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ B @ A )
= ( plus_p6165643967897163644ring_a @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_34_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_35_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_36_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_37_add__left__cancel,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ A @ B )
= ( plus_p6165643967897163644ring_a @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_38_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_39_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_40_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_41_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_42_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_43_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_44_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_45_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_46_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_47_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_48_add__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add_0
thf(fact_49_add__0,axiom,
! [A: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ zero_z7902377541816115708ring_a @ A )
= A ) ).
% add_0
thf(fact_50_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_51_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_52_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_53_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_54_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_55_add__cancel__right__right,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ A @ B ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_right
thf(fact_56_add__cancel__right__right,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( A
= ( plus_p6165643967897163644ring_a @ A @ B ) )
= ( B = zero_z7902377541816115708ring_a ) ) ).
% add_cancel_right_right
thf(fact_57_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_58_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_59_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_60_add__cancel__right__left,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ B @ A ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_left
thf(fact_61_add__cancel__right__left,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( A
= ( plus_p6165643967897163644ring_a @ B @ A ) )
= ( B = zero_z7902377541816115708ring_a ) ) ).
% add_cancel_right_left
thf(fact_62_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_63_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_64_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_65_add__cancel__left__right,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_right
thf(fact_66_add__cancel__left__right,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ A @ B )
= A )
= ( B = zero_z7902377541816115708ring_a ) ) ).
% add_cancel_left_right
thf(fact_67_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_68_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_69_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_70_add__cancel__left__left,axiom,
! [B: complex,A: complex] :
( ( ( plus_plus_complex @ B @ A )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_left
thf(fact_71_add__cancel__left__left,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ B @ A )
= A )
= ( B = zero_z7902377541816115708ring_a ) ) ).
% add_cancel_left_left
thf(fact_72_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_73_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_74_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_75_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_76_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_77_add_Oright__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.right_neutral
thf(fact_78_add_Oright__neutral,axiom,
! [A: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ A @ zero_z7902377541816115708ring_a )
= A ) ).
% add.right_neutral
thf(fact_79_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 ) ) ) ).
% add_gr_0
thf(fact_80_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_81_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_82_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_83_trans__less__add2,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_less_add2
thf(fact_84_trans__less__add1,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_less_add1
thf(fact_85_add__less__mono1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_86_not__add__less2,axiom,
! [J2: nat,I3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I3 ) @ I3 ) ).
% not_add_less2
thf(fact_87_not__add__less1,axiom,
! [I3: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ I3 ) ).
% not_add_less1
thf(fact_88_add__less__mono,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_89_add__lessD1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K )
=> ( ord_less_nat @ I3 @ K ) ) ).
% add_lessD1
thf(fact_90_less__imp__add__positive,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I3 @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_91_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_92_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_93_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_94_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_95_zero__reorient,axiom,
! [X: finite_mod_ring_a] :
( ( zero_z7902377541816115708ring_a = X )
= ( X = zero_z7902377541816115708ring_a ) ) ).
% zero_reorient
thf(fact_96_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_97_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_98_mem__Collect__eq,axiom,
! [A: complex,P: complex > $o] :
( ( member_complex @ A @ ( collect_complex @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_99_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_100_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_101_Collect__mem__eq,axiom,
! [A2: set_complex] :
( ( collect_complex
@ ^ [X2: complex] : ( member_complex @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_102_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_103_Collect__cong,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X3: complex] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_complex @ P )
= ( collect_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_104_add__right__imp__eq,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ B @ A )
= ( plus_p6165643967897163644ring_a @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_105_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_106_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_107_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_108_add__left__imp__eq,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ A @ B )
= ( plus_p6165643967897163644ring_a @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_109_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_110_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_111_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_112_add_Oleft__commute,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ B @ ( plus_p6165643967897163644ring_a @ A @ C ) )
= ( plus_p6165643967897163644ring_a @ A @ ( plus_p6165643967897163644ring_a @ B @ C ) ) ) ).
% add.left_commute
thf(fact_113_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_114_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_115_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_116_add_Ocommute,axiom,
( plus_p6165643967897163644ring_a
= ( ^ [A3: finite_mod_ring_a,B2: finite_mod_ring_a] : ( plus_p6165643967897163644ring_a @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_117_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_118_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_119_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A3: real,B2: real] : ( plus_plus_real @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_120_add_Oright__cancel,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ B @ A )
= ( plus_p6165643967897163644ring_a @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_121_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_122_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_123_add_Oleft__cancel,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ A @ B )
= ( plus_p6165643967897163644ring_a @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_124_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_125_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_126_add_Oassoc,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ ( plus_p6165643967897163644ring_a @ A @ B ) @ C )
= ( plus_p6165643967897163644ring_a @ A @ ( plus_p6165643967897163644ring_a @ B @ C ) ) ) ).
% add.assoc
thf(fact_127_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_128_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_129_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_130_group__cancel_Oadd2,axiom,
! [B3: finite_mod_ring_a,K: finite_mod_ring_a,B: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( B3
= ( plus_p6165643967897163644ring_a @ K @ B ) )
=> ( ( plus_p6165643967897163644ring_a @ A @ B3 )
= ( plus_p6165643967897163644ring_a @ K @ ( plus_p6165643967897163644ring_a @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_131_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_132_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_133_group__cancel_Oadd2,axiom,
! [B3: real,K: real,B: real,A: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_134_group__cancel_Oadd1,axiom,
! [A2: finite_mod_ring_a,K: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( A2
= ( plus_p6165643967897163644ring_a @ K @ A ) )
=> ( ( plus_p6165643967897163644ring_a @ A2 @ B )
= ( plus_p6165643967897163644ring_a @ K @ ( plus_p6165643967897163644ring_a @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_135_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_136_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_137_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_138_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ( I3 = J2 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I3 @ K )
= ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_139_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: int,J2: int,K: int,L: int] :
( ( ( I3 = J2 )
& ( K = L ) )
=> ( ( plus_plus_int @ I3 @ K )
= ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_140_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: real,J2: real,K: real,L: real] :
( ( ( I3 = J2 )
& ( K = L ) )
=> ( ( plus_plus_real @ I3 @ K )
= ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_141_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ ( plus_p6165643967897163644ring_a @ A @ B ) @ C )
= ( plus_p6165643967897163644ring_a @ A @ ( plus_p6165643967897163644ring_a @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_142_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_143_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_144_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_145_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_146_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_147_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_148_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_149_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_150_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_151_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_152_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_153_sum_Oreindex__bij__witness,axiom,
! [S2: set_real,I3: nat > real,J2: real > nat,T2: set_nat,H: nat > nat,G: real > nat] :
( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( member_nat @ ( J2 @ A4 ) @ T2 ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T2 )
=> ( ( J2 @ ( I3 @ B4 ) )
= B4 ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T2 )
=> ( member_real @ ( I3 @ B4 ) @ S2 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ S2 )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_154_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I3: complex > nat,J2: nat > complex,T2: set_complex,H: complex > complex,G: nat > complex] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( member_complex @ ( J2 @ A4 ) @ T2 ) )
=> ( ! [B4: complex] :
( ( member_complex @ B4 @ T2 )
=> ( ( J2 @ ( I3 @ B4 ) )
= B4 ) )
=> ( ! [B4: complex] :
( ( member_complex @ B4 @ T2 )
=> ( member_nat @ ( I3 @ B4 ) @ S2 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups2073611262835488442omplex @ G @ S2 )
= ( groups7754918857620584856omplex @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_155_sum_Oreindex__bij__witness,axiom,
! [S2: set_real,I3: complex > real,J2: real > complex,T2: set_complex,H: complex > complex,G: real > complex] :
( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( member_complex @ ( J2 @ A4 ) @ T2 ) )
=> ( ! [B4: complex] :
( ( member_complex @ B4 @ T2 )
=> ( ( J2 @ ( I3 @ B4 ) )
= B4 ) )
=> ( ! [B4: complex] :
( ( member_complex @ B4 @ T2 )
=> ( member_real @ ( I3 @ B4 ) @ S2 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups5754745047067104278omplex @ G @ S2 )
= ( groups7754918857620584856omplex @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_156_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I3: real > nat,J2: nat > real,T2: set_real,H: real > nat,G: nat > nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( member_real @ ( J2 @ A4 ) @ T2 ) )
=> ( ! [B4: real] :
( ( member_real @ B4 @ T2 )
=> ( ( J2 @ ( I3 @ B4 ) )
= B4 ) )
=> ( ! [B4: real] :
( ( member_real @ B4 @ T2 )
=> ( member_nat @ ( I3 @ B4 ) @ S2 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S2 )
= ( groups1935376822645274424al_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_157_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I3: nat > nat,J2: nat > nat,T2: set_nat,H: nat > nat,G: nat > nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( member_nat @ ( J2 @ A4 ) @ T2 ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T2 )
=> ( ( J2 @ ( I3 @ B4 ) )
= B4 ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T2 )
=> ( member_nat @ ( I3 @ B4 ) @ S2 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S2 )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_158_sum_Oreindex__bij__witness,axiom,
! [S2: set_complex,I3: nat > complex,J2: complex > nat,T2: set_nat,H: nat > complex,G: complex > complex] :
( ! [A4: complex] :
( ( member_complex @ A4 @ S2 )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: complex] :
( ( member_complex @ A4 @ S2 )
=> ( member_nat @ ( J2 @ A4 ) @ T2 ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T2 )
=> ( ( J2 @ ( I3 @ B4 ) )
= B4 ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T2 )
=> ( member_complex @ ( I3 @ B4 ) @ S2 ) )
=> ( ! [A4: complex] :
( ( member_complex @ A4 @ S2 )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7754918857620584856omplex @ G @ S2 )
= ( groups2073611262835488442omplex @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_159_sum_Oreindex__bij__witness,axiom,
! [S2: set_complex,I3: real > complex,J2: complex > real,T2: set_real,H: real > complex,G: complex > complex] :
( ! [A4: complex] :
( ( member_complex @ A4 @ S2 )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: complex] :
( ( member_complex @ A4 @ S2 )
=> ( member_real @ ( J2 @ A4 ) @ T2 ) )
=> ( ! [B4: real] :
( ( member_real @ B4 @ T2 )
=> ( ( J2 @ ( I3 @ B4 ) )
= B4 ) )
=> ( ! [B4: real] :
( ( member_real @ B4 @ T2 )
=> ( member_complex @ ( I3 @ B4 ) @ S2 ) )
=> ( ! [A4: complex] :
( ( member_complex @ A4 @ S2 )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7754918857620584856omplex @ G @ S2 )
= ( groups5754745047067104278omplex @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_160_sum_Oreindex__bij__witness,axiom,
! [S2: set_complex,I3: complex > complex,J2: complex > complex,T2: set_complex,H: complex > complex,G: complex > complex] :
( ! [A4: complex] :
( ( member_complex @ A4 @ S2 )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: complex] :
( ( member_complex @ A4 @ S2 )
=> ( member_complex @ ( J2 @ A4 ) @ T2 ) )
=> ( ! [B4: complex] :
( ( member_complex @ B4 @ T2 )
=> ( ( J2 @ ( I3 @ B4 ) )
= B4 ) )
=> ( ! [B4: complex] :
( ( member_complex @ B4 @ T2 )
=> ( member_complex @ ( I3 @ B4 ) @ S2 ) )
=> ( ! [A4: complex] :
( ( member_complex @ A4 @ S2 )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7754918857620584856omplex @ G @ S2 )
= ( groups7754918857620584856omplex @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_161_sum_Oreindex__bij__witness,axiom,
! [S2: set_real,I3: nat > real,J2: real > nat,T2: set_nat,H: nat > finite_mod_ring_a,G: real > finite_mod_ring_a] :
( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( member_nat @ ( J2 @ A4 ) @ T2 ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T2 )
=> ( ( J2 @ ( I3 @ B4 ) )
= B4 ) )
=> ( ! [B4: nat] :
( ( member_nat @ B4 @ T2 )
=> ( member_real @ ( I3 @ B4 ) @ S2 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups465414945397457501ring_a @ G @ S2 )
= ( groups3558780024651037881ring_a @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_162_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I3: real > nat,J2: nat > real,T2: set_real,H: real > finite_mod_ring_a,G: nat > finite_mod_ring_a] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( I3 @ ( J2 @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( member_real @ ( J2 @ A4 ) @ T2 ) )
=> ( ! [B4: real] :
( ( member_real @ B4 @ T2 )
=> ( ( J2 @ ( I3 @ B4 ) )
= B4 ) )
=> ( ! [B4: real] :
( ( member_real @ B4 @ T2 )
=> ( member_nat @ ( I3 @ B4 ) @ S2 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( H @ ( J2 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups3558780024651037881ring_a @ G @ S2 )
= ( groups465414945397457501ring_a @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_163_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > real,A2: set_real,H: real > nat,Gamma: nat > nat,Phi: real > nat] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ( ( member_real @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups1935376822645274424al_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_164_sum_Oeq__general__inverses,axiom,
! [B3: set_complex,K: complex > nat,A2: set_nat,H: nat > complex,Gamma: complex > complex,Phi: nat > complex] :
( ! [Y2: complex] :
( ( member_complex @ Y2 @ B3 )
=> ( ( member_nat @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_complex @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups2073611262835488442omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_165_sum_Oeq__general__inverses,axiom,
! [B3: set_complex,K: complex > real,A2: set_real,H: real > complex,Gamma: complex > complex,Phi: real > complex] :
( ! [Y2: complex] :
( ( member_complex @ Y2 @ B3 )
=> ( ( member_real @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_complex @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups5754745047067104278omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_166_sum_Oeq__general__inverses,axiom,
! [B3: set_real,K: real > nat,A2: set_nat,H: nat > real,Gamma: real > nat,Phi: nat > nat] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B3 )
=> ( ( member_nat @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_real @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups1935376822645274424al_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_167_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > nat,A2: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ( ( member_nat @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_168_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > complex,A2: set_complex,H: complex > nat,Gamma: nat > complex,Phi: complex > complex] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ( ( member_complex @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups2073611262835488442omplex @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_169_sum_Oeq__general__inverses,axiom,
! [B3: set_real,K: real > complex,A2: set_complex,H: complex > real,Gamma: real > complex,Phi: complex > complex] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B3 )
=> ( ( member_complex @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( member_real @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups5754745047067104278omplex @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_170_sum_Oeq__general__inverses,axiom,
! [B3: set_complex,K: complex > complex,A2: set_complex,H: complex > complex,Gamma: complex > complex,Phi: complex > complex] :
( ! [Y2: complex] :
( ( member_complex @ Y2 @ B3 )
=> ( ( member_complex @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( member_complex @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_171_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > real,A2: set_real,H: real > nat,Gamma: nat > finite_mod_ring_a,Phi: real > finite_mod_ring_a] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ( ( member_real @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups465414945397457501ring_a @ Phi @ A2 )
= ( groups3558780024651037881ring_a @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_172_sum_Oeq__general__inverses,axiom,
! [B3: set_real,K: real > nat,A2: set_nat,H: nat > real,Gamma: real > finite_mod_ring_a,Phi: nat > finite_mod_ring_a] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B3 )
=> ( ( member_nat @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_real @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3558780024651037881ring_a @ Phi @ A2 )
= ( groups465414945397457501ring_a @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_173_sum_Oeq__general,axiom,
! [B3: set_nat,A2: set_real,H: real > nat,Gamma: nat > nat,Phi: real > nat] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups1935376822645274424al_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_174_sum_Oeq__general,axiom,
! [B3: set_complex,A2: set_nat,H: nat > complex,Gamma: complex > complex,Phi: nat > complex] :
( ! [Y2: complex] :
( ( member_complex @ Y2 @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_complex @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups2073611262835488442omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_175_sum_Oeq__general,axiom,
! [B3: set_complex,A2: set_real,H: real > complex,Gamma: complex > complex,Phi: real > complex] :
( ! [Y2: complex] :
( ( member_complex @ Y2 @ B3 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_complex @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups5754745047067104278omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_176_sum_Oeq__general,axiom,
! [B3: set_real,A2: set_nat,H: nat > real,Gamma: real > nat,Phi: nat > nat] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_real @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups1935376822645274424al_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_177_sum_Oeq__general,axiom,
! [B3: set_nat,A2: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A2 )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_178_sum_Oeq__general,axiom,
! [B3: set_nat,A2: set_complex,H: complex > nat,Gamma: nat > complex,Phi: complex > complex] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ? [X4: complex] :
( ( member_complex @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: complex] :
( ( ( member_complex @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups2073611262835488442omplex @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_179_sum_Oeq__general,axiom,
! [B3: set_real,A2: set_complex,H: complex > real,Gamma: real > complex,Phi: complex > complex] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B3 )
=> ? [X4: complex] :
( ( member_complex @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: complex] :
( ( ( member_complex @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( member_real @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups5754745047067104278omplex @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_180_sum_Oeq__general,axiom,
! [B3: set_complex,A2: set_complex,H: complex > complex,Gamma: complex > complex,Phi: complex > complex] :
( ! [Y2: complex] :
( ( member_complex @ Y2 @ B3 )
=> ? [X4: complex] :
( ( member_complex @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: complex] :
( ( ( member_complex @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( member_complex @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups7754918857620584856omplex @ Phi @ A2 )
= ( groups7754918857620584856omplex @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_181_sum_Oeq__general,axiom,
! [B3: set_nat,A2: set_real,H: real > nat,Gamma: nat > finite_mod_ring_a,Phi: real > finite_mod_ring_a] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups465414945397457501ring_a @ Phi @ A2 )
= ( groups3558780024651037881ring_a @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_182_sum_Oeq__general,axiom,
! [B3: set_real,A2: set_nat,H: nat > real,Gamma: real > finite_mod_ring_a,Phi: nat > finite_mod_ring_a] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_real @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3558780024651037881ring_a @ Phi @ A2 )
= ( groups465414945397457501ring_a @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_183_sum_Ocong,axiom,
! [A2: set_nat,B3: set_nat,G: nat > finite_mod_ring_a,H: nat > finite_mod_ring_a] :
( ( A2 = B3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B3 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups3558780024651037881ring_a @ G @ A2 )
= ( groups3558780024651037881ring_a @ H @ B3 ) ) ) ) ).
% sum.cong
thf(fact_184_sum_Ocong,axiom,
! [A2: set_nat,B3: set_nat,G: nat > nat,H: nat > nat] :
( ( A2 = B3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B3 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= ( groups3542108847815614940at_nat @ H @ B3 ) ) ) ) ).
% sum.cong
thf(fact_185_sum_Ocong,axiom,
! [A2: set_complex,B3: set_complex,G: complex > complex,H: complex > complex] :
( ( A2 = B3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ B3 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7754918857620584856omplex @ G @ A2 )
= ( groups7754918857620584856omplex @ H @ B3 ) ) ) ) ).
% sum.cong
thf(fact_186_sum_Oswap,axiom,
! [G: nat > nat > finite_mod_ring_a,B3: set_nat,A2: set_nat] :
( ( groups3558780024651037881ring_a
@ ^ [I: nat] : ( groups3558780024651037881ring_a @ ( G @ I ) @ B3 )
@ A2 )
= ( groups3558780024651037881ring_a
@ ^ [J: nat] :
( groups3558780024651037881ring_a
@ ^ [I: nat] : ( G @ I @ J )
@ A2 )
@ B3 ) ) ).
% sum.swap
thf(fact_187_sum_Oswap,axiom,
! [G: nat > nat > nat,B3: set_nat,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( groups3542108847815614940at_nat @ ( G @ I ) @ B3 )
@ A2 )
= ( groups3542108847815614940at_nat
@ ^ [J: nat] :
( groups3542108847815614940at_nat
@ ^ [I: nat] : ( G @ I @ J )
@ A2 )
@ B3 ) ) ).
% sum.swap
thf(fact_188_sum_Oswap,axiom,
! [G: complex > complex > complex,B3: set_complex,A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [I: complex] : ( groups7754918857620584856omplex @ ( G @ I ) @ B3 )
@ A2 )
= ( groups7754918857620584856omplex
@ ^ [J: complex] :
( groups7754918857620584856omplex
@ ^ [I: complex] : ( G @ I @ J )
@ A2 )
@ B3 ) ) ).
% sum.swap
thf(fact_189_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_190_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_191_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_192_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_193_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_194_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_195_add_Ogroup__left__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add.group_left_neutral
thf(fact_196_add_Ogroup__left__neutral,axiom,
! [A: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ zero_z7902377541816115708ring_a @ A )
= A ) ).
% add.group_left_neutral
thf(fact_197_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_198_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_199_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_200_add_Ocomm__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.comm_neutral
thf(fact_201_add_Ocomm__neutral,axiom,
! [A: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ A @ zero_z7902377541816115708ring_a )
= A ) ).
% add.comm_neutral
thf(fact_202_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_203_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_204_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_205_comm__monoid__add__class_Oadd__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_206_comm__monoid__add__class_Oadd__0,axiom,
! [A: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ zero_z7902377541816115708ring_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_207_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_208_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_209_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_210_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_211_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_212_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_213_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_214_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_215_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_216_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_217_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_218_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_219_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_220_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_221_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_222_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J2 )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_223_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: int,J2: int,K: int,L: int] :
( ( ( ord_less_int @ I3 @ J2 )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_224_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: real,J2: real,K: real,L: real] :
( ( ( ord_less_real @ I3 @ J2 )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_225_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ( I3 = J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_226_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: int,J2: int,K: int,L: int] :
( ( ( I3 = J2 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_227_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: real,J2: real,K: real,L: real] :
( ( ( I3 = J2 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_228_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_229_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: int,J2: int,K: int,L: int] :
( ( ( ord_less_int @ I3 @ J2 )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_230_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: real,J2: real,K: real,L: real] :
( ( ( ord_less_real @ I3 @ J2 )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_231_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A2: set_real] :
( ( ( groups1935376822645274424al_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_232_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > int,A2: set_nat] :
( ( ( groups3539618377306564664at_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_233_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > int,A2: set_real] :
( ( ( groups1932886352136224148al_int @ G @ A2 )
!= zero_zero_int )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_int ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_234_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A2: set_nat] :
( ( ( groups6591440286371151544t_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_235_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A2: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_236_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > complex,A2: set_nat] :
( ( ( groups2073611262835488442omplex @ G @ A2 )
!= zero_zero_complex )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_237_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > complex,A2: set_real] :
( ( ( groups5754745047067104278omplex @ G @ A2 )
!= zero_zero_complex )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_238_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A2: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_239_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > complex,A2: set_complex] :
( ( ( groups7754918857620584856omplex @ G @ A2 )
!= zero_zero_complex )
=> ~ ! [A4: complex] :
( ( member_complex @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_complex ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_240_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > finite_mod_ring_a,A2: set_real] :
( ( ( groups465414945397457501ring_a @ G @ A2 )
!= zero_z7902377541816115708ring_a )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_z7902377541816115708ring_a ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_241_sum_Oneutral,axiom,
! [A2: set_nat,G: nat > finite_mod_ring_a] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_z7902377541816115708ring_a ) )
=> ( ( groups3558780024651037881ring_a @ G @ A2 )
= zero_z7902377541816115708ring_a ) ) ).
% sum.neutral
thf(fact_242_sum_Oneutral,axiom,
! [A2: set_nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_243_sum_Oneutral,axiom,
! [A2: set_complex,G: complex > complex] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_zero_complex ) )
=> ( ( groups7754918857620584856omplex @ G @ A2 )
= zero_zero_complex ) ) ).
% sum.neutral
thf(fact_244_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_245_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_246_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_247_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_248_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_249_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_250_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_251_sum_Odistrib,axiom,
! [G: nat > finite_mod_ring_a,H: nat > finite_mod_ring_a,A2: set_nat] :
( ( groups3558780024651037881ring_a
@ ^ [X2: nat] : ( plus_p6165643967897163644ring_a @ ( G @ X2 ) @ ( H @ X2 ) )
@ A2 )
= ( plus_p6165643967897163644ring_a @ ( groups3558780024651037881ring_a @ G @ A2 ) @ ( groups3558780024651037881ring_a @ H @ A2 ) ) ) ).
% sum.distrib
thf(fact_252_sum_Odistrib,axiom,
! [G: nat > nat,H: nat > nat,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( plus_plus_nat @ ( G @ X2 ) @ ( H @ X2 ) )
@ A2 )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H @ A2 ) ) ) ).
% sum.distrib
thf(fact_253_sum_Odistrib,axiom,
! [G: complex > complex,H: complex > complex,A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [X2: complex] : ( plus_plus_complex @ ( G @ X2 ) @ ( H @ X2 ) )
@ A2 )
= ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H @ A2 ) ) ) ).
% sum.distrib
thf(fact_254_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_255_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_256_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_257_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_258_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_259_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_260_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_261_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_262_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_263_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_264_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_265_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_266_homomorphism__add,axiom,
! [X: int,Y: int] :
( ( plus_p6165643967897163644ring_a @ ( finite8272632373135393572ring_a @ X ) @ ( finite8272632373135393572ring_a @ Y ) )
= ( finite8272632373135393572ring_a @ ( plus_plus_int @ X @ Y ) ) ) ).
% homomorphism_add
thf(fact_267_sum__in,axiom,
! [F: nat > finite_mod_ring_a,Y: finite_mod_ring_a,X: nat] :
( ( groups3558780024651037881ring_a
@ ^ [I: nat] : ( times_5121417576591743744ring_a @ ( F @ I ) @ Y )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ X ) )
= ( times_5121417576591743744ring_a @ ( groups3558780024651037881ring_a @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ X ) ) @ Y ) ) ).
% sum_in
thf(fact_268_sum__diff__in,axiom,
! [F: nat > finite_mod_ring_a,X: nat,G: nat > finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ ( groups3558780024651037881ring_a @ F @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ X ) ) @ ( groups3558780024651037881ring_a @ G @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ X ) ) )
= ( groups3558780024651037881ring_a
@ ^ [I: nat] : ( minus_3609261664126569004ring_a @ ( F @ I ) @ ( G @ I ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ X ) ) ) ).
% sum_diff_in
thf(fact_269_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( P @ M3 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_270_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_nat @ M3 @ N )
& ( P @ M3 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_271_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_272_add__less__zeroD,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_273_sum_Oshift__bounds__nat__ivl,axiom,
! [G: nat > finite_mod_ring_a,M: nat,K: nat,N: nat] :
( ( groups3558780024651037881ring_a @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups3558780024651037881ring_a
@ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_274_sum_Oshift__bounds__nat__ivl,axiom,
! [G: nat > nat,M: nat,K: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K ) )
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_275_atLeastLessThan__eq__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_276_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_277_atLeastLessThan__eq__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_278_Ico__eq__Ico,axiom,
! [L: real,H: real,L2: real,H2: real] :
( ( ( set_or66887138388493659n_real @ L @ H )
= ( set_or66887138388493659n_real @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_real @ L @ H )
& ~ ( ord_less_real @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_279_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_280_Ico__eq__Ico,axiom,
! [L: int,H: int,L2: int,H2: int] :
( ( ( set_or4662586982721622107an_int @ L @ H )
= ( set_or4662586982721622107an_int @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_int @ L @ H )
& ~ ( ord_less_int @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_281_atLeastLessThan__inj_I1_J,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_282_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_283_atLeastLessThan__inj_I1_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_284_atLeastLessThan__inj_I2_J,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( set_or66887138388493659n_real @ A @ B )
= ( set_or66887138388493659n_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_285_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_286_atLeastLessThan__inj_I2_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_287_mult__zero__left,axiom,
! [A: complex] :
( ( times_times_complex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_288_mult__zero__left,axiom,
! [A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ zero_z7902377541816115708ring_a @ A )
= zero_z7902377541816115708ring_a ) ).
% mult_zero_left
thf(fact_289_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_290_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_291_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_292_mult__zero__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_293_mult__zero__right,axiom,
! [A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A @ zero_z7902377541816115708ring_a )
= zero_z7902377541816115708ring_a ) ).
% mult_zero_right
thf(fact_294_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_295_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_296_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_297_mult__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_298_mult__eq__0__iff,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A @ B )
= zero_z7902377541816115708ring_a )
= ( ( A = zero_z7902377541816115708ring_a )
| ( B = zero_z7902377541816115708ring_a ) ) ) ).
% mult_eq_0_iff
thf(fact_299_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_300_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_301_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_302_mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( times_times_complex @ C @ A )
= ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_303_mult__cancel__left,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ C @ A )
= ( times_5121417576591743744ring_a @ C @ B ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_304_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_305_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_306_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_307_mult__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( times_times_complex @ A @ C )
= ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_308_mult__cancel__right,axiom,
! [A: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A @ C )
= ( times_5121417576591743744ring_a @ B @ C ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_309_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_310_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_311_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_312_diff__self,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% diff_self
thf(fact_313_diff__self,axiom,
! [A: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ A @ A )
= zero_z7902377541816115708ring_a ) ).
% diff_self
thf(fact_314_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_315_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_316_diff__0__right,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_0_right
thf(fact_317_diff__0__right,axiom,
! [A: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ A @ zero_z7902377541816115708ring_a )
= A ) ).
% diff_0_right
thf(fact_318_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_319_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_320_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_321_diff__zero,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% diff_zero
thf(fact_322_diff__zero,axiom,
! [A: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ A @ zero_z7902377541816115708ring_a )
= A ) ).
% diff_zero
thf(fact_323_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_324_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_325_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_326_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_327_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ A @ A )
= zero_z7902377541816115708ring_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_328_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_329_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_330_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_331_add__diff__cancel,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ ( plus_p6165643967897163644ring_a @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_332_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_333_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_334_diff__add__cancel,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ ( minus_3609261664126569004ring_a @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_335_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_336_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_337_add__diff__cancel__left,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ ( plus_p6165643967897163644ring_a @ C @ A ) @ ( plus_p6165643967897163644ring_a @ C @ B ) )
= ( minus_3609261664126569004ring_a @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_338_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_339_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_340_add__diff__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_341_add__diff__cancel__left_H,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ ( plus_p6165643967897163644ring_a @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_342_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_343_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_344_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_345_add__diff__cancel__right,axiom,
! [A: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ ( plus_p6165643967897163644ring_a @ A @ C ) @ ( plus_p6165643967897163644ring_a @ B @ C ) )
= ( minus_3609261664126569004ring_a @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_346_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_347_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_348_add__diff__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_349_add__diff__cancel__right_H,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ ( plus_p6165643967897163644ring_a @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_350_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_351_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_352_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_353_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_354_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_355_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_356_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_357_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_358_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ C )
= ( times_5121417576591743744ring_a @ A @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_359_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_360_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_361_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_362_eq__add__iff1,axiom,
! [A: finite_mod_ring_a,E: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a,D: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A @ E ) @ C )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ B @ E ) @ D ) )
= ( ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ ( minus_3609261664126569004ring_a @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_363_eq__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_364_eq__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_365_eq__add__iff2,axiom,
! [A: finite_mod_ring_a,E: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a,D: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A @ E ) @ C )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ B @ E ) @ D ) )
= ( C
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ ( minus_3609261664126569004ring_a @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_366_eq__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_367_eq__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( C
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_368_diff__eq__diff__eq,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a,D: finite_mod_ring_a] :
( ( ( minus_3609261664126569004ring_a @ A @ B )
= ( minus_3609261664126569004ring_a @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_369_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_370_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_371_mult_Oassoc,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ C )
= ( times_5121417576591743744ring_a @ A @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% mult.assoc
thf(fact_372_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_373_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_374_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_375_mult_Ocommute,axiom,
( times_5121417576591743744ring_a
= ( ^ [A3: finite_mod_ring_a,B2: finite_mod_ring_a] : ( times_5121417576591743744ring_a @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_376_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_377_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_378_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_379_mult_Oleft__commute,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ B @ ( times_5121417576591743744ring_a @ A @ C ) )
= ( times_5121417576591743744ring_a @ A @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_380_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_381_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_382_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_383_square__diff__square__factored,axiom,
! [X: finite_mod_ring_a,Y: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ X @ X ) @ ( times_5121417576591743744ring_a @ Y @ Y ) )
= ( times_5121417576591743744ring_a @ ( plus_p6165643967897163644ring_a @ X @ Y ) @ ( minus_3609261664126569004ring_a @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_384_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_385_square__diff__square__factored,axiom,
! [X: real,Y: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
= ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_386_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ ( minus_3609261664126569004ring_a @ A @ C ) @ B )
= ( minus_3609261664126569004ring_a @ ( minus_3609261664126569004ring_a @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_387_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_388_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_389_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_390_right__diff__distrib_H,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A @ ( minus_3609261664126569004ring_a @ B @ C ) )
= ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ ( times_5121417576591743744ring_a @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_391_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_392_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_393_right__diff__distrib_H,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_394_left__diff__distrib_H,axiom,
! [B: finite_mod_ring_a,C: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( minus_3609261664126569004ring_a @ B @ C ) @ A )
= ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ B @ A ) @ ( times_5121417576591743744ring_a @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_395_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_396_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_397_left__diff__distrib_H,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
= ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_398_right__diff__distrib,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A @ ( minus_3609261664126569004ring_a @ B @ C ) )
= ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ ( times_5121417576591743744ring_a @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_399_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_400_right__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_401_left__diff__distrib,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( minus_3609261664126569004ring_a @ A @ B ) @ C )
= ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ A @ C ) @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_402_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_403_left__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_404_mult__not__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
!= zero_zero_complex )
=> ( ( A != zero_zero_complex )
& ( B != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_405_mult__not__zero,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A @ B )
!= zero_z7902377541816115708ring_a )
=> ( ( A != zero_z7902377541816115708ring_a )
& ( B != zero_z7902377541816115708ring_a ) ) ) ).
% mult_not_zero
thf(fact_406_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_407_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_408_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_409_divisors__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
=> ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_410_divisors__zero,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A @ B )
= zero_z7902377541816115708ring_a )
=> ( ( A = zero_z7902377541816115708ring_a )
| ( B = zero_z7902377541816115708ring_a ) ) ) ).
% divisors_zero
thf(fact_411_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_412_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_413_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_414_no__zero__divisors,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( times_times_complex @ A @ B )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_415_no__zero__divisors,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( A != zero_z7902377541816115708ring_a )
=> ( ( B != zero_z7902377541816115708ring_a )
=> ( ( times_5121417576591743744ring_a @ A @ B )
!= zero_z7902377541816115708ring_a ) ) ) ).
% no_zero_divisors
thf(fact_416_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_417_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_418_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_419_mult__left__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ C @ A )
= ( times_times_complex @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_420_mult__left__cancel,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( ( times_5121417576591743744ring_a @ C @ A )
= ( times_5121417576591743744ring_a @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_421_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_422_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_423_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_424_mult__right__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ C )
= ( times_times_complex @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_425_mult__right__cancel,axiom,
! [C: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( ( times_5121417576591743744ring_a @ A @ C )
= ( times_5121417576591743744ring_a @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_426_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_427_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_428_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_429_ring__class_Oring__distribs_I2_J,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( plus_p6165643967897163644ring_a @ A @ B ) @ C )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A @ C ) @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_430_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_431_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_432_ring__class_Oring__distribs_I1_J,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A @ ( plus_p6165643967897163644ring_a @ B @ C ) )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ ( times_5121417576591743744ring_a @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_433_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_434_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_435_comm__semiring__class_Odistrib,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( plus_p6165643967897163644ring_a @ A @ B ) @ C )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A @ C ) @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_436_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_437_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_438_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_439_distrib__left,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A @ ( plus_p6165643967897163644ring_a @ B @ C ) )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A @ B ) @ ( times_5121417576591743744ring_a @ A @ C ) ) ) ).
% distrib_left
thf(fact_440_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_441_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_442_distrib__left,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% distrib_left
thf(fact_443_distrib__right,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( plus_p6165643967897163644ring_a @ A @ B ) @ C )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A @ C ) @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% distrib_right
thf(fact_444_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_445_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_446_distrib__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_447_combine__common__factor,axiom,
! [A: finite_mod_ring_a,E: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A @ E ) @ ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ B @ E ) @ C ) )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ ( plus_p6165643967897163644ring_a @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_448_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_449_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_450_combine__common__factor,axiom,
! [A: real,E: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_451_less__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_452_less__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_453_less__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_454_less__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_455_lambda__zero,axiom,
( ( ^ [H3: complex] : zero_zero_complex )
= ( times_times_complex @ zero_zero_complex ) ) ).
% lambda_zero
thf(fact_456_lambda__zero,axiom,
( ( ^ [H3: finite_mod_ring_a] : zero_z7902377541816115708ring_a )
= ( times_5121417576591743744ring_a @ zero_z7902377541816115708ring_a ) ) ).
% lambda_zero
thf(fact_457_lambda__zero,axiom,
( ( ^ [H3: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_458_lambda__zero,axiom,
( ( ^ [H3: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_459_lambda__zero,axiom,
( ( ^ [H3: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_460_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: complex,Z: complex] : ( Y3 = Z ) )
= ( ^ [A3: complex,B2: complex] :
( ( minus_minus_complex @ A3 @ B2 )
= zero_zero_complex ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_461_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: finite_mod_ring_a,Z: finite_mod_ring_a] : ( Y3 = Z ) )
= ( ^ [A3: finite_mod_ring_a,B2: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ A3 @ B2 )
= zero_z7902377541816115708ring_a ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_462_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A3: int,B2: int] :
( ( minus_minus_int @ A3 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_463_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: real,Z: real] : ( Y3 = Z ) )
= ( ^ [A3: real,B2: real] :
( ( minus_minus_real @ A3 @ B2 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_464_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_465_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_466_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_467_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_468_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_469_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_470_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_471_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_472_group__cancel_Osub1,axiom,
! [A2: finite_mod_ring_a,K: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( A2
= ( plus_p6165643967897163644ring_a @ K @ A ) )
=> ( ( minus_3609261664126569004ring_a @ A2 @ B )
= ( plus_p6165643967897163644ring_a @ K @ ( minus_3609261664126569004ring_a @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_473_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_474_group__cancel_Osub1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_475_diff__eq__eq,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( minus_3609261664126569004ring_a @ A @ B )
= C )
= ( A
= ( plus_p6165643967897163644ring_a @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_476_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_477_diff__eq__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( minus_minus_real @ A @ B )
= C )
= ( A
= ( plus_plus_real @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_478_eq__diff__eq,axiom,
! [A: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( A
= ( minus_3609261664126569004ring_a @ C @ B ) )
= ( ( plus_p6165643967897163644ring_a @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_479_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_480_eq__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( A
= ( minus_minus_real @ C @ B ) )
= ( ( plus_plus_real @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_481_add__diff__eq,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ A @ ( minus_3609261664126569004ring_a @ B @ C ) )
= ( minus_3609261664126569004ring_a @ ( plus_p6165643967897163644ring_a @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_482_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_483_add__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_484_diff__diff__eq2,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ A @ ( minus_3609261664126569004ring_a @ B @ C ) )
= ( minus_3609261664126569004ring_a @ ( plus_p6165643967897163644ring_a @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_485_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_486_diff__diff__eq2,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_487_diff__add__eq,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ ( minus_3609261664126569004ring_a @ A @ B ) @ C )
= ( minus_3609261664126569004ring_a @ ( plus_p6165643967897163644ring_a @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_488_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_489_diff__add__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_490_diff__add__eq__diff__diff__swap,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ A @ ( plus_p6165643967897163644ring_a @ B @ C ) )
= ( minus_3609261664126569004ring_a @ ( minus_3609261664126569004ring_a @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_491_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_492_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_493_add__implies__diff,axiom,
! [C: finite_mod_ring_a,B: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ C @ B )
= A )
=> ( C
= ( minus_3609261664126569004ring_a @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_494_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_495_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_496_add__implies__diff,axiom,
! [C: real,B: real,A: real] :
( ( ( plus_plus_real @ C @ B )
= A )
=> ( C
= ( minus_minus_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_497_diff__diff__eq,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ ( minus_3609261664126569004ring_a @ A @ B ) @ C )
= ( minus_3609261664126569004ring_a @ A @ ( plus_p6165643967897163644ring_a @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_498_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_499_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_500_diff__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_501_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_502_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_503_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: real,B: real] :
( ~ ( ord_less_real @ A @ B )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_504_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_505_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_506_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_507_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_508_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_509_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_510_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_511_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_512_mult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_513_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_514_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_515_mult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_516_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_517_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_518_mult__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_519_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_520_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_521_mult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_522_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_523_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_524_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_525_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_526_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_527_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_528_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_529_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_530_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_531_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_532_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_533_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_534_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_535_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_536_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_537_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_538_mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_539_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_540_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_541_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_542_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_543_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_544_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_545_mult__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_546_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_547_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_548_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_549_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_550_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_551_sum__subtractf,axiom,
! [F: nat > finite_mod_ring_a,G: nat > finite_mod_ring_a,A2: set_nat] :
( ( groups3558780024651037881ring_a
@ ^ [X2: nat] : ( minus_3609261664126569004ring_a @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_3609261664126569004ring_a @ ( groups3558780024651037881ring_a @ F @ A2 ) @ ( groups3558780024651037881ring_a @ G @ A2 ) ) ) ).
% sum_subtractf
thf(fact_552_sum__subtractf,axiom,
! [F: complex > complex,G: complex > complex,A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [X2: complex] : ( minus_minus_complex @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ A2 ) ) ) ).
% sum_subtractf
thf(fact_553_sum__distrib__left,axiom,
! [R: finite_mod_ring_a,F: nat > finite_mod_ring_a,A2: set_nat] :
( ( times_5121417576591743744ring_a @ R @ ( groups3558780024651037881ring_a @ F @ A2 ) )
= ( groups3558780024651037881ring_a
@ ^ [N3: nat] : ( times_5121417576591743744ring_a @ R @ ( F @ N3 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_554_sum__distrib__left,axiom,
! [R: nat,F: nat > nat,A2: set_nat] :
( ( times_times_nat @ R @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3542108847815614940at_nat
@ ^ [N3: nat] : ( times_times_nat @ R @ ( F @ N3 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_555_sum__distrib__left,axiom,
! [R: complex,F: complex > complex,A2: set_complex] :
( ( times_times_complex @ R @ ( groups7754918857620584856omplex @ F @ A2 ) )
= ( groups7754918857620584856omplex
@ ^ [N3: complex] : ( times_times_complex @ R @ ( F @ N3 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_556_sum__distrib__right,axiom,
! [F: nat > finite_mod_ring_a,A2: set_nat,R: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( groups3558780024651037881ring_a @ F @ A2 ) @ R )
= ( groups3558780024651037881ring_a
@ ^ [N3: nat] : ( times_5121417576591743744ring_a @ ( F @ N3 ) @ R )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_557_sum__distrib__right,axiom,
! [F: nat > nat,A2: set_nat,R: nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R )
= ( groups3542108847815614940at_nat
@ ^ [N3: nat] : ( times_times_nat @ ( F @ N3 ) @ R )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_558_sum__distrib__right,axiom,
! [F: complex > complex,A2: set_complex,R: complex] :
( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R )
= ( groups7754918857620584856omplex
@ ^ [N3: complex] : ( times_times_complex @ ( F @ N3 ) @ R )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_559_sum__product,axiom,
! [F: nat > finite_mod_ring_a,A2: set_nat,G: nat > finite_mod_ring_a,B3: set_nat] :
( ( times_5121417576591743744ring_a @ ( groups3558780024651037881ring_a @ F @ A2 ) @ ( groups3558780024651037881ring_a @ G @ B3 ) )
= ( groups3558780024651037881ring_a
@ ^ [I: nat] :
( groups3558780024651037881ring_a
@ ^ [J: nat] : ( times_5121417576591743744ring_a @ ( F @ I ) @ ( G @ J ) )
@ B3 )
@ A2 ) ) ).
% sum_product
thf(fact_560_sum__product,axiom,
! [F: nat > nat,A2: set_nat,G: nat > nat,B3: set_nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B3 ) )
= ( groups3542108847815614940at_nat
@ ^ [I: nat] :
( groups3542108847815614940at_nat
@ ^ [J: nat] : ( times_times_nat @ ( F @ I ) @ ( G @ J ) )
@ B3 )
@ A2 ) ) ).
% sum_product
thf(fact_561_sum__product,axiom,
! [F: complex > complex,A2: set_complex,G: complex > complex,B3: set_complex] :
( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ B3 ) )
= ( groups7754918857620584856omplex
@ ^ [I: complex] :
( groups7754918857620584856omplex
@ ^ [J: complex] : ( times_times_complex @ ( F @ I ) @ ( G @ J ) )
@ B3 )
@ A2 ) ) ).
% sum_product
thf(fact_562_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_563_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_564_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_565_not__sum__squares__lt__zero,axiom,
! [X: real,Y: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_566_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_567_diff__less__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_568_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_569_less__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_570_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_571_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_572_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_573_sum__squares__eq__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 ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_574_sum__const,axiom,
! [C: finite_mod_ring_a,X: nat] :
( ( groups3558780024651037881ring_a
@ ^ [I: nat] : C
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ X ) )
= ( times_5121417576591743744ring_a @ ( finite8272632373135393572ring_a @ ( semiri1314217659103216013at_int @ X ) ) @ C ) ) ).
% sum_const
thf(fact_575_mult__hom_Ohom__zero,axiom,
! [C: complex] :
( ( times_times_complex @ C @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_hom.hom_zero
thf(fact_576_mult__hom_Ohom__zero,axiom,
! [C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ C @ zero_z7902377541816115708ring_a )
= zero_z7902377541816115708ring_a ) ).
% mult_hom.hom_zero
thf(fact_577_mult__hom_Ohom__zero,axiom,
! [C: int] :
( ( times_times_int @ C @ zero_zero_int )
= zero_zero_int ) ).
% mult_hom.hom_zero
thf(fact_578_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_579_mult__hom_Ohom__zero,axiom,
! [C: real] :
( ( times_times_real @ C @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_580_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_581_sum__squares__gt__zero__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
= ( ( X != zero_zero_real )
| ( Y != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_582_of__int__mod__ring__hom_Ohom__zero,axiom,
( ( finite8272632373135393572ring_a @ zero_zero_int )
= zero_z7902377541816115708ring_a ) ).
% of_int_mod_ring_hom.hom_zero
thf(fact_583_mult__diff__mult,axiom,
! [X: finite_mod_ring_a,Y: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ ( times_5121417576591743744ring_a @ X @ Y ) @ ( times_5121417576591743744ring_a @ A @ B ) )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ X @ ( minus_3609261664126569004ring_a @ Y @ B ) ) @ ( times_5121417576591743744ring_a @ ( minus_3609261664126569004ring_a @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_584_mult__diff__mult,axiom,
! [X: int,Y: int,A: int,B: int] :
( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_585_mult__diff__mult,axiom,
! [X: real,Y: real,A: real,B: real] :
( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_586_mult__hom_Ohom__add__eq__zero,axiom,
! [X: complex,Y: complex,C: complex] :
( ( ( plus_plus_complex @ X @ Y )
= zero_zero_complex )
=> ( ( plus_plus_complex @ ( times_times_complex @ C @ X ) @ ( times_times_complex @ C @ Y ) )
= zero_zero_complex ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_587_mult__hom_Ohom__add__eq__zero,axiom,
! [X: finite_mod_ring_a,Y: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ X @ Y )
= zero_z7902377541816115708ring_a )
=> ( ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ C @ X ) @ ( times_5121417576591743744ring_a @ C @ Y ) )
= zero_z7902377541816115708ring_a ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_588_mult__hom_Ohom__add__eq__zero,axiom,
! [X: int,Y: int,C: int] :
( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
=> ( ( plus_plus_int @ ( times_times_int @ C @ X ) @ ( times_times_int @ C @ Y ) )
= zero_zero_int ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_589_mult__hom_Ohom__add__eq__zero,axiom,
! [X: nat,Y: nat,C: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
=> ( ( plus_plus_nat @ ( times_times_nat @ C @ X ) @ ( times_times_nat @ C @ Y ) )
= zero_zero_nat ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_590_mult__hom_Ohom__add__eq__zero,axiom,
! [X: real,Y: real,C: real] :
( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
=> ( ( plus_plus_real @ ( times_times_real @ C @ X ) @ ( times_times_real @ C @ Y ) )
= zero_zero_real ) ) ).
% mult_hom.hom_add_eq_zero
thf(fact_591_add__scale__eq__noteq,axiom,
! [R: complex,A: complex,B: complex,C: complex,D: complex] :
( ( R != zero_zero_complex )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_complex @ A @ ( times_times_complex @ R @ C ) )
!= ( plus_plus_complex @ B @ ( times_times_complex @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_592_add__scale__eq__noteq,axiom,
! [R: finite_mod_ring_a,A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a,D: finite_mod_ring_a] :
( ( R != zero_z7902377541816115708ring_a )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_p6165643967897163644ring_a @ A @ ( times_5121417576591743744ring_a @ R @ C ) )
!= ( plus_p6165643967897163644ring_a @ B @ ( times_5121417576591743744ring_a @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_593_add__scale__eq__noteq,axiom,
! [R: int,A: int,B: int,C: int,D: int] :
( ( R != zero_zero_int )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_594_add__scale__eq__noteq,axiom,
! [R: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_595_add__scale__eq__noteq,axiom,
! [R: real,A: real,B: real,C: real,D: real] :
( ( R != zero_zero_real )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_real @ A @ ( times_times_real @ R @ C ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_596_mult__less__iff1,axiom,
! [Z2: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_597_mult__less__iff1,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_598_homomorphism__mul__on__ring,axiom,
! [X: int,Y: int] :
( ( times_5121417576591743744ring_a @ ( finite8272632373135393572ring_a @ X ) @ ( finite8272632373135393572ring_a @ Y ) )
= ( finite8272632373135393572ring_a @ ( times_times_int @ X @ Y ) ) ) ).
% homomorphism_mul_on_ring
thf(fact_599_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_600_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_601_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_602_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_603_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_604_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_605_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_606_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( semiri5074537144036343181t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_607_diff__diff__left,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K )
= ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_608_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri9180929696517417892ring_a @ ( times_times_nat @ M @ N ) )
= ( times_5121417576591743744ring_a @ ( semiri9180929696517417892ring_a @ M ) @ ( semiri9180929696517417892ring_a @ N ) ) ) ).
% of_nat_mult
thf(fact_609_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_610_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_611_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_612_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_613_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 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_614_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_615_of__nat__sum,axiom,
! [F: complex > nat,A2: set_complex] :
( ( semiri8010041392384452111omplex @ ( groups5693394587270226106ex_nat @ F @ A2 ) )
= ( groups7754918857620584856omplex
@ ^ [X2: complex] : ( semiri8010041392384452111omplex @ ( F @ X2 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_616_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3539618377306564664at_int
@ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_617_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [X2: nat] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_618_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri9180929696517417892ring_a @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3558780024651037881ring_a
@ ^ [X2: nat] : ( semiri9180929696517417892ring_a @ ( F @ X2 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_619_of__nat__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( semiri1316708129612266289at_nat @ ( F @ X2 ) )
@ A2 ) ) ).
% of_nat_sum
thf(fact_620_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_621_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri8010041392384452111omplex @ M )
= zero_zero_complex )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_622_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_623_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_624_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_625_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_complex
= ( semiri8010041392384452111omplex @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_626_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_627_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_628_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_629_of__nat__0,axiom,
( ( semiri8010041392384452111omplex @ zero_zero_nat )
= zero_zero_complex ) ).
% of_nat_0
thf(fact_630_of__nat__0,axiom,
( ( semiri9180929696517417892ring_a @ zero_zero_nat )
= zero_z7902377541816115708ring_a ) ).
% of_nat_0
thf(fact_631_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_632_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_633_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_634_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_635_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_636_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri9180929696517417892ring_a @ ( plus_plus_nat @ M @ N ) )
= ( plus_p6165643967897163644ring_a @ ( semiri9180929696517417892ring_a @ M ) @ ( semiri9180929696517417892ring_a @ N ) ) ) ).
% of_nat_add
thf(fact_637_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_638_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_639_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_add
thf(fact_640_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_641_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_642_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_643_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_644_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_645_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_646_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_647_zmult__zless__mono2,axiom,
! [I3: int,J2: int,K: int] :
( ( ord_less_int @ I3 @ J2 )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I3 ) @ ( times_times_int @ K @ J2 ) ) ) ) ).
% zmult_zless_mono2
thf(fact_648_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M4: nat,N2: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_649_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_650_int__sum,axiom,
! [F: nat > nat,A2: set_nat] :
( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3539618377306564664at_int
@ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
@ A2 ) ) ).
% int_sum
thf(fact_651_zmult__zless__mono2__lemma,axiom,
! [I3: int,J2: int,K: nat] :
( ( ord_less_int @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I3 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J2 ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_652_mult__of__nat__commute,axiom,
! [X: nat,Y: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( semiri9180929696517417892ring_a @ X ) @ Y )
= ( times_5121417576591743744ring_a @ Y @ ( semiri9180929696517417892ring_a @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_653_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_654_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_655_mult__of__nat__commute,axiom,
! [X: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_656_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 ) ) ) ).
% diffs0_imp_equal
thf(fact_657_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_658_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_659_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_660_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_661_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_662_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_663_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_664_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_665_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_666_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_667_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_668_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_669_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_670_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_671_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_672_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_673_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_674_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_675_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_676_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_677_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_678_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_679_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_680_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_681_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_682_less__diff__conv,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_683_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_684_mult__less__mono2,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_685_mult__less__mono1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_686_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_687_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 ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_688_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 ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_689_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_690_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_691_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_692_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_693_add__0__iff,axiom,
! [B: complex,A: complex] :
( ( B
= ( plus_plus_complex @ B @ A ) )
= ( A = zero_zero_complex ) ) ).
% add_0_iff
thf(fact_694_add__0__iff,axiom,
! [B: finite_mod_ring_a,A: finite_mod_ring_a] :
( ( B
= ( plus_p6165643967897163644ring_a @ B @ A ) )
= ( A = zero_z7902377541816115708ring_a ) ) ).
% add_0_iff
thf(fact_695_crossproduct__noteq,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a,D: finite_mod_ring_a] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A @ C ) @ ( times_5121417576591743744ring_a @ B @ D ) )
!= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ A @ D ) @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_696_crossproduct__noteq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
!= ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_697_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_698_crossproduct__noteq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
!= ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_699_crossproduct__eq,axiom,
! [W: finite_mod_ring_a,Y: finite_mod_ring_a,X: finite_mod_ring_a,Z2: finite_mod_ring_a] :
( ( ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ W @ Y ) @ ( times_5121417576591743744ring_a @ X @ Z2 ) )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ W @ Z2 ) @ ( times_5121417576591743744ring_a @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_700_crossproduct__eq,axiom,
! [W: int,Y: int,X: int,Z2: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z2 ) @ ( times_times_int @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_701_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z2: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z2 ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z2 ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_702_crossproduct__eq,axiom,
! [W: real,Y: real,X: real,Z2: real] :
( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X @ Z2 ) )
= ( plus_plus_real @ ( times_times_real @ W @ Z2 ) @ ( times_times_real @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_703_mult__hom_Ohom__add,axiom,
! [C: finite_mod_ring_a,X: finite_mod_ring_a,Y: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ C @ ( plus_p6165643967897163644ring_a @ X @ Y ) )
= ( plus_p6165643967897163644ring_a @ ( times_5121417576591743744ring_a @ C @ X ) @ ( times_5121417576591743744ring_a @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_704_mult__hom_Ohom__add,axiom,
! [C: int,X: int,Y: int] :
( ( times_times_int @ C @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_int @ ( times_times_int @ C @ X ) @ ( times_times_int @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_705_mult__hom_Ohom__add,axiom,
! [C: nat,X: nat,Y: nat] :
( ( times_times_nat @ C @ ( plus_plus_nat @ X @ Y ) )
= ( plus_plus_nat @ ( times_times_nat @ C @ X ) @ ( times_times_nat @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_706_mult__hom_Ohom__add,axiom,
! [C: real,X: real,Y: real] :
( ( times_times_real @ C @ ( plus_plus_real @ X @ Y ) )
= ( plus_plus_real @ ( times_times_real @ C @ X ) @ ( times_times_real @ C @ Y ) ) ) ).
% mult_hom.hom_add
thf(fact_707_add__diff__add,axiom,
! [A: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a,D: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ ( plus_p6165643967897163644ring_a @ A @ C ) @ ( plus_p6165643967897163644ring_a @ B @ D ) )
= ( plus_p6165643967897163644ring_a @ ( minus_3609261664126569004ring_a @ A @ B ) @ ( minus_3609261664126569004ring_a @ C @ D ) ) ) ).
% add_diff_add
thf(fact_708_add__diff__add,axiom,
! [A: int,C: int,B: int,D: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
= ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% add_diff_add
thf(fact_709_add__diff__add,axiom,
! [A: real,C: real,B: real,D: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
= ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% add_diff_add
thf(fact_710_mult__hom_Ohom__sum,axiom,
! [C: finite_mod_ring_a,F: nat > finite_mod_ring_a,X5: set_nat] :
( ( times_5121417576591743744ring_a @ C @ ( groups3558780024651037881ring_a @ F @ X5 ) )
= ( groups3558780024651037881ring_a
@ ^ [X2: nat] : ( times_5121417576591743744ring_a @ C @ ( F @ X2 ) )
@ X5 ) ) ).
% mult_hom.hom_sum
thf(fact_711_mult__hom_Ohom__sum,axiom,
! [C: nat,F: nat > nat,X5: set_nat] :
( ( times_times_nat @ C @ ( groups3542108847815614940at_nat @ F @ X5 ) )
= ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( times_times_nat @ C @ ( F @ X2 ) )
@ X5 ) ) ).
% mult_hom.hom_sum
thf(fact_712_mult__hom_Ohom__sum,axiom,
! [C: complex,F: complex > complex,X5: set_complex] :
( ( times_times_complex @ C @ ( groups7754918857620584856omplex @ F @ X5 ) )
= ( groups7754918857620584856omplex
@ ^ [X2: complex] : ( times_times_complex @ C @ ( F @ X2 ) )
@ X5 ) ) ).
% mult_hom.hom_sum
thf(fact_713_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_714_ex__less__of__nat__mult,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N2: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_715_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_716_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_717_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_718_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_less_as_int
thf(fact_719_diff__commute,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_720_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_721_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_722_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_723_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_724_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_725_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_726_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_727_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_728_left__add__mult__distrib,axiom,
! [I3: nat,U: nat,J2: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I3 @ J2 ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_729_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_730_nat__int__comparison_I1_J,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A3: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_731_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_732_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_733_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_734_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_735_verit__sum__simplify,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% verit_sum_simplify
thf(fact_736_verit__sum__simplify,axiom,
! [A: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ A @ zero_z7902377541816115708ring_a )
= A ) ).
% verit_sum_simplify
thf(fact_737_reals__Archimedean2,axiom,
! [X: real] :
? [N2: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% reals_Archimedean2
thf(fact_738_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_739_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_740_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_741_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_742_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_743_plusinfinity,axiom,
! [D: int,P2: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [X_1: int] : ( P2 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_744_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_745_inf__period_I1_J,axiom,
! [P: finite_mod_ring_a > $o,D3: finite_mod_ring_a,Q: finite_mod_ring_a > $o] :
( ! [X3: finite_mod_ring_a,K2: finite_mod_ring_a] :
( ( P @ X3 )
= ( P @ ( minus_3609261664126569004ring_a @ X3 @ ( times_5121417576591743744ring_a @ K2 @ D3 ) ) ) )
=> ( ! [X3: finite_mod_ring_a,K2: finite_mod_ring_a] :
( ( Q @ X3 )
= ( Q @ ( minus_3609261664126569004ring_a @ X3 @ ( times_5121417576591743744ring_a @ K2 @ D3 ) ) ) )
=> ! [X4: finite_mod_ring_a,K3: finite_mod_ring_a] :
( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P @ ( minus_3609261664126569004ring_a @ X4 @ ( times_5121417576591743744ring_a @ K3 @ D3 ) ) )
& ( Q @ ( minus_3609261664126569004ring_a @ X4 @ ( times_5121417576591743744ring_a @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_746_inf__period_I1_J,axiom,
! [P: int > $o,D3: int,Q: int > $o] :
( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ! [X4: int,K3: int] :
( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) )
& ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_747_inf__period_I1_J,axiom,
! [P: real > $o,D3: real,Q: real > $o] :
( ! [X3: real,K2: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D3 ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D3 ) ) ) )
=> ! [X4: real,K3: real] :
( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D3 ) ) )
& ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_748_inf__period_I2_J,axiom,
! [P: finite_mod_ring_a > $o,D3: finite_mod_ring_a,Q: finite_mod_ring_a > $o] :
( ! [X3: finite_mod_ring_a,K2: finite_mod_ring_a] :
( ( P @ X3 )
= ( P @ ( minus_3609261664126569004ring_a @ X3 @ ( times_5121417576591743744ring_a @ K2 @ D3 ) ) ) )
=> ( ! [X3: finite_mod_ring_a,K2: finite_mod_ring_a] :
( ( Q @ X3 )
= ( Q @ ( minus_3609261664126569004ring_a @ X3 @ ( times_5121417576591743744ring_a @ K2 @ D3 ) ) ) )
=> ! [X4: finite_mod_ring_a,K3: finite_mod_ring_a] :
( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P @ ( minus_3609261664126569004ring_a @ X4 @ ( times_5121417576591743744ring_a @ K3 @ D3 ) ) )
| ( Q @ ( minus_3609261664126569004ring_a @ X4 @ ( times_5121417576591743744ring_a @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_749_inf__period_I2_J,axiom,
! [P: int > $o,D3: int,Q: int > $o] :
( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D3 ) ) ) )
=> ! [X4: int,K3: int] :
( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) )
| ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_750_inf__period_I2_J,axiom,
! [P: real > $o,D3: real,Q: real > $o] :
( ! [X3: real,K2: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D3 ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D3 ) ) ) )
=> ! [X4: real,K3: real] :
( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D3 ) ) )
| ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K3 @ D3 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_751_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_752_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_753_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_754_pinf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_755_pinf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_756_pinf_I1_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_757_pinf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z3 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_758_pinf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ Z3 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_759_pinf_I2_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_760_pinf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_761_pinf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_762_pinf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_763_pinf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_764_pinf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_765_pinf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_766_pinf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T ) ) ).
% pinf(5)
thf(fact_767_pinf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ~ ( ord_less_int @ X4 @ T ) ) ).
% pinf(5)
thf(fact_768_pinf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ~ ( ord_less_real @ X4 @ T ) ) ).
% pinf(5)
thf(fact_769_pinf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ord_less_nat @ T @ X4 ) ) ).
% pinf(7)
thf(fact_770_pinf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ord_less_int @ T @ X4 ) ) ).
% pinf(7)
thf(fact_771_pinf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ Z4 @ X4 )
=> ( ord_less_real @ T @ X4 ) ) ).
% pinf(7)
thf(fact_772_minf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_773_minf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_774_minf_I1_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_775_minf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_776_minf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z3: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_777_minf_I2_J,axiom,
! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z3: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_778_minf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_779_minf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_780_minf_I3_J,axiom,
! [T: real] :
? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_781_minf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_782_minf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_783_minf_I4_J,axiom,
! [T: real] :
? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_784_minf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ord_less_nat @ X4 @ T ) ) ).
% minf(5)
thf(fact_785_minf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ord_less_int @ X4 @ T ) ) ).
% minf(5)
thf(fact_786_minf_I5_J,axiom,
! [T: real] :
? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ( ord_less_real @ X4 @ T ) ) ).
% minf(5)
thf(fact_787_minf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ~ ( ord_less_nat @ T @ X4 ) ) ).
% minf(7)
thf(fact_788_minf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ~ ( ord_less_int @ T @ X4 ) ) ).
% minf(7)
thf(fact_789_minf_I7_J,axiom,
! [T: real] :
? [Z4: real] :
! [X4: real] :
( ( ord_less_real @ X4 @ Z4 )
=> ~ ( ord_less_real @ T @ X4 ) ) ).
% minf(7)
thf(fact_790_is__num__normalize_I1_J,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ ( plus_p6165643967897163644ring_a @ A @ B ) @ C )
= ( plus_p6165643967897163644ring_a @ A @ ( plus_p6165643967897163644ring_a @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_791_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_792_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_793_sum_Oop__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > int] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= zero_zero_int ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_794_sum_Oop__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > real] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_795_sum_Oop__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > complex] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= zero_zero_complex ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_796_sum_Oop__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > finite_mod_ring_a] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups3558780024651037881ring_a @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= zero_z7902377541816115708ring_a ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups3558780024651037881ring_a @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= ( plus_p6165643967897163644ring_a @ ( groups3558780024651037881ring_a @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_797_sum_Oop__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > nat] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or4665077453230672383an_nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_798_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_799_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_800_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_801_sum__bounded__above__strict,axiom,
! [A2: set_real,F: real > nat,K4: nat] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_nat @ ( F @ I2 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A2 ) )
=> ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_real @ A2 ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_802_sum__bounded__above__strict,axiom,
! [A2: set_int,F: int > nat,K4: nat] :
( ! [I2: int] :
( ( member_int @ I2 @ A2 )
=> ( ord_less_nat @ ( F @ I2 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A2 ) )
=> ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A2 ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_803_sum__bounded__above__strict,axiom,
! [A2: set_complex,F: complex > nat,K4: nat] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A2 )
=> ( ord_less_nat @ ( F @ I2 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A2 ) )
=> ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_complex @ A2 ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_804_sum__bounded__above__strict,axiom,
! [A2: set_real,F: real > int,K4: int] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_int @ ( F @ I2 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A2 ) )
=> ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_real @ A2 ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_805_sum__bounded__above__strict,axiom,
! [A2: set_nat,F: nat > int,K4: int] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_int @ ( F @ I2 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A2 ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_806_sum__bounded__above__strict,axiom,
! [A2: set_int,F: int > int,K4: int] :
( ! [I2: int] :
( ( member_int @ I2 @ A2 )
=> ( ord_less_int @ ( F @ I2 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A2 ) )
=> ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_int @ A2 ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_807_sum__bounded__above__strict,axiom,
! [A2: set_complex,F: complex > int,K4: int] :
( ! [I2: complex] :
( ( member_complex @ I2 @ A2 )
=> ( ord_less_int @ ( F @ I2 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A2 ) )
=> ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_complex @ A2 ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_808_sum__bounded__above__strict,axiom,
! [A2: set_real,F: real > real,K4: real] :
( ! [I2: real] :
( ( member_real @ I2 @ A2 )
=> ( ord_less_real @ ( F @ I2 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A2 ) )
=> ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A2 ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_809_sum__bounded__above__strict,axiom,
! [A2: set_nat,F: nat > real,K4: real] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_less_real @ ( F @ I2 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_nat @ A2 ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_810_sum__bounded__above__strict,axiom,
! [A2: set_int,F: int > real,K4: real] :
( ! [I2: int] :
( ( member_int @ I2 @ A2 )
=> ( ord_less_real @ ( F @ I2 ) @ K4 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A2 ) )
=> ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A2 ) ) @ K4 ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_811_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_812_diff__nat__eq__if,axiom,
! [Z5: int,Z2: int] :
( ( ( ord_less_int @ Z5 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) )
= ( nat2 @ Z2 ) ) )
& ( ~ ( ord_less_int @ Z5 @ zero_zero_int )
=> ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) )
= ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z2 @ Z5 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z2 @ Z5 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_813_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_814_int__exp__hom,axiom,
! [X: nat,I3: nat] :
( ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ I3 )
= ( semiri1314217659103216013at_int @ ( power_power_nat @ X @ I3 ) ) ) ).
% int_exp_hom
thf(fact_815_exp__homo,axiom,
! [X: int,I3: nat] :
( ( finite8272632373135393572ring_a @ ( power_power_int @ X @ I3 ) )
= ( power_6826135765519566523ring_a @ ( finite8272632373135393572ring_a @ X ) @ I3 ) ) ).
% exp_homo
thf(fact_816_exp__rule,axiom,
! [C: finite_mod_ring_a,D: finite_mod_ring_a,E: nat] :
( ( power_6826135765519566523ring_a @ ( times_5121417576591743744ring_a @ C @ D ) @ E )
= ( times_5121417576591743744ring_a @ ( power_6826135765519566523ring_a @ C @ E ) @ ( power_6826135765519566523ring_a @ D @ E ) ) ) ).
% exp_rule
thf(fact_817_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_818_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_819_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_820_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_821_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_822_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_823_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_824_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_825_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_826_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_827_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_828_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_829_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_830_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_831_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_832_add_Oinverse__neutral,axiom,
( ( uminus1482373934393186551omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% add.inverse_neutral
thf(fact_833_add_Oinverse__neutral,axiom,
( ( uminus3100561713750211260ring_a @ zero_z7902377541816115708ring_a )
= zero_z7902377541816115708ring_a ) ).
% add.inverse_neutral
thf(fact_834_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_835_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_836_neg__0__equal__iff__equal,axiom,
! [A: complex] :
( ( zero_zero_complex
= ( uminus1482373934393186551omplex @ A ) )
= ( zero_zero_complex = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_837_neg__0__equal__iff__equal,axiom,
! [A: finite_mod_ring_a] :
( ( zero_z7902377541816115708ring_a
= ( uminus3100561713750211260ring_a @ A ) )
= ( zero_z7902377541816115708ring_a = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_838_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_839_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_840_neg__equal__0__iff__equal,axiom,
! [A: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% neg_equal_0_iff_equal
thf(fact_841_neg__equal__0__iff__equal,axiom,
! [A: finite_mod_ring_a] :
( ( ( uminus3100561713750211260ring_a @ A )
= zero_z7902377541816115708ring_a )
= ( A = zero_z7902377541816115708ring_a ) ) ).
% neg_equal_0_iff_equal
thf(fact_842_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_843_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_844_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_845_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_846_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_847_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_848_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_849_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_850_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_851_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_852_mult__minus__right,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A @ ( uminus3100561713750211260ring_a @ B ) )
= ( uminus3100561713750211260ring_a @ ( times_5121417576591743744ring_a @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_853_mult__minus__right,axiom,
! [A: int,B: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_854_mult__minus__right,axiom,
! [A: real,B: real] :
( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_855_minus__mult__minus,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( uminus3100561713750211260ring_a @ A ) @ ( uminus3100561713750211260ring_a @ B ) )
= ( times_5121417576591743744ring_a @ A @ B ) ) ).
% minus_mult_minus
thf(fact_856_minus__mult__minus,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A @ B ) ) ).
% minus_mult_minus
thf(fact_857_minus__mult__minus,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( times_times_real @ A @ B ) ) ).
% minus_mult_minus
thf(fact_858_mult__minus__left,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( uminus3100561713750211260ring_a @ A ) @ B )
= ( uminus3100561713750211260ring_a @ ( times_5121417576591743744ring_a @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_859_mult__minus__left,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_860_mult__minus__left,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_861_add__minus__cancel,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ A @ ( plus_p6165643967897163644ring_a @ ( uminus3100561713750211260ring_a @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_862_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_863_add__minus__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_864_minus__add__cancel,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ ( uminus3100561713750211260ring_a @ A ) @ ( plus_p6165643967897163644ring_a @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_865_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_866_minus__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_867_minus__add__distrib,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( uminus3100561713750211260ring_a @ ( plus_p6165643967897163644ring_a @ A @ B ) )
= ( plus_p6165643967897163644ring_a @ ( uminus3100561713750211260ring_a @ A ) @ ( uminus3100561713750211260ring_a @ B ) ) ) ).
% minus_add_distrib
thf(fact_868_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_869_minus__add__distrib,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_870_minus__diff__eq,axiom,
! [A: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( uminus3100561713750211260ring_a @ ( minus_3609261664126569004ring_a @ A @ B ) )
= ( minus_3609261664126569004ring_a @ B @ A ) ) ).
% minus_diff_eq
thf(fact_871_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_872_minus__diff__eq,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
= ( minus_minus_real @ B @ A ) ) ).
% minus_diff_eq
thf(fact_873_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 ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_874_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_875_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 ) ) ) ).
% nat_zero_less_power_iff
thf(fact_876_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_877_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri8010041392384452111omplex @ X )
= ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_878_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_879_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri5074537144036343181t_real @ X )
= ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_880_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
= ( semiri1316708129612266289at_nat @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_881_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
= ( semiri8010041392384452111omplex @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_882_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
= ( semiri1314217659103216013at_int @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_883_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
= ( semiri5074537144036343181t_real @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_884_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_885_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri9180929696517417892ring_a @ ( power_power_nat @ M @ N ) )
= ( power_6826135765519566523ring_a @ ( semiri9180929696517417892ring_a @ M ) @ N ) ) ).
% of_nat_power
thf(fact_886_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
= ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% of_nat_power
thf(fact_887_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% of_nat_power
thf(fact_888_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
= ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% of_nat_power
thf(fact_889_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_890_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_891_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_892_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_893_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_894_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_895_ivl__subset,axiom,
! [I3: real,J2: real,M: real,N: real] :
( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ I3 @ J2 ) @ ( set_or66887138388493659n_real @ M @ N ) )
= ( ( ord_less_eq_real @ J2 @ I3 )
| ( ( ord_less_eq_real @ M @ I3 )
& ( ord_less_eq_real @ J2 @ N ) ) ) ) ).
% ivl_subset
thf(fact_896_ivl__subset,axiom,
! [I3: nat,J2: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I3 @ J2 ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J2 @ I3 )
| ( ( ord_less_eq_nat @ M @ I3 )
& ( ord_less_eq_nat @ J2 @ N ) ) ) ) ).
% ivl_subset
thf(fact_897_ivl__subset,axiom,
! [I3: int,J2: int,M: int,N: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I3 @ J2 ) @ ( set_or4662586982721622107an_int @ M @ N ) )
= ( ( ord_less_eq_int @ J2 @ I3 )
| ( ( ord_less_eq_int @ M @ I3 )
& ( ord_less_eq_int @ J2 @ N ) ) ) ) ).
% ivl_subset
thf(fact_898_ivl__diff,axiom,
! [I3: real,N: real,M: real] :
( ( ord_less_eq_real @ I3 @ N )
=> ( ( minus_minus_set_real @ ( set_or66887138388493659n_real @ I3 @ M ) @ ( set_or66887138388493659n_real @ I3 @ N ) )
= ( set_or66887138388493659n_real @ N @ M ) ) ) ).
% ivl_diff
thf(fact_899_ivl__diff,axiom,
! [I3: nat,N: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I3 @ M ) @ ( set_or4665077453230672383an_nat @ I3 @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% ivl_diff
thf(fact_900_ivl__diff,axiom,
! [I3: int,N: int,M: int] :
( ( ord_less_eq_int @ I3 @ N )
=> ( ( minus_minus_set_int @ ( set_or4662586982721622107an_int @ I3 @ M ) @ ( set_or4662586982721622107an_int @ I3 @ N ) )
= ( set_or4662586982721622107an_int @ N @ M ) ) ) ).
% ivl_diff
thf(fact_901_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_902_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_903_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_904_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_905_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_906_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( minus_minus_nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_907_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_908_card__atLeastLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card_int @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% card_atLeastLessThan_int
thf(fact_909_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_910_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_911_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_912_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_913_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_914_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_915_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_916_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_917_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_918_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_919_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_920_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_921_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_922_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_923_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_924_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_925_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_926_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_927_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_928_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_929_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_930_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_931_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_932_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_933_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_934_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_935_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_936_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_937_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_938_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_939_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_940_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_941_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_942_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_943_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_944_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_945_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_946_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_947_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_948_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_949_add_Oright__inverse,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
= zero_zero_complex ) ).
% add.right_inverse
thf(fact_950_add_Oright__inverse,axiom,
! [A: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ A @ ( uminus3100561713750211260ring_a @ A ) )
= zero_z7902377541816115708ring_a ) ).
% add.right_inverse
thf(fact_951_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_952_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_953_ab__left__minus,axiom,
! [A: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
= zero_zero_complex ) ).
% ab_left_minus
thf(fact_954_ab__left__minus,axiom,
! [A: finite_mod_ring_a] :
( ( plus_p6165643967897163644ring_a @ ( uminus3100561713750211260ring_a @ A ) @ A )
= zero_z7902377541816115708ring_a ) ).
% ab_left_minus
thf(fact_955_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_956_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_957_verit__minus__simplify_I3_J,axiom,
! [B: complex] :
( ( minus_minus_complex @ zero_zero_complex @ B )
= ( uminus1482373934393186551omplex @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_958_verit__minus__simplify_I3_J,axiom,
! [B: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ zero_z7902377541816115708ring_a @ B )
= ( uminus3100561713750211260ring_a @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_959_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_960_verit__minus__simplify_I3_J,axiom,
! [B: real] :
( ( minus_minus_real @ zero_zero_real @ B )
= ( uminus_uminus_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_961_diff__0,axiom,
! [A: complex] :
( ( minus_minus_complex @ zero_zero_complex @ A )
= ( uminus1482373934393186551omplex @ A ) ) ).
% diff_0
thf(fact_962_diff__0,axiom,
! [A: finite_mod_ring_a] :
( ( minus_3609261664126569004ring_a @ zero_z7902377541816115708ring_a @ A )
= ( uminus3100561713750211260ring_a @ A ) ) ).
% diff_0
thf(fact_963_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_964_diff__0,axiom,
! [A: real] :
( ( minus_minus_real @ zero_zero_real @ A )
= ( uminus_uminus_real @ A ) ) ).
% diff_0
thf(fact_965_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
= zero_zero_complex ) ).
% power_0_Suc
thf(fact_966_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_967_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_968_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 ) ) ) ) ).
% mult_eq_1_iff
thf(fact_969_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_970_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_971_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_972_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_973_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_974_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I3 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_975_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_976_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_977_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_978_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 ) ) ) ).
% one_le_mult_iff
thf(fact_979_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_980_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_981_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I3 @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_982_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I3 )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_983_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_984_zless__nat__conj,axiom,
! [W: int,Z2: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ( ord_less_int @ zero_zero_int @ Z2 )
& ( ord_less_int @ W @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_985_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_986_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_987_card__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
= ( nat2 @ U ) ) ).
% card_atLeastZeroLessThan_int
thf(fact_988_nat__one__le__power,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I3 )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I3 @ N ) ) ) ).
% nat_one_le_power
thf(fact_989_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_990_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_991_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_992_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_993_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M4: nat] :
( M5
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_994_le__trans,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_995_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_996_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_997_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_998_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_999_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1000_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1001_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1002_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1003_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1004_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1005_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1006_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z4: nat] :
( ( R2 @ X3 @ Y2 )
=> ( ( R2 @ Y2 @ Z4 )
=> ( R2 @ X3 @ Z4 ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1007_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M6: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M6 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1008_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1009_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1010_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1011_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1012_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1013_inc__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( P @ J2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% inc_induct
thf(fact_1014_dec__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( P @ I3 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_1015_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1016_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1017_int__cases,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_1018_int__of__nat__induct,axiom,
! [P: int > $o,Z2: int] :
( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
=> ( P @ Z2 ) ) ) ).
% int_of_nat_induct
thf(fact_1019_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1020_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 ) ) ) ) ).
% Suc_diff_le
thf(fact_1021_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_1022_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1023_card__less,axiom,
! [M6: set_nat,I3: nat] :
( ( member_nat @ zero_zero_nat @ M6 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K5: nat] :
( ( member_nat @ K5 @ M6 )
& ( ord_less_nat @ K5 @ ( suc @ I3 ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_1024_card__less__Suc,axiom,
! [M6: set_nat,I3: nat] :
( ( member_nat @ zero_zero_nat @ M6 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K5: nat] :
( ( member_nat @ ( suc @ K5 ) @ M6 )
& ( ord_less_nat @ K5 @ I3 ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K5: nat] :
( ( member_nat @ K5 @ M6 )
& ( ord_less_nat @ K5 @ ( suc @ I3 ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_1025_card__less__Suc2,axiom,
! [M6: set_nat,I3: nat] :
( ~ ( member_nat @ zero_zero_nat @ M6 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K5: nat] :
( ( member_nat @ ( suc @ K5 ) @ M6 )
& ( ord_less_nat @ K5 @ I3 ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K5: nat] :
( ( member_nat @ K5 @ M6 )
& ( ord_less_nat @ K5 @ ( suc @ I3 ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_1026_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_1027_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1028_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1029_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1030_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1031_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1032_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1033_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P @ X3 @ Y2 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1034_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1035_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1036_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1037_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1038_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_1039_Nat_OlessE,axiom,
! [I3: nat,K: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( ( K
!= ( suc @ I3 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1040_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1041_Suc__lessE,axiom,
! [I3: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_1042_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1043_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1044_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1045_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_1046_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1047_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1048_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_1049_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1050_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1051_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1052_less__trans__Suc,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I3 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1053_less__Suc__induct,axiom,
! [I3: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ( P @ I2 @ J3 )
=> ( ( P @ J3 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I3 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_1054_strict__inc__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ! [I2: nat] :
( ( J2
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I3 ) ) ) ) ).
% strict_inc_induct
thf(fact_1055_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1056_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1057_card__sum__le__nat__sum,axiom,
! [S2: set_nat] :
( ord_less_eq_nat
@ ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
@ ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ S2 ) ) ).
% card_sum_le_nat_sum
thf(fact_1058_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1059_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1060_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1061_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1062_int__cases2,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_1063_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1064_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1065_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1066_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I3: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I3 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1067_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1068_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1069_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1070_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1071_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1072_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1073_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J2: nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1074_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1075_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1076_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1077_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1078_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1079_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1080_add__le__mono,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_1081_add__le__mono1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_1082_trans__le__add1,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_1083_trans__le__add2,axiom,
! [I3: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_1084_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K5: nat] :
( N3
= ( plus_plus_nat @ M3 @ K5 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1085_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1086_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1087_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1088_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1089_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1090_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1091_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1092_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1093_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1094_mult__le__mono,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1095_mult__le__mono1,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).
% mult_le_mono1
thf(fact_1096_mult__le__mono2,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1097_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1098_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N2: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_1099_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1100_nat__power__less__imp__less,axiom,
! [I3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I3 )
=> ( ( ord_less_nat @ ( power_power_nat @ I3 @ M ) @ ( power_power_nat @ I3 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1101_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1102_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1103_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1104_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1105_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J: nat] :
( ( M
= ( suc @ J ) )
& ( ord_less_nat @ J @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1106_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 ) ) ) ) ) ).
% one_is_add
thf(fact_1107_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 ) ) ) ) ) ).
% add_is_1
thf(fact_1108_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1109_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
? [K5: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M3 @ K5 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1110_less__add__Suc2,axiom,
! [I3: nat,M: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ M @ I3 ) ) ) ).
% less_add_Suc2
thf(fact_1111_less__add__Suc1,axiom,
! [I3: nat,M: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ I3 @ M ) ) ) ).
% less_add_Suc1
thf(fact_1112_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1113_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1114_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1115_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1116_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1117_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1118_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_1119_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1120_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1121_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1122_diff__less__mono,axiom,
! [A: nat,B: nat,C: 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 ) ) ) ) ).
% diff_less_mono
thf(fact_1123_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I3 )
= K )
= ( J2
= ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1124_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I3 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1125_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1126_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1127_le__diff__conv,axiom,
! [J2: nat,K: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I3 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I3 @ K ) ) ) ).
% le_diff_conv
thf(fact_1128_nat__mono__iff,axiom,
! [Z2: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_1129_zless__nat__eq__int__zless,axiom,
! [M: nat,Z2: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z2 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1130_nat__int__add,axiom,
! [A: nat,B: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
= ( plus_plus_nat @ A @ B ) ) ).
% nat_int_add
thf(fact_1131_int__minus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% int_minus
thf(fact_1132_int__cases4,axiom,
! [M: int] :
( ! [N2: nat] :
( M
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_1133_diff__Suc__less,axiom,
! [N: nat,I3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1134_one__less__mult,axiom,
! [N: nat,M: 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 ) ) ) ) ).
% one_less_mult
thf(fact_1135_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1136_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1137_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1138_less__diff__conv2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I3 )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I3 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1139_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z6: int] :
? [N3: nat] :
( Z6
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1140_nat__eq__add__iff1,axiom,
! [J2: nat,I3: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J2 @ I3 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J2 ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1141_nat__eq__add__iff2,axiom,
! [I3: nat,J2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I3 ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1142_nat__le__add__iff1,axiom,
! [J2: nat,I3: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J2 @ I3 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J2 ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1143_nat__le__add__iff2,axiom,
! [I3: nat,J2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I3 ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1144_nat__diff__add__eq1,axiom,
! [J2: nat,I3: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J2 @ I3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J2 ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1145_nat__diff__add__eq2,axiom,
! [I3: nat,J2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I3 ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1146_nat__times__as__int,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_1147_nat__plus__as__int,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_1148_nat__minus__as__int,axiom,
( minus_minus_nat
= ( ^ [A3: nat,B2: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_1149_split__nat,axiom,
! [P: nat > $o,I3: int] :
( ( P @ ( nat2 @ I3 ) )
= ( ! [N3: nat] :
( ( I3
= ( semiri1314217659103216013at_int @ N3 ) )
=> ( P @ N3 ) )
& ( ( ord_less_int @ I3 @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1150_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_1151_nat__less__add__iff2,axiom,
! [I3: nat,J2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I3 ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1152_nat__less__add__iff1,axiom,
! [J2: nat,I3: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J2 @ I3 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J2 ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1153_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_1154_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A: nat > nat,B: nat > nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J3 ) ) ) )
=> ( ! [I2: nat,J3: nat] :
( ( ord_less_eq_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ N )
=> ( ord_less_eq_nat @ ( B @ J3 ) @ ( B @ I2 ) ) ) )
=> ( ord_less_eq_nat
@ ( times_times_nat @ N
@ ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( times_times_nat @ ( A @ I ) @ ( B @ I ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
@ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_1155_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_1156_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_1157_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ Z2 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1158_nat__0__iff,axiom,
! [I3: int] :
( ( ( nat2 @ I3 )
= zero_zero_nat )
= ( ord_less_eq_int @ I3 @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1159_int__nat__eq,axiom,
! [Z2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_1160_imp__le__cong,axiom,
! [X: int,X6: int,P: $o,P2: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_1161_conj__le__cong,axiom,
! [X: int,X6: int,P: $o,P2: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_1162_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1163_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1164_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1165_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1166_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_1167_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1168_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_1169_eq__nat__nat__iff,axiom,
! [Z2: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ( nat2 @ Z2 )
= ( nat2 @ Z5 ) )
= ( Z2 = Z5 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1170_nat__power__eq,axiom,
! [Z2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( nat2 @ ( power_power_int @ Z2 @ N ) )
= ( power_power_nat @ ( nat2 @ Z2 ) @ N ) ) ) ).
% nat_power_eq
thf(fact_1171_all__nat,axiom,
( ( ^ [P3: nat > $o] :
! [X7: nat] : ( P3 @ X7 ) )
= ( ^ [P4: nat > $o] :
! [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P4 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_1172_ex__nat,axiom,
( ( ^ [P3: nat > $o] :
? [X7: nat] : ( P3 @ X7 ) )
= ( ^ [P4: nat > $o] :
? [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
& ( P4 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_1173_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z6: int] :
? [N3: nat] :
( Z6
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1174_subset__card__intvl__is__intvl,axiom,
! [A2: set_nat,K: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
=> ( A2
= ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_1175_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1176_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1177_subset__eq__atLeast0__lessThan__card,axiom,
! [N4: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ N4 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_1178_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1179_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1180_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_1181_nat__0__le,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) ) ).
% nat_0_le
thf(fact_1182_int__eq__iff,axiom,
! [M: nat,Z2: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z2 )
= ( ( M
= ( nat2 @ Z2 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).
% int_eq_iff
thf(fact_1183_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_1184_nat__eq__iff,axiom,
! [W: int,M: nat] :
( ( ( nat2 @ W )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1185_nat__eq__iff2,axiom,
! [M: nat,W: int] :
( ( M
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1186_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_1187_nat__less__eq__zless,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_1188_nat__le__eq__zle,axiom,
! [W: int,Z2: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_1189_nat__mult__distrib,axiom,
! [Z2: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( nat2 @ ( times_times_int @ Z2 @ Z5 ) )
= ( times_times_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) ) ) ) ).
% nat_mult_distrib
thf(fact_1190_nat__add__distrib,axiom,
! [Z2: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( nat2 @ ( plus_plus_int @ Z2 @ Z5 ) )
= ( plus_plus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1191_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1192_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1193_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
= ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_1194_nat__diff__distrib,axiom,
! [Z5: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ord_less_eq_int @ Z5 @ Z2 )
=> ( ( nat2 @ ( minus_minus_int @ Z2 @ Z5 ) )
= ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_1195_nat__less__iff,axiom,
! [W: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_1196_nat__mult__distrib__neg,axiom,
! [Z2: int,Z5: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ ( times_times_int @ Z2 @ Z5 ) )
= ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z5 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_1197_real__add__minus__iff,axiom,
! [X: real,A: real] :
( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X = A ) ) ).
% real_add_minus_iff
thf(fact_1198_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 ) ) ).
% real_0_le_add_iff
thf(fact_1199_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 ) ) ) ).
% real_add_le_0_iff
thf(fact_1200_complete__real,axiom,
! [S2: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S2 )
=> ( ? [Z3: real] :
! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z3 ) )
=> ? [Y2: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Y2 ) )
& ! [Z3: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ord_less_eq_real @ Y2 @ Z3 ) ) ) ) ) ).
% complete_real
thf(fact_1201_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_1202_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y5: real] :
( ( ord_less_real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_1203_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 ) ) ).
% real_minus_mult_self_le
thf(fact_1204_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 ) ) ) ).
% real_add_less_0_iff
thf(fact_1205_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 ) ) ).
% real_0_less_add_iff
thf(fact_1206_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_1207_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X2: real,Y5: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y5 ) ) ) ) ).
% minus_real_def
thf(fact_1208_fib_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ( ( X
!= ( suc @ zero_zero_nat ) )
=> ~ ! [N2: nat] :
( X
!= ( suc @ ( suc @ N2 ) ) ) ) ) ).
% fib.cases
thf(fact_1209_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N )
= A )
& ! [Y4: real] :
( ( ( ord_less_real @ zero_zero_real @ Y4 )
& ( ( power_power_real @ Y4 @ N )
= A ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1210_card__nth__roots,axiom,
! [C: complex,N: nat] :
( ( C != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z6: complex] :
( ( power_power_complex @ Z6 @ N )
= C ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_1211_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1212_Bernoulli__inequality,axiom,
! [X: real,N: nat] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_1213_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N2: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ).
% real_arch_pow
thf(fact_1214_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X @ N2 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1215_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N3: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).
% nat_less_real_le
thf(fact_1216_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N3: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1217_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 ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1218_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 ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1219_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1220_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1221_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1222_zle__diff1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1223_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1224_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1225_one__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% one_less_nat_eq
thf(fact_1226_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X2: complex] : X2
@ ( collect_complex
@ ^ [Z6: complex] :
( ( power_power_complex @ Z6 @ N )
= one_one_complex ) ) )
= zero_zero_complex ) ) ).
% sum_roots_unity
thf(fact_1227_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1228_int__less__induct,axiom,
! [I3: int,K: int,P: int > $o] :
( ( ord_less_int @ I3 @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_less_induct
thf(fact_1229_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 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1230_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1231_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1232_int__le__induct,axiom,
! [I3: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_le_induct
thf(fact_1233_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1234_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1235_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1236_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1237_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1238_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1239_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1240_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1241_int__gr__induct,axiom,
! [K: int,I3: int,P: int > $o] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_gr_induct
thf(fact_1242_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1243_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
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1244_int__ge__induct,axiom,
! [K: int,I3: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_ge_induct
thf(fact_1245_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1246_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1247_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1248_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1249_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1250_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1251_zless__imp__add1__zle,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1252_add1__zle__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 )
= ( ord_less_int @ W @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1253_int__induct,axiom,
! [P: int > $o,K: int,I3: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_induct
thf(fact_1254_sum__nth__roots,axiom,
! [N: nat,C: complex] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X2: complex] : X2
@ ( collect_complex
@ ^ [Z6: complex] :
( ( power_power_complex @ Z6 @ N )
= C ) ) )
= zero_zero_complex ) ) ).
% sum_nth_roots
thf(fact_1255_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1256_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1257_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1258_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1259_Suc__as__int,axiom,
( suc
= ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_1260_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1261_Suc__nat__eq__nat__zadd1,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( suc @ ( nat2 @ Z2 ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1262_card__roots__unity__eq,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z6: complex] :
( ( power_power_complex @ Z6 @ N )
= one_one_complex ) ) )
= N ) ) ).
% card_roots_unity_eq
thf(fact_1263_finite__number__segment,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
& ( ord_less_nat @ M3 @ N ) ) ) )
= ( minus_minus_nat @ N @ one_one_nat ) ) ).
% finite_number_segment
thf(fact_1264_nat__exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X7: nat] : ( P3 @ X7 ) )
= ( ^ [P4: nat > $o] :
? [N3: nat] :
( ( P4 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P4 @ M3 ) ) ) ) ) ).
% nat_exists_least_iff
thf(fact_1265_card__atMost,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_1266_polynomial__product__nat,axiom,
! [M: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ M @ I2 )
=> ( ( A @ I2 )
= zero_zero_nat ) )
=> ( ! [J3: nat] :
( ( ord_less_nat @ N @ J3 )
=> ( ( B @ J3 )
= zero_zero_nat ) )
=> ( ( times_times_nat
@ ( groups3542108847815614940at_nat
@ ^ [I: nat] : ( times_times_nat @ ( A @ I ) @ ( power_power_nat @ X @ I ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups3542108847815614940at_nat
@ ^ [J: nat] : ( times_times_nat @ ( B @ J ) @ ( power_power_nat @ X @ J ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [R4: nat] :
( times_times_nat
@ ( groups3542108847815614940at_nat
@ ^ [K5: nat] : ( times_times_nat @ ( A @ K5 ) @ ( B @ ( minus_minus_nat @ R4 @ K5 ) ) )
@ ( set_ord_atMost_nat @ R4 ) )
@ ( power_power_nat @ X @ R4 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product_nat
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
ord_less_nat @ r1 @ r2 ).
thf(conj_1,conjecture,
( ( plus_p6165643967897163644ring_a @ ( groups3558780024651037881ring_a @ f @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ r1 ) ) @ ( groups3558780024651037881ring_a @ f @ ( set_or4665077453230672383an_nat @ r1 @ r2 ) ) )
= ( groups3558780024651037881ring_a @ f @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ r2 ) ) ) ).
%------------------------------------------------------------------------------